Anuncio

09069-Prelims.qxd 8/12/03 2:07 PM Page I FLAC AND NUMERICAL MODELING IN GEOMECHANICS Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-Prelims.qxd 8/12/03 2:07 PM Page III PROCEEDINGS OF THE THIRD INTERNATIONAL FLAC SYMPOSIUM, 21–24 OCTOBER 2003, SUDBURY, ONTARIO, CANADA FLAC and Numerical Modeling in Geomechanics Edited by Richard Brummer & Patrick Andrieux Itasca Consulting Canada Inc., Sudbury, Ontario, Canada Christine Detournay & Roger Hart Itasca Consulting Group Inc., Minneapolis, Minnesota, USA A.A. BALKEMA PUBLISHERS Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands LISSE / ABINGDON / EXTON (PA) / TOKYO 09069-Prelims.qxd 8/12/03 2:07 PM Page IV Cover picture: The “Big Nickel” is a famous landmark and symbol of Sudbury, the “Nickel Mining Capital of the World”. Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system,or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: A.A. Balkema, a member of Swets & Zeitlinger Publishers www.balkema.nl and www.szp.swets.nl ISBN 90 5809 581 9 Printed in the Netherlands Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-Prelims.qxd 8/12/03 2:07 PM Page V FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Table of contents Preface IX Organisation XI Constitutive models Compensation grouting analysis with FLAC3D X. Borrás, B. Celada, P. Varona & M. Senís 3 An automated procedure for 3-dimensional mesh generation A.K. Chugh & T.D. Stark 9 A new constitutive model based on the Hoek-Brown criterion P. Cundall, C. Carranza-Torres & R. Hart 17 A study of compaction band formation with the Double-Yield model C. Detournay, P. Cundall & J. Parra 27 A new viscoplastic model for rocks: application to the Mine-by-Test of AECL-URL F. Laigle 35 Prediction of deformations induced by tunneling using a time-dependent model A. Purwodihardjo & B. Cambou 45 Modeling of anhydrite swelling with FLAC J.M. Rodríguez-Ortiz, P. Varona & P. Velasco 55 Scenario testing of fluid-flow and deformation during mineralization: from simple to complex geometries P.M. Schaubs, A. Ord & G.H. German Constitutive models for rock mass: numerical implementation, verification and validation M. Souley, K. Su, M. Ghoreychi & G. Armand 63 71 Slope stability A parametric study of slope stability under circular failure condition by a numerical method M. Aksoy & G. Once 83 Numerical modeling of seepage-induced liquefaction and slope failure S.A. Bastani & B.L. Kutter 91 Complex geology slope stability analysis by shear strength reduction M. Cala & J. Flisiak 99 Analysis of hydraulic fracture risk in a zoned dam with FLAC3D C. Peybernes 103 Mesh geometry effects on slope stability calculation by FLAC strength reduction method – linear and non-linear failure criteria R. Shukha & R. Baker 109 3D slope stability analysis at Boinás East gold mine A. Varela Suárez & L.I. Alonso González 117 V Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-Prelims.qxd 8/12/03 2:07 PM Page VI Underground cavity design The effect of tunnel inclination and “k” ratio on the behavior of surrounding rock mass M. Iphar, M. Aksoy, M. Yavuz & G. Once 127 Numerical analysis of the volume loss influence on building during tunnel excavation O. Jenck & D. Dias 135 Application of FLAC3D on HLW underground repository concept development S. Kwon, J.H. Park, J.W. Choi & W.J. Cho 145 Numerical simulation of radial bolting: Application to the Tartaiguille railway tunnel F. Laigle & A. Saïtta 153 Recent experiences of the prediction of tunneling induced ground movements C. Pound & J.P. Beveridge 161 Numerical modeling of remedial measures in a failed tunnel Y. Sun & P.J.N. Pells 169 Mining applications Sill pillar design at the Niobec mine using FLAC3D P. Frenette & R. Corthésy 181 Stability analyses of undermined sill mats for base metal mining R.K. Brummer, P.P. Andrieux & C.P. O’Connor 189 FLAC numerical simulations of tunneling through paste backfill at Brunswick Mine P. Andrieux, R. Brummer, A. Mortazavi, B. Simser & P. George 197 FLAC3D numerical simulations of ore pillars at Laronde Mine R.K. Brummer, C.P. O’Connor, J. Bastien, L. Bourguignon & A. Cossette 205 Modeling arching effects in narrow backfilled stopes with FLAC L. Li, M. Aubertin, R. Simon, B. Bussière & T. Belem 211 FLAC3D numerical simulations of deep mining at Laronde Mine C.P. O’Connor, R.K. Brummer, P.P. Andrieux, R. Emond & B. McLaughlin 221 Three-dimensional strain softening modeling of deep longwall coal mine layouts S. Badr, U. Ozbay, S. Kieffer & M. Salamon 233 FISH functions for FLAC3D analyses of irregular narrow vein mining H. Zhu & P.P. Andrieux 241 Soil structure interaction A calibrated FLAC model for geosynthetic reinforced soil modular block walls at end of construction K. Hatami, R.J. Bathurst & T. Allen 251 Three-dimensional modeling of an excavation adjacent to a major structure J.P. Hsi & M.A. Coulthard 261 Pile installation using FLAC A. Klar & I. Einav 273 Axial tension development in the liner of a proposed Cedar Hills regional municipal solid waste landfill expansion F. Ma VI Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 279 09069-Prelims.qxd 8/12/03 2:07 PM Page VII The usability analyses of HDPE leachate collection pipes in a solid waste landfill F. Ma 287 FLAC numerical simulations of the behavior of a spray-on liner for rock support C.P. O’Connor, R.K. Brummer, G. Swan & G. Doyle 295 A numerical study of the influence of piles in the passive zone of embedded retaining walls T.Y. Yap & C. Pound 301 Dynamic and thermal analysis A practice orientated modified linear elastic constitutive model for fire loads and its application in tunnel construction E. Abazović & A. Amon 313 Seismic liquefaction: centrifuge and numerical modeling P.M. Byrne, S.S. Park & M. Beaty 321 Modeling the dynamic response of cantilever earth-retaining walls using FLAC R.A. Green & R.M. Ebeling 333 VII Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-Prelims.qxd 8/12/03 2:07 PM Page IX FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Preface The first two International FLAC Symposia were held in Minneapolis (USA) in September 1999, and in Lyon (France) in October 2001. In 2003, the third International Symposium on FLAC and Numerical Modeling in Geomechanics returned to North America and was held in Sudbury (Ontario, Canada) from October 21 to October 24, 2003, with two days of short courses before the symposium. Technical contributions to the conference were received from a wide range of different disciplines, representing virtually the entire globe. A volunteer Technical Committee reviewed the papers, and where necessary clarifications were suggested to the authors prior to finalization of their manuscripts. The contributions in this volume cover seven main topics: • • • • • • • Constitutive Models Slope Stability Underground Cavity Design Mining Applications Soil Structure Interaction Dynamic Analyses Thermal Analyses The FLAC conferences provide all FLAC and FLAC3D users with an opportunity to meet and learn from each other and from the people who develop the code. Conversely, they also allow Itasca staff members to learn from the practical experiences of code users “out there in the real world”. These interactions improve our collective knowledge and allow us to improve the performance of these numerical models in simulating the behavior of geomaterials. These proceedings contain a comprehensive collection of FLAC & FLAC3D applications – case studies as well as research presentations. We believe that this publication will help users by documenting a valuable resource for the solution of geomechanical problems. The compilation presented here would not have been possible without the efforts of our authors and our Technical Committee, and we thank them. We particularly thank and recognize the efforts of Michele Nelson, who served as an extremely capable and efficient Technical Editor. Richard Brummer Patrick Andrieux Itasca Consulting Canada Inc. Roger Hart Christine Detournay Itasca Consulting Group Inc. IX Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-Prelims.qxd 8/12/03 2:07 PM Page XI FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Organisation Conference Technical Committee: The following individuals provided technical input to the conference, and scientific overview and reviews of the abstracts and papers. Patrick Andrieux, Itasca Consulting Canada Inc. Daniel Billaux, Itasca Consultants SA Richard Brummer, Itasca Consulting Canada Inc. Peter Cundall, Itasca Consulting Group Inc. Christine Detournay, Itasca Consulting Group Inc. Samantha Espley, INCO Limited Roger Hart, Itasca Consulting Group Inc. Ugur Ozbay, Colorado School of Mines Chris Pound, Mott MacDonald Limited Graham Swan, Falconbridge Limited XI Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-01.qxd 08/11/2003 20:13 PM Page 1 Constitutive models Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-01.qxd 08/11/2003 20:13 PM Page 3 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Compensation grouting analysis with FLAC3D X. Borrás Gestió D’Infrastructures SA (GISA), Barcelona, Spain B. Celada Geocontrol SA, Madrid, Spain P. Varona & M. Senís Itasca Consultores SL, Asturias, Spain ABSTRACT: The Barcelona Metro Line 3 extension was excavated crossing 6.5 meters below a main water supply pipe. Compensation grouting was used to minimize the deformations in the gallery. A FLAC3D model was developed to investigate the efficiency of this process. The model was first calibrated to reproduce the extensometer measurements and was later re-run without the compensation grouting in order to assess the effectiveness of such treatment. 2.2 1 INTRODUCTION Excavation sequence The numerical model considers the sequential excavation of the metro tunnel: The Barcelona Metro Line 3 extension was excavated crossing 6.5 meters below a main water supply pipe (Borrás et al. 2001). This is one of the two pipes of Aigües Ter-Llobregat (ATLL) which supply water to the city of Barcelona. Due to the importance of this water pipe, during the construction of the tunnel, compensation grouting was used to minimize the deformations induced by the excavation process in the existing gallery that contains the water pipe. A FLAC3D model simulating the whole excavation sequence and the compensation grouting process has been developed in order to evaluate the effects of the construction and the treatment in the pipe, calibrating the model with the actual instrumentation results obtained during the excavation. The instrumentation installed consisted of 17 rod extensometers located 0.5 m below the bottom of the ATLL gallery. – Excavation of the heading in steps of 1 m length RodExtensometer t er location i 20 20m m 0.5 m 0.5 m 6.5 6.5m m 2 2.1 FLAC 3D In Injection depth MODEL Geometry of the model Figure 1 presents the problem geometry with the dimensions of both tunnels and their location. The FLAC3D model (Figs. 2 & 3) reproduces this geometry. In plan view the pipe gallery forms a 35° angle with the axis of the tunnel (Fig. 4). Figure 1. Problem description. 3 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 4.7 m 4.7 m 09069-01.qxd 08/11/2003 20:13 PM Page 4 Figure 2. FLAC3D model. General view. Figure 4. Figure 3. FLAC3D model. Plan view. FLAC3D model. Tunnels geometry. Figure 5. Excavation sequence. – Installation of the support: shotcrete with a thickness of 30 cm and TH-29 steel arches. – Installation of a 15 cm thick shotcrete lining and provisional invert, 10 m behind the excavation face. Figure 5 shows a detail of the excavation sequence followed in the model. 2.3 Material properties The geological profile assumed is shown in Figure 6. The Mohr-Coulomb constitutive model has been assigned to all the soils. The properties assumed are shown in Table 1. Both the shotcrete and the lining have been modeled using regular elements with an elastic constitutive model. The aging of the shotcrete has been simulated by the Young Modulus evolution law shown in Figure 7 (based on Estefanía 2000). Figure 6. Geological profile. 4 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-01.qxd 08/11/2003 20:13 PM Page 5 Table 1. Geotechnical properties assigned to the soils. Quaternary Natural fills Weathered Granite (V) Weathered Granite (IV) Weathered Granite (II/III) E (MPa) v (°) c (t/m2) (t/m3) 50 30 75 100 300 0.33 0.35 0.30 0.30 0.25 25 32 37 37 37 1.5 1.0 1.5 2.8 7.5 2.1 2.0 2.2 2.3 2.6 4.0E+04 3.5E+04 E (MPa) 3.0E+04 y = 6644.4Ln(x) + 11076 R2 = 0.9262 2.5E+04 2.0E+04 1.5E+04 1.0E+04 5.0E+03 0.0E+00 0 5 10 15 20 25 30 35 Distancia al frente (m) Figure 7. Hardening law applied to the concrete. Figure 9. Location of drills and sleeves in the model. 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 The methodology used for the simulation of the injection process is based in the bulb expansion model proposed by Buchet et al. (1999). According to them the injection effect can be modeled by increasing the volume of the elements in which the injection is made. This volume increment is carried out applying some “fictitious” hydrostatic stresses in the element, which makes it expand. These stresses are applied instantaneously, initializing an hydrostatic stress increment of the element as pulses and then reaching a mechanical equilibrium. This process is repeated until the volumetric strain induced in the element is the one corresponding to a fraction of the volume injected. The volumetric strain increment due to the injection is defined by: 20 03 21 02 01 Figure 8. Drills and sleeves location. 2.4 Simulation of the compensation grouting Compensation grouting injections consist of a mixture of cement and bentonite. To simulate these injections a methodology which reproduces the injection process carried out during the real construction has been developed. Figure 8 shows a plan view of the treatment area with the location of the drills and sleeves used. This real geometry has been reproduced in the FLAC3D model. Figure 9 shows the location of all the drills considered and the sleeves used in one of the injection cycles. (1) where Vi is the injected volume, V0 is the initial volume of the element and is the efficiency of the injection. The process followed during the injection modeling is shown in Figure 10. Figure 11 shows, as an example of the process, the increments applied to the vertical stress and their later 5 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-01.qxd 08/11/2003 20:13 PM Page 6 T v Vi V0 FLAC 3D 2.00 Step 9271 18:48:40 Mon Nov 13 2000 Deformacion volumetrica ⴛ10^-2 5.0 History INI Hydrostatic stress 1Volumetric Strain Inc. Zone 113 Line style 1.347e-003 <-> 5.443e-002 4.5 Vs. 4.0 Step 1.000e+001 <-> 9.270e+003 Calculation of v produced 3.5 3.0 2.5 2.0 no T v v 1.5 1.0 yes 0.5 End of injection Itasca Consulting Group, Inc. Minneapolis, MN USA 1.0 2.0 3.0 4.0 5.0 6.0 NⴗPasos ⴛ10^3 7.0 8.0 9.0 Figure 10. Modeling the injection process. Figure 12. Evolution of the volumetric strain. FLAC3D 2.00 Step 9271 18:50:04 Mon Nov 13 2000 Tension Vertical ⴛ10^5 FLAC 3D 2.00 1.394 Step 54106 Model Projection 09:45:05 Tue Nov 14 2000 History Rev 5 tension_media (FISH function) Line style 1.392 1.375e+005 <-> 1.396e+005 Vs. Step 1.390 1.000e+001 <-> 9.270e+003 1.388 1.386 Rotation: X : 90.000 Y : 0.000 Z : 0.000 Size: 4.830e+001 Plane Origin: X: 0.000e+000 Y: 0.000e+000 Z: 1.077e+001 Plane Orientation: Dip: 0.000 DD: 0.000 Contour of Z-Displacement Plane: on 0.0000e+000 to 5.0000e-004 5.0000e-004 to 1.0000e-003 1.0000e-003 to 1.5000e-003 1.5000e-003 to 2.0000e-003 2.0000e-003 to 2.5000e-003 2.5000e-003 to 3.0000e-003 3.0000e-003 to 3.5000e-003 3.5000e-003 to 4.0000e-003 4.0000e-003 to 4.5000e-003 4.5000e-003 to 5.0000e-003 5.0000e-003 to 5.5000e-003 5.5000e-003 to 5.6775e-003 Interval = 5.0e-004 1.384 1.382 1.380 1.378 1.376 Itasca Consulting Group, Inc. Minneapolis, MN USA Center: X: 0.000e+000 Y: 4.000e+001 Z: 4.885e+000 Dist: 2.964e+002 Taladros 1.0 2.0 3.0 4.0 5.0 6.0 NⴗPasos ⴛ10^3 7.0 8.0 Itasca Consulting Group, Inc. Minneapolis, MN USA 9.0 Figure 11. Vertical stresses during the injection process. Figure 13. Vertical displacement increments in one of the injection cycles, at extensometer depth. relaxation until the equilibrium is reached. Figure 12 presents the evolution of the volumetric strain, showing the successive increments produced until the strain corresponding to the injected volume is reached. Following the real injection scheme, the drills and sleeves that are injected every cycle are reproduced in the FLAC3D model, finding the closest element to the position of the sleeve and proceeding in the way described above. As an example of the modeling, Figure 13 shows the increment of the vertical displacements (at extensometer depth) produced during one of the injection cycles (the location of the sleeves injected in the cycle is shown too). Figure 14 shows, for the same cycle, the heave produced at the ground surface. These two figures show how the heave is less pronounced but affects a larger area as the distance from the sleeves increases. FLAC 3D 2.00 Step 54106 Model Projection 09:46:43 Tue Nov 14 2000 Rotation: X: 90.000 Y: 0.000 Z: 0.000 Size: 4.830e+001 Plane Origin: X: 0.000e+000 Y: 0.000e+000 Z: 2.470e+001 Plane Orientation: Dip: 0.000 DD: 0.000 Contour of Z-Displacement Plane: on 0.0000e+000 to 2.5000e-005 2.5000e-005 to 5.0000e-005 5.0000e-005 to 7.5000e-005 7.5000e-005 to 1.0000e-004 1.0000e-004 to 1.2500e-004 1.2500e-004 to 1.5000e-004 1.5000e-004 to 1.7500e-004 1.7500e-004 to 2.0000e-004 2.0000e-004 to 2.2500e-004 2.2500e-004 to 2.5000e-004 2.5000e-004 to 2.7500e-004 2.7500e-004 to 3.0000e-004 3.0000e-004 to 3.2500e-004 3.2500e-004 to 3.5000e-004 3.5000e-004 to 3.5961e-004 Itasca Consulting Group, Inc. Minneapolis, MN USA Figure 14. Vertical displacement increments in one of the injection cycles, at ground surface. 6 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Center: X: 0.000e+000 Y: 4.000e+001 Z: 4.885e+000 Dist: 2.964e+002 09069-01.qxd 08/11/2003 20:13 PM Page 7 3.2 3 RESULTS An average grouting efficiency can be defined as the ratio of difference between the volume of the settlement trough without (VNI s ) and with compensation VsWI to the injected volume: Comparison with instrumentation Figure 15 shows the location of the 17 rod extensometers used to monitor the compensation grouting process. The model was first calibrated varying the grout efficiency , in order to match the actual measurements with the calculated values, achieving a good fit. A second run was made without the compensation grouting in order to calculate what deformations would have been induced without any treatment. Figure 16 shows the evolution of extensometer E4 located outside the treated area. A vertical displacement of 10 mm was measured and without grouting, 12 mm are predicted. Figure 17 shows the evolution of extensometer E6 located in a relatively stiff material within the treated area. Here 8 mm settlement was recorded compared to 18 mm predicted without compensation grouting. In a softer material, extensometer E10 (Fig. 18), the difference between measured (8 mm) and predicted without treatment (22 mm) is even larger. FLAC3D2.00 Step 17228 Model Projection 10:34:27 Wed Nov15 2000 Rotation: X: 90.000 Y:0.000 Z: 0.000 Size: 4.830e+001 Plane Origin: X: 0.000e+000 Y: 0.000e+000 Z: 1.077e+001 Plane Orientation: Dip: 0.000 DD: 0.000 In the present analysis the efficiency obtained has been: (3) This empirical parameter is crucial for predictive studies, and a sufficient database for a given soil type is necessary before such analysis should be attempted. 3.3 E1 E4 Calculation of volume loss The volume loss can be defined as the ratio of the volume of the settlement trough to the excavated volume. Figure 19 shows the volume loss calculated along the tunnel axis for both hypotheses (with and without grouting). In both cases the volume loss depends on E2 E3 E5 E13 Geología Plane: on 1.111000e+007 1.880000e+007 2.884500e+007 Extensometer E6 Date E6 E11 E9 E10 Taladros 0.002 E7 Vertical displacement (m) Center: X: 0.000e+000 Y: 4.000e+001 Z: 4.885e+000 Dist: 2.964e+002 (2) E8 Extensometros E12 E1B Tunel GaleriaATLL E2B E14 E3B 10/9 11/9 12/9 13/9 14/9 15/9 16/9 17/9 18/9 19/9 20/9 21/9 22/9 23/9 24/9 25/9 26/9 27/9 28/9 29/9 30/9 1/10 2/10 3/10 4/10 5/10 6/10 7/10 8/10 9/10 10/10 11/10 12/10 13/10 14/10 15/10 16/10 17/10 18/10 19/10 20/10 21/10 22/10 23/10 3.1 Calculation of the efficiency 0.000 -0.002 -0.004 -0.006 -0.008 -0.010 -0.012 -0.014 -0.016 -0.018 Itasca Consulting Group, Inc. Minneapolis, MN USA -0.020 -0.022 Measured Figure 15. Location of the rod extensometers. FLAC FLAC No Injections Figure 17. Results obtained for extensometer E6. Extensometer E4 Date 10/9 11/9 12/9 13/9 14/9 15/9 16/9 17/9 18/9 19/9 20/9 21/9 22/9 23/9 24/9 25/9 26/9 27/9 28/9 29/9 30/9 1/10 2/10 3/10 4/10 5/10 6/10 7/10 8/10 9/10 10/10 11/10 12/10 13/10 14/10 15/10 16/10 17/10 18/10 19/10 20/10 21/10 22/10 23/10 10/9 11/9 12/9 13/9 14/9 15/9 16/9 17/9 18/9 19/9 20/9 21/9 22/9 23/9 24/9 25/9 26/9 27/9 28/9 29/9 30/9 1/10 2/10 3/10 4/10 5/10 6/10 7/10 8/10 9/10 10/10 11/10 12/10 13/10 14/10 15/10 16/10 17/10 18/10 19/10 20/10 21/10 22/10 23/10 Extensometer E10 Date 0.002 0.000 0.000 - 0.002 -0.004 Vertical displacement (m) Vertical displacement (m) 0.002 -0.002 -0.006 -0.008 -0.010 -0.012 -0.014 -0.016 -0.018 - 0.004 - 0.006 - 0.008 - 0.010 - 0.012 - 0.014 -0.016 - 0.018 -0.020 - 0.020 -0.022 - 0.022 Measured FLAC FLAC No Injections Measured FLAC No Injections Figure 18. Results obtained for extensometer E10. Figure 16. Results obtained for extensometer E4. 7 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands FLAC Vsubsidence/Vexcavated (%) 09069-01.qxd 08/11/2003 20:13 PM Page 8 maximum values of 19 mm in the hypothesis without treatment and 15 mm in the one with compensation grouting. In the same figure the corresponding horizontal strains have been represented too. These horizontal strains have been calculated as: 0.0 -0.3 No Noinjections injections Injections Injections -0.5 -0.8 -1.0 -1.3 -1.5 -1.8 -2.0 -2.3 -2.5 1668 1673 1678 1683 1688 1693 1698 1703 1708 1713 1718 1723 1728 1733 1738 1743 1748 Chainage (4) Quaternary Quaternary where L is the initial distance between two points along the gallery, and L is the distance once the displacement has occurred. The strains show low values in both cases, although the induced tensile strains are higher in the hypothesis with injections. Natural Natural Fills Fills A. V A.Granite GraniteV A.A.Granite GraniteIIVV A.A.Granite Granite anit II–III III 4 CONCLUSIONS Figure 19. Volume loss analysis. 0 0.050 -2 0.025 -4 0.000 -6 -0.025 -8 -0.050 -10 -0.075 -12 -0.100 -14 -0.125 -16 -0.150 -18 -0.175 -20 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 – The expected grout efficiency (30%) was much higher than the actual efficiency (10%). – The expected volume loss (0.07–0.2%) was much lower than the actual volume loss (1–2%). – The expected volume to inject (13.7 m3) was much lower than the actual volume injected (68 m3). Still only partial compensation was achieved. – According to the comparison between the model with compensation grouting and the model without the treatment area has been insufficient. – Numerical models should play an important role in the design of compensation grouting providing accurate estimates of the ground loss and of the required treatment area. Strain (mm/m) Vertical displacement (mm) The main conclusions that can be obtained from this analysis are: -0.200 0 5 10 15 20 25 30 35 40 45 50 Distance to the cross measured on the ATLL (m) Injections No Injections Strain I. Strain N.I. Figure 20. Horizontal strains in the ATLL gallery bottom. REFERENCES the characteristics of the soil above the tunnel. So in the area in which the tunnel is excavated in natural fill the volume loss is about 2.4%; and as the tunnel runs towards stiffer soils (weathered granite) this relation decreases to 0.8%. These values agree with the ones described in the literature. For example, Oteo (2000) reports values of 1–2% for stiff clays and 1–5% for granular soils above the water table. The maximum effect of the compensation grouting is a reduction of the volume loss of 0.4% from 1.2% to 0.8% at chainage 1708. 3.4 Borrás, X., Pérez, A., Magro, J.A., Celada, B. & Varona, P. 2001. Construcción del tramo Montbau-Canyelles de la Línea 3 del Metro de Barcelona. In Ingeopres N° 92, Abril 2001, Madrid: 54–64. Buchet, G. & Van Cotthem, A. 1999. 3D “Steady State” numerical modeling of tunneling and compensation grouting. In Detournay & Hart (eds), FLAC and Numerical Modeling in Geomechanics; Proc. intern. symp., Minneapolis, MN, 1–3 September 1999: 255–261. Rotterdam: Balkema. Estefania, S. 2000. Utilización de Métodos Numéricos en el Proceso Constructivo. Proc. III Curso sobre Ingeniería de Túneles. Madrid, 12–14 June 2000. Oteo, C. 2000. Subsidencia producida por los túneles. In Jimeno (ed), Manual de túneles y obras subterráneas. U.D. Proyectos, E.T.S.I. Minas, U.P.M., Madrid. Effects on the gallery Figure 20 shows the vertical displacements produced on the bottom of the ATLL gallery along its axis, with 8 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 9 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 An automated procedure for 3-dimensional mesh generation A.K. Chugh Bureau of Reclamation, Denver, CO, USA T.D. Stark University of Illinois, Urbana, IL, USA ABSTRACT: An automated procedure is presented to generate a 3-dimensional mesh for numerical analysis of engineering problems. The procedure is simple, effective and efficient, and can be applied to represent complex geometries and material distributions. A listing of the program that was used for the sample problem of a landfill slide is included. 1 INTRODUCTION 2 CONCEPTUAL MODEL One of the essential tasks in a 3-dimensional (3-D) numerical analysis is to represent the geometry and distribution of materials in the numerical model. FLAC3D provides means to facilitate mesh generation and the built-in programming language FISH can be used to develop and implement additional program instructions during execution of a data file. In geotechnical engineering, surface geometry, distribution of materials, and water table conditions usually vary from one location to the next and pose a difficult set of conditions to represent in a numerical model. In order to facilitate the analysis of landslides, a simple procedure was devised to represent complex surface geometry, subsurface material horizons, and water table conditions. The objectives of this paper are to present: The conceptual model for the generation of a 3-D mesh follows the conventional procedure of portraying spatial variations of materials in 3-D via a series of 2-dimensional (2-D) cross-sections. This technique is commonly used by engineers and geologists in constructing visual models of complex geologic sites where a number of 2-D cross-sections are used to represent the field conditions. In these representations, linear variations between material horizons in consecutive 2D cross-sections are used to depict the 3-D spatial variability of a site. The accuracy of the representation is improved by using closely spaced 2-D cross-sections. The 3-D mesh generation procedure presented herein follows the conventional practices used by engineers in constructing 2-D numerical meshes by hand for geotechnical problems to be solved using methods other than FLAC3D. For example, in the creation of a 2-D numerical model of a slope to be analyzed using a limit-equilibrium based procedure, it is a common practice to define profile lines via a set of data points followed by specifications of their connectivities. Also, in the creation of a 2-D model of a continuum to be solved by a finite-element based procedure, it is a common practice to discretize the continuum into a network of zones; assign identification numbers to the grid points; define the coordinates of the grid points; and then specify the connectivity of grid points. 1 a simple method to describe field geometry and conditions for a 3-D numerical model of a slope problem; 2 a simple procedure for automatic generation of a 3-D mesh; and 3 an illustration of the use of the procedure for analysis of a large slide in a landfill. A listing of the program for the landfill slide is included in the paper. This program listing is in the FISH language and uses some of the functions available in the FISH library. 9 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 10 each profile line and the water table (for parallel 2-D cross-sections, y-coordinate shall have same constant value between two consecutive crosssections). 2 The following steps are used for creating similar sets of data at each of the 2-D cross-sections: a From the data in step 1(c) above, select control points that are of significance in defining the profile lines in all of the 2-D cross-sections. Tabulate the x-coordinates of these control points in increasing order. For reference purposes, this table is referred to as Table 100. b Use of the “Interpolate” function expands the 2-D cross-sectional data of step 1(c) by linear interpolation for all of the control points listed in Table 100 for all of the profile lines and stores the data in separate tables; assigns Table numbers in increasing order starting with the user specified starting number and incrementing it by 1; assigns an identification number to each point; and positions the points in the 3-D model space. These tables contain the (x,z) coordinates of expanded 2-D cross-sectional data. A sample listing of the “Interpolate” function and its dependency function “zz” in FISH language is given in Figure 1. The starting table number used in the sample problem data file is 200. 3 The following steps are used for creating zones in the 3-D model space: Thus, in the conceptual model for the generation of a 3-D mesh in FLAC3D, use is made of defining a series of 2-D cross-sections at representative locations of a site; defining each of the 2-D sections as an assemblage of data points with line-segment connections; and organizing the data for an efficient and effective discretization of the volume. 3 WATER TABLE The water table surface is specified using the water table data of individual 2-D cross-sections and through the use of 3-point planar polygons between consecutive 2-D cross-sections. This scheme allows incorporation of non-coplanar variations in the water table surface in the entire 3-D model. 4 DESCRIPTION OF THE PROCEDURE In geotechnical engineering, the ground-surface geo-metry is obtained using contour maps that are prepared from land or aerial survey of the area. The subsurface material horizons are estimated from geologic data and information obtained from exploratory boring logs. The subsurface water conditions are estimated from field observations, piezometers installed at various depths, and/or from water levels in borings. Subsurface data are used to develop contour maps of the subsurface geology and water conditions. From these contour maps, the region-of-interest, and the locations of significant cross-sections are identified; information for 2-D cross-sections are read and tabulated; and 2-D cross-sections are drawn for an understanding of the site details and preparation of input data for a 2-D analysis. In general, the crosssectional data for a site varies from one location to the next. These variations may be caused by changes in the ground surface and (or) in subsurface material horizons, discontinuity of some materials, or a combination of these or some other variations. In the proposed procedure, the following steps are followed: (For ease of presentation, 2-D cross-sections are assumed to lie in x-z plane and the x,y,z coordinate system follow the right hand rule.) def zz zz=table(t_n,xx) end def interpolate loop j (js,je); profile line #s ; js is for the bottom, je is for top dt_n=dt_n_s+j; dt_n is destination table number loop i (is,ie); is is the first interpolation #, ; ie is the last interpolation # xx=xtable (100,i); x-coordinate of the ;interpolation point command set t_n=j end_command table(dt_n,xx)=zz id_pt=id_pt+1 x_pt=xtable(dt_n,i) y_pt=y_pt z_pt=ytable(dt_n,i) command generate point id id_pt x_pt y_pt z_pt end_command endloop endloop end 1 The following steps are used for creating an orderly assemblage of field data for 3-D discretization of the continuum of the region-of-interest: a On the site map, select values of x, y, and z coordinates that completely circumscribe the 3-D region-of-interest; b Mark locations of all significant 2-D crosssections oriented in the same and preferably parallel direction; c For each 2-D cross-section, tabulate (x,y,z) coordinates of end-points of all line segments for Figure 1. Listing of the “Interpolate” function and its dependency function “zz” in FISH language. 10 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 11 zones desired for each interval in the x-direction. Tabulate these values for all of the intervals in the increasing x-direction. For reference purposes, this table is referred to as Table 102. The number of entries in Table 102 should be one less than those in Table 100. c Considering the spacing between the 2-D crosssections in the y-direction, select the number of zones desired for each interval in the y-direction. Tabulate these values for all of the intervals in the increasing y-direction. For reference purposes, this table is referred to as Table 103. The number of entries in Table 103 should be one less than the number of 2-D cross-sections. d Considering the spacing of the profile lines in the z-direction, select the number of zones desired for each material horizon in the z-direction. Tabulate these values for all of the intervals in the increasing z-direction. For reference purposes, this table is referred to as Table 104. The number of entries in Table 104 should be one less than the number of profile lines. e Use of the “Fill_grid” function generates a brick mesh and assigns a group name to each 3D volume zone. A sample listing of the “Fill_grid” function in FISH language is given in Figure 2. a Tabulate the y-coordinates of the 2-D crosssections in increasing y-direction. For reference purposes, this table is referred to as Table 101. The number of entries in Table 101 should equal the number of 2-D cross-sections marked in step 1(b). b Considering the spacing of x-coordinates of the control points in step 2(a), select the number of def fill_grid i_n=table_size(102) j_n=table_size(103) k_n=table_size(104) loop jy (1,j_n) ny=xtable(103,jy) p0_d=(jy-1)*(i_n+1)*(k_n+1) loop kz (1,k_n) nz=xtable(104,kz) if kz=1 then material='shale' endif if kz=2 then material='ns'; native soil endif if kz=3 then material='msw'; municipal solid waste x_toe=xtable(105,jy) endif loop ix (1,i_n) if kz=3 then xx_toe=xtable(100,ix) if xx_toe < x_toe then material='mswt' endif endif nx=xtable(102,ix) p0_d=p0_d+1 p3_d=(p0_d+i_n+1) p6_d=(p3_d+1) p1_d=(p0_d+1) p2_d=((i_n+1)*(k_n+1)+p0_d) p5_d=(p2_d+(i_n+1)) p7_d=(p5_d+1) p4_d=(p2_d+1) command generate zone brick size nx,ny,nz ratio 1,1,1 & p0=point (p0_d) p3=point (p3_d) & p6=point (p6_d) p1=point (p1_d) & p2=point (p2_d) p5=point (p5_d) & p7=point (p7_d) p4= point(p4_d) group material end_command if kz=3 then material='msw' endif end_loop p0_d=p0_d+1 end_loop end_loop end 5 COMMENTS 1 Use of a Brick mesh with an 8-point description is versatile and allows for creation of degenerated brick forms through the use of multiple points with different identification numbers occupying the same (x,y,z) coordinate location in the 3-D model space. 2 During the development of the grid, it is possible to assign group names to different segments of the model. This information can be useful in modifying the generated grid. 3 Expanding the (x,y,z) location data for all 2-D crosssections to a common control number of locations via interpolations facilitates the programming of the automatic grid-generation procedure. 4 In engineering practice, it is generally desirable to analyze a few 2-D cross-sections at select locations prior to conducting a 3-D analysis. Because development of data for 2-D cross-sections is one of the steps for use of the proposed procedure, it is relatively easy to conduct a 2-D analysis using the 2-D cross-sectional data and the program FLAC. 5 The program instructions listed in Figures 1 and 2 can be modified to accommodate geometry and other problem details that are different or more complex than those encountered in the sample problem described in Section 6. Figure 2. Listing of “FILL_GRID” function in FISH language. 11 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 12 6 SAMPLE PROBLEM The problem used to illustrate the proposed 3-D mesh generation procedure is the 1996 slide in a waste containment facility near Cincinnati, Ohio (Stark & Eid 1998, Eid et al. 2000). Figure 3 is an aerial view of the slide. Figure 4 is the plan view of the landfill and shows the location of the sixteen cross-sections used to construct a FLAC3D model of the site (the project data shown are in Imperial units). There are three material horizons bounded by four profile lines, and a liquid level present at this site. Figure 5 shows the 2-D cross-sectional views of the site at the 16-locations prior to failure (the available project data were converted to SI units and this conversion lead to numerical values with fractional parts). Figure 6 shows a partial listing of the data file for the sample problem with the following details: – Table 100 lists the x-coordinates of the 22 control points considered significant from the sixteen 2-D cross-sectional data. – Table 101 lists the y-coordinates of the sixteen 2-D cross-section locations. – Table 102 lists the number of zones desired in each of the 21 segments in the x-direction. – Table 103 lists the number of zones desired in each of the 15 segments in the y-direction. – Table 104 lists the number of zones desired in each of the 3 material horizons at the site. – Table 105 lists the x-coordinates of the toe locations of the top profile line in the 2-D cross-sections in the increasing y-direction. Figure 3. Sample problem – aerial view of Cincinnati landfill failure (from Eid et al. 2000). (Reproduced by permission of the publisher, ASCE). For each cross-section, x- and z-coordinates for data points defining the profile lines are recorded in individual tables numbered as Table 1 for profile line 1 data, Table 2 for profile line 2 data, Table 3 for profile line 3 data, and Table 4 for profile line 4 data in the data file shown in Figure 6. Profile lines are numbered from 1 to 4 in the increasing z-direction and each profile line uses a different number of data points to define the line. For cross-sections where the top profile line terminates in a vertical cut at the toe, the top profile line was extended to x 0. For each cross-section and for each of the four profile lines, the x-coordinate locations identified in Table 100 are used to create data by interpolation at each of the 22 control points. For the sample problem, this amounts to 88 pairs of (x,z) coordinates per crosssection, and the y-coordinate of the data points is read from Table 101. Thus, the x-,y-, and z-coordinates for all of the points defined and (or) interpolated are known. Each point is assigned a numeric identity number (id #) starting with one and incrementing by one. The data points are located in the 3-D model space using their id # and x-,y-, z-coordinates. This task is accomplished using the “Interpolate” function Figure 4. Plan view of the sample problem showing locations of selected 2-D sections. and its listing in FISH language is given in Figure 1. At the end of this task, all of the defined and (or) interpolated points with an assigned id # have been located in the 3-D model space. The connectivity of data points to define volume discretization is accomplished in the function named 12 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 13 Figure 5. 2-D cross-sectional views of the sample problem. “Fill_grid”. For each interval in the location of crosssections in the y-direction (Table 103), and for each material horizon between the profile lines in the zdirection (Table 104), and for each interval in the x-direction (Table 102), the values of number of zones desired in the x, y, and z-direction and the id #s of points in the 3-D model space are used in the “GENERATE zone brick p0, p1, … p8” command of 13 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 14 table 4 0,251.46 91.14,280.42 107.90,283.46 table 4 144.48,286.51 169.77,289.56 table 4 194.46,292.61 332.54,338.33 table 4 348.08,338.33 interpolate . . . ; station at y=307.85 m set y_pt=307.85 table 2 erase table 3 erase table 4 erase set dt_n_s=dt_n table 2 0,254.08 348.08,254.08 table 3 0,259.08 348.08,259.08 table 4 0,261.08 29.87,265.18 185.93,268.22 table 4 348.08,307.24 interpolate ; Rumpke landfill site; Data are in metric units set g=0,0,-9.81 ; table 100 is for the x-coordinates of ; the desired 3-D grid table 100 0,1 13.11,2 15.54,3 22.86,4 34.75,5 table 100 42.67,6 49.07,7 57.61,8 63.70,9 table 100 64.92,10 72.54,11 78.94,12 92.66,13 table 100 100.89,14 107.90,15 115.21,16 table 100 158.50,17 199.64,18 284.38,19 table 100 318.52,20 337.72,21 348.08,22 ; table 101 is for y-coordinates of the ; 2-D cross-section locations table 101 0,1 15.24,2 20.73,3 28.96,4 42.06,5 table 101 62.48,6 96.93,7 138.07,8 164.29,9 table 101 201.47,10 234.09,11 253.29,12 table 101 268.83,13 287.43,14 293.83,15 table 101 307.85,16 fill_grid ; table 102 is for the number of zones ; desired in the x-direction table 102 2,1 1,2 1,3 2,4 1,5 1,6 1,7 1,8 1,9 table 102 1,10 1,11 2,12 1,13 1,14 1,15 5,16 table 102 5,17 10,18 4,19 2,20 2,21 delete range group mswt ; water surface water den=1 table & face 0,0,228.60 0,15.24,228.60 & 332.54,15.24,268.22 & face 0,0,228.60 332.54,15.24,268.22 & 348.08,15.24,268.22 & face 0,0,228.60 348.08,15.24,268.22 & 348.08,0,268.22 & ;interval # 1 face 0,15.24,228.60 0,20.73,228.60 & 340.77,20.73,268.22 & face 0,15.24,228.60 340.77,20.73,268.22 & 348.08,20.73,268.22 & face 0,15.24,228.60 348.08,20.73,268.22 & 332.54,15.24,268.22 & face 332.54,15.24,268.22 348.08,20.73,268.22 & 348.08,15.24,268.22 &;interval # 2 . . . face 0,293.83,259.08 0,307.85,259.08 & 63.70,307.85,259.08 & face 0,293.83,259.08 63.70,307.85,259.08 & 348.08,307.85,268.22 & face 0,293.83,259.08 348.08,307.85,268.22 & 63.70,293.83,259.08 & face 63.70,293.83,259.08 348.08,307.85,268.22 & 348.08,293.83,268.22;interval # 15 ; table 103 is for the number of zones ; desired in the y-direction table 103 2,1 1,2 1,3 2,4 2,5 3,6 4,7 3,8 4,9 table 103 3,10 2,11 2,12 2,13 1,14 2,15 ; table 104 is for the number of zones ; desired in the z-direction table 104 5,1 3,2 10,3 ; table 105 is for the x-coordinates of the ; receding toe table 105 0,1 0,2 15.54,3 22.86,4 34.75,5 table 105 49.07,6 57.61,7 64.92,8 78.94,9 table 105 92.66,10 100.89,11 107.90,12 table 105 115.21,13 63.70,14 0,15 set is=1 ie=22 set js=1 je=4 set id_pt=0 set dt_n_s=200 ; Station at y=0 set y_pt=0 table 1 -100,200 500,200 table 2 0,223.60 154.23,223.60 307.24,238.84 table 2 348.08,239.14 table 3 0,228.60 154.23,228.60 307.24,243.84 table 3 348.08,244.14 table 4 0,260.00 66.45,280.42 98.15,283.46 table 4 156.67,286.51 187.15,289.56 table 4 348.08,332.54 interpolate save cin_3D_grid.sav ; station at y=15.24 m set y_pt=15.24 table 2 erase table 3 erase table 4 erase set dt_n_s=dt_n table 2 0,223.60 163.07,223.60 306.02,238.84 table 2 348.08,240.67 table 3 0,228.60 163.07,228.60 306.02,243.84 table 3 348.08,245.67 Figure 6. Partial listing of the data file for the sample problem for FLAC3D. 14 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-02.qxd 08/11/2003 20:14 PM Page 15 2 3 4 5 6 7 8 9 Figure 7. 3-D mesh for the sample problem. FLAC3D for a regular 8-noded brick mesh. The material between the profile lines is assigned a group name for ease of modifying the grid and for convenience in assigning material properties and/or addressing them for some other reason. This task is also accomplished in the function named “Fill_grid” and its listing in FISH language is given in Figure 2. Table 105 data are used to assign a group name “mswt” to the zones past the vertical cut which are later deleted using the DELETE command with the range defined by the group name “mswt”. At the end of this task, a 3-D grid of specification exists in the region-of-interest. For the sample problem, the generated 3-D grid is shown in Figure 7. The representation of continuity of the vertical cut at the toe of the slope (as seen in 2-D cross-sections, Figure 5) in the 3-D model can be improved by increasing the number of 2-D cross-sections. 8 SUMMARY To facilitate 3-D analyses using FLAC3D or other software, an automated procedure is presented to create a 3-D mesh. The procedure utilizes commonly used techniques for drawing 2-D cross-sections and interpolation between 2-D cross-sections to portray spatial variations of geometry and distribution of materials in 3-D. REFERENCES Eid, H.T., Stark, T.D., Evans, W.D. & Sherry, P.E. 2000. Municipal solid waste slope failure. II Stability analyses. Journal of Geotechnical and Geoenvironmental Engineering 126(5): 408–419. Stark, T.D. & Eid, H.T. 1998. Performance of threedimensional slope stability methods in practice. Journal of Geotechnical and Geoenvironmental Engineering 124(11): 1049–1060. 7 ADVANTAGES OF THE PROPOSED PROCEDURE 1 The proposed procedure for describing 3-D field conditions utilizes 2-D cross-sections, which are essentially the same as commonly used by geologists and engineers to describe the field conditions. 15 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Linear variation in geometry, material horizons, and groundwater descriptions between known data points is generally accepted. Changes in field data can be incorporated in the numerical model by updating the affected tables. New cross-sections can be introduced or old crosssections deleted and a new discretization of the continuum made quickly. Describing the spatial location of data in a 3-D space followed by descriptions of their connectivity is a simple yet powerful way of constructing a 3-D numerical model for analysis purposes. The proposed procedure produces regions with acceptable geometries, i.e. no conflicts in connectivity. Changes in discretization due to changes in field data or due to numerical considerations can be included in the proposed procedure efficiently and a new discretization accomplished. Number of discretized volume units in different parts of the numerical model is estimated at the start of the problem solving effort. If it becomes necessary to change or refine the discretization, very little effort is needed to change the tabular data and the procedure is then rerun to obtain an updated 3-D mesh. A complete brick element is used to generate other degenerated volume element shapes. Because the proposed procedure is based on simple and commonly used ideas, it should be adaptable when using computer programs or procedures other than FLAC3D to perform numerical analysis work. The program instructions can be rewritten in other programming languages. 09069-03.qxd 08/11/2003 20:14 PM Page 17 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A new constitutive model based on the Hoek-Brown criterion P. Cundall, C. Carranza-Torres & R. Hart Itasca Consulting Group, Inc., Minneapolis, MN, USA ABSTRACT: A new constitutive model is proposed based on the Hoek-Brown failure criterion. This model incorporates a plasticity flow rule that varies as a function of the confining stress level. For a low confining stress, at which a large rate of volumetric expansion at yield is anticipated, an associated flow rule is applied. For high confining stress, at which the material no longer dilates at failure, a constant-volume flow rule is prescribed. A composite flow rule, which provides a linear variation from associated to constant-volume limits, is used between the low and high confining stress states. Using an appropriate softening relation, the model can also represent the transition between brittle and ductile rock behavior. The new model is programmed in C and compiled as a DLL file (dynamic link library) that can be loaded directly into either FLAC or FLAC3D. This paper describes the model and its implementation as a DLL. Physical justification is provided for the formulation and, specifically, the representation of the volumetric behavior during yield, which depends on confining stress. A verification example is provided. failure of the material. However, numerical simulations of elasto-plastic problems allow continuing the solution after failure has taken place, and the failure condition itself may change as the simulation progresses (by either hardening or softening). In this event, it is more reasonable to speak of yielding rather than failure. There is no implied restriction on the type of behavior that is modelled – both ductile and brittle behavior may be represented, depending on the softening relation used. 1 INTRODUCTION The Hoek-Brown failure criterion is an empirical relation that characterizes the stress conditions that lead to failure in intact rock and rock masses. It has been used very successfully in design approaches that use limit equilibrium solutions, but there has been little direct use in numerical solution schemes. Alternatively, equivalent friction and cohesion values have been used with a Mohr-Coulomb model that is matched to the nonlinear Hoek-Brown strength envelope at particular stress levels. Numerical solution methods require full constitutive models, which relate stress to strain in a general way; in addition to a failure (or yield) criterion, a flow rule is also necessary, in order to provide a relation between the components of strain rate at failure. There have been several attempts to develop a full constitutive model from the Hoek-Brown criterion: e.g. Pan & Hudson (1988), Carter et al. (1993) and Shah (1992). These formulations assume that the flow rule has some fixed relation to the failure criterion, and that the flow rule is isotropic, whereas the Hoek-Brown criterion is not. In the formulation described here, there is no fixed form for the flow rule; it is assumed to depend on the stress level, and possibly on some measure of damage. In what follows, the failure criterion is taken as a yield surface, using the terminology of plasticity theory. Usually, a failure criterion is assumed to be a fixed, limiting stress condition that corresponds to ultimate 2 GENERAL FORMULATION The generalized Hoek-Brown criterion (Hoek & Brown 1998), adopting the convention of positive compressive stress, is (1) where 1 and 3 are the major and minor effective principal stresses, and ci, mb, s and a are material constants that can be related to the Geological Strength Index (GSI) and rock damage (Hoek et al. 2002). For interest, the unconfined compressive strength is given by c ci sa and the tensile strength by t ci s/mb. Equation (1) and the stresses c and t are represented in Figure 1. 17 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-03.qxd 08/11/2003 20:14 PM Page 18 (2) where E1 K 4G/3 and E2 K 2G/3 and (e1, e2, e3) is the set of principal strain increments. If the yield criterion (equation 1) is violated by this set of stresses, then the strain increments (prescribed as independent inputs to the model) are assumed to be composed of elastic and plastic parts, i.e., (3) Note that plastic flow does not occur in the intermediate principal stress direction. The final stresses (1f , 2f , 3f ) output from the model, are related to the elastic components of the strain increments; hence, (4) Eliminating the current stresses, using equations (2) and (4), (5) We assume the following flow rule, (6) where the factor depends on stress, and is recomputed at each time step. Eliminating ep1 from equation (5) Figure 1. Graphical representation of the generalized Hoek-Brown failure criterion (equation 1) in the (a) compressive and (b) tensile region of the principal stress space (1, 3). (7) It should be noted that the failure criterion (equation 1) does not depend on the intermediate principal stress, 2; thus, the failure envelope is not isotropic. Assume that the current principal stresses are (1, 2, 3) and that initial trial stresses (1t , 2t , 3t ) are calculated by using incremental elasticity, i.e., At yield, equation (1) is satisfied by the final stresses; that is, (8) 18 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-03.qxd 08/11/2003 20:14 PM Page 19 By substituting values of 1f and f2 from equation (7), equation (8) can be solved iteratively for ep3, which is then substituted in equation (7) to give the final stresses. The method of solution is described later, but first the evaluation of is discussed. 3.2 Constant-volume flow rule As the confining stress is increased, a point is reached at which the material no longer dilates during yield. A constant-volume flow rule is therefore appropriate when the confining stress is above some userprescribed level, 3 cv3. This flow rule is given by (11) 3 FLOW RULES We need to consider an appropriate flow rule, which describes the volumetric behavior of the material during yield. In general, the flow parameter will depend on stress, and possibly history. It is not meaningful to speak of a dilation angle for a material when its confining stress is low or tensile, because the mode of failure is typically by axial splitting, not shearing. Although the volumetric strain depends in a complicated way on stress level, we consider certain specific cases for which behavior is well known, and determine the behavior for intermediate conditions by interpolation. Three cases are considered below. 3.1 The constant-volume flow rule defined by equation (11) is represented graphically by point C in Figure 1a. The normal to the vector ep at point C has a slope equal to unity, and therefore the rate of volumetric expansion in the plastic regime is null. 3.3 Radial flow rule Under the condition of uniaxial tension, we might expect that the material would yield in the direction of the tensile traction. If the tension is isotropically applied, we imagine (since the test is practically impossible to perform) that the material would deform isotropically. Both of these conditions are fulfilled by the radial flow rule, which is assumed to apply when all principal stresses are tensile. For a flow-rate vector to be coaxial with the principal stress vector, we obtain Associated flow rule It is known that many rocks under unconfined compression exhibit large rates of volumetric expansion at yield, associated with axial splitting and wedging effects. The associated flow rule provides the largest volumetric strain rate that may be justified theoretically. This flow rule is expected to apply in the vicinity of the uniaxial stress condition (3 ≈ 0). An associated flow rule is one in which the vector of plastic strain rate is normal to the yield surface (when both are plotted on similar axes). Thus, (12) The radial flow rule defined by equation (12) is represented graphically by points D1, D2 and D3 in Figure 1b. The directions of vectors ep at these points intercept all the origin of the diagram. (9) 3.4 Composite flow rule We propose to assign the flow rule (and thus, a value for ) according to the stress condition. In the fully tensile region, the radial flow rule, rf, will be used. For compressive 1 and tensile or zero 3 the associated flow rule, af, is applied. For the interval 0 3 cv 3, the value of is linearly interpolated between the associated and constant-volume limits, i.e., where the subscripts denote the components in the principal stress directions, and F is defined by equation (8). Differentiating this expression, and using equation (6), (10) (13) The associated flow rule used in the constitutive model is graphically represented in Figure 1a. The normal to the plastic strain-rate vector ep at point A is tangent to the yield envelope (equation 1) at 3 0. The slope of the normal to ep, denoted as k0 in the figure, is inversely related to the coefficient af defined by equation (10) – i.e., k0 1/af at 3 0. Finally, when 3 3cv, the constant-volume value, cv, is used. It is noted that if cv 3 is set equal to zero, then the model uses a non-associated flow rule with a zero dilation angle, for 3 0. If 3cv is set to a very high value relative to ci, the model uses an associated-flow rule for 3 0. 19 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-03.qxd 08/11/2003 20:14 PM Page 20 high confining stresses, the iteration converges in one step, but at low confining stresses, up to ten steps are necessary (the limit built into the code is presently 15). The composite flow rule defined by equation (13) in the case of compressive stresses is represented graphically by point B in Figure 1a. The slope of the normal to ep at point C is the linear interpolation of the slopes at points A and B. 5 MATERIAL SOFTENING In the Hoek-Brown model, the material properties, ci, mb, s and a, are assumed to remain constant, by default. Material softening, after the onset of plastic yield, can be simulated by specifying that these mechanical properties change (i.e., reduce the overall material strength) according a softening parameter. The softening parameter selected for the Hoek-Brown p model is the plastic confining strain component, e3. p The choice of e3 is based on physical grounds. For yield near the unconfined state, the damage in brittle rock is mainly by splitting (not by shearing) with crack normals oriented in the 3 direction. The parameter ep3 is expected to correlate with the microcrack damage in the 3 direction. The value of ep3 is calculated by summing the strain increment values for ep3 calculated by equation (16). Softening behavior is provided by specifying tables that relate each of the properties, ci, mb, s and a, to ep3. Each table contains pairs of values: one for the ep3 value and one for the corresponding property value. It is assumed that the property varies linearly between two consecutive parameter entries in the table. A multiplier, (denoted as mult in FLAC and FLAC3D), can also be specified to relate the softening behavior to the confining stress, 3. The relation between and 3 is also given in the form of a table. To illustrate the definition of softening parameters in the constitutive model proposed in this paper, we analyze the idealized response of a cylindrical sample of homogeneous-isotropic material in a typical triaxial experiment – as represented in Figure 2a. For example, Figure 2b shows a piecewise-linear stress–strain relationship expressed in terms of the deviator 1 3 and the shear strain e1 e3. The different curves in the diagram correspond to increasing values of confinement 3 in the triaxial experiment of Figure 2a. Two cases of practical interest will be considered here. The first case assumes that the slope of the softening branch is maintained for increasing values of confinement 3. In Figure 2b the case is represented by continuous curves (e.g., the line OPR). The second case assumes that the slope of the softening branch decreases (in absolute value) as confinement increases, and that the material behaves in a ductile manner (i.e., the slope of the softening branch becomes zero) for a confinement level 3 dc 3. In Figure 2b this case is represented by the dashed curves (e.g., the line OPR ). To illustrate the definition of input parameters in the constitutive model we need to consider in some 4 IMPLEMENTATION The equations presented above are implemented in a DLL (dynamic link library) written in C , with the model name hoekbrown. One difficulty with the failure criterion (equation 8) is that real values for F do not exist if 3 sci/mb. During an iteration process, this condition is likely to be encountered, so it is necessary that the expression for F, and its first derivatives, be continuous everywhere in stress space. This is fulfilled by adapting the following composite expressions: • if 3 sci /mb then (14) • if 3 sci /mb then (15) To initialize the iteration, a starting value for, ep3 is taken as the absolute maximum of all the strain increment components. This value, denoted as 1, is inserted into equation (7), together with the value for found from the flow-rule equations, and the resulting stress values inserted into equations (14) and (15). The resulting value of F is denoted by F1. Taking the original value of F as F0 (and the corresponding plastic strain increment of zero as 0), we can estimate a new value of the plastic strain increment, using a variant of Newton’s method, (16) From this, we find a new value of F (that we call F2), and if it is sufficiently close to zero, the iteration stops. Otherwise, we set F0 F1, F1 F2, 0 1 and 1 2, and apply equation (16) again. Tests show that the iteration scheme converges for all stress paths tried so far, including cases in which s 0 (material with zero unconfined compressive strength), which led to problems in previous implementations. For 20 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-03.qxd 08/11/2003 20:14 PM Page 21 (17) (18) The residual parameters Rci and mRb in equation (18) are decreased in the same proportion, multiplying the initial (peak) parameters by the factor (1 ), i.e., (19) (20) The parameter in the equations above, that lie in the range 0 1, controls the jump of strength from the peak to residual stages. If 0, then the peak and residual strength are the same and the material behaves in a ductile manner (see Figure 2b). If 1 then the material behaves in a brittle manner, with the minimum possible value for the residual strength (i.e., 1 3 in equation 18). In the simplest case we can consider that the loss of strength in the softening branch in Figure 2b is linearly related to the plastic shear-strain p as follows, (21) Note that in the equation above, crp is the critical value of plastic shear-strain for which the residual stage is reached (see point R in Figure 2b). The loss of strength can also be expressed in terms of the drop modulus 2G. indicated in Figure 2b (the definition of drop modulus used here is as in Linkov, 1992). This parameter controls the ductile/brittle behavior of the material. For example, when 0 the material behaves in a perfectly-plastic manner and when → the material behaves in a perfectlybrittle manner. The relationship between cr p and is, Figure 2. (a) Idealized triaxial experiment of a cylindrical sample of isotropic-homogeneous Hoek-Brown material. The diagrams (b) and (c) represent an idealized piecewise linear response obtained from the triaxial experiment. (22) In the constitutive model discussed in this paper, the plastic strain ep3 (rather than the plastic shear-strain p) is taken as a softening parameter. The relationship between ep3 and p can be constructed from the analytical-solution of the triaxial experiment of Figure 2a. This relationship, that is represented in detail the relationships that govern the response of the material represented in Figure 2. We assume that the peak and residual strength of the material are given by the following equations, 21 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-03.qxd 08/11/2003 20:14 PM Page 22 From equation (23), again considering 3 0, the critical value of plastic strain is, Figure 2c, depends on the flow rule assumed for the material as follows, ep3 cr (0) In the FLAC model, the tables for the softening parameters should be defined as follows: (23) In the equation above the parameter K is related to the instantaneous dilation angle as (24) For interest, we list here the expressions for the slopes corresponding to the elastic, softening and residual branches in the e3 vs. e1 – e3 diagram of Figure 2c, e3p ci [MPa] mb s a 0.000 0.013 0.10 0.05 0.05 5.0 2.5 2.5 1.0 1.0 1.0 0.5 0.5 0.5 In addition to the table above, a table defining the relationship between the multiplier and the confining stress 3 will be normally defined. The type of relationship to consider depends on how the drop modulus of the softening branch is assumed to vary with the level of confinement. To illustrate the definition of the multiplier we consider first the case in which the drop modulus of the softening branch, 2G, is maintained for increasing values of confinement 3 (see line OPR in Figure 2b). For this case, the multiplier is defined as follows, (25) (26) (27) (28) We consider now a practical case of definition of softening parameters in a FLAC model. Let us assume the following values for the parameters that control the response of the material in Figure 2: Assuming an upper limit for the confining stress equal to 10 ci, and taking 5 points to represent this relationship, the definition of the multiplier in FLAC will be as follows: ci 0.1 MPa mb 5 s 1 a 0.5 0.5 0.2 (for 3 0) E 100 ci 0.3 0o [Note that the condition 0o implies that the material does not dilate in the plastic regime; in the FLAC model this condition is satisfied by specifying cv3 0.] For the value of defined above, the residual parameters Rci and mRb are computed with equations (19) and (20) and result to be, 3 [MPa] 0.00 0.25 0.50 0.75 1.00 1.0000 3.6742 5.0990 6.2048 7.1414 Note that in the table above, the second column is computed using equation (28). As a second example of the definition of the multiplier we consider now the case for which the drop modulus of the softening branch, 2G, decreases (in absolute value) for increasing values of confinement 3 (see line OPR in Figure 2b). To achieve the ductile behavior ( 0) at the confinement level 3 dc 3, we can use the following relationship between the multiplier and the confining stress 3, Rci 0.05 MPa mRb 2.5 From equation (23), and considering 3 0 the critical value of plastic shear-strain for which the residual stage is achieved is, (29) cr p (0) 0.039 22 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0.013 09069-03.qxd 08/11/2003 20:14 PM Page 23 (30) Assuming then a value of dc 3 ci, and an upper limit for the confining stress equal to 10 ci, the definition of the multiplier in FLAC will be as follows: 3 [MPa] 0.000 0.033 0.067 0.100 1.000 1.0 1.5 3.0 Note that in the table above, the second column is computed using equations (29) and (30). 6 VERIFICATION Stresses and displacements are calculated for the case of a cylindrical tunnel in an infinite brittle Hoek-Brown medium subjected to an in-situ stress field. A uniform compressive stress of 0 15 MPa is assigned as the far-field stress, and an internal pressure pi 2.5 MPa is applied inside the tunnel (see Figure 3a). The problem is based on an example posed by Hoek & Brown (1980). The closed-form solution in that example only provided the stress distribution calculation, and is extended here to include the displacement solutions for both associated and non-associated plastic flow. (A description of the equations that summarize the solution is provided in the Appendix A.) The properties and conditions selected for this test are also listed in Figure 3a. Both initial rock and broken rock properties for the Hoek-Brown model are specified. The brittle behavior of the rock is simulated by instantaneous softening – i.e. the Hoek-Brown properties are changed from initial values at e p3 0 to broken values at a small value of e p3 0 (an arbitrarily small value for e p3 equal to 10 20 is assumed). The comparison of the results from the FLAC model using model hoekbrown to the analytical solution (given in the Appendix A) is shown in Figure 3b for the calculation of hoop stress and radial stress around the tunnel, and in Figure 3c for the calculation of radial displacements for both the associated and non-associated flow cases. In all cases, the agreement between FLAC and analytical results is characterized by an error of less than 1%. Figure 3. Elasto-plastic solution for excavation of a cylindrical tunnel in a brittle generalized Hoek-Brown material. FLAC3D. The flow rule is based on general knowledge of the volumetric behavior of rock, which usually exhibits large dilation at low confining stresses and small or zero dilation at large confining stresses, as the failure condition is approached. Although this assumption conforms to practical experience, it will be 7 CONCLUDING REMARKS A full constitutive model based on the Hoek-Brown criterion has been implemented for use in FLAC and 23 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-03.qxd 08/11/2003 20:15 PM Page 24 necessary to compare the response of the new model with actual measurements of rock behavior, both in the laboratory and in the field, and calibrate the parameter cv 3 from observations of volumetric strain. Further, the softening behavior is assumed to depend on the confining stress, not on deviatoric stress, which is the more usual assumption. This decision was made on a general knowledge of rock behavior, but it will need to be verified (or falsified) by comparing model predictions with measurements of rock response under post-peak conditions. A.1 Plastic region, r Rpl The critical internal pressure below which the failure zone develops is computed from the following transcendental equation, (A.1) The extent Rpl of the failure zone is, REFERENCES Carranza-Torres, C. & Fairhurst, C. 1999. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion. International Journal of Rock Mechanics and Mining Sciences 36(6), 777–809. Carter, T., Carvalho, J. & Swan, G. 1993. Towards the practical application of ground reaction curves. In W.F. Bawden & J.F. Archibald (Eds), Innovative mine design for the 21st century, pp. 151–171. Rotterdam: Balkema. Hoek, E. & Brown, E.T. 1980. Underground Excavations in Rock. London: The Institute of Mining and Metallurgy. Hoek, E. & Brown, E.T. 1997. Practical estimates of rock mass strength. International Journal of Rock Mechanics and Mining Sciences 34(8), 1165–1186. Hoek, E., Carranza-Torres, C. & Corkum, B. 2002. HoekBrown failure criterion – 2002 edition. In H.R.W. Bawden, J. Curran & M. Telesnicki (Eds), Proceedings of the 5th North American Rock Mechanics Symposium and the 17th Tunnelling Association of Canada Conference: NARMSTAC 2002. Mining Innovation and Technology. Toronto – 10 July 2002, pp. 267–273. University of Toronto. Linkov, A.M. 1992. Dynamic phenomena in mines and the problem of stability. MTS System corporation. 14000 Technology Drive, Eden Praire, MN 55344, USA. Notes from a course of lectures presented as MTS visiting professor of Geomechanics at the University of Minnesota, Minneapolis, MN, USA. Pan, X.D. & Hudson, J.A. 1988.A simplified three dimensional Hoek-Brown yield condition. In M. Romana (Ed.), Rock Mechanics and Power Plants. Proc. ISRM Symp., pp. 95–103. Balkema. Rotterdam. Shah, S. 1992. A study of the behaviour of jointed rock masses. Ph. D. thesis, Dept. Civil Engineering, University of Toronto. (A.2) The solution for the radial stress field is (A.3) The solution for the hoop stress field is (A.4) For the case of non-associated flow rule (with dilation angle equal to zero) the solution for the radial stress field is computed from integration of the following second-order differential equation, APPENDIX A. CLOSED-FORM SOLUTION FOR A CYLINDRICAL HOLE IN AN INFINITE BRITTLE HOEK-BROWN MEDIUM (A.5) The solution presented in thisAppendix is based on a scaled solution for cylindrical tunnels in Hoek-Brown media discussed in Carranza-Torres and Fairhurst (1999). Analytical expressions to compute the field quantities r, and ur are presented here for the plastic and elastic regions around the tunnel. where (A.6) 24 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands is defined as, 09069-03.qxd 08/11/2003 20:15 PM Page 25 the following non-linear second-order differential equation, The functions dr/d and d/d in the differential equation above are (A.14) (A.7) The only difference with the case of non-associated flow rule (zero dilation angle) is that the coefficients A1, A2 and A3 depend now on the solution of the stress field r as follows, and (A.15) (A.16) (A.8) (A.17) while the coefficients A1, A2 and A3 are A.2 Elastic region, r Rpl The solution for the radial stress field is (A.9) (A.18) (A.10) (A.11) The solution for the hoop stress field is The boundary conditions to integrate the differential equation (A.5) above are (A.19) (A.12) The solution for the radial displacement field is and (A.20) The plastic and elastic solutions for the field quantities r and ur presented above are continuous at the elasto-plastic boundary (i.e., at r Rpl). The solution for the field quantity is discontinuous when there is a jump of strength from peak values (ci, mb, s and a) to residual values (Rci, mbR, sR and aR) – see Figure 3b. (A.13) In the case of associated flow rule, the solution for the radial stress field is obtained from integration of 25 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 27 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A study of compaction band formation with the Double-Yield model C. Detournay & P. Cundall Itasca Consulting Group, Inc., Minneapolis, MN, USA J. Parra PDVSA – Intevep, S.A., Los Teques, Venezuela ABSTRACT: The occurrence of thin localized bands associated with concentration of compressive strain has recently been reported in very porous rocks, both in field and laboratory settings. These structures exhibit a reduction of porosity, and are of importance to the petroleum industry because they can impact reservoir permeability. Compaction bands have been the object of both theoretical and experimental studies by Olsson (1999), Issen & Rudnicki (2000), Bésuelle (2001) and others. In this paper, we examine, in a simple theoretical framework, the basic conditions for a band to appear. We consider the case of the Double-Yield model, identify conditions for localization related to the volumetric cap, and give examples of numerical simulations that illustrate band formation. with the presence of a cap in the yield surface. We consider the case of a strain softening/hardening cap, normal to the mean pressure axis in effective stress space, and examine the conditions on the cap for compaction bands to appear. For numerical investigation with FLAC, we use the Double-Yield constitutive model. This model is characterized by a strain softening Mohr Coulomb behavior for shear yielding, and by an independent strain hardening cap behavior for volumetric yielding. The theoretical conditions for compaction band formation associated with stress states on the volumetric cap of the Double-Yield model are derived in section 2. The results of numerical experiments are presented in section 3. Conclusions for the work are given in section 4. 1 INTRODUCTION Mollema & Antonellini (1996) recently identified the presence of thin compacted bands in porous sandstone, and made reference to these features as “compaction bands”. Although these structures are sometimes associated with the presence of shear bands, they have individual characteristics, which are outlined in these definitions, found in the literature: – Compaction bands are narrow planar zones of localized compressive deformation perpendicular to the maximum compressive stress (Issen & Rudnicki 2000). – A compaction band is a tabular zone that exhibits normal closure but no shear offset (Olsson 1999). – Pure compaction bands are bands that exhibit a normal compacting strain and a zero shear strain (Bésuelle 2001). 2 EXISTENCE CRITERIA FOR COMPACTION BAND Compaction bands have attracted attention because of the potential impact that the reduced porosity of these features may have on oil reservoir exploitation. The authors cited above are among those who have investigated the condition for their formation in theoretical, laboratory and field settings. In this paper, we work in a basic theoretical framework. The change of porosity localized in the band is interpreted as an inelastic volume deformation, which can occur as an alternative to the homogeneous mode. Irrecoverable volumetric deformations are associated As a convention in this paper, tension and extension are positive for stress and strain, respectively, compression is positive for pressure, and effective stresses are denoted without a dash. Stresses are denoted as ij, and strains as ij with i 1, 2, 3 and j 1, 2, 3. Volumetric strain, ev, is defined by ev 11 22 33, and mean pressure, by p (11 22 33)/3. Rates are denoted by a superscript dot. First, we derive the stress-rate/strain-rate relations 27 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 28 where for evolution of a stress point on the Double-Yield volumetric cap, and then, we express the condition for volumetric localization. (5) 2.1 Cap constitutive relations and Kc, Gc are current values of tangent bulk and shear modulus. The cap pressure is a function of plastic volumetric strain, evp, and the hardening rule is: In the Double-Yield model, the volumetric yield function is: (1) (6) where pc is the cap pressure, and F v 0 for elastic conditions. The cap F v 0 is represented by a straight line in the plane of shear stress, q, versus mean pressure, p, shown in Figure 1. The flow rule for volumetric yielding is associated; thus, the potential function is: The coefficient a is the hardening modulus (positive for softening) which is a function of total plastic volumetric strain. An example of volumetric hardening behavior is represented in Figure 2. The flow rule gives the direction of plastic strain rate, which is parallel to the gradient of Gv in stress space: (2) . The plastic multiplier , gives the magnitude of plastic strain rate. It may be found from the consistency condition: (7) The total strain rates are partitioned into elastic and plastic parts: (8) (3) Substitution of the expression 2 for plastic potential in Equation 7 gives, after differentiation: The stress-strain relations are, in rate form: (9) (4) Using Equation 9 for the plastic strain rate, the hardening rule in Equation 6 takes the form: (10) Figure 1. Volumetric cap for the Double-Yield model. Figure 2. Example of volumetric hardening rule. 28 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 29 met within the material sample. First, equilibrium at the interface of the band only allows a stress discontinuity for the direct stress parallel to the band. Second, the direct strain parallel to the band must remain continuous. We analyze the situation at the onset of band formation, and denote as ni the unit normal to the potential planar band in which localized deformation occurs. Mathematically, the condition for non-uniqueness translates as (see e.g. Issen & Rudnicki 2000): After substitution of this expression in the consistency condition, we obtain: (11) The plastic strain increments in Equation 9 may now be expressed as: (12) (17) From Equation 3, elastic strain rate may be expressed as total rate minus plastic rate: where the components of the stiffness matrix may be found in Equation 14. We look at the case when the out of plane component of the normal to the band is zero, or n3 0. Band formation is predicted to occur when the condition (13) Finally, after substitution of Equation 13 in Equation 4, using Equation 12 and some manipulations, the cap constitutive behavior may be written in the form: (18) is first met in a program of deformation. By using the relations (14) for unit length of the vector ni, and 2 1 2Gc in Equation 5, the condition may be expressed as: where (19) (15) A symbolic way to write this expression is: (16) where Lijkl is the stiffness matrix, and Einstein summation convention on repeated indices is used. 2.2 Condition for localization We consider the deformation of a homogeneous sample of material under gradual application of prescribed uniform stresses, as shown in Figure 3. We will assume that the major compressive stress, 1 is vertical. Prior to the occurrence of a band, the sample deforms uniformly. After appearance of the band, stress and strain rates will be uniform inside and outside the band, but they will be different from each other. A bifurcation from homogeneous deformation has occurred. The non-homogeneous solution can only exist provided that some continuity requirements on stress and strain are Figure 3. Material sample, prescribed stresses, and potential compaction band. 29 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 30 Figure 5. Plateau corresponding to pore collapse in a schematic porous material stress–strain curve. Figure 4. Example of cap softening behavior. The possible band orientations, given by the roots of a quadratic equation, are given by: for the occurrence of compaction band. A relation between hardening modulus and stiffness properties must also be satisfied for the bands to appear: according to Equation 25, the hardening modulus, (which is positive for softening of the cap) must be equal or larger than 4Kc/(4 3Kc/Gc). In a uniaxial compression test (oedometric test), . . with 11 33 . 0, the constitutive relation, . Equation 14, yields 22 (1 b) 22, so the localization condition, Equation 24, which may also simply be derived using the compliance approach of Vermeer (1982), corresponds to the first occurrence of a plateau in the plot of vertical stress versus strain, see Figure 5. It is interesting to note that, according to Equation 19, the condition for band formation in the direction parallel to the maximum compressive stress is also given by Equation 24. So the same condition predicts band formation in two perpendicular directions. (20) where (21) For a real solution, we must have: (22) Using the definition of b, c, and 1 given above, the condition takes the form: (23) 3 NUMERICAL EXPERIMENTS This condition can only be satisfied if a 0, that is when softening of the cap occurs, see Figure 4. Physically, cap softening can correspond to grain collapse or breakage of cemented grains. By definition, a pure compaction band is oriented perpendicular to the maximum compressive stress. According to our convention, we must have: n1 0, in which case Equation 19 implies: Our theoretical derivation shows that cap softening is a necessary but not a sufficient condition for the formation of compaction bands. A relation between hardening modulus (a) and stiffness properties (Kc, Gc) must also be satisfied for the bands to appear. In an oedometric test, two sets of bands (one horizontal, and one vertical) are predicted to occur for (24) (26) Finally, using Equation 5 for 1, and Equation 15 for b, the localization condition may be expressed as: Numerical experiments are carried out to validate this prediction, and illustrate band formation. Numerical simulations of an oedometric test are performed using the finite difference code FLAC. The configuration is axi-symmetric. The grid contains a total of 400 elements. The boundary conditions correspond to (25) From the above consideration it follows that softening of the cap is a necessary but not sufficient condition 30 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 31 Figure 7. Seed 1 – Contours of 11 at 7000 steps. roller boundaries at the bottom and lateral side of the model. A Double-Yield constitutive model is assigned to the zones in the grid. Friction is zero, and cohesion is assigned a large value (compared to maximum mean stress in the simulation). We consider the state of the model at the onset of band formation. The initial stress field is isotropic, and the material is normally consolidated. Several cases are considered, corresponding to hypothetical values of current hardening modulus. The test is strain-controlled: a compressive velocity is applied at the top of the model. The stiffness properties for the simulation are chosen such that Kc/Gc 2, and Equation 26 translates to a 0.4 Kc. To trigger the localization process, the material bulk modulus is given a random deviation of 1%. The simulations are performed using the data file cb.dat, listed in the Appendix. In cases when the initial hardening modulus is equal to 0.2Kc or 0.3Kc, (softening of the cap occurs, but the criteria for band formation is not met), no band is observed. In the case when a 0.405Kc (a value slightly larger than the threshold for band formation), two sets of bands develop in the model: one normal, and one parallel to the major compressive stress. Simulation results at three different stages, and for two different random seeds (used for assignment of a small deviation around an average value for the bulk property) are presented below. Figure 6 show the results obtained for seed 1: a first horizontal band appears, then a second one starts to grow, and a third one develops. At each step, the additional deformation is seen to localize in the new band. The behavior of the normal stress parallel to the bands, at the end of the simulation is shown in Figure 7. When the simulation was repeated, this time with another seed, the results in Figure 8 were obtained. The first band to appear is vertical and it grows across the Figure 6. Seed 1 – Contours of volumetric strain increments and displacement vectors at: a) 4000, b) 5000 and c) 7000 steps. 31 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 32 Figure 9. Seed 2 – Contours of 22 at 7000 steps. grid before a second horizontal compaction band appears. Contours of vertical stress at the end of the simulation are shown in Figure 9. Apparently it is random whether horizontal or vertical bands appear first. 4 CONCLUSIONS A simple theoretical framework has been adopted to derive the basic conditions for compaction band formation. The Double-Yield constitutive model in FLAC was considered, and conditions for localization related to the volumetric cap were identified using the approach of Issen & Rudnicki (2000). It was found that softening of the cap, which can correspond to grain collapse or breakage of cemented grains, was a necessary condition for the occurrence of compaction band. But the condition is not sufficient; in addition, the hardening modulus (positive for softening of the cap) must exceed a critical value, which is a function of material bulk and shear moduli. The critical value, which may also be derived using the compliance approach of Vermeer (1982), corresponds to the first occurrence of a plateau in a plot of major compressive stress versus strain. The analysis predicts the occurrence of two sets of bands, normal and parallel to the direction of major compressive stress. Examples of numerical simulations are given that illustrate band formation. ACKNOWLEDGEMENTS The work related in this paper was performed as part of a research project carried out for INTEVEP. Chad Sylvain is thanked for editing of the figures and Michele Nelson for her help in formatting the manuscript. Figure 8. Seed 2 – Contours of volumetric strain increments and displacement vectors at: a) 4000, b) 5000 and c) 7000 steps. 32 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-04.qxd 08/11/2003 20:16 PM Page 33 REFERENCES prop bul 1110e6 rdev 1110e4 ; --- 1% deviation ; case 1 a0.2K ;table 1 0 5e6 1e-3 47.78e5 1e-2 2.78e6 1 0.5e5 ; no cb ;ini yvel 0 var 0 -1e-6 ; case 2 a0.3K ;table 1 0 5e6 1e-3 46.67e5 1e-2 1.70e6 1 0.5e5 ; no cb ;ini yvel 0 var 0 -1e-6 ; case 3 a0.405K table 1 0 5e6 1e-3 45.5e5 1e-2 0.5e6 1 0.5e5 ; cb ini yvel 0 var 0 -1e-7 ; case 4 a0.5K ;table 1 0 5e6 1e-3 44.45e5 0.8e-2 0.56e6 1 0.5e5 ; cb ;ini yvel 0 var 0 -1e-7 Bésuelle, P. 2001. Compacting and dilating shear bands in porous rock: Theoretical and experimental conditions. Journal of geophysical Research, 106(B7): 13,435–13,442. Issen, K.A. & Rudnicki, J.W. 2000. Conditions for compaction bands in porous rock. Journal of Geophysical Research 105(B9): 21,529–21,536. Itasca Consulting Group, Inc. 2000. FLAC Ver. 4.0 User’s Guide. Minneapolis: Itasca. Mollema, P.N. & Antonellini, M.A. 1996. Compaction bands: A structural analog for anti-mode I cracks in Aeolian sandstone. Tectonophysics 267: 209–228. Olsson, W.A. Theoretical and experimental investigation of compaction bands in porous rock. Journal of Geophysical Research 104(B4): 7219–7228. Vermeer, P.A. 1982. A simple shear-band analysis using compliances. IUTAM Conference on Deformation and Failure of Granular Materials, Delft. 31Aug–3 Sept, 1982. fix x i21 fix y j1 fix y j21 ini sxx -5e6 syy -5e6 szz -5e6 step 4000 APPENDIX A: DATA FILE save cb1.sav plot hold vsi fill step 1000 save cb2.sav plot hold vsi fill new title Oedometric test with DY model config axi g 20 20 gen 0 0 0 1 1 1 1 0 mo dy pro bu 1110e6 sh 555e6 cap_pressure 5e6 cptable 1 mul 10 pro den 1000 coh 1e10 ten 1e10 step 2000 save cb3.sav plot hold cap_pressure fill plot hold vsi fill plot hold sxx fill ret 33 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 35 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A new viscoplastic model for rocks: application to the Mine-by-test of AECL-URL F. Laigle Electricité de France, Hydro Engineering Centre, France ABSTRACT: A new viscoplastic constitutive model has been developed by EDF-CIH. Its aim is able to take into account delayed behavior of rock materials in the framework of nuclear waste repository studies. In this case, it’s important to predict the rock damage evolution in time in the neighboring storage tunnels. The main assumptions of the constitutive model are presented in this paper. One application to the Mine-by-Test done by the AECL in the Lac de Bonnet granite is shown. The low field strength of the granite in comparison with laboratory measurements may be justified by the delayed behavior of this granite. Failure with v-shape notches is well shown by the simulation. A prediction of the hydraulic permeability increasing around the tunnel versus time is presented. The Mine-by test has been done at the level – 420, in an undamaged granite mass (Lac de Bonnet Granite). This experiment consists of digging a gallery in wellknown conditions in a previously monitored part of the rock mass. The major aim is to observe the behavior of the granite during the excavation phase and at long-term. The direction of the gallery has been defined in accordance with initial stresses in the ground. At this depth, the major compressive principal stress is more or less horizontal, and is about 55 MPa. This stress is 3.9 times the vertical stress corresponding at the weight of overburden. The intermediate principal stress is about 48 MPa. 1 INTRODUCTION In the framework of studies of underground nuclear waste storage, it’s important to predict the evolution in time and at very long term of the ground surrounding the excavations. One objective of these studies is to assess the evolution of the EDZ (Excavation Damage Zone) in the time. This EDZ is assumed to be the zone where rock is fractured and where the permeability increasing is large. In this aim, a viscoplastic constitutive model has been developed by EDF and integrated in FLAC. This model will be briefly presented in the paper. The application is about the Mine-by-Test done in the AECL-URL. The evolution in time of the failure zone in the roof and invert of the gallery is well simulated. Assumptions of this model allow to assess the damage zone (fissured rock) and the fractured zone (continuous fissure) associated with a strong permeability increasing. 2 THE MINE-BY TEST EXPERIMENT The Underground Research Laboratory (URL) of the Atomic Energy of Canada Laboratory (AECL) has been dug in the framework of the Canadian nuclear waste management program launched in the 70’s. This underground laboratory is located in the state of Manitoba. It’s composed of a main shaft of 443 m depth, reaching two experimental levels excavated at 240 m depth and 420 m depth. Figure 1. Location and view of the URL. 35 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 36 Figure 2. Location of v-shaped notches in the Mine-by test tunnel. 2.1 Figure 3. Stress–strain curve for Lac de Bonnet granite. Bonnet Granite, it seems that the cracking and existing damage is not enough to justify a sufficient decrease in the strength. – A “structural effect” as suggested by Diederich (2002), taking into account some geometrical and shape differences between laboratory and field conditions. – A softening effect due to the stress path generated by the digging process. Some 3D simulations (Eberhardt 2001) show that there is a stress rotation forward the face heading which can induce additional damage in rock not taken into account in a 2D approach. – A progressive damage of the rock structure in time. Some creep tests and UCS tests have been done by Schmidtke & Lajtai (1985) showing that this granite presents an apparent “viscous” behavior. This phenomenon corresponds to a decrease in the strength in accordance with the loading rate. So, Martin shows that the UCS can decrease 30% if the loading rate reduces from 0.75 MPa/s to 0.0075 MPa/s. At this low loading rate, the measured strength is about 150 MPa. This does not seem enough to justify the field failure, however, we have to be conscious that field loading rates are much lower than those applied in laboratory conditions. The previous strength measured at 0.0075 MPa/s is still not representative of the in situ characteristic. Observed and monitored behavior In these conditions, during digging process and after that, a stress-induced fractured zone has been observed above the crown and below the floor of the gallery, corresponding to a brittle failure mechanism generating a classical v-shape notches. Outside of these highly stressed zones, some Microseismic events could be monitored (Cai et al. 2001) but no major damage could be observed. 2.2 Laboratory tests Some usual unconfined compressive tests have been done by Read et al. (1998) on the Lac du Bonnet granite. These tests provided following mechanical characteristics (Fig. 3): – short term UCS strength: f 213 MPa – compressive stress corresponding to the volumetric strain reversal: cd 160 MPa – compressive stress corresponding to the initiation of the crack growth: ci 90 MPa. If we only consider the short term strength, it’s impossible to justify the appearance of the broken zone in highly compressive zones, considering initial state of stresses. Martin shows if we want to find by simulation the occurring of the failure, it’s necessary to consider a limit strength threshold around 100 MPa. So some reasons have to be found to explain this strong decreasing of the strength in field in comparison with laboratory measurements. These explanations could be: Some other observed phenomena on site suggest that there is a significant time behavior of the ground. As we see on the Figure 2, the roof spalling failure appeared progressively during several months. Similarly, some acoustic emissions have been registered several years after the digging of this gallery. Time behavior seems to be the major phenomenon, which can explain and justify the reduction of laboratory – The well-known “scale effect”. A decrease in strength is generally observed with increasing elementary size of rock. In the case of the Lac de 36 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 37 strength from 213 MPa to a field strength less than 100 MPa. 2.3 Definition of the in situ strength The elastoplastic strain component ensues from the following non-associated rule function: In the case of an undamaged rock like the Lac de Bonnet Granite, the apparent in situ strength under a specific confinement state is function of the loading rate. As this loading rate is much lower than these applied in laboratory conditions, it seems reasonable to assume that the in situ strength could be assimilated to the long term strength of the rock. From laboratory tests, this “long term strength” could have several definitions: (3) The hardening of the elastoplastic mechanism is only negative. Some specific hardening laws were defined for each of the internal parameters m, s and a, allowing to describe the evolution of the rock sample strength from maximum peak value to the residual state. The softening behavior domain reached after the peak strength, is assumed to be divided in three phases: – From Sangha et al. (1972), the long term strength corresponds to the volumetric strain reversal. Above this threshold, the crack growth is assumed to be “unstable”. This notion of “instability” seems totally subjective because is related to the delay allowed by experimental testing in laboratory conditions. – From Morlier (1966) or Wiid (1970), the long term strength of rocks is assumed to be correlated to the beginning of the crack initiation. Assuming that the time behavior of cohesive materials like rocks is associated with a crack growth, this definition appears more physical. It’s in accordance with experimental results carried out by Schmidtke which show that the long term strength of the Lac de Bonnet granite could be estimated to 90 to 100 MPa. This threshold corresponds to the crack initiation under unconfined conditions. 1. The first phase of softening corresponds to a deterioration of the rock’s cementation illustrated by a progressive disappearance of the cohesion at the macroscopic scale. This first phase is associated with an increasing of the dilatancy. 2. The second phase corresponds to the shear of an induced fracture. It’ s associated with a decreasing of the dilatancy at the macroscopic scale. 3. Finally, the last domain corresponds to a purely frictional behavior, which defines the residual strength. The shear occurs without any volumetric strain. A viscoplastic version of the model has been developed after that. This version is based on the Perzyna’s theory, which assumes that the viscoplastic strain rate is a function of the distance between the loading point representative of the state of stresses and a yield viscoplastic surface, in accordance with the following flow rule: 3 CONSTITUTIVE MODEL 3.1 General principles A new constitutive model has been suggested by Laigle (2003) aiming at accurate simulation of the rock behavior in the averaged and large strains domain. A first version of the model has been initially developed in the framework of the elastoplastic theory. In this case, the yield surface corresponds to a generalized form of the Hoek-Brown criterion. Internal parameters “m”, “a” and “s” change according to an hardening variable p: (4) O(F) is a flow function and F is the overstress function. Their expressions are followings: (5) (1) (2) (6) 37 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 38 Figure 7 shows simulations of a triaxial test with 10 MPa of confinement at several strain rates. In accordance with the increasing of the strain rate, we may observe both an increasing of the peak strength and a changing of the behavior. It appears that the lower the rate, the more the rock behaves like a ductile material. An important assumption is to assume that the hardening parameter rate p corresponds to the total irreversible strain as follows: (7) and the global strain rate is the following: 1000 (8) Deviatoric stress (MPa) 900 el is the elastic strain rate. ep is the elastoplastic strain component. vp is the viscoplastic strain component. 3.2 Identification of parameters 800 Initial elastoplastic loading surface 700 600 500 Residual strength criterion Strength criterionfor a -5 Strain rate of 10 /s 400 300 200 Viscoplastic yield surface 100 0 Three main sets of rheological parameters are needed for the constitutive model: 20 0 40 60 80 100 Minimal principal stress (MPa) 1. Parameters describing the elastic reversible behavior which is assumed to not be time dependent. 2. Parameters affected to the elastoplastic instantaneous mechanism: Four variables c, mpeak, speak and apeak are needed to describe the initial elastoplastic loading surface. This yield surface characterizes the rock strength for a very large loading rate. An additional parameter defines the residual strength criterion. Some few additional parameters describe the hardening kinetic of the loading surface from its initial position to the residual state. Another set of variables describes an intermediate criterion corresponding to the stress threshold when the apparent cohesion vanishes. 3. Parameters affected to the viscoplastic time dependent mechanism: Two parameters are needed to describe the yield viscoplastic surface. This yield corresponds to the damage criterion, which is the initiation of cracking and so of the dilatancy. Up to now, in this constitutive model, it has been assumed to be a Tresca surface described by the parameters c, m0, and s0. Figure 4. Stress criteria held for the Lac de Bonnet granite. 300 UCS (MPa) Experimental results 200 150 100 50 0 1,E-12 1,E-10 1,E-08 1,E-06 1,E-04 1,E-02 1,E+00 -1 Strain rate (s ) Figure 5. UCS vs. strain rate – comparison simulation– experience. 1hour 1day 1month 1year 100 Deviatoric stress level (%) 90 The kinetic of viscous behavior is adjusted by 2 parameters “n” and “A” intervening in the flow rule (equation 5). These 2 parameters are identified using 2 types of experimental results: – Evolution of the strength in accordance with the loading rate (Martin & Read 1992). Comparison of these experimental results and simulations is shown in Figure 5. – Evolution of the creep time to failure versus the deviatoric creep level (Schmidtke & Lajtai 1985). The adjustment of the theoretical curve is shown in Figure 6. 80 70 Experimental results 60 50 Simulation 40 30 20 10 0 1 10 100 1000 10000 100000 1000000 10000000 1E+08 1E+09 Time to failure (s) Figure 6. Time to failure vs. deviatoric stress level – comparison simulation–experience. 38 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Simulation 250 09069-05.qxd 08/11/2003 20:16 PM Page 39 ε =1.5.10-4/s 250 Deviatoric stress (MPa) Table 1. Definition of domains in accordance with the rock damage. . 300 . ε =1.5.10-5/s . 200 ε. =1.5.10-6/s ε =1.5.10-7/s . ε =1.5.10-8/s . ε =1.5.10-9/s 150 100 50 0 0 0,1 0,2 0,3 0,4 0,5 Axial strain(%) Figure 7. Simulation of triaxial tests at several strain rates (confinement: 10 MPa). 3.3 Results interpretation The formulation of the constitutive model and some initial assumptions allow interpretation of computational results in accordance with physical criteria. For example, it’s not always interesting and useful to estimate accurately ground displacements induced by a tunnel excavation in a hard rock. In this case, the collapse mechanism is a brittle failure, which occurs violently and rapidly without any significant advance movements. In the framework of studies for underground nuclear waste repositories in hard rock masses, it seems more accurate to interpret numerical simulations using 2 kinds of results: of the dilatancy. As long as the rock stays only fissured, the dilatancy increases. As soon as the rock is fractured, the dilatancy starts to decrease. It’s an important aspect because we assume that the evolution of the rock mass water permeability is a function of this irreversible volumetric strain, in accordance with the following equation: 1. The physical state of the rock. 2. The evolution and increasing of the rock mass permeability. Depending on the hardening level, which is characterized by the hardening variable p, it’s possible to estimate qualitatively the local damage of the rock (see Table 1). (9) – If p 0, the rock is assumed to be intact. – During the first phase of softening, as long as the parameter s(p), so the cohesion, is not null, the rock is assumed to be fissured. This cracking may be generated by a stress variation (activating of the plastic mechanism) or/and the delayed behavior (activating of the viscoplastic mechanism). – As soon as the parameter s(p) becomes equal to zero, it’s assumed that the cohesion at a macroscopic scale is null. Physically, this corresponds to the creation of a continuous fracture crossing the elementary volume of rock. The global mechanical behavior is governed by the mechanical response of the fracture under a shear loading. During the fracture slide, the dilatancy will involve until a residual state is reached. where k is the current permeability, k0 is the ground’s initial permeability, pv is the volumetric plastic strain induced by the load, and is an adjustment parameter for the model. Figure 8 shows the theoretical evolution of the relative permeability in accordance with the deviatoric stress for a triaxial test under 10 MPa of confinement. The strain rate is assumed to be 1.5 10 5/s. 4 SIMULATION OF THE GALLERY The excavation of the gallery is simulated in two phases: – Phase 1: Simulation of the short term behavior during excavation phase using the elastoplastic The main difference between these two last physical configurations of the rock is the associated evolution 39 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 40 Two domains may be observed: version. The excavation is simulated by a progressive reducing of initial stresses existing on the gallery perimeter. – Phase 2: Simulation of the long term behavior of the gallery, using viscoplastic version of the constitutive model. Initial state of stresses and hardening parameters at the beginning of this phase are issued from the previous elastoplastic step. This long term analysis is continued until a stabilization of mechanical phenomena. 4.1 1. An intact zone rock: In this area, the stress level is enough low to be under the damage criterion. No viscoplastic strains will appear in this domain without significant change of the stress diagram around the tunnel. 2. A fissured rock zone: At the roof and under the invert, one part of the rock mass is fissured. In this domain, the state of stresses is sufficient to be above the damage criterion. Some viscoplastic strains may be created in time in these zones. With the time, viscoplastic strains will appear in these zones, generating a negative hardening of the peak criterion. If this hardening is sufficient, the rock will locally loose its cohesion. At this moment, it could assume that a macroscopic induced fracture has been created. Damage of rock at short term The Figure 9 shows the state of the rock around the gallery at the end of the excavation process. Deviatoric stress (MPa) 350 300 Simulation 4.2 Evolution of the damage in time 250 Figures 10–13 show the increasing of a fractured zone above and below the gallery in time. There is an Experimental results 200 150 Strain rate: 1.5e-5/s Confinement: 10MPa 100 Fractured rock 50 0 0,1 1 10 100 1000 10000 Relative permeability k/k0 Figure 8. Lac de Bonnet granite – evolution of the permeability during a triaxial test in the pre-peak domain (confinement: 10 MPa). Tension zone Fissured rock 2 months Figure 10. Theoretical damage state of rock after 2 months. Fractured rock Fissured rock Tension zone Intact rock Fissured rock 2 years Figure 9. Damage state of rock around Mine-by test tunnel at the end of the excavation. Figure 11. Theoretical damage state of rock after 2 years. 40 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 41 the fracturing process over several years. From these theoretical results, it’s only after several hundred years that this fracturing will stop. From available information, it seems that the expansion rate of the fractured zone is too slow and that this evolution has been more rapid (apparently, few months). However, we have to precise that only one simulation has been done using set of parameter coming from an adjustment with few laboratory tests. It could be now possible to do a back analysis and to adjust some parameters like “n” and “A” which govern the kinematic viscoplastic strains creation. Despite this, after several years, when the stabilization is reached, the shape of the fractured zone is similar to these which has been observed on site (Fig. 14). Figure 15 shows the evolution of the parameter “s” versus time. This variable represents the cohesion of expansion of the fracturing in the vicinity of the tunnel in time. This expansion is located in highly stressed zone. Table 2 presents the depth of the fractured zone at the crown versus time. In accordance with rheological parameters estimated before, the simulation shows an evolution of Fractured rock Tension zone Fissured rock 4 years Figure 12. Theoretical damage state of rock after 4 years. Fractured rock Tension zone Fissured rock 10 years Figure 14. Theoretical damage state of rock at long term. Figure 13. Theoretical damage state of rock after 10 years. 1 Parameters ⇔ Damage indicator 0,9 Table 2. Depth of the fracture zone versus time. Time Thickness of the fractured zone in roof (cm) 2 months 1 year 2 years 4 years 10 years 100 years 500 years 1,000 years 10,000 years 15 20 25 35 43 54 65 65 65 0,7 0,6 0,5 0,4 0,3 Distance to the crown 71 cm 0,2 Distance to the crown 60 cm 0,1 0 50 cm 0 100 200 300 400 500 600 700 800 900 1000 Years Figure 15. Evolution of the damage indicator versus time above the crown. 41 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0,8 09069-05.qxd 08/11/2003 20:16 PM Page 42 4.3 the rock. As long as it isn’t null, the rock is intact or only fissured. So, this value is a damage indicator. According to rheological parameters retained, Figure 15 show that the damage continues to involve during a long time. This is the case at a distance of 71 cm above the crown. Even if the kinematic is not correct in our simulations, the results are similar to monitoring results. Several years after the end of excavation, some acoustic emissions have still been registered while fracturing process is stopped. Evolution of the permeability in time Damage of the rock results in a local increasing of the rock macroscopic hydraulic permeability. This increasing is in accordance with the suggested Equation (9). Considering the parameter , which has been adjusted on triaxial test results, this increasing could be about several order of magnitude in the fractured zone. Figures 16–19 show increasing of the permeability around the mine-by test tunnel. 20 days 2 ans Log(k/k0) Log(k/k0) Figure 16. Increasing of the hydraulic permeability after 20 days. Figure 18. Increasing of the hydraulic permeability after 2 years. 1000 years 2 months Log(k/k0) Log(k/k0) Figure 19. Increasing of the hydraulic permeability after 1000 years. Figure 17. Increasing of the hydraulic permeability after 2 months. 42 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-05.qxd 08/11/2003 20:16 PM Page 43 Considering there is little published information about creep behavior of the Lac de Bonnet granite, a simulation has been done using this model. It seems that this time behavior could justify the apparent low field strength of the granite in comparison with the laboratory strength. Only one computation has been done and parameters of the model have not been adjusted after this first simulation. However, even if the kinetic seems too slow, this computation allows us to find the observed failure mechanism in high stressed zones, associated with an increase of hydraulic permeability. k/k0=10 k/k0=100 Fractured zone REFERENCES Cai, M., Kaiser, P.K. & Martin, C.D. 2001. Quantification of rock mass damage in underground excavation from microseismic event monitoring. Int. J. Rock Mech. & Min. Sci. 38, 1135–1145. Diederichs, M.S. 2002. Stress induced damage accumulation and implications for hard rock engineering. In Hammah et al. (eds), NARMS-TAC 2002, 7–10 July 2002. University of Toronto press. Eberhardt, E, 2001. Numerical modelling of three-dimension stress rotation ahead of an advancing tunnel face. Int. J. Rock Mech. & Min. Sci. 38, 499–518. Laigle, F. 2003. Modélisation rhéologique des roches adaptée à la conception des ouvrages souterrains. Ph.D. Ecole Centrale de Lyon, in prep. Morlier, P. 1966. Le fluage des roches. Annales de l’institut technique du bâtiment et des travaux publics: 80–111. Read, R.S., Chandler, N.A. & Dzik, E.J. 1998. In situ strength criteria for tunnel design in highly-stressed rock masses. Int. J. Rock Mech. & Min. Sci. 35, 261–278. Sangha, C.M. & Dhir, R.K. 1972. Influence of time on the strength, deformation and fracture properties of a lower Devonian sandstone. Int. J. Rock Mech. & Min. Sci. 9, 343–354. Schmidtke, R.H. & Lajtai, E.Z. 1985. The long-term strength of Lac du Bonnet Granite. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 22, N°6, 461–465. Wiid, B.L. 1970. The influence of moisture on the prerupture fracturing of two rock types. Proc. 2nd Cong. Int. Soc. Rock Mech., Belgrade. 239–245. Figure 20. Increasing of the hydraulic permeability at very long term. Figure 20 shows that the increasing of hydraulic permeability is not only located in the fractured zone, but also in the fissured zone. Possible permeability changes in tension zones are not taken into account in the presented approach. 