elahi2017

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
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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
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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
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(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
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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
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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
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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
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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
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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
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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.
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