Geotech Geol Eng DOI 10.1007/s10706-017-0360-3 ORIGINAL PAPER Pseudostatic Seismic Response Analysis of a Pile Group in a Soil Slope H. Elahi . H. G. Poulos . H. Hajimollaali . A. Elahi Received: 11 April 2016 / Accepted: 1 September 2017 Springer International Publishing AG 2017 Abstract This paper presents a simple approximate pseudostatic method for estimating the maximum internal forces and horizontal displacements of a pile group located in a soil slope. The method is extension of an existing similar method developed by the authors for the case of a horizontal ground surface. The method employed for horizontal ground case involves two main steps: first, the free-field soil movements caused by the earthquake are computed; Then, the response of the pile group is analyzed based on the maximum free-field soil movements as static movements, as well as a static loading at the pile head, which depends on the computed spectral acceleration of the structure being supported. This newly developed methodology takes into account the effects of group interaction and soil yielding. Simple modifications are applied to take into account the effect of slope on seismic deformations of the pile group, making use of the Newmark sliding block method. The applicability of the approach and the developed program is verified by comparisons made with both experimental shaking table tests and the results of a more refined analysis of a pile-supported wharf. It is demonstrated H. Elahi H. Hajimollaali (&) A. Elahi Department of Civil Engineering, University of Science and Culture, Tehran 11365-4563, Iran e-mail: [email protected] H. G. Poulos Coffey Geotechnics, 8/12 Mars Road, Lane Cove West, NSW 2066, Australia that the proposed method yields reasonable estimates of the pile maximum moment and horizontal displacement for many practical cases, despite its relative simplicity. The simplifying assumptions and the limitations as well as reliability of the methodology are discussed, and some practical conclusions on the performance of the proposed approach are suggested. Keywords Pile group Soil slope Simplified boundary element method Pseudo static analysis Newmark method 1 Introduction Several analytical, numerical and experimental methods have been applied to investigate the seismic behavior of pile groups, but there appears to be little or no research taking into account the seismic behavior of a pile group located in a soil slope. The positioning of a pile within a soil slope mostly has been considered for stabilization of unstable slopes in the absence of seismic activity (Poulos 1999; Hayward et al. 2000). Since the approach in these studies has focused primarily on the increase in safety factor of the slope through the application of piles, the group effect of piles has not often been considered. A number of studies can be found in the literature regarding the effects of inclined geometry of the ground on the lateral behavior of piles under static 123 Geotech Geol Eng loading (Chen and Poulos 2001; Mezazigh and Levacher 1998).These studies have usually examined the effect of ground inclination on the lateral bearing capacity of piles. Also, numerous studies have been reported on the subject of seismic behavior of soil slopes (Rathje and Bray 2000; Wartman et al. 2005). Such analytical and/or experimental studies have focused on both the stability and the seismic deformations of slopes, and most involve the use of Newmark’s sliding block theory (Newmark 1965). Pseudostatic approaches for the seismic analysis of pile foundations have emerged in recent years. For example, Abghari and Chai (1995) developed a pseudostatic procedure using a beam on nonlinear Winkler foundation model (BNWF) to evaluate the soil-pile-superstructure interaction. Following the pseudostatic approach, Tabesh and Poulos (2001) presented a method based on simplified boundary element models for seismic analysis of a single pile with linear soil behavior. A procedure similar to that of Tabesh and Poulos is considered here with some modifications and extensions involved. As piles are mostly installed in weak soils, one of the main factors affecting the pile internal response is possible soil yielding, especially near the top of the pile where the super structure is installed. Also, often piles are constructed in groups to support the structures and the behavior of pile groups differs substantially from that of single piles due to group interaction effects. In this respect, the modifications and extensions made, include both soil yielding and group effects which can be regarded as the advantages of this proposed method over the existing pseudostatic approaches. Elahi et al. (2010) have extended the method developed by Tabesh and Poulos (2001) to consider pile groups and soil yielding effects and they managed to produce a computer program named PSPG for pseudostatic analysis of pile group in horizontal ground. The current paper studies the seismic behavior of a pile or pile group in a soil slope, using the pseudostatic analysis method developed by Elahi et al. (2010) and employing a relatively simple modification to it in order to include the effects of a soil slope. Applicability of the approach and the program developed is verified by comparisons made with both experimental shaking table tests as well as the results of a more refined numerical analysis of a pile-supported wharf. It 123 is demonstrated that the proposed method yields reasonable estimates of the pile maximum moment and horizontal displacement for many practical cases, despite its relative simplicity. Simplifying assumptions, limitations and reliability of the methodology are discussed and some practical conclusions on the performance of the proposed approach are outlined. 