5 CONCLUSION Even in very hard rock like granites, a delayed behavior could exist. This phenomenon could result in a progressive damage of rock and delayed failure. Several microscopic theories have been suggested to physically justify this delayed behavior in cohesive rocks. One explanation could be a “stress corrosion” in high stressed zones around existing and induced cracks. The aim of the presented work here was to suggest a macroscopic and phenomenological model taken into account this behavior. This model has been integrated in EDF’s local version of FLAC. 43 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-06.qxd 08/11/2003 20:17 PM Page 45 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Prediction of deformations induced by tunneling using a time-dependent model A. Purwodihardjo & B. Cambou Laboratoire de Tribologie et Dynamique des Systèmes, Ecole Centrale de Lyon, France ABSTRACT: Since the past 30 years, the research for estimating an accurate prediction of deformations induced by tunneling has been a major engineering challenge all around the world. The in situ measurements have shown that deformations of the soil on the vicinity of a tunnel show a strong evolution with time. Three essential phenomena, actually, can be related to this evolution: the evolution with time of the distance to the working face, the distance of the lining to the working face and the viscous effects occurring in the soil. The objective of this paper is to propose a procedure for predicting the deformations induced by tunneling, by taking into account these three essential phenomena, particularly the third phenomenon. Therefore, in this study, a constitutive model for the time-dependent behavior of cohesive soil has been developed within the framework of elastoplasticity–viscoplasticity and critical state soil mechanics. The consideration of viscoplastic characteristic sets the current model apart from the CJS model, and introduces an additional viscous mechanism. The evolution of the viscous yield surface is governed by a particular hardening called “viscous hardening” with a bounding surface. To describe this procedure and the capability of the model, a comparison between numerical calculations and monitoring the progressive closure of tunnel conducted in the TGV tunnel of Tartaiguille, is performed. The finite difference software, Fast Lagrangian Analysis of Continua (FLAC), has been used for the numerical simulation of the problems. The comparison results show that the observed deformations could have been reasonably predicted by using the proposed excavation model. been developed in the Ecole Centrale de Lyon, to analyze the influence of these essential phenomena in the prediction of deformations induced by tunneling by using numerical methods. This model is within the framework of elastoplasticity–viscoplasticity from the basic elastoplastic model (CJS model) including an additional viscous mechanism. 1 INTRODUCTION The behavior of a tunnel is greatly influenced by the characteristics of the soils and the tunneling procedure. They will give a strong influence to the initial and long term deformations on the vicinity of a tunnel and on the ground surface, particularly when the ground traversed by tunnels has poor geotechnical characteristics: little or no cohesion, medium-high deformability and high viscosity. In this area, more considerations should be taken because deformations of the soil on the ground surface and on the vicinity of a tunnel show generally a strong evolution with time. This evolution is essentially related to three phenomena, i.e. the evolution with time to the distance from the working face (the advance rate of tunneling), the distance of the lining to the working face and the viscous effects occurring in the soil. To predict the deformations induced by tunneling by considering these three phenomena, a better understanding of these influences and proper considerations of their effects on the support design and installation are required. Therefore a time-dependent model has 2 DESCRIPTION OF THE MODEL The CJS model is a constitutive model with different hierarchical levels which has been developed 16 years ago in the Ecole Centrale de Lyon (Cambou & Jafari 1987, Maleki 1998). This model is based on nonlinear elasticity and two mechanisms of plasticity. It also takes into account the dependency on density of geomaterials through the critical state. The rate of the strain tensor can be decomposed into an elastic part and a plastic part. The plastic deformations consist of an isotropic and a deviatoric mechanism. Figure 1 shows the two plastic mechanisms in the CJS model. 45 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-06.qxd 08/11/2003 20:17 PM Page 46 The yield surface’s evolution is defined by an isotropic hardening mechanism depending on a scalar variable Q and Tr is a parameter of the model to take into account the cohesion. The hardening rule has the form: (5) The isotropic flow rule is described as: (6) Figure 1. Plastic deviatoric mechanism and plastic isotropic mechanism in CJS. Kp is the plastic bulk modulus and n is a parameter of the model which can be determined by experimental test. i is a plastic multiplication for the isotropic plastic mechanism. The total strain of the model is decomposed in four parts: (1) 2.1.3 Deviatoric plastic mechanism In the deviatoric plastic mechanism, for the sake of simplicity no kinematic hardening but only isotropic hardening is taken into account (CJS level 2). The yield surface can be written as: The first part is an elastic mechanism, the second part is an isotropic plastic mechanism, the third part is a deviatoric plastic mechanism and the last part is concerned with an added viscous mechanism. 2.1 (7) Brief description of the basic elastoplastic model of CJS 2.1.1 Elastic mechanism The elastic law is given by the following incremental nonlinear relation: where is a parameter of the model and Tr is a parameter of the model to take into account the cohesion. The evolution of the yield surface is characterized by the evolution of R with the internal variable p. The relationship between R and p is written as: (2) where I1 and S are the first invariant and the deviatoric part of stress tensor while K and G are the bulk and shear modulus, respectively, which depend on the stress state through a power law: (8) (3) where Rm is a parameter that corresponds to a radius of the rupture surface and A is a parameter of the model. The evolution of p is defined by: K oe , Go and n are parameters while Pa is a reference pressure which equals to 100 kPa. (9) 2.1.2 Isotropic plastic mechanism The yield surface associated to this mechanism is a plane perpendicular to the hydrostatic axis. The yield surface is given by: The deviatoric flow rule is given by: (10) (4) 46 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-06.qxd 08/11/2003 20:17 PM Page 47 where d is a plastic multiplication for the deviatoric plastic mechanism. The deviatoric potential function (gd) used in relation (10) corresponds to a non-associated plastic mechanism. Tensor nij is a symmetrical tensor so that tr(n2) 1 and is a tangential tensor to the surface corresponding to the potential function. It is defined by: S11 rupture surface (Rm) characteristic surface (Rc) yield surface (R) (11) S22 S33 Figure 2. Different surfaces in the deviatoric mechanism of the CJS model. c where SII represents the second invariant of the deviatoric part of stress in the characteristic state and is a parameter of the model. The characteristic surface is defined by: SII (12) e°II where Rc represents the radius of the characteristic surface. Figure 2. shows the deviatoric mechanism in the CJS. eII Figure 3. Strain softening behavior in the CJS model. 2.1.4 Critical state Two important phenomena can be noted from the drained triaxial tests: 2.1.5 Strain softening model The CJS model takes into account the strain softening behavior of the soil which depends on the accumulated deviatoric strain. This model is made up of three portions, an elastoplastic portion up to the peak strength, a softening portion in which the strength (Rc and Tr) reduces from the peak to residual, and finally, a constant residual strength portion. Figure 3 shows the strain softening behavior in the CJS model. – an increase of the peak resistance with the initial density of material – the material tends to the state called the critical state characterized by a null volume variation and a ratio q/p constant independent to the initial density. To take into account these phenomena and be inspired by the formulation developed by the Cambridge University (Roscoe et al. 1968), in this model, the radius of rupture surface varies as a function of the mean effective stress and the density of material. For simplifying the problem, we take the critical state similar to the characteristic state. Hence, the evolution of rupture surface is defined by: 2.2 Viscous hardening with a bounding surface The viscous effect of the soil is connected with an internal characteristic. This internal characteristic is represented by a creep surface which is bounded by a (current) state of stress surface defined by Equation 20. It means that the creep surface can evolve but the evolution is limited by the state of stress surface. So we call this function as a viscous hardening with a bounding surface, where the bounding surface in this case is the state of stress surface. Meanwhile, the evolution of state of stress surface is limited by the yield surface (elastoplastic concept). The evolution of the yield surface is limited by the rupture surface. Figure 4 shows an illustration of viscous evolution concept with a bounding surface. The basic formulation for this viscous mechanism is inspired by the overstress model of Perzyna (1966). (13) where is a parameter of the model, and pc is a critical pressure which is defined by: (14) where c is a parameter of the model, pco is a critical pressure corresponding to the initial density and v is an accumulated volume strain. 47 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands e IIf 09069-06.qxd 08/11/2003 20:17 PM Page 48 reduced as a passing point between the primary creep and the tertiary creep. This idea has been selected for the sake of simplicity in the measurement of the distance in the stress space. This function is defined by: S11 rupture surface (Rm) yield surface (R) state of stress artificial state of stress surface (Re) creep surface S22 (17) where Rv corresponds to the radius of the creep surface. The power m in relation (16) is defined as: S33 (18) Figure 4. Illustration of viscous evolution concept with a bounding surface. where m1 and m2 are parameters of the model. Lade (1998) has shown in his laboratory test results that the potential plastic surfaces for the elastoviscoplastic and the elastoplastic are homothetic. Based on this idea, it means that the direction of the plastic strain in the elastoplastic is similar to the viscoplastic one. Thus, the direction of viscoplastic strain is defined as: To keep on near to the framework of elastoplasticity is the reason of the use of this formulation. The idea is, then, starting from the general framework of elastoplasticity and introducing the viscosity of material and the retardation function. Many authors, (Katona 1984, Adachi 1982, Sekiguchi 1984, etc.), have employed this formulation, and they have shown that this model is incapable to introduce the acceleration deformation phenomenon in the case of tertiary creep. Therefore the ambition of the proposed model is to take into account the tertiary creep. Thus, three important terms have to be defined in the framework of this model. The first one is the viscosity of the material, the second one is the function of retardation and the last one is the direction of the viscoplastic strain. The function is as follows: (19) where f e is the artificial state of stress surface which is homothetic to the yield surface for the deviatoric mechanism. It is defined by: (20) The rupture surface is defined by: (15) (21) The creep surface is defined by: where 1/ is the viscosity of the material, (v/r) is the function of retardation and Gijvd is the direction of the viscoplastic strain. The viscosity of the material in this model is a function of the distance of the state of stress surface (Re) to the rupture surface (Rm). This function is defined by: (22) The evolution of the creep surface is given by: (23) (16) Av is a parameter of the model and eIIvd is an accumulated deviatoric viscoplastic strain, which is defined by: where 0 is a parameter of the model, Re is the radius of the current state of stress surface, Rm is the radius of the rupture surface and k is a parameter of the model. The function of retardation, (v/r), is inspired by the bounding surface theory (Kaliakin & Dafalias 1990). This function will drive the evolution of the three types of creep. The secondary creep will be (24) where eII is the deviatoric viscoplastic strain rate. 48 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-06.qxd 08/11/2003 20:17 PM Page 49 2.3 Parameter identification The identification of elastoplastic mechanism parameters can be determined by using classical laboratory tests. The procedure for calibrating the model parameters is briefly defined by Maleki (1998). Drained or undrained creep tests with different levels of stresses have to be achieved, for identifying the viscoplastic mechanism parameters. 3 CALCULATIONS 3.1 Presentation of the case studied The tunnel of Tartaiguille is located on the new Méditerranée high-speed line between Valence and Montélimar (France). All these structures are double track single tube, allowing a speed of 300 km/h. It crosses fractured limestones on the north sides, stiff marls and sandstones in the south and stampian clays in the central parts. Convergence measurement devices had been installed by CETU (Centre d’Etudes des Tunnel) during construction processes in this tunnel. PM 1168 has been selected as the simulation data in the deformation analysis. From the cross section of the tunnel, we can see that the support of the tunnel is a shotcrete (thickness 300 mm) and a steel frame every 1.5 m. Five samples of soil blocks have been obtained and these blocks have been studied in detail by Serratrice (1999) (Laboratoire Regional des Ponts et Chaussées d’Aix en Provence). From the five samples of soil blocks, three layers of soil can be concluded at that section. The upper one is the black marl, the middle one is the calcareous marl and the lower one is the grey marl. The soil characteristics for the black marl and the grey marl are almost similar, on the other hand the soil characteristics of the calcareous marl is significantly different. The calcareous marl is stiffer than the black marl and the grey marl. For the sake of simplicity, only two types of soil will be considered, for the upper one and the lower one, we will use the same parameters of soil. Figure 5 shows the dimensions of the tunnel and Figure 6 shows the soil stratigraphy at section PM 1168. 3.2 Figure 5. Dimensions of tunnel and the measurement position. Figure 6. Soil stratigraphy at section PM 1168 (Lunardi 2000). the elastoplastic and viscoplastic behavior of the soils quite satisfactory. 3.3 Model parameters Elastoplastic and viscoplastic model parameters identified in this analysis can be seen in Table 1. Figure 7 shows the simulation results of triaxial tests by using elastoplastic soil parameters. Figure 8 shows the simulation results of creep tests by using elastoviscoplastic soil parameters. It can be seen from Figures 7 & 8 that the simulation results are closely match to the experimental results. It means that this model can take into account 49 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Plane strain calculations The convergence curves of the unsupported tunnel are derived by using virtual support pressures in plane strain calculations (Panet et al. 1982, AFTES 2002). Two types of calculation have been achieved. The first one is by using the circular shape of tunnel with R 7.65 m, Ko 1.0, only used one soil layer (black and grey marl parameters) and the second one is by using the actual shape of the tunnel, the actual value of Ko 1.2 and the actual soil layers. The objective of this calculation is to get the result comparison between those two shapes for the reason that the calculation taking into account the distance from the working face and the progressing of the tunneling 09069-06.qxd 08/11/2003 20:17 PM Page 50 Table 1. Parameters of the model. Parameter Black and grey marl Calcareous marl Density (kN/m3) 22.15 Elastoplastic parameters Go (MPa) 27 K oe (GPa) 139 Rm 0.103 0.3616 Rc pic(Rc res 0.0784 A (kPa 1) 2 n 0.7 p K o (MPa) 139 c 60 0.033 pco (MPa) 17 0.005 Tr pic (MPa) 11.825 Tr res (MPa) 7.112 e°II 0.02 ef 0.065 96.15 208.33 0.2661 0.7852 0.213 25 0.6 208.33 75 0.05 40 0.38 8.768 5.273 0.02 0.065 Viscoplastic parameters Av 125 0 108 k 6.0 m1 0.3 m2 0.0 450 106.69 30.673 0.4083 8.0214 II 24.34 Figure 8. Simulation results of drained creep tests. On the other hand, in the second calculation, the two type parameters of soil have been employed. The overburden pressure height is 100 m from the crown of the tunnel and the ground water table is 6.6 m from the ground level. 3.4 This computation is performed by using the sequential excavation method (SEM) in the axisymmetric condition. Distance to the working face is defined by d and the advance rate of the excavation is defined by p. Figure 12 shows the geometry of the tunnel in the axisymmetric calculation and Figure 13 shows the mesh used in the axisymmetric calculation. Figures 9 and 10 show the mesh used in the plane strain calculations and Figure 11 shows the comparison results of the convergence analysis between the two shapes. From that figure, the ratio between the actual shape and the circular shape can be determined, and we get the ratio of convergence (RT) at position F-G equal to 0.885. This ratio will be used for adjusting the axisymmetric calculations, since in the axisymmetric calculations, we can only use the circular shape, Ko 1.0 and one layer of the soil. The lining support in this tunnel is a combination of shotcrete ring and steel frame. So for simplifying the analysis, the equivalent stiffness of the lining of Figure 7. Simulation results of triaxial tests. will be performed using an axisymmetric calculation (see section 3.4). The result comparison obtained in the plane strain calculation allows the results obtained in the axisymmetric condition to be transformed to take into account the actual conditions of the tunnel section. This approximation has been taken because those two shapes are almost similar and the calcareous marl is not dominant. In the first calculation, a quarter of tunnel geometry has been modeled, and for the soil parameters, the black and grey marl parameters have been employed. 50 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Axisymmetric calculation 09069-06.qxd 08/11/2003 20:17 PM Page 51 Figure 11. Comparison of convergence analysis. Figure 9. Mesh used in the circular shape. Figure 12. Geometry of the tunnel in the axisymmetric calculation. In this calculation, the black and grey marl parameters have been used because they are more dominant than the calcareous marl in the soil stratigraphy. For the SEM analysis, we use d 1.5 m and p 1.5 m (the advance length of tunneling). The convergence of the tunnel is determined by: Figure 10. Mesh used in the actual shape. the combined lining support has been used. The elastic model is used for this lining. For modeling the ground anchor on the working face, the equivalent pressure on the working face has been used (Peila 1994). This pressure is determined by: (26) where U(x) is the deformation of the tunnel as a function of the distance from the working face, U(o) is the deformation of the tunnel on the working face and RT is the shape ratio of the tunnel. In the first simulation, the influence of the advance rate of the tunneling is illustrated. Three types of the advance rate are used, i.e. 1.5 m per 0.5 day, 1.5 m per 1.0 day and 1.5 m per 2.0 day. The elastoplastic calculation is used to represent the infinite advance rate of tunneling. The tunneling simulation results can be seen in Figure 14. From that figure, we can see that the convergence of the tunnel can be reduced by increasing the (25) where n is the number of the anchor, Tb is the tensile strength of the anchor, Sb is the shear strength of the anchor, and S is the working surface area. In this tunnel, 120 fiberglass anchors with 800 kN tensile strength have been installed on the working face to stabilize the working face. 51 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-06.qxd 08/11/2003 20:17 PM Page 52 Figure 13. Mesh used in the axisymmetric calculation. Figure 15. Convergences of the tunnel as a function of the distance from the working face. also in the case of viscous materials, a time-dependent model is very essential. 4 CONCLUSIONS Analysis of deformations due to tunneling using the elastoplastic–viscoplastic constitutive model has been performed in this study. It has been demonstrated that the influence of viscous effects cannot be neglected in the soil which has been analyzed. It means that the role of the timedependent model in this case is very important and a necessity. The influence of viscous effects can be reduced by increasing the advanced rate of tunneling but an attention to the lining should be taken because the load transfer to the lining will be higher. This becomes significant when there is a large distance between the installation point of the lining and the working face, and could induce plastic deformation around the tunnel. The calculation procedure proposed has provided an effective approach for analyzing the ground-structure interaction situation and offers a systematic way of optimizing lining design. This kind of calculation can be improved by using a complete 3D approach. However, this is a rather difficult calculation and the computation time will be long. In practice, the 2D and the axisymmetric analysis can be successfully used to develop a pragmatic solution. The constitutive model, which has been presented, is quite satisfactory to model the elastoplastic– viscoplastic behavior of the soils. The parameters of the model can be identified by using the classical laboratory tests such as, triaxial tests and creep tests. Figure 14. Tunneling simulation results in elastoplastic– viscoplastic calculation. advance rate of tunneling. Actually, in this case, we prevent the evolution of the creep deformation. If we only use the elastoplastic constitutive model, we cannot illustrate this phenomenon. The total elastoviscoplastic deformation of the tunnel could be two or three times bigger than the elastoplastic deformation. In the second simulation, the actual advance rate of tunneling at section PM 1168 is used. The sequences of the actual excavation are as follows: 1. Excavating the upper section with the advance rate: 1.5 m per 0.5 day until 6 m. 2. Stopping for one day (to represent the excavation of the lower section). 3. Continuing the excavation of the upper section with the advance rate: 1.5 m per 0.5 day until 6 m. 4. Stopping for three days (to represent the excavation of the lower section and the installation time of the anchors on the working face). 5. Continuing the four sequences above. The measurements at position F-G (see Figure 5) are started after 6 m from the working face. Figure 15 shows the comparison results between the calculations and the measurements started from the first measurement (6 m from the working face). From those figures we can see that the model proposed can model acceptably the viscous behavior of the soil in the tunneling. It has been demonstrated ACKNOWLEDGEMENTS The authors gratefully acknowledge the information on the geotechnical data and the convergence data provided 52 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-06.qxd 08/11/2003 20:17 PM Page 53 Lunardi, P. 2000. The design and construction of tunnels using the approach based on the analysis of controlled deformation in rocks and soils, ADECO-RS. Maleki, M. 1998. Modélisation hiérarchisée du comportement des sols, Phd. Thesis, École Centrale de Lyon. Panet, M. & Guenot, A. 1982. Analysis of convergence behind the face of a tunnel, Tunneling’ 82: 197–204. Peila, D. 1994. A theoretical study of reinforcement influence on the stability of a tunnel face, Geotechnical and Geological Engineering, 12. Perzyna, P. 1966. Fundamental Problems in viscoplasticity. Advances in Applied Mechanics, Vol. 9: 243–377. Roscoe, K. H. & Burland, J. B. 1968. On the Generalised Stress-Strain Behavior of ‘Wet Clay’, Engineering Plasticity, J. Heyman and F. A. Leckie (Eds). Cambridge: Cambridge University Press: 535–609. Sekiguchi, H. 1984. Theory of undrained creep rupture of normally consolidated clays based on elastoviscoplasticity, Soils and foundations, Vol. 24, No. 1: 129–147. Serratrice, J.F. 1999. Tunnel de Tartaiguille (Drôme) TGV Méditerranée, Essais de laboratoire sur la marne, LRPC d’Aix en Provence. by Mr. Alain Robert and Mr. Adrien Saïtta from CETU (Centre d’Etudes des Tunnel), Lyon, France. REFERENCES Adachi, T. & Oka, F. 1982. Constitutive equations for normally consolidated clays based on elasto-viscoplasticity, Soils and foundations, Vol. 22, No. 4: 57–70. AFTES. 2002. La méthode convergence-confinement, Tunnels et ouvrages souterrains, No 170: 79–89. Cambou, B. & Jafari, K. 1987. A constitutive model for granular materials based on two plasticity mechanisms. Constitutive equations for granular non-cohesive soils, Saada & Bianchini (Eds), Balkema, Rotterdam: 149–167. Kaliakin, N. & Dafalias, F. 1990. Theoretical aspects of the elastoplastic-viscoplastic Bounding surface model for cohesive soils, Soils and foundations, Vol. 30, No. 3: 11–24. Katona, M. G. 1985. Evaluation of viscoplastic cap model, Journal of Geotechnical Engineering, Vol. 110, No. 8: 1107–1125. Lade, P. V. 1998. Experimental Study of Drained Creep Behavior of Sand, Journal of Engineering Mechanics, Vol. 124, No. 8, August: 912–920. 53 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-07.qxd 08/11/2003 20:43 PM Page 55 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Modeling of anhydrite swelling with FLAC J.M. Rodríguez-Ortiz Gamma Geotécnica SL, Madrid, Spain P. Varona & P. Velasco Itasca Consultores SL, Asturias, Spain ABSTRACT: Anhydrite and rocks containing argillaceous minerals experience swelling phenomena when they come into contact with water. In tunneling, this can lead to a strong heave of the floor and to a high level of stresses in the lining. Although characterization of swelling potential, monitoring of swelling process, and a lot of relevant case histories of tunnel construction in swelling rocks are currently available, the design of the support in swelling rocks usually do not consider an accurate stress–strain relationship for the swelling. Current trend in tunneling design considers numerical modeling of the rock-support interaction, but the available geotechnical codes do not include the swelling formulation. This paper presents the implementation in FLAC (via FISH routines) of the analytical stress–strain formulation for the swelling presented by Wittke (1999) and the validation of this algorithm against the swelling tests carried out by different authors and presented by Wittke (1999). created at the surface of the grains inhibits the water penetration stopping the process. In the case of interbedded anhydrite-mudstone the swelling process leads into the disintegration of the rock, reducing its strength. Steiner (1993) quantifies this reduction of strength with an angle of friction of 20°. 1 DESCRIPTION OF THE ANHYDRITE SWELLING PHENOMENA 1.1 Chemical description Calcium sulphate appears naturally as two different minerals: gypsum (CaSO4.H2O), in which water appears within the crystalline structure, and anhydrite (CaSO4). The hydration of anhydrite is a complex process that depends on the pressure and the temperature. For ambient conditions the chemical reaction is illustrated in Table 1. With an external inflow of water, the volumetric increment associated to this process is presented in Equation 1: 1.2 The International Society of Rock Mechanics has proposed a set of tests to quantify the swelling of argillaceous rocks: the Maximum Axial Swelling Stress test, the Axial and Radial Free Swelling Strain test, and the Axial Swelling Law test (axial swelling stress as a function of axial swelling strain, or Huder-Amberg swelling test). An illustration of the results from the HuderAmberg swelling test (total vertical strain of the sample, z in %, versus vertical load, z in kPa) is presented in Figure 1, taken from Wittke (1999). Stages 1, 2 and 3 correspond to the initial loading phase with 2 load cycles; stage 4 corresponds to the watering of the sample (no stress increment but strain increment), and finally stage 5 corresponds to the different points of the unload-swelling process. The swelling strain equals the total strain (stage 5 of the test) minus the elastic strain (stages 2 and 3 of the test). If the strain due to swelling is plotted against the stress in a semi-logarithmic scale (Figure 2) a (1) Transformation of anhydrite into gypsum can be inhibited at 20°C with a pressure of 1.6 MPa; this pro cess is reversible, being necessary a pressure of 80 MPa at 58°C to transform gypsum into anhydrite. Previous data refer to pure anhydrite, but in case of interbedded mudstone-anhydrite the maximum swelling volume is lower but the swelling stress is larger (in the order of 2 to 5 MPa). In the case of pure anhydrite, as the hydration pro cess begins, the thin layer of impervious gypsum 55 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Characterization of the swelling behavior 09069-07.qxd 08/11/2003 20:43 PM Page 56 swelling process); Kq swelling deformation parameter; z axial effective stress; and 0 axial swelling stress (as showed in Figure 2, it is intersection of the straight line with qz 0). Table 1. Transformation of anhydrite into gypsum. Equation Mass (gr) Density (gr/cm3) Volume (cm3) Anhydrite water Gypsum CaSO4 H2O 136.14 36 2.96 1 46.2 36 CaSO4.2H2O 172.14 2.32 74.3 2 ANHYDRITE SWELLING LAW The following description of swelling law is taken from Wittke (1999), and starts from the axial stress–strain relationship previously presented (Equation 2). The swelling law only applies for compressive effective stresses, where c (Figure 2) is a minimum stress representing the lower limit of validity of the swelling law. As the swelling strain equals to zero for compressive stresses larger than the swelling stress, the swelling law can be finally formulated as Equation 3: (3) where qi final axial strain due to swelling in the direction i; Kq swelling deformation parameter; i axial stress in the direction i; 0 axial swelling stress; and c minimum limit for the axial stress. Equation 3 represents the 3D (i 1,2,3, means the 3 directions in the space) isotropic (the same Kq parameter is considered for the 3 directions) swelling law, where the final axial strain is reached at the end of the swelling process. This swelling law considers that the principal directions of swelling qi (i 1,2,3) are coaxial with the principal stresses i (i 1,2,3), and therefore the value of the swelling principal strain depends only on the value of its coaxial principal stress. Previous relations refer to the strain reached at the end of the swelling process. Furthermore, for the kinetics of the process Wittke (1999) presents the following Equation 4 for the swelling strain rates at time t: Figure 1. Swelling test of an interbedded anhydritemudstone sample (Wittke 1984, in Wittke 1999). (4) Figure 2. Axial swelling law (Grob 1972, in Wittke 1999). where q swelling time parameter; qi principal swelling strains for t ; and qi(t) principal swelling strains which already occurred until time t. According to Wittke (1999), the time dependence of swelling is adequately described by Equation 4 as long as the strength of rock is not exceeded. The plastic deformations occurring if the rock strength is exceeded lead to a volume increase and to an increase of permeability that accelerates the penetration of water, increasing the swelling strain rate. straight line is obtained which may be described by the relationship presented in Equation 2. (2) where qz final axial strain due to swelling (final deformation means deformation at the end of the 56 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-07.qxd 08/11/2003 20:43 PM Page 57 To consider this effect in the formulation, Wittke (1999) includes the following relationship for the swelling time parameter q, Equation 5: t=0 σi=1,2 = f(σxx ,σyy ,τxy,θ) σ3 = σZZ (5) Principal stresses t< T_fin where a0, ael, avp constant values. The parameter a0 represents the dependence of the swelling velocity on the anhydrite content, regardless of whether a strain occurred before or not; elv is the elastic volumetric strain occurred prior to the beginning of swelling that also influence the permeability; pl v is the volumetric plastic strain; and maxEVP represents an upper limit of the plastic volume strain with regard to an eventual acceleration of swelling. According to Wittke (1999), plastic volumetric strains larger than maxEVP do not lead to a further increase of the swelling velocity because the penetration of water into the rockmass cannot be further accelerated by these. Following Equation 5 the swelling time parameter is no longer constant but dependent on time as elastic – plastic volumetric strains varies during the swelling process. εqi` = Kq log σi ∆σi = f(εqi` ;λ,G) 1 ηq Swelling strains Associated stress increment =a +a . εelv +a . min{εplv , maxEVP} Swelling time 0 el vp parameter 10%. σi ∆σi { { ∆t = ηq . min i=1,2,3 ∆t = min{∆t} min σi = σi + ∆σi ∆tη Minimum timestep min q σxx ,σyy , τxy = f(σi=1,2,θ) 3 FINITE DIFFERENCES CALCULATION ALGORITHM σzz = σ3 The swelling law presented in previous paragraphs has been implemented in FLAC, coupling the swelling phenomena with the built-in elastic–plastic constitutive relationships via FISH routines. The principal concept of this algorithm is that the volumetric strain is reached in the zones of the model introducing of small increments of isotropic stress within them, Noorany et al. (1999). The sketch of the algorithm is to calculate the final swelling strain tensor for all the zones of the model, transform the strains into an increment of stresses, and then “inject” the stresses in small increments into the zones. The flowchart of the calculation algorithm is presented in Figure 3, and can be resumed in the following points: Mechanical equilibrium t = t + ∆t min ε = ε + (ε − εiq) ∆t ηq q i q i q i` min “Injection” of a fraction of the stress increment Solve to mechanical equilibrium Accumulation of swelling time and swelling strains END Figure 3. Flowchart of the calculation algorithm. – Solve to mechanical equilibrium of the current timestep. – Accumulation of swelling strains and time. – Repetition of the algorithm until the expected age of the simulation is reached. – Determination of the principal effective stresses. 1 and 2 are principal stresses in the calculation plane, and 3 is the out-of-plane stress. – Determination of the swelling principal strains and of the stress increments associated with these strains. – Determination of the swelling time parameter. – Determination of the minimum timestep necessary for numerical convergence of the algorithm and to synchronize the swelling rate of all the elements in the model. – “Injection” of a fraction of the stress increment associated to the swelling strain. After initiation of swelling time to zero, the principal stress tensor for all the elements is calculated from the current stress state. As the routine has been implemented in a 2D model, the out-of-plane stress, zz in FLAC, is a principal stress (3 in the formulation) but not strictly the minor principal stress. Then, a loop is performed until the swelling time reaches the expected simulation age. Within this loop, the strain tensor due to complete swelling is calculated according to Equation 3. The stress increments 57 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands σ0 09069-07.qxd 08/11/2003 20:43 PM Page 58 associated to these strains are calculated with the following lineal elastic relationship, Equation 6: To determine a value of t small enough, only a fraction of the stress increment associated to the remaining swelling strain should be “injected” in the elements of the model. A criterion of a maximum of 1% of the current stress state has been adopted to determine de fraction of i to “inject”. The minimum fraction obtained from the 3 principal directions in each element is adopted. These relationships are illustrated in Equation 11 for every element in the model. (6) where and G are constants known as Lamé’s parameters ( K 2/3G; K is the bulk modulus and G is the shear modulus). Previously to the calculation of the swelling time parameter the plastic component of the volumetric strain has to be determined. The total volumetric strain, addressed in FLAC with a FISH variable, is the sum of the following components, Equation 7: (11) where ri fraction of the stress increment i; i stress state; and r minimum fraction of the 3 principal directions. From Equations 6 and 11 the fraction r of the stress increment that are going to be “injected” in the elements of the model can be expressed with the following Equation 12: (7) el0 where tot v total volumetric strain; v elastic volumetric strain produced in the model previous to any calculation; elv elastic volumetric strain produced during the calculation; it can be calculated as vel (1 2 3)/(3 K), being K the bulk modulus; qv swelling volumetric strain accumulated during calculation, qv q1 q2 q3; therefore, the plastic volumetric strain, plv, can be calculated with the following Equation 8: (12) and therefore, the timestep for each element can be obtained as, Equation 13, (8) (13) The swelling strain does not occur instantaneously but following the kinetics formulated with Equation 4. Expressing this differential equation in finite differences we obtain the following Equation 9: It is necessary to synchronize the rate of swelling for all the elements of the model adopting the same timestep for all; the minimum timestep of all the elements is the searched, (9) (14) and therefore, The “injection” of stresses associated to the swel ling behavior can be expressed with the following Equation 15, (10) As in all finite difference algorithm schemes, this equation applies only for values of t that are significantly low. This means that the swelling strain at time t that still remains to produce, [qi(t) – qi(t)] cannot be induced in the model instantaneously because the model would degenerate. Thus, the next phase is to determine a critical value of t to use in the finite difference scheme. (15) that gives, the relationship between the current stress state – i(t), the total increment of stresses due to swelling – i, the minimum timestep – tmin, and the swelling time parameter – q. 58 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-07.qxd 08/11/2003 20:44 PM Page 59 These stress increments have to be transformed from the principal axes reference to the coordinated axes reference, assuming that the principal stresses have not rotated during the swelling processes, and the angle between the principal stresses and the horizontal remains the same. Finally, once the mechanical equilibrium has been reached for this fraction of stresses injected into the elements of the model, it is necessary to actualize the accumulated swelling strain of each element of the model, Equation 16: (16) and for the swelling time, Equation 17. (17) This procedure is repeated until the accumulated swelling time reaches the expected age of the swelling simulation. Figure 4. Swelling pressure test (in Wittke 1999). 500 400 2 (kN/m ) 4 VALIDATION OF THE ALGORITHM The algorithm introduced in previous paragraphs has been implemented in FLAC via FISH routines. Now, the validation of this algorithm against the swelling tests presented in Wittke (1999) is presented. 4.1 Swelling pressure test 0 0 5 10 15 t (horas) sx sy sz FLAC Figure 5. Swelling pressure test simulation with FLAC. v 0.33. The results from FLAC simulation are presented in Figure 7 (strain in % versus time in days) against the results of the test. 4.3 Huder-Amberg swelling test in plasticity The objective now is to validate the algorithm against a test in which the strength of the sample is exceeded and therefore, plastic strains develop in addition to the elastic and swelling deformations. The test was carried out following the HuderAmberg procedure. The initial vertical load (applied in two cycles) is 15 MPa; the sample is then flooded and unloaded to a vertical pressure of 6.5 MPa. The results for this test are presented in Figure 8. The elastic constants of the material are E 2800 MPa and v 0.33, and swelling parameters are Kq 6.4% and 0 89.2 MPa. Huder-Amberg swelling test in elasticity Figure 6 shows the swelling strain-time curves obtained for the different stages of loading in a Huder-Amberg swelling test on an anhydritic mudstone from de Gypsum Keuper. A FLAC model with 1 element has been set up; the constitutive model is elastic with E 1000 MPa and 59 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 200 100 The first validation test is a swelling pressure test of a cubic sample of swelling mudstone carried out in a triaxial test apparatus. After a load–unload cycle the sample was flooded. Preventing the strains in the 3 directions the swelling pressures were measured in function of time. Figure 4 presents the evolution of the swelling pressure, in the 3 directions, versus time, and the parameters for the swelling law. To simulate this test, a FLAC model has been set up; the constitutive model is elastic with the same properties presented in Figure 4. The results from FLAC simulation are presented in Figure 5 (stress in kPa versus time in hrs). As the model is isotropic, the 3 components of the stress are identical in the simulation. 4.2 300 09069-07.qxd 08/11/2003 20:44 PM Page 60 Figure 8. Huder-Amberg swelling test in plasticity (in Wittke 1999). -24 -20 ez (%) -16 Figure 6. Huder-Amberg swelling test in elasticity (in Wittke 1999). -12 -8 -4 0 -5 4 0.1 ε zq (%) 1 10 100 sz (MPa) -4 Ensayo: Carga-descarga inicial Wittke elástico FLAC elástico -3 Ensayo: hinchamientos Wittke plástico FLAC plástico Figure 9. Huder-Amberg swelling test in plasticity simulated with FLAC. -2 -1 with the results from the tests and the values fitted by Wittke (1999). 0 0 5 10 15 t (días) sz = 520 kN/m2 FLAC sz=520 kPa sz = 260 kN/m2 sz = 130 kN/m2 sz = 65 kN/m2 FLAC sz=260 kPa FLAC sz=130 kPa FLAC sz=65 kPa sz = 32.2 kN/m2 FLAC sz=32.2 kPa 4.4 Figure 7. Huder-Amberg swelling test in elasticity simulated with FLAC. This is a swelling test on a sample taken from Gypsum Keuper. The test was carried out in a confined compression test apparatus (horizontal strains of the sample were prevented during the test) with boundary conditions, which were variable with time for a period of more than 14 years. The test sequence and the test results are presented in Figure 10. The description of the test, taken from Wittke (1999) is as follows. Phase 1 may be divided into 4 partial stages, form 1a to 1d. In phase 1a strains in the vertical direction were also prevented and the vertical stress was monitored. After 2.7 years, a vertical stress of 4.2 MPa Wittke (1999) suggests that it is necessary to consider the plastic strain of the sample to reproduce this test, as it is shown in Figure 8 where the back-analysis with elastic–plastic stress–strain relationship fits better with the measured values than the back-analysis with elastic stress–strain relationship. The plastic constants are c 0, 11° and 5.5°. Figure 9 presents the results from the FLAC model, also for elastic and elastic–plastic behaviors, together 60 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Combined swelling pressure and swelling strain test 09069-07.qxd 08/11/2003 20:44 PM Page 61 Table 2. Elastic–plastic parameters used in Wittke (1999) to reproduce the test. Elastic Plastic Parameter E (MPa) v c (MPa) Gypsum keuper 4000 0.2 0.65 (°) 30 (°) 30 Table 3. Swelling kinetics parameters used in Wittke (1999) to reproduce the test. 1a 2 3 Figure 10. Combined swelling pressure and swelling strain test on a sample from Gypsum keuper (in Wittke 1999). a0 (year 1) ael (year 1) avp (year 1) maxEVP (%) 0.0018 0.0018 0.0 0.0 40.0 2.0 0.1 0.1 5 was reached. Starting phase 1b it was allowed for a small vertical strain (that is not recognizable in Figure 10 because of the chosen scale) that results in a reduction of the vertical stress to approximately 3.8 MPa. Following to this, the vertical deformation of the sample was again prevented, and consequently, the vertical stress increased again to the same value of 4.2 MPa. The course of phase 1c was equivalent to the one of phase 1b. During phase 1d it was allowed for a vertical strain slightly larger than during the preceding phases. At the beginning this led to decrease the vertical stress to less than 0.5 MPa. Subsequently, the vertical stress was increased to 2.5 MPa over a period of 0.3 year without stabilization of the vertical stress. During phase 2 of the test, the vertical stress was lowered to 0.5 MPa keeping it constant for more than 5 years. The vertical strain was measured as a function of time. Phase 2 was stopped after a vertical strain of approximately 28% had occurred without stabilization of the deformations. During the phase 3 of the test, a further increase of the vertical strain was prevented and the increase of the vertical stress was registered as a function of time. The slope of the stress–strain curve decrease continuously with time and after a period of 5.7 years a vertical stress of 4 MPa was measured. Wittke (1999) reproduced this test with the elastic– plastic properties presented in Table 2. For the swelling parameters, Wittke (1999) uses the following values, 0 16 MPa and Kq 15%. Nevertheless, regarding on the kinetics of the swelling, Wittke found necessary to change the swelling time parameter during the course of the test. The parameters proposed are presented in Table 3. As Wittke (1999) refers, to reproduce accurately the phase 3 of the test it is necessary to reduce the value z(MPa) 4 2 1 0 0 5 t (años) Ensayo 10 15 FLAC Figure 11. Combined swelling pressure–strain test simulated with FLAC; comparison of stresses. of the coefficient avp from 40 year 1 to 2 year 1, that is equivalent to a reduction of the permeability of the sample during phase 3 due to the increment of the vertical stress in this phase of the test. Changes in permeability of the sample during the load process are not taken into account in the formulation of the kinetic, thus Wittke (1999) suggests that the coefficient avp should vary during the calculation for an accurate simulation of the swelling process. The test described in this paragraph has been simulated with FLAC, considering the same parameters (in Table 2 and Table 3). Figure 11 presents the evolution of the vertical stress (in MPa) versus time (in years), comparing the results from FLAC simulation against the test. Figure 12 presents the evolution of the vertical strain (in %) versus time (in years), comparing the results from FLAC model against the test. 61 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 3 09069-07.qxd 08/11/2003 20:44 PM Page 62 – All the elements of the model are susceptible to swell; this means that the whole rockmass is saturated and the penetration of water is enough to permit the complete swelling of the anhydrite. – The proposed kinetics describe adequately the swelling process when the strength of the rock is not exceeded, but the parameters of the formulation need to be changed in case of large plastic deformations. 40 εz (%) 30 20 10 0 0 5 10 15 REFERENCES t (años) Ensayo Cálculo FLAC ISRM 1989. Suggested Methods for Laboratory Testing of Argillaceous Swelling Rocks. In Int. J. Rock Mech. Min. Sci. & Geomech. Abstr, Vol. 26, No. 5: 414–426. Huder J. & Amberg G.1970. Quellung in Mergel, Opalinuston und Anhydrit. Schweizer, Bauzeit, 83: 975–980. Noorany I., Frydman S. & Detournay C. 1999. Prediction of soil slope deformation due to wetting, In Detournay & Hart (eds), FLAC and Numerical Modeling in Geomechanics: 101–107. Rotterdam: Balkema. Saïta A., Robert A. & Le Bissonnais H. 1999. A Simplified Finite Element Approach to Modeling Swelling Effects in Tunnels. In Alten et al. (eds), Challenges for the 21st Century: 171–178. Rotterdam: Balkema. Steiner W. 1993. Swelling Rock in Tunnels: Rock Characterization, Effect of Horizontal Stresses and Construction Procedures. In Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 30. No. 4: 361–380. Wittke W. 1999. Stability Analysis for Tunnels. Fundamentals. Geotechnical Engineering in Research and Practice. WBI-Print 4. Ed. WBI Prf.Dr.Ing. W. Wittke. Consulting engineers for Foundation and Construction in Rock Ltd. Verlag Glückauf GmbH. Essen. Figure 12. Combined swelling pressure–strain test simulated with FLAC; comparison of strains. 5 CONCLUSIONS The formulation for the swelling behavior presented by Wittke (1999) has been reviewed and a calculation algorithm, based in this formulation, has been implemented in FLAC, via FISH routines. This algorithm allows the simulation of the swelling behavior with FLAC code. The algorithm has been checked against different swelling tests presented by Wittke (1999), and the results from the model fit quite well to the results of the different tests. Therefore, these routines can be used to simulate the swelling behavior of expansive grounds in real engineering problems. Nevertheless, when using these routines to simulate a swelling behavior, the following limitations of the formulation have to be remembered: – The direction of the principal stresses does not change during the swelling process. 62 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-08.qxd 08/11/2003 20:18 PM Page 63 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Scenario testing of fluid-flow and deformation during mineralization: from simple to complex geometries P.M. Schaubs, A. Ord & G.H. German CSIRO Exploration and Mining, Bentley, Western Australia, Australia ABSTRACT: We present the use of FLAC3D in conjunction with Gocad and the CSIRO-developed software 3DMACS to model deformation – fluid processes during mineralization. Simple idealized geological models containing one planar fault are used to determine the effects of a number of deformation scenarios on volume strain, pore pressure and resultant fluid flow patterns. Our results show that whether fluid flows up or down the fault is strongly controlled by the dip of the fault; steep faults cause fluids to flow up the fault, while more shallow faults dilate and draw in fluid from the overlying sedimentary unit as well as the surrounding host rocks. Geometrically complex models, which more closely resemble the geology surrounding an ore deposit, are aimed at determining how the shape of a doubly plunging dome affects fluid flow patterns and the location of sites of dilation. Areas of maximum dilation occur on the flanks of the dome near its crest. Complex meshes are constructed using Gocad, which is then translated into FLAC3D using 3DMACS. This software may also be used to set model parameters and properties, and for coupling FLAC3D to other numerical codes. number of parameters and only require one or two models. Simple models are aimed at determining what causes fluid to flow up or down faults and where areas of high positive volume strain and low fluid pressure occur. This has implications for the location of fluid mixing and mineralization if it is assumed that there are two distinct fluid reservoirs within the model. The geometrical complex model is concerned more with the effect of the irregular shape of the geological units on fluid flow patterns and the location of sites of dilatancy. 1 INTRODUCTION An understanding of the relationship between fluid flow and deformation is important for determining how hydrothermal ore deposits form. Deformation may lead to the development, or reactivation, of structures such as faults, fractures and veins which may host ore deposits or may act as conduits for mineralizing fluids. Deformation may also induce volume changes (dilatancy) that further affect rock permeability and pore pressure gradients. Here we present two methodologies for determining the relative importance of certain parameters and processes during deformation and mineralization. The first method involves simplifying the geometry of the structures involved and is aimed at determining the effects of various parameters. In this way we are able to narrow down the effects of one parameter and reduce the uncertainty caused by geometrically complex models. For this reason the geometry of the model is rather simple and number of zones in the model is low (12500). This allows us to run a large number of models with different parameters in a short period of time. The second type of model is aimed at testing the effects of complex geometry, which more closely approximates that of the geology we see in the field. Here we are not concerned with changing a large 2 MODEL BUILDING AND VISUALIZATION THROUGH THE USE OF ADDITIONAL SOFTWARE 2.1 63 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Simple and complex model generation Simple single fault models are constructed using FLAC3D “generate” commands. The use of FISH allows for the rapid construction of models with different fault dips and strikes. Geometrically complex meshes are constructed using Gocad. This is done first by building or importing tri-surfaces, which represent the contacts of the various geologic units. These surfaces are then used to “distort” the initially orthogonal and regular Gocad mesh (stratigraphic grid) 09069-08.qxd 08/11/2003 20:18 PM Page 64 Figure 1. Screen shot of web-browser interface of 3DMACS showing how properties and boundary conditions are applied. – Allows the user to set group properties imported from an external properties database, which can then be edited by the user. These properties can be from any of the 4 domains above. – Allows the user to set model parameters and choose visualization outputs. – Due to its underlying XML character, users can use a web browser (or the built-in 3DMACS GUI) from any machine connected to the internet and run their simulations remotely via 3DMACS. Multiple processes can be distributed amongst various machines. – Allows for the storage of all user parameters and selected properties within a nominated repository, so that the user can re-run prior defined problems. The above functionality allows the simulation to be fully specified within the user-domain, rather than the process domain, which normally requires specialized knowledge of syntax and macro languages such as FISH. By providing basic problem “templates” for scenarios such as mechanical/fluid, mechanical/fluid/ thermal and mechanical/fluid/chemical modeling, so that the zones become parallel to the surfaces. The CSIRO-developed software 3DMACS (Fig. 1) is then used to import this model, along with a set of properties, into FLAC3D. 2.2 The 3DMACS software suite Primarily, 3DMACS is used for the importation, parameter-selection/editing and running of 3D geological models. It is a suite of software modules that at its core, leverages an XML data model. It harnesses various vendor-provided software, such as FLAC3D, to provide the background simulation capabilities. Overall, it provides the following functionality: – Allows for the importation of 3D models such as those produced by Gocad or FracSIS. – Can “couple” a simulation across 4 phenomenologically distinct domains: mechanical/deformation, thermal, fluid and chemical. Currently FLAC3D and FastFlo (a CSIRO package for partial differential equation solving) are used to provide modeling across these domains. 64 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-08.qxd 08/11/2003 20:18 PM Page 65 users without particular expertise in FLAC3D can still build and run a model. However, expert users can interact directly with the underlying processes and override any presets set in the templates. As 3DMACS has the ability to couple FLAC3D to other software packages, we are able to create models which simulate deformation, fluid flow, thermal and chemical processes, all of which may be important for mineralization. 2.3 Figure 2. Typical simple fault model. Arrows indicate applied fluid discharge. Visualizing results 3DMACS provides for the visualization of FLAC3D results in Gocad or the commercial software FracSIS, via the export of scalar and vector data as 3D pointcloud sets. Gocad allows for the creation of isosurfaces from scalar point data. Both Gocad and FracSIS can be used for volume rendering of scalar data. FracSIS also allows the user to control the opacity of certain color values in both scalar and vector data. By “hiding” certain values we are able to see inside the FLAC3D model more easily and are not required to use cross-sections or cut planes. Using FISH from within FLAC3D, we are also able to create VTK files of scalar and vector data, which are used by the freeware software MayaVi. MayaVi is able to visualize isosurfaces, and scalar and vector cut planes as well as fluid flow vectors so that their color varies with magnitude. All of these software packages are able to create VRML files which, given the appropriate plug-in, allows one to use a web-browser to view results. Figure 3. Examples of initial geometries of simple fault models. 3 SIMPLE FAULT MODELS 3.1 Model setup, properties and boundary conditions In this group of models we present a number of scenarios with a simple geometry. The initial model is made up of a simple fault region bounded by steeply dipping hangingwall and footwall rocks. These rocks are truncated by a horizontal interface and flat-lying sedimentary unit (Fig. 2). We test different orientations of far-field stresses, various dip and strike angles, for the fault and different hanging wall and footwall permeabilities (Fig. 3). The types of deformation applied include (Fig. 4): – compression and extension, where the bottom boundary is fixed and initial velocities are horizontal and perpendicular to the left and right boundaries, – strike slip, where initial velocities are horizontal and parallel to the left and right boundaries but in opposite directions, – reverse and normal movement, where initial velocities are parallel to the dip of the fault and the base Figure 4. Different styles of deformation applied to simple fault models. a) compression, b) extension, c) dextral strikeslip, d) sinistral strike-slip, e) reverse movement, f) normal movement, g) transpression, and h) transtension. 65 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-08.qxd 08/11/2003 20:18 PM Page 66 of the model is allowed to move in the vertical direction, and – transpression and transtension, which are similar to the reverse and normal models but contain a strike slip component of movement. of extension pore pressure is 30 MPa lower than those, which are essentially compressional. In all models, contours of pore pressure in the sedimentary unit slope towards the right due to the application of a fluid flux (discharge) at the left boundary. By changing the dip and strike of the fault we have a number of scenarios which range from a model with a shallow dipping faults with a dip of 30° and compression at right angles to the strike of the fault to a model with a steeply dipping fault (60°) where the compression direction is at 45° to the strike of the fault. Constant fluid fluxes of 1m/yr are applied at the base of the fault and the left boundary of the sedimentary unit. Permeability is isotropic and remains constant during deformation. Mechanical anisotropy is modeled using the ubiquitous joints constitutive model. In the sedimentary unit these are oriented horizontally and represent bedding, while in the basement units they are oriented roughly parallel to the contacts of the units and represent a pervasive cleavage. These fabrics are given 90% of the strength (cohesion, tensile strength) of the rock type. Mechanical properties are listed in Table 1. The size of the model varies depending on the dip of the fault. All models are 2 km tall (z-direction) and 2 km deep (y-direction) but the width (x-direction) changes. In all cases the bottom of the fault is a minimum of 1500 m away from either boundary. All models are deformed to 5% shortening or the equivalent amount of displacement for those models with a strike-slip component. 3.2.2 Volume strain In all models the fault region is an area of high positive volume strain (dilation) and a zone of significant dilation propagates from the tip of the fault into the sedimentary (Fig. 5). In the compression, reverse and transpression models this zone is oriented roughly parallel to the strike of the fault. In the extension, normal and transtension models this zone is much steeper, and in the strike-slip models it is close to vertical. Positive volume strain in the fault is greatest in models with an extensional component. In the reverse model only the fault is a region of significant positive volume strain and therefore is also a region of low pore pressure relative to the other basement units. The transpression model is similar; however, it contains regions of dilation in both the hangingwall and footwall. In the normal and transtension models the fault is also a region of significant dilation (higher than the reverse and transpression models). As with volume strain (dilation/contraction) the fault region records the highest shear strain in all models. The location and orientation of the zones of high shear strain are coincident with those of significant dilation in all models. 3.2 Results 3.2.1 Pore pressure Models which have an extensional component of deformation (extension – pure shear, normal faulting, transtension) cause the greatest decrease in pore pressures because they have the greatest dilation (positive volume strain). The normal and transtension models also have the steepest pore pressure gradients and therefore fluid flow rates in the fault are highest in these models. Pore pressure at the bottom of all models is similar; however, in the models with a component Figure 5. Volume strain increment and fluid flow vectors in compression model (cross-section view through middle of model). Maximum fluid flow velocities are 2.02 10 8 m/s. Table 1. Mechanical and fluid flow properties used in simple fault models. Property Density Bulk modulus Shear modulus Cohesion Tensile strength Friction angle Dilation angle Permeability Units 3 kg/m Pa Pa Pa Pa deg deg m2 Sandstone Granite 2400 2.40E 2.60E 2.70E 1.20E 28 4 1.00E 2700 5.0E 3.0E 4.0E 2.0E 30 6 1.01E 10 10 07 06 14 66 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Fault 10 10 07 07 16 2600 9.5E 9.6E 1.8E 2.8E 15 5 2.02E Pelitic gneiss 09 09 07 06 15 2600 1.9E 2.0E 3.5E 5.5E 20 5 2.02E 10 10 07 06 15 09069-08.qxd 08/11/2003 20:18 PM Page 67 more because they are oriented at an angle which is parallel to the direction of maximum compression. Steeply dipping faults and those oriented normal to the maximum compression direction are more likely to contract. In models where the compression direction is normal to the strike of the fault, the dip of the fault has little affect on the orientation of the high strain zone, which propagates into the sedimentary unit. In all models where the compression direction is normal to the strike of the fault, this zone of high strain takes on a dip of 45°. In models where the fault dips 60° and the hanging and footwall are both pelitic gneiss the orientation of the high strain zone in the basement is also 45°. When the footwall rock type is made for rigid, the high strain zone in the basement is nearly parallel to the dip of the fault. In models where the fault dips 30° and the high strain zone and the footwall is more rigid the high strain zone in the basement is parallel to the fault. This high strain zone becomes steeper (close to 45°) in the sedimentary unit. Changing the orientation of the fault with respect to the model boundaries and direction of compression results in different orientations for the high strain zones. In the models where the strike of the fault is oriented at 45° to the maximum compression direction (and the dip is 60°), close to the fault the high strain zone is parallel to both the dip and strike of the fault in both the basement units and the sedimentary unit. Away from the fault zones, high strain zones form with a strike normal to the maximum compression direction and a dip of 45°. When the angle between the strike of the fault and the maximum compression direction is increased to 67.5°, the orientation of the high strain zone in the basement is nearly parallel to the fault. As this zone propagates into the sedimentary unit its orientation rotates towards a strike which is normal to the maximum compression direction and a dip closer to 45°. Therefore, both the strike and dip of the fault, with respect to the maximum compression direction as well as the strength of the rocks, may control the orientation of the high volume and shear strain zones. In models where the direction of maximum compression is oriented less than 90° to the strike of the fault, fluid flow vectors change along a line stretching from the top west end to the bottom east end of the fault. On the west side flow is directed up and out into the hanging wall, while on the east side fluid flows down and into the footwall side of the fault. Increasing the permeability of the fault marginally does not change the values of volume strain or pore pressure in a significant manner. Fluid flow patterns remain the same however fluid flow velocities are increased slightly. Decreasing the strength of bedding and cleavage fabrics from 90 to 75% of the strength of the host 3.2.3 Fluid flow vectors Fluid flows towards the center of the fault in the compression, reverse and transpression models (Fig. 6). This is a result of the fault being an area of significant dilation, and low fluid pressure, relative to the surrounding rocks. In the extension, normal and transtension models, fluid flows up and out of the fault into the hanging wall. This occurs despite the fault being an area of higher positive volume strain (dilation) than the surrounding rocks. Fluid flows up because pore pressure gradients are much steeper than in the compressional models. In the strike-slip models fluid flows up and parallel to the boundaries of the fault. In the extension, normal and transtension models fluid on the right side of the sedimentary unit flows to the right towards the zone of significant dilation which propagates from the tip of the fault. 3.2.4 Dip and strike of fault The dip angle of the fault strongly controls whether fluid flows up and parallel to the fault boundaries. Models with the fault dipping 60° allow fluid to flow up the fault. Fluid will flow up a steep fault whether the footwall rocks are quartzite or pelitic gneiss. Similarly when the fault dips 45°, fluid flows into the fault from the foot and hanging walls whether the footwall rocks are quartzite or pelitic gneiss. In models where the fault is steep (60°), the fault does not dilate as much and therefore pore pressure does not decrease in the fault as much as it does in models where the fault has a shallower angle. Decreasing the dip angle of the fault to 30° causes the fault to dilate more than models where the fault dips at 45°. This however does not significantly affect the pore pressure values in the fault and fluid still flows towards the center of the fault from the hanging and footwalls. Moderate to shallowly dipping faults are able to dilate Figure 6. Fluid flow vectors in and around the fault (compression model). 67 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-08.qxd 08/11/2003 20:19 PM Page 68 rocks has little effect on the fluid flow patterns. In the model where the joint strength is 90% of the host rocks strength, slip along bedding planes occurs within the sedimentary unit above the fault and may indicate that some flexural slip has occurred. In the model where joint strength is 75% of the host rocks strength, slip also occurs along bedding planes in the sedimentary unit as well as along cleavage planes within the fault. 4 COMPLEX GEOMETRIES 4.1 Model setup, properties and boundary conditions The geometry of the model is reasonably complex (Fig. 7) and contacts between different units are smooth curved surfaces. The objective of this model is to determine what affect the shape of a basalt dome has on fluid flow patterns and the position of regions of dilation in relationship to the formation of gold deposits. The model is made up of rigid doubly plunging basalt dome which is blanketed by a thin weak altered metasedimentary unit and surrounded by a moderately stiff metamorphic rocks (Fig. 8). Mechanical properties are listed in Table 2. Deformation is applied so as to simulate horizontal compression perpendicular to the long axis of the dome. 4.2 Figure 7. Outline of basalt unit in geometrically complex model (in FLAC3D). Results The altered metasedimentary unit contains regions of negative volume strain (contraction) on the flanks of the basalt dome where the dip is steep and at a high angle to the compression direction. Towards the top of the dome (but not at the crest) the weak altered metasedimentary unit contains regions of high positive volume strain (dilation) above the areas of contraction (Fig. 9). This causes fluid flow rates to be highest close to the top of the dome where areas of contraction and maximum dilation are in close proximity (Figs. 10 & 11). Contraction occurs within the matrix above the highest point of the dome. Regions of high positive volume strain are also regions that have failed in tension. These areas are more likely to have formed quartz veins, which commonly host gold. 5 CONCLUSIONS Figure 8. Cross-section through center of model showing outline of the main basalt dome in light grey, the thin altered metasedimentary unit in dark grey and the surrounding metamorphic matrix in white. FLAC3D has been used to test the effects of fault and far-field stress orientation and the shape of irregularly shaped bodies on fluid flow in regions of mineralization. In geometrically simple models with a single planar fault the results of the models show that a low angle fault with permeability similar to the surrounding host rocks causes the fault to dilate and fluid to flow down from the sandstones into the fault. Steeply oriented faults, strike-slip deformation and high permeability faults cause fluid to flow up the fault. This has implications for the location of fluid mixing and 68 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-08.qxd 08/11/2003 20:19 PM Page 69 Table 2. Mechanical and fluid flow properties used in geometrically complex model. Property Units Basalt Density Bulk modulus Shear modulus Cohesion Tensile strength Friction angle Dilation angle Permeability kg/m3 Pa Pa Pa Pa deg deg m2 2700 5.00E 3.00E 4.00E 2.00E 30 2 1.00E Matrix 10 10 07 07 16 2700 4.0E 2.0E 3.0E 1.0E 25 3 1.00E Altered unit 10 10 07 07 15 2700 3.0E 1.0E 2.0E 9.0E 20 3 5.00E 10 10 07 06 15 Figure 9. Isosurface of high positive volume strain (black) occurs on flanks of the basalt within the weak altered metasedimentary unit. Surface of basalt exported from Gocad is shown in grey. Visualized in MayaVi. Figure 11. Cut plane of fluid flow vectors through highest portion of the basalt dome. Surface of basalt is shown in grey. Visualized in MayaVi. mineralization, if it is assumed that the horizontal interface represents a boundary between two distinct fluid reservoirs. In the geometrically complex model of a basalt dome, the area of maximum dilation occurs on the flanks of the dome near its crest. Areas of maximum positive volume strain are coincident with maximum fluid flow velocities and occur within the weak altered metasedimentary unit which blankets the dome. Sites Figure 10. Fluid flow vectors as visualized in FracSIS. Vectors are shaded using greyscale where black is highest and white is lowest fluid velocity. Only the highest values are shown (others remain transparent). The highest values are coincident with areas of high positive volume strain on flanks of the basalt within the weak altered metasedimentary unit. Surface of basalt is shown in grey. 69 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-08.qxd 08/11/2003 20:19 PM Page 70 ACKNOWLEDGEMENTS of high volume strain or dilation are likely sites of quartz vein formation and gold mineralization. GoCAD has been used to construct models of significant geometrical complexity and the CSIROdeveloped software 3DMACS has been used to translate the resultant mesh to FLAC3D. Numerical modeling results are visualized in either Gocad, MayaVi or FracSIS any of which allow for the creation of isosurfaces of scalar data and the export of VRML files. We would like to thank Irvine Annesley, Michel Cuney, Jon Dugdale, Nick Fox, Peter Hornby, Catherine Madore, Phillipe Portella, Dave Quirt, Tim Rawling, Dave Thomas, Chris Wilson and Rob Woodcock for their advice and input into the models presented here and with help visualizing the results. 70 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-09.qxd 08/11/2003 20:19 PM Page 71 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Constitutive models for rock mass: numerical implementation, verification and validation M. Souley INERIS, Ecole de Mines de Nancy, Nancy Cedex, France K. Su ANDRA, Châtenay-Malabry, France M. Ghoreychi INERIS, Parc Technologique ALATA, Verneuil-en-Halatte, France G. Armand ANDRA, Laboratoire de Recherche Souterrain Meuse/Haute-Marne, Bure, France ABSTRACT: This paper deals with numerical implementation of non-linear constitutive models of rock mass and its verifications and validations. In the 3-dimensionnal code, FLAC3D, an elasto-damage-plastic model (damage is approached through the theory of plasticity) for Hoek-Brown media has been implemented. Simulations of triaxial compression tests provide a verification of the numerical implementation with a good agreement between predictions and theoretical values of peak and residual strengths. The applicability of the implemented model to predict the damage and/or failure development around a circular opening is checked. Finally a validation of poroplastic calculations based on the drainage of a cylindrical hole in poroplastic media is achieved by comparison to an existing semi-analytical solution. progressive damage as microcracks initiate and grow at small scale and coalesce to form large-scale fractures and faults. The involved mechanisms include sliding along pre-existing cracks and grain boundaries, pore collapse, elastic mismatch between mineral grains, dislocation movement, etc. In the model considered in this study, the initiation and growth of cracks as well as failure and the postpeak behavior are approached through the theory of plasticity. Furthermore, the transition between the brittle failure and the ductile behavior depending on the mean stress is generally observed on rock samples. The purpose of this paper is to present: (a) a numerical implementation of an elasto-damage–plastic model obeying to the Hoek-Brown criterion and taking into account the brittle/ductile transition, (b) the corresponding verification based on simulation of triaxial compression tests and the prediction of the extent of damaged/failed zone around a hypothetical circular opening, (c) validation of poroplastic calculations based on a variant of the previous implemented model and an existing semi-analytical solution. 1 INTRODUCTION Analysis of stresses and displacements around underground openings in rock mass is required in a wide variety of civil and geotechnical, petroleum and mining engineering problems such as tunnels, boreholes, shafts, disposal of radioactive waste and mines. In addition, an excavation damaged zone (EDZ) is generally formed around underground openings excavated in rocks in relation to high in situ stresses and/or high anisotropic stress ratios even without blasting. The mechanical and hydraulic properties are then changed within EDZ. The failure mechanism in the damaged zone is the initiation and growth of cracks and fractures, and is directly related to the constitutive behavior of the rock mass. Several experimental studies on rocks have shown that there are many different mechanisms through which cracks can be initiated and grown under compressive stresses (Wong 1982, Steif 1984, Martin & Chandler 1994, etc.). Indeed, irreversible deformations and failure of rocks subjected to compressive stresses occur through 71 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-09.qxd 08/11/2003 20:19 PM Page 72 For instance, the initiation of damage (Fsend), the peak (Fsrup) and residual (Fsres) strengths are given by: 2 MECHANCICAL SHORT TERM BEHAVIOR 2.1 Brief mathematical description Based on several triaxial laboratory tests performed on the argillite rock samples, the typical characteristics of stress–strain curves are displayed in Figure 1: Phase 1: linear isotropic and elastic behavior after a short non-linear phase corresponding to the closure of microcracks; Phase 2: strain-hardening in the pre-peak region corresponding to the initiation and the growth of microcracks assumed to be described by plasticity, contrary to the concept of effective stress and the hypothesis of strain equivalence (Lemaitre 1995, Ju 1989) that is generally used; Phase 3: softening after the peak (failure) associated with a progressive loss in material cohesion and then a decrease in strength; Phase 4: residual phase where the rock strength remains practically constant. Based on these observations, a constitutive model for this material was firstly developed in the framework of the European project: EURATOM MODEX-REP and recently compiled by Su (2003). The features of this model are: (1) where mend, send and mrup, srup are Hoek-Brown constants respectively corresponding to onset of damage and the peak; cend and crup are uniaxial compressive strength at the onset of damage and peak; uniaxial residual strength; 3b d confining pressure for brittle/ductile transition; 1 and 3 major and minor principal stresses (compressive stress is negative and 1 2 3). 2.2 (a) linear elasticity to model the Phase 1; (b) damage initiation and growth are approached by a strain-hardening based on Hoek-Brown criterion where the Hoek-Brown constants and the uniaxial compressive strength are plastic strain dependent; (c) the peak, post-peak (Phase 3) and residual (Phase 4) are also based on Hoek-Brown criterion with respect to brittle/ductile transition in accordance with the experimental data. 1 2 3 (2) where softening flow function (parabolic form with respect to the internal plastic variable, in phase 3, and null elsewhere); m, s Hoek-Brown constants (linearly varying with in phase 2, and constants in phase 3 and 4); c uniaxial compressive strength (linearly varying with in phase 2, and constant in phase 3 and 4). It is assumed that the material damage (hardening) and failure (up to the peak) depend on the generalized plastic strain, : 4 Peak Deviatoric stress (σ1-σ3) Constitutive equations In order to obtain a simple but general constitutive model, an extended Hoek-Brown yield function is used. The general form of the yield function is expressed in the following equation: Onset of damage Residual (3) p where d increment of total damage/plastic strain tensor in phases 2, 3 and 4. For simplicity, an associated flow rule is used (the plastic potential is identical to the yield function given in Equation 2). In addition, in order to take into account the geometry of stresses (compression differing to extension), the previous yield function is generalized Axial strain( ε1) Figure 1. A typical stress–strain curve provided by a triaxial test. 72 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-09.qxd 08/11/2003 20:19 PM Page 73 By substituting Equations (4), (7) and (5) in Equation (8), we can express the plastic multiplier: in terms of the three stress invariants (J1, J2 , J3 ). Then, principal stresses are expressed in terms of mean stress (p), generalized deviatoric stress (q) and Lode’s angle () according to: (9) (4a) and then, the elasto-damage-plastic behavior: where (10) 2.3 (4b) Numerical implementation In the three-dimensional explicit finite-difference code, FLAC3D, we have implemented the elasto-damageplastic model described above. The main procedure is summarized below. – The first approximation of stress tensor I, is evaluated by adding to the previous stress tensor the stress increments computed from the total strain increments and the Hooke’s law. – Computation of the corresponding mean stress pI, deviatoric stress qI and Lode’s angle I corresponding to I. – Compute the generalized yield function, Fs(pI, qI, I I). If I verifies the yield function (Fs( ) 0), the derivatives of Fs with respect to and of m, s and c (phase 2) or (phase 3) with respect to , are evaluated, and then Equation 10 is used to compute the current increment of stress tensor. – Current stress tensor, generalized plastic strain and flow functions are updated. It should be noted that, in FLAC3D, zones are internally discretized into tetrahedra and the current flow functions (damage/ plastic) and, stress and strain tensors for each zone are evaluated as a volumetric average for the zone. stress tensor. Assuming that only small strain occurred, the total strain increment, d , can be subdivided in elastic part, e p d and damage/plastic part, d : (4c) (5) where plastic multiplier. This leads to: (6) This routine has been written in C and compiled as DLL file (dynamic link library) that can be loaded whenever it is needed. The incremental expression of Hooke’s laws in terms of generalized stress and strain tensors has the form: (7) 2.4 In order to verify the implemented model, seven triaxial compression tests with confining pressures of 2, 5, 10, 12, 16, 20 and 25 MPa have been simulated. They are the part of the wide number of triaxial compression tests used to characterize the non-linear behavior of the studied materials. where C isotropic linear tensor. The consistency condition, dFs() 0 leads to: (8) 73 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Verification and validation 09069-09.qxd 08/11/2003 20:19 PM Page 74 Table 1. Values of input parameters. send cend (MPa) mrup srup crup (MPa) 3b 1 0.9 15 0.43 2.5 33.5 20.01 20 MPa 16 MPa 12 MPa Lateral strain [-] -0.02 Residual mend 25 MPa -0.03 Peak -0.01 55 50 45 40 35 30 25 20 15 10 5 0 (MPa) d 3 80 70 Principal major 1 (MPa) Deviatoric stress 13(MPa) Onset of damage 10 MPa 5 MPa 2 MPa 0.01 0.02 0.03 -10 40 30 20 Principal minor stress 3 (MPa) 0 10 20 30 Peak [Eq. 1] Residual [Eq. 1] Onset of damage [Eq. 1] Peak - Flac3d Residual - Flac3d Onset of damage - Flac3d Figure 2. Numerical result of triaxial compression tests. The input parameters needed are derived from standard laboratory tests and based on the identification of model parameters. These parameters are summarized in Table 1. The isotropic elastic characteristics used are E 4500 MPa and 0.3. The generalized peak and residual plastic strain were also identified from triaxial laboratory tests. These are: rup 0.0063 and res 0.0175. FLAC3D simulations are carried out on a single zone of unit dimensions: fixed normal displacements are applied to 3 perpendicular planes (bottom and two perpendicular vertical faces). First, the three other faces of model are subjected to an isotropic stress state corresponding to the given confining pressure. Secondly, the deviatoric stress is exerted by applying a constant displacement rate at the top of model. Figure 2 presents the deviatoric stress–axial and lateral strain curves for different confining pressure. From this figure, we note that the post-peak behavior is confining pressure dependent: the transition stress between brittle failure and ductile behavior is clearly marked and the numerical transition stress, b3, d is approximately about 20 MPa. These curves are qualitatively similar to the experimental ones (not reported herein). Figure 3 shows a comparison in terms of the damage threshold, the peak and residual strengths between the predictions (corresponding values in Fig. 2) and theory (Eq. 1). The match is very good as may be seen in this figure, where numerical and analytical solutions coincide. More precisely, the relative error for peak and residual strengths is less than 0.3%, and 0.9% for the onset of damage (dependent on the magnitude of loading at Figure 3. Onset of damage, peak and residual strengths: numerical and analytical solutions. the beginning of phase 2). This validates the numerical implementation of the elasto-damage-plastic model in FLAC3D. 2.5 Application to a circular opening The aim of this section is to provide a verification of the implementation for non-triaxial stress paths and to show numerically the ability of the implemented model to evaluate the extent of damaged and/or failed zones around a circular underground excavation. We then consider an infinite circular opening in an infinite elasto-damage-plastic medium initially subjected to an anisotropic initial stress in order to maximize the deviatoric stress and then, the risk of damage and/or failure. The axis of gallery is parallel to the horizontal minor stress leading to a maximum deviatoric stress in the gallery section. The 2D-plane strain geometry as well as the initial in situ stresses and model geometry including a circular gallery are plotted in Figure 4. The modeling sequence was performed as follows: (1) the model without excavation was consolidated under the previous in situ stresses, and (2) the circular excavation was carried out using roller boundaries to the model sides respectively parallel to x- and z-axis for seeking symmetry. 74 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 50 10 0 Axial strain [-] 0 60 09069-09.qxd 08/11/2003 20:19 PM Page 75 v v radius of gallery: 3 m model length : 30 m model heigth : 30 m model thickness : 1 m gallery axis : // à h(// à Oy) v= h= 10.8 MPa H= 15.1 MPa 0 -2 0 -4 -6 -8 -10 -12 -14 -16 -18 -20 v H H 2 4 6 8 0 Figure 4. Model geometry, initial stress state and boundary conditions. 10 axial orthoradial (a) Stresses (MPa) 0 -5 r* radial 2 4 6 8 r* 10 radial -10 -15 axial -20 -25 -30 -35 orthoradial Stresses (MPa) (b) Figure 6. Radial, orthoradial and axial stresses along radial lines (a) at 4.5° (b) at 85.5° (elastic lines; elastoplastic circles). Damaged Failed to damage models based on the concept of effective stress and assuming the strain equivalence (Shao et al. 1998, Souley et al. 1998, Homand et al. 1998, Souley et al. 1999, etc.). Figure 6 shows the profiles of radial, orthoradial and axial stresses along two radial lines at 4.5° and 85.5° with respect to x-axis as a function of the adimensional radial distance (r* r/a; where r radial distance and a gallery radius). In addition, the corresponding stresses for elastic calculations are also plotted. From the profile of orthoradial and axial stresses, one can distinguish three different regions (elastic, damaged and failed) through the slopes of curves. At 4.5°, only one loss of slope can be noticed along the profile of orthoradial stress: the correponding radial distance (approximately 3.2 m) is in accordance with the previous investigation of damage extent. Up to this radial distance, the orthoradial stress profile is qualitatively similar to the elastic ones. At 85.5°, the first failure of curve slopes is noted at a radial distance of 3.3 m from the gallery wall, as well as for orthoradial stress profile (major principal stress) than axial stress (intermediate principal stress). This radial distance corresponds to the extent of failed zones in the direction of initial principal minor stress. The second failure of orthoradial and axial stress slopes can be shown at a radial distance of 1.5 m from the gallery wall. This corresponds to the damaged region located between the failed zone and the elastic zone. Figure 5. Extent of damaged and failed zones around circular opening. Figure 5 shows the extent of damaged and failed zones around the circular opening. Damaged zones correspond to the set of elements where the behavior is associated within the pre-peak region; the peak strength is not yet reached. Failed zones correspond to the model region where the peak strength was reached: it should be noted that, in the direction of the initial major principal stress (Ox), the extent of damage is about 17 cm: the radius of damaged zone in this direction is 1.06 times greater than the gallery radius. The extent of failed areas is concentrated in the compressive region where the maximum deviatoric stress is located. Then, in the direction of the initial minor principal stress (Oz), the maximum extent of failed zone reaches 26 cm, whereas the damaged areas are ranged between the failed and elastic regions with an extension about 1.5 m. Finally, in the case of a circular opening created in an infinite elasto-damage-plastic medium initially subjected to an anisotropic initial stress, the damaged zone has an elliptical form (big axis is parallel to the direction of the initial minor principal stress) similarly 75 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-09.qxd 08/11/2003 20:19 PM Page 76 Deviatoric stress 13(MPa) 3 HYDROMECHANICAL VALIDATION Generally, validation of poroelastic calculations with FLAC3D for which the formulation of hydromechanical coupling is used within the framework of the quasistatic Biot theory, has been undertaken by several examples and studies: – one-dimensional filling of an initially dry porous media compared with the analytical solution developed by Voller et al. (1996); – one-dimensional consolidation compared with the analytical solution developed by Detournay & Cheng (1993); – two-dimensional consolidation of a borehole in an elastic medium compared with the analytical solution developed by Detournay & Cheng (1988). -0.03 3 = 20 MPa 3 = 15 MPa 20 3 = 10 MPa 15 3 = 5 MPa 10 3 = 1 MPa 5 Lateral strain [-] -0.02 -0.01 3 = 0.5 MPa Axial strain [-] 0 0 0.01 0.02 0.03 where c uniaxial compressive effective strength of the intact rock; m peak value of Hoek-Brown constant; residual strength parameter. For softening phase, the yield function is assumed to be: (13) where softening internal variable, representing the opposite value of the plastic strain 1p associated with the major principal stress 1; R (0 R) value of the softening internal variable for which residual phase is reached. Finally, the potential function is given by: (14) Brief mathematical description of mechanical model This formulation slightly differs from the elastodamage-plastic model detailed in section 2 by the absence of hardening in the pre-peak region and brittle/ductile transition. Based on the previous implementation, this variant of the elasto-damage-plastic model is implemented in FLAC3D. As verification, triaxial com-pression tests were simulated. The results are shown in Figure 7. In addition, the corresponding numerical residual and peak strengths are represented in Figure 8 and compared with the analytical expressions (Eq. 12 & 13). From Figure 7, it should be noted that for a given level of confining pressure, the three phases (elastic before failure, softening for post-peak behavior and perfect plastic for residual behavior) are clearly distinguished. The match is very good as may be seen in Figure 8, where numerical and analytical solutions coincide. The relative error for strengths is less than 0.5%. The mechanical behavior is described by an elastoplastic model with a post failure softening phase. The model is based on the Hoek-Brown criterion with associated plastic potential. The main characteristics of this model are: (a) linear and isotropic behavior in the pre-peak region; (b) peak strength governed by the Hoek-Brown criterion; (c) a softening phase based on a Hoek-Brown yield function and an associated flow rule; (d) a perfectly plastic behavior in the residual phase. Assuming that compressive stress is negative and 1 2 3, the peak strength and residual strength are given by: (11) 3.2 Definition of hydromechanical problem Problem definition consists of a cylindrical hole created in an infinite poroelastoplastic medium initially (12) 76 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 25 30 Figure 7. Verification – simulated triaxial compression tests. Validation of hydromechanical coupling in the framework of poroplasticity is specific to each nonlinear model and each numerical code. The aim of this section is to provide an example of validation of poroplastic calculations in Hoek-Brown media. This example concerns the drainage of an infinite medium by a cylindrical hole for which a semi-analytical solution is developed by Vouille et al. (2001). In this solution, the mechanical model is based on the Hoek-Brown yield function and can be viewed as an extension of Carranza-Torres and Fairhust works and, also a particular case of the previously described constitutive model in the sense that neither damage (hardening in pre-peak region) nor brittle/ductile transition are considered. 3.1 35 Major principal stress 1 (MPa) 09069-09.qxd 08/11/2003 20:19 PM Page 77 60 Table 2. Initial and boundary conditions of the 1D problem. 50 Initial conditions Total stresses (MPa) Pore pressure (MPa) Boundary conditions Normal displacement (P9; PY; PZ0; PZ1) Hydraulic flux (P9; PY; PZ0; PZ1) Radial total stress at the outer radius r30 (MPa) Pore pressure at the outer radius r30 (MPa) Radial total stress at the inner radius Pore pressure at the inner radius 40 Peak [Eq. 11] 30 Residual [Eq. 12] 20 Peak - Flac3d Residual - Flac3d 10 Minor principal stress 3 (MPa) 0 -10 -5 0 5 10 15 20 25 30 11.5 ij 4.7 null null 11.5 4.7 Eq. 15 Eq. 16 Figure 8. Peak and residual strengths: numerical and analytical solutions. Table 3. Hydromechanical properties used in poroplasticity validation. r30 9º 3m 5800 0 0.3 c (MPa) 14.8 m 2.62 h (m/s) 10 12 b 0.8 M (MPa) 6000 w (kN/m3) 10 E0 (MPa) P9 R 0.015 0.01 PY 30 m and pore pressure p, along the inner wall are expressed as follows: PZ1 r3 0,1 m Figure 9. PZ0 (15) FLAC3D geometry of the 1D problem. (16) subjected to a uniform and isotropic stress state and a uniform pore pressure. The induced mechanical and hydraulic perturbations are examined during and after excavating. The main assumptions are: where t time; T 1.5 106 s represents the excavation duration. A semi-analytical solution of this H-M 1D problem has been developed in the framework of the European project: EURATOM MODEX-REP (Su 2002). Finally, the geometry shown in Figure 9, initial and boundary conditions reported in Table 2 are used in our FLAC3D model. Hydromechanical properties are shown in Table 3, where E0 and 0 denote the drained elastic properties; h is the hydraulic conductivity; b the Biot coefficient; M the Biot modulus and w the specific weight of water. For both semi-analytical and numerical solutions the required results are: – gravity forces are neglected; – mechanically, the medium behaves as an isotropic and elastoplastic material according to the model described in §3.1; – hydromechanical coupling process is expressed by Biot’s theory; – hydraulic and mechanical boundary conditions at the hole walls are time-dependent: continuous reduction of normal stress and pore pressure at the hole boundaries from their initial values to zero. The geometry of this 1D problem is shown in Figure 9. It consists of a thick wall cylinder with internal radius of 3 m and external radius of 30 m. The initial and boundary conditions are summarized in Table 2. As previously mentioned, the hydraulic and mechanical boundary conditions along the inner wall are timedependent. More precisely, the total radial stress r – the radial displacement; – the pore pressure; – the radial; orthoradial and axial effective stresses. as a function of radial distance from the hole center (r ranged from 3 to 30 m) and time (ranged from 0 to 100 Ms) in this paper. 77 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-09.qxd 08/11/2003 20:19 PM Page 78 to negative pore pressure) are well reproduced by numerical results for radial distance and time ranged respectively from 3.05 to 3.2 m, and from 1.5 to 1.6 Ms (corresponding to the start of full drainage). Due to null and negligible values of pore pressure in the vicinity of the inner radius; relative errors between semi-analytical and numerical solutions are not evaluated for radial distance inferior to 3.7 m; so the difference in results of pore pressure does not exceed 0.02 MPa. For radial distance superior to 3.7, the maximum relative error between semi-analytical and numerical solutions is about 1.2%. Comparison of radial and orthoradial effective stresses between the semi-analytical and numerical solutions is plotted in Figures 12 & 13. It should be noted that the profiles of principal effective stresses are qualitatively returned. From a quantitative point of view and for a radial distance superior to 3.1 m; the absolute error on the Height radial distances are considered for output. They are: 3, 3.05, 3.1, 3.2, 3.5, 3.7, 5, and 10 m. Ten time periods are also considered for result output. The involved times are: 1.2, 1.5, 1.6, 2.5, 10, 50, and 100 Ms (million of seconds). In the case of the semi-analytical, all of these required results are given at the previous radial distances. Because of displacements and pore pressure are gridpoint variables whereas stresses are zone variables and evaluated at the zone centroid, numerical solutions are checked at the following set of radial distances: – 3, 3.05, 3.1, 3.2, 3.5, 3.7, 5, and 10 m for radial displacement and pore pressure; – 3.0125, 3.0625, 3.1125, 3.2125, 3.5625, 3.725, 5.05, and 10.05 m (centroid of the closest zone) for stresses. Therefore, small differences in the results of stresses compared to the semi-analytical solution are to be expected. Comparison with the semi-analytical solution 4 Pore pressure (MPa) 3.3 Figure 10 presents a comparison of normal displacement between the semi-analytical solution and the numerical ones. This shows a very good agreement between both the solutions. In particular, the maximum of relative error between semi-analytical solution and FLAC3D results is about 0.7% and corresponds to radial distance inferior to 3.2 m and t 5 Ms. In the other cases, the relative error is about 0.2%. Figure 11 illustrates the comparison of pore pressure between the semi-analytical solution and the numerical ones. For a given radial distance, both numerical and semi-analytical solutions are quantitatively and qualitatively similar. In particular, it should be noted that some underpressures (i.e. “unsaturated” zones corresponding -2 -3 -4 -5 -6 -7 -8 -9 -10 2 1 0 0 40 20 r=3 - Anal 20 r=3,05 - Anal r=3,1 - Anal r=3,2 - Anal r=3,55 - Anal r=3,7 - Anal r=5 - Anal r=10 - Anal r=3 - Flac3D r=3,05 - Flac3D r=3,1 - Flac3D r=3,2 - Flac3D r=3,55 - Flac3D r=3,7 - Flac3D r=5 - Flac3D r=10 - Flac3D 100 -2 -4 -6 -8 -10 Figure 10. Numerical and semi-analytical solutions: radial displacement. 60 80 100 Time (Ms) 0 20 40 60 80 100 r=3 - Anal r=3,05 - Anal r=3,1 - Anal r=3,2 - Anal r=3,55 - Anal r=3,7 - Anal r=5 - Anal r=10 - Anal r=3,013 - Flac3D r=3,063 - Flac3D r=3,113 - Flac3D r=3,213 - Flac3D r=3,563 - Flac3D r=3,725 - Flac3D r=5,05 - Flac3D r=10,05 - Flac3D Figure 12. Numerical and semi-analytical solutions: radial effective stress. 78 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 40 Figure 11. Numerical and semi-analytical solutions: pore pressure. 0 20 20 -1 Radial effective stress (MPa) Radial displacement (mm) -1 0 3 Time (Ms) 0 r=3 - Anal r=3,05 - Anal r=3,1 - Anal r=3,2 - Anal r=3,55 - Anal r=3,7 - Anal r=5 - Anal r=10 - Anal r=3 - Flac3D r=3,05 - Flac3D r=3,1 - Flac3D r=3,2 - Flac3D r=3,55 - Flac3D r=3,7 - Flac3D r=5 - Flac3D r=10 - Flac3D 5 09069-09.qxd 08/11/2003 20:19 PM Page 79 Orthoradial effective stress (MPa) -8 0 20 40 60 80 – The plastic radius rp for the semi-analytical solution is about 3.55 m whereas rp equals 3.7 in numerical model. However, for numerical solution the orthoradial and radial plastic strain are respectively 5 and 12 , and reached at the beginning of full drainage (t 1.5 Ms). Note that similarly to the stress tensor, strain tensor and principal plastic strains are zone variable. – For radial distance ranged between the hole wall and the elastic/plastic transition region, the maximum relative error between the semi-analytical and numerical solution is about 8%. 100 r=3 - Anal r=3,05 - Anal r=3,1 - Anal r=3,2 - Anal r=3,55 - Anal r=3,7 - Anal r=5 - Anal r=10 - Anal r=3 - Flac3D r=3,05 - Flac3D r=3,1 - Flac3D r=3,2 - Flac3D r=3,55 - Flac3D r=3,7 - Flac3D r=5 - Flac3D r=10 - Flac3D -10 -12 -14 -16 -18 -20 Time (Ms) 4 CONCLUSION Figure 13. Numerical and semi-analytical solutions: orthoradial effective stress. This paper presents numerical implementation of non-linear constitutive model of rock mass in the three-dimensional code FLAC3D, as well as its verification and validation. Firstly, a non-linear elastodamage-plastic model based on the Hoek-Brown failure criterion and for which hardening in pre-peak (characterizing the material damage), softening (characterizing the post-peak behavior and the failure of sample) is implemented in FLAC3D. Simulation of triaxial compression tests at different level of confining pressure provides a verification of the implemented model. The resulting curves display four regions (elastic, damage in pre-peak, softening in post-peak and residual phase) when the confining pressure is below the transition stress, and three regions (elastic, damage and perfect plastic phase) under high confining pressure. In addition, the onset of damage (limit between elastic/damage region), the peak and residual strengths derived from these simulations are compared with the theoretical envelops: the corresponding relative error does not exceed 0.3%. The ability of the implemented model to predict the damaged and failed regions around an underground excavation is successfully tested. In this verification, a circular and an initial anisotropic stress (in order to maximize the extent of damage and failure) are considered. The extent of failed areas is concentrated at the gallery wall in the compressive region where the maximum deviatoric stress is prescribed (direction of the initial minor principal stress), whereas the damaged areas are ranged between the failed and the elastic regions. As a result, the damaged zone has an elliptical form similarly to the prediction of damage models based on the concept of effective stress (damage theory). Secondly, a variant of the elasto-damage-plastic model, for which a semi-analytical solution of drainage of an infinite medium by a cylindrical hole exists, is used in order to validate the poroplastic calculations in FLAC3D. The previous implementation has been slightly modified for the variant version, and firstly tested on triaxial compression tests with a good orthoradial and radial is respectively about 0.3 and 0.2 MPa; that corresponds to a relative error of 2%. However in the vicinity of the inner wall (3 m for the semi-analytical solution and 3.0125 m in FLAC3D), the maximum difference between both solutions is 0.6 MPa. In order to capture the magnitude of error in terms of stresses due to the difference in the radial distances where the principal effective stresses were computed respectively for semi-analytical and numerical solutions, the closed-form solution for prediction displacements and stresses around circular openings in elasto-brittle-plastic rock (based on Hoek-Brown criterion) recently developed by Sharan (2003) is used. This closed-form solution is only valid for the mechanical configuration. For Sharan closed-form solution, the previous hole geometry, mechanical properties, mechanical initial and boundary conditions are used. In addition, it was assumed that the 3 m-radius hole is instantaneously excavated. Under these conditions, the orthoradial and radial stresses are calculated based on the closed-form solution for these pairs of radial distances (in meters): 3–3.0125, 3.05–3.0625, 3.1–3.1125, 3.2–3.2125, 3.5– 3.5625, 3.7–3.725, 5–5.05 and 10–10.05. The maximum of difference for each pair is about 0.2 MPa. In relation to the previous discussion on the radial and orthoradial effective stresses, we can say that the difference between the semi-analytical and numerical solutions for radial distance superior to 3.1 m remains in an acceptable order of magnitude while in the vicinity of the inner radius, numerical results derived from FLAC3D can be ameliorated by increasing the mesh density (unfortunately, this will considerably decrease the FLAC3D hydraulic characteristic time, and then increase the calculations duration). Finally, the investigation of orthoradial and radial plastic strains (not reported herein) between the semianalytical and numerical solutions leads to the following comments. 79 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-09.qxd 08/11/2003 20:19 PM Page 80 Martin, C.D. & Chandler, N.A. 1994. The progressive failure of Lac du Bonnet granite. International Journal of Rocks Mechanics and Mining Sciences. 31(6): 643–659. Shao, J.F., Chiarelli, A.S. & Hoteit, N. 1998. Modeling of coupled elastoplastic damage in rock materials. International Journal of Rocks Mechanics and Mining Sciences. 35(4–5): Paper No. 115. Sharan, S.K. 2003. Elastic-brittle-plastic analysis of circular openings in Hoek-Brown media. to appear in International Journal of Rocks Mechanics and Mining Sciences & Geomechanics Abstracts. Souley, M., Homand, F., Hoxha, D. & Chibout, M. 1999. Damage around a keyed URL excavation: change in permeability induced by microcracks growth. In Detournay & Hart (eds), FLAC and Numerical Modeling in Geomechanics: 205–213. Rotterdam: Balkema. Souley, M., Hoxha, D. & Homand-Etienne, F. 1998. Distinct element modelling of an underground excavation using a continuum damage model. International Journal of Rocks Mechanics and Mining Sciences. 35(4–5): Paper No. 6. Steif, P.S. 1984. Crack extension under compressive loading. Engineering Fracture Mechanics. 20(3): 463–473. Su, K. 2002. Analysis of the capacity of numerical models to simulate excavation in deep argillaceous rock, 5th EURATOM framework programme, MODEX-REP project contract FIKW-CT2000-00029 – Deliverable 1, August 2002. Su, K. 2003. Constitutive models of the Meuse/HauteMarne Argilites, MODEX-REP project contract FIKWCT2000-00029 – Deliverable 2&3, February 2003. Voller, V., Peng, S. & Chen, Y. 1996. Numerical Solution of Transient, Free Surface Problems in Porous Media. International Journal of Numerical Methods in Engineering. 2889–2906. Vouille, G., Tijani, M. & Miehe, B. 2001. Hydro-mechanical theoretical problem: Drainage of an infinite medium by a cylindrical hole. In EC-5th EURATOM framework programme 1998–2000 MODEX-REP project: contract FIKW-CT-200-00029, NOT-EMP-01-02, Technical Note, fevrier 08. Wong, T.F. 1982. Micromechanics of faulting in Westerly granite. International Journal of Rocks Mechanics and Mining Sciences. 19(1): 49–62. agreement between predicted peak and residual strengths and theoretical ones. Comparison of normal displacement between the semi-analytical solution and the numerical ones indicates a very good agreement: the relative error is globally about 0.2%. Also, for a given radial distance, both numerical and semi-analytical solutions are quantitatively and qualitatively similar in terms of pore pressure results with a neglected difference (0.02 MPa) compared to the initial field of pore pressure (4.7 MPa). In terms of radial and orthoradial effective stresses, the semi-analytical and numerical solutions are qualitatively the same. Far to the gallery wall, the stress difference does not exceed 0.3 MPa (that corresponds to a relative error of 2%). REFERENCES Carranza-Torres, C. & Fairhurst, C. 1999. The elasto-plastic response of underground excavations in rocks masses that satisfy the Hoek-Brown failure criterion. International Journal of Rocks Mechanics and Mining Sciences. 36(6): 777–809. Detournay, E. & Cheng, A.H.-D. 1993. Comprehensive Rock Engineering. Pergamon Press Ltd. Detournay, E. & Cheng, A.H.-D. 1988. Poroelastic Response of a Borehole in a Non-Hydrostatic Stress Field. International Journal of Rocks Mechanics and Mining Sciences. 25(3): 171–182. Homand-Etienne, F., Hoxha, D. & Shao, J.F. 1998. A continuum damage constitutive law for brittle rocks. Computers and Geotechnics. 22(2): 135–151. Ju, J.W. 1989. On the energy based on coupled elastoplastic damage theories: constitutive modeling and computational aspects. International Journal of Solids Structures. 25(7): 803–833. Lemaitre, J. 1985. A course on damage mechanics. 2nd edition. Springer. 80 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-10.qxd 08/11/2003 20:20 PM Page 81 Slope stability Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-10.qxd 08/11/2003 20:20 PM Page 83 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A parametric study of slope stability under circular failure condition by a numerical method M. Aksoy & G. Once Osmangazi University, Mining Engineering Dept., Eskisehir, Turkey ABSTRACT: Slope failures can cause delay in the production schedule and the loss of life and equipment. In this study, slopes excavated in very weak rock masses where expected failure mode is circular failure have been studied. The effects of slope height, slope angle, water saturation, cohesion, internal friction angle and density on slope stability under circular failure conditions have been investigated by three methods: Hoek and Bray stability diagrams, Bishop’s simplified method of slices, and finite difference numerical code, FLAC3D (Itasca 1997). Safety factor calculations have been carried out for the various values of parameters and obtained values are compared with each other. However, the main focus is on the results of the numerical modeling. The presence of correlation between the studied parameters and the factors of safety obtained from numerical models has been searched and the fitted equation has been given. Factor of safety is used as an index to define the slope stability and it can be simply described as the ratio of the total resisting force to the total inducing force. In this study, the investigation of circular (rotational shear) failure usually observed in the altered rock or soil slopes has been based on the effects of geomechanical properties of rock or soil and the shape of the slope on the slope stability. How the safety factor values are affected with the variation of the parameters values have been searched by three methods and calculated safety factors have been compared. 1 INTRODUCTION Slope stability is one of the most important subjects in mining and civil applications. In open pit mining, especially, the design of a stable slope has become important to meet the safety regulations in addition to the profitable extraction of the deposit. This can be achieved by the proper selection of slope angle, shape and height. The factors governing the stability of an open pit slope can be listed as follows (Stacey 1968): – – – – Geological structure Rock stresses and ground water conditions Strength of discontinuities and intact rock Pit geometry including both slope angles and slope curvature – Vibrations from blasting or seismic events – Climatic conditions – Time 2 METHODS APPLIED IN THE STUDY As mentioned before, safety factors have been calculated by three different approaches: 1. Hoek and Bray stability diagrams 2. Bishop’s simplified method of slices 3. Numerical modeling in FLAC3D The failure mode of a pit slope is also determined by these factors. It can be said that a pit slope is designed according to the failure mode expected to occur (Sjöberg 1999). The main failure modes observed in slopes can be listed as: – – – – It should be emphasized that the assumptions of each of these three methods are quite distinct and clearly stated in the literature. In fact, one of the main differences of these methods is that Hoek and Bray and Bishop’s simplified method of slices are based on 2-dimension limit equilibrium analysis whereas FLAC3D is based on 3-dimension numerical analysis. Therefore, in order to compare the results of these methods, the location of critical failure surface Plane failure Wedge failure Circular failure Toppling failure 83 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-10.qxd 08/11/2003 20:20 PM Page 84 Table 1. Parameter values used in all approaches. determined from the chart given by Hoek and Bray (1981) was chosen as a basis for comparison purpose. In other words, safety factors have been calculated for this critical failure surface by these three methods. However, the main focus of this study is on the results of numerical modeling. Geomechanical properties of rock and the shape of the slope used in this study can be listed as: – – – – – – Values Parameters Values Cohesion (kPa) 50 90* 130 170 20 25* 30 35 40 Cohesion fully saturated (kPa) 50 90 130 170 30 40 50 60* 70 1.6 1.9 2.2 2.5* 2.8 Slope height (m) Internal friction angle (°) Cohesion Internal friction angle Density Water condition Slope angle Slope height Density (t/m3) All these factors have been taken as parameters and safety factors have been calculated. For all three methods, the value of the parameter whose effect on safety factor will be investigated has been changed while the other parameters have been kept constant. The parameters and values used are given in Table 1. 2.1 Parameters Slope angle (°) 20 50* 80 140 200 *Constant values. Hoek and Bray stability diagrams Hoek and Bray have adopted an approach in which a series of the slope stability charts have been presented for circular failure (Hoek & Bray 1981). These stability diagrams have been used to find safety factor values. 2.2 Bishop’s simplified method of slices For the safety factor calculations, models have been formed in SLOPE/W program (Geo-soft, student edition). The search for the critical failure surface could be carried out in the program. But as mentioned before, instead of this, these calculations have been done for the critical failure surfaces whose locations have been determined from the Hoek’s chart. A model used in the analysis is shown in the Figure 1. 2.3 Figure 1. A model formed in the SLOPE/W. numerical models have been performed by means of a fish function written for this purpose (Aksoy 2001). The safety factor definition used has been based on the first approach proposed by Kourdey et al. (2001), but it has been modified and these modifications are as follows: Numerical modeling in FLAC3D – The mohr-coulomb failure criteria is directly used – The state of stress of zones are obtained from elastic, isotropic models – Normal and shear stresses are calculated on the critical plane of each zone According to the methodology proposed by Starfield & Cundall (1998), numerical modeling can be used to determine how different variables affect the slope stability. In this study, FLAC3D, a commercial finite difference code by Itasca, was selected for the purpose of numerical modeling and analysis. It is a threedimensional explicit finite difference program for engineering mechanics computations and it offers an ideal analysis tool for the solution of three-dimensional problems in geotechnical engineering (Itasca 1997). The stresses developed on the any zone in the numerical model can be expressed in terms of 1 and 3 and these stresses can be plotted on the mohr diagram as seen in Figure 2. To make safety factor definition clear, it is explained below in detail for the case in which the value of cohesion is changed while the value of internal friction angle is kept constant. There will be a critical plane on which the available shear strength will be first reached as 1 is increased. The orientation of this critical plane for 2.3.1 Safety factor calculations in FLAC3D FLAC3D does not calculate factor of safety directly (in version 2.0). However, it can be done by writing a fish function. In this study, safety factor calculations for 84 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-10.qxd 08/11/2003 20:20 PM Page 85 Figure 3. The general slope model and initial vertical stress state in FLAC3D. Figure 2. Mohr diagram. where si shear strength; n normal stress; internal friction angle. Local safety factor is described as the ratio of shear strength to shear stress developed on the critical plane for each zone: each zone can be calculated from the Equation 1 (Brady & Brown 1993): (1) where angle between critical plane and the horizontal; internal friction angle. Normal and shear stresses developed in this plane can be expressed as follows: (7) where F1 local safety factor; si shear strength; st shear stress. For the calculation of general safety factor Fg, the zones on the critical failure surfaces whose locations have been determined from the Hoek’s chart are used and general safety factor defined as: (2) (3) where n normal stress; 1 major principal stress; 3 minor principal stress; angle between critical plane and the horizontal; st shear stress. For the critical plane, these equations are rewritten due to sin2 cos and cos2 sin (Brady & Brown, 1993): (8) where Fg general safety factor; Fli local safety factor of the zone I; and vi volume of the zone i. This approach is used for all parametric studies. But, in the case of different internal friction angle values, the orientation of critical plane for which shear and normal stresses are calculated is taken as constant at the value found for internal friction angle 25°. In other words, it is assumed that the orientation of critical plane has not been affected by the change of internal friction angle value. The reason for this is to compare safety factor values at the same normal stress level. (4) (5) where st shear stress; 1 major principal stress; 3 minor principal stress; internal friction angle; n normal stress. As it can be seen on Figure 3, mohr failure envelopes that have different cohesion values with the same internal friction angle are drawn. Shear stress values at the intersection points (A1, A2) of mohr failure envelopes with the A2 D line are accepted as the shear strength values (s1, s2) of the zone depending on the value of cohesion and internal friction angle. And these shear strengths can be calculated from the Equation 6: 3 NUMERICAL MODEL STUDIES AND PARAMETRIC ANALYSIS Rock mass has been assumed as isotropic and homogeneous material through the study and the stresses in the numerical models have been initialized by taking the slope geometry into consideration. The general slope model and initial stress state is given in the Figure 3. In addition to cohesion, internal friction angle and density properties given in Table 1, the other material (6) 85 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-10.qxd 08/11/2003 20:20 PM Page 86 models has been searched. For this purpose, a R factor has been proposed and calculated from the Equation 11: properties used in numerical modeling are given in Table 2. k ratio, the ratio of horizontal stress to vertical stress, has been found from the Equation 9: (11) (9) where c cohesion; internal friction angle; H slope height; slope angle; rock mass density; and v Poisson ratio. where h horizontal stress; v vertical stress; Poisson ratio. The procedure followed in numerical modeling can be described as follows; first, material model for all numerical models has been selected as elastic, isotropic model. Safety factors have been calculated by using the results of these model runs. After safety factor calculations, all numerical models have been modified in such way that their material models have been changed from elastic, isotropic model into the mohr-coulomb plasticity model. Modified models have been run again and evaluated to determine the failure condition (Aksoy 2001). As it can be seen in Table 2, it has been assumed that rock mass has no tensile strength. However, in order to observe the effect of tensile strength on the safety factor values of numerical models, new models have been formed. In these models, tensile strength of rock mass has been calculated from the Equation 10 (Brady & Brown 1993). 4 RESULTS Results of three methods and failure conditions of numerical models are summarized in Tables 3 & 4. During the evaluation of the numerical models in terms of failure, it should be noted that FLAC3D does not produce a solution at the end of its calculation. However, several indicators such as unbalanced force, gridpoint velocities, plastic indicators and histories are used to asses the state of the numerical model in terms of stable, unstable, or in steady-state plastic flow (Itasca 1997). Table 3. Safety factors by Bishop and Hoek and Bray. Safety factor (10) where t tensile strength; c cohesion; internal friction angle. It has been also considered that changing the Poisson ratio taken as 0.25 for all models will change the magnitude of the horizontal stress and this will differentiate the stress state developed within the slope. As a result, this will affect the safety factor values of the slopes and to observe this effect, new models having different Poisson ratios have been run for different cohesion values. At the final stage of this study, the presence of a relationship between the parameters with the addition of Poisson ratio and safety factors in the numerical Values Elastic modulus Bulk modulus Shear modulus Tensile strength Poisson ratio k Ratio Dilation angle 1.70 GPa 1.13 GPa 0.68 GPa 0.00 Pa 0.25 0.33 0.0 Bishop Hoek and Bray Cohesion (kPa) 50 90 130 170 20 25 30 35 40 1.6 1.9 2.2 2.5 2.8 50 90 130 170 30 40 50 60 70 20 50 80 140 200 0.752 0.952 1.156 1.361 0.849 0.952 1.088 1.222 1.371 1.210 1.097 1.015 0.952 0.904 0.364 0.561 0.751 0.943 1.866 1.375 1.114 0.952 0.802 1.589 0.952 0.781 0.666 0.614 0.695 0.895 1.070 1.236 0.800 0.895 1.014 1.152 1.263 1.108 0.990 0.960 0.895 0.859 0.498 0.695 0.848 0.990 1.630 1.306 1.070 0.895 0.760 1.485 0.895 0.733 0.593 0.522 Density (t/m3) Cohesion (fully saturated) (kPa) Slope angle (°) Slope height (m) 86 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Values Internal friction angle (°) Table 2. Material properties. Properties Parameters 09069-10.qxd 08/11/2003 20:20 PM Page 87 These indicators are shown in Figure 4 for one of the numerical models. The displacement vectors and the contours of shear strain increment of the same numerical model are given in Figure 5. In the light of the evaluation of the numerical models having the mohr-coulomb plasticity model, safety factors of numerical models have been classified in terms of failure, and given in Figure 6. Safety factor values obtained from the models having different Poisson ratios for cohesion parameter are given in Figure 7. Results of the safety factor calculation for numerical models with different tensile strengths calculated according to Equation 10 are shown in Figure 8 for the cohesion parameter. A correlation has been established between R factor and safety factors of numerical models. According to this, the obtained linear regression model and correlation coefficient are as follows: Table 4. Safety factors and failure conditions of numerical models in FLAC3D. Parameters Values Safety factor Failure Cohesion (kPa) 50 90 130 170 0.551 0.705 0.859 1.013 YES YES NO NO Internal friction angle (°) 20 25 30 35 40 0.641 0.705 0.750 0.788 0.821 YES YES NO NO NO Density (t/m3) 1.6 1.9 2.2 2.5 2.8 0.900 0.815 0.753 0.705 0.668 NO NO YES YES YES Cohesion (fully saturated) (kPa) 50 90 130 170 0.103 0.141 0.180 0.218 YES YES YES YES Slope angle (°) 30 40 50 60 70 1.450 1.038 0.856 0.705 0.551 NO NO NO YES YES Slope height (m) 20 50 80 140 200 1.094 0.705 0.597 0.533 0.472 NO YES YES YES YES (12) where F safety factor; r correlation coefficient. It can be said that there is a very strong positive linear relationship between R factor and safety factor and it is shown in the Figure 9. 5 CONCLUSIONS For all parameters high safety factor values have been given by Bishop Approach and it has been followed by Hoek and Bray and FLAC3D approaches. But in full saturated condition, high safety factor values are given by Hoek and Bray approach and it is followed by Bishop and FLAC3D approaches. It was considered that the reason for low safety factor values in numerical models was the assumption of no tensile strength. Then new numerical models in which tensile strength was calculated depending on the cohesion and internal friction angle have been run and Figure 4. Indicators used to assess the state of the numerical model. 87 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-10.qxd 08/11/2003 20:20 PM Page 88 Figure 5. The displacement vectors and the contours of shear strain increment of the numerical model. 1.6 1.2 Safety Factor Safety Factor 1.4 1.0 0.8 0.6 0.4 Failure No failure 0.2 0.0 1 2 3 4 5 6 7 8 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 στ = 0 στ # 0 50 9 10 11 12 13 90 130 Cohesion (kPa) 170 Figure 6. Safety factor values classified in terms of failure. Figure 8. Safety factors at two different tensile strength conditions. 1.4 2 1.0 0.8 0.15 0.6 0.20 0.4 0.25 0.2 0.30 0.0 Safety Factor Safety Factor 1.2 90 130 Cohesion (kPa) 0.5 0 0.005 0.01 0.015 0.02 R Factor 170 Figure 9. The relationship between R factor and safety factor. Figure 7. Safety factors calculated for different cohesion values with the different Poisson ratio. When safety factor values of numerical models given in Figure 6 are considered, no failure has been observed in the models whose safety factors are higher than 0.8. Finally, a preliminary estimate value of the safety factor can be obtained before numerical modeling by using proposed regression model (equation 12). It can safety factor values have been calculated. As it can be seen in Figure 8, the new values are higher. It can be also said that another reason for low safety factor values in numerical models can be the numerical discretization chosen in this study. 88 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 1 0 0.35 50 1.5 09069-10.qxd 08/11/2003 20:20 PM Page 89 Hoek, E. and Bray, J.W. 1981. Rock Slope Engineering. London Institution of Mining and Metallurgy, 358 p. Itasca Consulting Group, Inc. 1997. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 2.0 User’s Manual. Minneapolis, MN: Itasca. Kourdey, A., Alheib, M. and Piguet, J.P. 2001. Evaluation of Slope Stability by Numerical Methods, 17th Int. Mining Congress and Exhibition of Turkey, IMCET 2001. Ankara. Sjöberg, J. 1999. Analysis of Large Scale Rock Slopes, Doctoral Thesis, Lulea University of Technology. Stacey, T.R. 1968. Stability of Rock Slopes in Open Pit Mines. National Mechanical Engineering Research Institute. Council for Scientific and Industrial Research, CSIR Report MEG 737, Pretoria, South Africa, 66 p. Starfield, A.M. and Cundall, P.A. 1988. Towards a Methodology for Rock Mechanics Modeling. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 25(3): 99–106. be useful during the design stage. But it should be noted that this regression model should be used with great caution. The reason for this is that safety factors calculated by using equation 12 will depend on numerical models formed in this study. Because, equation 12 has been obtained from regression analysis carried out on the results of numerical modeling studies. These results are greatly affected by numerical discretization chosen for numerical models in this study. REFERENCES Aksoy, M. 2001. A Study on the Effect of Parameters Affecting safety Factor of Slopes under Circular Failure Condition, MSc Thesis, Osmangazi University, Turkey. Brady, B.H.G. and Brown, E.T. 1993. Rock Mechanics for Underground Mining. London Second Edition, Chapman & Hall. 89 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-11.qxd 08/11/2003 20:20 PM Page 91 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Numerical modeling of seepage-induced liquefaction and slope failure S.A. Bastani Leighton Consulting, Inc., Irvine, California, USA B.L. Kutter University of California, Davis, California, USA ABSTRACT: Several earth dams, tailings dams, and slopes failed or were severely damaged due to liquefaction during or after earthquakes. In seismic areas, earth structures such as embankments may be subjected to two forces: the static loads due to gravity and the inertia forces caused by earthquakes. In a significant number of cases, liquefaction-induced failure of embankments occurred from seconds to hours after an earthquake. In these cases, liquefaction reduced the material strength and the failure occurred only under static loads. A finite slope was modeled at UC Davis National Geotechnical Centrifuge to evaluate post-earthquake deformations with an injection-induced liquefaction system. A constitutive model was developed to capture the behavior of sands with a minimum number of physically meaningful parameters to enable prediction of post-earthquake liquefaction and/or seepage-induced liquefaction. This constitutive model is based on the Mohr-Coulomb constitutive model and the Critical State concept by adding three parameters to the conventional Mohr-Coulomb model. The constitutive model performed adequately for modeling the sand behavior under monotonic drained and undrained triaxial loading and water injection for a simple shear test under a constant shear stress. Using the new constitutive model, the failure mode of the centrifuge model due to seepage-induced liquefaction was studied utilizing FLAC. Stress and strain paths for specific elements in the embankment are studied and presented in this paper. The centrifuge test was modeled by Fast Lagrangian Analysis of Continua (FLAC) computer code utilizing a new constitutive model as presented in this paper. For more details on the centrifuge and numerical models refer to Bastani (2003). 1 INTRODUCTION Examples of post-earthquake liquefaction-induced failures of embankments are reported by Dobry & Alvarez (1967), Seed et al. (1975), Okusa et al. (1978), and Finn (1980). In these cases, liquefaction reduced the material strength and the failure occurred under static forces after the earthquake shaking. One mechanism for the delayed failure is the softening associated with redistribution of void ratio caused by gradients of pore water pressure in sloping ground with non-uniform permeability. This mechanism has been studied by Malvick et al. (2003) and Kokusho & Kojima (2002). For the present study, the post-earthquake liquefaction-induced failure of granular embankments was investigated by a static centrifuge test in which the water that might be produced during an earthquake due to densification of deep saturated soil was simulated by injecting a similar volume of water at the base of the model as presented in detail by Bastani (2003). This centrifuge model consisted of a coarse sand layer with a constant thickness at its base to spread the injected water beneath an embankment composed of a fine sand capped by a layer of low permeability clayey silt. 2 CENTRIFUGE MODEL The centrifuge model consisted of three layers: 1. A uniform 51 mm thick layer of Monterey Sand (mean grain size 1.25 mm); 2. A fine sand (Nevada Sand, mean grain size 0.12 mm) embankment with a minimum thickness of 102 mm at its toe and a maximum thickness of 356 mm at the slope crest; and 3. A uniform 51 mm thick layer of Yolo Loam that capped the Nevada Sand embankment. The horizontal lengths of the embankment toe, the slope, and the crest were 356, 584, and 533 mm, respectively. The slope angle was 23.5 degrees. The average void ratio of the Nevada Sand was 0.77 corresponding to a relative density of 33 percent; at this density, 91 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-11.qxd 08/11/2003 20:20 PM Page 92 (15,19) (15,16) (18,4) where e is the void ratio, (ecs)a is the critical state void ratio at one atmosphere, is the virgin compression slope, p is the mean effective stress, and pa is the atmospheric pressure. As explained by Been & Jefferies (1985), the gradual change from dilative to contractive behavior can be quantified in terms of the state parameter . The dilation angle (dilation) was modified based on the state parameter and its changes according to the equation: (40,16) (34,12) (28,16) (20,13) Yolo Loam (13,9) (18,6) Nevada Sand (47,6) Monterey Sand 0 0.5 Grid plot (2) Figure 1. FLAC grid. where dilation is the updated dilation angle, (dilation)0 is the initial dilation angle, is the state parameter, and is the variation of the state parameter. The soil’s bulk and shear moduli are also modified in the model. The bulk modulus (K) is evaluated using the relation: the sand was highly dilative at the confining pressures experienced in the experiment. The Yolo Loam had an average undrained shear strength of about 10 kPa and a water content of 33 percent. An additional overburden pressure equivalent to 90 mm of water head was applied over a plastic membrane on the Yolo Loam layer. All dimensions are provided in the model scale and the embankment’s configuration is presented in Figure 1. The centrifuge model was consolidated in several stages as the centrifuge speed was increased up to 37.9 g. (3) where is the unloading slope and p and e are defined as above. The shear modulus (G) was consequently determined based on the bulk modulus (K) and their elastic relationship: 3 CONSTITUTIVE MODEL (4) A constitutive model was developed to simulate behavior of Nevada Sand in the FLAC program framework, and it was incorporated in the numerical modeling of the centrifuge test. The purpose of this model was to predict the principal behavior of Nevada Sand with a minimum number of parameters that are physically defined and measurable. The failure envelope for this constitutive model corresponds to the Mohr-Coulomb constitutive model (shear yield function) with tension cutoff (tensile yield function). The shear flow rule is non-associated and the tensile flow rule is associated. The shear potential function corresponds to a non-associated flow rule. Details of Mohr-Coulomb model implementation are explained in the FLAC manual published by Itasca (2001). Several modifications are made to the MohrCoulomb model. The mobilized friction angle (mobilized cs dilation) is represented as a sum of the critical state friction angle (cs) and the dilation angle (dilation) as described by Bolton (1991). cs is considered constant, while, the dilation is assumed to be variable depending on the distance of the material state from the critical state line in e-log(p ) space, defined by the state parameter: where v is Poisson’s ratio. The behavior of the constitutive model under selected load paths are presented in Figure 2 conventional drained triaxial compression, conventional undrained triaxial compression, and a simple shear element subjected to a constant applied shear stress and water injection. Results are shown for deviator stress (q), p , volumetric strain (v), void ratio (e), and shear strain (). This constitutive model predicted the strain hardening behavior of the Nevada Sand during undrained shearing until cavitation occurred prior to reaching the critical state line. The undrained path in Figure 2 approximately simulated the triaxial test data. Bastani (2003) compared the calculated undrained stress paths with experimental data (not shown here). The model behaved more stiffly under the undrained condition, approximately 2 times more than what was observed in the triaxial experiments for the Nevada Sand with a relative density of 26%; but the model reasonably matched test results for the Nevada Sand with a relative density of 39.4%. The model behavior exhibited elastic contraction under the drained condition up to the peak shear stress. Dilation started after the peak shear stress and (1) 92 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-11.qxd 08/11/2003 20:20 PM Page 93 300 0.92 300 (')cs (')cs+(')dilation 250 200 200 0.88 150 0.84 0.8 q (kPa) q (kPa) e 250 150 0.76 100 1 100 10 100 p'(kPa) 50 50 0 0 50 100 150 200 p'(kPa) 250 300 150 0.04 100 0.03 Model Behavior: 50 εv Pore Water Pressure (kPa) 0 0 0.05 0.1 0.15 0.2 γ 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 γ 0.25 0.3 0.35 0.4 0.02 Drained Triaxial Test Undrained Triaxial Test 0 0.01 Simple Shear (Khc =0.6, τxy =62 kPa) -50 0 Cavitation -100 -0.01 0 0.05 0.1 0.15 0.2 γ 0.25 0.3 0.35 0.4 Figure 2. Behavior of the new constitutive model. Table 1. Model parameters. continued up to the critical state condition. Finally, the model behavior was studied under a constant shear stress and pore water pressure increase, modeling a simple shear test with pore fluid injection. The constitutive model slightly dilated prior to reaching the failure envelope; thereafter, the sample dilated with the increase of pore water pressure and the stress path approached the origin along the failure envelope in the p -q space until it reached the critical state condition similar to the stress path suggested by Boulanger (1990). The dilation rate was less than that shown by his experiment (Boulanger, 1990); however, the stress path, boundary condition, and initial condition of the experiments performed by Boulanger prior to water injection into his simple shear tests were not known, and therefore were not completely simulated by this calibration. As expected, the water injection to the element led to an unstable condition when the strength of the element dropped below the applied shear stress. Continued softening caused the stress path to drop toward the origin while the sample collapsed Value * ( /5) cs (degree) (dilation)max (degree) e0 (initial void ratio) (ecs)a* Atmospheric pressure, pa (kPa) 0.022 0.0044 32 10 0.25 0.77 0.809 101.2 * Archilleas et al. 2001. dynamically under the unbalanced external loads. Some oscillation is observed in the q– curves at shear strains greater than 0.12, but the softening behavior can still be clearly observed during the dynamic collapse. The parameters used for this calibration and later in the numerical modeling based on this constitutive model are provided in Table 1. 93 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Parameters 09069-11.qxd 08/11/2003 20:20 PM Page 94 4 for the two runs are plotted on Figure 3. This figure indicates the following behaviors: FLAC MODEL BEHAVIOR FLAC version 4.0 was utilized to model the centrifuge test. The numerical model was run twice. The first run used the conventional Mohr-Coulomb constitutive model, while the second run utilized the new constitutive model (discussed in Section 3) to model the Nevada Sand behavior. This numerical model was bounded with its and the constitutive model’s limitations; however, it was successfully used to observe the general mechanism of localized increase in void ratio just beneath the less permeable clayey silt layer, and the failure mechanism; exact predictions were not expected. The FLAC runs were performed with the large-strain mode. Figure 1 shows the grid utilized in this model. The grid nodes and elements are identified in the subsequent figures with their column and row numbers (i,j). The column and row numbers increase from left to right and bottom to top, respectively. Contours of mobilized friction angle, volumetric strain, and shear strain and grid deformation patterns 1. The mobilized friction angle was reduced along the Nevada Sand interface elements by the new constitutive model and along a deeper seated failure plane as shown by the new constitutive model; 2. Volumetric strains were concentrated along the interface of Nevada Sand and Yolo Loam in both numerical models. However, deeper volumetric strains were observed in the modified constitutive model, which coincided with the friction angle and shear strain patterns; 3. Shear strains were also concentrated at the slope interface within the Nevada Sand layer. Similarly a deeper shear zone was predicted by the new constitutive model matching the volumetric strain and mobilized friction angle reduction patterns; 4. Sand and clay layers moved downward at the slope, which was translated to vertical uplift at the toe. It is worthwhile to mention that the pore water pressure was mainly increased from the slope toe within Figure 3. Friction angle, volumetric and shear strains, and deformation patterns at 13 seconds of seepage. 94 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-11.qxd 08/11/2003 20:20 PM Page 95 the second run due to the ability of the modified constitutive model to withstand a mobilized friction angle greater than the critical state friction angle during dilation. The majority of volumetric strains of elements were induced when the mean effective stress (p ) became less than 10 kPa and close to zero. The volumetric strains were stabilized wherever the injection did not cause the strength to fall below the applied shear stress. Predicted pore water pressures and deformations are compared with the experimental results in Figures 5 & 6, respectively. In general the predicted pore water pressures are in good agreement with the experimental results for both constitutive models. However, the initial rate of pore water pressure increase is in better agreement with the new constitutive model. Similar trends were obtained by the numerical model, such as stabilization of pore water pressure at the beginning and its further increase for PPT# 5296 (Fig. 5). Other than deformation at the middle of the slope (LVDT #3), where the experimental result indicated bulging, deformation rates and magnitudes were predicted very well by both models. The numerical model successfully predicted the dilatancy to cause a very loose layer of sand below the Nevada Sand layer toward the slope crest and with a slower rate from the back of the slope crest toward the slope. It should also be noted that the development of a deep failure mechanism, or not, was affected by the rate at which the water was injected relative to the permeability of the soils. For somewhat slower injection, the deeper mechanism would disappear and sliding along the bottom interface of the Yolo Loam would be apparent. For much greater injection rates, a failure mechanism at the interface between the coarse Monterey Sand and the fine Nevada Sand was observed (Bastani 2003). Stress/strain paths of several elements at the toe, along the slope, and at the slope crest are plotted on Figure 4. Effective stresses of slope/leaning elements reduced while oscillating around constant shear stresses up to the failure envelope. However, shear stresses of carrying elements along the slope and its toe increased during the failure of leaning elements until reaching the failure envelope. Stress paths moved toward the origin after reaching the failure envelope and strain softening was observed. In general, the elements at the toe and along the slope showed higher shear strengths prior to their stress paths diving toward the origin in Figure 4. Stress path and behavior of elements. Solid and dashed lines refer to the results of the new and Mohr-Coulomb constitutive models, respectively. 95 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-11.qxd 08/11/2003 20:20 PM Page 96 Figure 6. Deformation time histories. the less permeable layer of Yolo Loam. The calculated volumetric strains of the dilated sand indicate a negligible residual strength after dilation. 5 CONCLUSION A modified Mohr-Coulomb constitutive model was developed based upon critical state theory in conjunction with a new expression for dilatancy that depends on the state parameter (the distance between the state and the critical state). The constitutive model was shown to enable calculation of strain-softening paths, and dilation due to water injection. The constitutive model was implemented in FLAC and used to analyze results of centrifuge model tests of layered sloping ground subject to pore fluid injection. The injection was intended to simulate the upward flow of water that might be generated by densification of deep soil deposits during earthquake shaking. In the past, embankments made of dilative material were considered to be safe, because the undrained strength is greater than the driving stress (Poulos et al. 1985). The centrifuge tests and the FLAC analyses Figure 5. Pore water pressure time histories. 96 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-11.qxd 08/11/2003 20:20 PM Page 97 Boulanger, R.W. 1990. Liquefaction Behavior of Saturated Cohesionless Soils Subjected to Uni-Directional and BiDirectional Static and Cyclic Simple Shear Stresses. Dissertation presented to University of California, at Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Castro & Poulos (ASCE paper circa 1984). Dobry, R. & Alvarez, L. 1967. Seismic Failure of Chilean Tailing Dams. Journal of Soil Mechanics and Foundations Division, Proceeding of the American Society of Civil Engineers 93(SM6): 237–260. Finn, W.D. 1980. Seismic Response of Tailing Dams. Presented at Seminar on Design and Construction of Tailing Dams, Colorado School of Mines, Denver, Colorado, pp. 76–97. Itasca Consulting Group, Inc. 2001. FLAC – Fast Lagrangian Analysis of Continua, Ver. 4.0 User’s Manual. Minneapolis, MN: Itasca. Kokusko, T. & Kojima, T. 2002. Mechanism for Postliquefaction Water Film Generation in Layered Sand. Journal of Geotechnical Engineering, ASCE 128(2): 129–137. Malvick, E.J., Kulasingam, R., Boulanger, R.W. & Kutter, B.L. 2003. Analysis of a Void Ratio Redistribution Mechanism in Liquefied Soil. To be Published in Proceedings of the June 2003 Soil and Rock America Conference. Okusa, S., Anma, S. & Maikuma, H. 1978. Liquefaction of Mine Tailing in the 1978 Izu-Ohshima-Kihkai Earthquake, Central Japan. Engineering Geology Vol. 16, pp. 195–224, Elsevier Scientific Publishing Co. Poulos, S.J., Castro, G. & France, J.W. 1985. Liquefaction Evaluation Procedure. Journal of the Geotechnical Engineering Division, ASCE 111(6): 772–792. Seed, H.B., Lee, K.L., Idriss, I.M. & Makdisi, F.I. 1975. The Slides in the San Fernando Dams during the Earthquake of February 9, 1971. Journal of the Geotechnical Engineering Division, ASCE 101(GT7): 651. presented here clearly demonstrate the possibility that layers that impede drainage may cause a significant localized zone of softened material that should be considered a possibility in seismic design. To determine induced deformations due to local drainage of a system, a material model that captures this process should be incorporated in the numerical model. The mode of failure and local drainage of the centrifuge test presented here was successfully predicted utilizing the modified Mohr-Coulomb constitutive model in conjunction with FLAC numerical framework. ACKNOWLEDGEMENT The authors would like to thank Dr. Ben Hushmand, James Ward, and Vivian Cheng for reviewing this paper and providing constructive comments. REFERENCES Archilleas, G.P., Bouckovalas, G.D. & Dafalias, Y.F. 2001. Plasticity Model for Sand Under Small and Large Cyclic Strains. Journal of Geotechnical Engineering, ASCE 127(11):.973–983. Bastani, S.A. 2003. Evaluation of Deformations of Earth Structures due to Earthquakes. Dissertation presented to University of California, at Davis, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Been, K. & Jefferies, M.G. 1985. A State Parameter for Sands. Geotechnique 35(2): 99–112. Bolton, M. 1991. A Guide to Soil Mechanics. Published by M D & K Bolton, Printed by Chung Hwa Book Company, pp. 63–92. 97 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-12.qxd 08/11/2003 20:21 PM Page 99 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Complex geology slope stability analysis by shear strength reduction M. Cala & J. Flisiak Dept. of Geomechanics, Civil Engineering & Geotechnics, AGH University of Science & Technology, Poland ABSTRACT: The stability of slopes may be estimated using 2D limit equilibrium methods (LEM) or numerical methods. Due to the rapid development of computing efficiency, several numerical methods are gaining increasing popularity in slope stability engineering. A very popular numerical method of slope stability estimation is the shear strength reduction technique (SSR). It’s a well known fact that for a simple slope factor of safety (FS) obtained from SSR is usually the same as FS obtained from LEM. However for slopes of complex geology, considerable differences between FS values may be expected. Application of SSR for such slopes is usually restricted to the weakest link estimation – that part of the slope with the lowest FS. Finite Difference Method code, FLAC (Itasca 2000), gives the opportunity to analyze several slip surfaces by using the modified SSR technique (MSSR). The method is based on reducing shear properties of soils after identification of the first slip surface. MSSR allows a complete estimation of stability for any type of slope. geometry (and geology) it’s not possible to analyze FS for other parts of the slope. This may sometimes lead to serious mistakes. 1 INTRODUCTION The stability of slopes may be estimated using 2D limit equilibrium methods (LEM) or numerical methods. Due to the rapid development of computing efficiency, several numerical methods are gaining increasing popularity in slope stability engineering. A very popular numerical method of slope stability estimation is shear strength reduction technique (SSR). In that procedure, the factor of safety (FS) of a soil slope is defined as the number by which the original shear strength parameters must be divided in order to bring the slope to the point of failure (Dawson & Roth 1999). It’s a well known fact that for simple slopes FS obtained from SSR is usually the same as FS obtained from LEM (Griffiths & Lane 1999, Cala & Flisiak 2001). However for complex geology slopes considerable differences between FS values from LEM and SSR may be expected (Cala & Flisiak 2001). Several analyses for the slope with weak stratum were performed to study the differences between LEM and SSR. It must be also stated that classical SSR technique has several limitations. Application of SSR requires advanced numerical modeling skills. Calculation time, in case of complicated models, can last as long as several hours. However, the most fundamental limitation of SSR is identification of only one failure surface (in some cases it may identify more than one surface, but with the same FS value). This is not a significant limitation in case of simple geometry slope. But in case with complex 2 STABILITY OF SLOPE WITH WEAK STRATUM 25 m 45° Figure 1. Slope with weak stratum. 99 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 25 m g h To investigate the influence of a weak stratum on FS some 350 models were analyzed. The thickness of the weak stratum was changed from 1.0 to 10.0 m and it was localized from 0 to 50 m from the top of the slope (Fig. 1). All slopes in this paper were simulated with FLAC/ Slope (Itasca 2002) or FLAC in plane strain, using small-strain mode. It was assumed that embankment is 25 m high and has a slope angle of 45°. It consists of two different geological units. The soil was given friction angle 30° 09069-12.qxd 08/11/2003 20:21 PM Page 100 2 2 1.9 1.9 1.8 1.7 1.6 1.7 1.5 FS FS 1.8 1.4 1.6 1.3 Weak layer 1 m thick FLAC Fellenius Bishop Janbu 1.5 Weak layer 5 m thick FLAC Fellenius Bishop Janbu 1.2 1.1 1.4 1 10 20 30 40 Distance of weak layer from slope crest 0 Figure 2. FS values for a 1.0 m thick weak layer. 20 30 40 50 Figure 3. FS values for a 5.0 m thick weak layer. FLAC FS = 1.54 Bishop FS = 1.731 20 m and cohesion c 75 kPa. The weak, thin layer had friction angle 10° and cohesion c 25 kPa. Both soils had unit weight 20 kN/m3. The thickness “g” of the horizontal weak layer was changed from 1.0 m to 10.0 m and its distance “h” from the top of the slope changed from 0 to 50 m. Figure 2 shows the FS values for a 1.0 m thick weak layer and Figure 3 for a 5.0 thick one. The decrease of FS is quite small if the thin weak layer is located close to the top of the slope. Increasing the weak layer thickness produces considerable decrease of FS. The differences in FS values are significant especially in case of small thickness (1 m–3 m) of weak stratum Increase of weak layer thickness (irrespectively of its localization) reduces differences between FS values from LEM and SSR. Especially FS values estimated with Bishop’s are within 8 % of the FS obtained from SSR. For the thickness of the weak layer less than or equal to 5 m SSR produces lower FS values than any of the LEM methods. For the weak layer 5 m thick Bishop’s method produces FS 1.114 and SSR shows FS 1.07. Further increase of weak layer thickness (7.5 m and 10 m) produces lowest FS values from Bishop’s method (FS 0.926 and FS 0.811 respectively). SSR technique shows respectively FS 0.95 and FS 0.87 in this case. It seems that application of Bishop’s method produces the most reliable results among LEM. These results are simultaneously closest to the FS values obtained from SSR. Application of Fellenius’s method produces unreliable FS values in case of weak layer Figure 4. Critical slip surfaces identified by SSR and LEM. localization below slope toe. It shows the influence of weak layer on FS values even if the roof of the stratum lays 15 m below the slope toe. It must be also pointed out that failure surfaces identified by SSR technique are sometimes considerably different than surfaces identified by LEM (Fig. 4). Figure 4 shows the situation when FS computed by SSR is considerably lower and unit volume of failed slope is significantly higher than estimated from LEM. 3 MODIFIED SHEAR STRENGTH REDUCTION TECHNIQUE (MSSR) 3.1 Benched slope stability case Application of SSR for complex geology slopes is usually restricted to the weakest “link” estimation – part 100 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 10 Distance of weak layer from slope crest 1m 0 09069-12.qxd 08/11/2003 20:21 PM Page 101 10 m of the slope with the lowest FS. However the Finite Difference Method code FLAC gives the opportunity to analyze several slip surfaces using modified shear strength reduction technique – MSSR (Cala & Flisiak 2003a, b). This method is based on reducing shear properties of soils after identification of first slip surface (FS1). It is simply the continuation of classic SSR, but after first instability occurrence. It is possible only using Finite Difference Method. The FLAC program uses the explicit, Lagrangian calculation scheme. The full dynamic equations of motion are used, even when modeling systems that are essentially static. This enables FLAC to follow physically unstable processes (i.e. several processes simultaneously) without numerical distress. In fact, FLAC is most effective when applied to nonlinear or large-strain problems, or to situations in which physical instability may occur. This may lead to identification of several other slip surfaces. The same criterion is used to identify secondary (and further) failure surfaces. The primary and the following identified failure modes are constantly active (not suppressed) during entire calculation process. Let’s consider benched slope stability (Fig. 5). Figure 6 shows the slip surfaces identified in benched slope by MSSR and LEM. Failure of the lower part of the slope was detected first. FS1 0.90 calculated by SSR is very close to FS 0.921 given by Bishop’s method. And precisely here ends the range of 10 m 40º 3 = 20 kN/m = 20º C = 10 kPa 15 m 45º 15 m 15 m 20.918 m Figure 5. Benched slope case geometry. Bishop FS = 1.228 Bishop FS3 = 1.24 FS = 0.921 Bishop FS = 1.008 FS2 = 1.00 classical SSR technique – especially with application of any Finite Element Method code. However FLAC is created especially for modeling physical instability (in this case – physical instabilities would be better term). This allows to continue shear strength reduction and to identify another possible slip surfaces. In analyzed case, next identified failure surface is located in the upper part of the slope. FS2 1.00 calculated by MSSR is again very close to FS 1.008 given by Bishop’s method. And finally application of MSSR allowed to evaluate FS for entire slope – FS3 1.24 is also very close to FS 1.228 given by Bishop’s method. It seems that FS calculated with MSSR are within a few percent of the FS obtained from LEM for simple cases. It must be however underlined that effectiveness of MSSR must be verified on real cases. 3.2 Large scale, complex geology slope stability case Let’s consider a slope consisted of eight different geological units (from a Polish lignite open pit mine). The mechanical properties of the soil units involved in the slope are given in Table 1. Figure 7 shows geometry and geology of the analyzed slope. The overall sloping angle was equal ! 7.477°. Figure 8 presents the slip surface identified by MSSR and LEM. Again SSR finds the location of the lowest safety factor FS1 0.67. Application of MSSR identifies four new slip surfaces in several parts of the slope. FS2 0.87 also shows the local failure surface which, in fact, does not affect the overall slope stability (precisely like previous one). Another possible failure surface with FS3 1.02 is based on layer 5 (very thin and weak one) and broken line upward. Further analysis showed development of previous failure surface with FS4 1.17 occurring mainly in layer 5. Bishop’s method applied to the upper part of the slope shows cylindrical failure surface with FS 1.351. It must be noted that due to cylindrical shape Bishop’s slip surface covers a little more soil volume. Table 1. Mechanical properties of soil units. Unit Cohesion c, kPa Friction angle , deg Unit weight , kN/m3 1 2 3 4 5 6 7 8 14.0 90.0 11.4 90.0 11.4 90.0 28.0 1000 6.5 10.9 7.9 10.9 7.9 10.9 8.5 30.0 18.3 19.5 19.5 19.5 19.5 19.5 20.0 23.0 FS1 = 0.90 Figure 6. Several slip surfaces identified in benched slope by MSSR and LEM. 101 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-12.qxd 08/11/2003 20:21 PM Page 102 63 m 2 4 5 6 7 8 168 m 1 3 800 m Figure 7. Slope geometry and geology. FS4 = 1.17 FS3 = 1.02 FS1 = 0.67 Bishop Bishop FS = 1.255 FS = 1.351 FS2 = 0.87 FS5 = 1.29 Figure 8. FS values and critical slip surfaces identified with MSSR and LEM. FS 1.351 is however considerably higher than FS4 1.17 from MSSR. And finally an overall slope failure surface with FS5 1.29 is identified. Bishop’s method shows FS 1.255, but it covers considerably lower soil volume. Generally, the results obtained from LEM are not that close to MSSR as in the simple case discussed before. It’s a well-known fact that application of LEM requires assumption about shape and location of slip surface. Circular failure surfaces were assumed here for calculation purposes. Critical slip surface with lowest FS value was estimated from 20,000 circles. In MSSR there is no need for such assumptions. Stress and strain field in analyzed soil determines the shape and location of the slip surfaces. 4 CONCLUSIONS For a simple, homogeneous slope FS calculated with SSR are usually the same as FS obtained from LEM. In the case of a simple geometry slope consisting of two geological units, FS calculated with SSR may be considerably different than FS from LEM. In the case of complex geometry and geology slopes SSR technique is much more “sensitive” than LEM. Another step forward is the modified shear strength reduction technique – MSSR. Application of SSR with FLAC may be recommended for the largescale slopes of complex geometry. Such a powerful tool as MSSR with FLAC gives the opportunity for the complete stability analysis for any slope. ACKNOWLEDGEMENTS Support for this research by the State Committee for Scientific Research (Project No. 5 T12A 022 24) is gratefully acknowledged. REFERENCES Cala M. & Flisiak J. 2001. Slope stability analysis with FLAC and limit equilibrium methods. In Billaux, Rachez, Detournay & Hart (eds) FLAC and Numerical Modeling in Geomechanics; Proc. Intern. Symp., Lyon, France, 29–31 October 2001: 111–114. Rotterdam: Balkema. Cala M. & Flisiak J. 2003a. Analysis of slope stability with modified shear strength reduction technique. XXVI Winter School of Rock Mechanics: 348–355. Wroclaw. IGiH, (in polish). Cala M. & Flisiak J. 2003b. Slope stability analysis with numerical and limit equilibrium methods. Computational Methods in Mechanics; Proc. Intern. Symp., 3–6 June 2003 (in press). Dawson E.M. & Roth W.H. 1999. Slope stability analysis with FLAC. In Detournay & Hart (eds) FLAC and Numerical Modeling in Geomechanics; Proc. intern. symp., Minneapolis, MN, 1–3 September 1999: 3–9. Rotterdam: Balkema. Itasca Consulting Group. 2000. FLAC – Fast Lagrangian Analysis of Continua, Ver. 4.0 User’s Manual. Minneapolis, Minnesota: Itasca. Itasca Consulting Group. 2002. FLAC/Slope Ver. 4.0 User’s Manual. Minneapolis, Minnesota: Itasca. Griffiths D.V. & Lane P.A. 1999. Slope stability analysis by finite elements. Geotechnique. 49(3): 387–403. 102 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-13.qxd 08/11/2003 20:24 PM Page 103 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Analysis of hydraulic fracture risk in a zoned dam with FLAC3D C. Peybernes Electricité de France, Centre d’Ingénierie Hydraulique, France ABSTRACT: The construction and first filling of a 150 m high zoned dam are modeled with FLAC. The site is a deep and curved canyon under the dam. The aim of this study is the understanding of the dam behavior and the assessment of the dam safety. A lot of attention is put on the hydraulic fracturing risk during construction or during first filling due to the core arching in the deep canyon. 2D and 3D models are compared. The 2D model is unable to explain the monitoring data, but the 3D model fits more accurately the monitoring measurement. Although a high contrast of modulus exists between core and shell, no hydraulic fracturing is observed in the core. 2.1 1 DESCRIPTION OF STRUCTURE The zoned dam has a clayey core with vertical downstream face, gravely downstream and upstream filters and shells. The canyon in the bottom of the valley is 60 m deep, narrow, and turns under the dam. Main features are (Fig. 1): – – – – Maximum height of dam: 137.00 m, Maximum elevation: 1000.00 m, Minimum elevation: 863.00 m, Slopes of the faces: H/V 2/1. Two geometrical models were meshed by Itasca Consultant Spain office. The strategy was to use the 2D model to set the characteristics of the materials, the loading scenario and the boundary conditions, and to use the 3D model to analyze the arching effects caused by the stiff banks. load transfer from dam body to the canyon, turn on the left of the valley (Fig. 3), dissymmetry between the banks (Fig. 4), plating of the filter against left bank (Fig. 5). 3 SETTING OF MATERIAL PROPERTIES The model is fitted on several indicators from the monitoring measurement. Figure 2. Two-dimensional mesh (“section”). 103 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Three-dimensional model The mesh has 66,054 3D elements and 71,828 nodes. The geometry of the contact dam foundation, in particular the canyon is rather faithful to reality. This model has great advantages over the 2D. It takes into account the following items: – – – – 2 TWO GEOMETRICAL MODELS Figure 1. Standard profile. The section is the deepest one in the canyon perpendicular to the dam axis (Fig. 2). It is 1 m wide modeled by FLAC3D. The mesh has 1861 3D elements and 3890 nodes. The constitutive equations of material are programmed for both 2D and 3D files. 2.2 The downstream toe is submerged by the reservoir of the downstream dam from elevation 890.0 to 906.0 m. Two-dimensional model 09069-13.qxd 08/11/2003 20:24 PM Page 104 – Settlements in the core measured at the end of the construction (up to 80 cm). – total stresses in the core – pore water pressures in the core – deformations in the downstream shoulder recorded by three tassometers and two extensometers. 4 CONSTITUTIVE EQUATIONS FOR SOIL Two different constitutive models are used: 1. elastic model, for first parameter setting, 2. plastic law: Mohr-Coulomb. 5 SCENARIO OF LOADING 5.1 Initialization of the model The initial stress state is calculated in the foundation, alluvium filling and excavation at the core location. The initial equilibrium calculation is only mechanical (zero pore pressure). 5.2 Figure 3. Global sight of the model. Construction The construction period is 4 years and the embankment is placed in 4 m layers. 5.2.1 Hydraulic boundary conditions At the boundary of the core, the water pressure (Pw) is fixed at zero to dissipate pore pressures in the core to allow drainage due to the filter and shell. 5.2.2 Mechanical boundary conditions With every placed layer, the increase of vertical and horizontal stresses is given by the weight of the layer, zz 0.5 * h * and xx 0.5 * zz. 5.2.3 Hydraulic-mechanic coupling The pore pressure is assumed to be generated by Skempton’s B coefficient via the water modulus, Kw (B * n * Kcore)/(1 – B). 5.3 First filling and steady state Figure 4. Sight of top of the valley, with studied sections. 5.3.1 Hydraulic boundary conditions The pore pressure is fixed by the value of the hydraulic load caused by the reservoir filling, Pw (Hw – h) * w, if Hw h where Hw storage level; h node level; and w unit weight of water. Figure 5. Section right bank – left bank. Figure 6. Boundary conditions to construction. 104 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-13.qxd 08/11/2003 20:24 PM Page 105 Two storage versus time curves are used: one for the downstream shell and one for the upstream storage (Fig. 7). 5.3.2 Mechanical boundary conditions On the upstream and downstream faces of the dam, mechanical pressure caused by the impounding is modeled by normal stresses on the dam faces. The specific weights of materials are modified when they are saturated. 1000 980 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 1986 1985 1984 1983 1982 1981 1980 1979 First filling dates 6 COMPARISON OF 2D AND 3D MODELS In this section, the results of the model analyses are compared to measurements at the end of construction (Fig. 9). 6.1 2D Model The calculated stresses are too high at the base of the core compared to the measured ones. A parametric study of mechanical properties could not solve the discrepancy. It is speculated that the arching effect unloads the central section and transfers stresses to the banks. No realistic calculation could be reached using the 2D model. cote (m) 960 940 6.2 920 For this model, the calculated settlements in the core are in good agreement with the measured ones. The stresses are smaller than the 2D problem, because of arching. Nevertheless, the deformation of the shells is still larger than measured. A parametric study of core and shell moduli was undertaken to reconcile the discrepancy. 900 880 upstream downstream Figure 7. Curves of upstream and downstream fillings. 3D Model 7 PARAMETRIC STUDY OF THE 3D MODEL DURING CONSTRUCTION Figure 8. Boundary conditions during the filling. Several simulations were studied varying the mechanical properties (moduli) and flow parameters (saturation). The initial and final mechanical properties are presented in Table 1. A comparison of results is made at the end of construction. Figure 9. Comparison between effective stresses with 2D and 3D models. 105 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-13.qxd 08/11/2003 20:24 PM Page 106 7.1 Displacements Vertical and horizontal displacements are in good agreement with the measurements. The shell modulus shown gave results that agreed with the deformations measured downstream, but decreased the core settlements. The simulation carried out using a decreased modulus of the core gave the closest values to the measurements. 7.2 Pore pressures Pore pressures are very low, like the measurements at the end of construction (Fig. 10). The clay compaction Table 1. Initial and final mechanical properties. Material C (kPa) Phi (°) E (Mpa) v B Core initial Core final Shell initial Shell final 20 20 0 0 25 25 40 40 40 20 120 240 0.35 0.35 0.30 0.30 0.6 0.1 – – carried out (dry or very dry of optimum) results in very low pore pressure generation. This lowers the risk of hydraulic fracturing during first filling. 7.3 The results that best agree with cell measurements were obtained with a relatively high shell modulus, and low core modulus. On the other hand, the calculated total vertical stresses near the left bank were far from the measured stresses. This phenomenon is not clearly understood. The most important conclusion is that hydraulic fracture was not indicated in any calculated case, even though the base of the core is heavily unloaded in the narrow canyon (Fig. 11 Area 1). The load transfer is clearly observed in the core section parallel to the dam crest (Fig. 11 Area 2). After completing the parametric study at the end of construction, calculation was continued for the first filling case. Figure 10. Pore pressure at the end of construction. Figure 11. Vertical effective stresses at the end of construction. 106 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Stresses 09069-13.qxd 08/11/2003 20:24 PM Page 107 8 3D MODEL FOR FIRST FILLING CASE 8.1 Pore pressures The pore pressures calculated in the core at the level 925 are close to the measured values (Fig. 13). It was difficult to get agreement between the measured and calculated values below this level. Variations of permeability in the core were not represented in the modeling. Spatial variation of the hydraulic properties in the core may improve the correlation, but this was not done. 8.2 Stresses Generally the shapes of the calculated and monitored stresses versus time were similar. The construction phase is apparent in the plots, then the filling of the two reservoirs, and finally a steady state was reached (Fig. 14). The values from the simulation and from measurements are rather close until the date 1981 for the cells to the level 925, and 1983 for the cells of level 905 (Figs. 15 & 16). These dates correspond to the sudden drop of measured stress values. Then the simulations and measured values disagree. It is speculated that this phenomenon was induced by water infiltration at the cell level, collapsing the clay and lowering the stress. This phenomenon should be integrated in the future and modeled by the clay collapse after wetting. Some hydraulic fracturing can be observed in the upstream shell, but this is not of concern. The shell is drained and the water tightness of the core is not altered. Figure 12. Pore pressures at steady state. 420 380 340 300 260 220 180 140 100 60 20 -20 01/01/76 31/12/77 First filling 01/01/80 31/12/81 01/01/84 31/12/85 125-E11-905 125-E15-914 125-E17-925 01/01/88 31/12/89 01/01/92 E11-calcul elastique E15-calcul elastique E17-calcul elastique Figure 13. Pore pressures in kPa in the core (calculation: full features). 107 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 31/12/93 01/01/96 31/12/97 01/01/00 31/12/01 09069-13.qxd 08/11/2003 20:24 PM Page 108 Figure 14. Vertical effective stresses during steady state. 10e2 kPa 11 10e2 kPa 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 2 3 1 1 126-T2-905 T2 Szz calcul 126-T3-905 127-T5-905 126-T4-905 T4 Srd-rg calcul 126-T6-905 T6 Sami-avl calcul Figure 15. Comparison between measured and calculated total stresses at level 905. 125-T7-925 125.5 T8-925 124.5-T9-925 126-T10-925 T7 Szz calcul T8 Srd-rg calcul T9 Samt-avl cal cul 125-T11-925 31/12/01 01/01/00 31/12/97 01/01/96 31/12/93 01/01/92 31/12/89 01/01/88 31/12/85 01/01/84 31/12/81 01/01/80 01/01/76 31/12/01 01/01/00 31/12/97 01/01/96 31/12/93 01/01/92 31/12/89 01/01/88 31/12/85 01/01/84 31/12/81 01/01/80 -1 31/12/77 0 -1 01/01/76 0 31/12/77 2 Figure 16. Comparison between measured and calculated total stresses at level 925. 9 CONCLUSION Detection of hydraulic fracturing in the core is the objective of the analysis of this zoned dam. The three-dimensional modeling appears to be the only way to model the problem. The real geometry of the foundation has to be carefully modeled to represent the phenomenon of stresses transferred to the banks and the unloading of the core in the canyon. The parametric study of the mechanical properties of the materials was required to fit the monitoring data and ensure the accuracy of the analysis. Finally, according to this calibration, hydraulic fracturing of the core was not indicated. REFERENCES Varona P. 2001. Curso de FLAC3D, Itasca Consultant Spain, January 15–19, 2001. Laigle, F. & Boymond, B. 2001. CERN-LHC Project – Design and excavation of Large-Span Caverns at Point 1, EDF-CIH France. 108 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 109 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Mesh geometry effects on slope stability calculation by FLAC strength reduction method – linear and non-linear failure criteria R. Shukha & R. Baker Faculty of Civil and Environmental Engineering, Technion I.I.T., Haifa, Israel ABSTRACT: Results of FLAC’s strength reduction technique are compared with conventional limit equilibrium analysis for both linear and non-linear strength criteria. The comparison includes both safety factors and failure modes (critical slip surfaces and normal stress functions). The collection of FLAC’s plastic points is not a reasonable criterion for estimating the potential failure zone and it is necessary to establish this zone by postprocessing FLAC’s results. It is shown that failure modes implied by FLAC analysis are sensitive to mesh geometry effects and, in order to obtain reasonable results, it is necessary to use meshes consisting of nearly square elements. Safety factors are much less sensitive to mesh geometry effects than failure modes. FLAC’s mesh sensitivity is more pronounced for non-linear failure criterion than in the linear case. Using acceptable mesh geometry, FLAC’s strength reduction technique and limit equilibrium procedures yield comparable results (failure modes and safety factors) for both linear and non-linear strength criteria. Engineering implications of linear and non-linear failure criteria in the context of slope stability analysis are presented and discussed. It is shown that equally valid interpretations of the same experimental information may, under certain conditions (e.g. steep slopes), lead to very different engineering implications. Under such conditions the choice between alternative strength models must be based on the practical implications of these laws. 1 INTRODUCTION Strm(f) which is defined as: Almost all practical slope stability calculations quantify the stability of a given slope using the notion of safety factor with respect to shear strength. This quantity is commonly defined as a reduction constant by which the available shear strength function of the soil needs to be factored down in order to bring the slope to failure. In conventional limit equilibrium (L-E) calculations, safety factors are associated with “test bodies” and it is necessary to search for the critical test body that yields the minimal safety factor for a given slope. Incorporation of safety factor, with respect to strength, in a general continuum mechanics framework results in a class of slope stability procedures known as strength reduction (S-R) methods. This approach was used as early as 1975 by Zienkiewics et al. (1975) and has since been applied by Naylor (1982), Donald & Giam (1988), Matsui & San (1992), Ugai (1989), Ugai & Leshchinsky (1995) and others. 2 COMPARISON OF THE L-E AND S-R FRAMEWORKS Both L-E and S-R techniques analyze an equivalent material characterized by a mobilized strength function (1) where f is the normal (in general effective) stress acting at failure on the failure plane, Str(f) is the strength function (Mohr envelope) of the material, and F is the slope’s safety factor. Equation 1 is used by both the S-R technique and conventional L-E procedures. However, the conceptual framework employed in these two approaches is not equivalent. In particular: 1. Application of the S-R technique requires complete specification of the soil’s constitutive relation while the L-E framework does not depend on deformation parameters. 2. L-E procedures include a minimization stage, which establishes the critical slip surface and its associated minimal safety factor. In S-R techniques on the other hand, existence of a unique slip surfaces is not a priori assumed, but such a surface can be established after completion of the basic analysis. 3. S-R methods define failure at a point (element). However yielding of a single element does not 109 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 110 mean global failure of the slope. The local definition of failure embedded in all S-R methods is probably the main disadvantage of these procedures compared with the inherently global L-E approach. The L-E and S-R techniques have their strengths and weaknesses. A number of previous studies (e.g. Naylor 1982, Dawson et al. 1999, Griffith & Lane 1999) showed that both methods yield approximately the same safety factors. The present work extends this comparison to critical slip surfaces and distribution of normal stress acting along such surfaces. Figure 1. Experimental strength functions for compacted Israeli clays. 3 STRENGTH FUNCTIONS FOR A GIVEN STATE OF INFORMATION Most practical slope stability calculations are based on the linear Mohr-Coulomb (M-C) strength function: (2) where {c, } are the conventional M-C strength parameters cohesion and angle of internal friction respectively. Experimental studies by Penman (1953), Bishop et al. (1965), Day & Axten (1989), and Maximovic (1989) have indicated that actual failure envelopes of most soils are not linear, particularly in range of small normal stresses. There exists a number of studies incorporating non-linear (N-L) failure criteria in conventional L-E calculations such as Maximovic (1979), Charles & Soares (1984), and Perry (1994). The N-L strength function used in most of these studies is the Mohr form of the Hoek-Brown (H-B) empirical failure criterion (Hoek & Brown 1980). This criterion can be written as: (3) where Pa stands for atmospheric pressure and {A,n,T} are non-dimensional strength parameters. This nondimensional form was introduced by Jiang et al. (2003), where it was shown that the parameters {A,n,T} must satisfy the requirements {A 0, 1⁄2 n 1, T 0} and T represents a non-dimensional tensile strength. Baker (2003a) demonstrated that Equation 3 provides a reasonable representation of experimental results for a wide range of different geological materials. It is important to realize that a physically significant assessment of the effect of strength functions non-linearity of the results of slope stability calculations is possible only if the linear and N-L strength functions are fitted to the same data set. Stated differently, in order to asses the effect of different strength functions (strength models) on results of slope stability calculations it is necessary to consider a given state of experimental information (given data set). Jiang et al. (2003) and Baker (2003b) performed such studies using approximate L-E procedures and showed that, under certain conditions, the strength functions non-linearity may have very significant effect on results of slope stability calculations. One of the purposes of the present work is to study the same effect using a FLAC based S-R slope stability analysis. The points in Figure 1 show results of 103 consolidated undrained triaxial tests with pore pressure. Measurements were performed on compacted Israeli clays. These tests were done as part of routine testing programs for design of small water reservoirs. Additional information about the clays and tests is given by Frydman & Samoocha (1984). The lines in Figure 1 are the M-C and H-B strength envelopes fitted to the experimental data set, using the iterative least square procedure described by Baker (2003a). The fitting process resulted in 25°, c 11.7 kPa, SOSMC 31908 kPa2 and A 0.58, n 0.86, T 0, SOSHB 30263 kPa2, where SOSMC and SOSHB are the sum of squares associated with the M-C and H-B models respectively. The following comments are relevant with respect to the above results: 1. T 0 is a result of the estimation process, not an a priori assumption. This result implies that the optimal H-B model represents a zero tensile strength material. The M-C model on the other hand predicts a non-negligible tensile strength of t c/Tan() 25 kPa. 2. The sum of squares associated with the M-C and H-B models are nearly equal SOSHB/SOSMC 0.95. This result implies that the M-C and H-B models provide equally valid descriptions of the experimental information (data points). 3. Inspection of Figure 1 shows that the M-C and N-L criteria predicts almost identical strength values in the range of experimental normal stresses 110 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 111 33.4 f 351.2 kPa. The predictions of these models differ from each other only at very low and very high normal stresses. In both of these ranges the H-B model predicts smaller strength values than M-C (convexity of the H-B criterion guaranties that this will always be the case). In fact the main practical significance of the H-B criterion is that it delivers conservative (compared with M-C) strength estimates in normal stress ranges in which there are no direct experimental information. 4 FLAC IMPLEMENTATION OF H-B CRITERION FLAC has a feature allowing a direct use of the H-B criterion in slope stability calculations. However in FLAC this criterion is formulated in the principle stress space and for the present purpose it is convenient to use the Mohr form of this criterion (Eq. 3). Formally this is done by considering a M-C model with the following stress dependent tangential strength parameters t and ct: (4.1) Figure 2. FLAC results for the M-C criterion. (a) Square mesh. (b) Inclined mesh. (4.2) Equations 4.1 & 4.2 were programmed as a simple FISH routine, and using the “whilestepping” option embedded in FLAC, this routine updates the tangential M-C parameters in each FLAC’s time step. The S-R technique was applied using the definition of mobilized strength function in Equation 1, i.e. FLAC was run with a sequence of progressively increasing trial safety factors until the slope failed, (i.e. until FLAC fails to converge to a static equilibrium configuration). Attempting to apply FLAC’s SOLVE FOS command with the mobilized H-B criterion we have encountered convergence difficulties, and all the following results were obtained by manual change of trial safety factors. 5 EFFECT OF MASH GEOMETRY ON RESULTS OF THE S-R TECHNIQUE The calculation framework presented above was applied to a simple homogeneous slope stability problem without pore pressure or external loads. The slope is defined by an inclination 30°, slope height H 6 m, unit weight 18 kN/m3, and the two strength functions shown in Figure 1. Figures 2a, b show FLAC results the for the M-C criterion using two different mesh geometries; a mesh including essentially square elements (Fig. 2a), and the inclined Figure 3. FLAC results for the H-B criterion. (a) Square mesh. (b) Inclined mesh. mesh shown in Figure 2b. Figures 3a, b show the corresponding results for the H-B model. The following observations are relevant with respect to the results in Figures 2 & 3: 1. The H-B model resulted in significantly lower safety factors then the linear M-C model (FHB 1.4 111 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 112 compared with FMC 1.95). The previous discussion showed that these two strength models are supported equally well by the available experimental information (Fig. 1). Faced with a situation in which two material models are equally justified by the data, the choice between these models must be based on their engineering consequences. In the present problem the H-B model delivered smaller safety factors than M-C and this model should be used in order to ensure a safe design. 2. For the M-C criterion the two meshes result in practically the same safety factor. The corresponding difference for the H-B model (F 0.1) is small, but it is not negligible. 3. Figures 2 & 3 show that the square and inclined meshes resulted in very different failure mechanisms. This difference is seen in terms of both distribution of FLAC’s plastic point, and contours of shear strain increments. Plastic points identify elements, which are at yield (failure). However yielding of a particular element does not imply that the slope as a whole is at failure. Consequently, the collection of FLAC’s plastic points does not provide a clear indication of the global failure mechanism. In particular, some of the plastic points for the inclined mesh are located deep in the interior of the slope, and they are obviously not relevant for the purpose of identifying the slope’s failure mechanism. The distribution of plastic points and shear strain increments in the square meshes appears to indicate that very large area of the slope is in a state of simultaneous failure. The inclined meshes imply failure mechanisms of the type postulated in conventional L-E calculations; namely an essentially rigid body sliding along a narrow transition zone. It is noted however that the “critical slip surface” implied by Figure 3b emerges above the toe of the slope. From a L-E perspective, such a surface cannot be critical, corresponding essentially to a slope with a “reduced height”. Yet, this “unreasonable” slip surface is associated with a smaller safety factor than the reasonable (but ill-defined) critical slip surface in Figure 3a. 6 FAILURE MODES IMPLIED BY FLAC’S S-R TECHNIQUE Following a FLAC run, the state of stress (Mohr circles) is known at each element of the mesh. The state of stress in failed elements satisfies Equation 1, and such stress states are represented by Mohr circles, which are tangential to the mobilized strength envelope. The tangency requirement has to be satisfied with a certain tolerance in order to prevent exclusion of all elements. Each tangential Mohr circle is associated with a certain mesh element, which can be identified; and the collection of all such elements represents a L-E definition of the critical slip surface function implied by the S-R technique. Mesh elements defined by the above process are shown as the open circles in Figures 2 & 3, and critical slip surfaces defined by this process are shown as the heavy dashed lines in those figures. In principle, the above identification of failed elements is not different from FLAC’s definition of plastic points. Nevertheless the set of failed elements shown in Figure 3b is quite different from the set of FLAC plastic points. The source of this difference is probably related to an internal programming detail in the FLAC program. More detailed investigation appears to indicate that the internal FLAC criterion used for definition of plastic points employs a too-large tolerance in the definition of these points, resulting therefore with inclusion of elements which are not really at failure. Controlling the accuracy with which the tangency requirement is enforced provides a convenient numerical mechanism eliminating at least some failed elements, which are not relevant for definition of global slope failure (critical slip surface). The following comments are relevant with respect to the process of identifying failed elements: 1. Inferred critical slip surfaces defined by the above process are consistent with the shear strain increment contours shown in Figures 2 & 3, but they provide a clearer definition of the global failure mode. 2. In some cases the set of failed elements includes a group of elements located in the vicinity of the high entry point of the critical slip surface. This group represents elements failing in tension rather than shear. The L-E critical slip surface is not well defined in such zones. 3. Tangency points between Mohr circles and the mobilized strength envelope define the normal stress acting on the critical slip surface passing through a given element. Consequently, the above process results with L-E definition of both critical slip surfaces and normal stress functions. The inferred L-E critical slip surfaces and normal stress functions resulting from the above process are shown in Figures 4 & 5, which correspond to Figures 2 & 3 respectively. In those figures we have superimposed also critical slip surfaces and normal stresses functions resulting from the following approximate L-E analyses: 1. Simplified Bishop’s method. The original formulation of this procedure was based on the linear M-C strength functions. For the present purpose we have modified this classical procedure incorporating in it also the H-B criterion. 2. The local linear approximation (LLA) technique presented by Baker (2003b). This approximation is 112 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 113 based on classical Taylor analysis in which the effect of strength functions non-linearity is accounted for by an iterative procedure which utilizes Janbu’s approximation (Janbu 1957) of the normal stress function. When applied to M-C material the LLA technique is identical to Taylor’s analysis. It is noted that both of the above L-E procedures are based on the a priori assumption that critical slip surfaces can be approximated by a circular arc. Critical slip surface inferred based on FLAC’s results do not include such a restriction. The following comments are relevant with respect to Figures 4 & 5: Figure 4. Normalized critical slip surfaces and normal stress functions for the M-C criterion. (a) Square mesh. (b) Inclined mesh. Figure 5. Normalized critical slip surfaces and normal stress functions for the H-B criterion. (a) Square mesh. (b) Inclined mesh. 1. Using the square mesh; FLAC’s S-R technique results with safety factors which are very close to those based on both Bishop’s analysis and the LLA technique. Critical slip surfaces and normal stress functions inferred based on the S-R technique are similar but not identical, to the corresponding L-E functions. The difference between the critical slip surfaces inferred based on FLAC analysis and the corresponding L-E surfaces is mainly due to the circular arc restriction used in the present approximate L-E methods. It is frequently stated that the circular arc restriction provides a reasonable approximation for homogeneous slopes. The results in Figures 4 & 5 do not support such a far-reaching conclusion. The variational formulation of L-E problems (Baker & Garber 1978, Baker 2003c) provides a means of avoiding a priori assumptions with respect to the form of critical slip surfaces. Such advanced L-E procedures are not widely used, and they are not considered in the present work. Both Janbu’s normal stress approximation and the normal stress function implied by Bishop’s analysis appear to be consistent with FLAC’s results. Those observations are valid for both M-C and H-B criteria. 2. Using the inclined mesh; there is a small but not negligible difference between safety factors based on the S-R technique and L-E safety factors. However the failure mechanisms implied by these two approaches are significantly different. Figure 6 illustrates the extent of FLAC’s mesh sensitivity with respect to inferred critical slip surfaces and normal stress functions for the case of H-B failure criterion. It is noted again that the inclined mesh resulted with an unreasonable critical slip surface emerging above the toe, while the squared mesh is consistent with the L-E based argument that slip surfaces emerging about the toe correspond in effect to a slope having a reduced height, and such surfaces cannot be critical. Using an inclined mesh the discrepancies between results based on S-R and L-E techniques are more pronounced for the H-B model than the M-C criterion, but they exist in both cases. Based on the above 113 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 114 the H-B strength law is significantly smaller than the safety factor FMC 1.96 associated with the M-C criterion. This result depends however on the slope stability problem under consideration and in the following section we establish a more general perspective for investigating the effect of strength functions on results of slope stability calculations. 7 THE CRITICAL HEIGHT FUNCTION Figure 6. Mesh effect on inferred critical slip surfaces implied by the S-R technique (H-B failure criterion). Figure 7. Critical failure mechanisms associated with the M-C and H-B criteria. discussion it is clear that FLAC results are sensitive to mesh geometries. Using a square mesh yields more consistent results than use of the inclined mesh. This is not really surprising; some elements in the inclined mesh have relatively high aspect ratios, and it is well known that results based on such meshes should be viewed with suspicion. Safety factors are relatively insensitive to mesh geometries. However mesh geometries have quite a significant effect on failure mechanisms. In the following we restrict attention to results obtained using only square meshes of the type shown in Figure 2a. Figure 7 compares FLAC’s failure mechanisms (critical slip surfaces and normal stress functions) obtained for the M-C and H-B strength functions in Figure 1. It is seen that the critical slip surface associated with the H-B strength function is shallower than the one associated with the M-C criterion. As a result, normal stresses acting on the H-B slip surface are smaller than those operating along the M-C slip surface. Inspection of Figure 1 shows that, in the range of small normal stresses, the H-B criterion predicts smaller strength than M-C. It is not surprising therefore that the safety factor FHB 1.45 obtained using Safety factors are practically useful abstractions. However, the physical significance of results obtained by S-R or conventional L-E techniques is clear only at failure when F 1. At any other value of F such calculations deal with failure conditions of an equivalent material with a reduced strength, rather than the actual physical problem. In order to avoid this conceptual difficulty it is convenient to study the effect of strength criteria on results of slope stability calculations in terms of critical heights rather than safety factors. The critical height of a slope is defined as a height for which the minimal safety factor is equal to one. Critical heights depend on the inclination of the slope, i.e. Hcr Hcr(). Figure 8 show critical height functions resulting from following analyses: 1. FLAC S-R technique based on the H-B criterion (triangles). 2. Bishop analysis based on the H-B criterion (crosses). 3. The LLA technique (Baker 2003b) (solid heavy line) based on the H-B criterion. 4. FLAC S-R technique based on M-C criterion. (open circles). 5. The LLA technique based on the linear M-C criterion (light solid line). It is noted that for homogeneous slopes and a M-C strength criterion this technique is reduced to classical Taylor analysis. It is noted that both the H-B and M-C criteria are fitted to the same experimental data set obtained for compacted Israeli clays (Fig. 1), and both models represent this data equally well (sum of squares ratio is equal to 0.95). The following comments are relevant with respect to the results in Figure 8: 1. In the linear M-C framework the safety factor of slopes with is always larger than 1. Consequently a critical height function based on this criterion is asymptotic to a vertical line located at . It can be verified that the non-linear H-B criterion implies finite critical heights for all nonzero slope inclinations. 2. Using the linear M-C criterion corresponding to the data set in Figure 1, Taylor’s analysis shows the critical height of a vertical slope is 3.9 m. On the other hand, analysis based on the H-B criterion fitted to the same data set implies that slopes steeper than 114 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 115 8. It is important to realize that the critical height functions shown in Figure 8 are relevant for the compacted Israeli clays data set (Fig. 1), and this figure does not represent a general relations. However, qualitatively similar results were obtained also for a number of other data sets. 8 SUMMARY AND CONCLUSIONS Figure 8. Critical slope heights for compacted Israeli clays. 3. 4. 5. 6. 7. approximately 55° are not stable. The very significant difference between prediction of the two models illustrate very clearly the importance of using non-linear failure criteria in stability calculations of steep slopes. Recalling that the M-C and H-B failure criteria provide equally valid descriptions of the compacted Israeli clay’s data set, the practical implications of Figure 8 are worrying. The figure shows that equally valid interpretations of the same experimental information may under certain conditions lead to diverging practical implications (i.e. HCrHB ( 60°) → 0, HCrMC ( 90°) 3.9 m. The physical basis of the above result is related to the following observations: a) Critical slip surfaces for steep slopes are shallow, resulting therefore with small normal stresses acting on this surface. b) A T 0 H-B model predicts significantly smaller shear strength at small normal stresses than a M-C criterion with a non-zero cohesion. There is a range of slope inclinations (28° 34°) in which the critical height predicted by the M-C criterion is slightly larger then those predicted on the basis of the H-B law. The physical reason for this behavior is discussed by Baker (2003b). The open circles in Figure 8 show results of FLAC analysis based on the M-C criterion. Those results are almost identical with results based on the classical Taylor analysis (light solid line). For the non-linear H-B criterion, the critical height functions based on Bishop analysis, and FLAC’s S-R technique are practically identical with results based on Baker’s (2003b) L.L.A. technique. This observation supports the validities of all three calculation methods. It is noted that Bishop’s analysis is restricted to circular slip surfaces and it is not expected to yield good results in nonhomogeneous problems. Both FLAC and LLA can be applied to non-homogeneous problems. Two general approaches (FLAC’s strength reduction technique and conventional limit equilibrium calculations) for analysis of slope stability are discussed and compared. The present work extends previous presentations on this subject in number of respects: 1. The comparison includes failure mechanisms (critical slip surfaces and normal stress functions) in addition to safety factors. 2. The comparison is done for both the linear MohrCoulomb failure criterion and the non-linear Hoek and Brown strength function. Mesh geometry effects on FLAC’s results are presented and discussed. It is shown that a mesh consisting of essentially square elements results in more consistent results than meshes including relatively slender elements. Safety factors are relatively insensitive to mesh geometries, however failure mechanisms depend very strongly on mesh geometry, and using a mesh which includes slender elements may lead to wrong conclusions with respect to critical slip surfaces and normal stress functions inferred on the basis of FLAC’s S-R technique. FLAC’s mesh sensitivity is more pronounced for non-linear strength functions than in the linear M-C case. It is noted that FLAC’s plastic points do not provide a reasonable measure of the potential failure zone. A procedure which identifies the critical slip surface based on FLAC’s calculated stresses is presented and discussed. Considering results obtained using square meshes, FLAC’s strength reduction technique and conventional limit equilibrium procedures yield similar failure mechanisms and safety factors. This conclusion is valid for both linear and non-linear failure criteria. Engineering implications of linear and non-linear failure criteria in the context of slope stability analysis are presented and discussed. In particular it is shown that equally valid interpretations of the same experimental information may under certain conditions (steep slopes) lead to very different engineering implications. Under such conditions the choice between strength functions must be based on the practical implications of these laws, which in most cases means using the Hoek-Brown criterion rather than the conventional linear Mohr-Coulomb strength function. 115 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-14.qxd 08/11/2003 20:25 PM Page 116 REFERENCES Baker, R. & Garber, M. 1978. Theoretical analysis of the stability of slopes. Geotechnique, 28(4): 395–411. Baker, R. 2003a. Inter-Relation between experimental and computational aspects of slope stability analysis. Inter. Jour. Numer. Anal. Meth. Geamech. 27(5): 379–401. Baker, R. 2003b. Non-linear strength envelopes based on triaxial data. Accepted for publication in J. Geotech. And Geoenvir Engrg., ASCE. Baker, R. 2003c. Sufficient conditions for existence of physically significant solutions in limiting equilibrium slope stability analysis. Accepted for publication in Inter. Jour. of Solids and structures. Bishop, A.W., Webb, D.L. & Lewin, P.I. 1965. Undisturbed samples of London clay from the Ashford common shaft: strength-effective normal stress relationship. Geotechnique, 15(1): 1–31. Charles, J.A. & Soares, M.M. 1984. The stability of slopes with nonlinear failure criterion. Cand. Geoth. J., 21(3): 397–406. Day, R.W. & Axten, G.W. 1989. Surficial stability of compacted clay slopes. J. Geoth. Eng. ASCE, 115(4): 577–580. Dawson, B.M., Roth, W.H. & Drescher, A. 1999. Slope stability factors of safety by strength reduction. Geotechnique, 49(6): 835–840. Donald, I.B. & Giam, S.K. 1988. Application of nodal displacement method to slope stability analysis. Proc. 5th Australia-New Zealand Conf. on Geomech., Sydney, Australia, 456–460. Frydman, S. & Samoocha, Y. 1984. Laboratory studies on Israeli clays for reservoir embankment design. Proc. 5th Inter. Conf. on Expansive soils, Adelaide, South Australia, 94–98. Griffith, D.V. & Lane, P.A. 1999. Slope Stability analysis by finite elements. Geotechnique, 49(3): 387–403. Hoek, E. & Brown, E.T. 1980. Empirical strength criterion for rock masses. ASCE Jour. Geotech. Eng., 106(9): 1013–1035. Jiang, J.C., Baker, R. & Yamagami, T. 2003. The effect of strength envelope nonlinearity on slope stability computations. Can. Geoteh. J., 40(2): 308–325. Matsui, T. & San, K.C. 1992. Finite element slope stability analysis by shear reduction technique. Soils and Foundations, 32(1): 59–70. Maximovic, M. 1979. Limit equilibrium for non-linear failure envelope and arbitrary slip surface. Proc. 3rd Intr. Conf. on Numerical Methods in Geomechanics, 769–777. Maximovic, M. 1989. Nonlinear failure criterion for soils. J. Geoth. Eng. ASCE, 115(4): 581–586. Naylor, D.J. 1962. Finite element and slope stability. Nume, Meth. un Geomech., Proc. NATO advanced study institute. Lisbon, Portugal, 229–244. Penman, A. 1953. Shear characteristics of saturated silt in triaxial compression. Geotechnique. 15(1): 79–93. Perry, J.A. 1994. A technique for defining non-linear shear strength envelopes and their incorporation in slope stability method of analysis. Quart. J. of Eng. Geology, 27(5): 231–241. Ugai, K. 1989. A method of calculation of total factor of safety slopes by elasto-plastic FEM. Soils and Foundations, 29(2): 190–195. Ugai, K. & Leshchinsky, D. 1995. Three-dimensional limit equilibrium and finite element analyses: a comparison of results, Soils and Foundations, 35(4): 1–7. Zienkiewicz, O.C., Humpheson, C. & Lewis, R.W. 1975. Associated and non-associated visco-plasticity and plasticity in soil mechanics. Geotechnique, 25(4): 671–689. 116 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-15.qxd 08/11/2003 20:25 PM Page 117 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 3D slope stability analysis at Boinás East gold mine A. Varela Suárez & L.I. Alonso González Río Narcea Gold Mines, S.A., Belmonte de Miranda, Principado de Asturias, Spain ABSTRACT: The Boinás East open pit mine has been exploited by the gold producer Río Narcea Gold Mines in Northwest of Spain. Considering the influence of the radius of curvature on the factor of safety for slope angle, and taking into account that a small increase in the overall angle will result in a very high increment in the total amount of the ore mined, the slope stability analysis was made using the finite difference code FLAC3D to calculate the factor of safety by reducing the rock shear strength. Due to the existing complicated geology and the complexity of the 3D geometrical modeling, a “FISH routine” was used to import the block model of the mine into the FLAC3D program. This block model is the database commonly used in the mine works and was generated with Datamine. This method is a very good tool to generate a complex model in FLAC3D. 1 INTRODUCTION Río Narcea Gold Mines has been operating from 1997 the gold deposit El Valle-Boinás, located in the Northwest of the Iberian Peninsula, within the wellknown Rio Narcea Gold Belt (Fig. 1). The deposit is in the environs of the town of Boinás, Belmonte de Miranda, in the Principality of Asturias. It’s formed by five separated bodies located around a granitic stock. The mineralization consists of various skarn types and zones with silicifications and significant epithermal oxidation. Of these five ore bodies, three have been operated (El Valle, Boinás East and Boinás west) by open pit Figure 1. 3D diagram of E1 Valle-Boinás deposit. mining techniques, being at the present time solely in production the deposit of El Valle. The present article tries to develop the methodology used for the design of stable slopes in the deposit of Boinás East, doing analysis of stability by means of FLAC3D software. 2 SITE GEOLOGY The Rio Narcea Gold Belt structure of 17 km in length has an approximate direction N 35° E and includes, in addition to the mentioned deposit Valle-Boinás, at least five other gold mineralizations, some of them widely operated during the rule of the Roman Empire. Geologically the gold belt consists of an anticline of Hercynian age, in the core of which there are the carbonate materials of the Láncara Formation (Middle Cambrian), above which are shale and sandstone of the Oville Formation (Middle-Upper Cambrian). Gold mineralization was initially deposited as calcic and magnesic copper-gold skarns at the contact between the Boinas granodiorite and limestone and dolomite of the Lancara Formation (Martin-Izard et al. 1998, Cepedal 2001). The auriferous mineralization mainly occurred during the phase of retro-gradation of the metamorphic process, to temperatures between 450°C and 250°C, separated in two stages (Cepedal 1998, 2001). During the Lower Permian, after an important dismantling of the hercynic relief, takes place the location of subvolcanic and porphyritic dikes that originate hydrothermal alterations with important silicification of pre-existing 117 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-15.qxd 08/11/2003 20:25 PM Page 118 Figure 2. Geological scheme of the Rio Narcea Gold Belt. rocks, to temperatures between 150°C and 250°C and smaller pressure of 0.2 kbar. This process gives rise to the formation of oxidized and very brecciated materials, with bad geotechnical quality and that approximately constitute 90% of the operated material in the open pits. The gold mineralization remained hidden by Tertiary lacustrine deposits, which as well were partially hidden by Alpine thrusts that placed an important repetition of limestone of the Láncara Formation above the Tertiary and the sandstone of the Oville Formation. Figure 1 shows a scheme of the zone of the deposit El Valle-Boinás and Figure 2 a scheme of the whole Rio Narcea Gold Belt. 3 PRELIMINARY ANALYSIS From the geotechnical data of the different materials taken in from the geological exploration holes, the shear strength of the different lithologies were determined following the Bieniawski (1989) classification and the Hoek-Brown 99 methodology. With the obtained values SRK Ltd. carried out the feasibility study in October 1996, updating it in 1999. Table 1. Feasibility study data. Factor of safety 55 50 45 0.91 1.03 1.15 The analyses were carried out using the XSTABL software over sections with simplified geology and considering different overall slope angles. Considering the slope totally drained, 330 meters of vertical height, the Bishop’s method and circular surfaces of failure, were obtained the factor of safety showed in the Table 1 (SRK 1996, 1999). The geology and the rock mass properties used by SRK in their analysis are summarized in Table 2. 4 NUMERICAL MODELING WITH FLAC The holes drilled during the year 1998 defined a new mineralized zone amenable to extraction by open pit 118 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Overall slope angle(°) 09069-15.qxd 08/11/2003 20:25 PM Page 119 Table 2. Rock mass properties in the SRK model. Lithology Good Quality Lancara Limestone Fair Quality Lancara Limestone Lower Oville Sandstone Fair Quality Granite Ore (MIN) Marble Table 3. Factor of safety obtained with FLAC. Cohesion Density (°) (kPa) (t/m3) 37 31 33 34 32 32 293 258 275 290 265 265 2.70 2.70 2.50 2.60 2.50 2.70 Slope angle Phase Type of analysis Upper* Lower* Factor of safety Final Final 1 Plane Axisymmetric Plane 65 65 50 45 45 50 1.00–1.05 1.65–1.70 1.20–1.30 * Upper/lower means above or below the main Alpine thrust. Table 4. Rock-mass properties. Lithology K G Cohesion Density (GPa) (GPa) (°) (kPa) (t/m3) Upper Sandstone Upper dolomite Lower Sandstone Alterd. Granite Ore (MIN) Fresh granite Fresh skarn Marble Hornfels Tertiary Black skarn 3.75 2.17 7.92 5.09 1.32 0.75 1.17 0.66 1.11 0.62 16.99 11.52 10.63 6.79 2.45 1.41 23.24 16.39 4.64 2.96 10.63 6.79 49 43 21 20 19 65 56 33 62 38 56 720 1120 270 250 250 4770 2430 500 11330 650 2430 2.65 2.70 2.29 2.28 2.21 2.67 3.17 2.43 2.77 2.25 3.17 Figure 3. Plane-strain and axisymmetric analysis with FLAC. provided it would be possible to increase the over all angle of the East slope of the pit. In the following paragraph will be described in detail the stability analysis carried out using more detailed geological models and two and three dimensional finite-difference programs. 4.1 Preliminary analysis Considering the effect of the radius of curvature of the pit in the global stability (Lorig 1999), different analysis were carried out using the program FLAC (Itasca 1998). In collaboration with Itasca Consultores S.L., we have made different analysis over a representative section, contemplating plane-strain and axisymmetric conditions. In first of them one assumes that the slope extends indefinitely in the perpendicular direction of the analysis plane and that deformations in the perpendicular direction of this plane do not exist. This type would be compatible with the one made in the feasibility study with the XSTABL software. In the case of axisymmetric analysis one assumes that the slope has truncated cone form. The graphical representation of both cases is indicated in Figure 3. Table 3 shows the factor of safety obtained for the same section and using the same strength parameters. In that table it shows also the factor of safety obtained in a back analysis of the slope already excavated in the pit as Phase 1. The strength parameters are those given in Table 4, and the slope was considered to be fully drained. The breakage mechanism that takes place is very similar in both types of analysis. It consists of the shear failure of the materials located below the main alpine thrust and tensile failure in the dolomite above that thrust. In the case of the axisymmetric analysis, the tensile failure is cushioned, increasing in this way the factor of safety of the slope. In following figures show a schematic of the geology (Fig. 4), the failure mechanism in the case of plane-strain geometry (Fig. 5), and the failure in the case of axisymmetric analysis (Fig. 6). Studying at the values of the obtained factors of safety it is clear that the analyses of three-dimensional geometry provide higher and more realistic factors of safety, considering the influence of the radii of curvature previously mentioned. In any case, the axisymmetric considerations are too optimistic, since in the reality the analyzed topography will not be totally 119 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-15.qxd 08/11/2003 20:26 PM Page 120 was used. This program allows define a more accurate topographical and geological model. 4.2 FLAC3D modeling Three-dimensional model construction of a geologically complex deposit such as El Valle-Boinás can be an arduous task. In this case a “FISH routine” was designed that allows the geological block model to be imported directly into FLAC3D. In a first step, using the DATAMINE program, different block models were created deactivating those blocks located over the topography that we try to analyze. The X, Y and Z coordinate of the center of each block is exported into a data file, as well as its lithology. In our particular case the mining blocks were 4 meters side bricks, that were rebuilt to 16 meters side bricks, with the intention of making the model usable. The blocks thus obtained were placed in the own local coordinates, and more blocks were defined around the model in order to avoid that the artificial contour condition do not affect the results of the stability analysis (Lorig et al 2000). Figure 7 shows the geology on the surface of the final pit plotting the DATAMINE block model, and the Figure 8 shows the geological model loaded into FLAC3D. The FISH routine was used to generate the 16 meter side blocks using the coordinates of the center of the blocks taken from DATAMINE (file “BE16X16A.txt”). The first line of this file has the identification of the rest of the parameters, and it will not be imported to the FLAC3D program. The whole routine is shown in the appendix. Figure 4. Summarized geology. Figure 5. Mode of failure in a plane-strain analysis. 4.3 Rock-mass properties The rock-mass properties used in all the stability analysis are indicated in Table 4. The tensile strength has been considered to be a tenth of the cohesion of each material. To obtain the factor of safety of the proposed model the shear-strength reduction technique was used. To perform slope-stability analysis, simulations are run for a series of increasing trial factors of safety, f, until the slope fails. At failure, the safety factor equals the trial safety factor (i.e. f F) (Lorig et al. 2000). Actual shear strength properties, cohesion (c) and friction (), are reduced for each trial according to the following equations: (1) Figure 6. Mode of failure in an axisymmetric analysis. conical and the different lithological units have a very marked dip towards the East. For the final design of the stable slope in the Boinás East pit, the FLAC3D software (Itasca 1997) (2) The reduction in the shear strength properties is made simultaneously for all materials. 120 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-15.qxd 08/11/2003 20:26 PM Page 121 Figure 7. Datamine block model. Figure 8. FLAC3D block model. 121 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-15.qxd 08/11/2003 20:26 PM Page 122 Table 5. Factor of safety for different slope angles and with different types of analysis. Face angle Phase A1 B2 Deepest level Program Type of Factor analysis of safety 1 6 6 6 6 6 6 50 45 45 45 50 50 50 404 340 340 340 350 340 300 Plane Plane Axis Axis 50 65 65 70 50 70 70 FLAC FLAC FLAC FLAC FLAC3D FLAC3D FLAC3D 1.25 1.05 1.70 1.50 1.45 1.503 1.35 1 A represents the face angle of the slope located above the main Alpine Thrust. It represents the slope angle in the Upper Limestones. 2 B represents the face angle in the materials located below the main Alpine Thrust. 3 The higher value of FoS in the steeper design is due to a change in the lithology present on that slope. Figure 9. Zone with the maximum displacement in the final Boinás East pit. 5 RESULTS AND CONCLUSIONS Once analyzed the different proposed models, it was proposed an open pit with lower level at 340 meters ASL, a maximum slope height (in the East wall) of 345 meters. A factor of safety of 1.45 was obtained with slope angles of 70° in the dolomite above the main thrust, 60° in upper sandstone, fresh granite, fresh skarn, tertiary, and black skarn, and 50° in the rest. A comparison between the factors of safety obtained for the different models and programs used are provided in Table 5. The minimum factor of safety values are obtained in the East wall of the pit. Figure 9 shows the FLAC3D model with the zone of maximum displacement. These zone correspond with a convex geometry in the slope, therefore steeper slopes could have been considered in areas where the pit slopes were concaves, but these was not our case. We only have changed the face angle according with the geology and not with its geometry. As we can see in the factor of safety obtained in the different model, the influence of the radii of curvature in the global stability of a pit is very considerable, specially taking into account that a small increasing in the overall face angle results in a very large amount of ore recovered, as it was in our case. On the other hand, the hard work required to design a model for a complex deposit in three dimensions is avoided when we import the block model into FLAC3D. With the routine described above, it is easy to create a block model with DATAMINE, or whatever other program, and delete the block above the surface we want to analyze, and them import all the model to FLAC3D and obtain a factor of safety. It must be take into account that with this system we are going to have free faces at 90º and the high of the block size, so we must confirm that the factor of safety obtained corresponds to the slope factor and not to the brick face. ACKNOWLEDGEMENTS The authors would like to thank Mr. Manuel G. Fernández of Río Narcea Gold Mines for his great work with Datamine, and Mr. Alan Riles, COO of Río Narcea Gold Mines Ltd. for the valuable help in the translation of the paper. Finally the authors are grateful to Mr. Pedro Varona of Itasca Consultores, S.L. for his technical support. Thanks are extended also to Mr. Pedro Velasco and Ms. Montserrat Senís for their help. REFERENCES Cepedal, A., Martin-Izard, A., Fuertes, M., Pevida, L., Maldonado, C., Spiering, E., Gonzalez, S. & Varela, A. 1998. Fluid Inclusions and Hydrothermal Evolution of the El Valle-Boinas Copper-gold Deposits. In Arias, A., Martin-Izard, A. & Paniagua, A. (eds), Gold Exploration and mining in NW Spain: 50–58. Oviedo. Cepedal, M.A. 2001. Geología, Mineralogía, Evolución y Modelo Genético del yacimiento de Au-Cu de El ValleBoinas. Belmonte (Asturias). Ph.D. thesis, University of Oviedo. Itasca Consulting Group, Inc. 1997. FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions), Version 2.0. Minneapolis: Itasca. Itasca Consulting Group, Inc. 1998. FLAC (Fast Lagrangian Analysis of Continua), Version 3.4. Minneapolis: Itasca. 122 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-15.qxd 08/11/2003 20:26 PM Page 123 Lorig, L. 1999. Lessons learned from slope stability studies. In Detournay & Hart (eds), FLAC and Numerical Modeling in Geomechanics: 17–21. Rotterdam: Balkema. Lorig, L. & Varona, P. 2000. Practical Slope-Stability Análisis Using Finite-Difference Codes. In Hustrulid, W.A., McCarter, M.K. & Van Zyl, D.J.A. (eds), Slope Stability in Surface Mining: 115–124. Colorado: Society for Mining, Metallurgy and Exploration, Inc. Martin-Izard, A., Cepedal, A., Fuertes, M., Pevida, L.R., Maldonado, C., Spiering, E., Varela, A. & Gonzalez, S. 1998. The El Valle Deposit: an example of Koper-gold Skarn Mineralization overprinted by late epithermal events. Cantabrian Mountains, Spain. In Arias, A., Martin-Izard, A. & Paniagua, A. (eds), Gold Exploration and mining in NW Spain: 43–50. Oviedo. Steffen, Robertson & Kirsten (UK) Ltd. 1996. Investigation into the Stability of Proposed Excavated Slopes and Excavatability of Materials at El Valle, Boinas West and Boinas East. Report to Rio Narcea Gold Mines, S.A. Report no. ADM/752MH001.REP, October 1996. Steffen, Robertson & Kirsten (UK) Ltd. 1999. Boinas East Open Pit Verification of Overall Slope Angles for pir Optimisation Studis. Report to Rio Narcea Gold Mines, S.A., January 1999. z1 zc z2 zc z3 zc 16 z4 zc z5 zc 16 z6 zc 16 z7 zc 16 command gen zon bri p0 x0 y0 z0 p1 x1 y1 z1 p2 x2 y2 z2 p3 x3 y3 z3 p4 x4 y4 z4 & p5 x5 y5 z5 p6 x6 y6 z6 p7 x7 y7 z7 size 1 1 1 group mat end_command end_loop status close end creamalla APPENDIX – FISH ROUTINE new def creamalla array aa(11488);(number of lines in file *.txt) status open(‘BE16x16a.txt’,0,1) status read(aa, 11488) loop k(2, 11488); no lee la primera línea xx parse(aa(k),1) yy parse(aa(k),2) zz parse(aa(k),3) xxmax max(xx,xxmax) mat parse(aa(k),4) xc (xx-1)*16 662;put blocks in x local yc (yy-1)*16 9745;put blocks in y local zc (zz-1)*16 x0 xc x1 xc 16 x2 xc x3 xc x4 xc 16 x5 xc x6 xc 16 x7 xc 16 y0 yc y1 yc y2 yc 16 y3 yc y4 yc 16 y5 yc 16 y6 yc y7 yc 16 z0 zc 123 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-16.qxd 08/11/2003 20:26 PM Page 125 Underground cavity design Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-16.qxd 08/11/2003 20:26 PM Page 127 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 The effect of tunnel inclination and “k” ratio on the behavior of surrounding rock mass M. Iphar, M. Aksoy, M. Yavuz & G. Once Osmangazi University, Mining Engineering Dept., Eskisehir, Turkey ABSTRACT: Rock behavior around tunnels excavated in the same rock with various inclinations and “k” ratios has been investigated by numerical analysis employing the FLAC3D finite difference code. Stress distributions and displacements at the critical points of the tunnels in underground mining have been examined. Observed stress and displacement values with respect to the change in tunnel inclination and “k” ratio have been analyzed by using statistical methods employing “multiple regression analysis” in order to find out a meaningful correlation between the stress, displacement values and the inclination and “k” ratios. Statistical analyses have presented meaningful correlations giving mathematical equations whose dependent variable is displacement or stress and independent variables are tunnel inclination and “k” ratio. 1 INTRODUCTION Numerical modeling is a very powerful and useful tool used widely in designing underground structures such as tunnels, roadways, caverns etc. The displacements and stresses around the underground openings can be predicted by employing numerical modeling in advance. During a project carried out for the GLI (Western Lignite Company in Turkey), a main roadway inclined at 8° dip has been designed down to a depth of 500 m in the underground coal colliery (Once et al. 2001a). The geomechanical properties of the rock mass where the roadway will be driven have been obtained from laboratory tests (Çekilmez 1988). In the light of this project, a new study has been carried out to investigate the effect of “k” ratio (the ratio of horizontal stress to vertical stress) and tunnel inclination on the rock mass behavior in terms of displacements and stresses. To achieve this goal, the FLAC3D finite difference code has been used. After the numerical modeling, statistical analyses have been performed to find out a meaningful correlation explaining the effects of “k” ratios and the inclination on the stresses and displacements. tunnel inclinations have been varied between 0 and 45 degrees in 5 degrees intervals while the “k” ratios have been varied between 0.5 and 2 in 0.5 intervals. Three points have been selected to observe the displacements around the tunnel. These history points have been located on the center-line of the tunnel, one at the roof and one at the floor, and one at axis level in the sidewall. The stresses in the zones adjacent to the history points have also been monitored. To evaluate the rock behavior properly, the coordinates of the history points have been kept at the same coordinates in each of the models although the tunnel inclination has been changed. Displacement values and maximum (1) and minimum (3) principal stresses observed in three directions at the roof, floor and sidewall of the tunnel have been subjected to the statistical analysis. After carrying out multiple regression analysis for displacements and 1, 3 principal stresses at the roof, floor and sidewall in x, y and z directions, regression models have been proposed to predict the displacements and stresses. Displacements in the x-direction at the history points in the roof and the floor have not been included because these points lie on the plane of the symmetry for the models. 