2 Pseudostatic Analysis of a Pile Group in a Horizontal Ground A procedure was presented by Elahi et al. (2010) for pseudo-static analysis of a pile group in horizontal ground, which takes into account soil yielding and group effects in a relatively simple manner. Their proposed method involves two main steps: 1. 2. First, the free-field soil movements caused by the earthquake are computed. Then, the response of the pile group is analyzed when subjected to the maximum free-field soil movements (which are considered as static movements) as well as a static loading at the pile head, which depends on the computed spectral acceleration of the structure being supported. In order to take into account both soil yielding and group effects, and to keep the analysis relatively simple, the present method adopts the following approach: • The earthquake which is assumed to consist of vertically incident shear-waves is applied at a level below the pile tip and the response of free-field (soil without piles) along the pile is obtained. • The piles are modeled as Eulerian beams and discretized and modeled by the finite difference method. • The soil is modeled as an elasto-plastic material whose elastic behavior is modeled via the Mindlin fundamental elastic solution (Mindlin 1936). • The maximum values of free-field motion obtained in the first step are applied to each pile as a static external soil movement profile. • A static lateral force is applied to the pile head, given by the spectral acceleration multiplied by the cap-mass (including superstructure mass). The above mentioned procedure is illustrated in Fig. 1. Geotech Geol Eng a Pile ‘m’ b Pile ‘k’ ‘r’ Rows 1 2 δ Columns Spacing ppi psi Rows Spacing L i ‘c’ Columns ppj psj j ue n+1 Shaking Fig. 1 Specifications for lateral analysis of pile group: a cross section, b plan view Each pile in a group is assumed to be a thin vertical strip of width d, length L and constant rigidity EpIp and is divided into n ? 1 elements, all elements being of equal length d, except those at the top and tip, which are of length d/2 (Fig. 1). The soil is first assumed to be an ideal isotropic, elastic material having a Young’s modulus Es and Poisson’s ratio ms that are unaffected by the presence of the piles. If purely elastic conditions prevail with in the soil, the horizontal displacements of the soil and the pile are equal. In this analysis, these displacements are equated at the element centers. In determining the pile displacements, the differential equation for bending of a simple thin beam is applied. This equation is written in finite difference form as follows: Ep Ip ½D up ¼ d Pp 4 d ð1Þ where {Pp} = vector of pressure that acts on pile, {up} = vector of pile displacements, [D] = matrix of finite difference coefficients. In the elastic analysis, the soil displacements can be calculated based on the Mindlin (1936) equation which gives the displacements within a semi-infinite elastic isotropic homogenous mass caused by a horizontal point load (Poulos and Davis 1980). The soil displacements for all points along pile ‘m’ in the group, which arises both from the external source of movement and the pressure caused by the soil-pile (from the same pile adjacent elements) and pile-soilpile (from adjacent piles elements) interaction, may be expressed as: fus gm ¼ fue gm þ ½Is mm fps gm þ rc X ½Is mk fPs gk k¼16¼m ð2Þ where {us} = vector of soil horizontal displacement, {ue} = vector of external soil movement, {ps} = vector of pressure acts on soil, [Is] = n ? 1 by n ? 1 matrix of soil-displacement influence factors, r = number of rows in group and c = number of columns in a group. ½Is mm Components (interaction factors from pile ‘m’ elements on each other) are evaluated by integration over a rectangular area of the Mindlin equation for the horizontal displacement of a point load within a semi-infinite mass while the ½Is mk component 123 Geotech Geol Eng (interaction factors from pile ‘k’ on pile ‘m’) are calculated directly from the Mindlin equation (Poulos and Davis 1980). A solution to the problem is obtained by imposing displacement compatibility between the pile and the surrounding soil, by combining Eqs. (1) and (2), the following equation is achieved: fugm þ rc E p Ip X ½Is mk ½Dfugk ¼ fue gm 4 dd k¼1 ð3Þ This equation results in n ? 1 equations for n ? 1 unknown displacements for pile ‘m’ in the group. Soil nonlinearity is considered by assigning an ultimate lateral pile-soil pressure (Py). A solution scheme can be introduced in which the value of the pressure at the soil-pile interface is monitored at each element. As long as this value is less than ultimate lateral soil pressure (Py), the elastic compatibility [i.e. Eq. (3)] is enforced at that element. If this value is larger than Py, the compatibility equation is replaced by the condition that the pressure at that element is equal to Py. Therefore, the pressure at all pile elements is recalculated and this is ensured by iteration in which at no element of each pile the pressure exceeds Py. The ultimate lateral pressure of soil for single pile is usually estimated based on simplified plasticity theory, for example, using the following simple equations: Py ¼ Nc Cu ðfor claysÞ ð4Þ Py ¼ Np Pp ðfor claysÞ ð5Þ in which Nc = bearing capacity factor (ranges between 8 and 12), and Cu = undrained shear strength, Np = factor which ranges between about 3 and 5, and Pp is the Rankine passive pressure [Pp ¼ r0v tan2 ð45 þ /=2Þ], where r0m is the vertical effective stress and / is the angle of internal friction]. Two types of modification can be applied to the above approximations to take into account the effects of group and cyclic loading on the Py value (Elahi et al. 2010). Based on the above framework, Elahi et al. (2010) developed a computer program named Pseudo Static analysis of Pile Group (PSPG) which can be used for elasto-plastic pseudostatic analysis of a pile group. Elahi et al. (2010) have shown that their proposed method is suitable for the analysis of a pile group in 123 horizontal ground by comparisons made with centrifuge data and an instrumented structure which experienced a real earthquake. 