2 NUMERICAL MODELING STUDIES 2.2 2.1 Applied method Forty different numerical models have been formed in FLAC3D (Itasca 1997). During the model formation, the The rock mass in which the tunnel will be driven is marl formation. The geomechanical properties of the marl were determined from MTA (General Directorate 127 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Rock mass properties 09069-16.qxd 08/11/2003 20:26 PM Page 128 Table 1. RMR calculation for marl. Geomechanical properties Uniaxial compressive strength RQD Spacing of joints Condition of joints Ground water Joint orientation RMR Values Rating 11.2 MPa 2 61.4% 80–200 mm Slightly rough surfaces, separation 1 mm, soft joint wall rock 115.2 lt/min Unfavorable 3.5 m 13 8 20 4.60 m 4 10 37 Figure 1. Tunnel geometry. Table 2. Rock mass properties used in modeling (Çekilmez 1988). Property Values Poisson’s ratio Bulk modulus (MPa) Shear modulus (MPa) Tensile strength (MPa) Internal friction angle (°) Cohesion (kPa) Density (kg/m3) 0.25 790 475 0.28 37 70 2500 of Mineral Research & Exploration Institution) drillings and discontinuity spacing were obtained using the approach proposed by Priest & Hudson (1976) because of the lack of information about discontinuities in the MTA report (Çekilmez 1987). The RMR value was calculated using the RMR classification system described by Bieniawski (1979) and their ratings are shown in Table 1. 2.3 Figure 2. The FLAC3D model with fixity condition and coordinate system. One of the models with fixity condition and coordinate system is shown in Figure 2. Numerical models in FLAC3D The rock mass has been assumed to be an isotropic, homogenous material. It has been modeled as a MohrCoulomb material through the study. The geomechanical properties used in the numerical modeling have been taken from the GLI project as mentioned before and the values of these properties are given in Table 2 (Once et al. 2001b). In-situ stresses have been calculated by using the following equations. z h x y k z (ton/m2) (ton/m2) where z vertical stress; density of marl; h depth, x and y lateral stresses, k “k” ratio (Hoek & Brown 1980). During the modeling, the presence of groundwater was ignored. The tunnel, whose dimensions are given in Figure 1, has been simulated as a single step excavation. 3 RESULTS Displacement and stress values obtained from the modeling studies have been evaluated separately considering “k” ratio and inclination of the tunnel. 3.1 The y and z displacements at the roof have been categorized in terms of “k” ratio and tunnel inclination. In Figure 3, it is observed that the magnitude of the y displacements was not significantly affected by varying the “k” ratio but was greatly affected by varying the tunnel inclination. It can be seen from the Figure 4 that displacements in the roof in the z-direction generally increase as the “k” ratio increases for the case where the inclination 128 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Evaluation of models in terms of displacements 09069-16.qxd 08/11/2003 20:26 PM Page 129 0.2 Predicted Values (m) 0.0 0.2 0.0 -0.2 -0.4 Ydisprf .6 -0 (m) -0.8 -1.0 -0.2 -0.4 -0.6 -0.8 -1.0 0 -1.2 -1.2 0 2. 5 0 1. k-ratio 0.5 10 15 20 25 30 35 40 45 50 5 1. -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 Observed Values (m) Incline (o) Figure 5. Observed vs predicted values in y-direction (roof ). Figure 3. y displacements vs “k” ratio and inclination (roof). 0.0 Predicted Values (m) -0.2 0.0 -0.3 -0.6 -0.9 -0.4 -0.6 -0.8 -1.0 -1.2 Zdisprf 1.2 (m) - -1.4 -1.4 -1.5 -1.8 0.5 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Observed Values (m) 50 45 40 35 30 25 20 15 10 5 1.0 k-ratio 1.5 Incline (˚) 0 2.0 Figure 6. Observed vs predicted values in z-direction (roof ). Figure 4. z displacements vs “k” ratio and inclination (roof ). is constant. On the contrary, displacements in the roof in z-direction generally decrease as the inclination increases for the case where the “k” ratio is constant. Moreover, it should be noted that increasing the inclination angle has a great effect on the amount of vertical displacement in the case of a high “k” ratio. The proposed regression equation for the displacements in the y-direction at the roof utilizing the data from the models studied is given in Equation 1: (1) where Ydisprf y-displacement in the roof; k “k” ratio and I inclination of the tunnel (°). The predicted values obtained from the proposed equation are plotted against the values from the numerical models in Figure 5. This regression model shows a strong correlation between the observed and predicted values (r 0.99). The proposed regression equation for the displacements in the z-direction in the roof is presented below: (2) where Zdisprf z-displacement at the roof; k “k” ratio and I inclination of the tunnel (°). The predicted values are plotted versus the observed values in Figure 6. This regression model also shows a strong correlation between the observed and predicted values (r 0.98). The magnitude of the y displacements for the history point in the floor was greatly affected by the variation 129 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-16.qxd 08/11/2003 20:26 PM Page 130 0.7 Predicted Values (m) 0.6 0.9 0.6 0.3 Ydispfl 0.0 (m) -0.3 -0.6 0.4 0.3 0.2 0.1 0.0 -0.1 2.0 -0.2 -0.1 0.0 1 .5 1 .0 0 k-ratio 0.1 0.2 0.3 0.4 0.5 Observed Values (m) 0.6 0.7 Figure 9. Observed vs predicted values in y-direction (floor). 0.5 50 5 4 0 4 5 3 0 3 5 2 0 Incline (˚) 2 15 10 5 Figure 7. y displacements vs “k” ratio and inclination (floor). Predicted Values (m) 0.7 1.8 1.5 1.2 Zdispfl (m) 0.9 0.6 0.5 0.4 0.3 0.2 0.6 0.1 0.1 0.3 50 2.0 0 0.3 0.4 0.5 Observed Values (m) 0.6 0.7 k-ratio 5.0 5 0.2 Figure 10. Observed vs predicted values in z-direction (floor). 1.5 1.0 45 0 4 35 30 5 2 0 Incline (˚) 2 5 1 0 1 0.5 Figure 8. z displacements vs “k” ratio and inclination (floor). of “k” ratio and tunnel inclination. As can be seen from Figure 7, the horizontal (y) displacements increase with an increase in both the “k” ratio and tunnel inclination. Figure 8 shows that the magnitude of the vertical displacements in the floor increased with an increase in the “k” ratio. On the contrary, the vertical displacements decrease as the tunnel inclination increases.(4) The regression equation proposed for predicting the displacements in the y-direction in the floor is given below: (3) where Ydispfl y-displacement at the floor; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values in the floor in the y-direction are shown in Figure 9. This regression model shows a strong correlation between the observed and predicted values (r 0.99). The regression equation proposed for predicting the z displacements in the floor is as follows: where Zdispfl z-displacement at the floor; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values in the floor in the z-direction are shown in Figure 10. This regression model also shows a strong correlation between the observed and predicted values (r 0.99). 130 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-16.qxd 08/11/2003 20:26 PM Page 131 0.0 0.60 -0.2 0.45 .4 Zdispsdw 30 0. (m) Xdispsdw -0 (m) -0.6 0.15 -0.8 0.5 1.0 k-ratio 1.5 2.0 0 2 1 0 10 5 5 4 50 4 5 35 0 2530 50 5 4 0 4 5 3 0 3 5 2 0 Incline (˚) 2 15 0 1 5 Incline (˚) Figure 11. x displacements vs “k” ratio and inclination (sidewall). 1.5 1.0 0 0.5 2.0 k-ratio Figure 13. z displacements vs “k” ratio and inclination (sidewall). 0.0 -0.1 Predicted Values (m) 0.20 0.15 0.10 Ydispsdw 0.05 (m) 0.00 0.05 - -0.3 -0.4 -0.5 2.0 1.5 -0.6 -0.6 1.0 k-ratio 5 0 Figure 12. y displacements vs “k” ratio and inclination (sidewall). The x displacements in the sidewall are not affected by the change in the tunnel inclination. But they are affected by the change in the value of “k” ratio as seen from Figure 11. In the case of the y displacements in the sidewall shown in Figure 12, they tend to increase as the tunnel inclination increases. When the change in the “k” ratio is considered, it should be noted that there is no significant increase in the y displacements up to 20° tunnel inclination. After this inclination, y displacements have increased as “k” ratio increased. Figure 13 shows that vertical displacements at the sidewall increase due to an increase in the “k” ratio. However, the vertical displacements decrease due to an increase in the tunnel inclination. -0.4 -0.3 -0.2 -0.1 0.0 Figure 14. Observed vs predicted values in x-direction (sidewall). The proposed regression equation for the displacements in the x-direction at the sidewall is given in Equation 5: (5) where Xdispsdw x-displacement at the sidewall; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values in the sidewall in the x-direction are shown in Figure 14. This regression model also shows a strong correlation between the observed and predicted values (r 0.99). 131 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands -0.5 Observed Values (m) 0. 50 5 4 0 4 35 30 5 2 0 2 Incline (˚) 1510 5 -0.2 09069-16.qxd 08/11/2003 20:26 PM Page 132 The regression model recommended to predict the y displacements in the sidewall is given below: (6) where Ydispsdw y-displacement at the sidewall; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values in the sidewall in the y-direction are shown in Figure 15. This regression model shows a strong correlation between the observed and predicted values (r 0.99). The regression model proposed for predicting the z displacements in the sidewall is as follows: (7) 0.20 where Zdispsdw z-displacement at the sidewall; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values in the sidewall in the z-direction are shown in Figure 16. This regression model also shows a strong correlation between the observed and predicted values (r 0.99). The correlation coefficients of all the proposed regression models for displacements have been summarized in Table 3. 3.2 Evaluation of models in terms of stresses As the observed principal stresses are examined, no strong relationship between the principal stresses and the “k” ratio or the tunnel inclination has been found except for the principal stresses in the tunnel floor. Therefore, only principal stresses in the floor are taken into consideration. The graphs of maximum and minimum principal stresses are given in Figures 17 and 18 respectively. The regression equation proposed for predicting the magnitude of 1 in the floor and the correlation coefficient are given below: Predicted Values (m) 0.15 0.10 (8) 0.05 r 0.95 0.00 Table 3. Correlation coefficients for displacements. -0.05 Correlation coefficient (r) -0.10 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 Direction Roof Floor Sidewall x y z – 0.99 0.98 – 0.99 0.99 0.99 0.99 0.99 Observed Values (m) Figure 15. Observed vs predicted values in y-direction (sidewall). Predicted Values (m) 0.5 0.4 -315 0.3 -330 -345 0.2 1 fl (kPa) 60 -3 -375 0.1 0.0 0.0 2.0 0.1 0.2 0.3 0.4 1.5 0.5 k-ratio 1.0 Observed Values (m) 0.5 Figure 16. Observed vs predicted values in z-direction (sidewall). Figure 17. 132 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0 5 1510 2 25 0 3 30 40 5 Incline (˚) 4 50 5 1 vs “k” ratio and inclination (floor). 09069-16.qxd 08/11/2003 20:26 PM Page 133 -10 Predicted Values (kPa) -15 -16 -20 -24 -28 3 fl (kPa) -32 -36 -40 0 2. 5 1. 0 k-ratio 1. 0. 5 0 5 1 15 0 2 2 0 30 5 3 Incline 4 5 4 0 50 5 -20 -25 -30 -35 -40 (˚) -45 -45 -40 -35 -30 -25 -20 -15 Observed Values (kPa) Figure 18. 3 vs “k” ratio and inclination (floor). Figure 20. Observed vs predicted 3 (floor). Table 4. Correlation coefficients for stresses. Predicted Values (kPa) -300 Correlation coefficient (r) -320 Stress Roof Floor Sidewall -340 1 3 0.75 0.63 0.95 0.96 0.68 0.68 -360 The correlation coefficients of all proposed regression models have been summarized in Table 4 for maximum and minimum principal stresses. As seen, there is only a meaningful correlation for the history point in the floor. -380 -400 -400 -380 -360 -340 -320 -300 Observed Values (kPa) Figure 19. Observed vs predicted 1 (floor). 4 CONCLUSIONS where 1fl max principal stress at the floor; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values of 1 at the floor are shown in Figure 19. This regression model shows a strong correlation between the observed and predicted values. The proposed regression equation and the correlation coefficient for 3 at the floor are as follows: (9) r 0.96 where 3fl min principal stress at the floor; k “k” ratio and I inclination of the tunnel (°). The predicted values versus observed values of 3 in the floor are shown in Figure 20. This regression model also shows a strong correlation between the observed and predicted values. When the displacements and stresses in the model studies have been examined in detail, the results can be stated as follows: 1. The y-components of the movements in the roof and the floor increase radially into the tunnel as the tunnel is inclined. This trend becomes more obvious with the increase of the horizontal stresses. 2. The z-components of the movements in the roof and the floor decrease as the tunnel is inclined. On the contrary, they increase as the horizontal stresses increase. 3. Displacements in the x-direction in the sidewall are not affected by the change of tunnel inclination. But, they increase towards the tunnel inside as the horizontal stresses increase. 4. The y-component of the movement in the sidewall increases due to the increase of tunnel inclination. This trend becomes more obvious with the increase of horizontal stresses. 5. The z-component of the movement in the sidewall tends to decrease with the increase of tunnel 133 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-16.qxd 08/11/2003 20:26 PM Page 134 inclination. The magnitude of this component increases as the horizontal stresses increase. 6. The maximum and minimum principal stresses in the floor generally increase as the tunnel is inclined. But, it should be noted that some model results are not consistent with this general trend. The results of the numerical model studies have been analyzed to find out the presence of correlation for the displacements and stresses at the tunnel boundary with the “k” ratio and tunnel inclination. As a result of multiple regression analyses carried out, strong relationships have been found between the “k” ratio, tunnel inclination and displacements. But the relationship between the “k” ratio, tunnel inclination and principal stresses are not as strong as those of displacements except for those of principal stresses in the floor. These proposed regression models can be used especially to predict the displacements around the boundary of tunnels which will be excavated in similar rock masses (marl formation). It should be noted that these predicted values should be used with great caution. Because they will be preliminary and rough estimates of displacements and principal stresses. Proposed regression equations can easily be affected by the chance of rock properties, size and shape of tunnel, groundwater etc. Similar analyses can be caried out for different rock masses. These regression models can be developed and they can be used for various rock mass conditions. REFERENCES Bieniawski, Z.T. 1979. The geomechanics classification in rock engineering applications. Proc. Xth. Congress Int. Soc. Rock Mech. Vol. 2: 41–48. Montreux. Çekilmez, V., Koç, S. & Alemdaroglu, T. 1987. The geotechnical research of the drills in Kütahya-Tavşanl1Tunçbilek District. M.T.A. Institute, Ankara/Turkey. Çekilmez, V. 1988. The geotechnical research of the JT4 drill in Kütahya-Tavşanl1-Tunçbilek district. M.T.A. Institute, Ankara. Hoek, E. & Brown, E.T. 1980. Underground excavations in rock. Institution of Mining Metallurgy, London. Itasca Consulting Group, Inc. 1997. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 2.0 User’s Manual. Minneapolis, MN: Itasca. Once, G., Iphar, M. & Yavuz, M. 2001a. Design of the main transport road of the deep coal seam panels of GLI Tunçbilek mine in Turkey, Osmangazi University Research Fund Project, Eskisehir, Turkey. Once, G., Iphar, M. & Yavuz, M. 2001b. Study of ground control of the main transport road of the deep coal seam panels of GLI Tunçbilek mine in Turkey, FLAC and Numerical Modeling in Geomechanics, Lyon, France, 29–31 October 2001. Rotterdam, Balkema. Priest, S.D. & Hudson, L. 1976. Discontinuity spacings in rock. International Journal of Rock Mechanics and Mining Sciences. Vol. 13: 135–148. 134 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-17.qxd 08/11/2003 20:27 PM Page 135 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Numerical analysis of the volume loss influence on building during tunnel excavation O. Jenck & D. Dias INSA Lyon, URGC Géotechnique, Villeurbanne, France ABSTRACT: Shallow tunneling performed by a Tunnel Boring Machine (TBM) induces volume loss, mainly due to the conical shape of the machine and the consolidation of the injected grout. This excavation volume loss causes ground movements at the surface that can induce damages to surrounding structures. However, to know the influence on structures, it is not sufficient to apply the Greenfield strains because of the influence of the structure’s stiffness. Therefore a computational three-dimensional soil-structure interaction analysis is required to take into account all the complexity of the problem. This paper presents a FLAC3D analysis of the interaction between shallow tunnel excavation and surface buildings, applied to the case of the Lisboa subway. A simplified simulation of TBM tunneling is adopted by imposing volume loss. To highlight the influence of settlements on a six-floor structure, different cases of volume loss are studied from 0.5 to 5 % of the total excavated volume. 1 INTRODUCTION During the construction of a tunnel at shallow depth in urban areas, prediction of the effects induced by the excavation on surrounding buildings is very important. In fact, the volume loss in tunnel generates soil displacements on surface that can cause damages to existing structures. The traditional design of the constructions doesn’t take into account this type of loading conditions. In order to control the volume loss in the tunnel and to limit the damages, the tunnel is excavated, when it is possible, with a TBM. A first approximation to predict the damages caused to surrounding structures is done by applying the soil deformations without any structure on surface, called Greenfield deformations, to the structure’s foundations. This method is recommended by AFTES (1999). The Greenfield deformations can be calculated by an empirical (O’Reilly & New 1982, Peck 1969), analytical (Panet 1995, Sagaseta 1987) or numerical method (Oteo & Sagaseta 1982, Swoboda et al. 1989). However, it is important to consider the structure to estimate the soil movements because it contributes to stiffen the ground and consequently to reduce the soil displacements. Then, the determination of the underground works influence on surrounding structures becomes very difficult with empirical or analytical methods. Only the numerical method is able to take account of all the complexity of this type of soilstructure interaction problem. Potts & Addenbrooke (1997) used two-dimensional numerical calculations considering the structure as an equivalent weightless beam with variable stiffness. They showed the structure’s rigidity influence on surface ground movements induced by tunneling. Franzius & Addenbrooke (2002) have then analyzed the influence of the structure’s weight. They showed that the weight has very low influence on ground movements when rigidity increases. Another two-dimensional calculation coupling the soil with a masonry building was performed by Miliziano et al. (2002). The threedimensional building is taken into account by an equivalent two-dimensional wall. They demonstrated the significant effect of the relative structure stiffness in reducing differential displacements, and on the predicted damage. Nevertheless, with 2D simulations it is worth noting that an empirical parameter such as the deconfinement ratio or the volume loss in tunnel has to be considered as remarked by Benmebarek et al. (1998). Dias et al. (1999) have compared results from 2D and 3D numerical simulations with experimental data. They showed that the surface settlement trough obtained with the 3D calculation is more realistic than the trough obtained with the 2D calculation, even with a simple constitutive model for the soil. Moreover, it is impossible to study the damages induced on the 135 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-17.qxd 08/11/2003 20:27 PM Page 136 2 EXPERIMENTAL DATA 2.1 Experimental section The studied model is based on the case of the Lisboa subway. The experimental section (Fig. 1) is located near the new Ameixoeira station. The geotechnical properties (Table 1) are given by Ribeiro e Sousa et al. (2003). The excavation is 26 m deep. The tunnel diameter is equal to D 9.8 m. The section is located in the silty sand layer, where the mechanical properties are relatively poor. 2.2 7m C2 26 m C3 14 m C4 17 m C5 Table 1. Geotechnical properties. Name: Soil type C1 Clay C2 Clay C3 C4 C5 Limestone Silty sand Clay E [MPa] c [kPa] " [°] [kN/m3] Ko 15 0.35 5 30 20.5 0.6 15 0.4 5 32 20.7 0.7 266.5 0.35 10 37 20.5 0.7 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 -40 Section S28 Section S29 Section S30 -20 0 44.7 0.35 0 35 20.65 0.8 S28 20 40 60 80 Distance to the section (m) 180 0.37 250 30 20.4 1.05 S29 S30 100 120 Figure 2. Longitudinal settlement troughs. section, the maximum measured settlement is equal to 0.3 cm. 3 NUMERICAL MODEL ADOPTED 3.1 Ground mass Figure 4 presents the numerical model of the groundmass. Due to the symmetry conditions, only half of the 136 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands C1 Figure 1. Experimental section. Measured surface settlements Measured settlements have been obtained on several sections near the Ameixoeira station (Ribeiro e Sousa et al. 2003). Results are shown in Figures 2 & 3. Figure 2 presents the settlements measured above the tunnel axis in three different sections, for different positions of the TBM. Figure 2 illustrates experimental longitudinal settlement troughs. The maximum observed settlement is about 0.4 cm settlements are observed even when the TBM has not already reached the instrumented section. On this underground works, an earth pressure shield is used. Figure 3 shows the observed surface settlements in a transversal section (S31), at the final state of the excavation (when the TBM is far away). This curve is called transverse settlements trough. In this instrumented 4m 9m Surface settlements (cm) structure in the tunnel axis direction when using 2D simulations. Some authors have already used three-dimensional numerical modeling. For instance Mroueh & Shahrour (2003) have compared the results of a soil-structure interaction calculation of tunnel excavation below a structure with the results obtained by imposing the Greenfield movements upon the structure. They showed that this last method is very severe in terms of induced forces in the structure. Netzel & Kaalberg (2000) have modeled the interaction between TBM digging and masonry structures in order to obtain specific damage criteria. This article presents a three-dimensional numerical analysis of the soil-structure interaction phenomenon during shallow tunneling. The tunnel excavation is a simplified simulation of the real phases of a TBM based on the concept of volume loss. The soil behavior is elastic perfectly plastic. The structure is composed of columns and floors founded on a raft. The parametrical study deals with the influence of volume loss in tunnel. Results are analyzed in terms of ground surface displacements and of stresses induced in the structure. 09069-17.qxd 08/11/2003 20:27 PM Page 137 Surface settlements (cm) 0 Step n Z -0.05 20 m -0.1 Y -0.15 -0.2 -0.25 Tunnel face -0.3 -0.35 -20 -15 -10 -5 0 5 10 Distance to tunnel axis (m) 15 20 Step n+1 20 m Figure 3. Transverse settlement trough. Z Y One element length X Figure 5. Longitudinal tunnel section – excavation process. have a linear variation on the distance of 20 m. After this distance, the lateral displacements are constant, as shown on Figure 5. This process simulates the principal excavation phases: – – – – Figure 4. Numerical model. ground mass is modeled (plane of symmetry Y–Z). The model is 100 m wide in the X direction, 90 m long in the Y direction (parallel to tunnel axis) and 51 m high (Z direction). The numerical model consists of approximately 85,000 nodes. 3.2 Ground behavior The behavior of the soil is set as elastic perfectly plastic with a Mohr-Coulomb failure criterion. The flow rule is non-associated and the dilatancy angle is set as # " 30°. Due to the very fast pore pressure dissipation observed, the calculation is done in drained conditions. 3.3 Simulation of excavation The adopted excavation process for the calculation is a simplification of the confinement-deconfinement phases induced by the boring machine. The hypotheses are as follows: the soil displacements at the tunnel face are blocked, simulating a perfect equilibrium of confinement pressures. The lateral soil displacements The initial position of the tunnel is Y 0 m and the numerical phases of excavation are as follows: – excavation on one element length, – fixation of the tunnel face nodes, – convergence of tunnel walls until reaching the given displacement shape, – if a node reaches the limit, it will be fixed, – when the model equilibrium is reached, all the nodes are freed, – translation of the loading system of one element length. Hence, there are as many excavation steps as there are elements on the model length. The displacement field imposed at the tunnel walls corresponds to a volume loss. This volume loss in tunnel normalized by the total excavated volume is called Vt. The excavation is ended when the model is entirely bored. Then the tunnel face is at Y 90 20 110 m which corresponds to the entire model length added with the distance between the tunnel face and the position where the lateral displacement is constant. 137 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Conical shape of TBM Grout injection Grout consolidation Setting of the concrete rings 09069-17.qxd 08/11/2003 20:27 PM Page 138 is the point of inflection of the transverse settlement trough. 4 REFERENCE CASE: EXCAVATION WITHOUT STRUCTURE ON SURFACE The reference case corresponds to an excavation of the numerical model with Vt 5%. According to Benmabarek et al. (1998), this value seems to be the higher value obtained with the TBM excavation method and do not correspond to the observed settlements. 4.1 5 INFLUENCE OF VOLUME LOSS IN TUNNEL ON GREENFIELD DISPLACEMENTS In order to study the influence of volume loss in tunnel on Greenfield soil displacements, several calculations are done with Vt from 0.5% to 5%. Surface settlements The surface settlement distribution in a transverse section at the final state is shown on Figure 6. The distance to tunnel axis is normalized by the tunnel diameter D. The maximum settlement is equal to 4 cm; it is about ten times higher than the observed settlements (see section 2.2). The adopted numerical process for simulating the tunnel excavation with a TBM is able to reproduce a surface displacement trough in agreement with the Gaussian distribution, which matches very closely experimental observations. This settlement distribution is given by Peck (1969) equation: 5.1 Surface settlements Figure 8 compares the transversal settlement troughs for the different values of Vt, at the final state. All the curves show a Gaussian distribution with the same value of i. The maximum surface settlements are: – for Vt 3%, Smax 2.5 cm – for Vt 1%, Smax 1 cm – for Vt 0.5%, Smax 0.5 cm These results are reported in Figure 9. A quasi-linear relation between tunnel volume loss and maximum surface settlement is observed. (1) 0.05 4.2 Horizontal surface displacement Figure 7 presents the horizontal surface soil strains in a transverse section, at the final state. In this figure, two distinct zones are observed. In the center of the trough, the soil is in compression and on the edges the soil is in extension. The limit between these two zones 0 0 1 Distance to tunnel axis (x/D) 2 4 3 5 6 8 9 -0.05 -0.10 -0.15 extension Figure 7. Horizontal strain in a transverse section. 0 7 -1 -2 -3 0 1 Distance to tunnel axis (x/D) 2 3 4 5 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -4 Figure 8. Transverse settlement trough. 138 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Distance to tunnel axis (x/D) 5 7 2 3 4 6 0.00 compression i Figure 6. Transverse settlement trough. 1 -0.20 Surface settlements (cm) Surface settlements (cm) Horizontal strain (%) Where S settlement at distance x of the tunnel axis, Smax maximum settlement obtained in the trough center, i distance from the inflection point of the trough to tunnel axis. In this case, i 1.8 D. 0 Vt 0.5% 0.5% 1% 1% 3% 3% 5% 5% 6 7 09069-17.qxd 08/11/2003 20:27 PM Page 139 D/2 Maximum settlement (cm) 5 Tunnel 4 18m Smax = 0.82Vt 4m 3 4m 2 Y 12m 1 0 0 1 3 4 2 Tunnel volume loss (%) X 5 Figure 9. Liner relation between maximum settlement and Vt. Symmetry axis Figure 11. Column position. Horizontal strain (%) 0.05 0 1 Distance to tunnel axis (x/D) 2 3 4 5 6 7 Z 8 Y X 0.00 -0.05 -0.10 -0.15 Vt 5% 3% 1% 0.5% -0.20 Figure 10. Horizontal strain in a transverse section. Figure 12. Numerical model coupling soil and structure. The tunnel volume loss of Vt 0.5% corresponds to the observed surface settlement (Figs. 2 & 3). 5.2 Horizontal surface displacement Figure 10 compares the horizontal surface displacements for the values of Vt, at the final state. The limit between the two zones is the same for all values of Vt. As in the previous paragraph, a linear relation is observed between Vt and the maximum soil compression. 6 COUPLED CALCULATION: MODELING OF THE STRUCTURE In order to study the soil-structure interaction during shallow tunneling, three-dimensional calculations coupling ground mass and structure are presented. 6.1 Geometry of the structure The studied structure is a simplification of existing buildings. It is composed of columns of square section (0.4 m 0.4 m), and slabs of 0.3 m thickness, founded on a 0.3 m raft, which size is 12 m 36 m. The building has seven levels of 4 m height. Figure 11 shows the column’s position. There is no eccentricity considered between the structure and the tunnel; therefore, only the half of the structure is modeled. 6.2 139 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Numerical model coupling soil and structure Figure 12 shows the numerical model coupling ground mass and structure. The structure is located in the middle of the model length (Y-axis). Columns are taken into account as beams, slabs and raft as shells. The structure behavior is elastic with properties of a reinforced concrete: E 19 GPa and 0.2. The soil nodes are bound with the structure nodes. The structure is disposed on the numerical model in one phase. Then equilibrium of the model is reached. The structure is only loaded with its own weight ( 25 kN/m3). After that the soil displacements are initialized in order to study only the tunneling effect. 09069-17.qxd 08/11/2003 20:27 PM Page 140 The forces in the structural elements are not initialized: this step represents the initial state in terms of internal forces in the structure. Finally, the obtained numerical model is excavated. Surface settlement (cm) 0 7 COUPLED CALCULATION: REFERENCE CASE, Vt 5% The reference case of the calculation coupling soil and structure is compared with the reference case without structure, presented in section 4. Surface soil displacements 5 Building -1 -2 Greenfield With structure -3 -4 0 0 Distance to tunnel axis (x/D) 2 3 4 5 6 7 1 8 9 -0.5 -1 Greenfield -1.5 With structure -2 Figure 14. Horizontal displacement with and without structure, Vt 5%. MY TY Tunnel axis Y Induced forces in the structure columns Tx Mx X During the excavation, the forces in the structure columns are analyzed. Figure 15 defines the shear forces and the bending moments in a column section. Figure 16 shows the maximum values of the internal forces normalized by the maximum initial values F/Fini, for different excavation length. The transverse shear forces increase considerably when the TBM passes below the structure. Then important values are kept until the end of the excavation. Longitudinal shear forces increase when TBM is located below and just next to the building, and decrease to their initial values when TBM moves off. The axial forces are very lightly affected by digging compared to the others forces. Then, they are not represented here. TY Tx MY Mx : longitudinal shear force : transversal shear force : transversal bending moment : longitudinal ending moment Figure 15. Forces in a column section. The forces induced in the columns are studied more in detail for four specific columns specified on Figure 17. These columns are chosen because they undergo a great stress increase or are representative of the general behavior. Only the evolution of the 140 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Distance to tunnel axis (x/D) 3 4 2 Figure 13. Surface settlement trough with and without structure, Vt 5%. Figure 13 compares the surface settlement troughs in the middle of the model length, at the final state. With a structure on surface, the settlements increase than with the Greenfield calculation. The maximum surface settlement increase from 4.1 cm to 4.8 cm is equal to 15%; and the volume trough increase is equal to 5%. Nevertheless, the settlement trough remains a Gaussian curve in the two cases, with appreciatively the same value for i. Figure 14 compares the horizontal surface displacements in a transverse section with and without structure, at the final state. In the first case, the horizontal soil displacements under the structure are negligible compared to the Greenfield case. This is due to the high axial stiffness of the raft and due to the fact that the raft is bound with the soil. This is a significant result, which has also been observed in field monitoring (Standing et al. 2002). Hence, it is very severe to apply the Greenfield movements of soil to this structure to estimate the induced damages, as recommended in first approximation by AFTES (1999). 7.2 1 -5 Horizontal displacement (cm) 7.1 0 09069-17.qxd 08/11/2003 20:27 PM Page 141 20 50 My Tx 40 Mx Ty Longitudinal bending moment (kN.m) F/Fini 60 30 20 10 15 10 5 0 0 10 20 0 30 40 50 60 70 80 90 100 110 Excavation length (m) Figure 16. Maximum beam forces. A D Column height (m) B X C Tunnel axis Transversal bending moment (kN.m) 30 20 10 0 0 2 20 40 60 80 Excavation length (m) 100 40 60 80 Excavation length (m) 20 Initial state Excavation of 45m Excavation of 77m Final state 16 12 8 4 -15 -10 -5 0 5 Longitudinal bending moment (kN.m) 10 Figure 20. Longitudinal bending moment (Mx) on column C. Column A B C D 40 24 0 -20 Figure 17. Position of studied columns. 50 0 Figure 19. Maximum longitudinal bending moment on the four columns. 28 Y Column A B C D 100 Figure 18. Maximum transversal bending moment in the four columns. bending moment is studied, because shear forces vary in the same way. Figure 18 presents the evolution of the maximum value of MY for the different studied columns. It can be noted that more columns are far from the tunnel axis, more columns are affected in the transverse direction. Figure 19 presents the evolution of the maximum value of Mx for the different studied columns. It seems that the longitudinal stresses are similar in all the building’s columns. The most affected column in both directions is column C. Bending moments and shear forces distribution in these columns are analyzed for different excavation lengths. Figures 19 and 20 present the repartition of the longitudinal forces in column C. The most prejudicial excavation step corresponds to the excavation length of about 80 m. The maximum longitudinal bending moment is 19 kN.m, reached between levels 3 and 4 (Fig. 19). For a length bored of 45 m, figure 19 shows that the column is affected in the opposite direction, with a value of 7 kN.m between levels 2 and 3. The maximum longitudinal shear force is equal to 2.8 kN, reached at level 5. Level 3 is also affected with TY 2.6 kN (Fig. 20). Figures 21 and 22 present the repartition of the transversal forces on column C. The most prejudicial excavation step is at the final state. The initial state is not represented because of negligible values. This figure show that the most affected levels in the transverse direction are levels 1 and 2. The maximum value of MY is 44 kN.m (Fig. 21) and the maximum value of TX is 11 kN (Fig. 22). 141 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 28 28 24 24 Column height (m) Column height (m) 09069-17.qxd 08/11/2003 20:27 PM Page 142 20 16 12 Initial state Excavation of 77.5m Final state 8 4 0 -4,0 -3,0 -2,0 -1,0 0,0 1,0 2,0 Longitudinal shear force (kN) Final state 20 16 12 8 4 0 -12,0 3,0 Figure 21. Longitudinal shear force (TY) on column C. Excavation of 45m -8,0 -4,0 0,0 4,0 8,0 Transversal shear force (kN) Figure 23. Transversal shear force (TX) on column C. 20 Excavation of 45m Final state Surface settlement (cm) Column height (m) 28 24 16 12 8 4 0 -10 0 10 20 30 40 Transverse bending moment (kN.m) 50 Figure 22. Transversal bending moment (MY) on column C. 0 Distance to tunnel axis (x/D) 3 5 2 4 1 6 7 Vt 0.5% 1% 3% 5% Building Ratio = Swith str./Swithout 1.2 Influence of tunnel volume loss on a structure is also studied. Calculations coupling soil and structure are done using different volume loss in tunnel. Influence of volume loss on surface settlements Figure 23 compares the surface settlement troughs obtained with a structure on surface for different volume loss imposed in tunnel. This figure can be compared with Figure 8. For a volume loss more than 1%, the surface settlements are greater in the coupled calculation than in the Greenfield case. All curves present a Gaussian distribution of surface settlements. Figure 24 presents the ratio between settlements with structure and without structure according to the position on the model length, above the tunnel axis, at the final state. A ratio greater than 1 corresponds to settlements higher than values without structure. It is always the case, except with a very low tunnel volume loss (Vt 0.5%). This figure illustrates that greater is the volume loss in tunnel, greater is the settlement increase compared to the Greenfield case. 0,5% 1% 3% 5% 1.15 1.1 1.05 1 0.95 0 10 20 30 40 50 60 70 Numerical model length (m) 80 90 Figure 25. Longitudinal settlements ratio. 8.2 Influence of volume loss on induced forces in columns Figure 25 presents the evolution of the maximum value of TX (same evolution for maximum MY) for different values of Vt. For each value of Vt, the transversal forces increase when TBM passes under the structure and keep important values at the final state. Figure 26 presents the evolution of the maximum value of TY (same evolution for maximum MX) for 142 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 Figure 24. Surface settlement trough with structure on surface for different volume loss in tunnel. 8 COUPLED CALCULATION: INFLUENCE OF VOLUME LOSS IN TUNNEL 8.1 12,0 09069-17.qxd 08/11/2003 20:27 PM Page 143 longitudinal forces are obtained for a length bored of about 80 m. 60 Vt F/Fini 50 5% 40 3% 30 1% 9 CONCLUSIONS 0.5% 20 10 0 0 10 20 30 40 50 60 70 80 Excavation length (m) 90 100 110 Figure 26. Evolution of maximum transversal forces in columns. 9 Vt 8 5% F/Fini 7 6 3% 5 1% 4 0.5% 3 2 This numerical analysis highlights the soil/structure interaction during shallow tunneling with TBM. The attention was focused on the influence of the volume loss in tunnel. The presence of the structure increases the surface settlement for a volume loss in tunnel higher than 1%, in comparison to the Greenfield case. Due to the presence of a raft with a great axial stiffness and the fact that the raft is bound with the soil, the horizontal displacements are negligible under the structure in comparison with the Greenfield case. This study has showed that the more affected direction of the building during tunneling is the transverse direction. Moreover, transversal loads are keeping important values at the final state whereas longitudinal loads are coming back to their initial values. The analysis of the volume loss influence showed a linear relation between volume loss in tunnel and forces induced in the structure columns during tunneling. 1 0 0 10 20 30 40 50 60 70 80 Excavation length (m) 90 100 110 Figure 27. Evolution of maximum longitudinal forces in columns. F/Fini 60 50 Tx - final state 40 Ty - excavation of 77 m y = 1032x 30 20 y = 171x 10 0 0 1 2 3 4 Volume loss in tunnel (%) 5 Figure 28. Maximum forces increase according to Vt. different values of Vt. For each Vt, the shape of the F/Fini evolution is similar but with higher values for higher Vt. When reporting the maximum values for the transversal and longitudinal forces according to the volume loss in tunnel, Figure 27 shows a linear relation. Maximum values for the transversal forces are obtained at the final state, whereas maximum values for the REFERENCES AFTES 1999. Recommandations pour les tassements liés au creusement des ouvrages en souterrain. Tunnels et Ouvrages Souterrains: 106–128. Benmebarek, S., Kastner, R. & Ollier, C. 1998. Auscultation et modélisation numérique du processus de creusement à l’aide d’un tunnelier. Géotechnique 48 (6): 801–818. Dias, D., Kastner, R. & Maghazi, M. 1999. Three dimensional simulation of slurry shield tunneling. In International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground, Tokyo, Japan, 6p. Franzius, J.N. & Addenbrooke, T.I. 2002. The influence of building weight on the relative stiffness method of predicting tunnelling-induced building deformation. In 4th Symposium Geotechnical Aspects of Underground Construction in Soft Ground, Toulouse, France, 1, 53–58. Miliziano, S., Soccodato, F.M. & Burghignoli, A. 2002. Evaluation of damage in masonry buildings due to tunnelling in clayey soils. In 4th Symposium Geotechnical Aspects of Underground Construction in Soft Ground, Toulouse, 3, 49–54. Mroueh, H. & Shahrour, I. 2003. A full 3-D finite element analysis of tunneling-adjacent structures interaction. Computers and Geotechnics 30: 245–253. Netzel, H. & Kaalberg, F.J. 2000. Numerical damage risk assessment studies on masonry structures due to TBMTunnelling in Amsterdam. In GeoEng 2000, Melbourne, Australia, 235–244. O’Reilly, M.P. & New, B.M. 1982. Settlements above tunnel in the United Kingdom – their magnitudes and prediction. In Tunelling 82’, London, IMM, 173–181. 143 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-17.qxd 08/11/2003 20:27 PM Page 144 Oteo, C.S. & Sagaseta, C. 1982. Prediction of settlements due to underground openings. In Int. Symp. On numerical Models in Geomechanics, Zurich, 653–659. Panet, M. 1995. Le calcul des tunnels par la méthode convergence-confinement. Paris: Presses de l’ENPC. Peck, R.B. 1969. Deep excavation and tunnelling in soft ground, State of the art report. In 7th International Conference on Soil Mechanics and Foundation Engineering, Mexico, 225–290. Potts, D.M. & Addenbrooke, T.I. 1997. A structure’s influence on tunnelling induced ground movements. In Instn Civil Engineers in Geotechnical Engineering. 125, 109–125. Ribeiro e Sousa, L., Dias, D. & Barreto, J. 2003. Lisbon Metro Yellow Line extension. Structural behaviour of the Ameixoeira Station. In 12ª Conferência Panamerican on Soil Mechanics and Geotechnical Engineering, Boston. Sagaseta, C. 1987. Evaluation of surface movements above tunnels, a new approach. In Colloque International ENPC Interactions sol/structure, Paris, Presses ENPC, 445–452. Standing, J.R., Gras, M., Taylor, G.R., Gupta, S.C., Nyren, R.J. & Burland, J.B. 2002. Building response to tunnel step-plate junction construction – the former Lloyds Bank building, St James’s, London. In 4th Symposium Geotechnical Aspects of Underground Construction in Soft Ground, Toulouse, France, 3. Swoboda, G., Mertz, W. & Schmid, A. 1989. Three dimensional numerical models to simulate tunnel excavation. Numerical Models in Geomechanics NUMOG III. Elsevier. 581–586. 144 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 145 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitling, Lisse, ISBN 90 5809 581 9 Application of FLAC3D on HLW underground repository concept development S. Kwon, J.H. Park, J.W. Choi & W.J. Cho Korea Atomic Energy Research Institute, Korea ABSTRACT: For the safe design of a deep underground high-level radioactive waste (HLW) repository, it is important to understand the thermal–mechanical behavior of the engineering barriers and rock mass around the repository influenced by the high stress and the heat generated from the waste. In this study, thermal–mechanical coupling analysis was carried out to investigate the reliability of the Korean HLW repository concept using FLAC3D with the thermal and mechanical properties of rock and rock mass measured at two drilling sites. For effective thermal–mechanical coupling, a FISH routine was developed and used for the modeling of different conditions. By using FLAC3D with the FISH routine, the evaluation of the thermal–mechanical stability of the preliminary disposal concept could be done successfully. 1 INTRODUCTION The Republic of Korea began operating commercial nuclear power plants in 1978. Now there are 17 operating plants, 4 CANDU (Canadian Deuterium Uranium Reactor) and 13 PWR (Pressurized Water Reactor). The current generating capacity is 14,720 MWe with a share of 39.3% of the total production of electricity. The total generating capacity is expected to be about 26.05 GWe by 2015. The cumulative amount of spent fuel from existing nuclear power plants reached 5,641 MTU by June 2002. It is expected that approximately 11,000 MTU and 19,000 MTU will be accumulated by the years 2010 and 2020, respectively. In Korea, a reference HLW disposal system is under development. According to the previously determined disposal concept, the PWR and CANDU spent fuel in corrosion resistant canister will be emplaced in a deep underground repository constructed in crystalline rock such as granite. To confirm whether the disposal concept is reliable or not under certain geological conditions, waste type, and operation procedure, computer simulations need to be carried out. FLAC3D had been widely applied in radioactive waste repository projects related to different rock types by many researchers (Johansson & Hakala 1995, Berge & Wang 1999, Fairhurst 1999, Francke et al. 2001, and Patchet et al. 2001). In Korea, FLAC3D had already been used for the thermal analysis (Park et al. 1998) and mechanical analysis (Park et al. 2001) for Korean reference repository design. In this study, FLAC3D was used for investigating the thermal–mechanical coupling behavior of rock, buffer, backfill, and canister. In order to carry out the thermal–mechanical coupling analysis, a FISH routine was developed and used for the modeling to investigate the coupling behavior of the rock mass around the disposal tunnel and deposition hole. 2 HLW REPOSITORY CONCEPT IN KOREA The Korea Atomic Energy Research Institute has been developing a reference HLW disposal system since 1997. According to the preliminary disposal concept, the repository is located in a crystalline rock mass at several hundred meters below surface. Like many other countries such as Sweden, Canada, Finland, Switzerland, and Japan, a multibarrier system consisting of canister, buffer, and backfill is supposed to be applied for safe containment of the radioactive waste. The buffer acts as a barrier to suppress the detrimental effects of the corrosive water in the host rock and to enhance the life of the container and serves as a geochemical filter for the sorption of radionuclides. The buffer dissipates the decay heat from the waste into the surrounding rock to avoid the possibility of thermal stress on the container. It also provides the mechanical strength to support the canisters and isolates the containers from detrimental rock mass movements (Selvadurai & Pang 1990). In many countries including Korea, bentonite is now considered as the buffer 145 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 146 40m TB: Backfill thickness(1.5m) BT: Upper buffer thickness(1m) FT: Bottom & side buffer thickness(0.5m) 6m 7m 6m 2.2m TB BT Backfill Buffer Outshell Canister Spent Fuel Deposition Hole FT Figure 1. Schematic drawing of the reference Korean repository design. material because of its low hydraulic conductivity, high sorption capacity, self-sealing characteristics, and durability in nature. In the Korean repository concept, the mixture of bentonite and crushed rock will be used as the backfilling material. The deposition tunnels are 6 m wide and 7 m high. The canister containing spent fuel is assumed to be emplaced in the vertical boreholes drilled along the center line on the floor as shown in Figure 1. 3 3.1 Table 1. Material properties of fuel part and outshell. Unit Material type Model type E Density Thermal conductivity Specific heat Thermal expansion GPa Kg/m3 W/m°K J/Kg°K /°K Fuel part Outshell Fuel cast iron Elastic 190 0.3 6500 43 424 1.2e-5 Stainless steel Elastic 200 0.3 8000 15.2 504 8.2e-6 FLAC3D MODELING Materials in the model 3.1.1 Fuel part and outshell Four PWR assemblies are inserted in a canister with outshell thickness of 5 cm. The mechanical and thermal properties of the fuel part, which represents the part inside of the outershell, were determined with the assumption of that the fuel and cast iron were uniformly mixed. The average properties of the fuel part were calculated based on volume ratio and listed in Table 1. Among the candidate material types for the outshell, stainless steel was considered in this study. The diameter of the canister is 1.22 m and the length is 4.78 m. The thermal–mechanical properties of stainless steel are also listed in Table 1. 3.1.2 Buffer and backfill Some of the thermal and mechanical properties of buffer and backfill material could be determined from laboratory tests using Korean bentonite (Kyungju bentonite), which is considered as a candidate buffer material for the Korean repository. The other material properties, which could not be determined from tests, were chosen from literature review and listed in Table 2. In this study, the buffer and backfill materials were modeled with a Drucker-Prager plastic model. The Drucker-Prager parameters in Eq. (1) for buffer and backfill could be determined from the triaxial compression tests under different confining pressures. (1) where, ! and kshear are material parameters, J1 is the first invariant of the stress tensor, and J2D is the second invariant of the deviatoric stress tensor (Desai & Siriwardane 1984). 146 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 147 Table 2. Material properties of buffer and back fill. Rock Unit Material type Model type Modulus GPa Density T. Conductivity Specific heat T. Expansion UCS Cohesion Friction angle Drucker-Prager parameters Kg/m3 W/m°K J/Kg°K /°K MPa MPa Degree Geothermal gradient °C/km Buffer Backfill Kosung Yusung Bentonite Drucker-Prager Bulk 0.345 Shear 0.258 Dry 1800 1.47 888 3.1e-4 7.66 1.1 50 Qvol 1.23 Crushed rock bentonite Drucker-Prager Bulk 0.038 Shear 0.029 Dry 1800 2.04 900 3.1e-4 0.93 1.1 17 Qvol 0.24 Granite Mohr-Coulomb E 56.6 0.25 2650 2.523 1576 19.244e-6 149.55 22.5 61 Granite Mohr-Coulomb E 46.8 0.28 2660 3.541 1212 19.312e-6 132.5 30.4 51 Kshear 944 Kshear 1472 37.5 25 3.1.3 Rock properties It is assumed that the underground repository is constructed in a granite body at 500 m below surface. The mechanical and thermal properties of the granites from two drilling sites, Kosung and Yusung, are listed in Table 2. The two sites are representing the east and west sides of Korean Peninsula. NX size rock cores were retrieved from the drill holes reached up to 500 m below surface. The influence of discontinuities is considered indirectly using the equations proposed by Fossum (1985) to calculated the modulus of randomly jointed rock mass. The effective bulk and shear moduli can be written in terms of the intact and joint properties. Effective bulk and shear moduli are (2) (3) where, E is Elastic modulus of rock, is Poisson’s ratio, S is joint spacing, and kn and ks are normal and shear stiffness of joint. Thermal logging was carried out to find the geothermal gradients at the two sites (Park et al. 2001). 3.2 Modeling method Thermal–mechanical coupling is important due to the thermal stress developed by the decay heat from the waste. Subsequent heating of the rock mass by the heat-generating waste would increase the stresses in the buffer, canister, and rock mass because of thermal expansion (Simmons & Baumgartner 1994). The thermal stress due to the thermal expansion can be calculated as follows: (4) where, is increase in stress due to the expansion of rock, is thermal expansion coefficient, T is temperature increase, E is Young’s modulus, and is Poisson’s ratio. FLAC3D has functions for coupling behaviors such as hydraulic–mechanical, thermal–mechanical, and thermal–hydraulic couplings. In FLAC3D, the thermal–mechanical coupling occurs only in one direction: temperature changes cause thermal strains to occur which influence the stresses, while the thermal calculation is unaffected by the mechanical changes taking place (Itasca 1996). As normal in most modeling situations, the initial mechanical conditions correspond to a state of equilibrium which must first be achieved before the coupled analysis is started. There are the following three suggestions for thermal–mechanical coupling in the FLAC3D manual. 1. A thermal only calculation is performed until the desired time and then the thermal calculation is to be turned off and the mechanical calculation is performed. When the mechanical equilibrium is reached, thermal calculation is performed again. 2. For each thermal time step, several mechanical steps are taken until detecting equilibrium condition. 3. The STEP command is used while both mechanical and thermal modules are on. In this approach, one mechanical step will be taken for each thermal step. 147 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 148 The first approach is useful for thermal–mechanical analysis of an elastic model. For non-linear models such as plastic models, the thermal change must be communicated to the mechanical module at closer time intervals to respect the path dependency of the system. In this case, a certain number of mechanical steps are taken for each thermal step to allow the system to adjust. In this approach, the transition from thermal to mechanical calculation is based on time instead of temperature variation. Since the heat generation is varying with time, the transition based on temperature variation is more reasonable in the early stage of the repository. The second and third approaches may be more accurate than the first approach, but the problem is that the calculation will take a long time to model the long-term behavior of repository. In order to overcome the disadvantages of the three approaches, another technique for thermal–mechanical coupling was developed. In the new approach, the transition from thermal to mechanical calculation is based on temperature change. A FISH program for the new approach was developed for PWR spent fuel. Figure 2 shows the flow chart of the thermal–mechanical coupling adapted in this study. 3.3 In the model, 5 different materials, rock, buffer, backfill, outshell, and fuel part, were included. In the Korean preliminary disposal concept, the backfill thickness L1 1 m, upper buffer thickness L2 1.5 m, bottom buffer thickness and side buffer thickness L3 0.5 m. 3.3.2 Initial and boundary conditions The in situ stress was assumed to be hydrostatic in this study based on the fact that the stress ratios in Yusung and Kosung sites are more or less 1.0 at 500 m depth. The initial temperature in the model was calculated with the geothermal gradients of Yusung and Kosung sites. It was assumed that the average surface temperature is 20°C. 3.4 Decay heat Decay heat is the thermal energy resulting from the radioactive decay of the radioactive materials in the spent fuel discharged from reactors. In Korea, the PWR spent fuel with 45,000 MWd/tHM is now considered as the reference PWR spent fuel, because that type of Model mesh and boundary conditions 3.3.1 Model mesh Figure 3 shows the model mesh around the deposition tunnel and deposition hole. The model mesh around the disposal tunnel and deposition hole located at the center of the whole model mesh, which covers from surface to 1000 m level. The backfilling material in the disposal tunnel is not shown in the figure to clearly show the model mesh of the floor and deposition hole. Figure 2. Flow chart of the TM coupling. Figure 3. Model mesh around the tunnel and deposition hole. 148 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 149 spent fuel occupies 64% of all spent fuel from Korean reactors. Even though significant amount of CANDU spent fuel with 7500 MWd/tHM is generated from CANDU reactors, CANDU spent fuel is not considered in this study, because of its much lower burnup compared to PWR spent fuel. For the spent fuel with 45,000 MWd/tHM, the heat decays exponentially with time as following: (5) where, t is time (year) after discharge from reactors. FLAC3D MODELING RESULTS 4 4.1 Temperature Temperature (deg. C) In order not to lose the required properties of bentonite buffer, the maximum temperature at the canister surface and throughout the buffer must not exceed 100°C. As shown in Figure 4, the highest temperature in the Kosung case was 92.97°C at 15 years after emplacement, while it was about 80.71°C in 20 years after emplacement in Yusung case. Since the highest temperatures in the model in both cases are lower than 100°C, the peak temperature in the buffer cannot 100 95 90 85 80 75 70 65 60 55 50 Kosung case Yusung case 0 50 100 150 200 Time after emplacement (year) 250 Figure 4. Variation of maximum temperature with time for Kosung and Yusung case. be higher than 100°C and thus the disposal design can satisfy the thermal criteria. Because of the higher geothermal gradient in Kosung area, the temperature around the repository in the Kosung case is higher than in the Yusung case. From Figure 4, it is possible to see when the transitions between mechanical and thermal steps had happened. At the 500 m deep location, the initial temperature in the Kosung case was 38.75°C while the temperature in the Yusung case was 32.5°C. The initial temperature difference due to the difference in geothermal gradient was about 6°C. Table 3 lists the temperatures at the checking points at 20 years and 200 years after the emplacement of canister. The difference in temperature after 20 years in the Kosung and Yusung cases ranges from 8 to 12.4°C, which is higher than the initial temperature difference. The increase of temperature difference is due to the lower thermal conduction in Kosung case, which has lower thermal conductivity than that in Yusung case. With increase in time, the temperature difference between the two cases decreased and it was about 7–9°C at 200 years after emplacement. 4.2 Displacement In the deposition hole, the heat from the waste will lead to thermal expansion of the canister, buffer, and backfill. The displacements around the tunnel will also be influenced by the heat generation from the deposition hole. Since the thermal and mechanical properties of rock are different in the Yusung and Kosung cases, the displacements around the disposal tunnel and deposition hole are different. Figure 5 shows the displacement plot around the deposition tunnel at 200 years after the emplacement of the canister and buffer. In Yusung case, the maximum displacement, which is recorded at the upper backfill, was about 19 cm, while it was about 23 cm in Kosung case. The upward displacement from the deposition hole to the tunnel is Table 3. Temperatures (°C) at the checking points and different time for Kosung and Yusung cases. 20 years 200 years Check points Kosung Yusung Difference Kosung Yusung Difference 1 2 3 4 5 6 7 8 9 92.84 92.51 80.62 72.77 65.72 69.42 89.46 89.44 76.42 80.72 80.39 68.19 62.45 57.60 60.35 77.73 77.53 64.76 12.12 12.13 12.43 10.32 8.12 9.07 11.74 11.91 11.66 81.95 81.83 77.70 74.97 72.32 73.66 80.78 80.79 76.29 72.68 72.56 68.32 66.33 64.52 65.51 71.65 71.59 67.17 9.27 9.27 9.38 8.65 7.80 8.15 9.13 9.20 9.12 149 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 150 of the outshell in 20 years is downward while the other parts show upward displacement. This can be explained by the tensile stresses developed in the outshell. The direction change of displacement at the outshell bottom to upward in 200 years does not mean that the tensile stresses disappeared, but the difference of displacement along the outshell shows that there are tensile stresses in the canister. In the Kosung case, the maximum displacement at the outshell is about 1.1 cm, which is a little larger than that in Yusung case. 4.3 Figure 5. Displacement plot around the excavation in Yusung and Kosung sites. Stress distribution Figure 7 shows the principal stress distribution at the canister in the Yusung case. The minimum principal stress was compressive and the magnitude was up to 18 MPa. The maximum principal stress was tensile and it was up to 28 MPa. The tensile stress needs to be considered as an important factor in the disposal concept design, since it may cause mechanical failure of canisters. It is important to check von-Mises stress, because the distribution of von-Mises stress is closely related to the mechanical stability of rock opening. Figure 8 shows the von-Mises stresses at the checking points. At the checking points, the calculated vonMises stresses from the Kosung case are higher than those from Yusung case. In the case of checking point 3, which represents the borehole surface, the von-Mises in Kosung case is highest up to 75 MPa, while that in the Yusung case is about 40 MPa. The higher stress distribution in the Kosung case might be due to the higher temperature as well as more stiff rock properties at the Kosung site. 5 CONCLUSIONS In this study, thermal–mechanical coupling analysis for the preliminary Korean disposal concept had been carried out using FLAC3D. In order to overcome the disadvantages of the previous approaches for thermal–mechanical coupling, a new method based on temperature variation was suggested and a FISH routine was developed. From the studies, the following conclusions could be drawn: Figure 6. Displacement plot at outshell. thought to be mainly from the thermal expansion of the materials inside of the deposition hole. Buffer movements can cause the canister to move in the deposition hole. In order to check the displacement pattern in the canister, the displacements at the outshell were plotted in Figure 6. The displacements in 20 years after emplacement were relatively smaller than those in 200 years. One interesting thing in the figure is that the displacement direction at the bottom – In both Kosung and Yusung cases, the maximum buffer temperature was found to be lower than 100°C, which is the most critical criteria for disposal concept design. – When using the geological information from the Kosung drilling site, it was found that the maximum temperature was 92.97°C in 15 years after emplacement, while it was 80.71°C in 20 years when the Yusung data were used. This could be explained with the higher geothermal gradient and lower thermal conductivity in Kosung site. 150 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 151 Figure 7. Principal stress contours at the canister for Yusung case, 200 years after emplacement. 80 Mises stress (MPa) 70 Kosung Yusung 60 50 40 30 20 10 0 1 2 3 4 6 5 Check Points 7 8 9 Figure 8. Comparison of von-Mises stress at different locations for Kosung and Yusung cases. – From the fact that the peak temperature around the repository is reached in several tens of years after the emplacement of canister, modeling up to several tens of years are good enough for the sensitivity analysis, which is mainly for investigation the relative influence of design parameters. – The stress distribution on the canister surface shows that the tensile stress is highest at lower part of the canister. The maximum principal stresses were tensile and it was up to 26 MPa and 28 MPa in Yusung and Kosung cases, respectively. Such a tensile stress may induce catastrophic failure of the outshell and thus needs to be carefully analyzed. – FLAC3D with FISH function could be successfully applied to evaluate the thermal–mechanical stability of the Korean preliminary repository design in deep underground rock. REFERENCES Berge, P.A. & Wang, H.F. 1999. Thermomechanical Effects on Permeability for a 3-D Model of YM Rock, Bernard Amadei et al. (eds), Proceedings of the 37th U.S. Rock Mech. Symp., Vail, Colorado, June 1999, Vol. 2: 729–749. Rotterdam: Balkema. 151 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-18.qxd 08/11/2003 20:28 PM Page 152 Desai, C.S. & Siriwardane, H.J. 1984. Constitutive laws for engineering materials, Prentice-Hall, Inc., Englewood Cliffs, NJ. Fairhurst, C. 1999. Rock Mechanics and Nuclear Waste Repositories. S. Saeb and C. Franke (eds), Proceedings of the International Workshop on the Rock Mechanics of Nuclear Waste Repositories, Vail, Colorado, June 1999: 1–43. Alexandria, Virginia: American Rock Mechanics Association. Fookes, P.G. 1995. Aggregates: a review of prediction and performance, Proceedings of STATS 21st Anniversary conference, London, UK: 91–170. Francke, C.T., Saeb, S. & Carrasco, R.C. 2001. ThreeDimensional Analysis of Nuclear Waste Disposal in Horizontal Boreholes, Proceedings of the 38th U.S. Rock Mechanics Symposium, Washington, D.C., July 2001, Vol. 1: 497–503. Lisse, The Netherlands: Swets & Zeitlinger B.V. Itasca Consulting Group, Inc. 1996. FLAC3D – Fast Lagrangian Analysis of Continua in Three-Dimensions, Ver 1.1 User’s Manual. Minneapolis, MN: Itasca. Johansson, E. & Hakala, M. 1995. Rock Mechanical Aspect on the Critical Depth for a KBS-3 Type Repository Based on Brittle Rock Strength Criterion Developed at URL in Canada, SKB, AR D-95-014, SKB. Kwon, Y.J., Kang, S.W. & Ha, J.Y. 2001. Mechanical structural stability analysis of spent nuclear fuel disposal canister under the internal/external pressure variation, KAERI/CM-440/2000, KAERI. Park, J.H., Kuh, J.E. & Kang, C.H. 1998. An examination of thermal analysis capability of FLAC3D on the near field of high level radioactive waste repository, KAERI/TR1187/98, KAERI Park, B.Y., Bae, D.S., Kim, C., Kim, K.S., Koh, Y.K. & Jeon, S.W. 2001. Evaluation of the Basic Mechanical and Thermal Properties of Deep Crystalline Rocks, KAERI/TR1828/2001, KAERI. Park, J.H., Kwon, S., Choi, J.W. & Kang, C.H. 2001. Sensitivity analysis on mechanical stability of the underground excavations for a high-level radioactive waste repository, KAERI/TR-1749/2001, KAERI. Patchet, S.J., Carrasco, R.C., Francke, C.T., Salari, R. & Saeb, S. 2001. Interaction Between Two Adjacent Panels at WIPP,” in Rock Mechanics in the National Interest, Proceedings of the 38th U.S. Rock Mechanics Symposium, Washington, D.C., July 2001), Vol.: 517–523. Lisse, The Netherlands: Swets & Zeitlinger B.V. Selvadurai, P.S. & Pang, S. 1990. Mechanics of the interaction between a nuclear waste disposal container and a buffer during discontinuous rock movement, Engineering Geology, Vol. 28: 405–417. Simmons, G.R. & Baumgartner, P. 1994. The disposal of Canada’s nuclear fuel waste: Engineering for a disposal facility, AECL Research, AECL-10715, AECL. 152 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-19.qxd 08/11/2003 20:28 PM Page 153 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Numerical simulation of radial bolting: Application to the Tartaiguille railway tunnel F. Laigle Electricité de France, Hydro Engineering Centre, France A. Saïtta Centre d’Etudes des Tunnels, Lyon, France ABSTRACT: In this paper, a numerical model is used to predict radial bolting performance in soft rocks crossed by the Tartaiguille TGV (high speed railway) tunnel located on the new TGV French southeast line. First, a brief description of results of field investigations and monitored displacements during excavation of this underground tunnel are done. The support system is constituted by shotcrete and radial grouted bars. Because of too high monitored displacements in some sections in these marls, a modification of the support system has been decided on site. This modification consists in an increasing of the density of bars in a specific zone. Back analysis confirmed the very strong efficiency of these additional bars in this case. However, conventional continuous modeling of the tunnel done up to now, strongly underestimated the real contribution of these extra bolts. Some new numerical simulations have been done using a constitutive model proposed by Laigle. This constitutive model integrated in FLAC focuses on the post-peak behavior of rocks. It’s based on a simple and physical description of the behavior of ground in this domain, with accurate evolutions of the cohesion and the dilatancy. This paper describes numerical results obtained using this new constitutive model applied to the Tartaiguille tunnel case. The significant monitored effect of additional grouted bars is well shown by this computation. 1 INTRODUCTION Bolting corresponding to grouted bars or friction bolts is a frequently used component in light support systems when driving underground galleries. This technique entails reinforcing a ring of ground around an excavation by introducing stiffer linear elements. This method, both effective and inexpensive, is the basis for methods of tunnel driving such as the new Austrian tunneling method. Bolt design has been based for a long time on empirical rules and on an optimization during the works themselves. At the present time, we notice a very clear evolution in design practices toward the frequent use of numerical methods to the detriment of empirical ones. However, there is considerable doubt about the ability of models now used to correctly simulate the effect of bolting. So we wanted to contribute to this reflection reporting, in an applicable way, the results obtained during the works on the Tartaiguille tunnel. The support system installed in this Aptian marls mainly consisted of grouted bars. In an initial stage, this study has made it possible to accurately quantify the influence of bolts on the deformations of the tunnel wall. Beginning with these conclusions, it then became possible to make a comparison with the results of a numerical modeling. In a second part of this paper, some numerical simulations are presented. These simulations are applied to the Tartaiguille tunnel. Using a new elastoplastic constitutive model developed by Laigle (2003), these simulations allow to find by computation the strong contribution of a grouted bars system on stability conditions of the tunnel. 2 EXPERIMENT FEEDBACK FROM THE TARTAIGUILLE TUNNEL 2.1 After the first “short” Paris-provinces lines, the highspeed train network has been extended in France and first provincial towns will soon be connected to each other. At the present time, the first line being completed 153 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands The new TGV southeast (high speed railway) line and the Tartaiguille tunnel 09069-19.qxd 08/11/2003 20:28 PM Page 154 stresses. This ratio has been estimated from 1.2 to 1.7 while the project had concluded on lower values. The major stress is horizontal which explains the crown breaking mechanism of the shotcrete shell, subjected to a lateral thrust. The support system had to be modified to control strains. The improvements, which made it possible to significantly reduce convergences, are an increase in the density of the radial bolting and their integral use at the face. Convergences were brought to a tolerable threshold for a double density of bolts when compared with the initial plan. Our study of radial bolting is restricted to the geological formation composed of Aptian marls. This is because of two advantages presented by this facies, one being the homogeneity of the geology and the other being the presence of a zone, the support system of which is composed only of shotcrete and bolts. This geological description comes from the geological and geotechnical wrap-up paper, prepared by the engineering firm Coyne and Bellier (1995). Figure 1. Layout of the Tartaiguille tunnel. 2.2 Figure 2. Cross section in Aptian marls. is that of the TGV Southeast which should make it possible to go from Paris to Marseilles in three hours. After Valence, the Tartaiguille tunnel is the first of underground structures in the southerly direction. It’s a 2340 m long tunnel which has been driven from the north and south extremities (Fig. 1). Excavations began in February 1996. In the Aptian marl geological formation, digging method retained was the upper half-bench cut method (Fig. 2). The support system is composed of radial bolts, associated with shotcrete and sometimes with yielding arches. The tunnel driving cycle was broken down into the excavation of the upper half-section and the laying of the support system. Then, about a hundred meters in back of the working face, a second station excavated the lower half-section. Finally, further in back of the face, a reinforced concrete invert, then the final concrete lining was poured. From the beginning of the driving in the Aptian marls, major convergences of the tunnel wall were measured. The alert thresholds were quickly exceeded and the phenomenon grew with the passing of the lower section. Strains of the wall resulted in a loading of the shotcrete shell that was greater than its breaking limit and a consistent cracking in crown, which generated safety problems for the worksite. Complementary tests then made it possible to estimate the ratio between horizontal and vertical initial For our study, the support zone is 335 meters length within which the support system is composed only of shotcrete and radial bolts composed of grouted bars 4 meters in length. The reinforcement of the bolting of the upper half-section takes place in several phases: – Zone 1: One ring of bars every 2 meters (Fig. 2). – Zone 2: 2 sets of bars inserted at the springing of sidewalls. – Zone 3: Return to the initial density (idem zone 1). – Zone 4: Re-establishing interposed ring (idem zone 2). In order to monitor the evolution of the ground and the efficiency of the support system, measurements of wall displacements were performed. Five measurement targets were used for each section, a target A, positioned at the crown, two targets B and C at the spring lines of the side walls of the upper half-section, and two targets D and E at the side walls of the lower half-section (Charmetton 2001). 2.3 Results of the measurements The effectiveness of the support system during the driving was essentially monitored beginning with two values out of the three targets, which comprise each testing section. These are the measurements of the leveling of point A and of the convergence of cord BC. The measurements sections called S07, S08, S09 and S10 were included in the Zone 1. The sections S11, S12, S13, S14, S15, S16 and S17 were in the Zone 2. The sections S18, S19 and S20 measured displacements of the Zone 3 while S21 and S22 were inside the Zone 4. 154 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Study zone and measures carried out 09069-19.qxd 08/11/2003 20:28 PM Page 155 parameters (excavation speeds, steps of advancement, cover height…) varied, but without our being able to establish direct links between their evolution and that of the two groups of curves of convergences. The alternation of the two bolt densities along our study zone is an argument which confirms the role of the bolts because this assures that the evolution of the convergences does not result from a modification of the ground. Number of days 0 50 100 150 200 250 300 0 S07 -20 S08 Convergences (mm) S09 S10 -40 S11 S12 S13 -60 S14 S15 S16 -80 S17 3 NUMERICAL SIMULATIONS S18 -100 S19 S20 The aim of these numerical simulations was to simulate in the framework of usual continuous modeling the strong effect of a grouted bars system on the stability conditions of the Tartaiguille tunnel. The particularity of this work is to use a new constitutive model developed by Laigle. These numerical simulations are done using the elastoplastic version. A more general elastoplastic–viscoplastic version has been developed and presented in another paper (2003). Following phenomena are studied in the framework of these simulations: S21 -120 S22 -140 Figure 3. Measurements of convergence of cords BC. 70 Convergences (mm) -46% : Average convergence 60 Zone 3 50 40 – What would be the behavior of the tunnel with an under-estimated support and what was the potential failure mechanism? – Was a support system needed? – What is the effect of a delay in the installation of the support system? – What is the effect of a local failure of the shotcrete layer? -37% Zone 1 Zone 4 30 Zone 2 20 S17 S21 S22 0 S07 S08 S09 S10 S11 S12 S13 S14 S15 S16 S18 S19 S20 10 3.1 Measurement sections Figure 4. Measurements of the convergences at 30 m from the face. So we interested ourselves, for each of the measurement sections, in these values, but also in the context in which they were obtained, that is, everything that could have an influence on the results of the measurements. The following graphs present the results, which come directly from the worksite of the convergences of the cords BC for sections S07 to S22 (Fig. 3). 2.4 Study of the results of the upper half-section The calculation of the average convergence at 30 m for each of the zones (see Fig. 4) shows a reduction of 37% between Zone 1 and Zone 2, and 46% between Zone 3 and Zone 4. The increase in the number of bolts at the sidewalls (from 1 bolt/3.3 m2 to 1 bolt/1.6 m2) therefore resulted in a reduction of at least 37% in short-term horizontal convergences of the BC cord. We may show that this reduction is directly due to the bolting. All the other 155 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands The constitutive model In general, at least for deep and no-urban tunnels, the goal of a support system composed with shotcrete and grouted bars is to prevent mechanical failure within a sufficient safety level. In cohesive rocks, which can be considered as a continuous material, this failure is associated with the development of a fracturing process induced by the excavation. So, it seems necessary to focus the simulation of the mechanical behavior both on the pre-peak behavior and the post-peak behavior. This new constitutive model has been written in the framework of the elastoplastic theory The expression of the yield surface corresponds to the generalized Hoek and Brown criterion. This surface is governed by 4 parameters, which are the unconfined compressive strength and three other parameters “m”, “s” and “a”. These 3 last parameters change in accordance with an internal variable p, which is the irreversible shear strain defined below: (1) Deviatoric stress 09069-19.qxd 08/11/2003 20:28 PM Page 156 (2) Several thresholds for the yield surface are proposed. Some specific hardening laws are suggested for each parameter “m”, “s” and “a” allowing to describe the evolution of the yield surface from one threshold to another (Laigle 2003). In the softening domain, the negative hardening is assumed to be divided into three phases: Domain 3: Fissured rock in post-peak domain Domain 2: Fissured rock in pre-peak domain Volumetric strain Domain 5: Fractured rock in a residual state Domain 4: Fractured rock Axial strain Figure 5. Schematic behavior of a rock sample during a triaxial test. 5 Maximal principal stress (MPa) – The first phase of softening corresponds to a deterioration of the rock’s cementation illustrated by a progressive disappearance of the cohesion at the macroscopic scale. This first phase is associated with an increasing of the dilatancy. – The second phase corresponds to the shear of an induced fracture. It’s associated with a decreasing of the dilatancy at the macroscopic scale. – Finally, the last domain corresponds to a purely frictional behavior, which defines the residual strength. The shear occurs without any volumetric strain. Figure 5 shows schematically various domains describing the physical state of a rock sample under a mechanical triaxial loading. In the domain 1, the behavior is elastic linear. Figure 6 presents thresholds retained for the Aptian marl of Tartaiguille. Major mechanical properties are as following: Domain1: Intact rock Axial strain – A first threshold corresponds to the damage criterion. This criterion is assimilated to the crack initiation, so to the beginning of the dilatancy. – The second threshold corresponds to the peak strength criterion. – The third threshold characterizes the strength of a damage rock sample crossed by an induced shear fracture. In these conditions, cohesion of the rock at a macroscopic scale is assumed to be null. – The last criterion corresponds to the residual strength criterion, which is reached at very large shear strains. 4.5 4 ual id Res 3.5 3 Peak criterion 2 rit kc 2.5 d ure roc age Dam on eri n erio crit rion crite ct Fra 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 Minimal principal stress (MPa) Figure 6. Threshold criteria for the Aptian marl. This constitutive model has been integrated in the EDF’s local version of FLAC V3.4, using FISH procedures. decreasing initial internal stresses applied to the tunnel perimeter. Excavation is simulated in two phases: the vault and the bench. Figure 7 shows the state of the rock mass at a decrease of 97.8% of these stresses during the vault excavation (100% corresponds to the end of excavation process of the vault). We may observe an important damage zone near the foot of the tunnel, progressing behind sidewalls towards the roof. Without any support, this mechanism will generate a global instability of the gallery in the short term. 3.2 3.3 – UCS 0.85 MPa – Young’s modulus: E 1000 MPa – Poisson ratio: 0.36 Behavior of the tunnel without support A first simulation has been done without considering any support system. The excavation is simulated by A second simulation has been done considering the initial support system (Zone 1, ring of grouted bars every 156 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Behavior of the tunnel with initial support 09069-19.qxd 08/11/2003 20:28 PM Page 157 Horizontal convergence of the tunnel (mm) Time (days) Fissured rock (Domain 2) Rock in tension Fractured rock (Domain 4) Fissured rock in postpeak domain (Domain 3) 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 With initial support Failure Without any support Figure 9. Horizontal convergence of the vault without and with an initial support system. Intact rock (Domain 1) Deconfinement: 97,8 % Development of the induced fractured zone Figure 7. Physical state of rock without any support. Fractured rock (Domain 4) Strains of bolts Fractured rock ( Domain 4) Fissured rock in the pre-peak domain (Domain 2) Fissured rock Fissured rock in ( Domain 3) the pre-peak domain ( Domain 2) Intact rock (Domain 1) Figure 10. Physical state of rock during the bench excavation and failure mechanism. Intact rock (Domain1) Figure 8. Physical state of rock at the end of the upper-half excavation. 2 meters). Bars are simulated using structural cable elements and a shotcrete layer using structural beam elements. These bars and beams are installed after a deconfinement of 70%, so approximately in the first 2 meters behind the face heading. Bars are linked to some beam nodes with the goal to simulate face plates. Despite this, we will observe that the maximum tensile stain is located in the ground and not near the wall (Fig. 8) With this support, the stability of the tunnel during the upper-half excavation can be theoretically justified, even if a damage zone exists near tunnel the foot of the tunnel (Fig. 8). Figure 9 presents horizontal convergences versus time, assuming an advancing rate of 2 m/day. These simulations don’t take into account the time. Convergences are drawn versus time only to compare with monitored values. Without any modification of this initial support, the excavation of the bench can’t be finalized. Because of the bench excavation, the fracturing phenomenon takes off again toward the roof. The initial support is not sufficient to stop the development of this damage zone. The stability cannot be demonstrated from this numerical simulation (Fig. 10). These results seem in accordance with the site engineer’s decision to adapt the support system. Considering high measured displacements and the development of a local failure of the shotcrete layer at the roof, he decided to reinforce this initial support system. 157 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Fissured rock in the post-peak domain (Domain 3) 09069-19.qxd 08/11/2003 20:28 PM Page 158 Horizontal convergence (mm) 0 Fractured rock (Domain 4) Intact rock (Domain 1) Rock at residual state (Domain 5) Fissured rock in post-peak domain (Domain 3) Fissured rock in the pre-peak domain (Domain 2) Figure 11. Physical state of rock at the end of the bench excavation, considering a reinforced support system. 3.4 Behavior of the tunnel with a reinforced support The reinforcement of the support system corresponds to the installation of additional bolts on sidewalls of the tunnel. In this zone, the bolting density is double. In the framework of numerical simulations, several patterns are considered: – 2 additional grouted bars are installed on each sidewall at 6 meters behind the face heading (90% of the vault deconfinement). – 1 additional grouted bar is installed on each side wall as soon as possible, about 2 meters from the face heading (70% of the vault deconfinement). In these 2 last cases, it appears that theoretically, the stability is demonstrated both during the vault and the bench excavation. The local increase of the bolting density prevents the development of induced fractures, which appear on the sidewall during the vault excavation. The precise location of these additional bolts is essential. We understand that an increase of the bolting density on the roof is not useful for the goal to delay and stop the observed mechanics on site. A second interesting aspect is the effect of the time at which additional bolts are installed. Figure 12 shows that the final horizontal convergence is smaller with only one additional grouted bar installed earlier rather than 2 bars added later. These simulations highlight very well what project engineers already knew but which has never been Time (days) 10 20 30 40 50 Excavation of vault 60 70 80 Excavation of bench -5 -10 -15 Without additional bolt -20 -25 With 1 additional bolt at 70% With 2 additional bolts at 90% Failure Figure 12. Influence of reinforced bolting pattern of horizontal convergences. shown by numerical computations. With the aim of reinforcing the rock mass, the grouted bars system has to be installed as early as possible, before any critical increasing of monitoring displacements. This is because the goal of this type of support is to protect and help the rock to keep a sufficient shear strength to ensure the global stability. 3.5 Effect of a local failure of the shotcrete lining During excavation of the vault, a crack appeared and developed in the shotcrete at the crown of the Tartaiguille tunnel. This same phenomenon has been observed during excavation of one large cavern of the CERN-LHC project in Geneva (Laigle 2002). Depending on the support design, a shotcrete failure could be critical for the global stability of the tunnel. However, in these two previous cases, the support has been designed with the consideration that one major component is the grouted bars system and not only the shotcrete layer. If the shotcrete keeps an essential function, it can’t be assimilated in these cases to a continuous shell like in the SCL approach. In the case of Tartaiguille tunnels, it was interesting to know if this crack in the shotcrete was really critical from a global stability point of view. A simulation has been done, considering the reinforced bolting system on sidewalls. Cracking of the shotcrete layer has been simulated by deleting some structural elements near the crown. This deletion is done at 90% of the vault deconfinement. Figure 13 shows the physical state of the rock mass at the end of the tunnel excavation. The local failure of the shotcrete generates new damage and a fractured zone above the tunnel roof. The growth of this new fractured zone is stopped by grouted bars. Figure 14 presents the evolution of the horizontal convergence in accordance with time, with and without failure of the shotcrete. An increase of displacement appears at the time of the failure but a new stable configuration is reached after that. 158 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0 09069-19.qxd 08/11/2003 20:28 PM Page 159 Time (days) 0 0 10 20 30 40 50 60 70 80 Horizontal convergence (mm) -5 Crack in the shotcrete Fractured rock (Domain 4) Fissured rock in the post-peak domain (Domain 3) Horizontal convergence (mm) 30 40 50 60 70 80 -5 -10 Without failure of the shotcrete layer -15 With failure of the shotcrete layer -20 -25 -30 -35 -40 Figure 14. Influence of a local shotcrete failure on horizontal convergences. 3.6 Justification of the length of bars Some computations have been done considering several lengths for grouted bars. Figure 15 shows horizontal convergences versus time, in accordance with these lengths. If the length is greater than 4 m, global stability is assured. On the contrary, if this length is 2 m or less, a stable configuration can’t be obtained and stability of the tunnel can not be justified during the bench excavation. From these results, we may conclude that there is an optimal length for grouted bars, depending on the potential failure mechanism of the tunnel during the excavation process. These numerical results confirm usual formulas, which provide an estimation of the -35 -40 Length: 2m Failure L 2 0.2D 4.7 m L 0.30D 4.0 m where D is the tunnel span in meters. 4 CONCLUSIONS The study of monitored convergences in the Aptian marls of the Tartaiguille Tunnel has made it possible to approach quantitatively the effect of bolting on the structure’s stability. We were able to confirm the very considerable efficiency that a few extra bolts bring to a mass of non-fractured soft rock. This back analysis has made it possible to quantify the effect of reinforcing the sidewalls on the reduction of convergences in the Tartaiguille tunnel. In the framework of a back analysis, some numerical computations have been done considering a new constitutive model well adapted to underground engineering expectations. Goals of these simulations were to find with a suited numerical tool major behaviors observed and monitored during the Tartaiguille tunnel digging. These simulations allow us to identify: – The potential major failure mechanism of the tunnel. The knowledge of this mechanism is essential both during the design phase and during excavation process. – The significant efficiency of a grouted bar system on tunnel stability conditions. – The effect of a local reinforcement of the bolting system on displacements and safety level during the digging. – The limited effect of a local shotcrete failure if the grouted bar system is sufficient and if the stability is not only ensured by a shotcrete shell. 159 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands -30 length in accordance with the span of the tunnel: Time (days) 20 Length 8m Length: 6m Length: 4m -25 Figure 15. Influence of bolt lengths on horizontal convergences. 0 10 -20 -50 Figure 13. Physical state of rock at the end of the bench excavation, considering a local failure of the shotcrete. 0 -15 -45 Fissured rock in the pre-peak domain (Domain 2) Intact rock (Domain 1) -10 09069-19.qxd 08/11/2003 20:28 PM Page 160 REFERENCES Charmetton, S. 2001. Reinforcement des parois d’un tunnel par boulons expansifs – retour d’expérience et étude numérique. Ecole Centrale de Lyon. Ph.D. Thesis, 2001 (In French). Coyne and Bellier. 1995. Geological and geotechnical wrapup paper. Mediterranean TGV. Tartaiguille tunnel (In French). Laigle, F. 2001. CERN-LHC Project – Design and excavation of Large-Span Caverns at point 1. Proc. of the IRSM regional Symposium Euorock 2001 – Rock Mechanics – a challenge for Society – Espoo – Särkkä & Eloranda (eds). Balkema Publishers. Laigle, F. 2003. Modélisation rhéologique des roches adaptée à la conception des ouvrages souterrains. Ph.D. Ecole Centrale de Lyon, in prep. Laigle, F. 2003. A new viscoplastic model for rocks – Application to the Mine-by-test of AECL-URL. Proc. Intern. Symp., Sudbury, Canada. To be published. 160 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-20.qxd 08/11/2003 20:29 PM Page 161 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Recent experiences of the prediction of tunneling induced ground movements C. Pound & J.P. Beveridge Mott MacDonald Ltd, Croydon, Surrey, UK ABSTRACT: The ability to predict ground movements caused by tunneling is becoming increasingly important as more tunnels are constructed in urban areas. It is generally recognized that the ground surface settlement trough above a tunnel is well represented by a gaussian curve. Data is available from many projects around the world, which provides guidance on the values to use in the gaussian curve. However, if novel forms of tunneling are used, if unusual ground conditions are present or if horizontal or subsurface ground movement predictions are required, then these empirical methods are not suitable. This paper presents the results of a suite of numerical analyses carried out to identify the most appropriate soil model to use for the prediction of surface settlement troughs. As shown by many other authors linear elastic or linear elastic perfectly plastic soil models are unsuited to the prediction of realistic surface settlements. Most of the analyses carried out in the study predict a settlement trough that is wider than observed despite modifications to the size of the model, the boundary conditions, the in-situ stress conditions and the initial small strain stiffness. The constitutive model that predicts the most realistic settlement trough was a non-linear anisotropic soil model with a higher horizontal than vertical stiffness. The soil non-linearity was based on the approach suggested by Jardine but modified for anisotropy. The shape of the trough was found to be sensitive to the value of the vertical to horizontal shear modulus and the ratio of the horizontal and vertical Young’s moduli. 1 INTRODUCTION The prediction of ground movements is very important during the planning and design phase of any tunnel construction project in an urban area. This prediction is used to identify the risk of damage to adjacent structures and utilities and to assess whether the proposed construction method needs to be modified. It can also be used to highlight where mitigation measures may be necessary in advance or during tunnel construction. Surface settlements caused by tunneling are normally assessed using empirical methods (O’Reilly & New 1982, Macklin 1999). The method was developed from review of settlement data from a large number of tunneling projects around the world. However, the method is difficult to apply when the ground conditions or construction method is unusual or where more than one tunnel is present. Many attempts have been made to use numerical methods to predict ground movements due to tunneling but almost without exception the analyses have predicted unrealistic surface settlement troughs. This paper presents the results of a numerical modeling study to identify the factors affecting the prediction of surface settlements above tunnels. 2 NUMERICAL ANALYSIS Numerical analysis is often used to predict the loads on tunnel linings using a variety of finite element and finite difference programs. However, unless the ground movements are predicted accurately it is difficult to be confident that the predicted ground load acting on the linings is correct. The prediction of surface settlement troughs caused by tunneling is difficult and even the adoption of sophisticated constitutive models for the soil rarely results in a realistic surface settlement trough. The following sections present a series of analyses carried out in an attempt to match the surface settlement troughs observed above the bored tunnels on the Heathrow Express project (Pound & Beveridge, in press). A section of single bored tunnel was considered where the volume loss was typically 0.8% with a 161 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-20.qxd 08/11/2003 20:29 PM Page 162 Table 1. Basic geotechnical parameters. Parameter Terrace gravel London clay Bulk Unit Weight (kN/m3) Porosity (%) Cohesion (kPa) Friction (°) 20.0 35 0 38 20.0 50 37.5 0 0 0.5 Earth pressure at rest, ko 1.5 1 2 2.5 3 10 Depth below ground level (m) 6z trough width factor of 0.5. The back analysis was carried out using the finite difference program FLAC. The ground conditions comprise 4.4 m of Terrace Gravel overlying London Clay. The base of the London Clay is at a depth of about 60 m below ground level. The water table was taken to be 2.1 m above the base of the Terrace Gravel and initial water pressures were taken to be hydrostatic through the Terrace Gravel and the London Clay. The tunnel axis was taken to be at a depth of 22.5 m below ground level and the tunnel diameter was taken as 6.115 m. The basic geotechnical parameters used in the analysis are given in Table 1. The strength properties for the London Clay represent the fissured undrained shear strength and the values vary with depth, z, below ground level. The stiffness adopted for the Terrace Gravel and London Clay is described in detail. The variation of coefficient of earth pressure at rest, ko, with depth is shown in Figure 1. The values in the London Clay were derived from assessment of the results of self-boring pressuremeter tests and pore suction measurements made on undisturbed samples. Only the short-term ground movements were modeled. Throughout the analysis the response of the Terrace Gravel was taken as drained whereas the response of the London Clay was taken to be undrained. This was achieved by setting the bulk modulus of the pore fluid to be zero in the Terrace Gravel and 2 GPa in the London Clay. The mesh for the modeling is shown in Figure 2 and comprises over 5000 elements. Advantage was taken of symmetry about a vertical plane through the tunnel axis. The far boundary is located 90 m from the tunnel centerline, which represents a distance of 4 times the tunnel depth. The base of the model was located at the base of the London Clay and was fixed against movement in both directions while in most analyses the vertical boundaries were fixed only in the horizontal directions. Tunnel excavation was modeled by first replacing the elements within the profile of the tunnel by equivalent grid-point forces and then by progressively reducing these grid-point forces. The volume loss was determined by integrating the vertical displacements at the ground surface. Once a volume loss of 0.8% was achieved the segmental concrete lining was installed and the remaining grid-point forces removed. The segmental lining was taken to have the 20 30 40 50 Test data Mayne and Kulhawy 60 Figure 1. K0 profile. Figure 2. Mesh. Table 2. Segmental lining properties. Young’s modulus (Gpa) Thickness (mm) Moment of inertia (m4) 20.0 225 3.1 10 4 properties given in Table 2. The moment of inertia was reduced to allow for the number of joints in the lining in accordance with Muir-Wood (1975). Soils are known to have a non-linear stress–strain behavior prior to peak with a high initial tangent stiffness at very small strains and reducing stiffness with strain. One set of equations often used to represent this decay of stiffness with strain was developed by 162 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0 09069-20.qxd 08/11/2003 20:29 PM Page 163 Jardine et al. (1986). The approximate tangential version of these equations were presented by Potts & Zdravkovic (1999) and are given below: Table 3. Jardine constants. Parameter Terrace gravel London clay A B C R S T ! smin smax vmin vmax 1104 1035 5.00E-06 275 225 2.00E-05 0.974 0.94 1.044 0.98 8.80E-06 3.50E-03 2.10E-05 2.00E-03 1260 1143 1.00E-06 618 570 1.00E-05 1.335 0.617 2.069 0.42 1.40E-05 2.00E-03 1.00E-04 2.00E-03 Distance from centre-line (m) and p is the current mean effective stress. Throughout the analysis the stiffness was continually updated. Up to a specified minimum strain (smin or vmin), the stiffness varies only with p , but thereafter the stiffness depends both on the current strain () and the mean effective stress (p ). It is considered that these equations lead to unrealistically low elastic moduli at very low stresses and therefore the minimum mean effective stress used in calculating the elastic moduli was 50 kPa. The constants used in the Jardine equations are given in Table 3. 2.1 Linear elastic analyses Initial analyses were carried out using linear elastic and linear elastic perfectly plastic soil models (analyses t1 and t2). The elastic moduli were taken as multiples of the mean effective stress in order to give a load in the lining of between 35 and 40% of overburden which is considered to be a typical short-term load on a bored tunnel lining in London Clay. To achieve this criterion the elastic model was taken to be 20% of the small strain stiffness for the linear elastic model and 35% of the small strain stiffness for the plastic analysis. Thus for the elastoplastic analysis the shear modulus for the London Clay was given by the following equation. Settlement / Maximum Settlement . where s is a generalized shear strain related to the octahedral shear strain, oct, by the following equation: 10 20 30 40 50 0.2 0.4 0.6 0.8 t1 t2 Gauss Curve 1 1.2 Figure 3. Linear elastic/elastoplastic analyses. This is significantly higher than is conventionally used in tunnel analyses even in overconsolidated materials. Figure 3 shows the surface settlement troughs for these two analyses and the corresponding gaussian curve for a volume loss of 0.8% and a trough width factor, K, of 0.5. The surface settlement troughs from the two analyses are clearly unrepresentative of the observed ground settlement showing a maximum settlement around 15 m from the tunnel centerline. The analysis with the elastoplastic model is worse because of the ground yielding that is predicted between the tunnel crown and the ground surface resulting from the high in-situ horizontal stresses. These results clearly show the limitations of using linear elastic ground models for the prediction of ground movement around tunnels. 2.2 Non-linear elastic analyses Figure 4 shows the surface settlement trough when the non-linear behavior given by the Jardine equations is adopted in the analysis (t3). The maximum 163 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0 0 09069-20.qxd 08/11/2003 20:29 PM Page 164 Distance from centre-line (m) 0 10 20 30 40 50 Settlement / Centre-line Settlement . Settlement / Centre-line Settlement . Distance from centre-line (m) 0 0.2 0.4 0.6 0.8 1 t1 t3 t4 Gauss Curve 1.2 0 0 10 20 30 40 50 0.2 0.4 0.6 0.8 t3 t5 t6 t7 Gauss Curve 1 1.2 1.4 Figure 4. Non-linear elastic analyses. Figure 5. Small strain model. settlement is still not located on the tunnel centerline, but is offset by about 10 m. The trough is significantly narrower than the linear elastic case, but is still much broader than the gaussian curve. To investigate the influence of the high horizontal stress on the shape of the settlement trough, an analysis was run using a k0 profile based on the approach suggested by Mayne & Kulhawy (1982). The k0 profile assumed that 170 m of overburden had been removed from the top of the London Clay prior to the deposition of the Terrace Gravel. The k0 was taken as 0.4 in the Terrace Gravel. The shape of the k0 profile is given in Figure 1 and shows lower k0 values particularly in the top ten meters of the London Clay than the profile used in analysis t3. The resulting settlement trough from analysis t4 is shown in Figure 4. Although the low point of the settlement trough is nearer to the tunnel centerline and the trough is generally narrower, the overall shape of the settlement trough is only slightly different. To consider the effect that fixity conditions on the far boundary have on the shape of the settlement trough, analyses were run with the far boundary fixed both horizontally and vertically and also with a stress boundary condition. Neither analysis gave an improved shape of settlement trough. Analyses were also carried out with wider meshes to see if a boundary width of 4 tunnel depths was inadequate. Analyses were carried out with a mesh width of 150 m and 1000 m. The effect of an increased mesh width was minor with a small reduction in the settlement at 50 m from the tunnel centerline, but a corresponding increase in the settlement 5 m from the tunnel centerline. Analyses were carried out to investigate the effect of modifying the shape of the non-linear model and the results are presented in Figure 5. In the first analysis (t5) the stiffness was increased by 50% at all strains compared to the model prediction. In the next analysis (t6) the strain limit for the plateau region of the model was extended to a higher strain level. In both of these analyses the increase in the soil stiffness made the shape of the settlement trough worse. In the third analysis (t7) the small strain stiffness was increased by 50%, but the shape of the stress–strain curve was the same after the end of the initial plateau region as in analysis t3. The modifications to the small strain stiffness had only a modest influence on the shape of the settlement trough. 2.3 A number of authors have indicated that only with an anisotropic soil model can a realistic shaped settlement trough be obtained (Simpson et al. 1996, Addenbrooke et al. 1997). There is good evidence that the behavior of London Clay is anisotropic with a higher horizontal than vertical Young’s modulus (Bishop et al. 1965). Data also exists for other overconsolidated clays (Lings et al. 2000). The anisotropic elastic model was modified to allow input of non-linear elastic behavior. No anisotropic elastoplastic soil model currently exists in FLAC, however with a volume loss of only 0.8%, the strains in the ground surrounding the tunnel are only sufficient for very local plastic yielding of the ground to occur and therefore there should be only a small error in the adoption of an elastic model. In the absence of any definitive anisotropic constitutive soil model for the London Clay, the basic Jardine equation was modified as follows: where X is defined as above. The values of the constants in the above equations are given in Table 3. 164 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Anisotropic soil model 09069-20.qxd 08/11/2003 20:29 PM Page 165 Distance from centre-line (m) 0 5 10 15 20 25 30 35 40 45 50 Settlement / Centre-line Settlement . Settlement / Centre-line Settlement . Distance from centre-line (m) 0 0.2 0.4 0.6 0.8 t8 t9 t10 Gauss Curve 1 1.2 Figure 6. Anisotropic model. 0 0 5 10 15 20 25 30 35 40 45 50 0.2 0.4 0.6 0.8 t9 t11 t12 t13 Gauss Curve 1 1.2 Figure 7. Effect of varying shear modulus. The small strain stiffnesses are slightly lower as they now relate to the vertical stiffness rather than an isotropic stiffness given previously. The adoption of the equations given above relate the variation of the stiffnesses only to shear strain and not to volumetric strain. This model is therefore only suitable for modeling shear deformations and would need to be modified to consider swelling or consolidation. The two independent Poisson’s ratios were taken as follows: Settlement / Centre-line Settlement . Distance from centre-line (m) 0 0 5 10 15 20 25 30 35 40 45 50 0.2 0.4 0.6 0.8 1 t9 t12 t14 t15 Gauss Curve 1.2 Figure 8. Effect of varying Poisson’s ratio. The analysis was first run with a pseudo-isotropic analysis with both Emul and Gmul set to 1.0. The results of the analysis are shown in Figure 6 (analysis t8). The shape of the curve is somewhat improved compared to the previous isotropic analyses. Two further analyses (t9 and t10) were carried out with higher horizontal stiffnesses by setting Emul to 1.6 and 2.0. The settlement troughs are also given in Figure 6. The shape of the settlement trough is significantly improved as the horizontal to vertical stiffness ratio is increased. The data by Bishop et al. (1965) and by Atkinson (1975) suggested that for London Clay the ratio of horizontal to vertical Young’s modulus is around 1.6. To investigate the effect of the value of the shear modulus on the shape of the settlement trough, a set of analyses were carried out with the value of Emul set to 1.6 and with values of Gmul of 0.5, 0.8 and 1.6 (Analyses t11, t12, t13). The results are compared against the analysis (t9) with a Gmul of 1.0 in Figure 7. The shape of the settlement trough is very sensitive to the value of the shear modulus. Generally as the shear modulus is increased the width of the settlement trough is also increased. The best fit to the middle part of the settlement trough is achieved when the shear modulus is only one third of the vertical Young’s modulus. However, this shape of settlement trough can also be achieved by setting the horizontal Young’s modulus to be equal to twice the vertical Young’s modulus. The settlement at the edges of the settlement trough is much greater than that suggested by the gaussian curve and is also greater than observed in practice. The settlement towards the boundary of the mesh results from the horizontal ground movements and a corresponding Poisson’s ratio effect. To reduce the vertical strains due to horizontal displacements, the Poisson’s ratio in the vertical plane was set to zero. Analyses t9 and t12 were repeated with a Poisson’s ratio of 0.0 as analyses t14 and t15 and the results are shown in Figure 8. There is a significant narrowing of the trough as well as a significant reduction in the farfield settlement. However, the settlement 50 m from the tunnel centerline is still 10% of the centerline settlement. Due to the assumed undrained response of the London Clay any horizontal movement will result in an equivalent vertical settlement. To prevent this surface settlement would require a volumetric change in the soil and thus a drained soil response. To investigate this effect, analysis t14 was repeated as analysis t16 with the soil more than 20 m from the tunnel centerline 165 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-20.qxd 08/11/2003 20:29 PM Page 166 Distance from centre-line (m) Settlement / Centre-line Settlement . 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 t14 t16 t17 t18 Gauss Curve 1.2 Figure 9. Effect of varying the bulk modulus of water. assumed to be fully drained. The predicted settlement trough is shown in Figure 9. As can be seen, the shape of the surface settlement trough predicted by this analysis is very similar to the shape of a gaussian distribution with only a very small surface settlement 50 m from the tunnel centerline. It is considered that the shape of this predicted curve is probably as good a match to observed tunnel settlements as a gaussian curve. 3 DISCUSSION The results of the series of analyses show that only with the adoption of an anisotropic model can a realistic settlement trough be predicted. This is similar to results found by others (Simpson et al. 1996). Threedimensional numerical analyses, which model the full construction sequence, have also been carried out and these also demonstrate that an isotropic soil model leads to wider surface settlement troughs than those observed in practice. Adoption of an anisotropic soil model results in surface settlement troughs very similar to those predicted by the two-dimensional analyses. The anisotropic behavior of stiff overconsolidated clays can be explained on the basis of the preferential alignment of clay particles. There is also some data from field and laboratory testing to indicate that stiff overconsolidated clays are anisotropic. However sands have an even narrower settlement trough than those of clays as indicated by the trough width factor normally adopted. There is less justification for the adoption of an anisotropic soil model for sands from field and laboratory testing data. Sands of course will not respond in an undrained manner during tunneling and it may be that the different pattern of groundwater pressures around the tunnel during excavation will result in a different pattern of surface settlements predicted by the isotropic soil model. The stress–strain behavior of most rocks is controlled not by the elastic behavior of the intact material but by the orientation and properties of the discontinuities. The presence of the discontinuity sets will inevitably impose an anisotropic response to the mass behavior of the rock which it is logical to suppose will influence the shape of the settlement trough. It is also reasonable to assume that where ground settlements are large, slip on discontinuities will occur. This could explain the narrow settlement troughs observed over many tunnels in rock. The in-situ stress conditions could also influence this behavior. The only way found to prevent the prediction of significant settlements at the boundary of the model was to assume drained behavior for the soil at a distance from the tunnel. The ratio of bulk stiffness of the water to that of the soil controls the drained or undrained behavior even where there is no groundwater flow. Because of the increased strains near to the tunnel and the formulation of the anisotropic model, the bulk stiffness of the soil model nearer to the tunnel is lower than that further from the tunnel. With a bulk stiffness for the water of 2 GPa, the water is at least one order of magnitude stiffer than even the small strain stiffness of the soil. Reducing the bulk modulus of the water has the effect of making the response of the soil apparently partially drained far from the tunnel and essentially undrained near to the tunnel. The effect was found to be modest with a bulk modulus of 0.2 GPa (analysis t17) but resulted in a realistic shaped settlement trough with a bulk modulus of 0.02 GPa (analysis t18). The results of these two analyses are plotted in Figure 9. Unfortunately a bulk modulus for the water of 0.02 GPa, is unrealistically small. It is possible that the apparent drained response of the ground far from the tunnel is due to a combination of a lower bulk modulus of water, a higher bulk stiffness of the soil than currently assumed and the effect of some drainage of the soil due to the slow small stress changes occurring in the soil far from the tunnel. 4 CONCLUSIONS The numerical analyses show that traditional linear elastic analyses with or without a yield criterion cannot predict settlement troughs similar to those observed. Even non-linear elastoplastic analyses with isotropic soil stiffnesses overpredict the width of the surface settlement trough. Only by adopting a nonlinear anisotropic elastic soil model can surface settlement troughs similar to those observed be predicted. To reduce the predicted settlements at the edges of the trough it is necessary to assume partially drained behavior of the soil. It is suggested that this could result from a lower bulk stiffness of the water. 166 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-20.qxd 08/11/2003 20:29 PM Page 167 REFERENCES Addenbrooke, T.I., Potts, D.M. & Puzrin, A.M. 1997. The influence of pre-failure soil stiffness on the numerical analysis of tunnel construction. Géotechnique 47(3): 693–712. Atkinson, J.H. 1975. Anisotropic elastic deformations in laboratory tests on undisturbed London Clay Géotechnique 25(2): 357–374. Bishop, A.W., Webb, D.L. & Lewin, P.I. 1965. Undisturbed samples of London Clay from the Ashford Common Shaft: strength-strain relationships. Géotechnique 15(1): 1–31. Jardine, R.J., Potts, D.M., Fourie, A.B. & Burland, J.B. 1986. Studies of the influence of non-linear stress–strain characteristics in soil-structure interaction. Géotechnique 36(3): 377–396. Lings, M., Pennington, L., Nash, D.S. & Poisson, D.F.T. 2000. Anisotropic stiffness parameters and their measurements in a stiff natural clay. Géotechnique 50(2): 109–125. Macklin, S.R. 1999. The prediction of volume loss due to tunneling in overconsolidated clay based on heading geometry and stability number. Ground Engineering, 32(4). Mayne, P.W. & Kulhawy, F.H. 1982. K0-OCR relationships in soil. Proc. ASCE, Journal of the Geotechnical Engineering Division, Vol. 108, No. GT6, 851–872. Muir-Wood, A.M. 1975. The circular tunnel in elastic ground. Géotechnique 25(1): 115–127. O’Reilly, M.P. & New, B.M. 1982. Settlements above tunnels in the United Kingdom–their magnitude and prediction. Tunnelling ’82, The Institution of Mining and Metalllurgy, 1982 pp. 173–181. Potts, D.M. & Zdravkovic, L. 1999. Finite element analysis in geotechnical engineering: theory. London Thomas: Telford. Pound, C. & Beveridge, J.P. 2002. Recent experiences of the measurement of ground movements around tunnels. In press. Simpson, B., Atkinson, J.H. & Jovicic, J.H. 1996. Geotechnical aspects of underground construction in soft ground. pp. 591–594. Balkema. 167 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:41 AM Page 169 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Numerical modeling of remedial measures in a failed tunnel Y. Sun & P.J.N. Pells Pells Sullivan Meynink Pty Ltd, Sydney, Australia ABSTRACT: A FLAC3D analysis was conducted for the investigation of the failure and the design of remedial works for one tunnel in Melbourne Australia. Failure of an approximately 8 m length of the sidewall of the formed concrete arch occurred in mid February 2000. The original design of the un-reinforced concrete lining was such that it just sits on the flat upper surface of the approximately 1.8 m-thick concrete invert. It was generally believed that the failure was primarily due to the compressive stresses across the arch/invert joint being substantially low, which means that the compressive stresses in the arch lining induced by the groundwater pressure at time of failure must have been transferred by 3D action to the west and east of the failure. The purpose of the numerical modeling is to return the failed section to a fully functional arch/invert concept. Key factors in the remediation are the width and sequence of removal of the panels, which were investigated in details in this paper. 1 INTRODUCTION The remedial concept is to remove the formed arch concrete within and immediately around the failure area and reinstate the original design. As a precursor to the 3D analysis, 2D analyses using Phase II was carried out in PSM office to assess the likely compressive and tensile stresses generated above and around panel cut-outs, and the effects of flat jack stressing. The results were used as a guide in selecting the 1.5 m panel width and the excavation sequence proposed in the design. A FLAC3D model includes a 36 m length of the tunnel and the surrounding rock, which contains interface elements between the arch lining and rock, between the arch lining and floor, and between arch pours A and B. The model allows an initial 3 mm gap to exist at the arch/invert interface within the modeled failure area, prior to application of groundwater pressures. 2 NUMERICAL MODELING 2.1 and 2 are 4 m high, Panels 3 and 4 are 3.2 m high and Panel 5 is 1.6 m high. The model geometry is shown in Figures 1a, b & 2. 2.2 Interfaces Interfaces are planes within a FLAC3D model along which sub-grids can interact, slip and/or separation is allowed. A total of eight interfaces shown in Figure 3 are modeled as: – Interface 1: between arch and invert. – Interface 2: between concrete arch/invert and rock simulating the membrane. – Interface 3: between arches A and B. – Interface 4: between panel 1 and arch B. – Interface 5: between panel tops and concrete arch A. – Interface 6: between back of panels 1, 3 and 5 and rock surface simulating the membrane. – Interface 7: between back of panels 2 and 4 and rock surface simulating the membrane. – Interface 8: between panels and concrete arch A. Geometry of model The model includes a 36 m length of the tunnel in the longitudinal direction that is divided into two equal parts, named as arches A and B. The depth of the tunnel is 60 m below the surface. Five vertical panels with a width of 1.5 m each from the contact between arches A and B were designed and named as Panels 1 through 5 sequentially along the arch A side. Panels 1 2.3 The initial stresses sxx, syy and szz of 60 m of rock load were applied to the model boundary with a gradient zero as required. A 3 mm gap from arch contact extending 6 m along arch A side and tapering off at 7 m was modeled to replicate the field observed initial stress conditions. 169 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Initial conditions 09069-21.qxd 8/26/03 10:41 AM Page 170 Figure 1a. Model geometry showing entire model. Figure 1b. Model geometry showing arch and invert. 2.4 Modeling sequence The following stages were developed: – Stage 0: Initial condition. A 3 mm gap was modeled. The pore pressure of pre-leak value of 470 KPa 170 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands was initially applied at the interface between the concrete arch/invert and rock surface. To do so, a virtual interface inside the tunnel surface has to be set up in order to store the face list for applying the equivalent normal stress to the rock surface. 09069-21.qxd 8/26/03 10:41 AM Page 171 Figure 2. Model geometry showing layout of panels. Figure 3. Plot showing the interfaces. – Stage 1: Reduce the pore pressure around the tunnel to pre-repairing condition that is zero behind the failure area and increases linearly to 470 KPa at 30 m away from the failure. – Stage 2: Remove the Panels 1, 3 and 5. The corresponding interface element at the back of the panels should be removed from the list and the same principal applies to the corresponding rock surface 171 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd – – – – – – – 8/26/03 10:41 AM Page 172 (virtual interface) that was in contact previously with the back of removed panels. Stage 3b: Cast concrete panels 1, 3 and 5 in place. The interfaces 1, 5, 6 and 8 for panels 1, 3 and 5 are established in the model in addition to the interface 4 for panel 1, and the bottom gap between new panels and invert is set to zero. Interface 6 has to be added into the face list for applying the normal stresses to the back of new panel concrete. The gravity stresses are initialized in the new panels and low strength properties were used for the interface 8 first (Stage 3a), and then the properties were returned to normal values. Stage 3d: apply flat jack load of 2 MPa at the top of Panels 1, 3 and 5. High strength properties were used (Stage 3c) for the arch including the new panels to get rid of the possible dynamic effect, and returned to normal afterwards. Stage 4: Remove Panels 2 and 4. Follow the similar procedure as described in Stage 2. Stage 5a: Cast concrete panels 2 and 4 in place. Follow the similar procedure as described in Stage 3b. Stage 5b: apply flat jack load of 2 MPa at the top of Panels 2 and 4. Follow the similar procedure as described in Stage 3d. Stage 6: Increase all flat jack loads equivalent to pressure of 4 MPa. Stage 7: Increase the hydrostatic load to 470 KPa. 2.5 Pore pressure The pore pressures are modeled explicitly by applying two opposite normal pressures that are equivalent to the pore pressures to the interface between the concrete arch/invert and rock (interfaces 2, 6 and 7) and to the corresponding rock surface. For the pre-repairing condition, it is assumed that the drain center is located at the top center of the panel 3. The region with a distance of less than 4 m from the drain center has zero pore pressure, while the region with a distance of more than 30 m from the drain center has a full pore pressure of 470 KPa. The region that falls in between has a linear distribution of pore pressure. 2.6 Parameters The concrete is assumed to have a Young’s modulus of 32000 MPa, an unconfined compressive strength of 50 MPa and a tensile strength of 2.5 MPa. The total zone elements of rock and concrete are 10624 and 12104, respectively. The surrounding rock is modeled as elastic material and the concrete arch and invert are modeled as Mohr-Coulomb material. The shear strength parameters adopted for the various interfaces are summarized in Table 1. Table 1. Interface properties. Type Interfaces ID No. Cohesion (KPa) Friction (deg.) Concrete/concrete Concrete/membrane Concrete/flat jack 1, 3, 4 & 8 2, 6 & 7 5 0 0 0 35 10 40 3 FISH CODING Various FISH codes were developed to perform the following functions as: – Storing all zone faces connected to the concrete/rock interfaces (2, 6 and 7) to create a list of all faces for “app nstr” late. Generally, there are two interface elements that are associated with one zone. We can pick up the first element and skip the second one to set up a list where the address of the zone to which the interface element is attached, and the corresponding face ID number are stored in a 2-dimensional array. – Removing faces from the list if zones are changed to a different model (here anisotropic) prior to being made “null”. – Applying the equivalent pore pressure to the zone faces in the current list. – Shifting the solid back to the tunnel for quick manipulation. – Adjusting the contact between the arch and the invert slab. – Setting the gap between arch A and the invert slab. – Calculating the pore pressure distribution at the pre-repairing condition. 4 RESULTS AND DISCUSSIONS Figure 4 shows contours of smin at the initial stage, where the 3 mm gap between the concrete arch A and invert slab was maintained. The majority area immediately adjacent to the gap has a compressive stress up to 5 MPa, while elsewhere has a notably higher compressive stress. Figures 5a & b show contours of major and minor principal stresses when panels 1, 3 and 5 were excavated. Figure 6 shows the tensile crack at the top of panel 1 and bottom between panel 3 and 5. Figures 7a & b show contours of major and minor principal stresses when panels 2 and 4 were excavated and the flat jacks above panels 1, 3 and 5 are stressed to 2 MPa. Tensile stresses above these panels were dropped from 1 MPa to less than 500 KPa. The maximum compressive stresses show quite a complex distribution with a maximum less than 8 MPa. Tensile failure (Fig. 8) remains at the top corner of panel 1 172 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:42 AM Page 173 Figure 4. Contour of smin at stage 0 showing the arching effect due to 3 mm gap between arch and invert. Figure 5a. Contour of smax at stage 2 where panels 1, 3 and 5 (from the right) were excavated. and develops between panels 4 and 5, where a stress concentration is noticed due to the difference in height. It should be noted that tensile cracking in these locations is not of a particular concern. Figure 9 shows the cracking pattern after all flat jacks are stressed to 4 MPa with groundwater pressure at low values, corresponding to the process during the repair work. In general, the stresses 173 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:42 AM Page 174 Figure 5b. Contour of smin at stage 2 where panels 1, 3 and 5 (from the right) were excavated. Figure 6. Plasticity plot at stage 2. are benign and there is no new cracking. Figure 10 shows the crack pattern when the groundwater pressures are returned to a high value at 470 KPa. A tensile crack is predicted on the rock side of the arch about 2.5 m above the panels vertically. This indicates the need to increase the flat jack pressures progressively as the groundwater pressures are allowed to recover. 174 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:42 AM Page 175 Figure 7a. Contour of smax at stage 4 where panels 2 and 4 (from the right) were excavated. Figure 7b. Contour of smin at stage 4 where panels 2 and 4 (from the right) were excavated. As an alternative, one more model was run from the end of stage 6. Instead of increasing the hydrostatic load to 470 KPa in one go, a progressive approach was adopted this time. First adjust the hydrostatic load around the tunnel to a lower and uniform load of 400 KPa. Then increase the hydrostatic load to 425 KPa, 450 KPa and finally to 470 KPa progressively. An improved cracking was noticed as 175 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:42 AM Page 176 Figure 8. Plasticity plot at stage 4. Figure 9. Plasticity plot at stage 6. shown in Figure 11. Clearly it indicates that both slowly and uniformly recovering of the groundwater pressure will reduce the final cracking on the concrete arch remarkably. 5 CONCLUSIONS The results provide confirmation that 1.5 m panel width, and the sequence of excavation of panels 176 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:42 AM Page 177 Figure 10. Plasticity plot at stage 7. Figure 11. Plasticity plot at stage 7 for the alternative approach. (excavate panels 1, 3 & 5 first and then panels 2 & 4) is reasonable design assumptions. Tensile cracking is predicted at the eastern top corner of Panel 1 adjacent to the frictional joint between arches pours A and B. It is recommended that the measures of progressively increasing the flat jack pressures as well as slowly recovering the groundwater pressure to a full uniform value of 470 KPa are to be taken to minimize the 177 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-21.qxd 8/26/03 10:42 AM Page 178 tensile cracks in the concrete arch region above the repaired section. It also demonstrates that the FLAC3D is a useful tool and can be well applied to solve the complicated engineering problem. ACKNOWLEDGEMENT The authors would like to thank Dr. Mike Coulthard from M.A. Coulthard & Associates Pty Ltd for his assistance in developing FISH coding. The authors also benefited from many discussions with him as well. REFERENCES Itasca Consulting Group, Inc. 1997. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 2.0 User’s Manual. Minneapolis: Itasca. Internal Report, PSM500.R3, April 2001. Design report for structural and water inflow remediation at CH 11945m, Appendix C, Three-dimensional analysis. 178 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-22.qxd 8/18/03 1:06 PM Page 179 Mining applications Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-22.qxd 8/18/03 1:06 PM Page 181 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Sill pillar design at the Niobec mine using FLAC3D P. Frenette & R. Corthésy Département des Génies Civil, Géologique et des Mines, École Polytechnique, Montréal, Canada ABSTRACT: The paper presents the numerical analyses performed with FLAC3D to study the stability of the rock mass surrounding the stopes at the Niobec mine in Chicoutimi, Québec. Since the mine expansion is done at depth, the stability of the planned stopes had to be evaluated in order to determine the support requirements linked to an increase of the in situ stresses. The paper focuses on the dimensioning of the sill pillar between mining blocks 3 and 4 using FLAC3D. A rock mass characterization of the site has been made prior to the numerical modeling. The characterization consisted of structural geological mapping, laboratory testing of rock samples and in situ stress measurement. All the stopes were then modeled and the parameters obtained from the site characterization were used in the FLAC3D model. Various alternatives have been analyzed, including the use of backfill and variations of pillar thicknesses for the third pillar. 1 INTRODUCTION Safe and economical dimensioning of underground excavations is often hard to achieve because of the numerous parameters involved. These parameters include rock mass characteristics, orientation and magnitude of stresses, excavation method and sequencing. Any combination of these factors may change from one point to another, requiring a reevaluation of the mine design. This is the case at the Niobec mine, located near Chicoutimi, Québec, were underground production is soon reaching the fourth mining block. The increase in stresses with depth requires calculating the dimension of the sill pillar between the third and fourth mining blocks. At the present time, Niobec mine has 3 horizontal pillars. The crown pillar with a thickness of 90 m (300 feet), the pillar between mining blocks 1 and 2 with a 30 m (100 feet) thickness and the pillar between mining blocks 2 and 3 being 45 m (150 feet) thick. These pillars are necessary for the stability of the excavations and absorb part of the stresses caused by the mining of the stopes which remain open after being mine out. As for any rock mechanics design, there is no direct method for dimensioning horizontal pillars in hard rock mines as each mine has its own geometric and geomechanical settings, which make it difficult to have a universal recipe that allows an optimal pillar design. Consequently, numerical modeling was considered the best tool for the project. Although the overall quality of the rock mass at the Niobec mine is good, the increasing stress levels with depth, as confirmed by in situ stress measurements, will increase the potential for failure which has to be investigated. Moreover, the stope geometry being relatively massive, a two-dimensional model was not considered realistic for the Niobec mine. Although the authors did not find applications of FLAC3D for the modeling of a complete mine in the literature, they found it would be interesting to use the software for that purpose, since it could efficiently model rock mass failure and, if required, the use of backfill in the open stopes. In order to gather the data for the numerical model, a rock mass characterization program including structural geological mapping, laboratory testing and in situ stress measurements was conducted. The work was facilitated by the fact that the rock mass including the ore bearing zones and host rock are relatively homogeneous and can be considered as a single zone. 2 SITE INVESTIGATION 2.1 Niobec mine is located near Chicoutimi in Québec and has been producing niobium since 1976. It is owned in equal part by Cambior and Mazarin. The mine produces 3500 tons of ore each day by long hole stoping. Each stope is 45 m deep, 25 m wide and 90 m high. Mining is done using primary and secondary stopes, creating openings up to 200 m wide. The mine has 3 mining blocks and 8 levels, the lowest production 181 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Niobec mine 09069-22.qxd 8/18/03 1:06 PM Page 182 Table 1. Principal stresses tensor. 1 2 3 Table 3. Mohr–Coulomb parameters for the rock substance. Stress Strike Plunge 29.5 MPa 16.0 MPa 9.1 MPa 45° 138° 310° 04° 38° 51° Parameter Unconfined compressive strength Tensile strength Young’s modulus Poisson’s ratio 124 MPa 8.1 MPa 55 GPa 0.254 Table 2. Stress gradients. 850 Level (Mpa/m) (hole #1) (hole #2) 1000 1450 Mean horizontal stress gradient Vertical stress gradient 0.0642 0.0454 0.0466 0.0484 0.0386 0.0194 0.0274 0.0267 along 1370 m of drift using the scanline method. Major joints of over 1m were plotted for both level 1150 and 1450. The results were compared with two other studies made on the previous levels and the comparison showed the persistence of two major families of joints on all levels with the appearance of a third family with increasing depth. 2.4 level being 1450 feet deep. A fourth mining block is scheduled to open in 2011. 2.2 In situ stress measurements Stresses were measured at the 1450 level (Corthésy 2000) using the modified doorstopper method (Leite et al. 1996). Unlike the conventional doorstopper method, the modified method allows continuous reading of the strains at the bottom of the hole including temperature readings. These continuous readings allow evaluating the quality of each measurement. These are performed in three differently oriented holes in order to obtain the three-dimensional stress tensor. Table 1 shows the principal stress tensor obtained by combining the data obtained from the three holes using the least squares approach. The stress calculation procedure allows considering both local anisotropy and heterogeneity. Those results were compared with another in situ stress measurement campaign made by Canmet (Arjang 1986) using the CSIR triaxial cell (Leeman 1967). In this earlier campaign, stress tensors were calculated on levels 850 and 1000. Two holes were used on level 850 and one on level 1000. Principal stress gradients on levels 1450 and 1000 are similar, but results from the 850 level are not, probably because the measurements were made in the influence zone of a stope. Table 2 shows the stress gradients for level 850, 1000 and 1450. 2.3 Structural geological mapping Structural geological mapping was conducted on levels 1150 and 1450. Over 8000 joints were identified 2.5 Laboratory testing Laboratory tests were conducted on rock samples. Seventeen samples were tested to obtain the unconfined compressive strength of the rock while seventeen other samples were tested to determine the tensile strength of the rock. Three triaxial compression tests were also conducted to verify the adjustment of the data to empirical strength criterion. The deformability parameters were obtained indirectly from the stress measurement campaign as they were required to interpret the results following a procedure suggested by Corthésy et al. (1993). Table 3 shows the mechanical parameters obtained for the rock substance. 3 NUMERICAL MODELING 3.1 Model geometry Autocad files representing all the stopes mined before 2000 were used as a database to build the geometry of the model. Unfortunately, no interface allowing the importation of dxf files is available with FLAC3D, so the dxf file containing all the stopes coordinates was used to build the model (see Fig. 1). The three existing mining blocks were divided into separate entities that were later merged. Each block 182 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Rock mass classification Once all the required parameters were obtained, the rock mass was classified according to the RMR and Q indexes. The RMR value was found to be 77 and the Q index was estimated at 40, which corresponds to a good rock mass in both cases. These results were also used to obtain the failure envelope of the rock mass for the Mohr–Coulomb criterion. 09069-22.qxd 8/18/03 1:06 PM Page 183 Figure 1. Isometric view of the three mining blocks. Figure 3. Perspective view showing the modelling of the walls surrounding a stope. Figure 2. View of part of the first mining block showing the 20 m 20 m grid and the simplifications made to the stopes. was subdivided into smaller sections to model the excavations using a 20 m 20 m 20 m mesh at the outer edge of the sections (Fig. 2). These sections were made from 7 different parts, the 4 walls along with the floor, the roof and the stope itself. This method allowed modeling all the excavations, but some simplifications were necessary in order to keep the number of zones and time spent to building the model geometry to reasonable values (Fig. 3). All the stopes were modeled this way and adjacent sections were merged to obtain a uniform mesh. A transition zone had to be inserted between each mining block to merge them together without creating a discontinuity in the model. 3.2 Boundary conditions Some problems were encountered for applying the stresses on the model. Since means of applying shear stress gradients on the boundary of the model were not found, the principal stresses with their orientations as shown in Table 1 could not be applied to Figure 4. View showing the zone added to the model so the stresses can be applied at 45°. the model. The solution was to make a simplifying assumption stating that the principal stresses 1 and 2 were horizontal with an azimuth of 45° and 135° respectively and that 3 was vertical (Fig. 4). The center part of Figure 4 (the small square) is the area containing the stopes while the rest of the model allows the boundaries to have the required orientation for applying the principal stress gradients and also insure these boundaries are not in the zone of influence of the excavations. This buffer zone was considered to have an elastic behavior since no failure around the excavations should extend that far and this would also 183 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-22.qxd 8/18/03 1:06 PM Page 184 speed up the run time of the analyses. In order to evaluate the influence of the principal stress tensor reorientation, two analyses were done using the Examine 3D boundary element program developed by Rocscience. This software allows us to apply the in situ major principal stresses with any orientation relative to the model. In one analysis, the measured in situ stress tensor (Table 1) was input and in the other, the modified stress tensor with 1 and 2 in the horizontal plane was used. As expected, the second model gave lower strength factor values, which confirms the simplifying assumption related to the reorientation of the stresses puts the model on the safe side. The strength factor given by Examine 3D compares the stress state at every point of the model with the strength envelope. A strength factor of 1.0 means the stress state is on the strength envelope while a lower value means the stress state exceeds the strength of the rock mass. These strength factors must be considered with care since they tend to underestimate the extension of potentially failed zones, since the elastic analysis the software performs does not consider the post failure stress redistribution. 4 FLAC3D SIMULATIONS Before running the analyses used to estimate what sill pillar size would be optimal, various scenarios were studied in order to perform the numerical analyses more efficiently and to verify if the use of certain support elements such as backfill would have an effect on the local and overall stability of the mine. It should also be stated that a validation of the model by comparing its results with in situ observations was difficult for various reasons. First, no in situ monitoring of displacements was available. Secondly, as the rare instabilities around the excavations in the mine are mostly controlled by the presence of discontinuities which are not considered in FLAC3D, it is difficult to perform a direct comparison between the extent of instabilities in the numerical model and the ones observed in the field. This only emphasizes the fact that in the absence of field monitoring and in the presence of a good quality rock mass, validation of numerical models is difficult. 4.1 3.3 Constitutive laws and material properties In the analyses shown in this paper, the rock mass was assumed to present an elastic perfectly plastic behavior, so no post peak strength values are given. Table 4 shows the strength and deformability parameters of the rock mass used in the model. These parameters were obtained by combining the laboratory test results with the rock mass classification parameters presented in section 2.4. The authors are aware that for fragile hard rock such as the one found at the Niobec Mine, perfect plasticity is a not realistic assumption, but using a strain softening constitutive law would have slowed down the runtime of the analyses which already took over 5 days to run on a 1.0 GHz Pentium PC. Nonetheless, now that the model is built, it would be a simple matter to implement the strain softening parameters and perform a sensitivity analysis by varying the postpeak strength parameters. The authors are also aware that perfect plasticity will underestimate the extension of eventual failure zones. Table 4. Material parameters used in the model to simulate the rock mass behavior. Uniaxial compressive strength Tensile strength Cohesion Friction angle Young’s modulus Poisson’s ratio 34.4 MPa 2.1 MPa 10.5 MPa 38.5° 47.3 GPa 0.254 Table 5. Comparison of the number of failed zones on different sections of the model for the analyses with and without sequential mining. Section Mining all at once (failed elements) Sequential mining (failed elements) Difference 4410 4465 4530 4575 4625 4675 4730 4795 4830 4900 36 74 65 70 58 45 41 87 37 14 37 80 64 81 71 35 37 92 43 15 1 6 –1 11 13 –10 –4 5 6 1 184 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Mining sequence The influence of the mining sequence (excavating the stopes in the same sequence they were mined out in blocks 1 and 2) on the results of the analyses was studied. This was an important point to verify since the mining sequence for the new mining blocks (3 and 4) was unknown and excavating the stopes all at once in the numerical model was an interesting alternative as it would allow important time savings. Consequently, two analyses were run, one by excavating the stopes one after the other and waiting for the unbalanced forces to stabilize in between and the other by nulling the elements in the stopes all at once. The comparison between the two runs is done by taking the number of failed elements in each simulation as shown in Table 5. 09069-22.qxd 8/18/03 1:06 PM Page 185 Most of the failed elements are located around the stopes and their dimensions are small compared to the 20 m 20 m 20 m mesh used in areas remote from the excavations. The section heading refers to various sections in the model that cut through the stopes in the model. It is considered that the excavation sequence has a negligible influence and that no clear pattern is observed. Consequently, it was concluded that for the following simulations, the stopes could be excavated all at once without much effect on the outcome of the runs. 4.2 Influence of backfill As the mine had never used backfill, it was decided to see if the use of such support would allow minimizing the occurrence of local failure around certain stopes. To estimate the influence of backfill on the stability of the openings, the same methodology as used to evaluate the influence of the mining sequence was adopted. In these simulations, after a stope was mined out and the unbalanced forces had stabilized, the null elements were replaced by zones having properties matching a backfill with 8% cement. The next stope was then excavated and the sequence was repeated for all the openings. The number of failed zones with and without backfill are presented in Table 6. These simulations showed the fill to have no significant influence on the stability of the mine. It is believed that the very low stiffness of the backfill compared to that of a good quality rock mass makes it almost impossible for it to absorb any significant stresses, so it would not serve the purpose of reducing the size of the third horizontal pillar. 4.3 The main objective of this project was to find the optimal thickness of the third horizontal pillar. The first pillar between mining blocks 1 and 2 is 30 m thick, the second between mining blocks 2 and 3 is 45 m thick and the third one between mining blocks 3 and 4 was also planned to be 45 m thick. Since the use of backfill was found of little use, only three simulations were made. One optimistic analysis with a 30 m thick pillar, another with a 45 m thick pillar identical Table 6. Comparison of the number of failed zones on different sections for the simulations with and without backfill. Section Without backfilling With backfilling Difference 4410 4465 4530 4575 4625 4675 4730 4795 4830 4900 37 80 64 81 71 35 37 92 43 15 37 77 61 79 67 33 34 88 43 15 0 3 3 2 4 2 3 4 0 0 Figure 5. Failure zones for the 30 m (100) pillar for section 4795. 185 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Design of the third horizontal pillar 09069-22.qxd 8/18/03 1:06 PM Page 186 Figure 6. Failure zones for the 45 m (150) pillar for section 4795. Figure 7. Failure zones for the 60 m (200) pillar for section 4795. 186 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-22.qxd 8/18/03 1:06 PM Page 187 to the one between blocks 2 and 3 and finally a last simulation with a 60 m pillar. As the stope layouts for the third and fourth mining blocks were not available at the time the simulations were run, the geometry of these blocks was assumed to be similar to that of blocks 1 and 2. The simulation with a 30 m pillar showed that the pillar would be stable, but that only a 20 m thickness would remain intact (Fig. 5). The two other simulations showed that a 40 m thick zone would remain intact with the 45 m pillar (Fig. 6) and 53 m would be free from failed zones for the 60 m pillar (Fig. 7). The number of failed zones in the pillars were then compared. The direct comparison can be done since the number of zones remained constant between simulations and only the zone thicknesses were changed to modify the pillar thickness. The analyses showed the 30 m pillar to have 657 failed zones on a total of 4425 zones in the pillar, while the 45 m pillar had 603 failed zones and the 60 m pillar showed 583 failed zones. Although the 30 m sill pillar showed an overall stability, the intact thickness is considered too small as the presence of planes of weakness not considered in the analyses may cause important instabilities. Bearing this in mind, the 45 m pillar would leave an intact rock section considered more adequate. The results show the 60 m thick pillar would not increase the overall safety factor significantly and the side effects of having a pillar which is too thick, is the ore loss and also the fact that a thicker pillar will expose the stopes in the fourth mining block to higher in situ stresses (due to their increased depth) causing unwanted dilution. 5 DISCUSSION AND CONCLUSIONS The proposed pillar design presented in this paper should, prior to accepting it, be analyzed using a more realistic constitutive law than perfect plasticity for hard rock. The strain-softening model available in FLAC3D should be tested with various post-peak strength parameters in order to perform a sensitivity analysis of the excavation response to these parameters. Also, if one wishes to fine-tune the model, an optimization of the element size around the excavations could be made. In conclusion, the work presented in this paper showed that modeling a complete mine with FLAC3D is quite an undertaking since there are no simple ways to create the geometry, although there is now an interface with AnsysCivilFEM which should facilitate this task (it was not available at the time the project started). There are also difficulties in the application of the boundary conditions as mentioned in section 3.2 since the authors we unable to apply shear stress gradients to the model boundaries. Besides these difficulties, once the model is built, it is interesting to be able to perform sophisticated sensitivity analyses by modifying the parameters of various constitutive laws. ACKNOWLEDGEMENTS The authors wish to acknowledge Martin Lancet and the personnel of the Niobec Mine who have contributed to the success of this M.A.Sc project. They also want to acknowledge the National Research Council of Canada for its financial support (grant # OGP0089752). REFERENCES Arjang, J. 1986. In situ stress measurement at Niobec Mine, Canmet Laboratory Report. Corthésy, R. 2000. Mesure des contraintes in situ, mine Niobec. CDT report, Ecole Polytechnique. Corthésy, R., Gill, D.E., Leite, M.H. 1993. An integrated approach to rock stress measurement in anisotropic non linear elastic rock, Int. J. Rock Mech. Min. Sci., Vol. 30, no. 3, pp. 395–411. Leeman, E.R., 1967. The doorstopper and triaxial rock stress measuring instruments developed by the CSIR, J. of the South Afr. Inst. of Mining and Metall., Vol. 69, no. 7, 1967, pp. 305–339. Leite, M.H., Corthésy, R., Gill, D.E., St-Onge, M., Nguyen, D. 1996. The IAM – A down-the-hole data logger conditioner for the modified doorstopper technique. 2nd North American Rock Mechanics Symposium, Montréal, pp. 897–904. 187 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-23.qxd 08/11/2003 20:31 PM Page 189 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Stability analyses of undermined sill mats for base metal mining R.K. Brummer, P.P. Andrieux & C.P. O’Connor Itasca Consulting Canada Inc., Sudbury, Ontario, Canada ABSTRACT: Mines are often faced with sill extraction situations, and one technique that can be used to extract a sill is to leave a consolidated fill mat in the sill cut. Several Canadian mines employ this sill extraction technique, and in this paper the stability of these sills was modeled using two-dimensional FLAC numerical simulations. The objective of this parametric study was to derive relationships between the required strength of the sill mats and the maximum stable unsupported undercut span for various orebody dips. The footwall-to-hangingwall spans described in this paper were 1.2 m (4 ft), 2.4 m (8 ft), 3.6 m (12 ft), 4.8 m (16 ft), 6.0 m (20 ft) and 10.5 m (35 ft), with mining dips of 60°, 70° and 80°. The range of fill cohesive strength used was from 100 to 500 kPa, a typical range for most hydraulic or paste fills. Stability charts were derived (one per ore body dip), that can be used to select the minimum fill strengths required (in terms of cohesion) to maintain stability for different combinations of spans and dips. 1 INTRODUCTION FLAC simulations were set up to examine the behavior of a typical backfill sill mat for sill extraction. The objective of this parametric study was to derive a relationship between the strength of the sill mat and its maximum stable unsupported span, for various different orebody dips. The footwall-to-hanging wall spans described here were 1.2 m (4 ft), 2.4 m (8 ft), 3.6 m (12 ft), 4.8 m (16 ft), 6.0 m (20 ft) and 10.5 m (35 ft). 2 2.1 range 5% to 12% for most fills in common use. A typical FLAC geometry is shown schematically in Figure 1. The thickness of the sill mat was assumed to be 3.3 m (10 ft), which is a typical thickness for a sill mat. A surcharge loading of up to 33.3 m (100 ft) above the mat was used. In order to reproduce this geometry, a 60 by 120 element grid was generated and the appropriate coordinates applied to its four corner nodes. As a result, each element, or individual zone, was 0.3 m by 0.3 m (1 ft by 1 ft) in size, which is sufficiently detailed for the problem considered. Interfaces were defined between the FLAC MODEL FLAC model geometry The stability of the fill mats was investigated by carrying out several FLAC analyses, using orebody widths of 1.2 m (4 ft), 2.4 m (8 ft), 3.6 m (12 ft), 4.8 m (16 ft), 6.0 m (20 ft) and 10.5 m (35 ft). The dips considered were 60°, 70°and 80° to give a realistic range of dips. It was assumed that 45° is too flat to allow for a stable unsupported backfill sill mat to be built because of slip on the hangingwall. For each combination of dip and span, the objective was to determine the cohesive strength required from the fill to ensure the stability of the sill mat, without additional support, when fully undercut. The range of cohesion values used was 100 to 500 kPa, which corresponds to cement contents in the Figure 1. Typical FLAC layout for stability analyses. (Schematic cross-section. Not to scale.) 189 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-23.qxd 08/11/2003 20:31 PM Page 190 fill and each wall, in order to allow movement of the backfill along this surface. Table 1. Mechanical properties used for the host rock and the ore. 2.2 Elastic modulus (GPa) Poisson’s ratio ( ) Shear modulus (GPa) Bulk modulus (GPa) Density (kg/m3) 40 0.25 16.0 26.7 2700 FLAC model run sequence The sequence of each FLAC run was as follows: 1. Define the grid to reproduce the required stope span and dip. At this point, consider the ore under the mat to still be in place (i.e. consider the sill mat to rest on solid rock). 2. Set up eight history points in the center of the backfill column, at 0.7 m (2 ft), 1.3 m (4 ft), 2.3 m (7 ft), 3.3 m (10 ft), 5.0 m (15 ft), 6.7 m (20 ft), 10.0 m (30 ft) and 13.3 m (40 ft) behind the back of the next cut. 3. Define materials properties with real elastic properties, but very high strengths for the fill, and zero friction and cohesion along the interfaces between the backfill and the two rock walls. 4. Cycle the model to equilibrium. This first part of the run is required to allow the fill to settle under its own weight, as would happen in real life. The artificially high strength of the fill ensures its elastic behavior, while the null friction at the interfaces prevents the development of artificial stresses during this gravity-driven compaction process. 5. Once at equilibrium, reset the fill material and wall contacts to realistic strength properties (these are discussed later). 6. Reset all the displacements tracked by the history points, in order to reflect only the changes subsequent to equilibrium. 7. Remove the restraint below the fill mat by “mining” the stope so that the fill mat takes load from its own weight and the waste fill column above. 8. Apply some convergence to the stope walls as a result of mining. Because the act of mining will involve some wall convergence a 10 mm incremental convergence was assumed to take place. Due to the explicit time-marching scheme used in FLAC, this movement had to be indirectly applied to the walls by applying a horizontal velocity to the model boundaries. To obtain the desired 10 mm closure, a horizontal velocity of 0.001 mm per time step was applied inwards on both the left and right model boundaries for 5,000 steps. 9. Remove the horizontal velocity applied on the model boundaries (as the desired closure has been reached), and cycle the model to equilibrium. 10. Check the history points and displacement results to see if the configuration is stable. 2.3 Constitutive models and material properties The constitutive model used for the host rock was elastic, while a strain-softening behavior was retained for the backfill. Elastic–plastic strain softening constitutive laws allow specifying a transition zone between the peak and residual mechanical properties of a material. In the cases where these mechanical properties decrease as the material yields (which is the case with typical backfills), a strain softening behavior was used to describe how the material’s strength is progressively decreased from its peak value to its residual one as irreversible/ plastic strain accumulates in it. 2.4 The properties retained for the rock mass (both the host rock and the ore) are shown in Table 1. The exact elastic and strength properties of a typical host rock are not important, because the behavior of a sill mat is not very sensitive to these properties, as long as they are orders of magnitude larger than those of the backfill. 2.5 Fill properties The fill properties used in the FLAC analyses were based on a large in-house database of fill properties. The main variable for the fill is its cohesive strength, which, as mentioned, was varied between 100 and 500 kPa. As the cohesive strength was changed, so were the elastic properties (even though not critical, this refinement was useful as a certain degree of convergence between the footwall and hanging wall was considered, which, in turn, induced stresses in the sill mat). In other words, as the cohesive strength of the fill was increased, so was its stiffness. The overall fill property setting process was carried out in the following methodology: 1. set the cohesive strength, for example 200 kPa; 2. set the tensile strength at half the cohesion – 100 kPa for our example; 3. multiply the cohesive strength by 4 (assuming a friction angle in the range 30° to 33°) to obtain the corresponding unconfined compressive strength – 800 kPa for the example considered; 4. derive the corresponding cement content using Figure 2 (to obtain an 800 kPa UCS, the required cement content would be around 6.4%); 5. derive the corresponding elastic modulus using Figure 3 (for a 6.4% cement content, the elastic modulus would be near 0.53 GPa); 190 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Host rock 09069-23.qxd 08/11/2003 20:31 PM Page 191 Table 2. Strength and elastic properties used for the backfill sill mat. 1,800 1,600 UCS (kPa) 1,400 1,200 1,000 Cement Cohesion UCS1 content (kPa) (kPa) (%) Elastic Shear Bulk modulus modulus2 modulus2 (GPa) (MPa) (MPa) 100 150 200 300 400 500 0.24 0.40 0.53 0.84 0.90* 0.90* 800 600 400 200 0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 400 600 800 1200 1600 2000 4.2 5.5 6.4 8.3 10.0 11.2 92 154 204 323 346 346 200 333 442 700 750 750 Cement Content (%) 1 Figure 2. Typical relationship between UCS and cement content for backfill. PASTE FILL PROJECT Paste Fill Moduli vs. Cement Content 1.00 Elastic Modulus (GPa) 0.90 0.80 0.70 Confinement during test (kPa) 0.60 0.50 800 400 Average 0.40 0.30 0.20 0.10 0.00 0.0 Assuming a 30° internal angle of friction. Assuming a Poisson’s ratio of 0.30. * Value outside of data range – elastic modulus fixed at 0.90 GPa. 2 1.0 2.0 3.0 4.0 5.0 Cement Content (%) 6.0 7.0 8.0 Figure 3. Typical stiffness properties for backfill (based on triaxial lab tests carried out on a typical paste fill). 6. calculate the corresponding shear modulus G and bulk modulus K, assuming a Poisson’s ratio of 0.30 (for our example G would be 206 MPa while K would be 442 MPa); and, 7. use these values as input to the FLAC model. For the example, these inputs would be: cohesion 200 kPa, tensile strength 100 kPa, G 206 MPa, K 442 MPa. The nominal friction angle was chosen at 30° (friction angle values will be discussed in more detail later). Table 2 summarizes the properties used for the various fill strengths considered. The density of the backfill was assumed to be 2.0 t/m3. The mechanical properties of the surcharge fill placed on top of the sill mats were lowered by 10% to account for the lower quality of fill typically placed on top of sill mats. The properties affected were the cohesion, tensile strength, shear modulus and bulk modulus – the friction angle was kept the same. As mentioned, the constitutive model used for the consolidated fill material was a strain-softening one. In order to simulate this behavior, the decrease in strength as a function of the plastic strain accumulated in the yielding material needs to be explicitly described. For the purpose of this study, it was assumed that the cohesion and, hence, tensile strength, would decrease linearly from their maximum value at zero plastic strain, down to zero at a cumulative plastic strain of 1.5%. The internal angle of friction was set to also vary linearly, but from its maximum value of 33° at zero plastic strain, down to 30° at a cumulative plastic strain of 1.5% and beyond. Neither the shear nor the bulk moduli are affected by plastic strain and were thus left unchanged. The older waste fill above the sill mat was subjected to the same plastic strain-dependent weakening process. Cohesion and tensile strength were also decreased linearly from their maximum value (set, as mentioned previously, at 90% of those of the sill mat) at zero plastic strain, down to zero, also at a cumulative plastic strain of 1.5%. Similarly to the sill mat, the internal angle of friction of the weaker fill material was set to decrease linearly from its maximum value of 33° at zero plastic strain, down to 30° at a cumulative plastic strain of 1.5% and beyond. 2.6 191 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Interface between rock and fill As previously discussed, interfaces between the host rock and the fill, along both the footwall and hanging wall, had to be specified due to the continuum nature of the finite difference approach used in FLAC. During the initial compaction stage of each run, both the internal angle of friction and the cohesive strength of these interfaces were set to zero in order to prevent artificial stresses from developing along them as the backfill settled. During the later stages of the runs these values were reset – the internal angle of friction was set equal to the internal angle of friction of the sill mat, and the cohesion was set equal to the cohesion of the sill mat. 09069-23.qxd 08/11/2003 20:31 PM Page 192 3 INTERPRETATION OF ANALYSES Several characteristics of the FLAC analyses were used to determine whether the various sill mats were stable or had failed. Figures 4 & 5 show the unbalanced force history for a stable layout and an unstable layout, respectively. The unbalanced force is a key element of the time-marching algorithm used in FLAC as it indicates the degree of static equilibrium reached within the model at any given cycle (as the unbalanced force diminishes, the degree of equilibrium increases). As can be seen, the unbalanced forces converge to zero for stable configurations, but continue to oscillate, or even increase, for unstable ones. Figures 6 & 7 show examples of the vertical displacement history for the control points located within the waste fill or backfill column, as described earlier. As can be seen, stable spans displace vertically by only a finite amount, whereas unstable spans continue to deform vertically as they fail. More crudely, the deformation of the FLAC grid can be examined, as can be seen in Figure 8. Unstable JOB TITLE : Undermined Sill Mat Stability Analyses FLAC (Version 3.30) JOB TITLE : Undermined Sill Mat Stability Analyses FLAC (Version 3.30) +05 (10 LEGEND step 9000 HISTORY PLOT Y-axis : Max. unbal. force X-axis : Number of steps ) LEGEND (10+05 ) step 9000 HISTORY PLOT Y-axis : Max. unbal. force X-axis : Number of steps 5.000 4.000 5.000 4.000 3.000 3.000 2.000 2.000 1.000 1.000 1 1 Itasca Consulting Canada Inc. 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 (10 9 +03 ) Itasca Consulting Canada Inc. (10 +03 ) Figure 4. Example of a FLAC unbalanced force history plot for a stable sill mat configuration. Note that the maximum unbalanced force stabilizes at zero, since the fill panel is stable. Figure 5. Example of a FLAC unbalanced force history plot for an unstable configuration. Note that the maximum unbalanced force is not zero and increases without bound as the fill panel fails. Figure 6. Example of a FLAC “y-displacement” (vertical) history plot at the various control points located within the backfill column, for a stable configuration. Note that the maximum displacement is finite at about 45 mm. 192 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-23.qxd 08/11/2003 20:31 PM Page 193 Figure 7. Example of a FLAC “y-displacement” (vertical) history plot at the various control points located within the backfill column, for an unstable configuration. Note that the maximum displacement increases without bound (up to 1.4 m at the end of 12,000 cycles in this case). Figure 8. Example of a FLAC grid plot showing failure of the sill mat at mid span, and especially at the hanging wall contact. Note that the fill displaces downward by up to 500 mm, indicating failure. configurations exhibit severe deformation of the grids. Stable spans also will deform to some extent due to settlement, as can be seen in Figure 9, but will stabilize and not continue to deform as the runs are further cycled. 4 SUMMARY OF RESULTS Figures 10 & 11 show the FLAC results obtained for a 60° dip and a 1.2 m (4 ft) mining width, with a 200 kPa cohesion backfill sill mat. 193 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-23.qxd 08/11/2003 20:31 PM Page 194 Figure 9. FLAC results for a 70° dip, a 400 kPa cohesion sill mat and a 6.7 m (20 ft) span. Most of the movement occurs in the lower part of the sill mat, and the whole arrangement remains stable. 600 500 Average Cohesion (kPa) Average Cohesion (kPa) 600 Stable 400 300 200 100 Failed 0 0 5 10 15 Stable 400 300 200 100 0 Stope Span (m) Stable 60 500 Failed 0 5 10 15 Stope Span (m) Failed 60 Stable 80 Failed 80 Figure 10. Stability Chart for 60° dipping ore showing failed cases and stope spans. Also shown is a crude contour separating the failed cases from the stable cases. Figure 11. Stability Chart for 80° dipping ore showing failed cases and stope spans. Also shown is a crude contour separating the failed cases from the stable cases. Back support in the form of mat reinforcing (e.g., screen placed on the floor of the stope to fill, together with vertical bolts) will stabilize the local back when the sill stopes are extracted and the fill mat is undermined. The aim of this study is limited to the overall stability of the backfill sill mats, and excludes minor falls of fill from the back that must be expected to occur unless appropriate mat reinforcing techniques are used. It is understood that the support of the immediate back will be ensured by mines through the appropriate use of this type of reinforcing. These charts can be used to select fill strengths (in terms of cohesion) for different combinations of spans and dips. Since fill strength is normally measured in terms of uniaxial compressive strength, this can be estimated by multiplying the cohesion by a factor of 4, i.e. for a cohesion of 200 kPa, a uniaxial compressive strength of 800 kPa will be necessary. Note that the FLAC analyses as presented do not incorporate any Factors of Safety – appropriate Factors of Safety must therefore be applied to the fill strength for design purposes. This will depend on the quality and degree of uniformity of the fill as placed. Note also that the charts suggest that it is not possible (for the fill strengths considered) to undermine a fill panel of 10.5 m width without some form of additional reinforcement. 194 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-23.qxd 08/11/2003 20:31 PM Page 195 5 CONCLUSIONS The FLAC model is capable of modeling both a “sag” mode of failure as well as a “rotational” mode of failure for a sill mat. For near-vertical or steep stope walls, the sag mode is observed to occur and is dominant. For flatter dips, it is possible to observe a rotational mode of failure, as the fill falls away from the hangingwall, and eventually rotates about the footwall support. The objective of this study was limited to the overall stability of the sill mats, and excluded consideration of minor falls of fill from the back that must be expected to occur unless appropriate mat reinforcing techniques (e.g. properly anchored screen or shotcrete) are used. For practical reasons, it is understood that the support of the immediate back will be ensured through the appropriate use of this type of reinforcing. All of the analyses presented show that it is usually possible, with sufficient binder, to create a stable mat back under a variety of geometric and loading conditions. However, this applies to the overall mat – not the immediate back. Even with a very strong mat, it is still possible to have falls of fill from the immediate back, unless some form of back support (e.g. screen on the back, with bolts or Splitsets) is used. This is equivalent to the screen commonly used on a rock back. This raises the possibility of incorporating the screen with the mat reinforcing (e.g., screen placed on the floor of the stope prior to filling). This screen will then be exposed on the fill back as the panel is undermined. If the screen is tied to the fill mat reinforcing, this will take the place of the bolts and will eliminate the support cycle when the mat is undermined. This has been successfully done at a number of mines, and is an economical way to reinforce the fill mat (and has the potential to save binder), as well as provide support for the back of the undermined fill panel. 195 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-24.qxd 08/11/2003 20:31 PM Page 197 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 FLAC numerical simulations of tunneling through paste backfill at Brunswick Mine P. Andrieux* & R. Brummer Itasca Consulting Canada, Inc., Sudbury, Ontario, Canada A. Mortazavi Previously with Itasca Consulting Canada, Inc., Sudbury, Ontario, Canada B. Simser* Falconbridge Ltd., Sudbury Mines/Mill Business Unit – Craig Mine, Onaping, Ontario, Canada P. George* New Brunswick, Canada *previously with Noranda, Inc., Brunswick Mine, Bathurst, New Brunswick, Canada ABSTRACT: In early 2001 Itasca Consulting Canada Inc. was contracted by Noranda Inc. to assist in the design of the first two drifts that were going to be driven through paste backfill at Brunswick Mine in the south end of the 1000 m Level in order to create alternate accesses to the western ore zones. A numerical stability analysis of the proposed tunnels was carried out by means of two-dimensional FLAC simulations, which took into account different fill strengths, alternate tunnel geometries, various floor conditions and the presence of unconsolidated plugs of waste rock within the paste backfill at close proximity to the tunnel in one area. The main objectives of this work were to investigate the self-standing characteristics of the exposed paste material, evaluate the deformations expected as a result of tunneling through it and recommend adequate ground support alternatives. This paper describes the modeling approach used, the results obtained and how they corresponded to the behaviors later encountered underground during the excavation of the tunnels. 1 INTRODUCTION AND BACKGROUND Itasca Consulting Canada Inc. (ICCI) of Sudbury, Ontario, was contracted to assist in the design of two drifts that were going to be tunneled through paste backfill in the south end of the 1000 m Level at the Noranda Inc. Brunswick Mine operation near Bathurst, New Brunswick. Drifting through the paste backfill was required in order to create alternate accesses to the western ore zones because some of the existing accesses were either in highly stressed ground that could burst, or were planned to be removed when future stopes were going to be mined. Tunneling through paste backfill being then a new procedure at Brunswick Mine, it was decided by senior engineering personnel at the site that a thorough numerical investigation was necessary to identify possible design limitations. The numerical analyses of the process of driving through paste backfill were carried out at the ICCI offices in Sudbury by means of two-dimensional FLAC simulations that used actual fill strengths, geometry, floor conditions and other expected field conditions. The main objectives of this work were to investigate the self-standing characteristics of the exposed paste material and the deformations expected as a result of tunneling through it. A series of numerical exercises were completed with the FLAC code to address these objectives for drifts of various shapes driven in paste materials of varying cohesive strength. Two situations were simulated: (1) the situation in the 236-8 Access on 1000-2 sub, where failed waste rock (which had caved from the back of the drift) and unconsolidated rockfill material (which had run from the 235-8 and 237-8 stopes above) ended below the paste material with a near-45° angle of repose; and, (2) the situation in the 129-7 Access on 1000-1 sub, where only paste material was present. 197 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-24.qxd 08/11/2003 20:31 PM Page 198 2 GENERAL APPROACH As mentioned, the FLAC numerical analyses performed focused primarily on investigating the free-standing characteristics of the paste backfill as it is excavated, without any support. This was deemed important for safety reasons (i.e. for assessing the level of risk the underground crews performing the excavation work were going to be exposed to), and for defining the long-term support requirements of the excavation. A sensitivity analysis was carried out for this aspect of the work, whereby the cohesive strength of the paste material was varied over a reasonable range, in order to identify threshold values with regard to instability. These threshold values were then compared to the values measured in the paste backfill in both the 129-7 and 236-8 excavation areas. This numerical sensitivity analysis also examined the effect the tunnel shape has on the transfer of the loads around it. Two profiles, a flat back and a pronounced horseshoe shape, were modeled, to determine the impact of shape on the self-standing stability of the tunnel. This analysis also produced deformation and convergence data for all the cases examined, N 16.00 15.00 14.00 13.00 12.00 11.00 10.00 9.00 8 9 10 11 12 13 14 15 16 17 18 19 Figure 1. Front view looking from the footwall into the 236-8 Access on 1000-2 sub. This photograph was used to build the numerical model. The mesh and drift outline visible in the foreground were generated by Microsoft Excel™, using the “digitizer mode”. The scale was obtained from the lines painted on the face visible on the background. which were important to derive adequate long-term support requirements. The two-dimensional approach was deemed adequate based upon the geometry of the tunnels, which were much longer in the third dimension. It however did not allow the examination of the actual driving process, whereby local stresses redistribute around and ahead of the tunneling front. The dimensions used in the FLAC model were based on measurements made on site at the beginning of the excavation process. Photographs were taken underground from which precise scaling was done in order to generate a very representative numerical mesh. Figure 1 shows a front view of the drift that was used to construct the numerical model. The FLAC strain-softening/hardening model was used to capture the non-linear behavior of the paste backfill material in its post-elastic range. This particular model considers the cohesion, tensile strength and friction angle to change as a function of the cumulative plastic strain within the material. In the numerical analyses performed, it was assumed that the cohesion and tensile strength of the paste backfill dropped to 25% of their original values after the material had experienced a cumulative plastic strain of 1.5%. (These settings were based upon previous Itasca modeling experience.) The simulations were designed such that they represented the actual sequence of events leading to the drift excavation. The models were initially cycled to equilibrium in order to simulate the various cured and hardened backfilling materials present. The drift was then excavated and the models cycled to equilibrium, with stresses and displacements being monitored throughout the cycling process. The failure mechanisms within the paste backfill and the stability of the drift were investigated as a function of the strength of the paste itself and of the various materials surrounding it. Table 1 shows a brief summary of the model input data used in these parametric analyses. The cohesive strength can be used to get an idea of the compressive strength of the material – assuming a 30° friction angle, cohesion is about 25% of the unconfined compressive strength (UCS). Hence, a 400 kPa cohesive strength paste material would have a UCS of about 1.6 MPa. Table 1. Input property data used in the FLAC numerical analyses. Material type Bulk modulus (MPa) Shear modulus (MPa) Material density (kg/m3) Cohesion (kPa) Tensile strength (kPa) Friction angle (degrees) Paste backfill Caved waste Rockfill 400 100 120 240 60 70 2000 2700 2700 50 to 400 zero 0.0 and 50.0 Half of cohesion Half of cohesion Half of cohesion 32 35 35 198 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-24.qxd 08/11/2003 20:31 PM Page 199 3 FLAC SIMULATIONS Simulations were run for the following cases: (1) a horseshoe shape drift driven with loose material on both sides; (2) a flat back drift also developed with loose material on both sides; (3) a horseshoe shape drift driven without any loose material on either side; and, (4) a horseshoe shape drift driven without any loose material on either side, but with a horizontal discontinuity in the paste material directly above it (in order to investigate potential large-scale slabbing of the paste into the tunnel). 3.1 Case 1 – arched back drift, with loose waste material on both sides This situation prevailed in the 236-8 Access on 1000-2 sub, where caving had occurred along a band of weak waste rock inside the original drift, and some of the Paste Fill Loose Waste Rock fill dry rockfill placed between 1000-3 sub and 1000-2 sub in the secondary stopes on both sides of this access had run into the area due to local caving on the 1000-2 sub horizon. Despite repeated attempts to remove this waste material, uncontrolled runs of fill had resulted in significant amounts of loose material being present above the 1000-2 sub elevation at the time the paste backfill was poured in the 236-8 Access. This, as shown in Figure 1, resulted in loose material being located on both sides of the future drift, at a repose angle of about 45°. Figure 1 also shows the numerical grid, geometry and boundary conditions of the model constructed for this first series of runs. Paste material cohesion values of 50, 100 and 400 kPa were investigated – the results indicated stable self-standing conditions for a paste material with a cohesive strength greater than 50 kPa. As shown in Figure 2, a 50 kPa cohesion resulted in the prediction of a maximum displacement of over 31 cm 12,000 cycles into the simulation. Further stepping of the model (to 15,760 cycles) confirmed the complete failure of the paste backfill material and clearly described the paste failure mechanism taking place under the simulated conditions. It is interesting to note that the back failure was not as pronounced as that of the walls. As intuitively expected, the simulation confirmed that, under vertical (gravity) loading conditions, most of the vertical load around the excavation is deflected and concentrated in the drift walls. The existence of weak contacts between the paste backfill and the loose waste material towards the bottom and on both sides of the drift initiated a deformation of the paste along this contact. This, in turn, led to the shearing of paste material on both sides of the drift. After the side wall failure, the process propagated upwards and led to the shearing of the paste material above the drift, as shown in Figure 3. A maximum displacement of as much as 1.20 m was observed at this later stage. Moreover the unbalanced force history within the model showed that after the initiation of failure the unbalanced force continued to increase, indicating that a progressive failure kept on evolving within the model as no state of equilibrium was being reached. 3.2 Figure 1. Material regions (top), and numerical grid, geometry and boundary conditions (bottom) of the model constructed for the first series of simulations. Following this first set of analyses it was decided to further investigate the failure mechanisms by considering a worst-case drift geometry, which would correspond to a flat back profile. In this case, one would intuitively expect significant roof deformation and failure. All the FLAC simulations done for this case were conducted using identical boundary conditions and input data as for the previous case, except for the geometry of the tunnel itself. 199 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Case 2 – flat back drift, with loose waste material on both sides 09069-24.qxd 08/11/2003 20:31 PM Page 200 Figure 2. Displacement vectors at step 12,000. A maximum displacement of over 31 cm was predicted in this case. Comparing Figure 4 and Figure 2, which both correspond to the same paste material cohesive strength of 50 kPa and the 12,000th analysis step, shows that the maximum displacement predicted is significantly larger in the case of the flat back geometry (1.31 m vs. only about 0.31 m) – this does highlight the improved stability the arched back geometry provides. The same failure mechanism (i.e. a side walls failure first, followed by a shearing effect through the overlying paste material) is however seen in both the flat and arched back arrangements. As in the case of the arched back profile, 100 and 400 kPa paste backfill cohesive strengths resulted in the material maintaining its integrity and the drift remaining stable regardless of its profile. With a 100 kPa paste material cohesion the same overall results were obtained as with the arched back case, but more displacement was predicted. The analyses also showed that if the area backfilled with paste material is damaged, due to dynamic loading from blasting, for example, then the underlying loose waste material does not offer much support against vertical movement in the paste backfill, which could potentially lead to large and even catastrophic failures in the paste material. 200 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-24.qxd 08/11/2003 20:31 PM Page 201 Figure 3. Deformed model geometry at step 15,760. 3.3 Case 3 – arched back drift, with no loose waste material on the sides The aim of this set of analyses was to investigate the stability of a drift driven through paste backfill under “normal” conditions, i.e., with no loose waste material in the pasted region and with the paste material poured directly on compacted rockfill. This situation, illustrated in Figure 5, corresponded to the situation in the 129-7 Access in the south end of the 1000 m Level #1 sub-level elevation. Three different cohesive strength values were again considered for the paste backfill, which were 25, 50 and 100 kPa in this case. No major displacement was predicted to occur in the drift for the cases of the 50 and 100 kPa cohesion. For the 50 kPa cohesion case a maximum displacement of just under 3.7 cm was predicted, whereas this maximum displacement was predicted at just over 3 cm for the 100 kPa cohesion case. 201 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-24.qxd 08/11/2003 20:32 PM Page 202 Figure 4. Displacement vectors at step 12,000 (50 kPa cohesion, flat back drift profile). 3.4 Paste Fill Rock fill Figure 5. View of the material regions modeled in FLAC for the case of the arched back drift, with no loose waste material around the drift. However, complete failure of the drift was predicted to occur when driven in 25 kPa cohesive strength paste backfill. Figure 6 shows the displacement vectors in this case, 8000 cycles into the simulation. Overall, and as intuitively expected, indications were that the absence of 45° piles of loose material underneath the paste fill helped significantly with respect to the stability of a tunnel driven through it. The objective of this analysis was to simulate the effects of a weak horizontal cold joint within the paste backfill, which could have been caused, for example, by interruptions during the pouring process. If sufficiently long interruptions occur in the normally continuous filling process, the previously placed material can cure sufficiently, eventually resulting in a strength discontinuity at the contact with more recently poured material. As shown in Figure 7, a horizontal interface element was thus incorporated at a distance of 1.5 m above the drift back, which represented a very adverse situation with regard to a potential layer of material in the back of the drift developing instability. Zero cohesion and zero tensile strength were assigned to the interface, in order to consider the worst-case scenario. The model was run using a 50 kPa cohesive strength paste material. The displacement results are presented in Figure 8. As shown, the presence of the horizontal discontinuity did not affect the overall behavior of the drift. Comparing Case 3 for a 50 kPa cohesion (which showed an identical situation, but without the horizontal discontinuity) and Case 4, the maximum displacement observed remains small (3.7 cm without the discontinuity vs. 3.6 cm with it). It should be noted that a fairly weak paste backfill (with only a 50 kPa cohesive strength) was used in Case 4. For the “ordinary” strength paste fill used at Brunswick Mine (with a 400 kPa cohesive strength), the 202 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Case 4 – arched back drift with no loose waste material on the sides, but with a horizontal discontinuity in the paste fill above it 09069-24.qxd 08/11/2003 20:32 PM Page 203 Figure 6. Displacement vectors for a cohesion of 25 kPa. This situation evolved into the complete collapse of the tunnel. Paste filled region Interface element Rock fill region Figure 7. Addition of an interface element to model a horizontal discontinuity in the paste fill material. effect of a horizontal discontinuity can, for all intended purposes, be ignored. 4 GENERAL OBSERVATIONS DURING THE FIELD WORK Two different excavation methods were tried underground during the development of these tunnels: a mechanical one, and a drilling and blasting one. Both methods showed promise, but initially required a subsequent finishing step to be carried out to smooth the final paste arch. As expected, the Brunswick Mine 400 kPa cohesive strength paste backfill stood well during the development phase. Based upon the observations made underground during a three-week period, the most significant improvements that could be made to the development procedure in paste backfill would be: (1) the development of a mechanical scraping machine, which would remove and trim all ridges and undulations along the initial excavation boundaries; or, (2) the adjustment of the drilling and blasting practice, in order to precisely cut the proper shape and eliminate damage to the excavation surface. A combination of both could potentially yield the best results, such as the rough mechanical excavation of a center cut, followed by the trimming to the proper shape using controlled blasting. This approach was implemented with success in February 2001. It consisted of excavating a center plug with a scooptram, and of trimming the tunnel to its final dimensions using lightly charged (with B-line detonating cord only) blastholes, including a series of trim blastholes drilled on a 20 cm (8 in) spacing directly along the planned periphery of the tunnel. Good results have also been reported when using a purely mechanical excavation approach, without any subsequent blasting. In these tests, a scooptram-mounted scaler normally used to scale unstable areas was used to trim the excavation to its final shape, after a center cut had first been exca- 203 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-24.qxd 08/11/2003 20:32 PM Page 204 Figure 8. Displacement vectors for a cohesion of 50 kPa. A maximum displacement of just over 3.6 cm was predicted in this case. N Figure 9. Photograph looking west in the 236-8 Access on 1000-2 sub on the footwall side showing the results of the third round blasted there in January 2001. vated with an 8 yd3 scooptram. Certain precautions were however required in order to attain these results, such as carefully leveling the floor beforehand, and ensuring no ground support elements were sticking out above the tunnel surface. This approach also reportedly provided an overall quick cycle time. Significant cycle time improvements can be achieved if a smooth initial arch profile is obtained on a “first pass” since this potentially allows one to defer the application of the required second layer of shotcrete until the end of the excavation process. The postponement of the second layer of shotcrete can however only be considered if a proper arch is created, if no cracks develop in the initial layer of steel fiber-reinforced shotcrete, and if no abnormal inclusions are encountered in the surrounding paste material during the bolting cycle. Figure 9 shows the type of results that were obtained with the blasting approach after it was optimized. The effective advance achieved with this particular shot was 4 m (13 ft), and, as can be seen, the blast was quite successful and produced a uniform fragmentation. ACKNOWLEDGEMENTS The authors would like to thank Noranda Inc. for the permission to present these data and publish this paper. 204 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-25.qxd 08/11/2003 20:32 PM Page 205 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 FLAC3D numerical simulations of ore pillars at Laronde Mine R.K. Brummer & C.P. O’Connor Itasca Consulting Canada Inc., Sudbury, Ontario, Canada J. Bastien, L. Bourguignon & A. Cossette Agnico-Eagle Mines – Laronde Mine, Cadillac, Quebec, Canada ABSTRACT: On November 27, 2002 Agnico Eagle’s Laronde Mine experienced a magnitude 2.6 seismic event. As part of the investigation into the cause of the burst, a high-resolution FLAC3D model was created to determine the effect of the mining sequence on stresses in the area of the burst. At this mine, secondary stopes are intended to fail following primary stope extraction. The numerical simulations showed that where remnants were left with non-ideal geometry (through unfortunate but necessary mining decisions), these remnants could be too strong to yield as intended. The FLAC3D model showed that one such 3-wide pillar centered at the location of the burst was subject to a local high stress concentration. This provided a unique opportunity to confirm calibration of the FLAC3D model. Further modeling also highlighted other areas of the mine where pillars were in a high stress state and recommendations were made to alter the mining sequence to prevent future events. 1 INTRODUCTION Agnico Eagle’s Laronde Mine is a high-tonnage underground mining operation in the Abitibi mining district in Northern Quebec. Currently the majority of the mining takes place at a depth of 1500 meters but a new mining horizon starting on 2150 meters has been in production since early 2002 and will become the major producing area of the mine as the upper levels become depleted. On November 27, 2002 the mine experienced a magnitude 2.6 rockburst between the main levels of 149 and 152 and centered along the main access into these levels. Damage on the 149 Level was light to moderate with some floor heaving and spalling along the footwall. On the 152 Level the damage was much more extensive and resulted in a large failure in the main intersection of the level. The burst occurred approximately 2 hours after a small slot blast in stope 146-20-62, a secondary stope expected to be carrying little stress. The slot blast was quite small and was not a likely trigger for the event although the timing of the burst in close proximity to the slot blast leaves this as a possibility. Fortunately no one was in the area at the time. Typically the mines in the area are seismically quiet which made the event that much more troubling and the cause of the burst needed to be found so similar situations could be avoided in the future. As part of the investigation into the cause of the burst, a high resolution FLAC3D model was built in order to examine the stresses throughout the region and the role of the mining geometry on the event. Other work being conducted at the mine provided a wellcalibrated set of properties to be used for this purpose. 2 MINE LAYOUT AND GEOLOGY Laronde Mine is currently producing 7000 tpd. Main production comes from the stopes in the 152 horizon while new production levels down on the 215 Level come online. In the coming years, the bulk of production is expected to come from the deeper levels as the upper levels become depleted. The geology of Laronde Mine is quite complex with multiple ore bodies spaced parallel to each other (of which 20-Zone is the major producer). The orebody is a gold-zinc massive sulphide with a thickness ranging from 10 to 30 meters. Along each contact of the orebody is a region of highly sheared schist material that can be up to 5 meters in thickness which can, at times, present hangingwall stability problems. There is also a regular banding of highly sheared material 205 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-25.qxd 08/11/2003 20:32 PM Page 206 Figure 1. A typical cross-section of the Laronde orebodies looking East showing the different rock elements. throughout the footwall that is more prevalent in the deeper sections of the mine. Figure 1 shows a typical cross section of the Laronde mine as constructed in the FLAC3D model. The mine uses a primary-secondary blasthole stoping mining method. Stopes are all sized at 15 meters along strike, 30 meters high, and the thickness of the orebody, which typically ranges from 15 to 25 meters. This stope dimension was chosen specifically so that secondary stopes will be in a post-failure state after primary mining. This helps to reduce stress problems by forcing the stresses to the abutments instead of secondary stopes and generally makes secondary mining easier. Pastefill is currently the backfill of choice for primary stopes in the upper levels. Rockfill is used in secondary stopes that will not be mined against. The deeper levels of the mine currently use cemented rockfill until the paste system is extended into this area. The mining sequence is based upon an expanding chevron extending upwards from the 149 Level with secondary stopes being mined first on the 152 Level. Secondary stopes typically lag behind the primaries by 2 stopes. For several reasons, the mining sequence had some instances in which the ideal mining shape could not be maintained. First, the main accesses to the 21-Zone run directly through the 64 and 66 series of stopes (refer to Fig. 2). This presented some stability concerns for these accesses if the 65 stope was brought up to its ideal position in the sequence. Additionally, on the 152 Level, the secondary (i.e. even numbered stopes) are mined first and were set up in a retreating fashion from each abutment back towards the main entrance of the level. These two scenarios combined to make a series of pillars three stopes wide. Unlike a single secondary stope, a three-wide stope is expected to be too large to fail. The burst appears to have been caused by a slippage along a foliated zone that runs parallel to the orebody and right through the back of the 152 Level. This same foliated zone also passes through the lower footwall of the 149 Level. The damage seen on both levels occurred along this contact. It was hypothesized Figure 2. Long section looking North of Laronde showing the current mining sequence. that the high stresses being forced through the threestope-wide pillar on 152 level was the driving force for the slippage along the foliated zone, but without a numerical model to determine the stresses in the region, no solid answers could be gleaned. 3 EVENT The seismic event measured 2.6 on the Nuttli scale and was centered on a foliated zone running between the 149 Level and the 152 Level. An investigation revealed what was thought to be a probably cause for the event. The three stope-wide pillar centered in the lower abutment on the 152 Level had been created through the mining sequence which likely provided the driving force for the event by concentrating stress through this region. At the same time, the foliated zone intersecting a long strike distance along both upper and lower levels provided a method of release by which the foliated zone was free to move. It was believed that the intense stress concentration wrapping around the lower abutment and through the three-wide pillar on the 152 Level provided enough of a driving force to cause a slip along the foliated zone. Most of the damage on 149 Level occurred on the footwall side along the floor with lots of displaced 206 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-25.qxd 08/11/2003 20:32 PM Page 207 Foliated Zone 152-20-64 Stope 20-North Orebody. Mined and Backfilled. 149 Level Shear stress on the foliations 152 Level Stress trajectories Figure 3. Simplified cartoon view of the intersection of the foliated zone with the levels involved in the burst. Figure 5. Looking East on the 152 Level. The damage is more severe especially to the east. A large amount of material up to 2 meters deep is seen in the main entrance to the level on the lower left. 4 Figure 4. Damage on the 149 Level. Most of the debris came from the base of the footwall (right side) with some additional secondary bagging of material in the screen (upper left). slabbing and some bagging of material in the screen (see Fig. 4). On the 152 Level, the most damage occurred right in the back of the stope with the large intersection failure being the dominant feature. Accessibility to the east was limited but most of the damage seemed to occur towards this direction (see Fig. 5). One of the other questions that arose in the aftermath of the burst was how to ensure that a similar situation did not occur again. Since the burst appeared to have been caused by a combination of the mining sequence and the unfortunate location of the foliated zone running directly through two main drifts, this particular mechanism might be a one-time event. However, this does not eliminate the possibility that pinch points in other areas could not cause seismicity through another mechanism. In order to try and determine the validity of this theory, a high resolution FLAC3D model of the area was created in order to determine the anticipated stresses and failure zones passing through the three-wide pillar. At the time of the event, Itasca was actively involved in modeling using FLAC3D at Laronde Mine on another project. From this other work and previous projects at the mine going back to 1997, a wellcalibrated set of material properties and stresses were available for the model. Previous model work however did not have sufficient resolution to be useful in such a specific case and so a more detailed model of the region was created. The model generated for the burst investigation focused on the 152 mining horizon with a block size through the area of interest of 3 meters on a side. With the stopes 15 meters wide this provided 5 blocks along the strike of the stope which was deemed important to ensure a proper modeling of pillar and confinement effects. 5 RESULTS The FLAC3D model provided evidence that the assumption of the three-wide pillar being created through the retreating extraction sequence resulted in a pillar that was too large to fail and hence became a stress concentrator. Figure 6 shows a principal stress plot on a long section looking North through the orebody. The location of the burst matches nearly perfectly with the high stress concentration predicted in the FLAC3D model. Figure 7 (which is the section marked as A-A in Figure 6 through the 64 stope) gives an indication of how the stress concentration in the pillar acts on the foliated zone some 30 meters into the footwall, causing it to slip. The stresses are deflected under the lower abutment and concentrated through the three-wide pillar, resulting in a vertical stress component. This 207 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands FLAC3D MODEL 09069-25.qxd 08/11/2003 20:32 PM Page 208 Figure 6. Maximum principal stress plot looking North showing the location of the three high stress pillars and the location of the burst. Figure 7. A cross section looking East through the region of the burst (section A-A in Fig. 6). vertical component appears to have provided enough localized stress on the foliated zone to cause the slip. From the mining plan, there are two other threewide pillars that appear to be concentrating stresses. A series of modeling runs was conducted in order to determine how the short-term mining plan needed to be adjusted to prevent additional stress building up in these areas. Figure 6 shows these areas above and to the East of the location of the burst. These two stopes, although not as critical as the one that caused the burst, were cause for concern. The short term mining plan did not include these particular stopes although after the modeling, recommendations were made to mine these stopes as early as possible. The additional scenarios showed that the mining of adjacent stopes created incremental increases in the stresses in these areas, so although they need not be mined immediately, a rapid development and 208 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-25.qxd 08/11/2003 20:32 PM Page 209 Figure 8. Maximum principal stress plot showing the changes caused by mining 143-20-65 stope. Figure 9. Maximum principal stress along a long section looking North following the mining of 155-20-59 stope in the bottom left. production schedule should be implemented in order to avoid further seismic events in the region. Using the simulations for the short term mining plan of the area, it was determined that the best scenario would involve mining the 143-20-65 pillar first. Figure 8 shows the stresses around the 152 mining horizon after mining out of 143-20-65 stope. The decision to recommend mining this stope first was it was made because it was postulated that the burst had likely dissipated some of the stored energy in 152-2064 stope and it was therefore unlikely that a second seismic event would occur in the short term. Also some significant rehabilitation on 152-level would delay the development and production of this stope by a couple of months. The mining of 143-20-65 would not shed much additional stress on the other stopes but 209 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-25.qxd 08/11/2003 20:32 PM Page 210 would take care of the most worrisome of the remaining three-wide areas. The mining of 143-20-65 stope would shed a small amount of additional stress on 143-20-69 but this stope was partially protected by the lead primary stope between them. By the time 143-20-65 was mined and filled, rehabilitation of the 152 Level should be nearly completed which would then be immediately developed to allow the mining of 152-20-64 (the location of the burst). This should push the entire lower abutment stress below the 152 level and relieve what stress was left after the burst. Finally, 143-20-69 does not appear to be critical in the short term as most of the mining in this area is to the west and this stope is well shielded from these stopes. In order to provide some short-term production while the 152 Level was closed for cleanup operations, the possibility of taking a stope down on 155 Level was examined. Figure 9 shows the results of the modeling of this particular scenario. A concentration of stresses appears to occur two stopes away on either side of the mined stope. The proximity of this increased stress to the location involved in the burst resulted in a recommendation not to mine this particular stope until 15220-64 was mined, relieving this stress concentration. 6 CONCLUSION Based on the on-site investigation and the FLAC3D modeling, the mechanism responsible for the burst appears to have been well established. A three stope wide pillar on the 152 Level resulted in a large stress concentration. This stress concentration resulted in an increased vertical stress component in the footwall of the orebody, allowing the foliated zone to slip between the 149 Level and the 152 Level. The FLAC3D model was also able to provide recommendation on which other areas might be of concern as well as the best sequence in which to take care of these problem areas. Finally, the modeling showed that mining of a stope down on the 155 Level would not be prudent at this stage until the three-wide pillar responsible for the burst was removed. ACKNOWLEDGEMENTS The authors thank Agnico Eagle Mines for permission to publish this paper. 210 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-26.qxd 08/11/2003 20:33 PM Page 211 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Modeling arching effects in narrow backfilled stopes with FLAC L. Li, M. Aubertin & R. Simon École Polytechnique de Montréal, Quebec, Canada B. Bussière & T. Belem Université du Québec en Abitibi-Témiscamingue, Quebec, Canada ABSTRACT: Numerical tools can be very useful to investigate the mechanical response of backfilled stopes. In this paper, the authors show preliminary results from calculations made with FLAC. Its use is illustrated by showing the influence of stope geometry. The results are compared with analytical solutions developed to evaluate arching effects in backfill placed in narrow stopes. Some common trends are obtained with the two approaches, but there are also some differences regarding the magnitude of the stress redistribution induced by fill yielding. 1 INTRODUCTION Even though backfill has been placed in underground stoping areas for many decades, it can be said that backfilling still remains a growing trend in mining operations around the world. This is particularly the case in Canada where significant efforts have been devoted, over the last 25 years or so, to improve our understanding of mining with backfill (e.g. Nantel 1983, Udd 1989, Hassani & Archibald 1998, Ouellet & Servant 2000, Belem et al. 2000, 2002). In recent years, the increased use of backfill in mining has been fuelled by environmental considerations (e.g. Aubertin et al. 2002). Many regulations now favor (and sometimes require) that the maximum quantity of wastes from the mine and the mill be returned to underground workings. This practice may induce significant advantages, as it can reduce the environmental impact of surface disposal and the costs of waste management during mine operation and upon closure. The first purpose of mine backfill is nevertheless to improve ground control conditions around stopes. Various types of fills can be used to reach this goal, each with its own characteristics. Backfill is often required to offer some self support properties, so it generally includes a significant proportion of binder such as Portland cement and slag. But even the strongest backfill is “soft” when compared to the mechanical properties of the adjacent rock mass. This difference in stiffness and yielding characteristics between the two materials can be the source of a stress redistribution in the backfill and surrounding walls, as deformation of the backfill under its own weight may create shear stresses along the interface. For relatively narrow stopes, the load transfer to the stiff abutments induces arching effects. When this phenomenon occurs, the vertical stress below the main arching area tends to become smaller than the backfill overburden pressure, as shown by in situ measurements (e.g. Knutsson 1981, Hustrulid et al. 1989). The same type of arching behavior is also known to occur in other types of structural systems, where a relatively soft material (like soil and grain) is placed between stiff walls; examples include silos and bins (Richards 1966, Cowin 1977, Blight 1986), ditches (Spangler & Handy 1984), and retaining walls (Hunt 1986, Take & Valsangkar 2001). Arching effects and load redistribution can be investigated using physical models, in situ measurements, analytical solutions, and numerical methods. The latter two approaches are particularly attractive to identify the main influence factors, and to evaluate how these may affect the load distribution in and around backfilled stopes. In a recent paper (Aubertin et al. 2003), the authors have proposed simple equations based on the Marston (1930) theory to evaluate the load distribution in stope backfill. The results of analytical calculations have been compared to numerical modeling performed with a commercially available finite element code. The calculation results highlighted some important differences between the two approaches, for the specific set of assumptions adopted. In this paper, the authors use FLAC (Itasca 2002) to further advance our understanding of the load transfer process in and around narrow backfilled stopes. In these calculations, some of the assumptions and input conditions differ from the previous FEM calculations, 211 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-26.qxd 08/11/2003 20:33 PM Page 212 including the use of a somewhat more representative constitutive model for the backfill. The mining sequence is also taken into account. It is shown that for specific cases amenable to analytical solutions, the calculated results from both approaches are fairly close to each other. rock mass layer element void space backfill stope h V dh 2 ARCHING EFFECTS H Arching conditions can occur in geomaterials such as soil, jointed rock mass and backfill, when differential straining mobilizes shear strength while transferring part of the overburden stress to stiffer structural components. Arching typically occurs when portions of a frictional material yield while the neighboring material stays in place. As the yielding material moves between stable zones, the relative movement within the former is opposed by shear resistance along the interface with the latter. The shear stress generated along the contact area tends to retain the yielding material in its original position. This is accompanied by a pressure reduction within the yielding mass and by increased pressure on the adjacent stiffer material. Above the position of the main arch, a large fraction of the total overburden weight in the yielding mass is transferred by frictional forces to the unyielding ground on both sides. Investigations on models and in situ measurements have shown that the magnitude of the stress redistribution depends to a large extent on the deformation of the walls confining the soft material (e.g. Bjerrum 1972, Hunt 1986). A few analytical solutions have been developed to analyze the pressure distribution since the pioneering work of Janssen (1895) (see Terzaghi 1943 for early geotechnical applications). Among these, the Marston (1930) theory was proposed to calculate the loads on conduits in ditches (see also McCarthy 1988). The authors have used this theory to develop an analytical solution for arching effects in narrow backfilled stopes (Aubertin 1999). Figure 1 shows the loading components for a vertical stope. On this figure, H is the backfill height, B the stope width, and dh the size of the layer element; W represents the backfill weight in the unit thickness layer. At position h, the horizontal layer element is subjected to a lateral compressive force C, a shearing force S, and the vertical forces V and V dV. The force equilibrium equations for the layer element provide an estimate of the stresses acting across the stope (Aubertin et al. 2003). From these, the vertical stress can be written as follow: C W S S B V + dV rock mass B Figure 1. Acting forces on an isolated layer in a vertical stope. with (2) where vh and hh are the vertical and horizontal stresses at depth h, respectively; represents the unit weight of the backfill; is the effective friction angle between the wall and backfill, which is often taken as the friction angle of the backfill, bf. Equations 1 and 2 constitute the Marston theory solution. In this representation, K is the reaction coefficient corresponding to the ratio of the horizontal stress hh to vertical stress vh. This reaction coefficient depends on the horizontal wall movement and material properties. When there is no relative displacement of the walls, the fill is said to be at rest, and the reaction coefficient is usually given by (Jaky 1948): (3) where bf is the friction angle of the backfill. For typical fill properties ( bf ≅ 30° to 35°), K0 is much smaller than unity. If the walls move outward from the opening, the horizontal pressure might be relieved, and the reaction coefficient tends toward the active pressure coefficient, which can be expressed as (Bowles 1988): (4) with Ka K0. If an inward movement of the walls compress the fill, it increases the internal pressure. Then, the reaction coefficient tends toward the passive condition, which becomes (Bowles 1988): (1) 212 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands dh C (5) 09069-26.qxd 08/11/2003 20:33 PM Page 213 shh Marston theory void space shh 0.3 overburden rock mass E = 300 MPa H = 45 m for B = 6m 0.2 0.1 (linear elastic) = 0.2 ã = 1800kg/m3 ' = 30° E = 30 GPa = 0.3 = 2700 kg/m3 c = 0 kPa 0 0 1 2 h/B 3 y 4 B=6m Figure 2. Overburden pressures are compared to vertical (vh) and horizontal (hh) stresses calculated with the Marston theory (Eqs. 1–2), with bf 30°, 0.02 MN/m3, and K K0 0.5. with Kp 1 K0. In the above equations, it is assumed that cohesion is low in the backfill; the fill then behaves as a granular material. Based on limit equilibrium, it can be anticipated that a cohesion would increase Kp but decrease Ka. However, more work is needed to investigate its influence on arching effects and stress distribution. Figure 2 shows values of vh and hh calculated with Equations 1 and 2 (with K K0 0.5), and calculated with the overburden pressure (i.e. vh h and hh K0 vh). It can be seen that the overburden stress represents the upper-bound condition, which is applicable for low fill thickness (or for wide stopes). Typically, when H 2 to 3B, the pressure near the bottom of the stope becomes more or less independent of the depth of the fill, in accordance with measurements in bins (Cowin 1977). 3 NUMERICAL CALCULATIONS 3.1 v natural stresses h = 2v backfill svh 0.4 stress (MPa) 0.5m svh 0.5 Vertical stope Recently, the authors have shown some preliminary calculation results obtained with a finite element code (Aubertin et al. 2003). Significant differences have been revealed between the Marston theory and these numerical calculations, which may be explained, in part, by different assumptions associated to the two approaches. In this paper, the same geometry and material properties (Fig. 3a) are used for the basic calculations made with FLAC. The dimensions of the opening are H 45 m and B 6 m, with a void of 0.5 m left at the top of the stope. The natural in situ vertical stress v in the rock mass is obtained x (a) 1 (b) 1 Figure 3. (a) Narrow stope with backfill (not to scale) used for modeled with FLAC; the main properties for the rock mass and backfill are given using classical geomechanical notations; (b) Schematic stress-strain behavior of the backfill (available as a material model in FLAC). by considering the overburden weight (for an overall depth of 250 m). The natural in situ horizontal stress h is taken as twice the vertical stress v, which is a typical situation encountered in the Canadian Shield. The rock mass is homogeneous, isotropic and linear elastic, while the granular backfill obeys a Coulomb criterion. Figure 3b shows the stress-strain relation used with the Coulomb plasticity model available in FLAC. This constitutive behavior is different from the one used for the finite element calculations presented in Aubertin et al. (2003), which was of the elastic-brittle type. There are 213 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands depth = 250 m rock mass 09069-26.qxd 08/11/2003 20:33 PM Page 214 modeling with FLAC-2D 0.8 overburden stress yy (MPa) 0.6 Marston theory K = 1/3 K = 1/2 K=3 0.4 0.2 0 0 9 18 (a) 36 45 modeling with FLAC-2D overburden stress Marston theory 0.3 xx (MPa) 27 h (m) 0.2 K=3 0.1 K = 1/2 K = 1/3 0 0 (b) 9 18 27 36 45 h (m) Figure 5. Comparison of the stresses calculated along the vertical central line, at different elevations h, with the analytical and numerical solutions: (a) vertical stress yy; (b) horizontal stress xx. Figure 4. Stress distribution in the backfilled stope calculated with FLAC: (a) vertical stress yy; (b) horizontal stress xx. no interface elements in the calculations made with FLAC (see discussion). The mining and filling sequence is considered as follow in the numerical modeling. The whole stope is first excavated, and calculations are then performed with FLAC to an equilibrium state. Backfill is placed in the mined stope afterward, with the initial displacement field set to zero when the calculation is performed. In this manner, wall convergence before backfilling is not included in the calculations (this assumption is discussed in Section 4). Figure 4 shows the vertical stress (Fig. 4a) and horizontal stress (Fig. 4b) distribution in the backfilled stope. As can be seen, the vertical and horizontal stresses show a non-uniform distribution. At a given elevation, both stresses tend to be lower along the wall than at the center. The stresses along the central line increase more slowly than the overburden pressures with depth. This indicates that arching does take place in this backfilled stope. Figures 5 and 6 present modeling results for stresses along the full height, with the overburden and the 214 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-26.qxd 08/11/2003 20:33 PM Page 215 modeling with FLAC-2D 0.8 modeling with FLAC-2D overburden stresses Marston theory 1.2 over burden stress Marston theory 0.8 σyy (MPa) σyy (MPa) 0.6 K = 1/3 K = 1/2 K=3 0.4 0.2 K = 1/3 K=3 0.4 K = 1/2 0 0 9 18 (a) 36 0 45 0 h (m) 2 (a) modeling with FLAC-2D 0.2 overburden stresses Marston theory Marston theory K=3 0.1 0.2 K = 1/2 K = 1/3 0 9 18 K=3 0.1 K = 1/3 0 6 modeling with FLAC-2D 0.3 K = 1/2 (b) 4 x (m) overburden stress σxx (MPa) σxx (MPa) 27 27 36 0 45 0 h (m) (b) Figure 6. Comparison of the stresses on the wall calculated at different elevations h, with the analytical and numerical solutions: (a) vertical stress yy; (b) horizontal stress xx. Marston theory solutions. As expected, the overburden stress is fairly close to analytical and numerical results when the backfill depth is small. At larger depth, arching effects become important and the vertical and horizontal stresses tend to be lower than those due to the overburden weight of the fill. However, the numerical results indicate that the Marston theory typically overestimates the amount of stress transfer, hence underestimating the magnitude of the vertical stress yy and of the horizontal stress xx along the stope central vertical line (Fig. 5). Along the walls (Fig. 6), the horizontal stress is also underestimated by the Marston theory, while the vertical stress component yy would be overestimated for the active and at rest cases, with K 1/2 or 1/3, respectively (and underestimated with K 3, but the passive case is not representative of this system behavior). Figure 7 shows the stress distribution on the floor of the stope, as obtained from the numerical and analytical solutions. It can be seen that the overburden pressure 4 6 x (m) Figure 7. Stresses calculated at the bottom of the vertical stope, with the analytical and numerical solutions; (a) vertical stress yy; (b) horizontal stress xx. exceeds the stress magnitudes given by the Marston theory (with K 1/2 and 1/3), which is in fair agreement with the numerical simulations. 