3 Development of Pseudostatic Method for a Pile Group in a Soil Slope The theory and basis of the proposed pseudostatic approach is precisely similar to the approach described by Elahi et al. (2010) for pile group in horizontal ground as summarized above. The following changes and modifications are applied in order to include the effects of slope angle on the analysis of pile group. 3.1 Modification of the Inclined Geometry of Ground Surface Three sets of parameters are modified for the inclined geometry of the ground surface as shown in Fig. 2. These include: (1) The profile of maximum free-field movement of soil (uef), (2) Elasticity modulus of soil (Esf), (3) yield pile-soil pressure (Pyf). The soil movement that is going to be statically applied to the embedded length of pile ‘i’ (uei) is considered equal to uef (maximum soil movement profile derived by dynamic analysis) at the corresponding depths. In other words, the uef profile influences the embedded length of pile ‘i’ and the values corresponding to the nodes on the free length of pile (Lti) are presumed to be zero. It is clear that this assumption does not account for the effects of changes taking place in the characteristics of free-field motion (amplitude and frequency) with respect to inclination as well as soil-pile stiffness difference. Hence, the aforesaid assumption may not be applicable in the case of steep slopes (typically steeper than 4H: 3 V) excited by an earthquake with a predominant period close to that of the natural period of soil profile (Elahi et al. 2012). Also for the condition in which very stiff piles are located in loose/soft soil slope, this assumption will not lead to correct results. The stiffness and strength of the soil at pile ‘i’ (Esi, Pyi) are equal to zero for nodes above the soil surface (Lti) and the corresponding values are considered for the embedded length of pile. Geotech Geol Eng Pile ‘i’ x Lti z uei Esi, Pyi uef Esf, Pyf Fig. 2 Modified parameters for inclined geometry of ground surface 3.2 Modifications for Probable Downward Movement of Slope the piles will also be modeled applying displacements derived from Newmark’s theory. As shown in Fig. 3, the (assumed) linear profile of seismic slope displacement is applied to each pile which is located within the range of a given sliding surface. In other words, the displacement profile from Newmark’s sliding block (ueNi), which is only applied to the pile length embedded in the critical sliding Elahi et al. (2012) suggested that there exists a correlation between the concepts and results of Newmark’s sliding block theory and the movement behavior of pile heads in a group embedded in a soil slope. Hence, the effect of soil slope displacement on Pile ‘i’ ueN ueNi Fig. 3 The effect of Newmark’s displacement of slope on pile group 123 Geotech Geol Eng wedge, is added to the free-field motion profile of the pile (uefi). Obviously, the piles that are not located within the sliding portion do not have this kinematic component of ground movement. It should be noted that the maximum value of ueNi for all the piles embedded in the sliding zone is equal to ueN and equivalent to the displacement of the slope which can be calculated via Newmark’s theory (Newmark 1965). The value of the yield acceleration of the slope (i.e. ay) is required for calculation of the values of ueN. The effect of the piles on ay may be overlooked in calculating ay for small groups with only a few piles, but as the number of piles in a group increases, their presence influences both the strength and deformation of the slope, and thus their effect should be included in the limit equilibrium calculations for the estimation of ay (McCulough et al. 2001). 3.3 Development of PSPG-Slope Program The PSPG program developed by Elahi et al. (2010) has been extended to consider the pseudostatic analysis of a pile group in a slope using the abovementioned modifications and a program named Pseudo-Static Analysis of Pile Group in Slope (PSGP-Slope) has been developed. The algorithm of this newly developed code is similar to that of PSPG, but some modifications have been made in the calculation and allocation of values of ue, Es and Py to the pile group. Moreover, the data entry stage also includes the geometry of critical sliding surface and the value of the yield acceleration of the slope. 4 Verification of the Proposed Method: Comparison with 3D Numerical Analysis Lu (2006) presented a 3D numerical model of the 100 container wharf of Los Angeles harbor with the aid of the powerful software, Parcyclic. This software employs parallel processing and is capable of elastoplastic dynamic analysis of large scale problems. However, it requires a great amount of computer memory and cannot run on conventional computers. For this reason, Lu (2006) utilized one of the largest parallel processing IBM computers in the world to solve his problems. A simplified numerical model of the wharf is shown in Fig. 4a–c and parameters used for numerical 123 analysis in Parcyclic program are presented in Table 1 (Lu 2006). The idealized model configuration is based on typical geometries (Lim et al. 2010) of pile-supported wharf structures (Berth 100 Container Wharf at the Port of Los Angeles). In Fig. 4, a 3D slice in this wharf system (central section) is shown, that exploits symmetry of the supporting pile-system configuration (Lu 2006). In this idealized model (Fig. 4), there are 16 piles in 6 rows. Each pile is 0.6 m in diameter, and 43 m in length (reinforced