3.2 Inclined stope Mining stopes are rarely vertical. The inclination of the foot-wall and hanging-wall may have a non-negligible effect on the load distribution. Figure 8 shows the geometry of an inclined backfilled stope modeled with FLAC (a similar stope was also modeled with the FEM code – see Aubertin et al. 2003). The rock mass and fill properties as well as the in situ natural stresses are identical to the previous case (see Fig. 3). Figure 9 shows numerical calculations and results based on overburden pressure and on the Marston theory solution (without modification for inclination). The horizontal stress calculated with FLAC along the 215 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 2 0.5m 09069-26.qxd 08/11/2003 20:33 PM Page 216 inclined central line of the stope is fairly close to the analytical solution (Fig. 9a), but the vertical stress is underestimated by the Marston theory (see Fig. 9b). Hence, modifications could be required to apply such analytical approach to the case of inclined stopes. void space backfill h 4 DISCUSSION H = 45 m rock mass B=6m 4.1 rock mass v stope y h = 2v 60˚ depth = 250 m x Figure 8. The inclined backfilled stope modeled with FLAC (properties are given in Fig. 3). modeling with FLAC-2D overburden Marston theory xx (MPa) 0.2 K =3 0.1 K = 1/2 K = 1/3 h (m) 0 0 (a) 0.4 9 18 27 36 45 modeling without FLAC-2D Influence of mining sequence In the numerical calculations presented in Aubertin et al. (2003), the mining sequence was not taken into account, so the wall convergence due to elastic straining of the rock mass was imposed on the fill. This created an increase of the mean stress in the fill, while vertical and horizontal stresses locally exceeded the overburden pressure and the Marston theory solution (near mid-height of the stope). Modeling in this manner implies that the backfill is placed in the stope before wall displacement takes place. For a single excavation stope, this is not a realistic representation (at least for hard rock masses). However, with a cut-and-fill mining method where the mining slices (or layers) are relatively small compared to the whole height of the stope, filling is usually made quickly after each cut. In this case, wall convergence after each cut compress the fill already in place (Knutsson 1981, Hustrulid et al. 1989). The inward movement of the walls may then create a condition closer to the passive pressure case. When a stope is excavated in a single step, wall convergence essentially takes place before any backfilling. If the rock mass creep deformation is negligible, the numerical modeling approach presented here is more appropriate. In this case, the Marston theory, with the “at rest” reaction coefficient (K K0) can be used to estimate the induced stresses in a narrow vertical backfill (see Figs. 5–7), at least for preliminary design calculations. overburden 4.2 yy (MPa) Marston theory 0.2 K=3 K = 1/2 K = 1/3 0 0 (b) 9 18 27 36 45 h (m) Figure 9. Comparison between stresses obtained with numerical and analytical solutions along the central line of the inclined stope: (a) horizontal stress xx; (b) vertical stress yy. 216 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Marston theory limitations Analytical solutions can be useful engineering tools as they are generally quick, direct and economic when compared to numerical methods. However, analytical solutions are only available for relatively simple situations and may involve strong simplifying hypotheses. For instance, with the Marston theory, the shear stress along the interface between the rock and fill is deduced from the Coulomb criterion (see details in Aubertin et al. 2003). Its value then corresponds to the maximum stress sustained by the fill material, as postulated in the limit analysis approach (e.g. Chen & Liu 1990). However, numerical simulations indicate that this assumption is not fully applicable. Figure 10 shows that for the vertical stope analyzed here the maximum shear stress is only reached near the bottom part of the 09069-26.qxd 08/11/2003 20:33 PM Page 217 modeling with FLAC-2D 0.04 0.5 at rest Marston theory K 0.02 h (m) σxy (MPa) 0 0 -0.02 9 18 27 36 0.3 45 -0.04 -0.06 0 2 x (m) 4 6 K = 1/2 -0.08 Figure 10. Comparison of shear stress distribution along the wall. 0.12 xx (MPa) modeling with FLAC 0.1 K = 1/3 K=3 active at floor at 1/2H at 1/2H at 3/4H uniformly distributed across the full width of the stope. Results shown in Figure 11 indicate that this is in accordance with numerical calculations for the horizontal stress component (Fig. 11a), but not for the vertical stress which shows a less uniform distribution (Fig. 11b). Also, this simplified theory considers that the reaction coefficient, K, depends exclusively on the fill property and not on the position in the stope. Results shown in Figure 12 indicate that this hypothesis is not too far from the numerical results. Near the boundary, the value of K would nevertheless be better described by a K value between Ka and K0. Work is underway to modify the analytical solution to extend the use of the Marston theory to more general cases. at 3/4H 0.1 at 1/2H at 1/4H 0.08 0.06 modeling with FLAC-2D 0.04 0 2 4 6 x (m) (a) Figure 12. Reaction coefficient K obtained with analytical and numerical solutions across the full width of the vertical stope at different elevations h. at 3/4H 0.3 4.3 Constitutive behavior yy (MPa) at 1/2H 0.2 at 1/4H 0.1 modeling with FLAC-2D 0 (b) 0 2 x (m) 4 6 Figure 11. Distribution of (a) lateral pressure xx and (b) vertical stress yy obtained with FLAC across the full width at different elevations of the vertical stope. stope. Hence, arching effect and stress redistribution are thus exaggerated. Another important assumption in the Marston theory is that both the horizontal and vertical stresses are The reliability of any numerical calculations depends, to a large extent, on the representativity of the constitutive models used for the different materials (and on the corresponding parameter values). In this paper, a Coulomb plasticity model (see Fig. 3) was employed for the fill material. This model is representative of some aspects of the mechanical behavior of backfill, such as the nonlinear relationship between the stress and strain (e.g. Belem et al. 2000, 2002). However, this simplified model neglects some important characteristics of the media, including its pressure dependent behavior under relatively large mean stresses. More representative models, such as the modified Cam-Clay model, are built in FLAC (e.g. Detournay & Hart, 1999). However, the application of such model is not straightforward because of the difficulties in obtaining the relevant material parameters. The influence of cohesion due to cementation and possible oxidation of the fill material may also be relevant to include in the analyses. 217 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-26.qxd 08/11/2003 20:33 PM Page 218 An interesting aspect of FLAC is that it allows userdefined models, which can be introduced with the language FISH. The authors are now working on introducing in FLAC a multiaxial, porosity dependent criterion (Aubertin et al. 2000, Li & Aubertin 2003) for the yielding and failure conditions of geomaterials. This aspect will be presented in upcoming publications. 4.4 ACKNOWLEDGEMENT Part of this work has been financed through grants from IRSST and from an NSERC Industrial Chair (http://www.polymtl.ca/enviro-geremi/). The authors would also like to thank the anonymous reviewers who provided valuable comments to improve the manuscript. Interface elements along the walls As was done with a finite element code in a previous investigation (Aubertin et al. 2003), some calculations were also performed with interfaces included in FLAC, to represent the contact between backfill and rock mass. Preliminary results (not shown here) indicate that the presence of interfaces along the walls and floor of the stope, which allow localized shear displacements, has relatively little influence on the stress distribution in the stope and at its boundary. Some differences between the cases shown here and models with interfaces nevertheless appear near the bottom and top of the stope where some stress reorientation and concentration seem to take place. This aspect however requires further investigation. The applicability of the (Coulomb) strength criterion and the numerical stability of the calculations along these elements also need more study. 5 CONCLUSION In this paper, numerical simulations have been performed with FLAC for a vertical and an inclined backfilled stope geometry. The results are compared to the Marston theory solutions. It is shown that the results obtained with the Marston theory can be considered as acceptable, especially for preliminary calculations. Nevertheless, the numerical results also reveal that the Marston theory tends to overestimate arching effect, and thus underestimate the stress magnitude near the bottom of backfilled stope. Also, the influence of the mining sequence can not be introduced in the Marston theory. The numerical results indicate that the filling sequence can significantly influence the stress distribution in and around filled stopes. For inclined stopes, the Marston theory is of limited use to estimate the stress magnitude. Additional work is underway into both analytical and numerical solutions to better describe the behavior of backfilled stope. More work is also needed to study the rock-fill interface behavior and the actual field response of backfill in stopes. Other important issues also remain to be resolved, including the possible degradation of the arch due to low pressure (and tensile stresses), the influence of water flow and distribution in backfilled stopes, the evolving properties of the fill material (particularly considering the action of cement in the presence of sulfide minerals), the dynamic response of the backfill, and the forces generated on retaining structures. REFERENCES Aubertin, M. 1999. Application de la Mécanique des Sols pour l’Analyse du Comportement des Remblais Miniers Souterrains. Short Course (unpublished notes), 14e Colloque en Contrôle de Terrain, Val-d Or, 23–24 mars 1999. Association Minière du Québec. Aubertin, M., Bussière, B. & Bernier, L. 2002. Environnement et Gestion des Rejets Miniers. Manual on CD-ROM, Presses Internationales Polytechniques. Aubertin, M., Li, L., Arnoldi, S., Belem, T., Bussière, B., Benzaazoua, M. & Simon, R. 2003. Interaction between backfill and rock mass in narrow stopes. In P.J. Culligan, H.H. Einstein & A.J. Whittle (eds), Soil and Rock America 2003, 1: 1157–1164. Essen: Verlag Glückauf Essen (VGE). Aubertin, M., Li, L. & Simon, R. 2000. A multiaxial stress criterion for short term and long term strength of isotropic rock media. International Journal of Rock Mechanics and Mining Sciences, 37: 1169–1193. Belem, T., Benzaazoua, M. & Bussière, B. 2000. Mechanical behavior of cemented paste backfill. Proceedings of the 55th Canadian Geotechnical Conference, 1: 373–380. Canadian Geotechnical Society. Belem, T., Benzaazoua, M., Bussière, B. & Dagenais, A.M. 2002. Effects of settlement and drainage on strength development within mine paste backfill. Tailings and Mine Waste ’02: 139–148. Swets & Zeitlinger. Blight, G.E. 1986. Pressure exerted by materials stored in silos. Part I: coarse materials. Géotechnique, 36(1): 33–46. Bjerrum, L. 1972. Embankments on Soft Ground. Proceedings of the ASCE Specialty Conference on Performance of Earth and Earth Supported Structures, 2: 1–54. New York: ASCE. Bowles, J.E. 1988. Foundation Analysis and Design. McGrawHill. Chen, W.F. & Liu, X.L. 1990. Limit Analysis in Soil Mechanics. Amsterdam: Elsevier. Cowin, S.C. 1977. The theory of static loads in bins. Journal of Applied Mechanics, 44: 409–412. Detournay, C. & Hart, R. (eds) 1999. FLAC and Numerical Modeling in Geomechanics – Proceedings of the International FLAC Symposium, Minneapolis, Minnesota, 1–3 September 1999. Rotterdam: Balkema. Hassani, F. & Archibald, J.H. 1998. Mine Backfill. CIM, CD-ROM. Hunt, R.E. 1986. Geotechnical Engineering Analysis and Evaluation. New York: McGraw-Hill Book Company. Hustrulid, W., Qianyuan Y. & Krauland, N. 1989. Modeling of cut-and-fill mining systems – Näsliden revisited. In F.P. Hassani, M.J. Scoble & T.R. Yu (eds), Innovation in Mining Backfill Technology: 147–164. Rotterdam: Balkema. 218 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-26.qxd 08/11/2003 20:33 PM Page 219 Itasca Consulting Group, Inc. 2002. FLAC – Fast Lagrangian Analysis of Continua, User’s Guide. Minneapolis, MN: Itasca. Jaky, J. 1948. Pressure in silos. Proceedings of the 2nd International Conference on Soil Mechanics and Foundation Engineering, 1: 103–107. Rotterdam: Balkema. Janssen, H.A. 1895. Versuche über Getreidedruck in Silozellen. Zeitschrift Verein Ingenieure, 39: 1045–1049. Knutsson, S. 1981. Stresses in the hydraulic backfill from analytical calculations and in-situ measurements. In O. Stephansson & M.J. Jones (eds), Proceedings of the Conference on the Application of Rock Mechanics to Cut and Fill Mining: 261–268. Institution of Mining and Metallurgy. Li, L. & Aubertin, M. 2003. A general relationship between porosity and uniaxial strength of engineering materials. Canadian Journal of Civil Engineering (in press). Marston, A. 1930. The theory of external loads on closed conduits in the light of latest experiments. Bulletin No. 96, Iowa Engineering Experiment Station, Ames, Iowa. McCarthy, D.F. 1988. Essentials of Soil Mechanics and Foundations: Basic Geotechnics. 4th edition, Prentice Hall. Nantel, J.H. 1983. A review of the backfill practices in the mines of the Noranda Group. In S. Granholm (ed), Mining with Backfill: Proceedings of the International Symposium on Mining with Backfill: 173–178. Rotterdam: Balkema. Ouellet, J. & Servant, S. 2000. In-situ mechanical characterization of a paste backfill with a self-boring pressuremeter. CIM Bulletin, 93(1042): 110–115. Richards, J.C. 1966. The Storage and Recovery of Particulate Solids. Institution of Chemical Engineers, London. Spangler, M.G. & Handy, R.L. (1984). Soil Engineering. Harper & Row. Take, W.A. & Valsangkar, A.J. (2001). Earth pressures on unyielding retaining walls of narrow backfill width. Canadian Geotechnical Journal, 38: 1220–1230. Terzaghi, K. 1943. Theoretical Soil Mechanics. John Wiley & Sons. Udd, J.E. 1989. Backfill research in Canadian mines. In F.P. Hassani, M.J. Scoble & T.R. Yu (eds), Innovation in Mining Backfill Technology: 3–13. Rotterdam: Balkema. 219 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:44 AM Page 221 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 FLAC3D numerical simulations of deep mining at Laronde Mine C.P. O’Connor, R.K. Brummer & P.P. Andrieux Itasca Consulting Canada Inc., Sudbury, Ontario, Canada R. Emond & B. McLaughlin Agnico-Eagle Mines – Laronde Mine, Cadillac, Quebec, Canada ABSTRACT: Agnico-Eagle’s Laronde Mine is currently investigating mining options down to 3000 meters below surface. FLAC3D was used to simulate the entire current mine from 1340 m to 2150 m below surface as well as the potential future expansion. FLAC3D allowed each of the three main ore lenses to be modeled together, providing information on the interaction between lenses which had never been available before. The model provides information useful for determining the ideal stope dimensions, mining method, mining sequence, support options for large excavations, as well as the best option for the shaft location. be encountered when placing infrastructure at depth. 1 INTRODUCTION Agnico Eagle’s Laronde Mine is a 7000 tpd underground operation located in Northern Quebec in the Abitibi Mining district near Rouyn-Noranda. Drilling from the bottom levels of the current mine have shown that there are significant reserves down to at least 3000 m below surface. Most of the tonnage has historically come from above the 152 Level but with a shift currently taking place to turn the 215 Level into the major source of ore as the upper levels are progressively becoming depleted. As part of the feasibility into the potential expansion down to 3000 m, a geomechanical review of the proposed expansion was conducted using FLAC and FLAC3D as numerical modeling tools to determine the anticipated response to mining at extreme depths. FLAC3D was used in several forms. First, it was used to model the entire mine from the top of the current mining horizon down to a depth of 3000 m. From this model, the in-situ and post-mining stresses were traced along with an analysis of the interaction between the different ore lenses in the upper levels. The second model was a high resolution mining method model used to determine the stresses and failure zones for different sized stopes and different ore thicknesses at extreme depths. FLAC was used in order to check shaft stresses at depth including post-mining stress changes. FLAC was also used to model the stresses around a hypothetical large excavation (conveyor drift) in order to investigate some of the issues that could 2 MINE GEOMETRY Laronde Mine is located in the Abitibi Mining district in northern Quebec. The orebody is a gold-zinc deposit that is part of an extensive intrusive complex that runs throughout the region. The 20-North deposit, which is the major producer, runs from a depth of 900 m down to at least 3000 m but currently mining is only taking place down to 2150 m. The orebody is steeply dipping to the South and raking towards the West. The current mining method used is primarysecondary stoping with high quality backfill (mainly pastefill). This method has been used since the mining of 20-North began and has proved to be successful in maintaining good stability in the hangingwall and minimizing stress related problems. Secondary stopes are designed to fail with the extraction of primary stopes, which results in a relatively low stress environment in which to mine (the secondary stopes are not large enough to carry significant stresses in post failure). Current stopes are 15 m along strike, 30 m high and the thickness of the orebody. With a few exceptions, this system has produced relatively trouble free mining at the current depths. The progression to greater depths however will result in greater stresses and more extensive failure zones that will cause greater difficulties than have been experienced to-date. 221 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:44 AM Page 222 Figure 1. Cross-section looking East of the Laronde orebody showing the different rock units included in the modeling. 180 MPa. The orebody itself is competent, also with a UCS of 180 MPa. Both of these values come from CANMET test results (Labrie 2000a). The stresses at Laronde were measured by CANMET on the 146 level and the 150 level (Labrie 2000b). The stress gradients are shown in Figure 2. CANMET Stress Model Used to Obtain Gradients for Flac 3D Model 0 S3 = 0.027 * Depth 200 S2 = 0.0362 * Depth 400 S1 = 0.0437 * Depth 3.1 Depth (m) 600 800 The overall mine model was used for a number of purposes. First, it was used as a calibration of the stress and material properties based upon information collected from site visits and previous experience at the mine. Secondly, it was used to determine how much of an interaction between ore zones was likely taking place. Finally, it was used to determine the in-situ and post-mining stresses along the potential shaft locations and the overall stress regime throughout the entire mine. CANMET @ 150 S1 Average = 70.2 S2 Average = 61.18 S3 Average = 47.61 1000 CANMET @ 146 S1 Average = 59.31 S2 Average = 48.13 S3 Average = 25.08 1200 1400 1600 0 20 40 60 80 100 Stress (Mpa) CANMET S1 CANMET S2 CANMET S3 Stress Gradient (S1) Stress Gradient (S2) Stress Gradient (S3) Figure 2. Stress gradients used in the FLAC modeling at Laronde Mine based upon CANMET stress measurements. 3 FLAC3D MODELING The use of FLAC3D in this project was a logical choice based on the geometry of the orebody and work previously performed by Itasca for the mine (a reasonably good calibration of the model, material properties, and stresses had already been performed). To extend upon this base information, new stress data and core testing by CANMET resulted in a more refined picture of the complex interactions between the different rock structures and stresses. Around each ore lens is a layer of highly sheared schist that varies in thickness up to 5 m on both the hangingwall and footwall. Surrounding the schist is a garnetiferous tuff material that is relatively strong and stiff with a uniaxial compressive strength (UCS) of 3.1.1 Calibration The calibration of the material properties became easier to perform after a seismic event occurred on the 152 Level of the mine. Using the existing model as a framework, an investigation showed – and the model confirmed – that there was a large stress concentration centered right around the location of the burst caused by a 3-wide pillar being formed by retreating stopes towards the central access for the level. In March 2003, a large fall of ground occurred in a stope on the 215 Level where a double width stope had been taken. Again, the model was able to show a similar pattern in the stresses and failure zones in this area. Between these two events, a comfortable degree of confidence was gained that the model reflects realistic stresses and failure zones based upon the known geology and geometry of the orebody at depth. 3.1.2 Ore lens interactions In all of the previous work performed, the investigation of the impact of the different ore lenses on each other was not considered relevant because of the distance 222 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Overall mine model 09069-27.qxd 8/26/03 10:44 AM Page 223 location for the shaft, a series of history points were taken along the length of five potential locations. Each shaft location represented a suitable surface location at which a shaft could be placed within the existing limits of the property. Tracking of the major and minor principal stresses along the length of each possible shaft location, a comparison between these locations became possible through the use of the stress ratio (SR). The SR (shown in Fig. 4) is calculated based upon the anticipated point at which damage will occur in an ideal circular opening and is defined as: (1) Figure 3. Surface contour plot generated by Surfer( on the 215 Level in the final year of the five-year plan. The upper right region is Zone 7, the middle is the main Zone 20, and the lower region shows Zone 21. Stress Ratio Versus Depth 1400 1600 Depth (m) 1800 2000 2200 Heavy Damage 2400 2600 Little to no Damage Moderate Damage 2800 3000 0 0.2 Shaft #1 0.4 Shaft #2 0.6 Shaft #3 0.8 1 1.2 Stress Ratio Shaft #4&6 Shaft#5 1.4 1.6 1.8 The stress ratio plots and an analysis of the maximum and minimum stresses showed that there was no single shaft location that stood out as being significantly better or worse than any other – rather it appeared that all of the shafts were likely going to experience similar stress levels with some variation in the timing and location of peak stresses depending on their proximity to the orebody. Based upon several factors, shaft location #3 was proposed as the best. It was located near the centroid of the orebody at depth, which will reduce haulage distances, and had the benefit of enjoying some level of stress shadowing from the orebody at the deepest levels. The geology of the shaft location was unknown at the time, as drilling had focused on delineating the orebody and not so much on investigating the footwall materials. Future drilling of this region could change the ideal location to avoid adverse geology. In Situ Stress Ratio Figure 4. Stress ratio plot for all shaft locations based upon FLAC3D results. between the mining fronts. As part of this project, it was decided to include the multiple ore lenses in order to determine just how small or large an interaction was likely to occur between these zones. From the model, it appeared that there is an interaction between the different ore zones as mining progresses into the future. Figure 3 shows a surface contour plot on the 215 Level with all three lenses being mined to the end of the five-year plan (as of November 2002). The mined out stopes show up as depressions while the abutment stresses appear as peaks in this perspective view of a surface plot. There is a definite bridging effect between the middle and upper zone abutment areas (shown as the raised region), as well as some stress shadowing occurring between the middle and lower zones (shown as a depression). 3.1.3 Shaft stresses The location of the shaft was one of the most important aspects of the project. In order to determine the best 3.1.4 Overall mine stresses The final purpose of the overall mine model was to examine the overall mine stresses over the entire life of the mine. In order to make a model that could be run within a reasonable amount of time, the resolution in the upper regions was reduced to allow for a higher resolution in the 2150 to 3000 m depth region. The mine stress model showed a number of interesting things. Firstly, the failure zones around a fully formed mining front are quite extensive depending on the ore thickness, and can even be greater than an entire stope width into the abutment. (see Fig. 5.) Stresses in the abutments and sill regions can exceed 250 MPa (as seen in Fig. 6). Also, even in failed ground very high stresses can be seen due to heavy confinement levels that will likely cause some significant issues in areas such as sills in which post-failure ground is subjected to very high stresses. 3.2 In order to take a closer look at the stope level stresses, another model was built with the sole purpose of running high resolution simulations at the maximum depth 223 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands High resolution mining method analysis 09069-27.qxd 8/26/03 10:44 AM Page 224 Figure 5. Failure plot on a long section looking North of the overall mine model as mining approaches a sill. Blue blocks are intact, red and light blue blocks have failed in shear and green blocks also have a tensile failure component. Figure 6. Maximum principal stress plot on a long section looking North through the orebody. Peak stresses in a couple of areas exceed 250 MPa. of the mine. The model used a simplified representation of the orebody to give a constant strike and thickness of the ore and schist zones to eliminate geometrical effects in the comparison. A total of four simulations were run, three with different ore thicknesses ranging from 10 m up to 30 m, and a fourth simulation in which the stope size was reduced to determine the impact on the stope stability. To simplify the running of multiple scenarios with similar geometry, a FISH function was written in which the thicknesses of the different units and their location in the model could be defined, as well as the depth at which the simulation was to occur. This automation reduced the turnaround time between model runs to only a few minutes. 3.2.1 Results Some results from the different ore thicknesses are shown below, in each case, the early stages of mining are shown when only four stopes have been 224 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:44 AM Page 225 Figure 7. Maximum principal stress plot on a long section looking North through the orebody with a 10 meter ore thick-ness showing some pinching of high stresses above the trailing primary stopes. Secondary stopes are in post-failure even in narrow ore. Figure 8. Maximum principal stress plot on a long section looking North through the orebody with a 30 meter ore thickness showing that the destressed zone is much larger due to the extra freedom provided by the larger stoping spans. Overall, the stresses are more spread out and peak stresses are predicted to be much lower in this case. mined.A 10 m and 30 m ore thickness are shown for comparison. Looking at the principal stress plots it can be seen that there is a definite pinching of the stresses with the narrower ore geometry Fig. 7) due to the stronger secondary stopes carrying more loading and the smaller failure zones. In contrast, the 30 m ore zones (Fig. 8) result in a very large and smooth stress distribution with lower peak stress levels. This pattern of pinched stresses and higher peak 225 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:44 AM Page 226 Figure 9. Cross-section looking North showing failed regions in the model with 10 meter ore thickness. The impact of individual stopes can be seen by the irregular shape of the failure zone. Most of the blocks are failed in shear (light blue and red blocks, although some tensile failure is evident along the stope boundaries (green blocks). Figure 10. Cross-section looking North showing failed regions in the model with a 30 meter ore thickness. The failure region is much larger and more even with the wider ore at these great depths. The color coding is the same as in Figure 9. levels continued throughout the simulation as mining proceeds. Looking at the failure plots (Figs. 9 & 10) the reason for the pinching of the stresses from the previous plots (Figs. 7 & 8) can be seen. The failure zones are much larger with the thicker orebody; this creates a more even shell of failure around the stopes which becomes spherical in shape. The impact of a single stope is lost in the overall picture. In the narrower orebody, the impact of individual stopes on the overall shape of the failure zone is still apparent. From these results it can seen that there is a significant impact on the stresses and failure zones with increasing ore thickness as would be expected with the change in the pillar width-to-height ratio. 226 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:45 AM Page 227 Figure 11. Grid used in the FLAC modeling of a 9 meter diameter shaft with liner support. Figure 12. Maximum principal stress plot around a 9 meter diameter shaft at a depth of 2000 meters. 4 FLAC MODELS Two other models were created in FLAC in order to give some measure to the stresses and anticipated failure zones around excavations at extreme depths. The first model was used to model the shaft in both in-situ and post-mining situations, whereas the second was used to model a hypothetical large excavation at 3000 m (9840 ft). 4.1 Shaft model The shaft model created in FLAC used the double donut FISH function provided with FLAC, which was modi- fied to provide a single circular opening with a liner component added. The advantage of using this scheme is that it allows for the modeling of both in-situ and postmining stresses in the same model, by adding the rotation to the stress tensor and observing the effects. Figure 11 shows the FLAC grid used for the shaft modeling. The liner was set up as a 12 inch concrete layer, which was added after the in-situ stresses had reached equilibrium in order to properly mimic the true response of the liner. The liner was assumed to respond only to post-mining stresses. The maximum principal stress was set to run North–South. The in-situ stresses and failure zones the shaft at a depth of 2000 meters are shown in Figures 12 & 13 respectively. 227 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:45 AM Page 228 Figure 13. Predicted failure envelope around a 9 meter diameter shaft at 2000 meters subjected to in-situ stresses. The depth of failure predicted from this model is around 3 m. Figure 14. Maximum principal stress plot at shaft location #3 at a depth of 2100 meters when subjected to mining induced stress changes. In order to determine the stresses around a large selection of shaft locations and depth/stress conditions, the stress ratio plot (see Fig. 4) was used to determine the best and worst conditions that may be expected from each shaft location. Six shaft models were run – this included three generic in-situ runs at 2000, 2500, and 3000 m. Another set of three models were used to represent post-mining conditions where the largest increases, decreases and rotations of the stress tensor were occurring. Figure 14 shows an example of the #3 shaft location at a depth of 2100 m (which corresponds to the largest increase in stress ratio). It can seen that the stresses have rotated clockwise about 45 degrees, which is shown in both the stress plot and the plasticity plot (Fig. 15). These results correspond very well with what was intuitively expected based on the geometry of the region. At this elevation, the shaft is just passing through the western abutment stress of the 215 mining horizon, which is reflected by the 228 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:45 AM Page 229 Figure 15. plot for shaft location #3 at a depth of 2100 m at the post-mining stage. The rotation of the stresses at this location has had a significant impact on the failure zones with an increased depth of failure along a line running North-West to South-East. Figure 16. Stresses in the 12 inch concrete liner for post-mining stresses of shaft #3 at 2100 m. Peak stresses approach 4 MPa in the liner in this case. slight increase in stresses and the general rotation of the stress tensor. Looking at the stresses in the liner (see Fig. 16), it can be seen that the stresses in the liner due to mining-induced stress are quite small with a peak stress of around only 4 MPa, which is well below the strength of the concrete to be used in the liner. From this analysis it was determined that there should be no excessive stresses or failure zones that cannot be designed for with current technology. Barring any poor geological horizons through the shaft locations, no significant difficulties are anticipated beyond those expected with mining at extreme depth. 4.2 The large excavation model provided some general guidelines that can be used in the design of infrastructure in the mine. To do this, an arched back drift was created in FLAC with a span of 11 m and a height of 6.5 m, as shown in Figure 17. The drift was set up at a 3000 meter depth using the in-situ stress. Post-mining 229 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Generic large excavation model 09069-27.qxd 8/26/03 10:45 AM Page 230 Figure 17. Grid used to create the large excavation model. Figure 18. Large excavation running parallel to the maximum principal stress. Peak stresses approach 200 MPa with a failure zone that extends around 5 m into the back of the drift. stresses were not considered since they are locationdependant and no information was available for placement of infrastructure in the deep mine. The first run of the model assumed that the drift ran parallel to the maximum principal stress (North–South), while the second model ran perpendicular to the maximum principal stress (East–West) in order to provide information on these two extreme situations. With the drift set up to run parallel to 1, the back is shielded from the effect of the highest stresses. As a result, the stress seen in the drift was a combination of 2 and the overburden-related vertical stress. This resulted in a peak stress of around 200 MPa and a failure zone that extended up to 5 meters into the back (Fig. 18). This represented a very extensive shell of failed material, which, depending on geology, could be difficult to support. The ground at Laronde tends to involve some significant displacements which make stiff support such as shotcrete a less attractive support system as it cannot accommodate much displacement. With the alternative scenario, which had the drift running perpendicular to 1, the drift was exposed to the full impact of the highest stress component (see Fig. 19). As a result, the peak stresses across the back of the 230 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-27.qxd 8/26/03 10:45 AM Page 231 Figure 19. Large excavation running perpendicular to the maximum principal stresses. In this case the peak stresses are about 25% higher and the failure zone extends nearly 6 m into the back. stope reached 250 MPa (which is about 25% higher than the previous case). The failure zones extend about 15% farther into the back that in the previous case, making for an even more difficult ground support requirement. were encountered. These problems will be magnified at depth. ACKNOWLEDGEMENTS The authors thank Agnico Eagle Mines for permission to publish this paper. 5 CONCLUSIONS The numerical modeling exercise using FLAC and FLAC3D enabled the known behavior of the mine in shallower areas to be extended to the planned deeper mining. With the knowledge thus gained, it is possible to estimate with some degree of confidence some of the issues that may come into play at the extreme depths involved in this mine expansion. Based upon experience gained in the upper levels of the mine, several important recommendations were made. Among other things, the mining sequence is critical to the stability of the mine at these depths. In instances in upper levels where deviations from the original plan created unfavorable geometry, problems REFERENCES Labrie, D. December 2000a. Strength and Elastic Modulus as Determined on the Drill Core HQ5 at #3 Shaft – AgnicoEagle Mines, Laronde Division, Cadillac, Quebec. (In French.), Technical Note from Laboratoires des mines et des sciences minérales, CANMET to Agnico Eagle Laronde Mine. Nepean, Ontario, Canada. Labrie, D. December 2000b. Laronde Mine (Project 610 660, Task B) – Evolution of the Stress Field as a Function of the Number of Mathematical Iterations Executed. (In French.), Technical Note from Laboratoires des mines et des sciences minérales, CANMET to Agnico Eagle Laronde Mine. Nepean, Ontario, Canada. 231 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-28.qxd 08/11/2003 20:36 PM Page 233 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Three-dimensional strain softening modeling of deep longwall coal mine layouts S. Badr, U. Ozbay, S. Kieffer & M. Salamon Colorado School of Mines, Golden, Colorado, USA ABSTRACT: This paper describes a FLAC3D model for a typical deep two-entry longwall coal mine. The coal seam is modeled as a strain softening material to attain a representative analysis of stresses and deformations experienced by the coal ribs and yielding chain pillars corresponding to various loading stages. The strain softening parameters are established by calibrating separate test pillar models to common empirical pillar strength formulas. The test pillar models showed that strain softening material behavior results in lower pillar strengths than the traditional Mohr–Coulomb models based on constant peak cohesion and friction values. The longwall model incorporates compaction simulations of the gob material in the back area. Two algorithms for representing gob compaction are described. 1 INTRODUCTION (4) Co Coal (1 (1) ( (2) ( (5) Gob (3) Figure 1. Simplified plan view of a two-entry longwall mine layout showing pillar loading stages. right to left as indicated, the chain pillars undergo five stages of loading. These stages are indicated in the diagram; the first three affect the pillars next to the head gate and the last two affect the pillars next to the tailgate. Stage 1 corresponds to the situation where the entry-pillar system is fully developed, but the extraction of the longwall panels has not yet affected the loading of the pillar. Stage 2 refers to the situation where the front and side abutments contribute to the pillar loading due to the approaching longwall face. In Stage 3, the gob on one side, and an unmined panel on the opposing side, affect the loading. The gob in the vicinity of the development is not fully compacted so it does not support the full weight of the 233 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands advance adv In mining practices, it is common for the induced loading to exceed the strength of the rock mass. Realistic representation of stresses and deformations in such situations requires use of constitutive laws that can account for the response of the rock mass in the postpeak state. Mohr–Coulomb (MC) and Hoek & Brown (HB) plasticity models are commonly used in these situations. Considering the brittle nature of many rock masses, strain softening type models, such as the Mohr–Coulomb Strain Softening (MCSS) option in FLAC3D (Itasca 2002), allow more realistic modeling of rock mass failure. A typical mining situation where the modeling of brittle behavior becomes important is the analysis of yielding chain pillars in deep longwall mines. At depths more than about 300 m, the vertical stress exceeds the strength of unconfined coal, resulting in failure of the excavation walls while they are being exposed. This can result in the sides of entry pillars failing before the pillars are fully isolated. Realistic estimation of the loads carried by these pillars during subsequent mining requires the use of a softening model. The longwall mining geometry and the sequence of excavation considered in this study are illustrated in a plan view in Figure 1. Three longwall panels are shown in this illustration. The upper panel is already extracted. The panel at the bottom of the illustration has been developed, but extraction has not yet commenced. As the longwall face in the middle panel moves from 09069-28.qxd 08/11/2003 20:36 PM Page 234 overburden. In Stage 4 on the tailgate side, as the face approaches, the front abutment increasingly contributes to loading of the pillar; hence the conditions around the tailgate pillars become progressively more adverse. Stage 5 corresponds to the situation where the influence of the face is no longer detectable and the chain pillars are surrounded on both sides by gobs. This paper describes a numerical model for assessing the longwall mining scenario described above. The coal seam is modeled as a MCSS material. A separate series of numerical analyses was carried out on a single pillar (test pillar model) to determine representative MCSS strength parameters for the coal seam. The test pillar model analysis was also performed with MC materials to permit comparison of the pillar res-ponse based on MC and MCSS behavior. Compaction of the fractured, particulate material, called the “gob”, created by the caving of the roof in the area from where the coal has been extracted, requires attention in the numerical modeling of longwall mining. With continuing extraction, the upper strata and the floor converge and gradually the vertical load on the gob material increases. Representation of this process requires consideration of the deformations of both the gob materials and the surrounding strata. This paper describes two alternative algorithms to simulate gob compaction. 240 m 1000 m 240 m Figure 2. The FLAC3D block model developed for longwall mining simulations. 2 LONGWALL MODEL The modeled longwall layout is similar to that shown in Figure 1. It represents a two-entry longwall mine located at a depth of 680 m below surface. The panel length is 220 m and the mining height is 3 m. The width of the entries and cross cut is 6.5 m. The chain pillars between the entries are 3 m high, 8 m wide and 26 m long. The mining geometry is built in a 1000 m long, 240 m high, and 240 m wide block with graded mesh, as shown in Figure 2. The bottom layer in this figure represents half of the 3 m thick coal seam. The meshing at the central portion of the base of the block is made finer in order to represent the entries and chain pillars in detail (Fig. 3). Within the fine meshed region, MC interface separates the coal seam from the roof strata. The roof and floor strata are assumed to remain elastic throughout all stages of mining. The vertical planes bounding the block are free of shear stresses and horizontal displacement. The horizontal plane at the base of the model, which is a plane of symmetry, is also free of shear stresses and subject to zero vertical displacement. The model is loaded at the top with a uniform vertical stress of 11 MPa to give a total overburden pressure of 17 MPa at the coal seam level. As seen in Figure 4, the element size in the chain pillars within the fine meshed central region is 1 m 3 m 0.5 m in the x, y and z-directions, respectively. Figure 3. Bottom view of the FLAC3D block model showing the fine mesh at the central area. 8m 26 m 6.5 m Figure 4. The entry system dimensions. 2.1 Determination of material properties In addition to the peak cohesion, friction angle, and dilation angle in the MC model, the MCSS model also requires parameters describing the rate of cohesion 234 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 6.5 m 1.5 m 09069-28.qxd 08/11/2003 20:36 PM Page 235 Loading 30 25 3.25 m Peak strength (MPa) Cross-cut 2.5 2.4 2.2 2 1.8 1.6 1.5 Salamon Bieniawski 0.5 m Coal Entry 6m 20 35 MPa/εp 15 10 5 0 1 0 3 2 5 4 w/h 30 2.5 2.4 2.2 2 1.8 1.6 1.5 Salamon Bieniawski 25 15 10 0 0 1 2 4 3 5 w/h 30 2.5 2.4 2.2 2 1.8 1.6 1.5 Salamon Bieniawski 25 20 15 100 MPa/εp 10 5 0 0 1 3 2 4 5 w/h Figure 6. Model pillar strength versus empirical pillar strength at cohesion drop rates of 35,50,100 MPa/p (Strength formulas: Salamon: 9(w0.46/h0.66), Bieniawski: 9(0.64 0.36 w/h) in MPa; assuming a coal cubic strength value of 9 MPa). of 2.2 MPa and cohesion drop rate of 50 MPa/p is considered suitable for modeling yielding of the chain pillars. The test pillar models were repeated using the MC failure criterion with the same peak cohesion, friction 235 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 20 5 Peak strength (MPa) and/or friction drop as a function of plastic strain in the post-peak region. The determination of the MC and MCSS parameters for a rock mass is a difficult task, but can be carried out empirically by performing back-analyses. In this study, the parameter determination is based on the two most commonly used empirical pillar strength formulas given by Salamon (1967) and Bieniawski (1984). A FLAC3D model of a single test pillar was developed to establish the most suitable combination of coal MCSS parameters for replicating pillar strength values based on empirical formulas. Figure 5 shows the FLAC3D model of the test pillar in a room and pillar environment. By considering symmetry conditions, one quarter of the pillar is modeled. The vertical walls of the model are set as frictionless by fixing the normal displacements on them, except for pillar sides when they are formed. The model is loaded along the top boundary using a constant displacement of 2 10 7 m per FLAC step. The floor material is modeled as an elastic layer having a 20 GPa elastic modulus. The MC interface between the pillar and floor has strength parameters of 0.5 MPa cohesion and friction angle of 23 degrees. For all pillar test simulations, the friction and dilation angles are held constant at 30 and 15 degrees, respectively. Four pillar width-to-height (w/h) ratios (1, 2, 3, and 4) were modeled. For each w/h ratio, the numerical model was run with different combinations of a peak cohesion and cohesion drop rate. The strengths established from the test pillar models are plotted against the empirical pillar strength formulas in Figure 6 for the cohesion drop rates of 35, 50, and 100 MPa per plastic strain (p) increment. Based on the trends of these plots, a peak cohesion Peak strength (MPa) Figure 5. Test pillar model geometry. 50 MPa/εp 09069-28.qxd 08/11/2003 20:36 PM Page 236 in Figure 8. The MC model strengths tend to increase rapidly while MCSS model strengths follow the empirical strength trends, indicating that MCSS models give more realistic pillar stress–deformation curves than MC models. 30 Legend 3 = w/h ratio 3. MC = Using MC model MCSS = Using MCSS model. 25 3 / MC Stress(MPa) 20 2 / MC 2.2 15 3 / MCSS 1 / MC 10 2 / MCSS 5 Gob compaction The gob compaction process is an essential part of the longwalling process since it can alter the pillar and abutment loads by acting as an additional support for the system. The gob behavior is based on the following “compaction” model: vertical stress (v) in the gob increases with increasing vertical strain (v) according to the relationship given by Salamon (1990), 1 / MCSS 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Strain (1) Figure 7. The vertical stress–strain curves of MC and MCSS pillars. 50 45 40 Peak Strength (MPa) Salamon's Bieniawski's MC models MCSS models MC models peak strength could not be defined beyond this point 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 w/h Figure 8. Pillar strength determination from numerical modeling and empirical formulas (refer to Figure 6 for empirical strength formulas). and dilation angle values as for the MCSS model. By averaging vertical stress and the vertical deformation histories across the top of the pillar, an overall stress–strain curve for an individual pillar could be obtained. Figure 7 shows such curves for pillar w/h ratios of 1, 2 and 3, using MC and MCSS criteria. The difference in pillar response is obvious; MC does not allow the true softening (no peak strength and no strength drop) and pillars maintain high residual strengths. On the other hand, MCSS models yield and reach much lower residual strengths. The pillar strength values, corresponding to both MC and MCSS materials, are plotted against the empirical pillar strength formulas of Salamon (1967) and Bieniawski (1984) where “a” is gob initial deformation modulus; and “b” is the limiting vertical strain. Based on studies carried out at the USBM on gob behavior, the values for the constants were taken as a 3.5 MPa and b 0.5 (Deno & Mark 1993). Two different algorithms are considered for implementation of the gob behavior of Equation 1 in the FLAC3D model. In the first algorithm, referred to as the “nodal force”, the compaction load is modeled as the sum of vertical forces applied at the grid points of the roof elements in the back area after mining. After each mining step, the vertical strain in a particular zone within the gob area is used to calculate the vertical stress according to Equation 1. Grid reaction forces are then calculated by multiplying vertical stress by the corresponding area of the roof element. In the second method, the gob is modeled as a non-linear elastic material. Its bulk modulus is continually increased as function of vertical strain within the gob area. The algorithm for this “modulus updating” method uses the bulk modulus K for each gob element: (2) where z is the vertical strain in the element (Badr et al. 2002). Implementation of these two methods makes use of the “linked list” concept in FLAC3D. The nodes (or zones) that will be replaced by gob material are defined by their addresses in a particular linked list. Then, using the FLAC3D programming language “FISH”, a function updates the forces (or bulk modulus) of each node (or zone) using Equation 1 or 2. After each mining step, the algorithm is executed in 50 step intervals until the model is brought to equilibrium (Badr 2003). 236 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-28.qxd 08/11/2003 20:36 PM Page 237 2.5 Analytical solution Nodel force method Modulus updating method 18 2 14 Cohesion (MPa) Gob stress (Mpa) 16 50 Cohesion Friction Dilation 12 8 6 40 35 1.5 10 45 30 25 1 20 15 4 0.5 2 10 0 Friction and dilation (degrees) 20 5 0 10 20 30 % closure 40 50 0 Figure 9. The gob stress-closure results from the analytical solution and two FLAC3D algorithms. The gob compaction curves for the analytic solution (Salamon 1990) and the two FLAC3D algorithms are compared in Figure 9. As shown, both nodal force and modulus updating algorithms compare well with the analytical model. Since the nodal force algorithm requires longer running time, the modulus updating method was embraced as the gob model for the FLAC3D longwall simulations performed in this study. 3 RESULTS Figure 10 defines the MCSS material parameters used in the model, which are also summarized in Table 1. For the coal seam, these parameters correspond to an MCSS material having a cubic strength of about 9 MPa, friction angle of 30 degrees, and cohesion drop rate of about 50 MPa/p. The model of the longwall layout described in Section 2 is brought to equilibrium elastically to horizontal and vertical virgin stress conditions of 17 MPa at the coal seam level. The elastic coal seam is then replaced by a MCSS material prior to development. The entries are developed with the right entry leading the left entry by 9 m. The entries advance by 3 m in each mining step. A cross-cut is then mined when the trailing entry is 9 m ahead. Mining of the longwalls is carried out starting at the right panel. The longwall advances initially in steps of 50 m and then the steps are reduced to 10 m in the fine-meshed central region of the model. After each longwall advance the area behind the longwall face is changed to “gob material” and the model is brought to equilibrium. The pillar response to mining is monitored using a FISH algorithm. The algorithm keeps a record of the vertical stress and vertical strain histories of all zones comprising the top of the pillar, 0.02 0.04 0.06 Plastic strain Figure 10. MCSS parameters used for modeling of the coal material. Table 1. Material properties used in longwall simulations. Property Miscellaneous Seam depth Stress gradient x, y and z Coal properties Coal elastic modulus Coal Poisson’s ratio Coal strength Coal density Roof properties Elastic modulus Poisson’s ratio Density Interface properties Type Cohesion Friction angle Values 680 m 0.025 MPa/m 17 MPa 3 GPa 0.25 7.6 MPa 1313 Kg/m3 20 GPa 0.25 2500 Kg/m3 Mohr–Coulomb 0.5 MPa 20° and then averages these values to produce an average vertical pillar stress–strain curve. Figure 11 shows a typical pillar stress–strain curve obtained from the FLAC3D simulation. The vertical dashed line on the left shows the pillar loading at the end of entry development. At this stage, the pillar is at or close to its peak capacity. The pre-peak stress drops indicate sidewall failures experienced by the pillar during entry development. As the longwall approaches, the pillar initially sheds load slowly and subsequently rapidly, eventually reaching eight per cent compression. At its residual strength, the pillar carries a vertical stress of 4 MPa, 237 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0 0 09069-28.qxd 08/11/2003 20:36 PM Page 238 25 22 Average Pillar Stress (MPa) 20 20 18 Gob Stress (MPa) Average pillar stress (MPa) 24 16 14 12 Dev. Longwalling 10 8 17 After second longwall 15 10 After first longwall 6 5 4 2 0 0 0 0.02 0.04 0.06 0.08 0 10 Figure 11. Complete average vertical stress–strain curve of the yielding chain pillar in modeled longwall layout. which is considered sufficient for supporting the roof in two entry systems. The pillar strength in the longwall model is more than that estimated by the test pillar model and empirical strength formulas. Further refinement of the strength parameters could be achieved by iterating on the contact and coal seam properties through parametric studies, which would involve six independent variables, not including parameters for the roof material. As was the case with the test pillar model, this iterative process would likely provide more than one set of parameters giving strength values similar to those predicted by the empirical strength formulas. Further studies in this area are needed to fine-tune the optimum parameter combination. Figure 12 shows the gob compaction as mining progresses, referenced to a point at the center of the first panel. After mining of the first panel, the vertical stress in the gob is 1.8 MPa. The gob stress increases to the virgin stress level of 17 MPa after the second panel is mined. The results from the longwall model are compared to in-situ measurements using borehole pressure cells (BPCs) from a mine with similar conditions (Schissler 2002). The FLAC3D model shows that the pillar hardens to 22 MPa while the in-situ pillar monitoring showed 16 MPa during entry development. This difference is probably partly due to the selection of the model parameters as discussed above, and partly due to the installation sequence of the BPCs, which occurred after the pillar was developed, and thus did not completely capture the side wall loading by the approaching development faces. When the pillar yielded in the model, the longwall face was approximately 150 m from the pillar centerline. Although there is no in-situ load measurement available in pillars under similar 30 40 50 Figure 12. Vertical stress and closure induced at a point in the gob. situations, the authors’ observations of intense pillar scaling in similar face positions in deep coal mines support the finding of the model. 4 CONCLUSIONS A three dimensional model of a coal longwall mine is developed using FLAC3D. The model incorporates mining stages, softening behavior of the coal seam, and gob compaction in the mined out area. The model results indicate that FLAC3D is a suitable tool to aid in the design, evaluation, and performance assessments for complex longwall layouts. The test pillar studies show that the Mohr–Coulomb Strain Softening model is more realistic than the traditional Mohr–Coulomb constitutive law for estimating the strength and post peak behavior of coal pillars. The strain softening parameters developed in this study could be used as a starting point for modeling of coal seams. However, due to more than one combination of strength parameters giving the same rock mass strength value and also mesh size dependency of the program, it is advised that the strength parameters for a particular coal seam be developed on a case bases, using a back-analysis process similar to that described in the paper. ACKNOWLEDGMENT This publication was supported by Cooperative Agreement number U60/CCU816929-02 from the Department of Health and Human Services, the Center for Disease Control and Prevention (CDC). Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the Department 238 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 20 % CLOSURE Strain 09069-28.qxd 08/11/2003 20:36 PM Page 239 of Health and Human Services, CDC. Support provided by Department of Health and Human Services, CDC, is greatly acknowledged. The work presented is part of the Health and Safety research activities currently carried out at Western Mining Resource Center (WMRC) at the Colorado School of Mines. REFERENCES Badr, S.A., Schissler, A., Salamon, M.D.G. & Ozbay, U. 2002. Numerical Modeling of Yielding Chain Pillars in Longwall Mines. Proc. of the 5th North American Rock Mechanics Symposium, Toronto, Canada, pp 99–107. Badr, S.A. 2003. Numerical Analysis of coal yield pillars at deep longwall mines. Ph.D. Thesis in preparation. Department of Mining Engineering, Colorado School of Mines, Golden, Colorado (To be submitted.). Bieniawski, Z.T. 1984. Rock Mechanics Design in Mining and Tunneling. A.A. Balkema, p. 1–272. Deno, M.P. & Mark, C. 1993. Behavior of Simulated Longwall Gob material. United States Department of the Interior, Bureau of mines, Report of investigation No. 9458. Itasca Consulting Group, Inc. 2002. FLAC3D – Fast Lagrangian Analysis of Continua in Three Dimensions, Ver. 2.1. Minnesota: Itasca. Salamon, M.D.G. 1990. Mechanism of caving in longwall coal mining. Paper in Rock Mechanics Contributions and Challenges Proceedings of the 31st US Symposium, Ed. W. Hustrulid and G. A. Johnson. Denver, Colorado, June 18–20, 1990. A.A. Balkema, 1990, p. 161–168. Salamon, M.D.G. 1967. A study of the strength of coal pillars. Journal of South Africa Institute of Mining and Metallurgy, v. 68, p. 55–67. Schissler, A. 2002. Yield pillar design in non-homogenous and isotropic stress fields for soft minerals. Ph.D. Thesis. Department of Mining Engineering, Colorado School of Mines, Golden, Colorado. 239 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-29.qxd 08/11/2003 20:36 PM Page 241 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 FISH functions for FLAC3D analyses of irregular narrow vein mining H. Zhu & P.P. Andrieux Itasca Consulting Canada Inc., Sudbury, Ontario, Canada ABSTRACT: FISH functions were developed to generate FLAC3D elements and to accurately present numerical simulation results for underground mining situations whereby the ore lenses have complex geometries and erratic distributions. The element-generating FISH functions for FLAC3D proved flexible and provided significant timesavings during the model construction stage. Furthermore, these functions made it easier to modify a model, to achieve a high degree of resolution in the domains of interest and to create a minimum number of elements in order to minimize the computational power required to run the model. Long, narrow and winding ore bodies with complex geometries do not lend themselves well to the representation on longitudinal sections of the modeling results, mainly because the rendering planes wander in and out of the ore body. Such ore bodies are however often visualized and managed based on their longitudinal appearance, which is typically projected and simplified on an idealized plane. This difficulty of longitudinally showing the predicted stresses and displacements within the ore lens can impair the full and clear understanding of the modeling results. This paper describes two means of solving this problem based on FISH functions. The FISH functions presented in this paper have been applied to, and validated by, a FLAC3D modeling exercise carried out at the Falconbridge Thayer Lindsley Mine near Sudbury, Ontario, Canada. 1 GENERAL INSTRUCTIONS There are essentially two ways to build a FLAC3D numerical model: one is to generate regular elements over the entire domain and then structure the desired geometry and geology around them, the other is to set FLAC-provided blocks for specific objects to simulate and to assemble these blocks into the model. In underground mining numerical modeling applications the first approach is generally used because there is usually no need to account for topographically irregular ground surfaces or very complex and precise excavations, as is often the case in civil engineering applications. This method, although quite versatile, can however result in the creation of a large number of elements in order to achieve the desired degree of resolution, particularly when the geometry of the ore lenses is complex, or when multiple independent ore lenses are present. A large number of elements can, in turn, result in excessively long running times and even prevent a model from running if the computer platform is insufficiently powerful. In such cases, the second strategy may not be adequate either, due to the irregularity of the geometry of the ore lenses. Furthermore, it is usually more timeconsuming to build a model using the second approach. Narrow and undulating ore lenses also make it difficult to represent the simulation results on longitudinal views. The existing FLAC3D “plot” command can prove inadequate to illustrate load and deformation results because longitudinal sections generated through the approximate center of a given narrow and undulating ore lens typically wanders in and out of it. This made it difficult to visualize the stress redistribution and deformation everywhere within the ore lens itself. Two approaches can be used to solve this problem. One consists of extracting from save files the stress and deformation data at each point along a curved surface centered in the middle of the undulating narrow body of interest, and to generate iso-contour plots with specialized software (such as Goldsoft Surfer®, for example). This approach has the advantage of allowing the user to extract and plot any desired parameter or criterion, such as factors of safety or custom-defined stress ratios, and to clearly represent their variation. Another way is to define a thin central zone in the middle of the undulating narrow body as a FLAC3D Group or a FLAC3D Range, which can subsequently be used to represent a true longitudinal section. This approach allows use of existing FLAC3D commands and functions to generate the plots. Both 241 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-29.qxd 08/11/2003 20:37 PM Page 242 methods require FISH functions that are quite similar. The first approach is however more flexible and more precise, but requires additional third-party software. Ore lens of #2 Zone (a) Plan view of Level 11-0. 2 GENERATION OF THE ELEMENTS FOR A FLAC3D MODEL Ore lens of #4 Zone Ore lens of #3 Zone (b) Cross-section looking East, showing (c) Plan view fo Level 13-1. the vertical extent of the ore bodies. Figure 1. Various views of the Thayer Lindsley Mine ore lenses showing the relatively narrow and winding #2 Zone, which was the ore lens of interest in the numerical exercise. The figure also shows the satellite lenses of the #3 and #4 zones, which had to be considered in the model. (The figures are at different scales.) dxyz(1,3) As shown in Figure 1, the geometry of the #2 Zone ore lens at the Falconbridge Thayer Lindsley (T.L.) Mine near Sudbury, Ontario, Canada, is quite complicated and resulted in difficulties being encountered when constructing a representative FLAC3D model. A series of FISH functions was therefore developed to generate the elements throughout the entire model. Also, the narrow and undulating geometry of this lens made it difficult to represent the simulation results on a true longitudinal section, i.e. on a longitudinal section that did not wander in and out of the ore material. The T.L. ore body is up to 500 meters in strike (East–West) and occurs as several distinct lenses below Level 13-2 as shown in Figure 1. The narrowest lens width encountered is approximately 5 meters. The maximum width is of the order of 20 meters. A 400 m-high, 240 m-thick (in the North–South direction) and 300 m-long (in the East–West direction) section of the mine, centered on the #2 Zone, represented the region of interest for the FLAC3D simulations. A resolution of 5 meters in the East–West direction, by 2 meters in the North–South direction, by 5 meters in the vertical direction was considered a minimum requirement within the domain of interest. Such a resolution would require as many as 960,000 elements for the inner domain, and close to 1.2 million elements for the entire model. This would make it almost impossible to run the model on even the most powerful personal computers currently available. From a geomechanics perspective it is not necessary to generate elements with the same resolution within the entire internal region of a model, as illustrated in Figure 2. As a result, this internal model can be divided into several blocks with different element resolutions, in order to end up with a reasonably sized model. This rationale lead to the development of a FISH functions-based approach for the construction of the FLAC3D model, which would be broadly applicable to many other geometries. For the T.L. Mine analyses, four different element resolutions were used, as shown in Figure 2. Block #1, which contained the #2 Zone of interest, was assigned the finest resolution of 5 meters in the East–West direction, by 2 meters in the North–South direction, by 5 meters in the vertical direction. Block #2, which had a different panel height, was assigned a coarser 10 m 2 m 10 m resolution. Block #3, which covers the satellite ore lenses and previously mined-out voids, was assigned a yet coarser 10 m 4 m 10 m dayz(i,1) Coordinates at the point O: dxyz(i,J) Index j (1,2,and3) corresponds to x,y, and z. Boundary block (outer model) Internal model Coordinates at point O: dxyz (i, j) Index j (1, 2, and 3) corresponding to x, y, and z. Figure 2. Schematic sketch showing the model structure and block parameters. resolution. Finally, Block #4, which encompassed the rest of the internal model, was fitted with the coarsest 20 m 8 m 20 m resolution. As a result, the number of elements inside the internal model dropped from 960,000 to 248,500, which allowed the model to be run on a personal (albeit upper end) computer. The FISH function developed and used to generate the elements in the five blocks within the internal model, as well as in the outer model, is shown in Appendix I. This FISH function can be used to modify the model and to create other models for situations with a generally similar geometry – different values simply have to be assigned to the variables. The function as it stands allows the user to break the internal model into up to ten blocks – this maximum number of blocks can also be customized if the user is familiar with the FISH language. A separate file was prepared to invoke each sub-routine in the function in order to assign initial values to the parameters. 242 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Illustration of the initial parameters of a block 09069-29.qxd 08/11/2003 20:37 PM Page 243 3 CREATION OF A NON-PLANAR LONGITUDINAL SECTION THAT FOLLOWS THE CENTRE OF A NARROW AND UNDULATING ZONE Longitudinal sections are well-suited to, and widely used for, the planning and sequencing of full-width open stopes in narrow ore lenses that are extracted by means of retreating mining methods. When longitudinal sections are used for planning and sequencing it is advantageous to also use them to display the stress and deformation results from numerical analyses. Not only do they represent a viewpoint familiar to the mine personnel, they also clearly show how stress is redistributed in workings ahead of the mining front as the extraction sequence progresses. Such longitudinal section views are being widely used at T.L. Mine. FLAC3D results would ideally have been shown on them. The difficulty, as described previously, is that – as is the case at T.L. Mine – it is impossible when the ore lens is narrow and undulates over an amplitude greater than its width for a true (planar) longitudinal section to remain entirely within it. (In general under these circumstances, the longitudinal sections used for planning purposes are composite simplified views, not true sections.) As a result, FISH functions had to be developed to create stress and displacement plots along the geometrical center of the lens. The approach is illustrated in Figure 3. The procedure can be summarized as follows: 1. select the groups that encase the area to be examined; 2. search the footwall and handing wall boundaries by element along the strike of the ore lens; 3. identify the ID of the elements where the boundaries are; 4. calculate the coordinates of the point between the two boundaries. This point needs to keep the desired hanging wall-to-footwall distance ratio to these boundaries; 5. if this point lies in an edge element along the strike direction, extend the extraction further outwards (into the surround rock mass) by a predetermined distance (represented by the lines of A—A, B—B, and C—C in Fig. 3); 6. trace the element ID to which this point belongs; 7. output the element stresses to a file for further analysis, or name this element in a new Range/Group; and, 8. repeat the procedure for each level. A continuous surface entirely comprised within the ore lens and following its center will be obtained by connecting all the points generated in this manner. The desired longitudinal section can thus be constructed either by projecting this surface onto a longitudinal plan, or by ignoring the y coordinate (as was done in the case of T.L. Mine). The FISH function developed to generate a longitudinal section that follows the center of a narrow and undulating zone is shown in Appendix II. The option of naming a new Group is recommended in order to avoid the need for external software packages, such as Surfer™ for example, to present the FLAC3D results. However, the alternate approach of extracting the element stresses from the middle of the ore zone was employed for the T.L. Mine, due to its higher precision and the need in this particular case to examine a userdefined stress ratio. 4 FURTHER APPLICATION OF CUSTOMISED FISH FUNCTIONS AT THE FALCONBRIDGE THAYER LINDSLEY MINE A user-defined stress ratio was also used at T.L. to evaluate the state of the rock mass throughout the #2 Zone of interest. The objective of this work was to not only assess which elements had started to undergo failure, but also determine how far elastic state elements were from. The stress ratio retained is defined as 1/ 1 , and is illustrated Figure 4. τ Element Strength envelope Hanging wall A A B B C C 3 Footwall Figure 3. Sketch illustrating the concept of constructing a non-planar longitudinal section from an arbitrary narrow and undulating ore lens. 1' Figure 4. Illustration of the user-defined stress ratio where 1 and 3 are the major and minor principal stress of rock mass, respectively, at a given point as computed by FLAC3D. 1 is the major principal stress at this point assuming that the rock mass is undergoing yielding under the same confinement conditions. 243 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 1 09069-29.qxd 08/11/2003 20:37 PM Page 244 Figure 5. Longitudinal section looking North illustration of the Surfer™-generated contours of a user-defined stress ratio based on the customized FISH functions described in the previous sections. Figure 5 shows a simplified composite longitudinal section of the T.L. Mine #2 Zone. The small squares along the edge of the excavations represent the state of the local rock mass at a given mining step, as inferred from the FLAC3D results. The overlaid contours of the user-defined stress ratio were generated based on the stresses at each element extracted with the FISH function mentioned previously. In the areas where the rock mass is still in its elastic state, which are beyond the small squares as shown in Figure 5, the low values of the stress ratio refer to a low likelihood that stressinduced problems will arise at this mining stage. With the help of the FISH function, the state of the rock mass can be illustrated quite precisely. In Figure 5, all the numerical elements from which the stresses were extracted are those located the closest to the middle of the #2 Zone ore lens at T.L. Mine, or those extending into the surrounding rock mass away from the east and west boundaries of the lens. These contours present quite a bit of information about the state of the rock mass, both in the main lens and the surrounding rock. Furthermore, how far away the elastic state rock is from the onset of yielding can be readily estimated and displayed by the contours generated by the customized FISH functions. Currently, this cannot be achieved with any built-in FLAC command. 5 CONCLUSIONS User-defined FISH functions can be a powerful tool to solve various FLAC or FLAC3D numerical modeling problems. The FISH function presented in this paper for the generation of regular elements is applicable to many scenarios where similar geometrical issues are present. The other FISH function discussed in this paper is a good example of how experienced FLAC and FLAC3D users can develop very specific functions to solve specific problems. ACKNOWLEDGEMENTS The authors would like to thank Scott Carlisle1 for reviewing, and Falconbridge Limited for granting permission to publish this paper and for the use of Thayer Lindsley data. REFERENCE Itasca Consulting Group, Inc. (1997) FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 2.0. Minneapolis, MN: Itasca. APPENDIX I – FISH FUNCTION FOR THE GENERATION OF THE ELEMENTS OF A FLAC3D MODEL ; Define blocks in the internal region of the model ; num_box10 blocks currently limited def Ore_box array xyz(10,3),dxyz(10,3),p_xyz(10,3) loop i (1,num_box) P0_xxyz(i,1) P0_yxyz(i,2) P0_zxyz(i,3) p_xyz(1,1)p0_x p_xyz(1,2)p0_y p_xyz(1,3)p0_z 1 Falconbridge Limited, Sudbury Mines/Mill Business Unit – Mining Services, Onaping, Ontario, P0M 2R0, Canada. 244 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-29.qxd 08/11/2003 20:37 PM Page 245 P_xyz(2,1)p0_x dxyz(i,1) P_xyz(2,2)p0_y P_xyz(2,3)p0_z P_xyz(3,1)p0_x P_xyz(3,2)p0_y dxyz(i,2) P_xyz(3,3)p0_z P_xyz(4,1)p0_x P_xyz(4,2)p0_y P_xyz(4,3)p0_z dxyz(i,3) P_xyz(5,1)p0_x dxyz(i,1) P_xyz(5,2)p0_y dxyz(i,2) P_xyz(5,3)p0_z P_xyz(6,1)p0_x P_xyz(6,2)p0_y dxyz(i,2) P_xyz(6,3)p0_z dxyz(i,3) P_xyz(7,1)p0_x dxyz(i,1) P_xyz(7,2)p0_y P_xyz(7,3)p0_z dxyz(i,3) P_xyz(8,1)p0_x dxyz(i,1) P_xyz(8,2)p0_y dxyz(i,2) P_xyz(8,3)p0_z dxyz(i,3) loop n (1,8) id_pn 8*(i-1) P_xp_xyz(n,1) P_yP_xyz(n,2) P_zP_xyz(n,3) command gen po id id_p p_x p_y p_z end_command end_loop end_loop end ; Building boundary blocks def right_box ; calculate the dimensions of this box in x, y and z ; xyz_ratio grid length ratio ; n_grid number of grids ; x_1, y_1 and z_1 element length along ; corresponding directions in boundary blocks ; x0, y0 and z0 coordinates of point O of the ; internal model as shown in Figure 2. x_lenx_1*(1-xyz_ratio^n_grid)/(1-xyz_ratio) y_leny_1*(1-xyz_ratio^n_grid)/(1-xyz_ratio) z_lenz_1*(1-xyz_ratio^n_grid)/(1-xyz_ratio) P0_xx0 dx P0_yy0 P0_zz0 P1_xx0 dx x_len P1_yy0-y_len P1_zz0-z_len P2_xx0 dx P2_yy0 dy0 P2_zz0 P3_xx0 dx P3_yy0 P3_zz0 dz P4_xx0 dx x_len P4_yy0 dy0 y_len P4_zz0-z_len P5_xx0 dx P5_yy0 dy0 P5_zz0 dz P6_xx0 dx x_len P6_yy0-y_len P6_zz0 dz z_len P7_xx0 dx x_len P7_yy0 dy0 y_len P7_zz0 dz z_len Y_S_boun p1_y Z_B_boun p1_z command gen po id 1011 p0_x p0_y p0_z gen po id 1012 p1_x p1_y p1_z gen po id 1013 p2_x p2_y p2_z gen po id 1014 p3_x p3_y p3_z gen po id 1015 p4_x p4_y p4_z gen po id 1016 p5_x p5_y p5_z gen po id 1017 p6_x p6_y p6_z gen po id 1018 p7_x p7_y p7_z end_command end def Back_box P0_xx0 P0_yy0 dy0 P0_zz0 P1_xx0 dx P1_yy0 dy0 P1_zz0 P2_xx0-x_len P2_yy0 dy0 y_len P2_zz0-z_len P3_xx0 P3_yy0 dy0 P3_zz0 dz P4_xx0 dx x_len P4_yy0 dy0 y_len P4_zz0-z_len P5_xx0-x_len P5_yy0 dy0 y_len P5_zz0 dz z_len P6_xx0 dx P6_yy0 dy0 P6_zz0 dz P7_xx0 dx x_len P7_yy0 dy0 y_len P7_zz0 dz z_len X_E_bounp7_x Y_N_bounp7_y Z_T_bounp7_z command gen po id 1021 p0_x p0_y p0_z gen po id 1022 p1_x p1_y p1_z gen po id 1023 p2_x p2_y p2_z 245 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-29.qxd 08/11/2003 20:37 PM Page 246 gen po id 1024 p3_x p3_y p3_z gen po id 1025 p4_x p4_y p4_z gen po id 1026 p5_x p5_y p5_z gen po id 1027 p6_x p6_y p6_z gen po id 1028 p7_x p7_y p7_z end_command end def Top_box P0_xx0 P0_yy0 P0_zz0 dz P1_xx0 dx P1_yy0 P1_zz0 dz P2_xx0 P2_yy0 dy0 P2_zz0 dz P3_xx0-x_len P3_yy0-y_len P3_zz0 dz z_len P4_xx0 dx P4_yy0 dy0 P4_zz0 dz P5_xx0-x_len P5_yy0 dy0 y_len P5_zz0 dz z_len P6_xx0 dx x_len P6_yy0-y_len P6_zz0 dz z_len P7_xx0 dx x_len P7_yy0 dy0 y_len P7_zz0 dz z_len X_W_bounp3_x command gen po id 1031 p0_x p0_y p0_z gen po id 1032 p1_x p1_y p1_z gen po id 1033 p2_x p2_y p2_z gen po id 1034 p3_x p3_y p3_z gen po id 1035 p4_x p4_y p4_z gen po id 1036 p5_x p5_y p5_z gen po id 1037 p6_x p6_y p6_z gen po id 1038 p7_x p7_y p7_z end_command end ; generate elements in the internal regions of the ; model def gen_ele array Len_xyz(10,3) ; reference point for reflected boundary boxes x_refx0 dx/2.0 y_refy0 dy0/2.0 z_refz0 dz/2.0 num_x1dx/x_1 num_y1dy0/y_1 num_z1dz/z_1 loop n (1,num_box) id_p11 8*(n-1) id_p22 8*(n-1) id_p33 8*(n-1) id_p44 8*(n-1) id_p55 8*(n-1) id_p66 8*(n-1) id_p77 8*(n-1) id_p88 8*(n-1) num_xdxyz(n,1)/Len_xyz(n,1) num_ydxyz(n,2)/Len_xyz(n,2) num_zdxyz(n,3)/Len_xyz(n,3) command gen zone bri & p0 po id_p1 p1 po id_p2 p2 po id_p3 p3 po id_p4 & p4 po id_p5 p5 po id_p6 p6 po id_p7 p7 po id_p8 & size num_x num_y num_z & group waste & rat 1 1 1 end_command end_loop ; generate elements in boundary blocks ; generate elements in one boundary block and then ; reflect to the opposite side command ; Left-hand and right-hand sides gen z brick & p0 po 1011 p1 po 1012 p2 po 1013 p3 po 1014 & p4 po 1015 p5 po 1016 p6 po 1017 p7 po 1018 & size n_grid num_y1 num_z1 & group right_side & ratio xyz_ratio 1 1 gen zone reflect dip 90 dd 90 ori x_ref y_ref z_ref & range group right_side ; Back and front gen z brick & p0 po 1021 p1 po 1022 p2 po 1023 p3 po 1024 & p4 po 1025 p5 po 1026 p6 po 1027 p7 po 1028 & size num_x1 n_grid num_z1 & group Back_side & ratio 1.0 xyz_ratio 1 gen zone reflect dip 90 dd 180 ori x_ref y_ref z_ref & range group back_side ; Top and bottom gen z brick & p0 po 1031 p1 po 1032 p2 po 1033 p3 po 1034 & p4 po 1035 p5 po 1036 p6 po 1037 p7 po 1038 & size num_x1 num_y1 n_grid & group Top_side & ratio 1.0 1.0 xyz_ratio gen zone reflect dip 0 dd 0 ori x_ref y_ref z_ref & range group top_side end_command command group external range group waste not end_command end 246 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-29.qxd 08/11/2003 20:37 PM Page 247 APPENDIX II – FISH FUNCTION FOR THE CREATION OF A LONGITUDINAL SECTION IN THE CENTRE OF THE UNDULATING ORE ZONE def get_profile x_0x_west ; x coordinates on west edge of area to be searched y_0y_south x_1x_east y_1y_north ; element length in the x direction x_dx_length y_dy_length ; group name of the main zone to be examined name_ggroup_name ; referring to the next FISH function for z_t and z_b z_po(z_t z_b)/2.0 n_x(x_1-x_0)/x_d n_y(y_1-y_0)/y_d if flag_boun0 then ; top element. Extend search 25m upwards z_1z_t 25.0 z_0z_b end_if ; bottom element if flag_boun1 then ; extend search 25m downwards to next level z_1z_t z_0z_b-25.0 end_if ; either top or bottom level if flag_boun2 then z_1z_t z_0z_b end_if kkn0 loop n1 (1,n_x) ;start point in the x direction x_pox_0 x_d*(n1-0.5) x_wx_po-0.5*x_d x_ex_po 0.5*x_d a_miny_north a_maxy_south kn0 ; searching FW and HW loop n2 (1,n_y) y_poy_0 y_d*(n2-0.5) p_zz_near(x_po,y_po,z_po) z_grz_group(p_z) if z_grname_g then g_p1z_gp(p_z,1) g_p2z_gp(p_z,3) y_mingp_ypos(g_p1) y_maxgp_ypos(g_p2) if y_min a_min then a_miny_min end_if if y_max>a_max then a_maxy_max knkn 1 end_if end_if end_loop ; main zone has been found if kn # 0 then kknkkn 1 x_kknx_e y_s(a_min+a_max)/2.0-0.55*y_d y_n(a_min+a_max)/2.0 0.55*y_d end_if ; on west edge of the main zone if kkn1 then x_wx_0 end_if if kn # 0 then command group profile ra x x_w x_e y y_s y_n z z_0 z_1 end_command end_if ; on the east edge of the main zone if n1n_x then command group profile ra x x_kkn x_1 y y_s y_n z z_0 z_1 end_command end_if end_loop end ; search loop on z direction in the next FISH function def get_para loop n (1,60) if n 40 then z_interval5 else z_interval10 end_if z_b2310-z_interval*(n-1) z_tz_b z_interval flag_boun2 if n1 then flag_boun0 ;top element end_if if n10 then flag_boun 1 ;bottom element end_if get_profile ; invoke the above FISH function end_loop end 247 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-30.qxd 08/11/2003 20:38 PM Page 249 Soil structure interaction Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-30.qxd 08/11/2003 20:38 PM Page 251 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A calibrated FLAC model for geosynthetic reinforced soil modular block walls at end of construction K. Hatami & R.J. Bathurst GeoEngineering Centre at Queen’s-RMC, Royal Military College of Canada, Kingston, Ontario T. Allen Washington State Department of Transportation, Washington, USA ABSTRACT: The paper describes a FLAC numerical model that was developed to simulate the construction and measured response of large-scale geosynthetic reinforced soil walls that were constructed at the Royal Military College of Canada (RMC). The reinforced soil structures were constructed with three different polymeric reinforcement configurations. The backfill strength properties and reinforcement material properties were determined from conventional laboratory tests. The soil elastic modulus values were back calculated from surcharge loading tests on the wall backfill. The numerical models were able to capture the observed differences in wall behavior due to different reinforcement configurations to within the accuracy of the measurements. Reinforcement strain magnitudes and distribution were more accurately predicted using a stress-dependent model for the soil backfill compared to a linear elastic model. 1 BACKGROUND A recent study by Allen et al. (2002) of the design, analysis and performance of instrumented full-scale geosynthetic reinforced soil walls constructed in the field has demonstrated that current design practice is excessively conservative. For example, they showed that most walls constructed to date could be expected to perform satisfactorily with as little as 50% of the reinforcement that has been used in the past. Nevertheless, the number of instrumented field walls reported in the literature is sparse and there is a requirement for better data and a wider range of case studies in order to refine current design methodologies that are based on conventional geotechnical limit equilibrium approaches. To fill this requirement, the GeoEngineering Centre at Queen’s-RMC at the Royal Military College of Canada (RMC) has been engaged in the construction, surcharge loading and monitoring of carefully instrumented large-scale geosynthetic reinforced soil retaining walls built within a controlled laboratory environment (Bathurst et al. 2001). This on-going research program has also been conceived to generate high-quality and comprehensive data that can be used to calibrate advanced numerical models of geosynthetic reinforced soil walls. The calibrated models can then be used to extend the database of physical tests to a wider range of reinforced soil wall types and configurations. The combination of physical and numerical test results can then be used to check or refine recently proposed analytical design methods for geosynthetic reinforced soil wall structures that hold promise to make these systems more cost effective (e.g. Allen et al. 2003). This paper is focused on the second step in this long-term research program, i.e. calibration of a FLAC numerical model for three recent large-scale test walls at RMC that were constructed with a polypropylene geogrid reinforcement material. This paper extends the results of an earlier paper by Hatami & Bathurst (2001) that was focused on a FLAC (Itasca 1998) numerical model for a single wall in the physical test program that was constructed using a polyester geogrid reinforcement material. In the current paper, the numerical model is calibrated against the end-of-construction stage for each of the walls, which represents a working stress condition. This is the operational condition that is of most interest to designers as opposed to an ultimate limit state or failure condition. The paper reports details of the constitutive models used for the component materials in the walls and compares selected measured and predicted responses for the three walls including facing horizontal displacements, horizontal and vertical toe boundary 251 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-30.qxd 08/11/2003 20:38 PM Page 252 reactions and reinforcement strain distributions. The comparisons are based on both linear elastic and nonlinear hyperbolic models for the soil backfill. 2 PHYSICAL TEST MODELS 24 rows of segmental blocks 3.60 m Figure 1 shows a front view of an RMC test wall with a modular block (segmental) facing. The walls were 3.6 m high with a facing batter of 8° from the vertical. The first wall (Wall 1 – control) was built with six layers of weak polypropylene geogrid (PP) reinforcement placed at a vertical spacing of 0.6 m. The second 8° 1.15 m 1.00 m 1.15 m Instrumented middle section of the wall Reinforcement layers Figure 1. Large-scale instrumented geosynthetic reinforced soil modular block retaining wall constructed in the RMC Retaining Wall Test Facility. wall was a nominally identical structure except that the reinforcement stiffness and strength of the geogrid were reduced by 50% by removing every other longitudinal member in each layer. Wall 3 was nominally identical to Wall 1 except that only four reinforcement layers were used in the wall at a vertical spacing of 0.9 m. In each structure, the wall facing consisted of a column of discrete, dry-stacked, solid masonry concrete blocks with continuous concrete shear keys. The wall facing was built with three discontinuous vertical sections with separate reinforcement layers in plan view. The width of the instrumented middle section was 1 m. The backfill was a clean uniform size rounded beach sand (SP) with a flat compaction curve. The sand was compacted to a unit weight of 16.7 kN/m3 using a lightweight vibrating mechanical plate compactor. The friction between the backfill soil and sides of the test facility was minimized by placing a composite arrangement of plywood, Plexiglas and lubricated polyethylene sheets over the sidewalls. The discontinuous wall arrangement and sidewall treatment were used to minimize the frictional effect of the lateral boundaries of the test facility and to thereby approach, as far as practical, a plane-strain test condition for the instrumented middle section of the wall structure. The reinforcement layers were rigidly attached to the facing using mechanical connections to simplify the interpretation of connection performance (i.e. this arrangement prevented any possibility of reinforcement slippage between the blocks). Figure 2 illustrates the test configuration for Walls 1 and 2 and the instrumentation that was used to record wall response. The horizontal movement of the wall facing was measured using displacement potentiometers mounted at different elevations against the 0.3 m Facing blocks 6 Facing potentiometer 5 Strain gauge Extensometer Connection load rings 4 3.6 m 3 0.15 m 2 Horizontal toe load ring Reinforcement layer 1 Vertical toe load cells 2.52 m Figure 2. Schematic instrumentation layout of the test walls used in calibrating the numerical model (Walls 1 and 2). Note: Wall 3 is constructed with four reinforcement layers with a vertical spacing Sv 0.9 m. 252 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-30.qxd 08/11/2003 20:38 PM Page 253 facing column. Horizontal toe loads were measured using load rings (a horizontal restrained toe boundary condition). Vertical toe loads were measured using load cells supporting a double row of steel plates, which were used in turn to seat the first course of modular block units. A set of steel rollers was located between the steel plates to de-couple the horizontal and vertical toe load reactions. Reinforcement strains in the wall were measured using strain gauges that were bonded directly to the polypropylene geogrid longitudinal members and extensometers attached to selected geogrid junctions. Backfill settlements were measured using tell-tales and settlement plates. Further details of the construction and monitoring techniques used in the RMC test walls have been reported by Bathurst et al. (2001). approaches: 1) a linear elastic (perfectly plastic) model, and 2) the stress-dependent hyperbolic model proposed by Duncan et al. (1980; also see Itasca 1998). Bathurst & Hatami (2001) and Hatami & Bathurst (2002) reviewed previous attempts reported in the literature to numerically model the response of reinforced soil structures. Their survey showed that the stress-dependent nonlinear elastic model (hyperbolic model) proposed by Duncan et al. (1980), or variants, was the most common constitutive model used to simulate the backfill response during construction and under surcharge loading. However, no comparisons have been reported for simulations using other models including a simple linear elastic model. The backfill material properties used in the current study are reported in Table 1. The values of soil hyperbolic parameters were determined by 3 NUMERICAL MODELING Table 1. Material properties for sand used in the numerical model. 3.1 General Value The finite difference-based computer program FLAC (Itasca 1998) was used to simulate the response of the reinforced soil test walls up to the end of construction. Figure 3 shows the numerical grid used for the segmental retaining walls. 3.2 Material mechanical models and properties 3.2.1 Soil The backfill in all simulations was modeled as a cohesionless granular soil with Mohr–Coulomb failure criterion and dilation angle. The backfill elastic response was simulated using two different 0.3 m 2000 2000 0.5 0.5 0.73 0–0.49 Strength properties (peak friction angle) (deg) c (cohesion) (kPa) (dilation angle) (deg) (density) (Kg/m3) 44 0 11 1730 Interfaces ° Concrete facing blocks Stiffness properties (Hyperbolic model) Kc (elastic modulus number) Kb (bulk modulus number) n (elastic modulus exponent) m (bulk modulus exponents) Rf (failure ratio) (range of permissible Poisson’s ratio values) 3.6 m ° Reinforcement ° 0.6 m 24 x 0.15 = 3.6 m Sand backfill ° 2.5 m 5.5 m Figure 3. Numerical model of the segmental retaining walls (Walls 1 and 2). Note: Wall 3 is constructed with four reinforcement layers with a vertical spacing Sv 0.9 m. 253 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-30.qxd 08/11/2003 20:38 PM Page 254 adjusting initial values taken from the results of triaxial compression tests on backfill sand specimens. The value of Poisson’s ratio for each soil zone during analyses was determined from the calculated values of the soil elastic modulus and bulk modulus from the hyperbolic model and hence was allowed to vary between values of 0 and 0.49 as noted in Table 1. The modulus numbers Ke and Kb were increased to match the measured settlement response of the backfill in the retained zone behind the reinforced soil zone during uniform surcharge loading. The backfill peak plane-strain friction angle value was taken as 44° (Bathurst et al. 2001). The backfill dilation angle value from direct shear tests was found to vary from 9° to 12° for the range of confining soil pressures in the test walls. The value 11° was used in the numerical models. Table 2. Reinforcement stiffness and strength properties. Wall Polymer type Number of layers Stiffness Jt () (kN/m)* Ty (kN/m) W1 W2 W3 PP PP PP 6 6 4 138–1698 69–845 138–1698 14 7 14 * Equations valid for 2.5% Facing Soil-Block Modular Interfaces Blocks Soil Column Behind Facing Nulled Zone (magnified) 3.2.2 Reinforcement The reinforcement layers were modeled with twonoded elastic-plastic cable elements with a straindependent tensile stiffness, J(), tensile yield strength, Ty and no compressive strength. The reinforcement load–strain response was modeled in parabolic form as: Block-Block Interfaces Connection Backfill Beam Numerical Elements Two-noded Reinforcement Grid Elements (1) where T is axial load and is axial strain. This equation is valid for 2.5% which captures the range of in-situ measured strains that correspond to the endof-construction working stress levels for the experimental walls and is well below the reinforcement strain at yield. The strain-dependent, secant tensile stiffness of the reinforcement, Js (), was calculated from Equation 1 as: (2) Parameter A in Equation 2 is the initial stiffness modulus and parameter B is the strain-softening coefficient, which is a positive value for polypropylene reinforcement prior to yield. The stiffness of the polypropylene geogrid reinforcement was determined from the constant rate of strain tests on virgin geogrid specimens tested in-isolation at a strain rate of 0.01%/min. The reinforcement material properties used in numerical simulations are presented in Table 2. The structural nodes of the reinforcement cable elements were rigidly attached to the gridpoints of the backfill numerical mesh. This was done to ensure compatibility of displacements between reinforcement structural nodes and backfill gridpoints. With this approach, the grout interface was not utilized in the numerical model. Therefore, pullout of the Figure 4. Details connection. of facing-backfill-reinforcement Table 3. Interface properties. Value Soil–Block sb (friction angle) (deg) sb (dilation angle)(deg) knsb (normal stiffness) (kN/m/m) kssb (shear stiffness) (kN/m/m) 44 11 0.1 106 103 Block–Block bb (friction angle) (deg) c (cohesion) (kPa) knbb (normal stiffness) (kN/m/m) ksbb (shear stiffness) (MN/m/m) 57 45.7 106 50 reinforcement from the backfill was prevented, which was consistent with measurements recorded in the physical tests. 3.2.3 Interfaces The concrete facing units in the test walls were modeled as linear elastic continuum zones separated by nulled zones of zero thickness that contained interfaces (Fig. 4). Table 3 summarizes the values for the 254 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Backfill (Continuum) Zones 09069-30.qxd 08/11/2003 20:38 PM Page 255 3.2.4 Construction and boundary conditions Fixed boundary conditions in horizontal and vertical directions were assumed in the numerical model for gridpoints at the rigid foundation level, and in the horizontal direction at the backfill far-end boundary. The toe boundary condition in the physical and numerical models is a reasonable approximation to the restraint that can be expected for the typical field case of a buried footing. The backfill and facing units were placed in lifts of 150 mm (i.e. the height of one modular block) and the reinforcement layers were numerically installed as each reinforcement elevation was reached. Backfill compaction during construction was modeled by applying a horizontal stress component on the back of the facing units as the soil layers and facing blocks were put in place and the model solved to reach equilibrium. However, as described in measured a) Wall 1 b) Wall 2 c) Wall 3 3 2 1 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Facing displacement (mm) Figure 5. Measured and predicted facing displacements at end of construction. Section 3.2.1, greater soil modulus values than those obtained from laboratory triaxial compression tests were used for the backfill model. With this approach, negligible horizontal stress was needed behind the facing panel to simulate backfill compaction in the walls reported in this paper. 4 RESULTS 4.1 Calibration results The response results for each of the three test walls in this investigation were obtained by changing the reinforcement stiffness (Table 2) or number of layers in the numerical model to match the physical test. The material properties for all other wall components were kept the same. 4.1.1 Facing displacements Figure 5 shows the measured and numerically calculated facing lateral displacement at potentiometer levels at the end of construction. The measured displacement results are readings from the potentiometers that were mounted against the facing blocks at reinforcement layer levels during construction. The predicted results are obtained using the material properties shown in Tables 1–3. The results of Figure 5 show satisfactory agreement between recorded and predicted facing lateral displacements for all three 255 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands predicted 4 Elevation (m) interface properties used in the wall simulations. The interfaces were modeled as a spring-slider system with constant strength and stiffness properties. The values of interface normal (kn) and tangential (ks) stiffness parameters were chosen after a parametric analysis to minimize computation time. The values reported in Table 3 were found to be smaller than the default values recommended in the FLAC manual (Itasca 1998) that were used as the starting point in the parametric analysis. The magnitude of the normal interface stiffness value was made as large as possible to avoid the intrusion of adjacent zones but not to cause excessive computation time. The wall deformation response was found to be relatively insensitive to the value of inter-block shear stiffness for ksbb 50 MN/m/m. Smaller values of ksbb together with material properties reported in Tables 1 to 3 were shown to over-predict measured wall deformation results. The value ksbb 50 MN/m/m gave the best overall agreement with the measured data. This value is also within the range of shear stiffness values backcalculated from load-displacement results of laboratory interface shear tests on the block units (Hatami et al. 2002). The interface shear strength was modeled with the Mohr–Coulomb failure criterion defined by interface cohesion and friction angle. The block– block interface peak friction angle and equivalent cohesion values were determined from the laboratory interface shear tests as bb 57° and cbb 45.7 kPa, respectively (Hatami and Bathurst 2001). The interface friction angle, sb, between the backfill and facing blocks was back-calculated from measured toe reactions and the sum of measured connection forces using the facing equilibrium analysis described by Hatami and Bathurst (2001). Their analyses demonstrated that the soil-facing interface friction angle value in the test walls was close to the magnitude of the backfill peak plane-strain friction angle (i.e. sb 44°). 09069-30.qxd 08/11/2003 20:38 PM Page 256 test walls. Both experimental and numerical results show greater facing displacement magnitudes for Walls 2 and 3 constructed with lower stiffness reinforcement and fewer layers, respectively, compared to the control wall (Wall 1). However, while not reported in this paper, the magnitudes of strain are very different between the three walls under surcharge loading at which time larger wall lateral deformations have occurred and the 4.1.2 Reinforcement strains Figures 6, 7 & 8 show the measured and predicted reinforcement strain distributions in the test walls at end of construction. The measured results are the data from the strain gauge readings. The predicted strain distributions for test walls show overall satisfactory agreement with the experimental results. The strain magnitudes at end of construction for all test walls are typically less than 1%. Measured strains of this magnitude for the polymeric reinforcement used in these walls have been calculated to have a standard deviation as large as $0.3% strain (Bathurst et al. 2003). Hence, within the accuracy of the physical measurements, the results shown in Figures 6–8 capture both the magnitude and distribution of strains in the measured data. The data show that strain magnitudes and distributions are similar for all three walls. This can be explained by the contribution of the very stiff concrete facing column that carries a large portion of the lateral earth forces at the end of construction. measured predicted 0.4 0.2 0.0 Layer 6 Strain (%) 0.4 0.2 0.0 Layer 5 0.6 0.4 0.2 0.0 Layer 4 1.0 Layer 3 0.5 0.0 0.8 Layer 2 0.4 0.0 0.8 Layer 1 0.4 measured 0.2 0.1 0.0 0.0 predicted 0.0 Layer 6 0.4 0.2 0.0 0.5 1.0 Distance (m) Layer 5 predicted 0.8 0.4 0.2 0.0 2.0 Figure 7. Measured and predicted strain distributions at end of construction using hyperbolic soil model (Wall 2). measured Layer 4 Layer 4 0.4 0.0 0.8 0.8 Layer 3 0.4 Strain (%) Strain (%) 1.5 0.0 0.8 Layer 2 0.4 0.0 Layer 3 0.4 0.0 0.8 Layer 2 0.4 0.0 0.8 0.8 Layer 1 0.4 Layer 1 0.4 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 Distance (m) Figure 6. Measured and predicted strain distributions at end of construction using hyperbolic soil model (Wall 1). 1.0 Distance (m) 1.5 2.0 Figure 8. Measured and predicted strain distributions at end of construction using hyperbolic soil model (Wall 3). 256 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0.5 09069-30.qxd 08/11/2003 20:38 PM Page 257 tensile load capacity of the reinforcement layers is mobilized (Bathurst et al. 2001). The calculated maximum reinforcement load in all wall models at end of construction was less than 1 kN/m, which was well below the yield strength of the reinforcement materials (Table 2). 4.1.3 Toe reactions Figure 9 shows the histories of the measured and calculated horizontal and vertical toe loads for the test walls during construction. The figure shows a satisfactory agreement between the predicted and measured horizontal and vertical toe reactions for the walls at the end of construction. The plots of measured horizontal toe load during construction of Walls 2 and 3 deviate from a smooth curve at early stages during construction. This is thought to be due to a local greater soil compaction effort at the back of the facing units during construction. Otherwise the plots of predicted and recorded horizontal toe loads are in close agreement. The results shown in Figure 9 indicate that wall reinforcement stiffness or number of layers has a negligible effect on the magnitude of toe loads during construction for the wall height and reinforcement configurations examined. The reason for this consistent response, particularly with respect to 50 vertical a) Wall 1 measured predicted 40 30 horizontal loads is the strong influence of the heavy facing column as noted previously. 4.2 Influence of soil model on predicted wall response Selected response features of Wall 2 were examined using a linear elastic soil model with the values of Young’s modulus and Poisson’s ratio taken as E 48 MPa and v 0.2, respectively. The value of Young’s modulus was back calculated from the measured pressure-settlement results of the backfill behind the reinforced soil zone during surcharge loading. Figure 10 shows the measured response and the predicted facing displacement results using the linear elastic and hyperbolic soil models for Wall 2. The plotted values are deformations with respect to the time of installation of each displacement device. Hence, these plots should not be confused with the actual wall deformation profiles at the end of construction. Both predicted curves capture the range of wall deformations recorded at the end of construction. The close agreement between the predicted facing displacement results in Figure 10 indicates that the values of soil hyperbolic model parameters reported in Table 1 are consistent with a constant soil modulus value that was determined from the measured loadsettlement response of the backfill. This result may not be unexpected since the hyperbolic modulus numbers Ke and Kb in Table 1 were independently 4 20 measured hyperbolic soil model linear elastic soil model horizontal 10 3 vertical b) Wall 2 40 Elevation (m) Toe reaction (kN/m) 0 50 30 20 horizontal 10 0 2 50 vertical c) Wall 3 40 1 30 20 horizontal 10 0 0 0 5 10 15 20 0 25 Number of facing units (blocks) Figure 9. Measured and predicted toe reaction forces during wall construction. 10 Figure 10. Measured and predicted facing displacements of Wall 2 at end of construction. 257 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 2 4 6 8 Facing displacement (mm) 09069-30.qxd 08/11/2003 20:38 PM Page 258 Toe reaction (kN/m) obtained by matching the load-settlement response of the backfill retained zone under surcharge loading as explained in Section 3.2.1. Figure 11 also shows that the predicted toe reaction responses were essentially identical using both soil models. Taken together, the data in Figures 10 and 11 suggest that the simpler elastic soil model is sufficient to model these performance features of the wall. However, the results using the two soil models shown in Figure 12 illustrate a subtle but important difference in the distribution and 50 linear elastic 40 vertical hyperbolic 30 20 horizontal 10 0 0 5 10 15 20 Number of facing courses placed 25 Figure 11. Comparison of predicted toe reaction forces using linear elastic and hyperbolic soil models during construction (Wall 2). linear elastic 0.2 0.1 0.0 Layer 6 0.4 0.2 0.0 Strain (%) hyperbolic Layer 5 0.6 0.4 0.2 0.0 Layer 4 magnitude of predicted strains in the reinforcement layers at the end of construction for Wall 2. Specifically, the predicted peak reinforcement strains using the soil linear elastic soil model are located farther back from the facing compared to the peaks from the hyperbolic model. Comparison with Figure 7 shows that the measured predicted peak strains are located close to the back of the facing column and not within the reinforced soil mass as predicted for all layers in Figure 12 with the exception of layer 1. The absence of peak strain values within the reinforced soil mass (as predicted using the linear elastic soil model) was corroborated by the lack of a visible shear zone in the backfill at the time of careful soil excavation of the wall. On the other hand, both the hyperbolic model results and the measured data show relatively high reinforcement strain magnitudes at the connections with the facing panel at end of construction, which are not captured using the linear elastic model. It can be argued that horizontal stresses in the soil decrease locally behind the facing due to the outward horizontal movement of the facing column during construction. As a result, the stress-dependent hyperbolic model predicts smaller soil stiffness values behind the facing compared to the constant stiffness model. Therefore, the predicted strain magnitudes at the reinforcement connections with the facing can be expected to be greater (and hence more accurate) using the hyperbolic model rather than the constant stiffness (linear elastic) soil model. Finally, the better match between the predicted and measured wall response using back-fitted modulus values from the measured load-settlement response of the backfill in the actual physical tests highlights the inability of conventional triaxial compression tests to capture the backfill plane strain stiffness in the largescale wall tests. 5 CONCLUSIONS 1.0 Layer 3 0.5 0.0 1.0 Layer 2 0.5 0.0 1.2 0.8 0.4 0.0 Layer 1 0.0 0.5 1.0 Distance (m) 1.5 2.0 Figure 12. Comparison of predicted strain distributions at end of construction using linear elastic and hyperbolic soil models (Wall 2). A numerical model has been developed using FLAC to predict the measured response of carefully instrumented, large-scale geosynthetic reinforced soil modular block retaining walls during construction. The numerical model accounts for staged construction of the retaining walls and incremental lateral displacement of the modular facing using FISH functions. Additional subroutines are included in the program to model the backfill stress-dependent stiffness properties and the nonlinear reinforcement strain-dependent axial stiffness. The measured and numerical results for the construction stage of each wall showed satisfactory agreement for different response parameters including facing displacements, reinforcement strains and history of toe forces. In particular, reinforcement strain 258 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-30.qxd 08/11/2003 20:38 PM Page 259 distributions using a hyperbolic soil model were found to be in good agreement with the measured data. ACKNOWLEDGEMENTS The financial support for this study has been provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada, 11 Departments of Transportation in the USA, and grants from the Department of National Defence of Canada. REFERENCES Allen, T.M., Bathurst, R.J. & Berg, R.R. 2002. Global Level of Safety and Performance of Geosynthetic Walls: A Historical Perspective. Geosynthetics International, (9): 395–450. Allen, T.M., Bathurst, R.J., Lee, W.F., Holtz, R.D. & Walters, D.L. 2003. A New Working Stress Method for Prediction of Reinforcement Loads in Geosynthetic Walls, Canadian Geotechnical Journal, (in press). Bathurst R.J. & Hatami, K. 2001. Review of numerical modeling of geosynthetic reinforced-soil walls. Proc. 10th Inter. Conf. Comp. Meth. Adv. Geomech. Invited Theme Paper, Tucson, AZ, USA, January 2001: (2) 1223–1232. Bathurst, R.J., Walters, D.L., Hatami, K. & Allen, T.M. 2001. Full-scale performance testing and modeling of reinforced soil retaining walls. Special Lecture, IS-Kyushu 2001. Fukuoka, Japan, November 2001. Duncan, J.M., Byrne, P., Wong, K.S. & Mabry, P. 1980. Strength, stress-strain and bulk modulus parameters for finite-element analysis of stresses and movements in soil masses. Report No. UCB/GT/80-01. University of California, Berkeley: Department of Civil Engineering. Hatami, K. & Bathurst, R.J. 2001. Modeling static response of a segmental geosynthetic reinforced soil retaining wall using FLAC. Proc. 2nd Int. FLAC Symp. Numerical Modeling in Geomechanics, Lyon, October 2001, 223–231. Hatami, K. Blatz, J.A. & Bathurst, R.J. 2002. Numerical modeling of geosynthetic reinforced soil retaining walls and embankments. Proc. 2nd Can. Spec. Conf. Comp. Appl. Geotech., Winnipeg, MB, Canada, April 2002. Hatami, K. & Bathurst, R.J. 2002. Numerical simulation of a segmental retaining wall under uniform surcharge loading. Proc. 55th Can. Geotech. Conf. Niagara Falls, ON, Canada, October 2002. Itasca Consulting Group, Inc. 1998. FLAC – Fast Lagrangian Analysis of Continua, Ver. 3.40. Minneapolis, MN: Itasca. 259 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 261 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Three-dimensional modeling of an excavation adjacent to a major structure J.P. Hsi SMEC Australia Pty Ltd, Sydney, Australia M.A. Coulthard M.A. Coulthard & Associates Pty Ltd, Melbourne, Australia ABSTRACT: An excavation adjacent to a major bridge was carried out for the construction of a cut and cover tunnel, Hawthorne Street Tunnel, as part of the South East Transit Project Section 2 in Brisbane, Australia. The bridge was founded on shallow foundations whilst the excavation extended to below the foundation level. There was limited tolerance for the bridge foundations to deflect resulting from the excavation, due to its structural articulation. The support system for the excavation consisted of contiguous and scallop bored piles and ground anchors. The excavation was carried out in stages taking advantage of the 3D effects to minimize ground deformation. To predict the ground performance during excavation and to optimize the design of the ground support, FLAC3D was employed to simulate the 3D effects, the construction sequence and the soil-structure interaction. Field monitoring results showed performance comparable with that predicted by FLAC3D. 1 INTRODUCTION The South East Transit Project Section 2 (SETP2) in Brisbane, Australia was recently constructed to provide a dedicated traffic corridor for public buses and emergency services vehicles. The project route, of a total length of 2.1 km, traversed well-developed areas, which imposed significant constraints on the construction work. One of the major challenges of the project was to provide a design for a transport corridor that passes through the inner urban zones of Brisbane, whilst minimizing the impact on adjacent properties, heavily trafficked arterial roads, public utility services and other infrastructure. The design therefore made substantial use of tunnels, bridges and retaining walls to minimize such impacts. A critical component of the project was to construct a cut and cover tunnel, Hawthorne Street Tunnel, below Hawthorne Street and closely adjacent to Hawthorne Street Bridge, which was an important bridge carrying through traffic between major roads. The bridge was supported on shallow foundations and was very sensitive to ground movement. Excavation for the tunnel construction in close proximity to the bridge was a major concern. A robust support system for the excavation was adopted to control ground movement and prevent damage to the existing bridge. This system involved installation of contiguous and scallop bored pile walls. As the excavation proceeded ground anchors were installed through the bored piles next to the bridge abutment. The excavation was staged and the bridge deflection was monitored to ensure that ground movements fell within the design limit. Due to the critical nature and complexity of the work detailed numerical modeling using FLAC3D (Itasca 1997) was carried out. The modeling considered the excavation and construction sequence, and the interaction between the ground, the bridge foundations, the bored piles, and the ground anchors. An optimized design of the support system was achieved via the use of FLAC3D. This paper presents the project overview, the construction constraints, the site geology, the design criteria, the work performance and, particularly, the FLAC3D modeling. 2 PROJECT DESCRIPTION The South East Transit Project (SETP), an initiative of Queensland Department of Transport, was developed to provide a state-of-the-art busway for public transport and emergency services vehicles. The route stretches from Brisbane’s Central Business District to Logan City, about 20 km to the south-east. 261 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 262 Figure 1. SETP2 route plan. In December 1998 Thiess Contractors Pty Ltd was appointed to design and construct a 2.1 km section of the Busway from Water Street in Woolloongabba to O’Keefe Street in Buranda, Brisbane. This section of the Busway is known as South East Transit Project Section 2 (SETP2); its route plan is shown in Figure 1. SMEC Australia Pty Ltd was appointed by Thiess as the Principal Designer of the project to provide detailed design of all civil engineering works. The SETP2 contract, valued at approximately $A70 million, included the construction of three bus stations, seven underpass structures, a 150 m long cut and cover tunnel, a 230 m long driven tunnel, a three span Super T bridge, and substantial retaining structures. Construction of the project commenced in April 1999 and was completed in November 2000. The Woolloongabba section of the project was completed in early August 2000 in readiness for the Olympic events to be held at the Woolloongabba Cricket Ground. One of the fundamental challenges of the project was to provide a design for a transport corridor that passes through the inner urban zones of Brisbane, whilst minimizing the impact on adjacent properties, heavily trafficked arterial roads, public utility services and other infrastructure (including the Cleveland Railway Line). The design therefore made substantial use of tunnels, bridges and retaining walls to reduce such impacts. 3 SITE GEOLOGY 3.1 General The SETP2 busway route passes through several geological formations, the oldest of which is the Devonian-Carboniferous low grade metasediments belonging to the Bunya Phyllites and Neranleigh Fernvale Beds. These rocks are overlain by the younger Tertiary volcanics of the Brisbane Tuff, and the sedimentary rocks of the Tingalpa Formation. In the northern part of the alignment the metasediments of the Bunya Phyllites and Neranleigh Fernvale form prominent topographic highs; the lows are generally infilled with Quaternary alluvial deposits and some fill. To the south the Brisbane Tuff trends north-east to south-west adjacent to the Neranleigh Fernvale Beds, both of which form undulating topographic highs. The metasediments mainly comprise fine and medium grained strongly foliated interbedded phyllites, argillites and greywacke with some thin quartzites, while the poorly bedded volcanics comprise welded ash flow, bedded tuffs, and breccia with some interbedded conglomerate and sandstone. The Tertiary sedimentary rocks mainly consist of conglomerate and sandstone. During construction these rocks were found to be mainly highly weathered or moderately weathered, of very low to low strength, with extremely weathered seams throughout. The extremely weathered seams contained extremely to very low strength material. Some slightly weathered rock was also encountered. 3.2 4 SITE CONDITIONS The SEPT2 route intersected Hawthorne Street at an angle of approximately 75° (see Figure 2). A cut and cover tunnel (Figure 3) was proposed to be built below Hawthorne Street to provide grade separated through traffics. The construction of this cut and cover tunnel was in the immediate proximity of the existing Hawthorne Street Bridge, which is a four lane, single span arch bridge supported on a strip footing at each abutment. Excavation for the cut and cover tunnel might cause movement of the foundation resulting in damage to the bridge structure. Structural analysis of Hawthorne Street Bridge indicated that the structure was very sensitive to any movement, as the bridge had a hinged mid-span, and therefore was held up by the horizontal support at the foundation level. 262 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Hawthorne Street tunnel site The area is underlain by Bunya Phyllite Formation comprising weathered fine grained phyllite. The soil profile consists of residual soil described as sandy silty clay and gravelly clay overlain by topsoil or a thin layer of fill in parts. The residual soil thins towards the south and west as the thickness of extremely and highly weathered rock increases. Moderately weathered rock was expected near the design level of the busway on the north-east side of Hawthorne Street Bridge but was expected to dip below this level towards the south and west. This change in level of moderately weathered rock is related to the contact with Brisbane Tuff about 15 m south of Hawthorne Street. The groundwater had been measured about 1 m below the busway design level. 09069-31.qxd 08/11/2003 20:39 PM Page 263 Figure 2. Site plan. Figure 3. Site elevation. A horizontal movement of the footing of 10 mm would result in a mid-span vertical movement of approximately 18 mm. Damage to the bridge structure was expected to occur should the foundation movement exceed 10 mm in the horizontal direction. The design of the support system therefore adopted a maximum lateral movement of the bridge foundation of 5 mm for conservative reasons. 5 STRUCTURAL SYSTEM The structural system for the cut and cover tunnel consisted of contiguous and scallop pile walls and a pre-stress concrete plank roof with a reinforced concrete deck slab on top. The planks were placed on a slope to match the slope of Hawthorne Street to reduce the amount of fill on top thereby reducing the 263 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 264 structural depth required. Fill has been maintained on the structure for the location of services. The western bored piles are located very close to the existing abutment of the Hawthorne Street Bridge. Each of the piles next to the bridge was to be fitted with permanent ground anchors to maintain the required horizontal bearing pressure and reduce foundation movements for the Hawthorne Street Bridge. Utilizing permanent ground anchors, the anchors can be stressed to counteract any predicted movement, and re-stressed if the measured movement is greater than that predicted. The eastern wall consisted of 0.9 m diameter piles at 1.5 m c/c spacing whilst the western wall comprised 1.2 m diameter piles at 1.77 m c/c spacing, except for the section within 3 m from the existing bridge foundation where the piles were at 1.25 m c/c spacing. All the piles were socketed 0.5 m into slightly weathered phyllite. The gaps between the piles were shotcreted with fibrecrete. Two VSL permanent ground anchors were installed on each of the piles adjacent to the bridge abutment. Each anchor was socketed 10 m into slightly weathered phyllite and prestressed to 1000 kN. During construction the horizontal movement of the abutment and vertical mid-span movement of the bridge were monitored. The measurements were compared to the estimated values and adjustments made to the construction method and program if required. Temporary struts and hydraulic jacks were specified as part of the contingency plan to help control pile movement at the most critical section. 6 CONSTRUCTION SEQUENCE To minimize ground movement associated with the construction work, excavation was carried out in stages, as follows: 1. Constructed bored pile walls and headstocks. 2. Installed instruments, with minimum accuracy of 1 mm, for monitoring deflection of headstocks, and abutments and mid-span joints of Hawthorne Street Bridge. 3. Undertook baseline readings of the instruments without live load on the bridge and at relatively similar climatic temperature during construction. 4. Excavated southern half of the tunnel to the level of the first (upper) row of ground anchors, with fibrecrete applied between piles progressively. 5. Installed and stressed the first (upper) row of round anchors on the western wall of the southern half of the tunnel. 6. As for 4, but for northern half of the tunnel. 7. As for 5, but for northern half of the tunnel. 8. Excavated southern half of the tunnel to the level of second (lower) row of ground anchors with fibrecrete applied between piles progressively. 9. Installed and stressed the second (lower) row of ground anchors on the western wall of the southern half of the tunnel. 10. As for 8, but for northern half of the tunnel. 11. As for 9, but for northern half of the tunnel. 12. Excavated to the tunnel floor level. 13. Installed the pre-stressed concrete planks over the headstocks. 14. Constructed cast in situ concrete slab over the tunnel floor. 15. Took deflection measurements frequently during the above construction stages with record of corresponding temperature and time. 16. At any time if the measured lateral deflection of the bridge foundation was greater than 5 mm, contingency measures including further stressing the ground anchors would be implemented. 7 NUMERICAL MODELING 7.1 The complex three-dimensional nature of the problem and the need to allow for possible yield of the various rock units and to account for a range of structural elements suggested that FLAC3D would be well suited to the modeling. The work was performed in 1999, using version 2.0 of that program. The existing bridge was included only as a loaded foundation on one side of the new tunnel then the construction sequence outlined above was represented in the numerical model. Coding in the in-built programming language FISH was used to manage grid generation, excavation stages, installation and linking of piles, struts, crossbeams and anchors, and many other aspects of the modeling. The task of development and testing would have been much more difficult without this powerful feature. 7.2 Geotechnical model The geotechnical model adopted for the numerical modeling included the subsurface stratigraphy and geotechnical parameters. Boreholes in this area indicated that the subsurface consisted of residual soils to 5–6 m depth underlain by extremely to highly weathered (EW/HW) phyllite to depths ranging from 9 m near the eastern wall to 12 m near the western wall, overlying moderately weathered (MW) phyllite. Slightly weathered (SW) phyllite occurred at depths between 14 m at the eastern wall and 16 m at the western wall. A uniform subsurface profile based on the more critical profile on the western side of the excavation was assumed and the Mohr-Coulomb soil/rock constitutive model was adopted for each rock and soil unit. The assumed geotechnical model and material 264 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Approach 09069-31.qxd 08/11/2003 20:39 PM Page 265 parameters are given in Table 1. The tensile strength for each rock unit was taken to be 25% of the cohesion, and zero for the soil. In situ horizontal stresses were assumed to be half the vertical in the soil layer and equal to the vertical in the rock units, where an initial approximation to the vertical stress was computed from the above layering and densities. 7.3 Bridge footing The Hawthorne Street Bridge was represented simply via loads applied to the footing shown in the central part of the grid in Figure 4. Two loading cases were considered, based on (a) SMEC’s independent structural 7.4 Table 1. Geotechnical model adopted for analysis. Soil/Rock Depth t c E Type (m) (kN/m3) (kPa) (deg.) (MPa) Residual EW/HW MW SW 0.35 0.30 0.25 0.20 0–6 6–12 12–16 16 18 20 22 25 10 500 750 2500 30 35 40 45 40 50 200 400 analysis of the existing bridge and (b) the bearing pressure specified on the original bridge drawing. Loading (a) was significantly smaller than (b), which is understandable as the foundation pressure shown in the drawing would generally include a factor of safety. For prudent and conservative reasons, the design of the supporting structures to the excavation was based on case (b). The total applied vertical and horizontal loads were 31.55 MN and 21.86 MN respectively, where the horizontal load was taken to act in the direction of the short axis of the footing, i.e. at 25° to the normal to the busway walls. These loads were converted to equivalent Cartesian stress components, which were applied to the upper surface of the footing. Retaining system The retaining system consists of a line of piles on each side of the busway and two rows of ground anchors attached to the piles on the western side to support the existing bridge footing. Temporary struts on the northern half of the busway were also modeled in some cases but proved to be ineffective. The piles on the eastern side are 0.9 m diameter, installed at 1.5 m center spacing. Those on the western Figure 4. Rock units and zoning in inner section of FLAC3D grid, before tunnel construction. 265 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 266 Table 2. Piles (pile structural elements). Parameter Value Young’s modulus Poisson’s ratio Density Shear/normal coupling spring stiffness Shear/normal coupling spring cohesion Shear/normal coupling spring friction angle 31 GPa 0.2 2400 kg/m3 50 MPa 1 MN/m 30° that a coarsely zoned region was attached at the base of the finely-zoned part shown, to provide a better representation of the rock mass response at depth. 7.6 The excavation and support sequence as modeled in FLAC3D was as follows, where the computation was stepped to equilibrium at each stage: Table 3. Ground anchors (cable structural elements). Parameter 8-strand 27-strand Young’s modulus Density Tensile/compressive yield strength Grout stiffness Grout cohesive strength Grout friction angle 200 7860 1.19 10 1.13 0° 200 GPa 7860 kg/m3 4.0 MN 9 GPa 1.7 MN/m 0° side are of 1.2 m diameter, at 1.25 m c/c spacing adjacent to the nearest corner of the bridge footing and 1.77 m c/c away from that corner. All piles are socketed 0.5 m into SW phyllite. The ground anchors for the tunnel as built are 8-strand cables of 15.2 mm diameter, installed in 0.145 m diameter holes and pre-tensioned to 1 MN after installation. They dip at 45° and are socketed 10 m into SW phyllite but are ungrouted over the remainder of their lengths. Other forms of anchor were considered in several of the numerical models, e.g. varying numbers of strand, hole diameter and pre-tension force. Non-geometric property values for the FLAC3D structural elements representing the piles and ground anchors, are given in Tables 2 and 3 respectively. 7.5 Finite difference grid As shown in Figure 2, the tunnel was to run obliquely under the existing bridge, with a bridge foundation immediately adjacent. In addition, a soil slope around the bridge foundation was to be replaced, as the tunnel was excavated, by a reverse-angle slope that was not constant in profile along the excavation. The grid was generated in sections, some of which had to be joined via “attach” commands, then the entire model was transformed to create the correct skew angle between the bridge and tunnel. A view of the inner part of the pre-construction grid is given in Figure 4. The cut-and-cover tunnel was to be constructed within the finer-zoned region to the left of, and parallel to, the slope shown in Figure 4. Note that the grid did not conform precisely to the assumed horizontal boundaries on either side of the MW rock unit, and 0. Apply boundary conditions to far boundaries (fixed horizontal displacements on vertical sides and fixed all displacement components on base), initialize approximate in situ stresses, apply gravity and footing load. 1. Excavate entire busway to 3 m depth. 2. Install both sets of piles then excavate more distant half of busway (relative to view in Figure 4) a further 3 m. 3. Install upper ground anchors, attached to piles between previous excavation and bridge footing, and excavate 2 m from nearer half of busway. 4. Install crossbeams along lines of piles and struts across excavation (not in all models), and excavate more distant half of busway a further 3 m. 5. Install lower ground anchors attached to piles between stage 2 excavation and bridge footing, and excavate further 3 m from nearer half of busway. 6. Remove struts (if installed at stage 4), install upper ground anchors attached to piles between stage 3 excavation and bridge footing, and excavate 2 m more from further half of busway. 7. Excavate another 2 m from nearer half of busway. 8. Install lower ground anchors attached to piles between stage 3 excavation and bridge footing and excavate final 1 m from nearer half of busway. The final model configuration is shown in Figure 5, where the view is as in Figure 4. Implementation of this construction sequence in a FLAC3D model was largely straightforward, with FISH routines controlling the various excavation stages and the placement of all types of structural elements. However, one aspect of the modeling proved to be unexpectedly complex, viz. the setting of links between the many structural nodes and rock zones or other structural nodes. The key constraint (D. Potyondy, private communication) is that each structural node in a FLAC3D model can only be the source of one link. This link may provide either a node-to-zone connection or a node-to-node connection. This means that, where cables or beams are linked to piles at sub-surface nodes, then multiple nodes must be created to manage the various links, and the direction of those links must be carefully controlled. For example, creation of a pile automatically creates links from each sub-surface 266 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Modeling strategy 09069-31.qxd 08/11/2003 20:39 PM Page 267 Figure 5. Inner section of FLAC3D model, showing structural elements at completion of excavation and support. structural node in the pile to the rock zone that contains it. If a ground anchor is to be connected to that pile then the pile must be formed in such a way that there is a node at the intended connection point. When the ground anchor is created as a series of cable elements, a duplicate node will exist at the same coordinates as the pile node to which it is to be connected. By default, each cable node will also have a node-tozone link. These must be deleted at all nodes above the grouted section of the cable, then a new node-to-node link must be created, emanating from the node at the top end of the cable and ending on the corresponding pile node (so that the pile node is the target of the node-to-node link from the cable node and the source of the node-to-zone link to the rock). A similar procedure had to be implemented to handle links between piles and crossbeams and between crossbeams and struts, in cases where the latter were modeled. In that case, it was critical that the final set of links be ordered thus: strut node → beam node → pile node → zone. Rigid links were used for all connections between structural elements. Further, when structural elements were deleted from a model, such as when struts were removed at computational stage 6 above, then the links from the associated structural nodes were not automatically deleted by FLAC3D; this also had to be done explicitly in the data and FISH files. A final complication arose from the fact that multiple links at a point in space can only be distinguished via the link number that is set within FLAC3D when each particular link is created. Management of the links therefore required careful monitoring of the numbers of active structural nodes, elements and links and the highest id number for each. Again, this was handled effectively using FISH coding. 7.7 After a great deal of development and testing several production analyses were performed. As indicated in section 7.4 above, temporary struts were predicted not to be effective, so the final analyses only included piles and ground anchors, thus: – bbhsu9c: 8-strand cables with 1 MN pre-tension; case (a) footing loading from section 7.3; – bbhsu9d: as for 9c except for higher case(b) footing loading; – bbhsu9e: as for 9d except 27-strand cables with 3 MN pre-tension. 267 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Cases modeled 09069-31.qxd 08/11/2003 20:39 PM Page 268 Figure 6. Induced horizontal displacements in footing at end of construction, from run bbhsu9d (upper plot) and bbhsu9e (lower plot). Piles and ground anchors are shown as solid and dashed lines respectively; busway is to left of piles. Figure 6 shows the predicted horizontal displacements of the bridge footing at the end of runs 9d and 9e respectively. These results suggested that the heavier ground anchors, with 3 MN pre-tension, would actually pull the footing away from the busway excavation, i.e. they would overdo the support. In contrast, the anchors in model 9d allowed the footing to relax towards the excavation, but the maximum horizontal displacement was constrained to be less than 5 mm, as required. Histories of x-, y- and z-components of 268 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 269 Figure 7. Histories of induced x-, y- and z-displacements at two points on footing, during the stages of construction, as computed in model bbhsu9d. Figure 8. Axial forces in one set of ground anchors at end of construction, as computed in model bbhsu9d. displacement of several points on the footing, for the same model 9d, showed that they would vary through the various computational stages but were also always predicted to be less than about 5 mm (see Fig. 7). Some further representative results from run bbhsu9d are presented and discussed below. The axial forces acting within one set of ground anchors at the completion of construction are shown in Figure 8; the vertical axis represents the vertical coordinate (RL) along the cables and the horizontal axis the axial force (note that the sign convention for forces in structural elements in FLAC3D depends 269 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 270 Figure 9. Axial forces in some of the piles on the footing side of the excavation, at the end of construction, as computed in model bbhsu9d. Figure 10. Moments (my – upper and mz – lower) in the same set of piles as in Figure 9. 270 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-31.qxd 08/11/2003 20:39 PM Page 271 upon the relative orientation of the nodes). In the grouted (lower) section, the developed forces are close to the pre-tension force of 1 MN but they are about 8% smaller in the ungrouted (upper) section. This is consistent with the response expected as the system is re-equilibrated after pre-tensioning of the anchors. Examples of forces and moments generated in piles adjacent to the bridge footing are given in Figures 9 and 10. In each case the vertical axis gives the RL (z-coordinate) of pile elements and the horizontal axis the force or moment, in SI units. The axial forces in Figure 9 clearly show the effects of the connection of ground anchors at two points in the upper sections of some of the piles. existing bridge were not disturbed. Analysis of the bridge indicated that lateral displacements of the footing must be constrained not to exceed 5 mm. Program FLAC3D was used to simulate the complex construction sequence, including the placement of piles and ground anchors. FISH programming was used extensively to assist in generating the grid and in managing the links between the various structural elements. The results from a series of production analyses indicated that a design based on 8-strand ground anchors, pre-tensioned to 1 MN, would be satisfactory. The predictions of FLAC3D have been confirmed by monitoring during construction. ACKNOWLEDGEMENTS 8 FIELD PERFORMANCE During the entire excavation process the measured lateral deflection of the abutment of Hawthorne Street Bridge was less than 5 mm. There was no distress of Hawthorne Street Bridge during and after construction of the Hawthorne Street Tunnel. The authors wish to acknowledge many valuable communications with Dr. D. Potyondy, Itasca, particularly about the linking of structural elements in FLAC3D. REFERENCES 9 SUMMARY Design and construction of the section of the Brisbane Busway had to ensure that the adjacent footings of an Itasca Consulting Group, Inc. 1997. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 2.0. Minneapolis, MN: Itasca. 271 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-32.qxd 08/11/2003 20:49 PM Page 273 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Pile installation using FLAC A. Klar Technion – Israel Institute of Technology, Haifa, Israel I. Einav Centre for Offshore Foundation Systems, UWA, Australia ABSTRACT: This paper presents a numerical simulation of pile installation using FLAC. A new contact formulation between rigid and deformable bodies is employed. This formulation utilizes equations of motion to describe the behavior of the deformable nodal point along the contact surface. Unlike FLAC’s own embedded interface formulation, the new formulation does not encounter discontinuities problem along nonlinear or piecewise linear surfaces. 1 INTRODUCTION The evaluation of pile installation has great significant in design, for two main reasons: 1. In saturated clay soils, a considerable change in pore pressure takes place due to the pile installation. This change of pore pressure and its subsequent dissipation process affect the pile capacity. 2. Simulation of pile installation allows for more accurate evaluation of the end bearing capacity. Over the last three decades, the problem of pile installation has been extensively researched by different analytical/numerical methods. These methods can be, generally, categorized into five groups: 1. 2. 3. 4. 5. Limit analysis approach, cavity expansion solution, strain path method, Eulerian large strain finite element analysis, and Lagrangian large strain analysis. The problem of pile installation is strongly related to the problem of cone penetration. As a result, advances in understanding were, and still are, strongly connected to research of cone penetrations. In the present work, simulation of pile installation is presented using the Lagrangian large strain analysis code FLAC. To understand the importance of using Lagrangian analysis, the following section overviews the different methods and their limitations. Since the problem of pile installation involves interaction between two bodies (pile and soil), there is a need to employ some kind of interface formulation. FLAC’s own interface formulation is associated with undesirable behaviors, especially along nonlinear convex surfaces. As a result, an alternative and simple approach is suggested. 2 REVIEW OF METHODS Generally, it may be said that five numerical/analytical methods exist for the evaluation of pore-pressure generation and/or end bearing capacity: 2.1 2.2 Cavity expansion solutions In this group, the stresses along the pile are related to solutions of cavity expansion. It is commonly assumed that the solution of cylindrical cavity expansion may represent the deformations and stresses along points, which are far from the end of the pile, and that solution of spherical cavity expansion may be used for approximation of field quantities near the pile lower tip. Figure 1 shows the different zones. Zone II and III 273 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Limit analysis approach This group of methods includes lower bound solution (or slip line method), upper bound solution and limit equilibrium analysis. Results obtained by this approach correspond to collapse mechanism. In these methods only the strength parameters of the soil are introduced; i.e. the stiffness of the soil has no influence on the results. One may refer to Durgunoglu & Mitchell (1975) for some examples of failure mechanisms. 09069-32.qxd 08/11/2003 20:49 PM Page 274 are the ones that may be represented by the cavity expansion solutions. The behavior in Zone I is highly affected by the soil surface. One of the purposes of the work presented here is to evaluate the surface effect on the solution, and to discover the required depth for which the evolution of the end bearing capacity factor Nc is redundant. To learn more about the use of cavity expansions in the solution of pile and cone penetration, one may refer to the excellent book by Yu (2000). 2.3 Large strain Lagrangian finite element analysis This method is the one employed in FLAC, in which the mesh is updated throughout the pile penetration process. This method is the only one that in theory can capture the installation process as it is, while including the influence of both soil surface and changing properties with depth. It should be noted, however, that this method is associated with numerous numerical problems, when simulating pile or cone penetrations, due to the great deformations involved in the problem. Strain path method The strain path method was first suggested by Baligh (1985). In this method, a flow field of soil is assumed to exist around the pile. From this assumed flow field, strains are derived, while stresses can then be determined according to a particular constitutive relation. Two main drawbacks are associated with this method: (a) equilibrium will not necessarily exist, and (b) the effect of the surface and/or changing properties with depth cannot be included, i.e. it can represent only deep steady penetrations. The first limitation may be partly overcome by the use of the iterative procedures suggested by Teh & Houlsby (1988). 2.4 2.5 Large strain Eulerian finite element analysis At this current stage, if one chooses to utilize FLAC’s embedded interface formulation to simulate the soilpile contact, he should acknowledge that on top of the problems due to the great deformations, he introduces new problem. As in many other codes, in FLAC’s interface formulation the two bodies are prevented from crossing each other. This leads to discontinuities in the contact between the bodies (Itasca 2000), if nonlinear or piecewise linear surfaces are involved. Figure 2 shows an example of the problem for piecewise linear rigid contact surface. In Figure 2a the contact formulation corresponds to that of FLAC; i.e. it does not allow for the deformable body (represented by the quadrilateral elements) to overlap the rigid body (represented by the thick black line), and therefore gaps between these two are developed near discontinuity points along of the rigid body. There are two kinds of gaps that may develop between the rigid body and the deformable one. The first kind, (noted as type I in Fig. 2), is a gap that will always result when the deformable body is in contact with a concave surface of a rigid body, and is fictional because the lamped grid points are still in contact. The second kind (noted as type II in Fig. 2) is true gap associated with zero forces acting on the grid points. Zone III Zone II Zone I In the Eulerian large strain finite element analysis, the material (soil) streams through fixed points in space [e.g. van den Berg 1994]. This method possesses the same limitation of the last methods; i.e. inability to model the soil free surface. 3 PENETRATION DIFFCULTIES IN FLAC Figure 1. Cavity expansion zones along the pile. Figure 2. Contact problems along piecewise linear surface. 274 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-32.qxd 08/11/2003 20:49 PM Page 275 This behavior, shown in Fig. 2(a) for piecewise linear surface, will be more pronounced in nonlinear surfaces, since, at least, every second grid point will be out of contact with the rigid body in a convex surfaces. To overcome this problem of undesirable gaps, the contact formulation must allow the deformable body to overlap the rigid body, as shown in Figure 2b. In this “overlapping” formulation, the grid points, from which the strain increments are derived, travel along the rigid body surface, unless some sort of contact logic that allows separation is included. In the following section, such a contact formulation is presented, and is used later for the analysis of pile installation. 4 NEW CONTACT FORMULATION FOR FLAC defined in the fixed coordinate system. If considering the moving coordinate system then the deformable body motion is defined by a velocity vector of vL vD vR and acceleration vector of aL aD aR. Note the rigid body is stationary in the moving coordinate system. Figure 3 shows velocity and acceleration diagrams of a grid point located on the rigid body. This body can be represented by a shape function x f (z) (x and z are the coordinates of the moving system). Since the deformable body cannot enter the rigid body nor departure from it (unless tensile failure is considered as will be discussed later) the motion of it can only be tangential to the rigid body; i.e. only the tangential components of both velocity vector and acceleration need to be introduced in to the equations of motion. If we consider an explicit time marching numerical scheme, the following expression can be written: The following contact formulation is applicable to the interaction between rigid and deformable bodies. In general, both the rigid and the deformable bodies are free to move in space. In the present formulation the rigid body motion is prescribed. However, it can easily be extended to a more general case where the motion of the rigid body is determined by the solution of its motion equations; this feature is studied these days and is being employed for the simulation of anchor installation. In the most degenerate way, the present formulation can also be used to create roller fixing in an inclined angle; an option that is absent from FLAC. 4.1 (1) Formulation A body may be defined as rigid if the distance between any two points of it is constant with time. The motion of a non-rotating rigid body can be described by two components, a velocity vector, vR and an acceleration vector aR. The motion of each lumped mass located on a grid point that represents the deformable body can also be described by two vectors, vD and aD for velocity and acceleration respectively. vD and aD are here Velocity Diagram where v Lx, vLzand aLx, aLzare the components of velocity and acceleration vectors in the moving coordinate x system (x, z) at time t. vxD, vDz , aD , aDz , vxR, vRz , axR, azR are the components of the motion vectors of the deformable and rigid body in the fixed coordinate system (x, z) at time t. If the motion of the rigid body is prescribed (i.e. know a priori) then the motion of the deformable body in the fixed coordinate system is as follows: Acceleration Diagram (2) νR Shape function X’ = f(Z’) a aL R νL νD aD Z' X' Figure 3. Velocity and acceleration diagrams. Note, that in explicit numerical scheme it is assumed that state variable are frozen at each step (dt); i.e. for each time step the rigid and deformable body are fixed in space, and therefore, all values in the right side of Equation 1 are known. The value of aD in Equation 1 is obtained from the assumption that the deformable body is not in contact with the rigid body; i.e. the acceleration is obtained from the forces acting on the grid point due to the deformation of the deformable body. To 275 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-32.qxd 08/11/2003 20:49 PM Page 276 introduce some friction between the rigid and the deformable body one can add frictional force in the tangential direction and re-obtain from it the value of aD. If a rigid-plastic tangential interface is desired then one can define the friction forces direction simply according to the relative velocity between the soil and the pile. If an elastic plastic tangential interface is desired a slightly more complicated formulation is required. Since in the current paper only smooth piles are considered, this kind of formulation is not presented, although written and verified by the writers. To consider possible separation between the rigid body and the deformable body, a contact logic must be introduced. If, for example, the contact logic considers zero tensile forces between the rigid and the deformable bodies as condition for separation, then it will occur once 180 a a. Whenever this condition is satisfied, the grid point is solved according to aD; i.e. vD(t dt) vD(t) aD(t)dt. If during one of the following steps the grid point comes in contact with the rigid body, Equations 1 & 2 are applied. Some small changes need to be introduced into Equation 1 if it is desired to apply one of FLAC’s damping schemes which operates on grid points mass. If damping results only from the constitutive model, then Equation 1 is satisfactory. One may refer to Einav & Klar (2003) where the above formulation is extended to a more general case of three-dimensional rotating rigid-deformable bodies in space. The described procedure is easily implemented in FLAC using a FISH function, which is called during each of the calculation cycles. Generally, since the motion of the contact grid points is solved independently (i.e. using Equations 1 & 2 rather than by FLAC), they need to be fixed in both directions. Quantities related to aD are extracted from FLAC’s gridpoint variable xforce and yforce. Quantities related to velocities, both readable and writeable, are manipulated using FLAC’s gridpoint variable xvel and yvel. 5.2 Assumptions The numerical analysis was conducted under the following assumptions: 1. The material behaves elastic perfectly plastic and satisfies the von-Mises failure criteria. Since the analysis is associated with undrained loading, and the volumetric stresses are decoupled from the deviatoric ones in the considered constitutive model, it is possible to perform a “Dry” simulation; i.e. to obtain the excess pore pressure value using Skempton’s parameter, B 1 1/(1 (Kw/n)/ Ks), utilizing the formula u B(ii ii0 )/3, where the superscript 0 denotes initial state. 2. The undrained strength of the soil is defined according to the relation Cu 0.25v0 OCR0.95, v0 is the initial vertical effective stress and OCR is the over consolidation ratio. 3. The shear modulus is taken proportional to the undrained strength Cu, and the bulk modulus was high enough so the material can be considered as incompressible. 4. The analysis presented herein considered a constant OCR with value that equals 2. 5. Initial stress condition corresponded to K0 of 0.7. 6. To avoid the kinematic constrains of a fixed boundary, and to allow approximate simulation of an infinite soil layer, a prescribed boundary condition was applied to the outer radius of the grid. The boundary condition corresponded to the cylindrical cavity expansion solution of an incompressible elastic material; i.e. the external pressure acting on the mesh was defined by the analytical solution of the internal 5 NUMERICAL ANALYSIS 5.1 Pile shape In the present numerical analysis the pile tip is represented by a continuous nonlinear shape function according to the following expression: (3) where r0 is the radius of the pile and and bc are parameters that define the shape of the pile tip. Figure 4 shows three examples of tip shapes once is set to give 95% of r0 at z 3r0. Due to space limitation, in the current paper we present results only for pile tip with bc equal to 2. bc = 1 Figure 4. Different pile tip shapes. 276 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands bc = 2 bc = 3 09069-32.qxd 08/11/2003 20:49 PM Page 277 pressure of cavity expansion with an identical radius to that of the outer boundary. It was verified that the plastic zone did not reach the outer boundary, thus the elastic cavity expansion solution was suited. 6 RESULTS As discussed in section 2, only Lagrangian large strain analyses can simulate the penetration of the pile through the surface, and therefore are the only ones that can evaluate the influence of the surface on the cone penetration. Figure 9 shows the cone resistance factor, Nc, for different rigidity indexes (G/Cu). The Nc factor was calculated according to: Figure 5 shows typical distortion of the mesh associated with steady penetration of the pile. It should be noted that analyses with much coarser mesh resulted with almost identical stress distributions and response, and that it was impossible for the pile to penetrate the surface without causing a bad geometry, unless more coarser mesh, than that shown in Figure 5, was prescribed near the soil surface. As a rough rule of a thumb, it was found that a contact soil element near the surface should have a radial dimension of about one pile radius, and this can be rapidly become finer as elements are deeper. Figure 6 shows normalized excess pore pressure associated with the state of Figure 5. Figure 7 shows the development of excess pore pressure at depth of 25 radiuses for different radial distances (xi is the initial distance from the axis-symmetric line). Figure 8 shows the changes of the second invariant of the stress during the installation of the pile for points located a depth of 25 radiuses. The y-axis is normalized such that it gives maximum value of one, in accordance to the von-Mises yield surface radius. Initial K0 conditions create initial value that is different than zero. Clearly, as the tip advances towards the checkpoints, the value of the second invariant increases until failure is reached; failure is reached quicker when the points are closer to the axis. (4) Figure 6. Normalized excess pore pressures (G/Cu 100). Figure 5. Distorted (G/Cu 100). mesh in steady penetration Figure 7. Development (G/Cu 100). 277 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands of excess pore pressure 09069-32.qxd 08/11/2003 20:49 PM Page 278 J2D0.5/(2/30.5Cu) 1.2 xi/r0=0.31 xi/r0=1.0 xi/r0=1.65 xi/r0=3.15 xi/r0=5.5 xi/r0=10.5 0.8 smaller than any reasonable slenderness ratio (L/D) associated with piles foundations. However, for G/Cu 100 the normalized steady state depth becomes in the same order of the piles slenderness ratios. For example, for G/Cu 1118 the pile slenderness ratio must be greater than 25 in order for the soil surface to have no effect. In case the pile slenderness is smaller than that value, the soil surface influences the Nc factor. In such case it is not legitimate to use cavity expansion solutions for obtaining the Nc factors, as they assume that there is no influence from surface. The same may be regarded to the solutions based on the strain path method, which also does not consider the soil surface. Points digned with cone tip 0.4 0 0 10 20 Penetration/r0 30 40 Nc Figure 8. Shear behavior (G/Cu) 100. 15 14 13 12 11 10 9 8 7 6 5 4 7 SUMMARY AND CONCLUSIONS G/Cu=1118 G/Cu=500 D Zc G/Cu=223 G/Cu=100 G/Cu=44.7 G/Cu=20 Steady state front 1 4 7 10 13 16 19 22 25 28 31 34 37 40 Zc/D Figure 9. Development of Nc factor with depth. where, Ftot is the total vertical force acting on the pile (considering a smooth pile), v(Zc) is the total vertical stress at depth Zc, and defined as the distance between the surface and the middle of the pile’s tip (and in our shapes, 1.5r0 above its tip). Note that in Figure 9 the horizontal axis Zc/D does not start at zero. This is due to the fact that the cone must be completely positioned inside the soil in order for the Nc factor to have a proper meaning, if it is obtained from the net vertical force, Fnet Nc · Cu · r20. As can be seen from Figure 9, as the rigidity index increases, both the Nc value and depth in which it becomes constant increase. The dashed line in Figure 9 represents a required depth to obtain 95% of the maximum Nc values. This depth is referred herein as the depth of the steady state front, Zss. It is obvious from Figure 9 that Zone III (see Fig. 3) can be associated with spherical cavity expansion solution, as was suggested by Yu (2000), only if some minimal pile slenderness ratio (L/D) is satisfied. It seems that for piles installed in soil with rigidity G/Cu 100 the requirements for minimum slenderness are irrelevant, since the normalized steady state penetration depth is A contact formulation for interaction between rigid and deformable bodies is presented. This approach overcomes some of the problems associated with FLAC’s own built-in interface formulation when it is applied to nonlinear or piecewise linear surfaces. The proposed formulation can easily be used to create rolling fixing along any line inclination, an option that is currently absent from FLAC. The contact formulation is employed in the large strain simulation of pile installation. A study on the generation of pore pressures and on surface effects is presented. There is a strong indication from the analysis results, that the use of cavity expansion or strain path method solutions should be carefully examined before employed in the estimation of end bearing capacity, specially for low slenderness driven piles in soils with high rigidity index. REFERENCES Baligh, M.M. 1985. Strain path method, J. Soil Mech. and Found. Div., ASCE, 111(9): 1108–1136. Durgunoglu, H.T. & Mitchell, J.K. 1975. Static Penetration Resistance of soil, I: Analysis. Proc. ASCE Spec. Conf. on In Situ Measurement of Soil Properties, New York, Vol. 1 151–171. Einav, I & Klar, A. 2003. An approach for nonlinear contact surface analysis and application to pile installation. BGA Int. Conf. On Foundations: “Innovations, Observations, Design and Practice”, Dundee, Scotland, Sept 2003. Itasca Consulting Group, Inc. 2000. FLAC (Fast Lagrangian Analysis of Continua) Ver. 4.0 User’s Manual, Minneapolis Minnesota: Itasca. Teh, C.I. & Houlsby, G.T. 1988. Analysis of the Cone Penetration Test by the Strain Path Method. Proceedings of the 6th International Conference on Numerical and Analytical Methods in Geomechanics, Innsbruck, April, Vol. 1, ISBN 90-6191-810-3, pp 397–402. van den Berg P. 1994. Analysis of soil penetration. Ph.D. thesis. The Netherlands: Delft University Press. Yu, H.S. 2000. Cavity Expansion Methods in Geomechanics. London: Kluwer Academic Publisher. 278 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-33.qxd 08/11/2003 20:51 PM Page 279 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Axial tension development in the liner of a proposed Cedar Hills regional municipal solid waste landfill expansion F. Ma Washington State Department of Ecology Solid Waste Program, Eastern Regional Office, Washington, USA ABSTRACT: A finite difference analysis using the computer code FLAC was conducted of a municipal landfill expansion proposal at the Cedar Hills Regional Landfill (CHRL), King County, Washington State. The main objective of the modeling efforts was to assess whether a standard design of a liner system would be adequate to withstand typical loading conditions for municipal solid waste landfills. The loading conditions were: (1) the gradual layered waste dumping up to 38 m (125) over the High Density Polyethylene (HDPE) liner; (2) dynamic loadings caused by a shallow earthquake and a deep subduction zone earthquake; and (3) the simulation of a cavity development in the old existing waste underneath the HDPE liner due to collapsing of some bulky items. The FLAC (2D) analyses have revealed (1) the developments of the axial tensile stress and displacement in the HDPE liner; (2) the stress and deformation developments in the municipal solid wastes; and, (3) the accumulative and separate developments of stress and displacement of the landfill system under waste dumping, earthquakes and cavity collapsing. The main conclusion was that the maximum axial tension in the 60 mil HDPE liner is higher than the yielding strength of a GSE 60 mil HDPE liner (HDR/Golder 2001) under the proposed site, operational and loading conditions. Thus some local reinforcement or stronger geomembrane liners will be needed. 1 INTRODUCTION A finite difference analysis using the computer code FLAC (Fast Lagrangian Analysis of Continua) was conducted of a municipal solid waste landfill expansion proposal at the Cedar Hills Regional Landfill (CHRL), King County, Washington State. The work was done to independently verify FLAC modeling conducted by King County Solid Waste Division’s Consultant HDR/Golder (2001). The main focus of the analyses was to predict axial tensions that could develop in the 60 mil HDPE geomembrane liner sandwiched between existing wastes and a future 38 m (125 foot) high waste pile. The analyses involved three loading conditions: 1986 lacked a bottom liner (Fig. 1). A portion of the proposed expansion at the CHRL will be located over the existing wastes. This portion of the expansion footprint was called liner-over-refuse (HDR/Golder 2001, Fig. 1). The largest axial stresses were expected to develop in the liner-over-refuse area due to anticipated excessive overall and differential settlements of the underlying wastes. The remainder of the expansion will be founded on a highly competent glacial till subgrade where settlements of the liner are anticipated to be minimal and thus settlement induced tensile stresses would not be of concern. 1. the future emplacement of wastes; 2. earthquakes; and 3. a cavity opening up in the existing waste at a shallow depth below the proposed liner due to the collapsing of some bulky items. 2 SITE CONDITIONS Landfilling commenced at the CHRL site in the early 60s. Those portions of the landfill started before Figure 1. Cross section of the landfill including location of existing wastes, liner, new wastes to be disposed and foundation soil. 279 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-33.qxd 08/11/2003 20:51 PM Page 280 3 MODEL CONFIGURATIONS Table 1. Stiffness of existing waste*. 3.1 Distance from left boundary of the model m (ft) Shear modulus G kPa (psi) Bulk modulus K kPa (psi) 0–73 73–88 88–104 104–119 119–134 134–149 149–165 165–180 180–195 195–210 210–226 226–241 241–256 256–271 271–287 287–293 1966.9 1985.5 1924.8 1829.0 1818.9 1737.5 1678.3 1562.1 1532.4 1590.9 1640.4 1684.7 1771.7 1948.3 2511.1 2670.3 5245.4 5294.4 5181.1 4877.4 4850.3 4633.4 4475.5 4165.6 4086.4 4242.3 3474.3 4492.6 4724.5 5196.0 6696.1 7120.8 Consultant’s study HDR/Golder (2001) predicted the axial tension in the geomembrane in the liner-over-refuse area using the FLAC code. This tension was induced by the overall settlement in the existing and future wastes. The geometry of the proposed landfill expansion was used in this load Case 1 modeling. Load Cases 2 and 3, i.e. the axial stresses induced by earthquake loading and the collapsing of a cavity in the existing waste, were analyzed parametrically using the simplifying assumption of a horizontally stratified site. The modeling assumed a rectangular mesh. The effects of the simplifying assumptions were unknown and were considered minor and probably conservative. Therefore it is probably reasonable to consider the predictions for load Cases 2 and 3 as upper bounds of the field behavior of the geomembrane. This will be further evaluated later in the paper. 3.2 The new models To further refine the understanding of the overall axial stress in the liner-over-refuse and settlements of the existing and future wastes, a single model (Fig. 1) is used in these analyses for the three different stages of loading as stated in the introduction. To make the comparison easier, the properties of existing, new wastes and interface used in the HDR/Golder (2001) study were adopted. i. The existing waste is modeled using FLAC’s MohrCoulomb option. Using the field measurement of the existing waste, stiffness parameters were developed by HDR/Golder (2001) as shown in Table 1. The relationship between unit weight and depth for municipal solid waste by Kavazanjian et al. (1995) is adopted. The friction angle of 35 degree is assumed. ii. The new waste is modeled using FLAC’s Modified Cam-Clay option. The model parameters are elastic shear modulus G 5.12 104 kPa (7430.6 psi), maximum elastic bulk modulus Kmax 1.53 105 kPa (22222.2 psi), density a variable with depth (Kavazanjian et al. 1995), slope of elastic swelling liner 0.03, slope of normal consolidation line 0.13, frictional constant M 1.418, preconsolidation pressure pc 71.8 kPa (10.4 psi), reference pressure p1 71.8 kPa (10.4 psi), and specific volume at reference pressure on normal consolidation lien v 1.75. iii. The 60 mil HDPE liner is connected with new waste above and existing waste or foundation soil below by FLAC interfaces. The interface allows the relative slip movements between the liner and wastes or foundation soil. The interface input parameters are normal stiffness kn 5.12 104 kPa (285.2) (287.9) (281.7) (265.2) (263.7) (251.9) (243.4) (226.5) (222.2) (230.7) (237.9) (244.3) (256.9) (282.5) (364.1) (387.2) (760.6) (767.7) (751.3) (707.2) (703.3) (671.8) (649.0) (604.0) (592.5) (615.1) (634.3) (651.4) (685.0) (753.4) (970.9) (1032.5) * Note: the values are obtained and deduced from HDR/Golder (2001). (7430.6 psi), shear stiffness ks 2.26 105 kPa (32777.8 psi), and friction angle 24°. The 60 mil HDPE liner is modeled with isotropicelastic beam segments; its material properties per unit length are area A 1.524 mm (0.06 inch), and elastic modulus 9.29 105 kPa (134722.2 psi) . The unit weight of the 60 mil HDPE geomembrane is 9.26 kN/m3 (59 pcf). iv. The foundation layer was modeled using FLAC’s Mohr-Coulomb option. The material parame-ters are elastic shear modulus G 5.12 104 kPa (7430.6 psi), bulk modulus Kmax 1.53 105 kPa (22222.2 psi) and frictional angle 35 degree. For static analyses, the left and right boundaries are constrained from horizontal movements and the bottom boundary is constrained from both horizontal and vertical movements. 4 MODELING RESULTS 4.1 Case 1 loading from up to 38 m (125) of solid waste The increase of tensile forces in the geomembrane in the liner-over-refuse area was modeled by simulating the time-history of waste emplacement as a sequence of some 3 m (10) thick layers. The axial stress in the linerover-refuse and the overall displacement in the solid waste are shown in Figures 2 & 3, respectively. 280 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands (0–240) (240–290) (290–340) (340–390) (390–440) (440–490) (490–540) (540–590) (590–640) (640–690) (690–740) (740–790) (790–840) (840–890) (890–940) (940–960) 09069-33.qxd 08/11/2003 20:51 PM Page 281 Figure 2. Distributions of axial tension in the geomembrane liner. Figure 3. Vertical-displacement contour. As shown in Figure 2, the maximum axial tensions in the geomembrane liner are predicted to occur at the left anchor trench (El 1 of Fig. 1) and the transition area (El 75 of Fig. 1) between the liner-over-refuse and regular liner. The axial tension for the portion of liner bearing on the competent foundation is minimal. The tension spike at the toe (El 75 of Fig. 1) of the liner-over-refuse is likely caused by the slope transition form 5H : 1V to 3H : 1V. Figure 3 shows that the center portion around the letter A of the new wastes physically displaced the most at approximate 3 m (10). 4.2 Representative earthquakes, damping and boundary conditions The USGS Probabilistic Seismic Hazard Deaggregation website (USGS 1996 maps) identifies two sources for the CHRLF site as the principal contributors to the earthquake hazard. The sources are: i. a crustal Moment Magnitude (Mw) 6.5–7.0 earthquake at a hypocentral distance within 20 km; and ii. a subduction zone earthquake Mw 8.3–9.0 at a hypocentral distance of approximately 135 km. The near field crustal earthquake has been associated with the potential rupture of the Seattle fault; and, the farther and larger quake represents an interface event on the Cascadia Subduction Zone along the Pacific Northwest Coast. In this analysis the near field quake was represented by the velocity time-history derived from the acceleration time-history recorded from the M7.3 Landers earthquake on June 28, 1992 in California (HDR/Golder 2001); and, the June 23, 2001 M7.9–8.4 Peru Earthquake was considered representative of the larger subduction earthquake. The acceleration time-history for this subduction earthquake was Figure 4. Distributions of axial tension in the geomembrane liner due to static and dynamic loadings. recorded at Moquegua, Peru, which is approximately 190 km southwest of the epicenter. The peak accelerations for the east-west and north-south time-histories are approximately 30 and 20% g, respectively. These accelerations fall within the ranges of predicted peak accelerations (mean plus one standard deviation) from attenuation relationships developed for the Pacific Northwest (Crouse 1991, Youngs et al. 1997). The Moquegua record was 200 seconds long. After the first 120 seconds the shaking produced minimal changes in axial tension. To facilitate further modeling, only the first 120 seconds of the record were used. As the site is asymmetric and the plastic nature of the wastes modeled, the time history was reversed (multiplied by 1) to account for directivity effects. Everything else remained the same. As the results showed almost no impact, further analyses were done using only the unmodified earthquake time history. Since peak acceleration in the E-W direction is about 50% higher that of the N-S direction, both records of the Moquegua time history were used in the analyses. Only 5% Raleigh damping is used for the dynamic analyses. Before running the dynamic analyses, the free field boundary condition of FLAC is applied to the numerical model. Then, either a velocity-time or acceleration-time history is applied from the bottom of the numerical model. 4.3 Using the N-S record, the cumulative development of axial tension from the dynamic loading on top of the static results (Fig. 2) is summarized in Figure 4. Under dynamic loading, axial tension increased approximately 70% in the geomembrane in the lower half of the liner-over-refuse segment, while the upper half stayed almost the same. As a result the maximum tension is predicted to shift to the break in slope (transition from the 5H : 1V to 3H : 1V grade). To further illustrate this effect, the axial tension histories at these two locations are shown in Figures 5 & 6. The very different predicted responses of the two locations are likely due to the different overburden conditions, the slope or slope change of the liner, etc. For example, the overburden for El 1 is only a couple 281 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Dynamic analyses under the subduction 2001 Moquegua, Peru, earthquake time-histories 09069-33.qxd 08/11/2003 20:51 PM Page 282 Figure 7. Distributions of axial tension in the geomembrane liner due to static and dynamic loadings. Figure 5. History of axial tension development in the linerover-refuse due to static and dynamic loadings at EL 75. Figure 8. History of axial tension development in the linerover-refuse due to static and dynamic loadings at El 75. Figure 6. History of axial tensions at EL 1 in response to the static and dynamic loadings. of meters; but El 75 is under approximately 23 m (75) of solid waste and the slope of the liner at this location changed from 5H : 1V to 3H : 1V. Thus, due to the plastic nature of the solid wastes, as they were modeled by the Cam-Clay Elasto-Plasticity Model, the axial tensions in El 75, were not released when the earthquake wave reversed its direction for the cases when the ground accelerations were relatively large. Similarly, the axial tensions predicted in the liner from the Moquegua, Peru 2001, E-W acceleration time history are summarized in Figures 7–9. Figure 9. History of axial tension development in the linerover-refuse due to static and dynamic loadings at El 1. As shown in Figures 7–9, the maximum axial tension occurred at the same location but is projected to be 70% larger when using the Moquegua, Peru E-W acceleration record rather than the N-S record. 282 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-33.qxd 08/11/2003 20:51 PM Page 283 Figure 10. Predicted distribution of axial tension in the geomembrane liner due to static and dynamic loadings. Figure 12. History of axial tension development in the liner-over-refuse due to static and dynamic loadings at EL 1. Figure 11. History of axial tension development in the linerover-refuse due to static and dynamic loadings at EL 75. The increase reflects the larger peak ground acceleration of the E-W record. 4.4 Dynamic analyses using the 1995 Landers, California, earthquake The cumulative development of axial tension on top of the static results (Fig. 2) in the liner from the 1995 Landers quake is summarized in Figure 10. Similarly the axial tension history at the two locations of El 1 and El 75 are shown in Figures 11 & 12. 4.5 Loading from a cavity collapsing in the existing wastes Since local different settlement can be very detrimental to the integrity of liner system, the effects of a void in the waste caused by deterioration of a large metallic object were investigated. Although the existing waste is covered by a 4- to 7-foot-thick interim soil cover, a geophysical survey was conducted and did not indicate any large metallic objects at depths down to about 10 feet. However, it is theoretically possible that collapse of such a void could cause a potential local settlement up to 0.9 m (3) (HDR/Golder 2001). It is more likely that a collapsing bulky item would cause a local settlement up-to 0.9 m (3) after an earthquake. Such localized collapsed items were assumed to occur at varying depths below the geomembrane in the liner-over-refuse area. The worst case scenario was assumed to be the collapse of the cavity following a major earthquake. Therefore, the local settlement in the old wastes was modeled after the dynamic analyses, i.e. after subjecting the model to the E-W Moquegua acceleration time-history. The original wastes are scheduled to be capped with a minimum of 3 m (10) of sand to act as a cushion. Thus, the cavity was placed at a minimum 3 m (10) below the liner. Cavities were simulated at greater depths in the existing waste at locations below the upper-middle part of the linerover-refuse as pointed in the Case B of Figure 13. However, the maximum tensile stress in the geomembrane resulted from the shallowest assumed cavity position. Thus, the results of the deeper cavities were not included here. Also, since the cavity collapsing was stress-controlled, the deformation at locations A, B and C in Figure 13 were only approximately 0.9 m (3). This deficiency would not have substantial impact on the results summarized in Figure 13. In Figure 13 as the location of the cavity changes from the toe of the existing waste (Case A) to the middle of the waste (Case B), the predicted axial tensions increase above the void by 3114 to 4448 N (700 to 1000 lbs). When the void was placed near the top of the slope (Case C of Fig. 13), the axial tension instead decreased by approximately one half of the original value. The reason for the decrease was that the void caused the anchor trench of the liner to move toward the collapsed hole and therefore relaxed the axial tension in the liner. Figure 13 shows that the maximum accumulated axial tension from the three loading scenarios will occur near the toe of the existing waste 283 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-33.qxd 08/11/2003 20:51 PM Page 284 Table 2. Maximum tensions in the geomembrane liner from FLAC models. Static Case tension No. N (lb) Dynamic tension N (lb) Cavity tension N (lb) Total tension N (lb) 11 12 13 204 5175(1165)1 3038(683)2 3519(791)3 890(200) 108(24) N/A N/A 4448(1000) 8301 (1867) 6056 (1361) 6537 (1469) 8231 (1850) 3018(678) 3018(678) 3018(678) 2893(650) 1 Note: Modeling results using Moquegua, Peru E-W Acc.-time history; 2 Note: Modeling results using Moquegua, Peru E-W Acc.-time history; 3 Note: Modeling results using Landers, California Vel.-time history; and 4 Note: Modeling results recommended for 100-foot-wide transition zone along lower edge of liner-over-refuse area (HDR/Golder, 2001) using Landers, California Vel.-time history and based on simplified assumption. Figure 13. Predicted axial tension in the geomembrane liner due to a cavity collapsing variously at locations A, B and C at a depth of 3 m (10) under the liner in addition to static and dynamic loadings. and the maximum axial tension in the geomembrane was only slightly impacted by the location of the collapsed void as in Case A of Fig. 13. 5 COMPARING CURRENT PREDICTIONS TO THOSE OF MODELING BASED ON SIMPLIFIED ASSUMPTIONS (HDR/GOLDER, 2001) The contributions to the maximum axial tension in the geomembrane liner from the three loading scenarios by current and simplified models (HDR/Golder 2001) are compiled in Table 2. Table 2 showed that the maximum tension at the lower edge of liner-over-refuse based on current model (Case 11) are very similar to that based on the simplified assumptions (Case 20, HDR/Golder 2001); and the tension caused by the subduction Moquegua quake represented by the E-W acceleration timehistory is about 50% larger than that based on the crustal near field quake using the Landers velocity time-history (Case 11 vs. 13). CONCLUSIONS First, the maximum axial tensions occur at different locations in the liner-over-refuse segment of the geomembrane depending on the loading condition. The maximum axial tension under three loading conditions at El 75 is 8301 N (1867 lb) per foot. This is higher than the yielding strength of 6938 N (1560 lb) per foot for a GSE 60 mil HDPE liner (HDR/Golder 2001). Thus some local reinforcement or stronger geomembrane liners will be needed. Although the current modeling results of maximum tensions are very similar to the interpretations of the HDR/Golder’s work (2001), the advantage of analyzing the three loading conditions on the same numerical model is that it provides the author with a clearer understanding of where and how the axial tensions in the liner developed as the waste pile rises and is subject to strong shaking and possibly the development of shallow voids. At the same time it also confirmed that the approach of superimposing tensions from individual, simplified loading mechanisms (HDR/Golder 2001) can yield reasonable results comparing with more complex modeling efforts. Secondly, for locations like Cedar Hills in western Washington State, both the crustal earthquake as represented by the Landers velocity records (HDR/Golder 2001); and, the subduction earthquake along the Pacific Northwest Coast need to be considered, since the tension caused by the subduction quake (Moquegua E-W acceleration time-history) was almost 50% larger than that caused by the near field crustal quake (Table 2). The selection of appropriate source zones is necessary to envelop the likely seismic response. This process is greatly aided by the USGS Probabilistic Seismic Hazard Deaggregation website. Further, the more severe responses to the strong distant subduction Moquegua E-W acceleration time-history may be due to its low frequency and long duration. A similar phenomenon has been observed by Matasovic et al. (1998). 284 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-33.qxd 08/11/2003 20:51 PM Page 285 ACKNOWLEDGMENTS The author wishes to sincerely thank the King County Solid Waste Division (KCSWD) for its permission to publish the paper. Special thanks to the reviews and comments of Dr. Victor Okereke of KCSWD and Mr. Frank Shuri of Golder Associates Inc. It is worth noting that the current work is a follow up to the HDR/Golder’s earlier work (HDR/Golder 2001). Special thanks also to Mr. Jerald LaVassor of Washington State Department of Ecology Water Resources Program Dam Safety Section. The author is indebted deeply to his direction, support and invaluable technical and editorial revisions of the paper. REFERENCES HDR Engineering, Inc. & Golder Associates Inc. 2001. Cedar Hills Regional Landfill Area 6 Development Draft Preliminary Design Technical Memorandum Lining System Over Unlined Waste Area. Seattle: King County Department of Natural Resources Solid Waste Division, Washington State. Kavazanjian, E. Jr., Matasovic, N., Bonaparte, R. and Schmertmann, G. R. 1995. Evaluation of MSW properties for Seismic Analysis. Geoenvironment 2000, ASCE Geotech. Spec. Publ. No. 46, 2, 1126–1141. Matasovic, N., Kavazanjian, E. Jr., and Anderson, R. 1998. Performance of solid waste landfills in earthquakes, Earthquake Spectra, Issue #2, Vol. 14, p. 319–334. Youngs, R. R., Chiou, S.-J., Silva, W. J. and Humphrey, J. R. 1997. Strong Ground Motion Attenuation Relationships for Subduction Zone Earthquakes. Seismological Research Letters. Vol. 68, No. 1, Jan./Feb. 58–73. USGS Probabilistic 1996. Seismic Hazard Deaggregation website (http://eqint1.cr.usgs.gov/eq/html/deaggint.shtml). Crouse, C. 1991. Ground-motion attenuation equations for Cascadia subduction zone earthquakes. Earthquake Spectra, 7, 201–236. 285 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-34.qxd 08/11/2003 20:40 PM Page 287 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 The usability analyses of HDPE leachate collection pipes in a solid waste landfill F. Ma Washington State Department of Ecology Solid Waste Program, Eastern Regional Office, Washington, USA ABSTRACT: The objective of the FLAC modeling effort was to analyze the stress–strain behavior of High Density Polyethylene (HDPE) leachate collection pipes proposed for an eastern Washington State solid waste landfill. Case one of the FLAC modeling assessed whether a 102 mm (4) diameter Standard Dimension Ratio (SDR) 9 perforated HDPE leachate collection pipe could withstand the load of up to 64 m (210) high column of solid waste. Similarly case two of the FLAC modeling predicted how a 305 mm (12) diameter perforated SDR 11 HDPE pipe would perform under a solid waste load of as much as about 26 m (85). The FLAC analyses allowed simulating the development of stresses and deformations in the HDPE leachate pipes as the solid waste column grows. The model predictions were compared with results from a methodology included in the Guidelines for HDPE Pipes in Deep Fills (Petroff 1998) used by CH2MHILL (2002) in their design. Using the industry standards of (1) ring compressive stress, (2) pipe deflection and (3) wall buckling, the FLAC results were very similar to the values in the CH2MHILL (2002) study. The main conclusion of the FLAC modeling is that the proposed HDPE leachate collection pipes will be adequate to withstand the loadings associated with the proposed solid waste column heights. overlies the bottom composite liner of the landfill (CH2MHILL 2002). 1 INTRODUCTION Finite difference analyses using the computer code FLAC (Fast Lagrangian Analysis of Continua) were conducted of a solid waste landfill expansion project at the Graham Road Landfill, Spokane County, Washington State. The analyses were performed to confirm the adequacy of modeling done by the project engineer, CH2MHILL. Specifically, the analyses focused on predicting the stress–strain responses of: i. a 102 mm (4) diameter SDR 9 perforated HDPE leachate collection pipe under solid waste load up to 64 m (210), and ii. a 305 mm (12) perforated SDR 11 HDPE pipe under a solid waste load of about 26 m (85) when the landfill is under final closure, respectively. 2 SITE CONDITIONS The landfill accepts solid wastes from industries and other sources, but it does not accept municipal solid wastes. The leachate collection and recovery system (LCRS) consists of a 0.3 m (12) thick granular drainage layer with embedded, perforated HDPE pipes to collect and remove leachate. This LCRS directly 3 MODEL CONFIGRATIONS 3.1 CH2MHILL (2002) evaluated the adequacy of the 102 mm (4) diameter SDR 9 LCRS header pipe and 305 mm (12) SDR 11 sump/pump access pipe by the procedures included in “Guidelines for HDPE Pipes in Deep Fills” (Petroff 1998). The methodology assesses (1) ring compressive stress, (2) pipe deflection, and (3) wall buckling. The evaluation confirmed that the 102 mm (4) SDR 9 and the 305 mm (12) SDR 11 were adequate for the anticipated 64 m and 26 m (210 and 85) of overlying fill, respectively. It should be noted that the analysis assumes the bedding encapsulating the pipe will be compacted to a minimum of 90 percent of the maximum density as determined by ASTM Procedure D698. This compacted zone must extend immediately above the pipe and for 5 feet on either side of the pipe. The specifications prepared by CH2MHILL (2002) accordingly required the compaction of the drain rock around the pipe to the above cited minimums. 287 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Consultant analyses 09069-34.qxd 08/11/2003 20:40 PM Page 288 3.2 This study’s current modeling The empirical equations provided in the Guidelines for HDPE Pipes in Deep Fills (Petroff 1998) predict the maximum ring compressive stress and the maximum pipe deflection. The FLAC modeling described herein went on to predict the distribution of the axial stress, shear stress and bending momentum in the pipes as well as the stress–strain or deformation relationship under plane strain conditions. The following simplifications or assumptions were made in the FLAC modeling. i. The HDPE pipe is modeled with isotropic-elastic beam segments; its material properties per unit length are area A t, moment of inertia I t3/12, elastic modulus under plane strain e ey / (1 2), where t is the pipe wall thickness, ey is the Young’s modulus and the Poisson ratio. The material properties of the 102 mm (4) SDR 9 and 305 mm (12) SDR 11 pipes are listed in Table 1. The effect of perforations in the HDPE pipes was modeled by reducing the wall thickness by one twelfth as typical perforations account for that much of the pipe mass. ii. The gravel drainage layer is simulated as a perfect plastic Mohr-Coulomb material. The material properties include bulk modulus K, shear modulus Gs, friction angle and density. The average values of density and friction angle of gravel are 2.16 g/cm3 (135 lb/ft3) and 40 degree, respectively. As adapted and extended from McGrath (1994) by Petroff (1998), the typical design values of one-dimensional constrained modulus Ms of soil increase linearly with the increase of the soil overburden pressure. This linear relationship (Petroff 1998) was used in the FLAC modeling and was related to bulk modulus K and shear modulus Gs by (K Ms (1 )/(3 (1 )) and Gs Ms(1 2)/ (2(1 ), respectively. The elastic modulus Es is related to the constrained modulus Ms of the soil by Es Ms(1 ) (1 2)/(1 )). The material properties of the gravel layer are listed in Table 2. iii. The waste behavior is simulated by the elastoplastic Modified Cam-Clay model in FLAC. The behavior of the HDPE pipes is the focus of the modeling effort here. The stress-strain response of the waste was of little interest. Thus, the waste properties approximated with values typical of soft clay at a density of 1441 kg/m3 (90 lb/ft3) were used as the input parameters of the wastes. The model parameters are elastic shear modulus G 5.12 104 kPa (7430.6 psi), maximum elastic bulk modulus Kmax 1.53 105 kPa (22222.2 psi), density a variable with depth (Kavazanjian et al. 1995), slope of elastic swelling liner 0.03, slope of normal consolidation line 0.13, frictional constant M 1.418, preconsolidation pressure pc 71.8 kPa (10.4 psi), reference pressure p1 71.8 kPa (10.4 psi), and specific volume at reference pressure on normal consolidation lien 1.75. Table 1. Material properties of HDPE pipes. Setting d kg/m3 t mm A mm2 I mm4 e kPa 4 SDR 9* 12 SDR 11* 955.2 955.2 12.7 29.4 322.6 746.8 4.34E 3 5.38E 4 2.14E 5 2.14E 5 * Note: the values are obtained or deduced and deducted from Driscoppipe data sheets and are values per 25.4 mm (1 inch) of the pipes. Table 2. Material properties of gravel layer.* Overburden kPa (psi) Ms kPa (psi) K kPa (psi) Gs kPa (Psi) 68.9 (10) 137.9 (20) 275.8 (40) 413.7 (60) 551.6 (80) 689.5 (100) 10,342.1 (1500) 11,721.1 (1700) 14,479.0 (2100) 17,236.9 (2500) 19,994.8 (2900) 22,063.2 (3200) 6405.2 (929) 7253.3 (1052) 8963.2 (1300) 10,673.1 (1548) 12,376.1 (1795) 13,658.5 (1981) 2957.9 (429) 3350.9 (486) 4136.9 (600) 4922.9 (714) 5715.8 (829) 6301.8 (914) *Note: the values are obtained or deduced and deducted from Petroff (1998) with Poisson ratio 0.3. 288 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-34.qxd 08/11/2003 20:40 PM Page 289 iv. The HDPE pipe is connected to the surrounding gravel layer using the FLAC interface. The interface allows the relative slip between the HDPE pipe and the gravel. The interface input parameters per unit length are normal stiffness kn 411.6 N/mm (2341 lb/in), shear stiffness ks 107.8 N/mm (619 lb/in), and friction angle 30°. Due to limited data availability on the interface properties, more studies will be done in some future researches. of the pipe response. When executing the models, the waste was added layer by layer to simulate a landfill operation. As the waste pile grew, increasing the vertical overburden pressure, the modulus of the gravel layer increased accordingly (Petroff 1998). Since the relatively small sizes of the HDPE pipes, no further refinements of the mesh around the pipe openings were done. The details of the model grids for the two scenarios are shown in Figures 1 & 2. The horizontal dimension of the grid was chosen such that a further increase in width had no material impact on the modeling results 4.1 4 MODELING RESULTS 305 mm (12 ) SDR 11 HDPE LCRS header pipe under solid waste load up to 26 m (85 ) The axial compression, shear and moment distribution for the 305 mm (12) SDR 11 HDPE LCRS sump/pump pipe under solid waste of 26 m (85) are shown in Figure 3. Since the results for pipes with or without perforations are very similar, only the latter are shown here. Numerical values of the data shown graphically in Figure 3 are presented in Table 3. Figure 1. Cross section of the 305 mm (12) SDR 11 HDPE pipe in the landfill leachate collection layer under the 26 m (85) of solid wastes. Figure 2. Cross section of the 102 mm (4) SDR 9 HDPE pipe in the landfill leachate collection layer under the 64 m (210) of solid wastes. Figure 3. Predicted distributions of axial compression, shear and moment of the 305 mm (12) SDR 11 HDPE pipe without perforation under a simulated solid waste loading of 26 m (85). 289 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-34.qxd 08/11/2003 20:40 PM Page 290 Table 3. Numerical results corresponding to Figure 3. Elem ID* Nod1* Nod2* 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 F-Shear N F-axial N 27.6 22.2 15.6 39.5 39.5 48.1 74.0 78.0 64.1 86.7 47.4 10.3 8.8 8.1 7.45 73.0 49.9 93.8 75.3 69.2 84.1 38.6 35.6 35.7 841.8 855.5 892.8 949.6 989.8 1068.2 1166.2 1254.4 1332.8 1411.2 1479.8 1489.6 1499.4 1506.3 1460.2 1381.8 1303.4 1244.6 1136.8 1048.6 989.8 912.4 860.4 837.9 Mom-1 N-m 4.60 4.15 3.82 2.98 2.16 1.15 0.39 2.03 3.36 5.17 6.16 5.95 6.13 5.96 4.44 3.20 2.17 0.21 1.36 2.81 4.57 5.37 6.13 6.87 Mom-2 N-m 5.18 4.61 4.15 3.82 2.98 2.16 1.15 0.39 2.03 3.36 5.17 6.16 5.95 6.16 5.96 4.44 3.20 2.17 0.21 1.36 2.81 4.57 5.37 6.13 *Note: Elem ID 1 corresponds to the pipe segment at pipe crown with nods 1 and 2. Elem ID 24 to the pipe segment at pipe invert. F-shear and F-axial are the shear and axial forces of the each pipe segment, respectively. Mom-1 and Mom-2 are the moments of both ends of each pipe segment. Figure 3a shows that the maximum hoop compressive force occurs near the pipe springline and is approximately 50% larger than that at the pipe crown and invert. The maximum shear force occurs at approximately 45 degrees below the pipe crown as in Figure 3b. Figure 3c shows that the moment near the pipe crown and springline are approximately 25% larger than that at pipe invert. 4.2 102 mm (4 ) SDR 9 HDPE LCRS header pipe under solid waste load up to 64 m (210 ) The axial compression, shear and moment distributions for the 102 mm (4) SDR 9 HDPE LCRS header pipe predicted for 64 m (210) of solid waste are shown in Figure 4. As in the earlier case, the results for pipes with or without perforations are very similar. Thus, only the latter are shown here. The numerical results underlying the graphical data in Figure 4 are cited in Table 4. Similar trends are evident in the loadings predicted in Figure 4 to those predicted for the 12 SDR 11 pipe case. However, the differences between the values of hoop compression and moments at the pipe crown, springline and invert are much smaller, see Table 4 and Figures 4a&c. As in the case of the 305 mm (12) SDR 11 pipe, the maximum shear force is predicted to occur at approximately 45 degrees below the pipe crown, see Figure 4b. 4.3 4.4 Comparing FLAC predictions to those of traditional empirical formulas (Petroff 1998) According to general thin beam theory, the normal stress in the pipe wall is the combination of normal stresses from the hoop force and the bending moments. It is expressed as follows: 290 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Pipe crown deflections The predicted crown deflection of a buried pipe is one of the key parameters in assessing the structural adequacy of a pipe in traditional pipe analyses. The predicted displacement histories of the pipe crown and invert are shown in Figures 5 & 6. (1) 09069-34.qxd 08/11/2003 20:40 PM Page 291 Table 4. Numerical results corresponding to Figure 4. Elem ID* Nod1* F-Shear Nod2* N F-axial N 12 11 10 9 8 7 6 5 4 3 2 1 12 11 10 9 8 7 6 5 4 3 2 1 13 12 11 10 9 8 7 6 5 4 3 2 661.5 698.7 751.7 792.8 852.6 941.8 933.0 895.7 861.4 789.9 708.5 672.3 33.0 65.9 80.7 54.1 70.2 58.2 40.1 38.5 95.1 100.4 70.7 46.1 Mom-1 N-m 2.07 1.11 0.07 0.87 1.89 2.74 2.16 1.59 0.20 1.26 2.30 2.97 Mom-2 N-m 2.56 2.07 1.11 0.07 0.87 1.89 2.74 2.16 1.59 0.20 1.26 2.30 *Note: Elem ID 1 corresponds to the pipe segment at pipe crown with nods 1 and 2. F-shear and F-axial are the shear and axial forces of the each pipe segment, respectively. Mom-1 and Mom-2 are the moments of both ends of each pipe segment. Figure 4. Predicted distributions of axial compression, shear and moment per 25.4 mm (1) of the 102 mm (4) SDR 9 HDPE pipe without perforation under solid waste loading of 64 m (210). where Normal stress, kPa (psi); N compressive hoop force, N (lb); A cross-sectional area of pipe wall, m2 (in2); M bending moment, N-m (lbin); I inertia of wall cross-section, m4 (in4); and t pipe wall thickness, m (in). For a pipe under plane strain conditions, only a unit length of pipe need to be considered in Equation 1. Using the modeling results of Tables 3 & 4 and the material properties and geometries of the pipes (Table 1), the normal stress distributions from inner to outer fibers in the pipe wall are shown in Figures 7 & 8 using Equation 1. Figures 7 & 8 clearly showed that there are only four locations where the pipe wall is under uniform compression. For the rest of the pipe section, larger compressive stresses exist either in the Figure 5. Displacement and crown deflection of the buried HDPE 305 mm (12) SDR 11 Pipe. outer or inner fibers of the pipe wall. The higher compressive stresses from both the hoop and bending effects are predicted to occur at the crown, invert or springline of the pipe. Furthermore, the maximum hoop and total compressive stresses occur at or near the springline of the pipe. The predicted maximum hoop and compressive stresses and the crown deflection of the pipe from the modeling results are summarized in Table 5 along with the results of CH2MHILL (2002). A comparison of data shows the CH2MHILL results based on Petroff (1998) are in relatively good agreement with the FLAC results. Note that the predicted displacements 291 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-34.qxd 08/11/2003 20:40 PM Page 292 Inner fiber Outer fiber 48 47 4.00E+03 3.00E+03 1 2 46 45 44 Hoop comp. 3 4 5 6 43 7 42 8 2.00E+03 9 41 40 1.00E+03 10 11 39 0.00E+00 38 12 37 13 -1.00E+03 36 14 35 15 34 16 33 17 18 32 19 31 30 29 Figure 6. Displacement and crown deflection of the buried HDPE 102 mm (4) SDR 9 Pipe. Inner fiber Outer fiber 24 8.00E+03 23 Hoop Comp. 28 27 26 25 24 23 22 Figure 8. Hoop compressive stress and normal stresses along the inner and outer fibers per 25.4 mm (1) of the 305 mm (12) SDR 11 HDPE pipe without perforation included (unit in kPa). 1 2 3 6.00E+03 Table 5. Maximum stresses from FLAC and empirical formulas. 4 22 20 21 4.00E+03 21 0.00E+00 20 Pipes Model SDR/D type 6 -2 .00E+03 19 -4 .0 0E+03 7 11/12 8 18 9/4 9 17 10 16 Pipe Max-comp crown stress deflection kPa % 2.01 103 2.16 103 1.73 103 2.94 103 3.10 103 3.78 103 3.68 103 3.6 3.72 103 4.0 3.5 6.93 103 6.7 6.99 103 7.7 6.8 Note: Modeling results ignoring pipe perforations; Note: Modeling results considering pipe perforations; and Note: See CH2MHILL (2002) study for details. 2 12 13 3 Figure 7. Hoop compressive stress and normal stresses along the inner, outer fibers per 25.4 mm (1) of the 102 mm (4) SDR 9 HDPE pipe without perforation (unit in kPa). of CH2MHILL’s (2002) modeling are similar to those of the solid pipe in the FLAC analyses. However, CH2MHILL’s prediction of crown deflections are lower than those of FLAC when modeling a perforated pipe. Overall the results of both methods are within or are very close to the allowable ranges of industry standards of 6.89 103 kPa ( 1000 psi) hoop compressive stress and 7.5% crown deflection (Wilson-Fahmy and Koerner 1994). 4.5 Buckling Ideally, a more sophisticated pipe model would be employed to account for the visco-plastic behavior of HDPE pipes. This would allow incorporating creep and pipe buckling effects into the FLAC modeling results. This was not done. Instead, the factor of safety against buckling was assessed by simply computing the ratio of computed stresses to the critical buckling stresses as per Wilson-Fahmy and Koerner (1994). According to their study, the critical buckling stresses for pipes of SDR 11 and 9 are 1.03 104 and 1.17 104 kPa (1500 and 1700 psi), respectively. 292 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands FLAC1 FLAC2 CH2MHILL3 FLAC1 FLAC2 CH2MHILL3 1 11 15 14 Max-hoop stress kPa 5 2.00E+03 09069-34.qxd 08/11/2003 20:40 PM Page 293 Dividing the critical buckling stresses by the pipe wall hoop stresses in Table 5, yields factors of safety of 4.79 and 3.79, respectively, for 12 SDR 11 and 4 SDR 9 pipes with perforation. These values exceed industry practice of a minimum factor of safety of 2. 5 CONCLUSIONS First, the results presented in Figures 7 & 8 demonstrate graphically the predicted stress distribution along the pipe wall. Based on these results, a stress induced failure of the pipe would be judged acceptably remote. If one where dealing with appreciably greater waste depths one would expect pipe overstressing to initiate near the springline of the pipe. Second, the results based on the “Guidelines for HDPE Pipes in Deep Fills” (Petroff 1998) are relatively close to the FLAC modeling results. Therefore, the more sophisticated FLAC modeling is likely unnecessary. It would seem warranted only when conventional empirical methods indicate the pipe stresses and crown displacements approach the minimum accepted factor of safety on a critical project. ACKNOWLEDGMENTS The author wishes to thank Waste Management (WM) Northwest for its permission to publish the paper. Special thanks to the review and comment of Mr. Rodger North of WM. Special thanks also to Mr. Jerald LaVassor of Washington State Department of Ecology Water Resources Program Dam Safety Section for his invaluable technical and editorial revisions of the paper. REFERENCES CH2MHILL. 2002. Cells 4 and 5 Design Report, Graham Road Recycling and Disposal Facility. Spokane, Washington: Waste Management, Inc. McGrath, T. 1994. Analysis of Burns & Richard Solution for Thrust in Buried Pipe. Simpson Gumpertz & Heger, Inc, Cambridge, Massachusetts. Petroff, L. 1998. Guidelines for HDPE Pipes in Deep Fills, (written under the employment of PLEXCO). Wilson-Fahmy, R.F. & Koerner, R.M. 1994. Finite Element Analysis of Plastic Pipe Behavior in Leachate Collection and Removal Systems. Geosynthetic Research Institute, Drexel University. 293 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-35.qxd 08/11/2003 20:40 PM Page 295 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 FLAC numerical simulations of the behavior of a spray-on liner for rock support C.P. O’Connor & R.K. Brummer Itasca Consulting Canada Inc, Sudbury, Ontario, Canada G. Swan Falconbridge Ltd, Sudbury Operations, Sudbury, Ontario, Canada G. Doyle 3M Canada Co., Mining Division, London, Ontario, Canada ABSTRACT: The current practice of bolting and screening of underground excavations is time consuming and labor intensive and requires extensive materials handling. In pursuit of alternative rock support systems, Falconbridge Ltd. has experimented with several different spray-on liners. In co-operation with 3M Canada, a thin spray-on liner was developed with the intention of replacing screen and reducing the cycle time in rapid development mining. FLAC was used on this project to provide an efficient method of investigating alternative material properties without the expense that is typically incurred in full scale testing. 1 INTRODUCTION 2 LINER SUPPORT SYSTEM The implementation of high speed development techniques necessitates the use of rapidly installed support in order to meet the desired cycle time. Current practices of bolting and screening are labor intensive and add significant time to the development cycle. The use of spray-on-liners to act as membrane support in place of screen is seen as the next step in the evolution of rapid development. The physical characteristics of membrane support need to be extensively tested. Full scale physical testing is expensive and time consuming and the number of such tests needs to be limited. In order to fill the gap between laboratory measurements of liner properties and full scale trials of the material, a numerical model of the testing apparatus was constructed. This provides a method by which a large number of potential liner formulations can be investigated without the expense of full scale testing. Using the cable element capability of FLAC, a model of the “baggage loading” testing apparatus was generated to allow for a large number of simulations to be carried out to cover the wide range of properties observed in the liner material. The spray on liner support system developed by 3M Canada is a polymeric compound that contains approximately 40% water when initially applied. As the liner dries out (the rate is dependant on the ambient temperature, humidity, and air flow) the strength builds until after 24 to 72 hours it approaches its ultimate tensile and adhesive strength. The time dependence of the strength of the liner is a critical aspect and one that is very difficult to define in full scale testing apparatus since there are so many variables involved. Shotcrete is often sprayed in excess of 4, this liner system only requires 2 to 3 mm final dried thickness to perform. When dealing with such a thin application, minor thickness variations become important. Another important characteristic of this liner compound is the elongation potential. Depending on the formulation in use, strains ranging from 100 to 600% are possible. Having such a large capacity to deform should help prevent violent failures due to stress build up. A stress–strain curve for some early liner formulations is shown in Figure 1. One final challenge in understanding this material is that it does not yield in a linear fashion. Instead, there 295 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-35.qxd 08/11/2003 20:40 PM Page 296 Stress–strain curve of System I and II Stress (MPa) System II with strong adhesion System I 0 100 200 300 400 500 Percent strain 600 700 800 Figure 1. Stress–strain curve for early revisions of the 3M spray on liner product. Figure 4. Grid used in the spray on liner modeling. modeled. There are currently two tests in use for the liner material. The first is the “dog bone” test, performed on small samples of material and used to generate the stress–stain curves. The second is the “baggage load” test in which a 1 m2 metal frame is filled with rock and sprayed with a coating of liner and tested to failure in a press (see Fig. 2). 3 Figure 2. Baggage load testing frame with a 2 to 3 mm sprayed thickness of liner product. Loading Platen Coarse Gravel Thin slabby granitic material 2 to 3 mm liner support Steel Loading Frame Figure 3. Geometry of the baggage loading test frame. is a well-defined initial yield point located near a strain of 30% after which the stiffness of the entire system reduces until the ultimate tensile strength is reached. All of these different parameters (thickness, time dependent strength, complex yield curve, and extensive deformations) must be accounted for and A FLAC model was built to replicate the baggage load testing apparatus. This involved a complex interaction between several different types of materials and cable elements. The actual grid used in the model is shown in Figure 4. The liner is represented by a string of cable elements across the bottom boundary of the slabby granitic material. A key challenge in the modeling was obtaining the correct response of the liner as the stiffness changes in response to plastic deformation. To do this, a FISH function was written in which the modulus of the material was dynamically adjusted within specific strain intervals defined in a table. This allows for automatic adjustments of the material properties during cycling ensuring an accurate response (see Fig. 5). The stiffness of the material depends on the amount of drying of the liner, and therefore the modulus of the material changes over time (see Fig. 7). In order to properly assess the effect of drying time on the performance of the liner, a sensitivity analysis was required in order to determine how the liner responds under different conditions of drying time and thickness. There is only a limited database of full scale testing on which to calibrate the model. Using this limited data and the material properties that were expected to be produced by the test (i.e. 4 hour drying time, 2 mm thickness), the model was calibrated as best as possible to the test results available (see Fig. 6). 296 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands FLAC MODEL 09069-35.qxd 08/11/2003 20:40 PM Page 297 FLAC Simulations of 3M Liner Systems in CANMET Test vs. Averaged Response Curves From Tensile Strength Testing (Liner Thickness = 2mm) 10 9 Tensile Stress (MPa) 8 7 6 3M System 1 - FLAC 3M System 2 - FLAC 5 3M System 1 Averaged 3M System 2 Averaged 4 3 2 1 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Strain Stress (MPa) Figure 5. A stress–strain plot created with the FLAC FISH function compared to the actual strengths reported in material tensile testing. Strain (mm/mm) Figure 6. Full scale baggage loading test stress–strain curve. 3M Mining Liner - 23˚C / 70% RH For <50fpm and 300fpm Air Flow DRYING TIME, hours 1000 Tensile Yield, <50fpm Tensile Ultimate, 300fpm Secant Modulus Yield, <50fpm 100 Tensile Yield, 300fpm Elongation Yield, <50fpm Adhesion, <50fpm 10 1 0.01 Tensile Ultimate, <50fpm Elongation Ultimate, <50fpm Adhesion, 300fpm 0.1 100 1 10 STRENGTH/STIFFNESS, MPa; ELONGATION,% Figure 7. Plot of adhesive and yield strength over time. 297 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 1000 09069-35.qxd 08/11/2003 20:40 PM Page 298 4 RESULTS Several model simulations were performed in order to calibrate the model against the observed and measured response of the liner. One of the first full scale tests carried out was an application of the liner after 4 hours with no adhesion to the rock (a silicone material prevented liner adhesion). Immediately it became apparent that the adhesion of the material was playing a much bigger role in the stability of the baggage loading test than would be anticipated. Figures 8 & 9 show the actual results and the FLAC modeling results respectively for a liner that has no adhesion to the rock. In both cases the liner quickly deforms due to the low strength observed at this early time after spraying. The second test in the series involved the baggage loading test with the material left to dry over a 24 hour period. In this case, adhesion was allowed to take place by lightly wetting the surface of the rock prior to application. With this second test, the liner was stable under gravity loading from the dead weight of the rock in the testing frame. The total displacement under this static loading was 24 mm. This test was also the source of the stress–strain data generated in Figure 6. This test provided valuable data for the calibration but also demonstrated that there was another effect at work that was not being accounted for in the modeling. The model predicted a total displacement for this particular test at 90 mm rather than the 24 mm observed in Figure 10. In order to make up for the difference in displacements observed between the baggage loading tests and the FLAC models, an investigation took place in which the stiffness of the material was increased until matching results were observed. Modifying the model properties resulted in a curious result. In order to match the performance of the baggage loading test, the stiffness of the material had to be increased to near 20 times its original value. When these modified properties were used, the results matched quite well with the observed testing (Fig. 12). Figure 10. Baggage load test with adhesion – total static displacement of 24 mm. Figure 8. Baggage loading test with no adhesion. JOB TITLE: Falconbridge-3M beggage Load Testing Model: 3M System 2 JOB TITLE: Falconbridge-3M Baggage Load Testing Model: 3M System 2 FLAC (Version 4.00) FLAC (Version 4.00) 0.900 27-Mar-02 14:56 step 19190 -3 860E-01 <x< 1.786E+00 -1.300E+00 <y< 8.715E-01 Grid plot 0 LEGEND 0.500 LEGEND 21-Mar-02 12:08 step 21000 -2.222E-02 <x< 1.422E+00 -4.222E-01 <y< 1.022E+00 0.200 5E-1 -0.200 Axial Force on Structure Max. value # 1 (Cable) -1.100E+02 0.700 Grid plot 0.500 2E-1 0 Axial Force on Structure Max. Value # 1 (Cable) -4.934E+03 0.300 0.100 -0.500 -0.300 -1.000 -0.100 Itasca consulting Group, Inc. Minneapolis, Minnesota USA Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 0.000 0.400 0.800 1.200 1.500 Figure 9. FLAC modeling results for the baggage loading tests with no adhesion. 0.300 0.500 0.700 0.900 1.100 1.300 Figure 11. FLAC results for the modeling of the baggage loading test with adhesion and modified material properties. 298 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0.100 09069-35.qxd 08/11/2003 20:40 PM Page 299 Figure 12. Calibration used on the FLAC model to match the observed baggage loading test results. JOB TITLE: Falconbridge-3M Baggage Load Testing Model: 3M System 2 FLAC (Version 4.00) 0.900 LEGEND 19-Mar-02 14:04 step 21000 -2.222E-02 <x< 1.422E+00 -4.222E-01 <y< 1.022E+00 0.700 0.500 Grid plot 0 2E-1 Axial Force on Structure Max. Value # 1 (Cable) -1.824E+03 0.300 0.100 -0.100 -0.300 Itasca Consulting Group, Inc. Minneapolis, Minnesota USA The adhesion certainly plays an important role in the strength of the material and the model is unable to fully account for the impact of adhesion because of the method by which the cable elements are used. Because the liner adheres to the slabs over most of its area, and only deforms over a relatively small “gauge length”, this would appear to be the largest difference between the actual liner and the model. This difference would have the effect of making the liner much stiffer than it appears in a “dog bone” test. The interlocking of the slabby blocks of material is even harder to quantify. Each frame of material is loaded by hand and as a result there is a complex interaction between the blocks supporting each other and being supported by the frame rails. It is difficult to try and quantify this behavior without additional testing being done to determine the sensitivity of the system to the loading of the frame. Despite the relatively small dataset used in the calibration, the model itself has proven quite useful for modeling the baggage loading tests. The next step in the evolution of this testing will be to use this information to develop a drift modeler that incorporates the information gathered from these trails. 0.100 0.300 0.500 0.700 0.900 1.100 1.300 Figure 13. FLAC output for baggage load testing with adhesion. Anticipated displacement was 90 mm instead of the actual 24 mm measured. Based upon the results observed in the full scale testing, it was obvious that some mechanism must be at work in order to account for the discrepancy in the modeling. There are three possible sources of uncertainty that would tend to artificially increase the stiffness of the material, the adhesion, and the impact of blocks interlocking with each other and the frame. Firstly, the FLAC model is a 2D model, while the actual baggage load test is three-dimensional. However, the actual discrepancy would appear to be too large to be accounted for simply by this difference. 5 CONCLUSIONS The FLAC model has provided a valuable tool for reducing the cost of full scale testing. With this model, it is possible to anticipate the response of different formulations of the liner at different thicknesses and time frames. Given the high cost of full scale tests, it can be used to narrow down the testing regime to the most promising combinations of thickness, time, and liner properties in order to maximize the data collected during full scale tests. Further calibration against fullscale tests will help to reduce some of the uncertainty involved with the model. ACKNOWLEDGMENTS The authors would like to thank all of those involved in the preparation of the baggage load tests including 3M Canada, Falconbridge Ltd, and CANMET. 299 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 301 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A numerical study of the influence of piles in the passive zone of embedded retaining walls T.Y. Yap & C. Pound Mott MacDonald Ltd, Croydon, Surrey, United Kingdom ABSTRACT: Piles are often placed in the base of deep excavations to carry future structure loads or to reduce base heave. Where these piles are located close to retaining walls, they can provide additional resistance to the movement of the embedded length of the retaining wall. This paper discusses a series two-dimensional and threedimensional analyses carried out using the finite difference programs FLAC and FLAC3D to investigate the increase in the passive resistance in front of the embedded retaining wall due to the presence of these piles. Two passive failure mechanisms were identified; the first involved squeezing of the ground upward between the wall and the piles and the second involved squeezing of the ground between the piles. The influence of pile and wall roughness, pile spacing and pile to wall separation was investigated in order to define which of the two passive failure mechanisms would govern and under what circumstances. Based on the results of the two-dimensional analyses a methodology was developed to determine the limiting passive resistance allowing for the presence of the piles. Three-dimensional analyses were carried out which showed a close agreement with the results of the two-dimensional analyses. 1 INTRODUCTION 2 TWO-DIMENSIONAL ANALYSES Excavations for building basements or transportation or utility tunnels are often carried out within retained cuts. Piles are often placed below the base of these excavations either to carry future structure loads or to reduce short-term or long-term ground heave. Often to ease pile construction or to reduce the overall construction program, these piles are installed from the ground surface. If these piles are located close to the retaining wall, they can reduce wall deflection over the embedded length, which can be beneficial when considering the effect of the construction works on adjacent structures. Normally design of embedded retaining walls is carried out using two-dimensional plane strain analyses. In such analyses piles would be represented as a wall with smeared structural properties. In reality, depending on the pile spacing, diameter and proximity of the wall, ground could be squeezed between the piles and the conventional analyses could significantly overestimate the benefit of the piles. This paper presents a two-dimensional numerical study carried out using the finite difference program FLAC to investigate the earth pressures developed between rough, partially rough and smooth walls and piles in close proximity. A three-dimensional analysis was also used to investigate the validity of adopting a smeared representation of the piles when the piles are installed in soft clay. 2.1 Figure 1 illustrates the geometry of the problem where a row of piles was located at a distance, d, from the embedded portion of a retaining wall of embedded length, H. The top of the model was taken to be the final excavation level for the construction. The boundary conditions were such that no displacement was allowed on the base of the model and no horizontal or vertical displacement was allowed of the piles. In order to H Embedded retaining wall Piles 1.5H d Figure 1. Passive earth pressure problem (d 20 m). 301 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Vertical section 09069-36.qxd 08/11/2003 20:41 PM Page 302 Table 1. Material properties. 6000 Drained Unit weight (kN/m3) Young’s modulus E (MPa) Poisson’s ratio v Cohesion c (kPa) Friction angle (degree) Dilation angle (degree) 15.5 6.0 0.49 20.0 0.0 0.0 18.0 25.0 0.25 0.0 30.0 0.0, 30.0 reduce boundary effects, the bottom of the soil mass was located at a depth of 1.5 H. Beam elements were used to model the embedded retaining wall and piles, with interfaces connecting the beam elements to the soil. The interface strength properties were varied to represent different values of wall roughness. A linear elastic perfectly plastic soil model was used throughout these analyses, with the soil stresses limited by the adoption of a Mohr-Coulomb failure criterion. The soil properties adopted for both undrained and drained materials are listed in Table 1. The undrained material properties are typical of a soft clay, whereas the drained properties are typical of a medium dense sand. For both materials the initial horizontal stress was generated using a coefficient of earth pressure at rest, ko, of 1.0, although the results obtained are not believed to be sensitive to the value of this parameter. To determine the passive pressure, the wall was forced towards the soil at a constant velocity and the reaction of the soil on the wall measured. Analyses were carried out for a range of values of d between 0.5 m and 20 m and for each analysis the limiting horizontal passive resistance was determined. The wall and pile friction was also varied between smooth and rough, with equal values of friction being used on both the wall and the piles in all cases. The limiting horizontal passive resistance forces, Pph, were used to back-calculate the mobilized passive earth pressure coefficients, for undrained and drained soil materials, from the following equations: (1) (2) where Kpc and Kp are the passive earth pressure coefficients associated with undrained and drained soil materials respectively. 5000 Horizontal Force (kN) Undrained cw = 0 cw = c/3 cw = c/2 cw = 2c/3 cw = c 4000 3000 2000 1000 0.1 1 10 Distance between wall and piles (m) 100 Figure 2. Influence of the distance of the walls on the mobilized ultimate load (undrained material). 25 20 cw = 0 cw = c/3 cw = c/2 cw = 2c/3 cw = c 15 Kpc Setting 10 5 0 0.1 1 10 Distance between wall and piles (m) 100 Figure 3. Influence of the distance of the walls on the mobilized ultimate passive earth pressure coefficient (undrained material). the undrained material. The limiting passive earth pressure coefficients are also given in Table 2. When the wall and piles were placed 20 m apart, the computed values of Kpc are very close to the theoretical values given in CP2 and reproduced in Table 2. BS8002 suggests that the passive resistance in a cohesive soil can be approximated by the following equation: (3) 2.1.1 Undrained material Figures 2 & 3 show the limiting horizontal passive resistance and the limiting passive earth pressure coefficients respectively for a 10 m deep wall embedded in The passive earth pressure coefficients predicted by this equation are also given in Table 2. The passive 302 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 303 Table 2. Passive earth pressure coefficients for an undrained material. Horizontal Force (kN) 0 100 200 300 400 0 d(m) 0 c/3 c/2 2c/3 c 20 10 5 3 2 1 0.5 BS8002 CP2 2.04 2.03 2.03 2.03 2.03 2.01 2.01 2.00 2.0 2.31 2.34 2.67 3.11 3.66 5.33 8.68 2.31 – 2.42 2.47 2.96 3.62 4.45 6.95 12.0 2.45 2.4 2.50 2.58 3.23 4.10 5.21 8.54 15.2 2.58 – 2.61 2.74 3.68 4.99 6.64 11.7 21.6 2.83 2.6 Depth below top of wall (m). Wall and pile friction, cw -2 -4 0.5m 1m 2m 3m 5m 10m 20m -6 -8 -10 Figure 5. Variation of earth pressure acting on a rough wall for different wall to piles spacings. Figure 4. Shear strain contours for a rough pile. earth pressure coefficient predicted by this equation are somewhat higher than the values predicted by FLAC or quoted in CP2. As the distance between the wall and the piles decreases, Pph and Kpc computed for the rough and partially rough walls increase. This is due to the frictional restraint developed on the piles. The higher the wall cohesion, cw, the larger the vertical restraint developed and hence the higher the values of Pph and Kpc. Conversely, for smooth walls Php and Kpc remain constant even though the distance between the walls decreases to as little as 0.5 m. Figure 4 shows a contour plot of shear strain increment for the analysis with a 20 m separation between a rough wall and piles. The failure surface is clearly shown comprising an arc adjacent to the wall and a straight portion up to the ground surface. There is a fan of intense shearing above the failure surface and adjacent to the wall, with the ground more distant from the wall comprising a passive block with little or no internal shearing. Figure 5 shows the influence of the distance d on the computed horizontal stress acting directly on the perfectly rough wall. The apparent localized reduction in the force at ground surface is due to the force acting over one half of the area represented by the other forces. When the piles are located 10 m or 20 m from the wall the horizontal force profiles are nearly identical although there is a slight divergence below 8 m. Inspection of Figure 4 shows that for the 20 m case the failure surface reaches the ground surface approximately 14 m from the wall. For the 10 m case this would not be possible and therefore the slightly higher horizontal forces below 8 m are indicative of the interaction of the failure surface with the piles. As the distance d reduces there is a progressive increase in horizontal force acting on the lower part of the wall although the forces at the top of the wall down to a depth of about 0.7 d remain unaffected by the presence of the piles. Inspection of other analyses indicates that the depth over which the forces on the wall remain unaffected by the presence of the piles is dependent on the wall roughness such that for a smooth wall and piles this depth is approximately equal to d. Further analyses were carried out and these analyses showed that the limiting passive earth pressure coefficients given in Table 2 were correct for different wall lengths provided the wall to pile separation, d, was normalized by wall embedment depth, H. The significant increase in horizontal force acting on the lower part of the embedded retaining wall is matched by an increase in the force acting on the piles at the same level. This suggests that the soil may be squeezed between the piles rather than forced upwards. On the other hand, the ground stresses over the top part of the wall and piles are limited by conventional passive failure. 