concrete). The cracked flexural rigidity (EI) of the pile is 159 MN-m2, with a moment of inertia (I) of 7.09 9 10-3 m4. Relative to the piles, the wharf deck is modeled to be an essentially rigid monolith (with a thickness of 0.8 m). Two soil layers were represented in this idealized model. The lower layer (25 m in height) was modeled as stiff clay with the upper layer being weaker medium-strength clay. The water table level was located at 16.6 m above the mud-line, and the slope inclination angle was 32. The base of the FE model was assumed to be rigid (the actual bedrock level is much deeper at this site). The applied motion is horizontal component of the 1994 Northridge earthquake, as recorded at the Rinaldi receiving station, which has been scaled down via a 0.5 coefficient. Due to the disordered positioning of piles in plan (with different spacing in the rows and columns of pile group along some axes), some simplifications were made by the authors to model this wharf in the developed program. In this regard, according to Fig. 4, piles in the rows of A, E and F were replaced by the piles having transverse spacing similar to the other axes (e.g. axis B) and on the other hand, longitudinal spacing of E and F axes was taken to be 7.33 m similar to other axes (e.g. D and E). According to geometrical definitions presented in Fig. 5, selected parameters for modeling the wharf in the program are presented in Table 2. It is noted that all parameters were assessed by Lu (2006) except Xc, Yc, Rc and ay that were calculated using SLIDE software which is a well known limit equilibrium based program. Lu (2006) has presented the results of his numerical analysis in the form of time history of acceleration and displacement of pile cap, displacement profile of the pile group and the deflection profile, shear and Geotech Geol Eng Fig. 4 Simplified pilesupported wharf model in Parcyclic software (Lu 2006). a Isometric view, b elevation view, c pile group layout (plan view) moment of piles in rows A and F. Examples of the outputs of Lu’s analyses are shown in Figs. 6 and 7. In order to study the results obtained from the analyses by PSPG-Slope program, two parameters, maximum displacement profile of the pile cap, and bending moment of piles in the group, were selected as the major parameters to be compared with the results presented by Lu. The simplified wharf model was simulated in this program and the results were presented for the following two cases: with and without Newmark-type displacement effects in the analysis. Figure 8 shows the deflection profile and bending moment for piles located in a group column for each of 123 Geotech Geol Eng Table 1 Specifications and parameters for 3D numerical analysis (Lu 2006) Section Specifications Soil Medium clay V. stiff clay Unit C 44 kPa / 0 degree c 15.5 kN/m3 G 8.00E?04 kPa K 9.73E?05 kPa m 0.46 – C 225 kPa / c 0 20 degree kN/m3 G 4.86E?05 kPa K 5.9LE?06 kPa m 0.46 – Group L9 Cap thickness (concrete) 0.8 m Piles Concrete Diameter 0.6L m Ip 7.09E-03 m4 Ep 2.237E?07 kPa c 24 kN/m3 m 0.3 – Xc , Yc Table 2 Parameters used for wharf modeling in PSPG-Slope program X0 , Y0 Section 0,0 Specifications Group 6 9 10 Rc Slope Critical Slip Circle Unit All – Spacing 7.33 9 6.67 Cap mass 3840 m ton X0, Y0 -7.33, 1.4 m m Xc 0.73 Yc 0.48 m Rc 76.56 m Slope angle ay 32 0.11 Degree g Fig. 5 Definition of geometrical parameters in PSPG-slope program the above cases, respectively. The values calculated by Lu (2006) are also presented in this figure as bold lines with discrete points. According to this figure, when Newmark-type displacement effects are considered in the analysis, maximum deflections differ less than 20% and for the case that these effects are not considered, the difference is approximately 40%. Additionally, comparing the bending moment profiles demonstrates that the occurrence of maximum bending moment along the 123 piles is at the same depth for the two cases and also the magnitude of maximum moments is quite proximate. Consequently, it can be concluded that the developed program estimates the maximum values of deflections and internal forces of pile groups with acceptable accuracy for engineering purposes, in spite of the simplified concepts applied in it. In addition to the simplification of hypotheses and analytical algorithm, some simplifications have also been made to the geometry of group and stratification geometry of subsurface soil. Therefore, not Geotech Geol Eng Fig. 6 The deflected model after the applied earthquake in W3N-F model (Lu 2006) the Newmark theory in the seismic analysis of pile group in slopes through the pseudostatic approach. 5 Assessment of Proposed Method: Comparison with Shaking Table Tests Fig. 7 Time history of pile cap deflection and Horizontal displacement profile of the pile group in W3N-F model (Lu 2006) surprisingly, some of the differences that exist between the results of the pseudostatic analysis and Lu’s (2006) more detailed numerical solution might be due to these simplifications. If the deflection effect pertinent to Newmark theory is removed from the analyses, the behavior totally changes and the results do not agree well with each other. This appears to confirm the concept of applying Elahi et al. (2012) conducted 16 tests of 1 g physical models on a pile group embedded in a soil slope with height of 1 m consisting of a sand layer using a 1:10 scale shaking table. The properties of the sand are presented in Table 3. All physical models were equipped with strain meter, accelerometer and displacement meter sensors, and were excited by nearly sinusoidal seismic motions. In order to examine the effects of various parameters on the seismic response of the pile group in a slope, the nature of the excitation, and the number and spacing of piles were changed. A single pile was considered, as well as 2 9 1 and 2 9 2 groups with spacing to diameter ratios of 3, 5.4 and 7.8. Some properties and a sample of the records from these tests are presented in Fig. 9 and Table 4. All physical pile group models that were used in the shaking table experiments, investigated by PSGPSlope program by applying the parameters presented in Table 5. All mentioned parameters were measured and/or calculated by Elahi except Xc, Yc, Rc and ay which were calculated using SLIDE program. The results of these analyses were compared with the 123 Geotech Geol Eng Row No.1 Col No.4 0.35 Row No.4 Col No.4 0.25 Row No.5 Col No.4 0.2 Row No.6 Col No.4 Pille F1 (Lu, 2006) 0.15 Pille A3 (Lu, 2006) 0.1 Row No.3 Col No.4 Row No.4 Col No.4 0.25 Row No.5 Col No.4 0.2 Row No.6 Col No.4 Pille F1 (Lu, 2006) 0.15 Pille A3 (Lu, 2006) 0.1 0.05 0.05 0 0 0 5 10 15 20 25 30 35 40 0 45 Pile Depth (m) With Newmark 2500 2500 2000 2000 1500 1000 500 0 -500 -1000 -1500 -2000 0 5 10 With Newmark 15 20 25 5 10 15 30 Pile Depth (m) Row No.1 Col No.4 Row No.2 Col No.4 Row No.3 Col No.4 Row No.4 Col No.4 Row No.5 Col No.4 35 No.640Col No.4 45 Row Pile F1 (Lu,2006) Pile A3 (Lu,2006) 20 25 30 35 40 45 Pile Depth (m) Without Newmark Momment (kN.m) Momment (kN.m) Row No.2 Col No.4 0.3 Row No.3 Col No.4 Deflection (m) Deflection (m) 0.3 Row No.1 Col No.4 0.35 Row No.2 Col No.4 1500 1000 500 0 -500 -1000 -1500 -2000 0 5 10 15 Without Newmark 20 25 30 Pile Depth (m) Row No.1 Col No.4 Row No.2 Col No.4 Row No.3 Col No.4 Row No.4 Col No.4 Row No.5 Col No.4 Row 35 No.640Col No.4 45 Pile F1 (Lu,2006) Pile A3 (Lu,2006) Fig. 8 Comparison of PSPG-Slope with Lu’s (2006) accurate numerical solution for deflection profile and bending moment of wharf piles Table 3 Properties of Firoozkooh 161 sand used in the shaking table tests Uses name D10 (mm) D30 (mm) D50 (mm) (mm) Passing #200 % Sand % Cu Cc u degree emax emin SP 0.16 0.21 0.27 0.3 1 99 1.87 0.88 40 0.874 0.548 measured responses. The results derived from some of the conducted analyses, including the distribution of the maximum deflection and bending moment of rear and front piles (Fig. 9a) are shown in Figs. 10, 11 and 12. The following conclusions may be drawn with respect to Figs. 10, 11 and 12: • • PSPG-Slope program is capable of estimating the maximum internal forces and pile deflections in a slope reasonably well, in spite of the use of simplified assumptions and analytical methods. The maximum deflections and bending moments calculated by PSPG-Slope are in good agreement with most of the records derived from the tests, in terms of their distribution along the pile, the 123 maximum values, and the positioning of the pile in the group (i.e. whether front or rear). 6 Discussion on Assumptions and Limitations Evidentially, while using simplified methods, careful attention should be paid to the assumptions involved and their limitations, as well as their reliability and their range of applicability. In what follows, some of the fundamental assumptions of the proposed method are addressed and their limitations are discussed. The major issues that include simplified and possibly limiting assumptions are as follows: • Free-field ground motion; Geotech Geol Eng • D3 D4 a D5 acc5 SG: Strain Gauge acc: accelerometer D: LVDT SG1.1 2.1 SG1.2 2.2 SG1.3 2.3 6.1 Free-Field Ground Motion acc4 SG1.4 2.4 SG1.5 2.5 acc3 SG1.6 2.6 acc2 The following assumptions have been employed to calculate the free field motion: 1. acc1 Front Pile Rear Pile 2. b Acceleration (g) 0.18 Amplitude (g) 0.16 0.14 0.12 0.1 0.4 0.3 6.1.1 One-Dimensional Calculation of Free-Field Motion 0 -0.1 -0.2 0 1 2 3 4 0.06 5 6 7 8 9 10 t (s) 0.04 0.02 0 0 5 10 15 20 25 30 35 40 Frequency (Hz) Fig. 9 a Schematic geometry of physical models (in cm), b a sample of the time history of arriving motions along the Fourier spectra Table 4 Specifications of physical models in different tests One-dimensional analysis is performed to calculate free field motions. The amount of this motion is assumed to be the same for all piles in the group. 0.2 0.1 -0.3 -0.4 0.08 The technique of employing the Newmark-type displacement estimation. It is obvious that the free field motion is affected by the two- or three- dimensional geometry of slope (Wolf 1985). There are different simple methods such as the equivalent linear method incorporated into the SHAKE program that can be employed to calculate the free surface motion. Furthermore, the case of inclined geometry of soil layers, subsurface rock or inclined geometry of ground surface, calculation of No. Test name amax (g) aRMS (g) Slope s/d Group configuration 1 1–1–1 0.22 0.08 1:2 0 Single 2 1–1–2 0.48 0.22 1:2 0 Single 3 2–1–1 0.15 0.04 1:2 5.4 2*1 4 2–1–2 0.46 0.20 1:2 5.4 2*1 5 2–2–1 0.22 0.08 1:2 7.8 2*1 6 2–2–2 0.46 0.23 1:2 7.8 2*1 7 2–3–1 0.11 0.09 1:2 3 2*1 8 2–3–2 0.39 0.20 1:2 3 2*1 9 3–1–1 0.16 0.07 1:2 3 2*2 10 3–1–2 0.44 0.21 1:2 3 2*2 11 3–2–1 0.20 0.08 1:2 5.4 2*2 12 13 3–2–2 3–3–1 0.43 0.18 0.21 0.06 1:2 1:2 5.4 7.8 2*2 2*2 14 3–3–2 0.43 0.16 1:2 7.8 2*2 15 3–4–1 0.17 0.06 Horizontal 7.8 2*2 16 3–4–2 0.50 0.23 Horizontal 7.8 2*2 123 Geotech Geol Eng X0, Y0 44.75,10 cm 2 3 1 (s/d) 5.4 – Cap mass 1.485 kg X0, Y0 2 3 1 (s/d) 41.15,10 7.8 cm – Cap mass 1.485 kg X0, Y0 37.55,10 cm 2 3 2 (s/d) 3 – Cap mass 3.015 kg X0, Y0 44.75,10 cm 2 3 2 (s/d) 5.4 – Cap mass 3.015 kg X0, Y0 41.15,10 cm 2 3 2 (s/d) 7.8 – Cap mass 3.015 kg X0, Y0 37.55,10 cm free field ground response by means of methods based on one-dimensional assumptions may lead to inaccurate results. Two or three dimensional equations of wave propagation in a half-space are required to be solved for the aforementioned conditions. Therefore, for the case where the subsurface soil is uniform or the slope is not steep, results derived from a one-dimensional solution of free field motion might be reasonable. Consequently, in cases of steep slopes, layered soils or proximity of the predominant frequency of input motion to that of soil slope, inaccurate results might be obtained using one-dimensional concepts of wave propagation. Concerning topography effects on the seismic movement of slope, Gatmiri et al. (2008) stated that as the slope inclination and frequency of input motion increase, the movements become greater. Research findings such as those expressed by Gatmiri et al. (2008) indicate that for common slopes in engineering applications (inclination angle between about 15 and 35), for typical earthquake frequencies (around 2–5 Hz) and for the soils with common shear wave velocity (ranging from 200 to 600 m/s), the ratio of maximum horizontal displacements of slope to the free field movement of horizontal ground, varies between 0.8 (at downslope) to 1.2 (at upslope). In order to examine effect of the abovementioned assumption, a numerical model of a pile-supported container wharf of the port of Los Angeles (Lu 2006) was solved under the three following conditions with PSPG-Slope: 2 3 2 (s/d)-Hor 7.8 – 1. Cap mass 3.015 kg X0, Y0 Diameter 37.55,10 15 cm mm Thickness 1 mm Table 5 Parameters used for shaking table tests modeling in PSPG-slope program (Elahi et al. 2012) Section Model Specifications Soil All C 0 kPa / 40 degree c 15.3 kN/m3 Py 4Pp (Eq. 3) – Es 10z MPa m 0.3 – Group 1–1 2–3 2–1 2–2 3–1 3–2 3–3 3–4 Piles Slope All All Single – Cap mass 0.755 kg X0, Y0 50,10 cm 2 3 1 (s/d) 3 – Cap mass 1.485 kg Ep 7.0E?07 kPa Ip 0.1083 cm4 Xc 142 cm Yc 160 cm Rc 210 cm Slope angle 26.6 degree 0.2 g ay 123 Unit 2. 3. Free-field displacement (Uef) is applied with a coefficient of 0.8 to all piles of the group. Free field displacement is applied with a coefficient of 1.0 to all piles of the group (main assumption used in the proposed procedure). Free-field displacement is applied with a coefficient of 1.2 to all piles of the group. Figure 13 shows the computed lateral deflection and bending moment of front and rear piles for the three abovementioned conditions. As it can be seen in this figure, a 20% change in the amount of free field ground displacements (along the length of all piles) leads to a change about 10–15% in the amount of the deflection and moment of piles. Thus, it may be acceptable to assume that free field displacements of horizontal Geotech Geol Eng Fig. 10 Comparison of PSPG-slope results with test records for 2–1 model Front Calculated Front Measured Rear Calculated 0.001 0.0009 2-1-1 0.0008 0.0007 0.0006 u (m) Rear Measured 0.006 2-1-2 0.005 0.004 0.003 0.0005 0.0004 0.0003 0.0002 0.002 0.001 0.0001 0 0 0 0.2 0.4 0.6 0.8 0 1 0.2 Depth (m) 2-1-1 M (kN.m) 2 0.2 0.2 0.4 0.6 0.8 1 0 -0.2 -1 -0.4 -2 -0.6 -3 -0.8 -4 0.2 0.4 0.6 1 Front Measured Rear Measured 0.0045 0.0009 0.004 0.0008 2-2-1 0.0007 u (m) 0.8 2-1-2 0 0 Front Calculated Rear Calculated 0.0035 0.0006 0.003 0.0005 0.0025 0.0004 0.002 0.0003 0.0015 0.0002 0.001 0.0001 0.0005 2-2-2 0 0 0 0.2 0.4 0.6 0.8 0 1 0.2 Depth (m) M (kN.m) 1 1 0 3 0.6 2 0.4 1 0.2 0 0 -1 0 0.2 0.4 0.6 0.4 0.6 0.8 1 0.8 1 Depth (m) 0.8 0.8 1 0 0.2 0.4 0.6 -2 -3 -0.4 -4 -0.6 -0.8 0.8 3 0.4 -0.2 0.6 Depth (m) 0.6 Fig. 11 Comparison of PSPG-slope results with test records for 2–2 model 0.4 2-2-1 -5 2-2-2 123 Geotech Geol Eng Fig. 12 Comparison of PSPG-slope results with test records for 3–2 model u (m) Front Calculated Rear Calculated Front Measured Rear Measured 0.0009 0.004 0.0008 0.0035 3-2-1 0.0007 0.003 0.0006 0.0025 0.0005 3-2-2 0.002 0.0004 0.0015 0.0003 0.001 0.0002 0.0005 0.0001 0 0 0 0.2 0.4 0.6 0.8 0 1 0.2 M (kN.m) Depth (m) 1 4 0.8 3 0.6 2 0.6 0.8 1 0.8 1 1 0.4 0 0.2 -1 0 -0.2 0.4 Depth (m) 0 0.2 0.4 0.6 -0.8 ground and common slopes in engineering applications are equal. It is obvious that in conditions contrary to the abovementioned, this simplifying assumption requires to be modified. 6.1.2 Distribution of Free-Field Movement Among the Piles of the Group 1 0.2 0.4 0.6 -2 -3 -0.4 -0.6 0.8 0 3-2-1 -4 -5 3-2-2 -6 engineering software such as GeoFEAP (Bray et al. 1995). As a result, this assumption appears to be justifiable at the present time for common engineering applications. 6.2 Utilization of the Newmark-Type Displacement 6.2.1 Amount of Applied Displacement The positioning of piles in the soil and the occurrence of pile-soil-pile interaction leads to changes in the free-field movement among the piles of the group and this phenomenon affects the distribution pattern of free-field displacement among the different piles of the group. However, many researchers working on the seismic behavior of pile groups based on simplified methods assume equal free-field displacements applied to different piles of group. This assumption is made with respect to the complexity and lack of knowledge towards this issue (Gazetas et al. 1993; Bray et al. 1995; Curras et al. 2001). Although the level of conservation has not been clearly demonstrated yet, many researchers have employed it in their simplified methods (particularly in the Winkler-type methods) and also it has been incorporated into the 123 Newmark’s sliding block approach has been used widely in various geotechnical issues, most notably soil slopes, in spite of its numerous simplifications, limitations and uncertainties lay in the accuracy of obtained results. Some codes (e.g. FEMA-273 and NAVFAC-1997) also recommend using this method to calculate seismic displacements of slopes for practical applications. Moreover, the Newmark concept has been incorporated into some engineering software (e.g. Geoslope). Consequently, as Elahi et al. (2012) illustrated through the results of shaking table tests, employing the displacements based on Newmark theory can be adopted a practical method to assess seismic displacements of piles located in soil slope. In the present study, the simplest technique has Geotech Geol Eng Fig. 13 piles response due to change in amount of applied free-field soil displacement been applied to calculate the Newmark-type displacements. Thus, the effects of the factors resulting from flexibility of the sliding block, inclination of the sliding surface and utilization of a corrected acceleration time-history for the geometry of sliding wedge, have not been taken into account. Evidently, modifications for each one of the abovementioned factors can be simply added with respect to the concepts presented in the technical literature. 6.2.2 Distribution of Applied Displacement in Depth The distribution pattern of the displacement of a failure wedge along the embedment length of piles is another assumption made in the proposed methodology to employ Newmark-type displacements. It is clear that the profile of distribution of slope displacements in depth is dependent on the amount of slope movements. In other words, this profile depends on the level of difference between slope stability and the entire failure condition (with respect to the utilized concepts in limit equilibrium method). Chen and Poulos (2001) carried out a survey on the distribution profile of the lateral soil displacements along the piles under different conditions. Examples of these conditions include piles adjacent to embankment, existing pile foundations adjacent to pile driving, excavation or tunneling operations, slope stabilizing piles and pile foundations in moving slopes. By employing the results of this survey as well as the results of field measurements and experimental centrifuge tests, they drew following conclusions: 1. 2. For piles adjacent to excavations or relatively small movements of soil slope movements, a linear profile for lateral soil movement can be considered. For landslides undergoing relatively large soil movements, a uniform distribution pattern in depth can be chosen for lateral movements in sliding zones. They compared their findings with the theories and methods similar to that of presented in this study and also with the analyzed results from the PALLAS software developed by 123 Geotech Geol Eng Hull (1999) and concluded that their obtained results are within a reasonable agreement. The abovementioned results are presented for lateral soil movement in a static condition, but Muraleetharan et al. (2004) obtained similar results for the centrifuge model of a sandy embankment under seismic condition. Figure 14 shows the model of the sandy embankment used in the analysis. As can be seen, the model includes an embankment with two different inclination angles (18 and 30) and the relative density of sand around 62%, subjected to an input base acceleration. Figure 15 shows the deformed shape of the slope at the end of the test as well as the displacements pattern (dotted lines). Two different distribution patterns of displacement in the upper part of the slopes are seen. One is a distribution close to linear at bottom of the slope with smaller inclination angle (due to smaller slope displacements) and the other one is a distribution closer to uniform at bottom with greater inclination angle (as a result of greater slope displacements). According to the abovementioned points, it can be inferred that depending on the rate of slope displacements and safety margin of slope stability from failure, it can be assumed that the distribution pattern of slope displacements varies from linear to uniform. The physical model tests conducted by Elahi and a numerical model presented by Lu (2006) are simulated assuming both linear and uniform distribution patterns and the results are compared so as to achieve a better understanding of the distribution pattern of Newmarktype displacements along the pile. The deflection and bending moment along piles are shown in different models, Figs. 16, 17 and 18. Models 2–1–1 and 3–1–1 are excited by lower amplitude of acceleration and consequently, smaller slope movements are obtained, but the other tests are subjected to higher amplitude of acceleration and greater movements are derived. In general, the model slopes in the experiments conducted, have not undergone large movements close to failure. Hence, as shown in these figures, typically the assumption of a linear distribution of slope displacements in depth leads to a closer agreement with the results of tests. As a result, in the case of lower amplitude of input acceleration the corresponding permanent displacements of slope are very small and the sensitivity of the behavior to the change in 123 displacement profile in depth is not significant. For greater base accelerations that cause a considerable amount of displacement the corresponding distribution pattern of displacement is more important and the assumption of uniform distribution pattern, irrespective of changing the deflection and bending moments of piles and transferring the maximum moment to the lower depths, increases the maximum displacements and bending moments by 1.2–1.5 and 2–4 times (compared with linear distribution pattern), respectively. Moreover, the more refined numerical analysis performed by Lu (2006) was examined more thoroughly concerning the profile of permanent slope displacements in depth. He presented slope displacements in three locations A, B and C (as defined in Fig. 19). Calculated displacements from his analysis and those using the proposed linear and uniform profiles are shown in Fig. 19. As can be seen, from the surface of the slope to a depth close to the middle of the height of the failure wedge, the displacements derived from accurate numerical analysis are closer to the uniform distribution, although in the lower half of the height of the failure wedge, the displacements decrease with depth and the assumption of a uniform profile is found to be conservative. The comparison of deflections and bending moments in rear and front piles (Piles F1 and A3, respectively) obtained by Parcyclic software, with those computed from the two distribution patterns (linear and uniform) are shown in Fig. 20. From this figure, the following conclusions can be drawn: 1. 2. 3. Displacements of free length of piles close to the surface are better estimated by applying the assumption of uniform distribution of soil displacement. A uniform distribution pattern of Newmark-type displacement transfers the location of maximum bending moment to the lower depths. In addition, the accuracy of calculated moments derived from this pattern is somewhat conservative as compared to the linear pattern. The shape of the bending moment profile of piles in upper depths corresponds to a uniform distribution of soil movement, whereas for lower parts, it corresponds more closely to a linear distribution of soil movement. Geotech Geol Eng Fig. 14 Specifications of centrifuge model of sandy slope (Muraleetharan et al. 2004) Fig. 15 Permanent displacements and deformation of sandy slope in centrifuge (Muraleetharan et al. 2004) Front Measured Front Calculated (Linear UeN) Front Calculated (Uniform UeN) 0.001 0.01 0.009 2-1-1 0.0008 u (m) Rear Measured Rear Calculated (Linear UeN) Rear Calculated (Uniform UeN ) 0.0009 2-1-2 0.008 0.0007 0.007 0.0006 0.006 0.0005 0.005 0.0004 0.004 0.0003 0.003 0.0002 0.002 0.0001 0.001 0 0 0 M (kN.m) Fig. 16 Comparison of deflection and bending moment along piles obtained from the assumption of linear and uniform Newmark-type displacement with the results of test models 2–1–1 and 2–1–2 (according to Elahi et al. 2012) 0.2 0.4 0.6 0.8 0 1 0.6 6 0.4 4 0.2 0.2 0.4 0.6 0.8 1 0.8 1 2 0 0 0.2 0.4 0.6 0.8 1 0 0 -0.2 -0.4 -0.6 -0.8 0.2 0.4 0.6 -2 -4 2-1-1 2-1-2 -6 123 Geotech Geol Eng Front Measured Front Calculated (Linear UeN) Front Calculated (Uniform UeN) Rear Measured Rear Calculated (Linear UeN) Rear Calculated (Uniform UeN) 0.012 0.0012 3-1-1 0.001 u (m) Fig. 17 Comparison of deflection and bending moment along piles obtained from the assumption of linear and uniform Newmark-type displacement with the results of test models 3–1–1 and 2–2–2 (according to Elahi et al. 2012) 0.0008 0.008 0.0006 0.006 0.0004 0.004 0.0002 0.002 0 0 M (kN.m) 0 0.2 0.4 0.6 0.8 1.5 15 1 10 0.5 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 5 0 0.2 0.4 0.6 0.8 1 0 -0.5 -5 -1.5 -10 3-1-1 2-2-2 -15 -2 Front Measured Front Calculated (Linear UeN) Front Calculated (Uniform UeN) Rear Measured Rear Calculated (Linear UeN) Rear Calculated (Uniform UeN) 0.005 0.0045 0.006 3-2-2 0.005 u (m) 0 0 -1 Fig. 18 Comparison of deflection and bending moment along piles obtained from the assumption of linear and uniform Newmark-type displacement with the results of test models 3–2–2 and 3–3–2 (according to Elahi et al. 2012) 2-2-2 0.01 3-3-2 0.004 0.0035 0.003 0.004 0.0025 0.002 0.0015 0.001 0.003 0.002 0.001 0.0005 0 0 0 0.2 0.4 0.6 0.8 1 8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 4 6 2 M (kN.m) 4 0 2 0 -2 -2 0 0.2 0.4 0.6 0.8 1 -4 -4 -6 -8 123 3-2-2 -6 -8 3-3-2 Geotech Geol Eng Location A Lateral Disp. (m) 0 0.5 Calculated (Lu, 2006) Linear Distribution 0 2 Uniform Distribution 4 Location B Lateral Disp. (m) 6 8 0 0.5 10 0 12 2 14 4 16 6 18 8 20 10 0 22 12 2 24 14 4 26 16 6 28 18 8 Location C Lateral Disp. (m) 0 0.5 Critical Slip Surface Fig. 19 Comparison of calculated displacement profiles by Lu (2006) with linear and uniform pattern of Newmark-type displacements 4. Apparently, in this particular problem, optimum results are obtained by applying a distribution pattern in-between linear and uniform (e.g. trapezoidal pattern). • 7 Conclusions This paper presents an approach for the pseudostatic analysis of a pile group in a soil slope subjected to seismic excitation. Comparisons with shaking table tests and a more refined numerical model, lead to the following conclusions: • Comparisons between the results of the relatively simple proposed method and those of pile groups in shaking table tests show reasonable agreement. Comparisons between the proposed method and a more complex and refined numerical analysis also show reasonably good agreement for a pilesupported wharf case study. The promising level of agreement achieved, suggests that the program developed, has some potential for practical application. Some consequences of the assumptions and limitations of the proposed pseudostatic approach have been investigated, and the following conclusions derived: • • The distribution with depth of Newmark-type displacements of slope in depth varies from linear to uniform, depending on the rate of slope movements and the stability condition of slope. For common slopes in engineering applications (with a slope angle between about 15 and 35), for typical earthquake frequencies (about 2–5 Hz) and 123 Geotech Geol Eng Pile F1 (Lu, 2006) Deflection (m) 0.3 Pile A3 (Lu, 2006) Row No.1 Col No.4 0.25 Uniform 0.2 Linear Row No.6 Col No.4 Row No.1 Col No.4 Row No.6 Col No.4 0.15 0.1 0.05 0 0 10 20 30 40 50 Pile Depth (m) 4000 Moment (kN.m) 3000 2000 1000 0 -1000 -2000 -3000 0 5 10 15 20 25 30 35 40 45 Pile Depth (m) Fig. 20 Comparison of accurate numerical solution by Lu (2006) with PSPG-Slope for deflection profile assuming linear and uniform pattern of Newmark-type slope movements for the soils within a certain range of shear wave velocity (between about 200 and 600 m/s), one dimensional calculation of the free field soil movements arising from seismic activity is acceptable. References Abghari A, Chai J (1995) Modeling of soil-pile-superstructure interaction for bridge foundations. 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