303 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 304 Table 3. Limiting passive earth pressure coefficients for a drained material with angle of friction of 30° and a dilation angle of zero. 6 5 Wall and pile friction, w 0 /3 /2 2/3 20 10 5 3 2 1 0.5 CP2 2.94 2.95 2.98 3.00 3.01 3.02 3.01 3.0 3.96 4.50 6.99 13.3 34.3 731 – 4.0 4.44 5.53 11.2 35.5 141 – – – 4.57 6.45 17.6 95.5 639 – – 4.9 4.70 7.09 36.2 354 – – – 5.8 Kp 4 d(m) 3 2 1 Associated Non-associated 0 0 50 40 40 60 80 100 Wall friction (phi %) phiw = 0 phiw = phi/3 phiw = phi/2 phiw = 2phi/3 phiw = phi Figure 7. Comparison of passive earth pressure coefficient for associated and non-associated materials. analyses with a 20 m separation between the walls and the piles were rerun assuming a dilation angle of 30° for the material. Note that the general trend of Kp is similar to that in Figure 5. The passive earth pressure coefficient values for the material with the associated flow rule are now equal to the values quoted in CP2. Kp 30 20 10 0 0.1 20 1 10 Distance between wall and piles (m) 100 Figure 6. Variation of wall to piles spacing on the passive earth pressure coefficient for drained material. 2.1.2 Drained material Figure 6 shows equivalent results for a drained material, with dilation angle, , of 0°. When the embedded retaining wall and the piles are placed far apart, Kp are somewhat below the values given by CP2, except for the smooth wall case. For a smooth wall and piles, the values of Kp remains constant at the theoretical value regardless of the distance d. However, for rough and partially rough walls, the values of Kp increase rapidly as d decreases. Note that in Figure 6 the lines are cut off at a point where back-calculated values of Kp are larger than 50. As for the results for an undrained material, an increase in the wall friction w, leads to significant increase in Kp. For high values of Kp it became increasingly difficult to obtain a reliable limit pressure since the compressive stresses developed in front of the wall became a significant proportion of the Young’s modulus. The effect of assuming an associated rather than a nonassociated flow rule is shown in Figure 7. The drained 2.1.3 Friction on pile surface The failure process describes above will lead to shearing along a vertical surface which passes around and between the piles. Around the piles the friction mobilized on this surface will be given by the piles shaft friction value whereas on the failure surface between the piles, the strength mobilized is given by the soil strength. The effective strength mobilized, c, on an equivalent planar surface is given by the following formula: (4) where D and S are the pile diameter and spacing respectively. The parameter k can be approximated by the following relationship: (5) For most normal situations, the value of c/c is close to unity suggesting that the piled wall can normally be considered as rough. Assuming that the embedded wall is not rough, it is suggested, though not proven, that the value of Kp can be obtained by averaging the Kp value for the wall and that for the piles. 304 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 305 16 14 12 Ncp 10 8 6 D 4 smooth, 0 kPa rough, 0 kPa smooth, 200 kPa rough, 200 kPa 2 S/2 0 1 2 3 4 5 6 7 8 9 10 S/D Figure 9. Comparison of passive resistance coefficient developed on smooth and rough piles for different pile diameter to pile spacing ratios. Figure 8. Horizontal slice model. 2.2 Horizontal section The analyses above show that the presence of a piled wall in close proximity to the embedded length of a retaining wall can have a significant effect on the mobilized passive resistance. However, piles are rarely installed as a continuous wall; more normally they are installed at a spacing of between two and five times their diameter. When the piles are not in contact it is possible for the ground located in front of the wall to be forced between the piles. This process cannot be modeled in a two-dimensional plane strain analysis of a vertical section and could therefore limit the applicability of this type of analysis. To investigate the probability of this form of behavior, a further set of two-dimensional analyses was undertaken. The analyses considered a horizontal slice through the piles. A diagram showing the configuration of the model is shown in Figure 8. Symmetry through the middle of the pile and the mid-point between the piles was assumed. The pile was prevented from moving in all directions. The upper boundary of the model had an applied pressure equal to the initial in situ stress. The lower boundary of the model was displaced at a constant rate towards the pile. The pile was connected to the ground through an interface which allowed both shear displacement of the ground around the pile and separation of the ground from the pile on the “back” side of the pile. Only undrained analyses were undertaken using the soil properties given in Table 1. The resistance provided by the pile was monitored in two ways; firstly by determining the reactions on the grid points around the pile and secondly by determining the reaction on the lower boundary. The difference in these two reactions at the limiting state after allowing for the magnitude of the in situ stress was no more than 1% of the measured value. Analyses were carried out for pile spacings ranging between 1.3 D and 10 D where D is the pile diameter, for different initial in situ stress conditions and with either smooth or rough pile interface properties. The force exerted on the pile, P, was converted to a bearing capacity factor, Ncp value as follows: (6) Figure 9 shows the effect on Ncp of varying pile spacing for two different in situ stress states of 0 kPa and 200 kPa and for a rough or smooth pile interface. For the 0 kPa analyses with a smooth pile interface the value of Ncp reaches a minimum for a pile spacing of about 1.6 D with rapidly increasing values of Ncp for smaller pile spacings and gradually increasing values for larger pile spacings. The minimum value of Ncp is about 4.20. The Ncp values are consistently higher for the rough pile interface analyses with a minimum Ncp value of about 5.6 occurring at a pile spacing of about 2.5 D. For the analyses with an in situ stress of 200 kPa the Ncp values are higher than for the corresponding analysis with an in situ stress of 0 kPa. The analyses with a smooth pile interface shows a minimum Ncp of 8.2 at a pile spacing of 2.0 D whereas the analyses with a rough interface show a minimum Ncp of 10.9 at a pile spacing of 2.5 D. Both sets of analyses indicate that at higher pile spacings the Ncp value becomes constant. For the 305 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 306 smooth and rough pile interface analyses the limiting Ncp values are 9.3 and 12.0 respectively. These are comparable to the results given by Chen & Martin (2002). Figure 10 shows the effect of varying the in situ stress state for a constant pile spacing of 3.0 D. For the smooth pile interface the Ncp value increases from 4.63 to 9.1 as the in situ stress is increased from 0 kPa to 70 kPa and is then constant for higher in situ stress states. For the rough pile interface the response is similar with the Ncp value increasing from 5.67 to 11.1 at about 70 kPa with the value constant at higher in situ stresses. The reason for the change from a gradually increasing Ncp values at low stress to a constant value at higher stresses can be resolved by inspection of the deformation pattern around the piles. At stresses less than 70 kPa the ground is not in contact with the back of the pile whereas above this stress the ground is in touch with the pile over the whole pile circumference. As the stress is gradually increased from 0 kPa to 70 kPa the length over which the ground is not in contact with the pile gradually reduces. These analyses would appear to suggest that the passive resistance provided by the pile would vary with depth down the pile. Near the surface the restraint provided by the pile would be least and the ground movement would lead to a gap developing on the side of the pile furthest from the wall. 2.3 Combined effect The two dimensional horizontal section analyses have shown that under high stress ground can be forced between the piles. However, the ground forced between 12 10 Nc 8 the piles is resisted by a passive wedge behind the piles. It is suggested that, except at very close pile spacings, the wedge mobilized behind the piles is identical to that which would have been mobilized if the piles had not been present. The restraint provided by the piles is therefore generally additive to the normal passive wedge. The piles provide resistance only in that portion of the passive wedge through which it passes. For a smooth wall the failure surface underlying the passive wedge rises at 45° from the toe of the wall. For a wall with friction the failure surface rises at a shallower angle (see Fig. 4). Conservatively it can be assumed that the pile penetrates through the passive wedge to a depth D–h. The total passive resistance, Ptotal, per meter run provided by this failure mode can therefore be expressed as follows: (7) To decide whether failure will occur by squeezing of ground between the piles or by failure of ground in front of the piles the mode with the lower failure load must govern. As an example the force required to develop the two failure mechanisms has been calculated for the soil conditions described above with a 10 m deep wall with 1.5 m diameter piles at 4.5 m centers located 2 m in front of the wall. The force for the combined failure mode is 1825 kN/m whereas for the failure mode in front of the piles, the force is 2103 kN/m. In this case failure by squeezing of the ground between the piles is more likely than failure by squeezing in front of the piles. It is illustrative to note that passive failure would have occurred at a force of 1297 kN/m if the piles had not been present. This illustrates the significant increase in the passive resistance caused by installing piles in this location. 3 THREE-DIMENSIONAL ANALYSES 6 4 2 Smooth pile Rough pile 0 0 50 100 150 In situ stress (kPa) 200 Figure 10. Comparison of passive resistance coefficient developed on smooth and rough piles for different in situ stress states. The analyses discussed above provide a basic understanding of the mechanisms involved with piles in the passive zone of embedded retaining walls, but the actual behavior is almost certainly more complex than the two dimensional analyses can show. It is conceivable that failure of a deeply embedded wall would occur by a combination of both mechanisms. It is also difficult to judge the apparent horizontal stress acting in the horizontal plane when assessing the resistance provided by the piles. To provide more guidance on the equivalence of discrete passive-zone piles compared to an equivalent continuous wall a pair of three-dimensional analyses have been carried out using FLAC3D. Both analyses considered clay with the properties given in Table 1. 306 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 307 The clay layer was taken to be 30 m thick overlying a hard stratum. To reduce the size of the model only that part of the construction below final excavation level was considered. The overall width of the model was 15 m which was deemed to be sufficiently wide to prevent interaction of the boundary with the wall or the pile. As in the two-dimensional analysis of the horizontal slice, symmetry was adopted on vertical planes perpendicular to the wall through the center of the piles and also through a point midway between the piles. The wall was embedded 10 m into the clay and was modeled using liner elements which comprise triangular plate elements connected to the ground through an interface with normal and shear elastic and plastic properties. Many propped embedded retaining walls undergo maximum horizontal displacement at around final excavation (formation) level and therefore the wall was moved towards the soil by applying a horizontal force to the wall at formation level while also preventing rotation about a horizontal axis at this point. In one of the analyses, 1.5 m diameter piles spaced at 4.5 m centers were modeled with the pile axes located 2.75 m in front of the wall, resulting in 2.0 m of soil between the wall and the nearest edge of the piles. A close-up of this model is shown in Figure 11. In the second analysis the pile was substituted by a continuous wall with equivalent smeared properties to that of the discrete piles. The centerline of the equivalent wall was also located 2.75 m from the embedded wall. Both the discrete piles and the equivalent wall were modeled using solid brick elements rather than structural elements. Both the discrete piles and the equivalent wall extended the full depth of the model and both were assumed to be rigidly fixed in a hard stratum at the base of the model. Rough interface properties were considered between the embedded wall, the pile, the equivalent wall and the ground. The wall and pile properties are given in Table 4. The equivalent wall properties were derived using the following formulae, which ensured that the equivalent wall had the same bending and axial stiffness as the discrete piles. (8) Where Ep and Es are the Young’s moduli of the pile and the equivalent wall respectively, and t is the equivalent smeared pile wall thickness. The model was initially brought to equilibrium under the in situ stress conditions and by fixing the horizontal movement of the embedded wall. The reactions developed on the embedded wall during this stage were then applied as a series of nodal forces acting on the wall for the remainder of the analysis. The analysis was continued by increasing the magnitude of the horizontal force at the top of the wall in increments. After each increase in the force the model was allowed to reach equilibrium. Figure 12 shows the results of the two analyses. The solid symbols show the displacement of the top of the wall versus applied force for the two analyses. In both cases the displacements are initially small as the force is increased. Up to an applied force of 1500 kN/m the wall movement is very similar in the two analyses. However as the force is increased above 1500 kN/m, the wall movement in the analysis with discrete piles increases rapidly and appears to become unlimited at an applied force of about 1850 kN/m. In the analysis with the piles represented by an equivalent wall, the wall movement does not increase as rapidly and only becomes unlimited as the applied force approaches 2000 kN/m. These limiting values are very similar to the theoretical values calculated in section 2.3 above. Also shown in Figure 12 as open symbols is the pile head or equivalent wall top movement versus applied force. The pile movement is very similar to the embedded wall movement up to an applied force of 1500 kN/m. In the analysis with discrete piles the pile starts to move less than the wall as the applied force is increased above 1500 kN/m. In the analysis with an equivalent wall, the equivalent wall only starts to move less than the embedded wall when the applied load Table 4. Structural properties. Figure 11. General view of the FLAC3D pile model. Setting Wall Pile Equivalent wall Thickness/diameter (m) Young’s modulus E (GPa) Poisson’s ratio v 1.0 28.0 0.2 1.5 28.0 0.2 1.35 8.46 0.2 307 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-36.qxd 08/11/2003 20:41 PM Page 308 0 1200 Depth below formation level (m). Displacement (mm) 1000 Discrete Piles Equivalent Wall Discrete Piles Equivalent Wall Structural Elements 800 600 400 200 0 500 1000 1500 Applied Force (kN/m) Relative displacement (mm) 350 300 Discrete Piles Equivalent Wall 250 200 150 100 50 0 -50 500 1000 1500 Applied Load (kN/m) 2000 Figure 13. Relative wall and pile displacement in the threedimensional analyses. exceeds 1850 kN/m. This effect is shown more clearly in Figure 13 which plots the differential movement between the top of the pile or equivalent wall and the top of the embedded wall for the two analyses. Positive relative movements indicate movement of the wall towards the piles. The differential movement is small in both analyses up to 1500 kN/m. In the analysis with discrete piles at higher applied forces the gap between the wall and the piles begins to close rapidly as the ground starts to squeeze between the piles. In the analysis with the equivalent wall, at applied forces greater than 1500 kN/m, the gap between the -15 -20 -25 Embedded wall Pile Ground 0 200 400 600 Horizontal displacement (mm) 800 Figure 14. Wall, pile and ground displacement profiles for an applied force of 1815 kN/m. embedded wall and the equivalent wall increases up to an applied force of about 1750 kN/m. This is believed to be due to high ground stresses in front of the toe of the embedded wall causing the equivalent wall to rotate forward more rapidly at formation level. At even higher applied forces the gap between the two walls reduces as ground starts to be squeezed upwards between the two walls. It is considered that the occurrence of significant differential movement between the embedded wall and the pile is indicative of the onset of passive failure of the ground in front of the wall. Figure 14 shows the horizontal displacement of the embedded wall and the piles at an applied force of 1815 kN/m for the analysis with the discrete piles. The horizontal wall displacement far exceeds the pile movement suggesting that there is failure of the ground past the pile. Also shown is the horizontal deflection of the ground mid-way between the piles at the same distance from the wall as the pile axis. The pattern of ground displacement is complex with the section near the ground surface moving significantly less than the wall and only slightly more than the pile. This is because a passive wedge develops near the ground surface which daylights in front of or between the piles. Between 3 m and 10 m below ground level the ground movement far exceeds the pile movement and is closer to the movement of the embedded wall. This clearly shows that the ground is being squeezed between the piles. Because of the restricted gap between the piles and the incompressible nature of the undrained material, under certain situations the ground displacement between the piles could actually exceed the embedded wall displacement. In two-dimensional analyses piles or walls are often represented using structural elements. These structural 308 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands -10 -30 -200 2000 Figure 12. Wall and pile displacements in the threedimensional analyses. -5 09069-36.qxd 08/11/2003 20:41 PM Page 309 elements have no physical thickness in the model although their axial and bending stiffness is modeled correctly by assigning appropriate values of area and moment of inertia. The structural elements are generally placed along the centerline of the piles or wall that they are intended to represent. To investigate the effect that modeling the piles as structural elements has on the predicted passive resistance the threedimensional model was rerun with the piles represented by shell elements with properties identical to those of the equivalent wall given in Table 4. The shell elements were rigidly connected to the mesh and therefore represented a rough wall. To be comparable to the other two analyses, the toe of the equivalent wall was fixed against horizontal movements and was prevented from rotating around a horizontal axis. Because the structural elements have no physical thickness in the model, 2.75 m of clay is now present between the embedded wall and the equivalent wall. The results of the analysis are shown in Figure 12 as crosses. The greater thickness of soil between the embedded wall and the equivalent wall gives a softer wall displacement response compared to the analysis with the equivalent wall modeled using solid elements. The limiting passive pressure is also lower because of the greater separation between the two walls. From Table 2 it can be seen that increasing the wall separation from 2 m to 2.75 m for a 10 m deep wall has the effect of reducing the passive earth pressure coefficient from 6.4 to about 5.3. This results in a reduction in the limiting passive resistance from 2100 kN/m to about 1835 kN/m which is very similar to the passive resistance predicted by this three-dimensional analysis. 4 DISCUSSION There appears to be a good match between the passive resistance obtained in the three-dimensional analysis and the predicted passive resistance made from the results of the two-dimensional analyses despite the obvious limitations of these analyses. A sensitivity study carried out using the results of the two-dimensional analyses for the undrained material shows that squeezing of the ground between the piles is more likely when: 1. The embedded retaining wall and piles are rough rather than smooth. 2. The piles are spaced more widely. 3. The piles are nearer to the wall. Using the two-dimensional analyses it is possible to identify the critical pile spacing defining the change in passive failure mechanism from squeezing of the ground between the piles to squeezing upwards in front of the piles. For a 10 m long embedded rough wall and piles the critical pile spacing appears to vary from 2 pile diameters when the piles are located 2 m in front of the wall to 4 diameters when the piles are located 5 m in front of the wall. 5 CONCLUSIONS The analyses have shown that piles installed in the passive zone of embedded retaining walls can significantly increase the passive resistance mobilized in front of the retaining walls. The passive resistance is sensitive to the distance of the piles to the wall and whether the piles and wall are rough or smooth. For granular deposits the passive earth pressure coefficient increases dramatically as the spacing between the walls and the piles reduces and it is suggested that passive failure is unlikely to occur in this material unless the piles are widely spaced or the wall has only a shallow embedment. Three-dimensional analyses showed a very similar limiting passive resistance to a calculation based on two-dimensional analyses. However, the deflections to mobilize this passive resistance are large and may imply unacceptable movement of the retaining structure. The piles undergo significant lateral deflection, localized bending and axial tension due to the movement of the embedded wall and therefore it is important that these piles are designed for these additional forces. Where the passive failure mechanism does not comprise squeezing of the ground between the piles, twodimensional plane-strain analyses in which the piles are represented by a wall with equivalent smeared properties will provide a safe estimate of the passive resistance. The analyses suggest that where piles are spaced at 3 diameters or less in a cohesive deposit, squeezing of the ground between the piles is unlikely to occur unless the piles are located nearer to the wall than 30% of the embedded length. Where the piles are represented by a structural element, the additional soil present in the model between the embedded wall and the structural member will lead to a conservative estimate of the passive resistance. REFERENCES BS8002: 1994. Code of Practice for Earth Retaining Structures. British Standards Institution. Chen C-Y & Martin, G.R. 2002. Soil-structure interaction for landslide stabilizing piles. Computers and Geotechnics 29: 363–386. Civil Engineering Code of Practice No. 2 1951. Earth Retaining Structures, Institution of Structural Engineers, London. 309 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-37.qxd 08/11/2003 20:48 PM Page 311 Dynamic and thermal analysis Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-37.qxd 08/11/2003 20:48 PM Page 313 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 A practice orientated modified linear elastic constitutive model for fire loads and its application in tunnel construction E. Abazović & A. Amon Geoconsult ZT GmbH, Salzburg, Austria ABSTRACT: A spate of fires in tunnels over the past decade, causing serious loss of life and significant structural damage, has led to new safety concepts in tunnel construction. Nowadays these concepts are already being considered during the design. In this context numerical methods represent a powerful tool for assessing the structural forces in the lining and the change of material properties caused by thermal effects. This article deals with the simulation of a fire within a tunnel by means of the program FLAC. The tunnel lining is modeled by four-node continuum elements for simulating non-linear and time dependent temperature variation within the lining. The thermal effect is applied according to the fire load curve of the BEG-project, a future major railway section between Italy and Austria passing the Alps, at the inside of the lining. The coefficient of thermal transmission between the thermal source and the lining is chosen such that the temperaturefield within the lining corresponds to experimental data. Non-linear material behavior due to thermal loading is implemented by varying the coefficient of thermal expansion. 1 INTRODUCTION Thermo mechanical processes are very complex and they are characterized by non-linear material behavior and transient heat transfer mechanisms. Mechanical processes are depicted through induced stresses as a result of mechanical loads. Alterations of the elastic properties, spalling and material failure are caused by fire loads. In a case of tunnel fire the heat between the heat source and the tunnel inner lining is transmitted by radiation and convection. These exchange mechanisms are dependent on various factors like brightness, air flow velocity, temperature difference and material conductivity. Convection is time dependent due to transient conditions of the temperature gradient. Heat interchange through radiation is characterized by the difference of the fourth power of the temperature quotient of the heat source and heat recipient. Heat interchange through radiation is time dependent too. The required calculation constants are difficult to determine because of the above mentioned reasons, and should therefore be determined by experiment. In addition in absence of a material law describing the complex thermo mechanical processes, numerical simulations are even more difficult. For this reason, this article is intended to describe thermo mechanical processes numerically through a modified linear elastic material model by means of a user-defined function (FISH-function). 2 ASSUMPTIONS It can be assumed that the major principal stress within a concrete tunnel lining tends to act in circumferential direction and the minor principal stress (radial direction) can be neglected. In this case a uniaxial state of stress within the tunnel lining prevails and the deviatoric stresses are negligibly small. The material behavior is determined by only the spherical tensor. Since temperature loads are also only influenced by the spherical tensor and the temperature load is linearly proportional to temperature increase it is possible to obtain a stress change by variation of the thermal coefficient of expansion. The thermal load is applied in terms of a temperature load at the inside of the tunnel lining. The applied temperature is equivalent to the fire load curve of the Brenner Eisenbahn Gesellschaft (BEG) project. The temperature increases linearly within seven minutes from the initial temperature to the maximum temperature of 1200°C which can be seen in Figure 1 (Gresslehner 2001). 313 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-37.qxd 08/11/2003 20:48 PM Page 314 s1 Temperature [°C] Fire load curve BEG-1 s1 1200 sq = −3K . aq . ∆q = −3K . q 1000 sM = E . 1 = 3K . (1 − 2v) . 1 800 600 − 400 ϑ1 elϑ ϑ2 + ϑ M 200 0 0 30 60 90 120 Time [minutes] 150 180 Temperature of concrete during BEG-1 fire 1200 1100 1000 180,ξ 900 360,ξ 800 540,ξ 700 1800,ξ 600 3600,ξ 500 5400,ξ 400 Figure 3. Material behavior. parameters, until no further stresses can be taken, i.e. material destruction. The presentation in form of a usual stress–strain diagram is insufficient and inappropriate as the process is still controlled by temperature. To be able to define a material-law dependent on temperature, the thermal process is depicted by analogy to the stress–strain behavior (dashed axes). Due to material warming, the elementary volume expands linearly proportional to the temperature increase (el) and the temperature expansion coefficient, whereby the material behavior is temperature independent (Fig. 3, 0 1 70°C). Due to restrained thermal expansion in the closed ring structure of a tunnel lining, the initial stress increase is calculated as: Figure 1. Fire load curve. Temperature of concrete [°C] ϑ0 (1) 7200,ξ 300 10080,ξ 200 (2) 100 0 0 5 10 15 20 25 30 35 40 45 50 55 60 ξ Depth of temperature penetration [cm] Figure 2. Depth of temperature penetration. The coefficient of heat transfer and the factor of thermal conductivity are chosen until the temperature gradients meet the experimental data, published by the University of Innsbruck (Kusterle & Waubke 2001) (Fig. 2). where El T elastic stress increase due to temperature increase; K compression modulus; 0 coefficient of linear thermal expansion; temperature increase. Within a temperature range between 70 and 700°C the elastic modulus decreases from 100% to 10% with a sudden drop to 0% thereafter (material destruction, spalling of concrete). The generated constraint and temperature stresses which are caused by the loss of the elastic modulus drop afterwards to zero. 3.2 Based on Hook’s law in the tensor form the following shall apply: 3 NON-LINEAR THERMAL CONSTITUTIVE MODEL 3.1 General (3) Due to mechanical loads the tunnel lining is under compression (M, Fig. 3) and normal stress (M). This state is the initial condition for the thermal calculations. In reality the normal stress will further increase without change in strain under temperature load and subsequently decrease due to the loss of the stiffness 314 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Governing equations (4) (5) 09069-37.qxd 08/11/2003 20:48 PM Page 315 where [Dij] deviatoric part of stress tensor; [0ij] volumetric part of stress tensor; [ij] total stress tensor; [Dij] deviatoric part of strain tensor; [0ij] volumetric part of strain tensor; G, K shear and compression modulus, respectively (Školska Knjiga 1996). Based on the assumption that the deviatoric part of the stress tensor is negligibly small in a closed tunnel lining the total stress then equals the volumetric part of the stresses: (6) (7) Stress increase due to temperature increase has an influence only on the volumetric part of the stress tensor. The volumetric strains are linearly proportional to the temperature increase and the thermal expansion coefficient: (8) where [ij ] thermal strain tensor; 0 coefficient of linear thermal expansion; 1, 0 temperature at time (1) and initial temperature; [ ij] Kronecker -tensor, respectively. If the volumetric displacements are restrained then stresses will be induced into the elementary volume as follows: (9) (10) The stresses within the elementary volume can be calculated as the sum of the stresses of the volumetric tensor (Fig. 4) and the stress increase as a result of the temperature load (Eq. 10). Figure 4. Cross section with surface load. where E t Young’s elastic modulus at time t for temperature ; E0 Young’s elastic modulus at time t 0; t actual temperature at time t; 2, 1 temperature at material failure and temperature at beginning of material softening respectively. The tensor form of Hook’s law written in incremental form for temperature loads gives: (14) The stress decrease observed in the temperature range between 1 and 2 can not be achieved by a reduction of the bulk modulus because a zero or negative increment characterized by a negative compression modulus is not possible. As the coefficient of temperature expansion (0) is a linear part of the stress increment it is possible to calculate a direct derivative of the equivalent thermal expansion coefficient which gives positive and negative increments. Analogous to Equation 13 we can write: (15) (11) (12) Within a temperature range between 1 70°C and 2 700°C the elastic modulus decreases approximately linearly from 100 to 10% so that the following relation can be written: (13) Until a relaxation occurs at a temperature greater than 1 70°C the state of stress is determined by mechanical and thermal stresses and therefore a negative equivalent coefficient of temperature expansion has to be recalculated. (16) where the term [Rij] the volumetric deformation at relaxation. 315 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-37.qxd 08/11/2003 20:48 PM Page 316 At a temperature 2 700°C the stress in the elementary volume has to be zero. (17) so that we can write: (18) (19) (20) where I 1 major principal stress at a temperature 1 70°C and 1 equivalent coefficient of thermal expansion for temperature between 70°C and 700°C. From Equation 20 we can easily obtain an equivalent coefficient of thermal expansion for the relaxation area: (21) Material Friction Angle [Degree] Young’s Modulus E [MPa] Poisson’s Ratio [–] Soil Concrete 22.0 25.0 38.0 – 65.0 30,000 0.30 0.20 (21) (22) 4 PRACTICAL APPLICATION OF A FIRE LOAD WITHIN A TUNNEL The investigated example is an NATM tunnel with an overburden of 12 m and a uniform surface load of 100 kN/m2. All calculations are performed with FLAC (Itasca 2000), a program for two-dimensional numerical calculations. The discrete model consists of a matrix of 70 110 four node continuum elements for the soil (Fig. 5) whereby the tunnel lining is also modeled by continuum elements to be able to implement a modified Table 1. Constitutive constants. Unit Weight [kN/m3] If the temperature in the elementary volume is greater than 700°C this effect is called as physical material destruction and the elastic modulus decreases to zero. As a zero value of the elastic constants within a numerical model is not possible a further correction of the coefficient of expansion is necessary in order to achieve a compensation of the increasing stresses, which would be caused by static loads. The external load would cause a negative extension (εM, Fig. 3) and therefore generate a compressive stress (M, Fig. 3) which should be compensated by thermal expansion. The maximum temperature in the element can theoretically reach the value of the temperature source (3 1200°C, fire load curve). Again, we can derive a temperature dependence of the coefficient of thermal expansion: (*10^2) JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes FLAC (Version 4.00) 0.300 LEGEND 0.100 19-May-03 8:03 step 45768 Thermal Time 7.2000E+03 -0.100 -3.000E+01 <x< 1.300E+02 -1.200E+02 <y< 4.000E+01 -0.300 Grid plot 0 2E 1 -0.500 -0.700 -0.900 -1.100 Geoconsult ZT GmbH Salzburg - Austria -0.200 0.000 0.200 Figure 5. Discretisized model. 316 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0.400 0.600 (*10^2) 0.800 1.000 1.200 09069-37.qxd 08/11/2003 20:48 PM Page 317 linear-elastic material model. The lining and the adjacent soil are linked with so called “Interface” elements. For the tunnel lining a Young’s modulus of E 30 GPa and a Poisson’s ratio of 0.2 has been chosen. The soil is modeled according to an elastic– plastic material model, the Mohr–Coulomb failure criterion. To keep the calculation time within reasonable time limits only one half of the system is modeled, introducing symmetry boundary conditions. The horizontal displacements at the symmetry axis at the right boundary of the mesh are fixed as well as the vertical displacements at the bottom boundary of the model. At the surface a constant uniformly distributed load of 100 kN/m2 is applied. The initial stress state is defined applying a lateral earth pressure coefficient of 40% of the vertical pressure. The implementation of the tunnel lining is performed without any relaxation immediately after excavation. After the static analysis, i.e. equilibrium within the system prevails, a temperature load as a function of time is applied at the inside of the tunnel lining (Fig. 6). The coefficient of heat transfer 0 160 W/m2K and the coefficient of thermal conductivity 1.6 W/mK were varied such that the temperature gradient and the velocity of thermal penetration corresponded to experimental data. The values for the specific heat CV 1000 Ws/kgK and the coefficient for thermal expansion 0 1 105 1/K were chosen according to the literature. The calculation was performed as a coupled timedependent mechanical analysis where mechanical and thermal time steps, which were calculated in real time, changed cyclically. 5 VERIFICATION OF MATERIAL-MODEL AND CALCULATION RESULTS The verification of the temperature fields at different time steps is done by comparison of the experimental data with the temperature pattern within the tunnel lining. As it can be seen in Figure 7 the numerical results match well with the experimental values. For verification of the material law a beam (1 m wide, 45 cm high) was modeled by using a 1 cm by 1 cm zone size. A linear-elastic material model was used with a Young’s modulus of 30 GPa and a Poisson’s ratio of 0.25. Normal pressure of 3.0 MPa was applied on the vertical boundaries and after static calculation a thermal load according to the fire load curve (Fig. 1) was applied on the bottom of the model. The “whilestepping” loop was used for calculating the equivalent JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes FLAC (Version 4.00) LEGEND 27-May-03 11:51 step 33549 Thermal Time 7.2000E+03 Table Plot 9 Minutes +03 (10 ) 1.000 0.800 30 Minutes 60 Minutes 90 Minutes 120 Minutes 0.600 0.400 0.200 5 10 15 20 25 30 JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes (*10^1) 0.900 LEGEND 19-May-03 8:03 45768 0.700 Thermal Time 7.2000E+03 -4.259E+00 <x< 1.019E+01 -4.270E+00 <y< 1.018E+01 0.500 Grid plot 0 2E 0 0.300 Applied Heat Sources O Max Value = 1.302E+03 0.100 -0.100 -0.300 Geoconsult ZT GmbH Salzburg - Austria -0.300 -0.100 0.100 Figure 6. Detail of tunnel lining with applied temperature load. 317 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 40 -02 ) Figure 7. Depth of temperature penetration in FLAC. FLAC (Version 4.00) step 35 (10 GEOCONSULT ZT GmbH Salzburg - Austria 0.300 0.500 (*10^1) 0.700 0.900 09069-37.qxd 08/11/2003 20:48 PM Page 318 JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes JOB TITLE : 90 minutes FLAC (Version 4.00) FLAC (Version 4.00) LEGEND 2-Jul-03 15:38 step 10200 Thermal Time 5.2000E+03 HISTORY PLOT Y-axis : Rev_Prin. stress 1( 50, 2) X-axis : ztemp2 (FISH) (10 LEGEND +07 ) 20-May-03 9:34 step 45768 Thermal Time 7.2000E+03 2.400 2.000 HISTORY PLOT Y-axis : m_his_239 (FISH) X-axis : Number of steps 1.600 1.200 0.800 -01 (10 ) 2.000 1.500 1.000 0.500 0.000 -0.500 -1.000 -1.500 0.400 -2.000 0.000 39 1 2 3 4 5 6 7 8 +02 (10 40 41 42 43 44 45 (10 Geoconsult ZT GmbH Salzburg - Austria 9 GEOCONSULT ZT GmbH Salzburg - Austria ) +03 ) Figure 11. Time progression of bending moment in roof. Figure 8. Reverse principal stress vs. zone temperature. JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes FLAC (Version 4.00) JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes LEGEND FLAC (Version 4.00) +01 LEGEND 20-May-03 9:34 step 45768 Thermal Time 7.2000E+03 HISTORY PLOT Y-axis : Prin. stress 1( 130, 116) X-axis : ztemp6 (FISH) (10 20-May-03 9:34 step 45768 Thermal Time 7.2000E+03 ) -2.000 -2.200 -2.400 -0.500 -2.600 HISTORY PLOT -1.000 Y-axis : n_his_130 -1.500 -2.800 (FISH) X-axis : Number of steps -2.000 -3.000 -3.200 -3.400 -3.600 -2.500 -3.800 -3.000 39 10 20 30 40 50 60 70 Geoconsult ZT GmbH Salzburg - Austria +01 (10 40 41 42 43 44 45 Geoconsult ZT GmbH Salzburg - Austria (10 +03 ) ) Figure 12. Time progression of normal force in shoulder. Figure 9. Major principal stress in dependence on temperature in element. JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes FLAC (Version 4.00) LEGEND 20-May-03 9:34 step 45768 Thermal Time 7.2000E+03 JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 12z0 minutes FLAC (Version 4.00) LEGEND 20-May-03 9:34 step 45768 Thermal Time 7.2000E+03 HISTORY PLOT Y-axis : n_his_239 (FISH) X-axis : Number of steps ) -2.000 -2.500 HISTORY PLOT Y-axis : m_his_130 (FISH) X-axis : Number of steps -1.600 -1.800 -01 (10 -1.400 -2.000 -2.200 -2.400 -2.600 -2.800 -3.000 -3.000 -3.500 -3.200 -4.000 39 Geoconsult ZT GmbH -4.500 40 41 42 43 44 45 (10 +03 ) Salzburg - Austria -5.000 39 Geoconsult ZT GmbH Salzburg - Austria 40 41 42 43 44 45 +03 (10 Figure 13. Time progression of normal force in roof. ) Figure 10. Time progression of bending moment in shoulder. thermal coefficient of expansion during the thermal analysis. Figure 8 shows the evolution of the reverse major principal stress of a zone depending on the temperature within the center of the zone. It is apparent that at the beginning the principal stress increases to a temperature of 70°C, and afterwards decreases linearly to zero at the temperature of 700°C. The same effect can be seen in Figure 9 where the evolution of the major principal stress in one of the elements of the tunnel lining boundary, depend on the temperature at the center of the element, is depicted. The effects of stress changes for the internal forces (axial force and bending moment) due to temperature 318 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-37.qxd 08/11/2003 20:48 PM Page 319 loading are presented for sections in the roof and shoulder area. Until application of a fire load a negative moment predominates (tension soil-sided) whereas in the roof a positive moment (tension cavity-sided) predominates. If the boundary elements do not reach the so-called “Creep-Temperature” it is apparent that at the beginning the negative moment in the shoulder increases significantly, but immediately afterwards the moment decreases because of the loss of stress reception. In the roof, constraint stresses develop from the temperature. On account of this reason the tensile stresses at the inside of the tunnel lining are reduced or turn into compressive stresses. As a result of fire loads the positive moment reduces as well until the elements achieve the relaxation temperature. Afterwards the moments change their direction in dependence on the temperature in the next element rows. The normal force in the lining is calculated over the projection of the stresses normal to the cross-section through integration over the thickness. From Figures 12 & 13, it is obvious that the normal force increases suddenly and decreases after failure of the material within a part of the lining elements. Afterwards the normal force levels around the initial value and stays more or less constant. 6 CONCLUSIONS From the calculation results it is apparent that a numerical simulation of thermo-mechanical processes is possible. The quality of the results is dependent on the quality of the implemented material-law and on the amount of experimental data on which the material model and the thermal process can be calibrated. The temperature pattern within the first centimeter of the section has the steepest temperature gradients and is highly non-linear. In comparison to experimental data the temperature interpolation in the center of the element (FLAC) of a discretisized model is linear. Because of the accuracy of the calculated bending moments and normal forces the relation of the height to the length of the element as well as the number of the Gauss integration points is a very important factor for the relation of the internal forces from the stresses of continuum elements. Therefore an optimum number of elements as well as geometry of the elements should be achieved in order not to falsify the calculation results. Local phenomena like spalling and local loss of thermal protection, which lead to irregular temperature variation within the cross-section of a concrete lining, are factors which influence the results considerably. These effects are not considered in this paper. The internal forces, especially the moments are considerably influenced by the rate of temperature spread. All calculations show that the moments are predominately effected within the first minutes of a fire case as soon the concrete has the full stiffness and stress reception capability. The negative moments (tensile stresses soil-sided) increase rapidly and afterwards decrease slowly because of the reduction of the stiffness and loss of stress reception capability within the heated zones of the lining. The positive moments (tensile stresses cavity-sided) show the same trends and can change their sign in dependence of the initial stress state. These extreme values can provide important information for dimensioning the lining. Within the first minutes of a fire load the normal forces in the tunnel lining show the same trend as the bending moments. As a result of fire loads a significant stress increase at the inside of the tunnel lining can speed up the spalling of concrete. REFERENCES Itasca Consulting Group, Inc. 2000. FLAC – Fast Lagrangian Analysis of Continua, Version 4.0 User’s Manual. Minneapolis: Itasca. Školska Knjiga, 1996. Inženjerski Priručnik, Zagreb: Stručno-Znanstvena Redakcija Biblioteke. Kusterle W., Waubke N.V., 2001. Baulicher Brandschutz – Betontechnologie, Innsbruck: Institut für Baustoffe und Bauphysik der Universität Innsbruck. Gresslehner K.H., 2001. Festlegung der BEG-1 Kurve. 319 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-38.qxd 8/11/03 9:20 PM Page 321 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Seismic liquefaction: centrifuge and numerical modeling P.M. Byrne & S.S. Park Department of Civil Engineering, University of British Columbia, BC, Canada M. Beaty Senior Engineer, Calif. Dept. of Water Resources, Sacramento, CA, USA ABSTRACT: A fully coupled effective stress dynamic analysis procedure for modeling seismic liquefaction is presented. An elastic plastic formulation is used for the constitutive model UBCSAND in which the yield loci are radial lines of constant stress ratio and the flow rule is non-associated. This is incorporated into the 2D version of FLAC by modifying the existing Mohr-Coulomb model. This numerical procedure is used to simulate centrifuge test data from the Rensselaer Polytechnic Institute (RPI). UBCSAND is first calibrated to cyclic simple shear tests performed on Nevada sand. Both pre- and post-liquefaction behavior is captured. The centrifuge tests are then modeled and the predicted accelerations, excess porewater pressures, and displacements are compared with the measurements. The results are shown to be in general agreement when stress densification and saturation effects are taken into account. The procedure is currently being used in the design of liquefaction remediation measures for a number of dam, bridge, tunnel, and pipeline projects in Western Canada. 1 INTRODUCTION Displacements arising from seismic liquefaction can be very large and are a major concern for earth structures located in regions of moderate to high seismicity. Liquefaction is caused by high porewater pressures resulting from the tendency for granular soils to compact when subjected to cyclic loading. Remedial measures typically involve attempts to prevent or curtail liquefaction so that displacements are reduced to tolerable levels. Modifications can also be made to the structure so that larger displacements can be tolerated. In either case, the rational design for remediation requires a reliable prediction of soil-structure response during the design earthquake. State-of-practice procedures for evaluating liquefaction typically use separate analyses for liquefaction triggering (e.g. Youd et al. 2001), flow slide (limit equilibrium with residual strength), and displacements (Newmark sliding block). While the results of the triggering evaluation are used as input into the flow slide and displacement evaluations, the analyses are otherwise independent. While this practice often provides a good screening level tool, these simplified total stress analyses cannot reliably predict excess porewater pressures, accelerations, or displacement patterns. State-of-art procedures involve dynamic finite element or finite difference analyses using effective stress procedures coupled with fluid flow predictions. These analyses can estimate the displacements, accelerations and porewater pressures caused by a specified input motion. Triggering of liquefaction, displacements and flow slide potential are addressed in a single analysis. Such analyses involve capturing the liquefaction behavior of a soil element as observed in laboratory tests, and then modeling the soil-structure as a collection of such elements subjected to the design earthquake base motion. It is vital that these sophisticated procedures be verified before they are used in practice. Instrumented centrifuge model tests can be used for verification and have some advantages over observed field behavior. Centrifuge tests allow the measurement of displacements, input and induced accelerations, and porewater pressures under field stress conditions. These tests can therefore provide a useful database for verification of numerical modeling. This approach is used below. 2 LIQUEFACTION Liquefaction is caused by the tendency of granular soil to contract when subjected to monotonic or cyclic shear loading. When this contraction is prevented or curtailed by the presence of water in the pores, normal stress is transferred from the soil skeleton to the water. This 321 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 8/11/03 9:20 PM Page 322 σv0 Shear Stress, τ (a) Shear Stress, τ Plastic Shear Strain Increment, dγ p 09069-38.qxd τ u Shear Strain, γ (b) Pore Pressure, u Contraction Strength Envelope f Elastic Normal Effective Stress, s' p Plastic Volumetric Strain Increment, dεv Dilation Figure 2. Classic Mohr-Coulomb model. τ , dγ p Shear Strain, γ (c) Effective stress, σv Plastic Strain Increment Vector σ'v0 f Plastic Strain Increment Vector φd B A Yield Locus p s', dεv Shear Strain, γ Figure 3. UBCSAND model. Figure 1. Undrained response of loose sand in simple shear: (a) stress–strain, (b) pore pressure, and (c) effective stress response. can cause high excess pore pressures resulting in a very large reduction in shear stiffness. Large shear strains may occur, and the soil will dilate with these strains unless the soil is very loose. This dilation causes the porewater pressure to drop and the stiffness to increase, which can limit the strains, induced by a load cycle. This behavior is illustrated in Figure 1 for monotonic loading. It is this tendency of the soil skeleton to contract and dilate that controls its liquefaction response. Once the skeleton behavior is modeled, the response under drained, undrained or coupled stress-flow conditions can be computed by incorporating the bulk stiffness and flow of the pore fluid. dilation angle, . This model is really too simple for soils since plastic strains also occur for stress states below the strength envelope. The UBCSAND stress– strain model described herein modifies the MohrCoulomb model incorporated in FLAC to capture the plastic strains that occur at all stages of loading. Yield loci are assumed to be radial line of constant stress ratio as shown in Figure 3. Unloading is assumed to be elastic. Reloading induces plastic response but with a stiffened plastic shear modulus. The plastic shear modulus relates the shear stress and the plastic shear strain and is assumed to be hyperbolic with stress ratio as shown in Figure 4. Moving the yield locus from A to B in Figure 3 requires a plasP tic shear strain increment, , as shown in Figure 4, and is controlled by the plastic shear modulus, GP. The associated plastic volumetric strain increment, dPv, is obtained from the dilation angle : 3 CONSTITUTIVE MODEL: UBCSAND (1) The simplest realistic model for soil is the classic MohrCoulomb elastic–plastic model as depicted in Figure 2. Soils are modeled as elastic below the strength envelope and plastic on the strength envelope with plastic shear and volumetric strains increments related by the The dilation angle is based on laboratory data and energy considerations and is approximated by 322 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands (2) 09069-38.qxd 8/11/03 9:20 PM Page 323 Stress Ratio, h (= τ /s’) 3.2 G p/ s' The plastic properties used by the model are the peak friction angle P the constant volume friction angle cv, and plastic shear modulus GP, where B A Plastic properties (5) Dg p GPi Ge and depends on relative density, is the current shear stress, f is the projected shear stress at failure, and Rf is the failure ratio used to truncate the hyperbolic relationship. The position of the yield locus d is known for each element at the start of each time step. If the stress ratio increases and plastic strain is predicted, then the yield locus for that element is pushed up by an amount d as given by Equation 6. Unloading of stress ratio is considered to be elastic. Upon reloading, the yield locus is set to the stress ratio corresponding to the stress reversal point. Plastic Shear Strain, g p Figure 4. Hyperbolic stress–strain relationship. Shear Stress, Dilation Contraction (6) Normal Effective Stress, Figure 5. Zones of shear-induced contraction and dilation. where cv is the phase transformation or constant volume friction angle and d describes the current yield locus. A negative value of corresponds to contraction. Contraction occurs for stress states below cv and dilation above as shown in Figure 5. Additional information on earlier but similar forms of UBCSAND is presented by Puebla et al. (1997) and Beaty & Byrne (1998). Elastic and plastic properties for the model are defined as follows. 3.1 Elastic properties The elastic bulk modulus, B, and shear modulus, Ge, are assumed to be isotropic and stress level dependent. They are described by the following relations where kB and kG are modulus numbers, PA is atmospheric pressure, and m is the mean effective stress: (3) (4) The elastic and plastic parameters are highly dependent on relative density, which must be considered in any model calibration. These parameters can be selected by calibration to laboratory test data. The response of the model can also be compared to a considerable database for triggering of liquefaction under earthquake loading in the field. This database exists in terms of penetration resistance, typically from cone penetration (CPT) or standard penetration (SPT) tests. A common relationship between (N1)60 values from the SPT and the cyclic stress ratio that triggers liquefaction for a magnitude 7.5 earthquake is given by Youd et al. (2001). Comparing laboratory data based on relative density to field data based on penetration resistance relies upon an approximate conversion, such as that proposed by Skempton (1986): (7) Model parameters based on penetration resistance and field observation may be useful for field conditions where it is very difficult to retrieve and test a representative sample. However, this indirect method is not appropriate for simulation of centrifuge models. Calibrations for this case should be based on direct laboratory testing of samples that are prepared in the same manner as the centrifuge model. 323 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-38.qxd 8/11/03 9:20 PM Page 324 Shear Stress (kPa) 20 15 10 5 0 -5 -10 -15 -20 Test 0 20 (b) 15 10 5 0 -5 -10 -15 -20 -20 -15 Test Shear Stress (kPa) Calibration 20 40 60 80 Vertical Effective Stress (kPa) 20 15 (b) 10 5 0 -5 -10 -15 -20 -20 -15 100 Drc=40% CSR=0.15 Test -10 -5 0 5 Calibration 10 15 20 Figure 7. Stress path and stress–strain relationship (CSR 0.15). 0.2 UBCSAND Dr=44% Dr=40% 0.15 0.1 0.05 Test: Dr=40% Test: Dr=44% 0 Calibration 20 40 60 80 Vertical Effective Stress (kPa) Drc=40% CSR=0.15 (a) Shear Strain (%) Drc=40% CSR=0.1 (a) 20 15 10 5 0 -5 -10 -15 -20 0 Cyclic Stress Ratio (CSR) Shear Stress (kPa) A number of cyclic simple shear tests have been conducted on Fraser River sand at the University of British Columbia. The samples were prepared by air pluviation with a target relative density Dr of 40% and tested at an initial vertical effective stress, v0, of 100 kPa. Samples were also tested at v0 of 200 kPa with a Dr of 44%. Samples were subjected to cyclic shear under constant volume conditions that simulate undrained response at a range of cyclic stress ratios. Typical examples of measured response are shown in Figures 6 & 7. From Figure 6a it may be seen that as the shear stress is cycled, the effective stresses decrease as the pore pressure ratio ru increases. This ratio ru is given by (u – u0)/v0, where u0 and u are the initial and current pore pressures. ru approaches unity after 5 cycles, which corresponds to a state of zero effective stress. Application of further cycles produce very large shear strains in the range of 10 to 15% or more as shown in Figure 6b. However, the strain per cycle is limited as the pore pressures drop with strain due to dilation. Figures 6 & 7 also show the response predicted using UBCSAND. The elastic and plastic parameters selected by the calibration were the same for both tests. The model gives a reasonable representation of the observed response, although the final predicted strains are less than measured for Figure 6. A summary Shear Stress (kPa) 4 SIMULATION OF CYCLIC ELEMENT TEST DATA 1 100 10 No. of Cycles to Liquefaction 100 Figure 8. Predicted and measured liquefaction response of Fraser River sand. Drc=40% CSR=0.1 of the test results and the UBCSAND calibration are shown in Figure 8. The predicted and measured liquefaction response for v0 of 100 and 200 kPa is in close agreement. Test -10 -5 0 Calibration 5 10 15 5 CENTRIFUGE TESTS 20 Shear Strain (%) Figure 6. Stress path and stress–strain relationship (CSR 0.1). A simulation using UBCSAND was made of 2 centrifuge tests carried out at RPI as described in Table 1. In the centrifuge test, a small model is used that is subjected to a high acceleration field during the test. 324 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-38.qxd 8/11/03 9:20 PM Page 325 This has the effect of increasing its stresses by the ratio of the induced acceleration divided by the acceleration of gravity. This ratio or factor is 120 for Model 1 and 60 for Model 2 as indicated by Table 1. The centrifuge model under the increased acceleration field can also be thought of as representing a prototype Table 1. Centrifuge model tests. Test condition Dr Centrifuge acc. Max. v Soil depth Fluid viscosity RPI Model 1 RPI Model 2 Level 55% 120 g 380 kPa 38 m 60 w Slope 40% 60 g 100 kPa 10 m 60 w Cyclic Stress Ratio (CSR) 0.5 UBCSAND Dr=44% Dr=62% Dr=88% Test: Dr=43-46% Test: Dr=60-63% Test: Dr=86-89% 0.4 0.3 that is 120 (Model 1) or 60 (Model 2) times larger than the actual model. Results from the centrifuge test can be presented at either the model or prototype scale. The prototype scale is used for this paper. While in flight, a motion simulating an earthquake time history is applied to the base of the model. For dynamic similitude at the model scale, the earthquake time scale must be decreased by a factor of 120 (Model 1) or 60 (Model 2), and the earthquake acceleration increased by the same factor. The engineering coefficient of permeability k will also increase by this same factor due to the increased unit weight of the fluid. k should be decreased for hydraulic similitude, although it is not necessary to model a specific k. It is common to use a fluid in the test that is 30 to 60 times more viscous than water to prevent rapid rates of dissipation that might unduly curtail liquefaction effects. Nevada sand was used for these centrifuge tests and its liquefaction and permeability (at 1 g using water as pore fluid) properties were obtained from laboratory tests (Arulmoli et al. 1992, Kammerer et al. 2000, Taboada-Urtuzuastegui et al. 2002). Its measured liquefaction resistance together with the UBCSAND prediction is shown in Figure 9. 5.1 0.2 0.1 0 1 10 No. of Cycles to Liquefaction 100 Figure 9. Liquefaction resistance of Nevada sand. Model 1 Model 1 comprises a uniform horizontal sand layer having a thickness of 37 m (prototype scale) and a placement density Dr of 55% as shown in Figure 10 (Gonzalez et al. 2002). After application of the 120 g acceleration field, Dr was estimated to increase to 63% near the base due to the increase in stresses. The amount of densification was estimated from one-dimensional compression tests. The applied base motion is shown (a) Centrifuge Model 1 (b) FLAC Model 1 120 g Measurements P7 Ac5 P8 Z = 0.0 m Z = 1.3 m Z = 6.3 m P5 Ac4 P6 Z = 13.1 m Ac3 P4 Z = 24.8 m Ac2 P3 Ac1 P2 Ac7 Ac6 Navada sand (Dr=55%) P1 Ac8 Input Motion: 50 cycles, 0.2g, 1.5Hz Pore Pressure Transducer Accelerometer Figure 10. Centrifuge Model 1 and FLAC Model 1. 325 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Z = 30.8 m Z = 37.0 m Z = 38.1 m 09069-38.qxd 8/11/03 9:21 PM Page 326 0.3 Centrifuge Input Numerical Input Acceleration (g) 0.2 0.1 0 -0.1 -0.2 -0.3 0 10 20 Time (Sec) 30 40 Figure 11. Based input motions of Model 1. Table 2. Key input for Model 1 numerical analysis. kB Top Middle Bottom 952 1020 1042 2856 3060 3126 0.44 0.61 0.67 Permeability (m/sec) 0.2 E5 0.6 E5 1.2 E5 5 E-5 5 E-5 5 E-5 EPP (kPa) kG Bf after spinup (kPa) 0.4 0.2 0 -0.2 -0.4 0.4 0.2 0 -0.2 -0.4 30 40 200 150 Depth = 24.8m 100 -- σ' vo 0 EPP (kPa) 0 P(depth=6.3m) 10 20 30 350 300 250 200 150 100 50 0 40 Depth = 30.8m -- σ' vo 0 M(depth=13.1m) 20 50 P(depth=1.3m) M(depth=6.3m) 10 250 P (prediction) M(depth=1.3m) Depth =13.1m 300 10 20 30 40 P(depth=13.1m) 400 EPP (kPa) Acc (g) Acc (g) 0.4 0.2 0 -0.2 -0.4 Acc (g) 0.4 0.2 0 -0.2 -0.4 Acc (g) Acc (g) Acc (g) M (measurement) 0.4 0.2 0 -0.2 -0.4 0.4 0.2 0 -0.2 -0.4 Prediction -- σ' vo 0 EPP (kPa) Layer Measurement 140 120 100 80 60 40 20 0 P(depth=24.8m) M(depth=24.8m) 300 200 Depth = 37.0m 100 0 -- σ' vo 0 10 20 30 40 Time (sec) 0 10 M(depth=30.8m) P(depth=30.8m) M(depth=37.0m) P(depth=37.0m) 20 Time (sec) 30 Figure 12b. Measured and predicted excess pore pressures of Model 1. 40 10 20 Time (sec) 30 40 Figure 12a. Measured (left) and predicted (right) accelerations of Model 1. in Figure 11 and consisted of 50 cycles with an amplitude of 0.2 g and a frequency of 1.5 Hz. The key inputs including water bulk stiffness (Bf) for different layers in the numerical model are listed in Table 2. The container for Model 1 consisted of slip “rings” that allowed differential horizontal displacements in the vertical direction but not in the horizontal. This was simulated in the FLAC model by “attaching” the vertical sides, Figure 10. The initial horizontal effective stresses were set to 0.5 times the vertical effective stresses. The measured and predicted excess pore pressures and accelerations for various depths are shown in Figure 12. The predicted accelerations are initially about the same at all depths and approximately equal to the base input value of 0.2 g. The accelerations decrease 326 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-38.qxd 8/11/03 9:21 PM Page 327 Figure 13. Cross section of Model 2 (Taboada-Urtuzuastegui et al. 2002). over much of the model as the shaking continues. The decay of acceleration is most rapid in the upper layers and can be explained in terms of the excess porewater pressures shown in Figure 12b. A large drop in acceleration response occurs when the measured excess pore pressure reaches the initial vertical effective stress v0, which corresponds to a liquefied state. Measurements show that liquefaction occurs first near the surface and then progresses downward. The accelerations and excess pore pressures predicted using UBCSAND are in generally good agreement with the measurements. The analysis described above incorporates the effect of densification due to the increased acceleration field. If this effect is not considered, and a uniform Dr of 55% is used in the analysis, then liquefaction is predicted to occur first at the base of the model rather than at the surface. The higher Dr at the base reverses this trend and indicates the importance of stress densification in centrifuge tests. Full saturation of the pores is difficult to achieve in a centrifuge test. The best fit with the data was obtained assuming an initial placement saturation, or Sr, of 98% at atmospheric pressure. The pore pressure will increase as the centrifuge acceleration is applied, and the resulting increase in Sr is modeled using the gas laws. In summary, (a) UBCSAND provides a reasonable agreement to the test results, (b) ru 1.0 and liquefaction can occur at depths of 40 m in medium dense sand strata, (c) a large reduction in the accelerations can occur upon liquefaction, (d) the effect of stress densification should be included, and (e) the degree of saturation, Sr, must be considered. Table 3. Key input for Model 2 numerical analysis. Layer kG kB Free field 867 2601 5.2 Permeability (m/sec) 1.0 E5 2.1 E-5 Model 2 The cross section for Model 2 is shown in Figure 13 and comprises a steep 1.5:1 slope in loose fine sand with Dr 40% (Taboada-Urtuzuastegui et al. 2002). The base motion consists of 20 cycles of 0.2 g at a frequency of 1 Hz. The container for model 2 was rigid and this was simulated in the FLAC model by applying the input motion to the vertical sides as well as the base. The key inputs for Model 2 are listed in Table 3. Pore pressures and accelerations were measured away from the face of the slope, approximating free field conditions, as well as adjacent to the slope. The predicted and observed accelerations and pore pressures in the free field are shown in Figures 14 & 15. As expected, similar trends are seen as for the level ground test of Model 1, i.e. ru of 100% and reduced accelerations. The accelerations and pore pressures near the slope are shown in Figures 16 & 17. It may be seen in Figure 16 that there is little or no reduction in the accelerations. Instead, large upslope acceleration spikes occur. Excess pore pressures are shown in Figure 17. Large negative excess pore pressure spikes occur that coincide in time with the upslope acceleration spikes. The slope is steep and the upslope acceleration of the base tends to induce failure of the slope and relative 327 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0.22 Bf after spinup (kPa) 09069-38.qxd 8/11/03 9:21 PM Page 328 0.2 0.2 0.0 -0.2 0 (a) Input 0.4 Input -0.2 AH1 0.4 Acceleration (g) 0.0 0 -0.4 (a) -0.4 0.8 0.4 0 -0.4 -0.8 0.8 0.4 0 -0.4 -0.8 AH1 0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8 (b) AH5 (Taboada-Urtuzuastegui et al. 2002) (c) AH6 0 5 10 Time (sec) 15 20 AH5 AH6 5 10 Time (sec) 15 20 Figure 14. Measured (left) and predicted (right) accelerations at free field. 80 80 Excess pore pressure (kPa) 40 40 PP1 0 (a) PP1 0 40 40 20 20 PP5 (b) 0 PP5 0 ru = 1.0 20 20 10 10 PP6 (Taboada-Urtuzuastegui et al. 2002) 0 -5 0 5 10 Time (sec) 15 (c) 0 PP6 5 20 10 Time (sec) 15 20 Figure 15. Measured (left) and predicted (right) excess pore pressures at free field. downslope movement. The soil dilates as it shears in the downslope direction, producing negative pore pressures which stiffen the shear modulus. Enough strength is mobilized through this dilation to arrest the downslope movement and gives rise to the acceleration spike (Taboada-Urtuzuastegui et al. 2002). UBCSAND provides a reasonable prediction of the accelerations and pore pressure response for the 328 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-38.qxd 8/11/03 9:21 PM Page 329 0.2 0.2 0.0 0 -0.2 (a) Input -0.2 AH2 0.4 0.4 Acceleration (g) Input 0.0 0 -0.4 -0.4 (b) AH2 0.2 0.0 -0.2 -0.4 -0.6 AH4 0.4 (Taboada-Urtuzuastegui et al. 2002) (c) 0.2 0 -0.2 -0.4 -0.6 0.0 0 -0.4 (d) AH7 0 5 10 Time (sec) 15 AH7 0.4 -0.4 -0.8 AH4 -0.8 5 20 10 Time (sec) 15 20 Figure 16. Measured (left) and predicted (right) accelerations near the slope. 20 20 PP2 10 10 0 PP2 Excess pore pressure (kPa) 0 30 (a) -10 30 20 20 10 10 PP3 0 PP3 (b) 20 0 20 0 0 PP4 -20 -20 PP4 (c) ru = 1.0 10 0 -10 -20 -30 10 0 -10 -20 -30 PP7 (Taboada-Urtuzuastegui et al. 2002) (d) 0 5 10 Time (sec) 15 PP7 5 20 Figure 17. Measured (left) and predicted (right) excess pore pressures near the slope. 329 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 10 Time (sec) 15 20 09069-38.qxd 8/11/03 9:21 PM Page 330 Displacement unit : m Test No.1 Test Conditions: 60g, 60 times viscosity of water Max. Acceleration: 0.25g Magnitude 1.91+ 1.74 to1.91 1.58 to1.74 1.41 to1.58 1.24 to1.41 1.08 to1.24 0.91 to1.08 0.75 to 0.91 0.58 to 0.75 0.41 to 0.58 0.25 to 0.41 0.08 to 0.25 10 10 8 8 6 6 4 4 2 2 0 50 0 45 40 35 35 30 25 20 15 10 5 0 Figure 18. Measured displacements for Model 2 from centrifuge test (Taboada-Urtuzuastegui et al. 2002). Maximum Displacement = 2.6 m Figure 19. Predicted displacements for Model 2 using UBCSAND. free field. More significant differences are observed for locations near the slope. Some of these differences are due to UBCSAND under predicting the dilative spikes. This requires further investigation. The measured and predicted displacements after shaking are shown in Figures 18 & 19. It may be seen that both the magnitude and pattern of displacements are in general agreement. In summary, (a) UBCSAND provides reasonable agreement with this centrifuge test, although further study is needed for locations close to the sloping face, (b) a decrease in accelerations after liquefaction was not observed near the slope, (c) a large upslope acceleration spikes occurred near the slope, (d) a decrease in pore pressure due to dilation corresponded with these upslope acceleration spikes, and (e) the dilative spikes prevented very large displacements from occurring in this homogeneous fine sand model. 6 SUMMARY A fully coupled effective stress dynamic analysis procedure has been presented. The procedure is first calibrated by comparison with laboratory element test data and then verified by comparison with two centrifuge model tests. Model 1 represented a deep sand layer with a level ground condition. This model showed that high excess porewater pressure and liquefaction can occur to depths of 40 m in medium dense sands. Liquefaction first occurred at the surface and progressed downward under continued shaking. Accelerations above 330 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-38.qxd 8/11/03 9:21 PM Page 331 the depth of liquefaction showed a significant decrease. The numerical model results were in good agreement with the measurement when stress densification and saturation effects were included. Model 2 represented a steep slope condition in homogeneous loose fine sand. The results showed that large upslope acceleration spikes occurred near the face of the slope after liquefaction. These acceleration spikes corresponded with large negative excess pore pressure spikes associated with dilation. It is the increase in effective stress associated with these negative pore pressure spikes that curtails the displacements and makes the slope more stable than might be expected under cyclic loading. The overall pattern of predicted response is in reasonable agreement with the measurements, although both the acceleration and pore pressure spikes are under predicted by the UBCSAND analysis. A new series of centrifuge tests are planned at CCORE (Centre for Cold Ocean Research), Memorial University, Newfoundland, which will permit further verification and refinement of the numerical model. REFERENCES Arulmoli, K., Muraleetharan, K.K., Hossain, M.M. & Fruth, L.S. 1992. VELACS laboratory testing program, soil data report. The Earth Technology Corporation, Irvine, California, Report to the National Science Foundation, Washington D.C., March. Beaty, M. & Byrne, P. 1998. An effective stress model for predicting liquefaction behaviour of sand. ASCE Geot. Special Pub. No. 75: 766–777. Gonzalez, L., Abdoun, T. & Sharp, M.K. 2002. Modeling of seismically induced liquefaction under high confining stress. Kammerer, A., Wu, J., Pestana, J., Riemer, M. & Seed, R. 2000. Cyclic simple shear testing of Nevada sand for PEER Center project 2051999. Geotechnical Engineering Research Report No. UCB/GT/00-01, University of California, Berkeley, January. Puebla, H., Byrne, P.M. & Phillips, R. 1997. Analysis of CANLEX liquefaction embankments: prototype and centrifuge models. Can. Geotech. Journal, Vol. 34, No. 5: 641–657. Skempton, A.W. 1986. Standard penetration test procedures and the effects in sands of overburden pressure, relative density, particle size, ageing and overconsolidation, Geotechnique 36, No. 3: 425–447. Taboada-Urtuzuastegui, V.M., Martinez-Ramirez, G. & Abdoun, T. 2002. Centrifuge modeling of seismic behavior of a slope in liquefiable soil, Soil Dynamic and Earthquake Engineering, Vol. 22: 1043–1049. Youd, T.L., Idriss, I. M., Andrus, R.D., Arango, I., Castro, G., Christian, J.T., Dobry, R., Finn, W.D.L., Harder Jr., L.F., Hynes, M.E., Ishihara, K., Koester, J.P., Liao, S., Marcuson III, W.F., Martin, G.R., Mitchell, J.K., Moriwaki, Y., Power, M.S., Robertson, P.K., Seed, R.B. & Stokoe, K.H. 2001. Liquefaction Resistance of Soils: Summary Report from the 1996 NCEER and 1998 NCEER/NSF Workshops on Evaluation of Liquefaction Resistance of Soils. ASCE J. of Geot. and Geoenv. Eng., Vol. 127, No. 10: 817–833. 331 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-39.qxd 08/11/2003 20:43 PM Page 333 FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9 Modeling the dynamic response of cantilever earth-retaining walls using FLAC R.A. Green Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, USA R.M. Ebeling Information Technology Laboratory, US Army Engineer Research and Development Center, Vicksburg, MS, USA ABSTRACT: A research investigation was undertaken to determine the dynamically induced lateral earth pressures on the stem portion of a concrete, cantilever, earth-retaining wall. In total, the wall-soil column system was 68.6 m in height, with the upper 6.1 m being composed of the cantilever wall retaining compacted backfill. A series of numerical analyses were performed using FLAC. The analyses consisted of the incremental construction of the wall and placement of the backfill, followed by dynamic response analyses, wherein the soil was modeled as elastoplastic. This paper outlines the details of the numerical model used in the analyses. Particular attention is given to how the ground motion was specified, determination of the wall and soil model parameters, and the modeling of the wall-soil interface. To benchmark the FLAC results, comparisons are presented between the FLAC results and the results from simplified techniques for computing dynamic earth pressures and permanent wall displacement. 1 INTRODUCTION 1.1 Scope A research investigation using FLAC was undertaken to determine the dynamically induced lateral earth pressures on the stem portion of a concrete, cantilever, earth-retaining wall. The analyses consisted of the incremental construction of the wall and placement of the backfill, followed by dynamic response analyses, wherein the soil was modeled as elasto-plastic with a Mohr-Coulomb failure criterion. The focus of this paper is to outline the details of the numerical model used in the analyses. Particular attention is given to how the ground motions were specified, the wall and soil model parameters were determined, and the wall-soil interface was modeled. To assess the validity of the proposed FLAC model, comparisons of the FLAC results are made with results from simplified analysis techniques for determining dynamic earth pressures (i.e., Mononobe-Okabe approach) and for determining permanent displacement of the wall (i.e. Newmark sliding block approach). 1.2 Description of wall-soil system The retaining wall analyzed was approximately 6.1 m in height, retaining medium-dense, cohesionless, compacted fill (total unit weight: t 19.6 kN/m3; effective angle of internal friction: 35°). Underlying the wall/backfill was approximately 62.5 m of naturally deposited dense cohesionless soil (t 19.6 kN/m3; 40°). The groundwater table was well below the base of the wall and was not considered in the analyses. The geometry and structural detailing of the wall were determined following the US Army Corps of Engineers static design procedures (Headquarters, US Army Corps of Engineers 1989, 1992), with the dimensions of the structural wedge (i.e. wall and contained backfill) depicted in Figure 1. The properties of the concrete and reinforcing steel used in the wall design are as follows: unit weight of concrete: c 23.6 kN/m3; compressive strength of concrete: fc 27.6 MPa; and yield strength of reinforcement: fy 413.4 MPa. Additional details about the wall design and soil profile are given in Green & Ebeling (2002). 2 NUMERICAL MODEL 2.1 The FLAC numerical model consisted of the upper 9.1 m of the wall-soil system, comprising the wall/backfill and approximately 3 m of the underlying natural deposit (foundation soil). Laterally, the FLAC model was approximately 22.9 m, to include approximately 7.6 m of the foundation soil in front of the wall and 333 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Overview of FLAC model 09069-39.qxd 08/11/2003 20:43 PM Page 334 be specified along the lateral edges of the model (freefield boundary conditions cannot be specified across the interface of two sub-grids). Sub grid four was included for symmetry, but its inclusion was not necessary. The sub-grids were “attached” at the soil-to-soil interfaces, as depicted by white lines in Figure 2, and interface elements were used at the wall-soil interfaces. The following sub-sections outline how the ground motions were specified and the procedures used to determine the various soil and wall model parameters. 0.2 m 0.5m 2.4m 0.9m Stem 6.1m Backfill 2.2 0.6 m Toe Heel Base 4m Figure 1. Dimensions of the structural wedge of the wall-soil system analyzed, wherein the term “structural wedge” refers to all that is shown above. 7.6 m 3m 2 3 4 9.1 m 6.1 m 15.3 m 1 22.9 m Figure 2. Annotated FLAC model of the wall-soil system. approximately 15.3 m of the backfill/foundation soil behind the wall (Fig. 2). An elasto-plastic constitutive model, in conjunction with Mohr-Coulomb failure criterion, was used to model the soil. Elastic beam elements were used to model the concrete retaining wall, with the wall/backfill being “numerically constructed” in FLAC similar to the way an actual wall would be constructed. The backfill was placed in 0.61 m lifts, for a total of ten lifts, with the model being brought to static equilibrium after the placement of each lift. Such placement allowed realistic earth pressures to develop as the wall deformed and moved because of the placement of each lift. The constructed retaining wall-soil model is shown in Figure 2. The model consists of four sub-grids, labeled one through four in Figure 2. The separation of the foundation soil and backfill into sub-grids one and two was required because a portion of the base of the retaining wall was inserted into the soil. Sub-grid three was included so that free-field boundary conditions could Dynamic analyses can be performed with FLAC, wherein user-specified acceleration, velocity, stress, or force time-histories can be input as exterior boundary conditions or as interior excitations. A parametric study was performed to determine the best way to specify the ground motions in FLAC for earthquake analyses. The parametric study involved performing a series of one-dimensional (1-D) site response analyses using consistently generated acceleration, velocity, and stress time-histories. Generally, earthquake ground motions are not defined in terms of force time-histories and therefore were not considered in the parametric study. The use of stress time-histories in FLAC has the benefit of allowing the time-history to be specified at “quiet boundaries,” thus simulating radiation damping. Using a free-field acceleration time-history recorded at the surface of a USGS site class B profile during the 1989 Loma Prieta earthquake, a 1-D site response analysis was performed using a modified version of SHAKE91 (Idriss & Sun 1992). The analysis was performed on a 68.6 m, 5% damped, non-degrading profile, wherein the acceleration time-history was specified as an outcrop motion. Interlayer acceleration and stress time-histories were computed at the profile surface and at depths of 7.6 m, 10.7, 15.2, and 68.6 m (i.e. bedrock). Interlayer velocity timehistories were computed by integrating the interlayer acceleration time-histories using the trapezoidal rule. The interlayer acceleration, velocity, and stress timehistories were used as base motions in a series of FLAC analyses, in which the acceleration timehistories at the surface of the FLAC profiles were computed. The profiles used in the FLAC analyses were comparable to the SHAKE profiles down to the depths corresponding to the interlayer motions. An elastic constitutive relation, with 5% Rayleigh damping, was used to model the soil layers in the FLAC profiles. The central frequency of the damping relationship was set to the fundamental frequencies of the respective FLAC profiles. Fourier amplitude spectra (FAS) and 5% damped, pseudo acceleration response spectra (PSA) were computed from the acceleration time-histories of the surface motions of the SHAKE and FLAC profiles. 334 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Specification of input motions 09069-39.qxd 08/11/2003 20:43 PM Page 335 Error analyses were performed on the spectra corresponding to the different profiles and different types of specified input motions. In the error analyses, the spectra for the SHAKE motions were used as the “correct” motions. The word “correct” does not imply that SHAKE precisely models the behavior of an actual soil profile subjected to earthquake motions. Rather, SHAKE gives the analytically correct motion for a visco-elastic profile with constant damping applied to all frequencies of motion. On the other hand, the FLAC models used in this study give numerical approximations of the correct analytical solution. The errors in the FLAC spectral values were computed at a spectrum of frequencies using the following expressions. (1a) (1b) From the results of the parametric study, it was determined that the specification of the input motion in FLAC in terms of stress time-histories gives the least accurate results, wherein the stress timehistories were applied at a “quiet boundary” along the base of the FLAC model. The errors corresponding to specifying the motions in terms of acceleration and velocity time-histories were essentially identical and considerably less than those associated with the stress time-histories. 2.3 had little energy at higher frequencies. The interlayer motion (at 9.1 m depth) computed using SHAKE was specified as an acceleration time-history along the base of the FLAC model. 2.4 Model parameters for soil The stress-strain behavior of the soil was modeled using the Mohr-Coulomb constitutive model. Four parameters are required for the Mohr-Coulomb model: effective internal friction angle (); mass density (); shear modulus (G); and bulk modulus (K). The first two parameters (i.e., and ) are familiar to geotechnical engineers, where mass density is the total unit weight of the soil (t) divided by the acceleration due to gravity (g), i.e. t/g. As stated previously, for the foundation soil was 40° and 35° for the backfill. These values are consistent with dense natural deposits and medium-dense compacted fill. G and K may be less familiar to geotechnical engineers; therefore, their determination is outlined below. Several correlations exist that relate G to other soil parameters. However, the most direct relation is between G and shear wave velocity (vs): (2) s may be determined by various types of site characterization techniques, such as cross hole or spectral analysis of surface waves (SASW) studies. Values for K can be determined from G and Poisson’s ratio (v) using the following relation: Development of input motions for wall analyses (3) As stated previously, the FLAC model of the soil-wall system consisted of only the upper 9.1 m of a 68.6 m profile. To account for the influence of the soil profile below 9.1 m on the ground motions, the entire 68.6 m profile, without the retaining wall, was modeled using a modified version of SHAKE91. The interlayer motion at the depth corresponding to the base of the FLAC model (i.e. 9.1 m) was computed. The input ground motion used in the SHAKE analysis was the same Loma Prieta motion used in the parametric study discussed above. The motion was specified as a rock outcrop motion at the base of the 68.6 m soil column. The small strain fundamental frequency of the retaining wall-soil system in the FLAC model was estimated to be approximately 6 Hz. At larger strains, the fundamental frequency of the system will be less than the small strain value. To ensure proper excitation of the wall, the cutoff frequency in the SHAKE analysis was set at 15 Hz. This value was selected considering both the fundamental frequency of the wall-soil system and the fact that the input motion v may be estimated using the following expression: (4) which was derived from the theory of elasticity (e.g. Terzaghi 1943), in conjunction with the correlation relating Ko and proposed by Jaky (1944), i.e. Ko 1 sin(). Using the above expression, v was determined to be 0.26 and 0.3 for the foundation soil and backfill, respectively. 2.5 335 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Model parameters for wall The concrete wall was divided into five segments having constant parameters, as illustrated in Figure 3, with each segment consisting of several 0.3 m elastic beam elements. Four parameters were required to define the mechanical properties of the elastic beam 09069-39.qxd 08/11/2003 20:43 PM Page 336 2.4 m 1.5 m 1.5 m Beam Elements 1 Beam Elements 2 1.5 m 3 Interface Elements 6.1m 6.1m 1.5 m 2.4 m 1.5 m No Interface Elements 1.5 m 4 5 4m 4m Figure 3. Numerical model of retaining wall using elastic beam elements. elements: cross sectional area (Ag); mass density (); elastic modulus (Ec); and second moment of area (I ), commonly referred to as moment of inertia. The basis for subdividing the wall into five segments was the variation of the mechanical properties in the wall. A wall having a greater taper or largely varying steel reinforcement along the length of the stem or base would likely require more segments. For each of the segments, Ag and were readily determined from the wall geometry and the unit weight of the concrete (i.e. 23.6 kN/m3). Ec was computed using the following expression (e.g. MacGregor 1992): (5) In this expression, f c is the compressive strength of the concrete (e.g. 4000 psi for the wall being modeled), and both Ec and f c are in psi. Because the structure is continuous in the direction perpendicular to the analysis plane, Ec computed using Equation 5 needed to be modified to account for plane-strain conditions. This modification was done using the following expression (Itasca 2000, FLAC Structural Elements Manual). (6) where 0.2 was assumed for Poisson’s ratio for concrete. I is a function of the geometry of the segments, the amount and location of the reinforcing steel, and the amount of cracking in the concrete, where the latter in turn depends on the static and dynamic load imposed Figure 4. Location of interface elements in the FLAC model. on the member. In dynamic analyses, it is difficult to state a priori whether the use of sectional properties corresponding to uncracked, fully cracked, or some intermediate level of cracking will result in the largest demand on the structure. However, I 0.4 Iuncracked was used as a reasonable estimate for the sectional properties (Paulay & Priestley 1992). 2.6 336 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Model parameters for wall-soil interface Interface elements were used to model the interaction between the concrete retaining wall and the soil. However, FLAC does not allow interface elements to be used at the intersection of branching structures (e.g. the intersection of the stem and base of the cantilever wall). Several approaches were attempted by the authors to circumvent this limitation in FLAC, with the simplest and best approach, as found by the authors, illustrated in Figure 4. As shown in this figure, three very short beam elements, oriented in the direction of the stem, toe side of the base, and heel side of the base, were used to model the base-stem intersection. No interface elements were used on these three short beam elements. However, interface elements were used along the other contact surfaces between the soil and wall, as depicted by the hatched areas in Figure 4. A schematic of the FLAC interface element is presented in Figure 5. As may be observed from this figure, the interface element has four parameters: S slider representing shear strength; T tensile strength; kn normal stiffness; and ks shear stiffness. The element allows permanent separation and slip of the soil and the structure, as controlled by the parameters 09069-39.qxd 08/11/2003 20:43 PM Page 337 Side A of Interface zone ks 600 τult 500 τf 400 grid point T τ (psf) grid point S 700 zone kn zone Side B of Interface Ksi ks 300 hyperbolic model FLAC model 200 100 Figure 5. Schematic of the FLAC interface element (adapted from Itasca 2000). T and S, respectively. For the cohesionless soil being modeled, T 0, while S was specified as a function of the interface friction angle ( ). For medium-dense sand against concrete, 31° (Gomez et al. 2000b). As a rule-of-thumb, the FLAC manual (Itasca 2000, Theory and Background Manual) recommends that kn be set to ten times the equivalent stiffness of the stiffest neighboring zone, i.e.: 0.000 r 0.002 0.008 0.010 Figure 6. Calibration of the FLAC interface model to the hyperbolic-type model proposed by Gomez et al. (2000a,b). where, (8b) (8c) Ksi dimensionless interface initial shear stiffness of the interface; n normal stress acting on the interface (determined iteratively in FLAC by first assuming a small value for ks and then constructing the wall); interface friction angle 31°; Rfj failure ratio 0.84; KI dimensionless interface stiffness number for initial loading 21000; nj dimensionless stiffness exponent 0.8; w unit weight of water in consistent units as r; and Pa atmospheric pressure in the same units as n. The values for Rfj, KI, nj, and were obtained from Gomez et al. (2000a). 2. ks was computed using the following expression: 1. Compute r using the following expression. (8a) 337 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 0.006 s (ft) (7) In Equation 7, K and G are the bulk and shear moduli, respectively, and zmin is the smallest width of a zone in the normal direction of the interfacing surface. The max[ ] notation indicates that the maximum value over all zones adjacent to the interface be used. The FLAC manual warns against using arbitrarily large values for kn, as is commonly done in finite element analyses, as this results in an unnecessarily small time step, and therefore unnecessarily long computational times. The determination of the ks required considerably more effort than the determination of the other interface element parameters. In shear, the interface element in FLAC essentially is an elasto-plastic model, with an elastic stiffness of ks and yield strength S. ks values were selected such that the resulting elasto-plastic model gave an approximate fit of the hyperbolic-type interface model proposed by Gomez et al. (2000a,b). A comparison of the two models for initial loading (i.e. construction of the wall) is shown in Figure 6. The procedure used to determine ks values for initial loading is outlined below. The reader is referred to Gomez et al. (2000a,b) for more details concerning their proposed hyperbolic-type model. 0.004 (9) The above computed ks values were used only for the initial construction of the wall. The ks values were changed after the construction of the wall and prior to the application of the earthquake loading to values consistent with the Gomez-Filz-Ebeling Version I load/unload/reload extended hyperbolic interface model (Gomez et al. 2000b). The procedure used to compute ks for the cyclic loading is outlined below. Again, the reader is referred to the cited report for more details concerning this model. 09069-39.qxd 08/11/2003 20:43 PM Page 338 (10a) where, with the earthquake-induced shear strains, frequently referred to as the “reduced” vs by FLUSH users. Assuming that the response of the retaining wall will be dominated by shear waves, substituting Equation 12 into Equation 11a gives: (10b) (10c) (13a) or Kurj unload-reload stiffness number for interfaces; and Ck interface stiffness ratio. Using the above expressions, the interface stiffnesses were computed for the interface elements identified in Figure 4. While the ks for unload-reload were higher than the corresponding values for initial loading (i.e., Equation 10a versus Equation 9), the values for kn were the same for both initial loading and unload-reload. 2.7 Dimensions of finite difference zones Proper dimensioning of the finite difference zones is required to avoid numerical distortion of propagating ground motions, in addition to accurate computation of model response. The FLAC manual (Itasca 2000, Optional Features Manual) recommends that the length of the element ( l) be smaller than one-tenth to oneeighth of the wavelength ( ) associated with the highest frequency (fmax) component of the input motion. The basis for this recommendation is a study by Kuhlemeyer & Lysmer (1973). Interestingly, the FLUSH manual (Lysmer et al. 1975) recommends l be smaller than one-fifth the associated with fmax, also referencing Kuhlemeyer & Lysmer (1973) as the basis for the recommendation, i.e.: (11a) (11b) is related to the shear wave velocity of the soil (vs) and the frequency (f) of the propagating wave by the following relation. (12) In a FLUSH analysis, it is important to note that the vs used in this computation is not that for small (shear) strains, such as measured in the field using cross-hole shear wave test. Rather, in FLUSH, the vs used to dimension the elements should be consistent (13b) As may be observed from these expressions, the finite difference zone with the lowest vs, for a given l will limit the highest frequency that can pass through the zone without numerical distortion. For the FLAC analyses performed in this investigation, 0.3 m by 0.3 m zones were used in sub-grids one and two; (refer to Figure 2). The top layer of the backfill has the lowest vs (i.e. 160 m/sec). Using Equations 13 and 0.3 m, the finite difference grid used in the FLAC analyses should adequately propagate shear waves having frequencies up to approximately 53 Hz. This value is well above the 15 Hz cutoff frequency used in the SHAKE analysis to compute the input motion for the FLAC analysis and well above the estimated fundamental frequency of the retaining wallsoil system being modeled (i.e. ≈6 Hz). 2.8 As stated previously, an elasto-plastic constitutive model, in conjunction with the Mohr-Coulomb failure criterion was used to model the soil. Inherent to this model, once the induced dynamic shear stresses exceed the shear strength of the soil, the plastic deformation of the soil introduces considerable hysteretic damping. However, for dynamic shear stresses less than the shear strength of the soil, the soil behaves elastically (i.e. no damping), unless additional mechanical damping is specified. FLAC allows mass proportional, stiffness proportional, and Rayleigh damping to be specified, where the latter provides a relatively constant level of damping over a restricted range of frequencies. For the analyses performed, Rayleigh damping was used, which required the specification of a damping ratio and corresponding central frequency. One- to twopercent damping ratio is commonly used as a lower bound for non-linear dynamic analyses to reduce highfrequency spurious noise (e.g. Finn 1988). However, it was found by the authors that considerable highfrequency noise may still exist even when one- to two-percent Rayleigh damping was specified; this is thought to be a numerical artifact of the explicit solution algorithm used in FLAC. The damping levels in 338 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands Damping the last iteration of the SHAKE analysis used to compute the FLAC input motion may be used as an upper bound of the values for Rayleigh damping. Judgment is required in selecting the damping ratio between the lower and upper bounds; three-percent Rayleigh damping was used for most of the retaining wall analyses performed by the authors. The central frequency corresponding to the specified damping ratio is typically set to either the fundamental period (small strain) of the system being modeled (an inherent property of the wall-soil system) or predominant period of the system response (an inherent property of the wall-soil system and the ground motion). For the FLAC analyses performed, the central frequency was set equal to the small strain fundamental frequency of the retaining wall-soil system (i.e. ≈6 Hz). Permanent relative displacement (m) 09069-39.qxd 08/11/2003 20:43 PM Page 339 5 10 15 20 25 Time (seconds) Permanent relative displacement (m) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 30 35 40 Newmark FLAC 0 5 10 15 20 25 Time (seconds) 30 35 40 Figure 7b. Comparison of the permanent relative displacements computed from the FLAC results and a Newmark sliding block analysis with N* g 0.27 g. Acceleration (g) 1.0 N*.g = 0.27g 0.5 N*.g = 0.22 g 0.0 0 -0.5 5 10 15 20 25 30 35 40 Time (seconds) -1.0 Figure 8. Acceleration time-history used in the Newmark sliding block analysis of the structural wedge. of acceleration imparted to the block resulting in a factor of safety against sliding equal to 1.0. Using the interface friction angle between the concrete wall and foundation soil (i.e. 31°) in conjunction with the weight of the structural wedge, N* g was determined to be approximately 0.22 g. The sliding block analysis resulted in dr 0.55 m, as shown in Figure 7a, which is considerably larger than that from the FLAC analysis. One possible reason for the difference in the dr values may be that the sliding block analysis did not account for additional sliding resistance resulting from the “plowing action” that occurs at the toe of the wall. 339 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands FLAC Figure 7a. Comparison of the permanent relative displacements computed from the FLAC results and a Newmark sliding block analysis with N*g 0.22 g. 3.1 Permanent wall displacement Comparisons of the permanent relative displacements (dr) of the wall computed from the FLAC results and computed by Newmark sliding block analyses (Newmark 1965) of the structural wedge (Fig. 1) are shown in Figure 7a,b. dr was not computed directly by FLAC, but rather was computed by subtracting the total displacement of the structural node at the intersection of the stem and base of the wall from the total displacement of the grid point at the free-field boundary at the same depth. As may be observed from Figure 7, dr computed from the FLAC results is about 0.33 m. Newmark sliding block analyses of the structural wedge (Fig. 1) were performed using the acceleration time-history shown in Figure 8. This time-history was computed by FLAC at the free-field boundary at a depth corresponding to approximately mid-height of the structural wedge. In order to perform a Newmark sliding block analysis, a maximum transmissible acceleration (N* g) has to be specified, which is the value Newmark 0 3 DISCUSSION Several analyses were performed using the model of the wall-soil system described above, scaling the input motion to different peak ground acceleration values. To assess the adequacy of the model, the results from the FLAC analyses were compared with the results from simplified techniques for estimating the permanent wall displacement and the dynamic earth pressures. The purpose of the comparisons was only to provide a reality check of the FLAC results, while true validation of the FLAC model would require a comparison with actual field observations. Comparisons of the results are discussed in the following sub-sections. However, the reader is referred to Ebeling & Morrison (1992) and Green and Ebeling (2002) for more detailed discussions about the simplified techniques used. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 09069-39.qxd 08/11/2003 20:43 PM Page 340 3.2 Dynamic earth pressures The dynamically induced lateral earth pressures acting on the stem of the wall were computed by FLAC. The corresponding lateral earth pressure coefficients (KFLAC) were computed from these stresses using the following expression (Green et al. 2003): 4.0 3.5 Lateral earth pressure coefficient (K) Although the wall is not embedded in the foundation soil in its initial, undeformed shape, the wall tends to rotate around the toe as it translates away from the backfill. As a result, the toe of the wall penetrates and plows through the foundation soil. Such a mechanism was observed in the deformed FLAC mesh. To account for this additional resistance to sliding, N* g was recomputed assuming a friction angle of 35°, which is between the interface friction angle (i.e. 31°) and the of the foundation soil (i.e. 40°), with the revised value of N* g 0.27 g. A comparison of the permanent relative displacements computed from FLAC and the sliding block analyses using the revised value of N* g is shown in Figure 7b. As may be observed from this figure, the predicted displacements are in very close agreement, thus giving credence to the validity of the proposed FLAC model. 2.5 2.0 1.5 1.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 kh Figure 9. Comparison of FLAC and Mononobe-Okabe dynamic lateral earth pressure coefficients. 1.0 N* = 0.27 0.5 kh 0.0 0 (14) 5 10 15 20 Time (seconds) 25 30 35 40 -1.0 Figure 10. kh at middle of structural wedge. The second item of particular note in Figure 9 is that FLAC predicts kh values as high as 0.5, while the upper bound value should be the maximum transmissible acceleration used in the sliding block analyses (i.e. N* 0.27). A plot of the kh time-history computed by FLAC at the approximate center of gravity of the structural wedge is shown in Figure 10. It can be observed from this figure that the kh values greater than 0.27 are associated with high-frequency motions (that contain little energy). There are two possible reasons for kh N*. First, this could simply be a numerical artifact of the explicit algorithm used in FLAC, rather than a physical phenomenon. However, the criterion that kh N* is based on the premise that the structural wedge is perfectly rigid. For a flexible structural wedge, such as the one modeled, higher modes of vibration could be excited in the structural wedge. This could result in high local kh values, while the global kh value for the structural wedge (i.e. that which contributes to base shear) is less than N*. Adding credence to the latter explanation is that Wartman et al. (2003) observed kh N* values in physical model tests of deformable blocks on an inclined plane. Additional analyses are underway to determine exactly the cause of the high kh values. 340 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands KAE 0.5 -0.5 where PFLAC the resultant of the FLAC computed stresses acting on the stem of the wall; t the total unit weight of the backfill; H the height of the wall; and kv vertical inertial coefficient (assumed to be zero). Equation 14 was used to compute KFLAC values at times corresponding to the peaks in the time-history of the horizontal inertial coefficient (kh) acting away from the backfill (i.e. active-type conditions). A plot of the computed KFLAC values versus kh is shown in Figure 9. Also shown in this figure are the lateral dynamic earth pressure coefficients (active: KAE; Passive: KPE) computed using the MononobeOkabe expressions for the wall-soil system (Okabe 1924; Mononobe & Matsuo 1929). The reader is referred to Green et al. (2003) and Ebeling & Morrison (1992) for details regarding the Mononobe-Okabe dynamic earth pressure coefficients. Two items are of particular note in Figure 9. First, in general, the KFLAC values are higher than the KAE for values of kh less than about 0.4. This phenomenon is discussed in detail in Green et al. (2003) and is due to the failure wedge in the backfill being composed of several failure wedges rather than a single rigid wedge, as assumed in the Mononobe-Okabe expressions. In short, the difference in the KFLAC and KAE values is attributed to a shortcoming in the Mononobe-Okabe expressions, rather than a shortcoming in the FLAC model. KPE 3.0 09069-39.qxd 08/11/2003 20:43 PM Page 341 4 SUMMARY AND CONCLUSIONS The authors outline the details of a numerical model and its calibration for use in computing the dynamic response of a cantilever retaining wall. The proposed model employs an elasto-plastic constitutive model for the soil in conjunction with the Mohr-Coulomb failure criterion. The wall is modeled with elastic beam elements using a cracked second moment of area (Icracked) equal to 0.4 Iuncracked. Interface elements are used to model the wall-soil interface, wherein the interface element parameters are those that give a best fit of the Gomez et al. (2000a,b) hyperbolic interface model. Based on comparisons with simplified techniques for dynamic lateral earth pressure and permanent relative displacement, the proposed wall model is believed to yield valid results. ACKNOWLEDGEMENTS A portion of this study was funded by the Headquarters, US Army Corps of Engineers (HQUSACE) Civil Works Earthquake Engineering Research Program (EQEN). Permission was granted by the Chief of the US Army Corps of Engineers to publish this information. During the course of this research investigation, the authors had numerous discussions with other FLAC users. Of particular note were lengthy conversations with Mr. C. Guney Olgun, Virginia Polytechnic and State University, Blacksburg, VA. Others who provided valuable insight into FLAC modeling were Mr. Nason McCullough and Dr. Stephen Dickenson, Oregon State University, Cornvallis, OR; Drs. N. Deng and Farhang Ostadan, Bechtel Corporation, San Francisco, CA; Mr. Michael R. Lewis, Bechtel Savannah River, Inc., Aiken, SC; Drs. Peter Byrne and Michael Beaty, University of British Columbia, Vancouver; and Dr. Marte Gutierrez, Virginia Polytechnic and State Uni-versity, Blacksburg, VA. Review comments by Dr. William F. Marcuson, III, Emeritus Director, Geotechnical Laboratory, US Army Engineers Waterways Experiment Station, were also greatly appreciated. REFERENCES Ebeling, R.M. & Morrison, E.E. 1992. The Seismic Design of Waterfront Retain Structures. US Army Technical Report ITL-92-11, US Navy Technical Report NCEL TR-939, US Army Engineer Waterways Experiment Station, Vicksburg, MS. (http://itl.erdc.usace.army.mil/ pdf/itl9211.pdf). Finn, W.D.L. 1988. Dynamic Analyses in Geotechnical Engineering. In J.L. Von Thun (ed), Earthquake Engineering and Soil Dynamics II – Recent Advances in Ground-Motion Evaluation, Geotechnical Special Publication 20, ASCE, 523–591. Gomez, J.E., Filz, G.M., & Ebeling, R.M. 2000a. Development of an Improved Numerical Model for Concrete-to-Soil Interfaces in Soil-Structure Interaction Analyses, Report 2, Final Study. ERDC/ITL TR-99-1, US Army Corps of Engineers, Engineer Research and Development Center. (http://libweb.wes.army.mil/uhtbin/hyperion/ITL-TR-991.pdf). Gomez, J.E., Filz, G.M., & Ebeling, R.M. 2000b. Extended Load/Unload/Reload Hyperbolic Model for Interfaces: Parameter Values and Model Performance for the Contact Between Concrete and Coarse Sand. ERDC/ITL TR-007, US Army Corps of Engineers, Engineer Research and Development Center. (http://libweb.wes.army.mil/uhtbin/ hyperion/ITL-TR-00-7.pdf). Green, R.A., Olgun, C.G., Ebeling, R.M., & Cameron, W.I. 2003. Seismically Induced Lateral Earth Pressures on a Cantilever Retaining Wall. Proceedings: The Sixth US Conference and Workshop on Lifeline Earthquake Engineering (TCLEE2003), ASCE, August 10–13, 2003, Long Beach, CA. Green, R.A. & Ebeling, R.M. 2002. Seismic Analysis of Cantilever Retaining Walls, Phase 1. ERDC/ITL TR-023, US Army Corps of Engineers, Engineer Research and Development Center. (http://libweb.wes.army.mil/uhtbin/ hyperion/ITL-TR-02-3.pdf). Headquarters, US Army Corps of Engineers. 1989. Retaining and Flood Walls. EM 1110-2-2502, Washington, DC. (http://www.usace.army.mil/inet/usace-docs/engmanuals/em1110-2-2502/toc.htm) Headquarters, US Army Corps of Engineers. 1992. Strength Design for Reinforced-Concrete Hydraulic Structures. EM 1110-2-2104, Washington, DC. (http://www.usace. army.mil/inet/usace-docs/eng-manuals/em1110-22104/toc.htm). Idriss, I.M. & Sun, J.I. 1992. User’s Manual for SHAKE91: A Computer Program for Conducting Equivalent Linear Seismic Response Analyses of Horizontally Layered Soil Deposits. Center for Geotechnical Modeling, Department of Civil and Environmental Engineering, University of California, Davis, CA. Itasca. 2000. FLAC (Fast Largrangian Analysis of Continua) User’s Manuals. Minneapolis: Itasca Consulting Group, Inc. Jaky, J. 1944. The Coefficient of Earth Pressure at Rest. Magyar Menok es Epitesz Kozloi (Journal of the Society of Hungarian Architects and Engineers). Kuhlemeyer, R.L. & Lysmer, J. 1973. Finite Element Method Accuracy for Wave Propagation Problems. Journal of the Soil Mechanics and Foundations Division 99(SM5): 421–427. Lysmer, J., Udaka, T., Tsai, C.-F., & Seed, H.B. 1975. FLUSH: A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problems. EERC Report No. EERC-75-30, Earthquake Engineering Research Center, University of California, Berkeley, CA. MacGregor, J.G. 1992. Reinforced Concrete Mechanics and Design. Englewood Cliffs: Prentice Hall. Mononobe, N. & Matsuo, H. 1929. On the Determination of Earth Pressures During Earthquakes. Proceedings: World Engineering Congress 9: 177–185. 341 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands 09069-39.qxd 08/11/2003 20:43 PM Page 342 Newmark, N.M. 1965. Effects of Earthquakes on Dams and Embankments. Geotechnique 15(2): 139–160. Okabe, S. 1924. General Theory of Earth Pressures. Journal Japan Society of Civil Engineering 10(6): 1277–1323, plus figures. Paulay, T. & Priestley, M.J.N. 1992. Seismic Design of Reinforced Concrete and Masonry Buildings. New York: John Wiley and Sons, Inc. Terzaghi, K. 1943. Theoretical Soil Mechanics. New York: John Wiley and Sons, Inc. Wartman, J., Bray, J.D., & Seed, R.B. 2003. Inclined Plane Studies of the Newmark Sliding Block Procedure. Journal of Geotechnical and Geoenvironmental Engineering 129(8): in press. 342 Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands