# Simplified Boundary Element Method for Kinematic Response of Single Piles in Two-Layer Soil

Authors: Fayun Liang; Haibing Chen; Wei Dong Guo
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

A simple approach is formulated to predict the elastic, kinematic pile bending during harmonic or transient excitation for a circular pile (rather than a simplified thin strip). The kinematic response of a pile embedded in two-layer soil is resolved in the frequency domain caused by the upward propagation of shear waves from the underlying bedrock. The simplified approach is generally valid to nonhomogeneous soil profiles, in light of the good comparison with the dynamic FE method and BDWF solution. It employs the soil-displacement-influence coefficientsIs to consider the pile-soil interaction (resembling the spring constant kx in the BDWF) and provides conservative estimations of maximum kinematic bending moments at the soil-layer interface (with a sharper stiffness contrast). The accuracy of the approach may be improved by incorporating the interaction of soil into the soil-displacement-influence coefficients Is for such cases with Vb/Va<3.


---

## Body

## 1. Introduction

Kinematic response is one of the key issues in seismic design of pile foundations [1–4], as the dynamic response of the pile structure largely differs from the response at the free field soil caused by seismic wave. In practice, the influence of inertial loading at the pile-head level has conventionally received sufficient attention by design engineers, but not the kinematic seismic response [5]. The existing earthquake investigations [6, 7] and experimental studies [8] demonstrate that at the interface of two-layer soils with a sharp stiffness contrast [2, 3, 9, 10], or at a pile-head with fixed constraints [11, 12], large kinematic bending moments may be induced to inflict damage to the pile. This problem has attracted the attention researchers [13–15] and is highlighted in some advanced seismic codes [16].The kinematic response in the pile-soil system has been analyzed by considering the effect of the passive pile using rigorous mechanical solutions [17, 18], numerical methods [19–25], and some simplified models [2, 3, 26–30]. Rigorous solutions and numerical methods for the kinematic pile bending are, however, not convenient to design purposes. Empirical formulas are developed for evaluating the bending moment at the pile-head or at the interface of two-layer soils [2, 3, 11]. Among them, a Beam-on-Dynamic-Winkler-Foundation (BDWF) formulation was used successfully in practice, although it was confined to the harmonic excitation at the pile head. The BDWF (or the Winkler model) is underpinned by a frequency-dependent impedance (=kx+iωCx), in which the continuously distributed springs kx(≅δEs) is empirically related to a dimensionless coefficient δ (used for a given pile-soil system, regardless of layers). Accurate selection of the value δ is not critical to calculating the pile-head deflection but is important to predicting the kinematic bending moment and shear force [19, 31]. An optimized δ is thus required to obtain correct kinematic pile bending at the interface of soil layers. On the other hand, a simplified boundary element formulation proposed by Poulos and Davis [32] offers good estimates of bending moment and shear force for static loading. It would be good to see its accuracy in predicting the kinematic seismic response of pile, especially at the sharp stiffness contrasts between adjacent soil layers.In this paper, the simplified approach [32] is employed to evaluate the kinematic pile bending during harmonic or transient excitation, concerning piles in two-layer soil. The pile is modeled as a circular shape rather than a thin strip adopted previously [32], and the soil displacement is given by the Mindlin equation with corresponding elastic modulus. A nodal relative displacement is obtained by one-dimensional site response and is then imposed on the pile. The solution was compiled into a program operating in Matlab platform. For some typical cases, a comprehensive study on the kinematic seismic response during harmonic or transient excitation has been carried out, and the results are compared with available dynamic FE method and the BDWF solutions. The study sheds new light on the kinematic bending moment at the interface of two-layer soil and at the pile head and may facilitate the use of the simplified boundary element method to predict the kinematic seismic response of a single pile.


## 2. Simplified Analysis Procedure

### 2.1. Basic Assumptions for Pile and Soil Model

The one-dimensional model for a floating or end-bearing single pile embedded in a two-layer soil is shown in Figure1. The circle pile is assumed as linearly hysteretic beam having a length L, a diameter d, a mass density ρp, and a bending stiffness EpIp. The pile is discretized into n+1 segments of equal lengths h, but for a length of h/2 for the top and the tip segments, respectively. Each segment is subjected to a uniformly distributed load pi over the semicircular area. The pile is head restrained (fixed head) or free to rotate (free head), and sits above a bedrock. The linearly hysteretic soil profile is characterized by an upper-layer of thickness Ha and a shear wave velocity Va, which is underlain by a lower layer of thickness Hb and shear wave velocity Vb. The two layers have damping ratios βb and βa, mass densities ρb and ρa and Poisson’s ratios μa and μb. A shear wave propagates vertically through the free field soil, which induces the horizontal harmonic motion and horizontal displacements. The motion at the bedrock surface is expressed by the amplitude of either bedrock displacement Ug or the bedrock acceleration ω2Ug.Figure 1
Analysis model of a pile in a two layer soil profile subjected to vertically-propagating seismic SH waves.


### 2.2. Calculation of the Horizontal Displacement of the Pile

Kinematic response of a single pile is induced by the free-field soil displacement shown in Figure1. The Mindlin hypothesis does not meet the needs of dynamic analysis. However, the Mindlin equation is still valid for calculating elastic displacement and stress fields caused by a dynamic loading, provided that the characteristic wavelength in the soil is sufficiently long in comparison with the horizontal distance across the zone of major influence [28, 33], as is noted for nonhomogeneous soil by Poulos and Davis [32]. In the current, simplified BEM formulation, the soil displacement us (due to the pile-soil interface pressure) is gained using the Mindlin solution [34], which is then added together with free-field soil displacement
(1)us=Ispi+ue,
where Is is the soil-displacement-influence coefficient; pi is the vector of soil-pile interface pressure over the semi-circular area; and ue is the free-field soil displacement estimated using one-dimensional site response for vertically propagating shear waves through an unbounded medium [35].The dynamic equilibrium under steady-state conditions for the pile may be written in the following form using the finite-difference method [28]:
(2)EpIph4Dup+Mu¨p+Cx(u˙p-u˙e)=-dpp,
where up is the horizontal displacement of the pile, with the cap “·” indicating differentiation with time; pp is the vector of soil-pile interface pressure; D is the matrix of finite difference coefficients; M is the pile mass; and Cx is the soil radiation damping. Here, the soil damping is the same as that of the simplified boundary element model [28], and
(3)Cx=5dρsVs,
where Vs is the shear wave velocity of soil and ρs is the density of soil.The displacement compatibility between the pile and the adjacent soil offersus=up. Taking the displacement as up and substituting (1) into (2) result in the following:
(4)Isd[Mu¨p+Cxu˙p+EpIph4Dup]+up=ue+IsdCxu˙e.
Equation (4), together with the pile-top and -bottom boundary conditions, leads to n+5 unknown displacements, which involve the pile nodes from 1 to n+1 (see Figure 1) and 4 additional nodes at the pile top and tip. Equation (4) is resolved in either frequency or time domain and may attain the required accuracy using 21 segments of the pile [32].


### 2.3. Soil-Displacement-Influence Coefficients

Poulos and Davis [32] obtained soil-displacement-influence coefficients Is by integrating the Mindlin equation over a rectangular plane and taking the pile as a thin rectangular vertical strip. Ideally, a circular pile (rather than a thin strip) should be used. In elastic, semi-infinite space, the force P in the horizontal direction at a depth c induces a displacement component ux, which at any other point (x,y,z) is given by
(5)ux=PGf(μs,x,y,z,c),
where G is shear modulus of soil and μs is the Poisson ratio.The coefficients of the proposed method are obtained in two steps: firstly, (5) is integrated over a rectangular area from a depth c of c1 to c2 and across the pile width from -d/2 to d/2 in Figure 2 which offers
(6)ux(μs,x,y,z,c,s)=PG∫-d/2d/2∫c2c1f(μs,x,y-s,z,c)ds.Integrated along the semi-circular pile surface.
(a)
Integrated area
(b)
Weighted average areaSecondly, (6) is reexpressed in a cylindrical coordinate, which is then integrated over the semi-circular pile surface with respect to the angle θ (see Figure 2) to gain the horizontal component of the displacement. The soil-displacement-influence coefficients, taking as weighted average of the integrated horizontal displacements, are deduced as(7)Is=PG∫0π/2∫-d/2d/2∫c2c1f(μs,d2cosθ,d2sinθ-s,z,c)cosθdcdsdθ.Equation (7) is solved by numerical integration using an adaptive Lobatto rule [36].


### 2.4. Harmonic Response in the Frequency Domain

Wave produces horizontal harmonic motion in the free field, as shown in Figure1. This is described by u(t)=Ugeiωt, and the associated free-field horizontal displacement is given by u(t)=Ueeiωt. One-dimensional site response analysis can be formulated as
(8)ρ∂2u∂t2=G∂2u∂z2+η∂3u∂z2∂t,
where ρ is the mass density, η is viscosity, and u(z,t) is displacement. Equation (8) is resolved in frequency domain analysis [37, 38], allowing the nodal relative displacement to be obtained.The current dynamic analysis employs time domain analysis and frequency domain analysis [21, 35, 39]. Frequency domain methods are widely used to estimate the dynamic impendences of the pile head. In the strong seismic motions, time domain method (involving the Newton-Raphson iteration and the Newmark method [40]) is used to obtain the nonlinear results. Equation (4) was resolved in time domain [28]. In contrast, to facilitate comparison with the rigorous FE method and BDWF model, (4) is resolved herein in the frequency domain by the following form:
(9)[E+Isd(EpIph4D-ω2M+iωCx)]Up=(E+iωIsCxd)Ue,
where i=-1; ω is the excitation frequency; Up is the amplitude of pile displacement; Ue is the amplitude of addition displacement in the free-field soil; and E is the identity matrix.A cut-off method [28, 33] is generally used to accommodate soil yield around the pile. If the pressure at the pile-soil interface exceeds the ultimate lateral pressure of soil, the excess pressure is redistributed to other segments through iteration until pressure at all pile nodes within the ultimate values. This study, however, will not consider this yield and will be confined to elastic analysis using the simplified approach for piles in two-layer soil.


## 2.1. Basic Assumptions for Pile and Soil Model

The one-dimensional model for a floating or end-bearing single pile embedded in a two-layer soil is shown in Figure1. The circle pile is assumed as linearly hysteretic beam having a length L, a diameter d, a mass density ρp, and a bending stiffness EpIp. The pile is discretized into n+1 segments of equal lengths h, but for a length of h/2 for the top and the tip segments, respectively. Each segment is subjected to a uniformly distributed load pi over the semicircular area. The pile is head restrained (fixed head) or free to rotate (free head), and sits above a bedrock. The linearly hysteretic soil profile is characterized by an upper-layer of thickness Ha and a shear wave velocity Va, which is underlain by a lower layer of thickness Hb and shear wave velocity Vb. The two layers have damping ratios βb and βa, mass densities ρb and ρa and Poisson’s ratios μa and μb. A shear wave propagates vertically through the free field soil, which induces the horizontal harmonic motion and horizontal displacements. The motion at the bedrock surface is expressed by the amplitude of either bedrock displacement Ug or the bedrock acceleration ω2Ug.Figure 1
Analysis model of a pile in a two layer soil profile subjected to vertically-propagating seismic SH waves.


## 2.2. Calculation of the Horizontal Displacement of the Pile

Kinematic response of a single pile is induced by the free-field soil displacement shown in Figure1. The Mindlin hypothesis does not meet the needs of dynamic analysis. However, the Mindlin equation is still valid for calculating elastic displacement and stress fields caused by a dynamic loading, provided that the characteristic wavelength in the soil is sufficiently long in comparison with the horizontal distance across the zone of major influence [28, 33], as is noted for nonhomogeneous soil by Poulos and Davis [32]. In the current, simplified BEM formulation, the soil displacement us (due to the pile-soil interface pressure) is gained using the Mindlin solution [34], which is then added together with free-field soil displacement
(1)us=Ispi+ue,
where Is is the soil-displacement-influence coefficient; pi is the vector of soil-pile interface pressure over the semi-circular area; and ue is the free-field soil displacement estimated using one-dimensional site response for vertically propagating shear waves through an unbounded medium [35].The dynamic equilibrium under steady-state conditions for the pile may be written in the following form using the finite-difference method [28]:
(2)EpIph4Dup+Mu¨p+Cx(u˙p-u˙e)=-dpp,
where up is the horizontal displacement of the pile, with the cap “·” indicating differentiation with time; pp is the vector of soil-pile interface pressure; D is the matrix of finite difference coefficients; M is the pile mass; and Cx is the soil radiation damping. Here, the soil damping is the same as that of the simplified boundary element model [28], and
(3)Cx=5dρsVs,
where Vs is the shear wave velocity of soil and ρs is the density of soil.The displacement compatibility between the pile and the adjacent soil offersus=up. Taking the displacement as up and substituting (1) into (2) result in the following:
(4)Isd[Mu¨p+Cxu˙p+EpIph4Dup]+up=ue+IsdCxu˙e.
Equation (4), together with the pile-top and -bottom boundary conditions, leads to n+5 unknown displacements, which involve the pile nodes from 1 to n+1 (see Figure 1) and 4 additional nodes at the pile top and tip. Equation (4) is resolved in either frequency or time domain and may attain the required accuracy using 21 segments of the pile [32].


## 2.3. Soil-Displacement-Influence Coefficients

Poulos and Davis [32] obtained soil-displacement-influence coefficients Is by integrating the Mindlin equation over a rectangular plane and taking the pile as a thin rectangular vertical strip. Ideally, a circular pile (rather than a thin strip) should be used. In elastic, semi-infinite space, the force P in the horizontal direction at a depth c induces a displacement component ux, which at any other point (x,y,z) is given by
(5)ux=PGf(μs,x,y,z,c),
where G is shear modulus of soil and μs is the Poisson ratio.The coefficients of the proposed method are obtained in two steps: firstly, (5) is integrated over a rectangular area from a depth c of c1 to c2 and across the pile width from -d/2 to d/2 in Figure 2 which offers
(6)ux(μs,x,y,z,c,s)=PG∫-d/2d/2∫c2c1f(μs,x,y-s,z,c)ds.Integrated along the semi-circular pile surface.
(a)
Integrated area
(b)
Weighted average areaSecondly, (6) is reexpressed in a cylindrical coordinate, which is then integrated over the semi-circular pile surface with respect to the angle θ (see Figure 2) to gain the horizontal component of the displacement. The soil-displacement-influence coefficients, taking as weighted average of the integrated horizontal displacements, are deduced as(7)Is=PG∫0π/2∫-d/2d/2∫c2c1f(μs,d2cosθ,d2sinθ-s,z,c)cosθdcdsdθ.Equation (7) is solved by numerical integration using an adaptive Lobatto rule [36].


## 2.4. Harmonic Response in the Frequency Domain

Wave produces horizontal harmonic motion in the free field, as shown in Figure1. This is described by u(t)=Ugeiωt, and the associated free-field horizontal displacement is given by u(t)=Ueeiωt. One-dimensional site response analysis can be formulated as
(8)ρ∂2u∂t2=G∂2u∂z2+η∂3u∂z2∂t,
where ρ is the mass density, η is viscosity, and u(z,t) is displacement. Equation (8) is resolved in frequency domain analysis [37, 38], allowing the nodal relative displacement to be obtained.The current dynamic analysis employs time domain analysis and frequency domain analysis [21, 35, 39]. Frequency domain methods are widely used to estimate the dynamic impendences of the pile head. In the strong seismic motions, time domain method (involving the Newton-Raphson iteration and the Newmark method [40]) is used to obtain the nonlinear results. Equation (4) was resolved in time domain [28]. In contrast, to facilitate comparison with the rigorous FE method and BDWF model, (4) is resolved herein in the frequency domain by the following form:
(9)[E+Isd(EpIph4D-ω2M+iωCx)]Up=(E+iωIsCxd)Ue,
where i=-1; ω is the excitation frequency; Up is the amplitude of pile displacement; Ue is the amplitude of addition displacement in the free-field soil; and E is the identity matrix.A cut-off method [28, 33] is generally used to accommodate soil yield around the pile. If the pressure at the pile-soil interface exceeds the ultimate lateral pressure of soil, the excess pressure is redistributed to other segments through iteration until pressure at all pile nodes within the ultimate values. This study, however, will not consider this yield and will be confined to elastic analysis using the simplified approach for piles in two-layer soil.


## 3. Validation of Simplified Method

### 3.1. Comparison with Dynamic Finite-Element Solution

The proposed simple approach is compared with dynamic FE results concerning a free head pile embedded in a two-layer soil deposit [19]. The pile-soil model is the same as that shown in Figure 1, except that the pile tip is extended into the underlying bedrock. The pile-soil system is featured by a ratio of soil layer thickness Ha/Hb of 1, a soil density ρa=ρb, a soil Poisson’s ratio μa=μb=0.4, and a soil damping coefficient βa=βb=10%. The pile has a slenderness ratio L/d of 20, a pile-to-soil stiffness ratio Ep/Ea of 5000, and a pile density ρp of 1.60 ρa (ρa=1900 kg/m3).A comparison between the simplified approach and FE solution [19] is presented in Figures 3(a) and 3(b), respectively, for the profiles of the pile deflection and bending moment amplitude at the natural frequency of soil deposit (ω=ω1). A good agreement is evident. The current, simplified approach can reveal the kinematic bending moments at the interface of the two layers, despite the ~20% overestimation of the maximum bending moment (against FE result) of the pile in Case D.Comparison of amplitude of pile deflections and bending moment between FE solution and simplified method in a two-layer soil (A:Vb/Va=0.58, B: Vb/Va=1, C: Vb/Va=1.73, and D: Vb/Va=3).
(a)
Deflections
(b)
Kinematic bending momentFigure4 shows the amplitude spectrum of maximum kinematic bending moment as a function of the frequency ratio ω/ω1. A good agreement is again observed between the simplified approach and the dynamic FE solution [19]. Both indicate that the peak kinematic bending moment occurs at the inherent frequency of the soil.Figure 4
Comparison of maximum bending moment amplitude between FE solution and simplified method in a two-layer soil (A:Vb/Va=0.58, B: Vb/Va=1, C: Vb/Va=1.73, and D: Vb/Va=3).


### 3.2. Comparison with BDWF Formulation

The proposed approach for kinematic loading along the pile depth during the lateral ground movements is compared with the BDWF solution [19]. The pile-soil system is the same as the case just discussed in the last section. To examine the sensitivity of the parameters, four groups of 12 cases (Table 1) were studied, by maintaining soil density ρa=ρb, soil Poisson’s ratio μa=μb=0.4, soil damping coefficient βa=βb=10%, and pile density ρp=1.60ρa.Table 1
Comparison of FE, BDWF, and the present simplified approach for dimensionless maximum kinematic pile bendingMmax(ω1)/ρdd4U¨g during the harmonic excitation.
Case
L
/
d
E
p
/
E
a
H
a
/
H
b
V
a
/
V
b
Previousω1d/Va
Computedω1d/Va
FE method
δ
comp
BDWF method
Error1: %
Simplified results
Error2: %
1
10
500
1
0.58
0.0964
0.0968
709
2.21
760
7.2
752
6.0
2
20
0.0482
0.0484
1253
2.41
1282
2.3
1326
5.8
3
40
0.0241
0.0242
2088
2.63
2196
5.2
2313
10.8
4
10
5000
1
0.0964
0.0968
1854
1.66
1623
-
12.5
1921
3.6
5
20
0.0482
0.0484
7718
1.81
8019
3.9
7735
0.2
6
40
0.0241
0.0242
13120
1.97
13320
1.5
13406
2.2
7
20
1000
3
1.5
0.0916
0.0910
508
2.27
515
1.4
555
9.2
8
20
10000
1/3
0.1164
0.1159
1113
1.42
1097
-
1.4
1119
0.6
9
20
1
0.1058
0.1056
931
1.55
1070
14.9
663
-
28.8
10
20
3
0.0913
0.0910
1482
1.70
1433
-
3.3
1547
4.4
11
10
500
1
1.73
0.2310
0.2293
86
2.05
79
-
8.1
96
11.6
12
20
0.1155
0.1147
269
2.24
253
-
5.9
315
17.2
13
40
0.0578
0.0573
656
2.44
577
-
12.0
778
18.7
14
10
5000
1
0.2310
0.2293
273
1.54
266
-
2.6
265
-
2.9
15
20
0.1155
0.1147
998
1.68
860
-
13.8
963
-
3.5
16
40
0.0578
0.0573
3054
1.83
2825
-
7.5
3144
3.0
17
20
1000
3
3
0.1017
0.1009
1046
2.17
1075
2.8
1331
27.3
18
20
10000
1/3
0.2117
0.2094
875
1.35
896
2.4
1217
39.1
19
20
3
0.1415
0.1403
2949
1.48
3042
3.2
3470
17.7
20
20
3
0.1019
0.1009
4867
1.63
4911
0.9
5608
15.2
21
20
1000
3
6
0.1042
0.1037
1486
2.07
1480
-
0.4
2077
39.7
22
20
10000
1/3
0.2897
0.2849
1128
1.29
1274
12.9
1827
62.0
23
20
1
0.1535
0.1526
3809
1.42
4193
10.1
5073
33.2
24
20
3
0.1045
0.1037
6730
1.55
6831
1.5
8675
28.9
(1) Values of previousω1d/Va, FE,δcomp, and BDWF computed form a proposed simple analytical expression [19].
(2) Error1: percentage error in the BDWF moment compared with the FE moment. Error2: percentage error in the elastic moment compared with the FE moment.
(3) The values of computedω1d/Va are associated with the computed time interval.As shown in Table1, the BDWF method adopts an optimized δ to obtain kinematic pile bending at the interface of two-layer soil. In contrast, the current method uses the displacement-influence coefficients Is to consider the pile-soil interaction (resembling the spring constant kx in the Winkler model) and may incorporate the interaction of soil along the pile to improve the accuracy.  Nevertheless, when the ratio of the shear wave velocities Vb/Va of two soil layers exceeds 3, a larger than 15% error (compared with the FE method) in maximum kinematic bending moment may be seen using the simplified method. This is discussed next concerning  Case 12 for kinematic bending at the two-layer interface and at the pile head (at the natural frequency of soil deposit).


#### 3.2.1. Kinematic Pile Bending

Figure5 shows the amplitude distributions of kinematic pile bending and shear force from the simplified method and the BDWF solutions. The kinematic pile bending and shear force profiles are slightly sensitive to the value δ. At δ=2.5, the distribution profiles of pile moment and shear force agree with each other between the simplified results and the BDWF solution, although the maximum moment of the simplified method is 17.2% larger than the BDWF solution.Comparison of amplitude of (a) bending moment profile and (b) shear force profile for two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).
(a)
(b)Figure6 provides the corresponding amplitude spectrum of maximum kinematic pile bending moment in the two-layer soil, owing to variation in the frequency ratio ω/ω1, among the FE, the BDWF, and the simplified approaches. It shows a consistent trend of variation in bending moment among various approaches.Figure 6
Comparison of maximum kinematic pile bending amplitude between FE, BDWF, and simplified approaches in a two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).


#### 3.2.2. Kinematic Response of Pile Head

Kinematic responses are obtained in form of the ratio of the amplitudes of pile-head displacementup(0) over the excitation motion ug or the ratio of the head displacement up(0) over the free field surface displacement ue(0) [17, 26, 39]. The responses for the free-head pile are plotted in Figure 7 for a spectrum of the frequency ratio ω/ω1. The good agreement of the factors up(0)/ug and up(0)/ue(0) among the simplified approach, the FE method, and the BDWF solution has been attributed to the predominant effect of the free field soil displacement.Normalized kinematic response of pile head to excitation among FE, BDWF, and simplified methods in a two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).
(a)
Amplitude of pile-top deflection
(b)
Pile-head to ground-surface displacementsAs for fixed-head piles, equally successful prediction is seen in Figure8, concerning the kinematic response of pile to soil displacement up(0)/ue(0) among the simplified method, the rigorous boundary integral method (Fan et al. [18]), and the BDWF (Makris and Gazetas [26]). Nevertheless, the current approach offers slightly larger ratio for Ep/Es=1000 than the other two solutions at high frequencies.Figure 8
Normalized response of a fixed-head pile among simplified approach, BDWF [26], and rigorous results [18] (L/d=20, ρs/ρp=0.7, βs=0.05, and μs=0.4).


## 3.1. Comparison with Dynamic Finite-Element Solution

The proposed simple approach is compared with dynamic FE results concerning a free head pile embedded in a two-layer soil deposit [19]. The pile-soil model is the same as that shown in Figure 1, except that the pile tip is extended into the underlying bedrock. The pile-soil system is featured by a ratio of soil layer thickness Ha/Hb of 1, a soil density ρa=ρb, a soil Poisson’s ratio μa=μb=0.4, and a soil damping coefficient βa=βb=10%. The pile has a slenderness ratio L/d of 20, a pile-to-soil stiffness ratio Ep/Ea of 5000, and a pile density ρp of 1.60 ρa (ρa=1900 kg/m3).A comparison between the simplified approach and FE solution [19] is presented in Figures 3(a) and 3(b), respectively, for the profiles of the pile deflection and bending moment amplitude at the natural frequency of soil deposit (ω=ω1). A good agreement is evident. The current, simplified approach can reveal the kinematic bending moments at the interface of the two layers, despite the ~20% overestimation of the maximum bending moment (against FE result) of the pile in Case D.Comparison of amplitude of pile deflections and bending moment between FE solution and simplified method in a two-layer soil (A:Vb/Va=0.58, B: Vb/Va=1, C: Vb/Va=1.73, and D: Vb/Va=3).
(a)
Deflections
(b)
Kinematic bending momentFigure4 shows the amplitude spectrum of maximum kinematic bending moment as a function of the frequency ratio ω/ω1. A good agreement is again observed between the simplified approach and the dynamic FE solution [19]. Both indicate that the peak kinematic bending moment occurs at the inherent frequency of the soil.Figure 4
Comparison of maximum bending moment amplitude between FE solution and simplified method in a two-layer soil (A:Vb/Va=0.58, B: Vb/Va=1, C: Vb/Va=1.73, and D: Vb/Va=3).


## 3.2. Comparison with BDWF Formulation

The proposed approach for kinematic loading along the pile depth during the lateral ground movements is compared with the BDWF solution [19]. The pile-soil system is the same as the case just discussed in the last section. To examine the sensitivity of the parameters, four groups of 12 cases (Table 1) were studied, by maintaining soil density ρa=ρb, soil Poisson’s ratio μa=μb=0.4, soil damping coefficient βa=βb=10%, and pile density ρp=1.60ρa.Table 1
Comparison of FE, BDWF, and the present simplified approach for dimensionless maximum kinematic pile bendingMmax(ω1)/ρdd4U¨g during the harmonic excitation.
Case
L
/
d
E
p
/
E
a
H
a
/
H
b
V
a
/
V
b
Previousω1d/Va
Computedω1d/Va
FE method
δ
comp
BDWF method
Error1: %
Simplified results
Error2: %
1
10
500
1
0.58
0.0964
0.0968
709
2.21
760
7.2
752
6.0
2
20
0.0482
0.0484
1253
2.41
1282
2.3
1326
5.8
3
40
0.0241
0.0242
2088
2.63
2196
5.2
2313
10.8
4
10
5000
1
0.0964
0.0968
1854
1.66
1623
-
12.5
1921
3.6
5
20
0.0482
0.0484
7718
1.81
8019
3.9
7735
0.2
6
40
0.0241
0.0242
13120
1.97
13320
1.5
13406
2.2
7
20
1000
3
1.5
0.0916
0.0910
508
2.27
515
1.4
555
9.2
8
20
10000
1/3
0.1164
0.1159
1113
1.42
1097
-
1.4
1119
0.6
9
20
1
0.1058
0.1056
931
1.55
1070
14.9
663
-
28.8
10
20
3
0.0913
0.0910
1482
1.70
1433
-
3.3
1547
4.4
11
10
500
1
1.73
0.2310
0.2293
86
2.05
79
-
8.1
96
11.6
12
20
0.1155
0.1147
269
2.24
253
-
5.9
315
17.2
13
40
0.0578
0.0573
656
2.44
577
-
12.0
778
18.7
14
10
5000
1
0.2310
0.2293
273
1.54
266
-
2.6
265
-
2.9
15
20
0.1155
0.1147
998
1.68
860
-
13.8
963
-
3.5
16
40
0.0578
0.0573
3054
1.83
2825
-
7.5
3144
3.0
17
20
1000
3
3
0.1017
0.1009
1046
2.17
1075
2.8
1331
27.3
18
20
10000
1/3
0.2117
0.2094
875
1.35
896
2.4
1217
39.1
19
20
3
0.1415
0.1403
2949
1.48
3042
3.2
3470
17.7
20
20
3
0.1019
0.1009
4867
1.63
4911
0.9
5608
15.2
21
20
1000
3
6
0.1042
0.1037
1486
2.07
1480
-
0.4
2077
39.7
22
20
10000
1/3
0.2897
0.2849
1128
1.29
1274
12.9
1827
62.0
23
20
1
0.1535
0.1526
3809
1.42
4193
10.1
5073
33.2
24
20
3
0.1045
0.1037
6730
1.55
6831
1.5
8675
28.9
(1) Values of previousω1d/Va, FE,δcomp, and BDWF computed form a proposed simple analytical expression [19].
(2) Error1: percentage error in the BDWF moment compared with the FE moment. Error2: percentage error in the elastic moment compared with the FE moment.
(3) The values of computedω1d/Va are associated with the computed time interval.As shown in Table1, the BDWF method adopts an optimized δ to obtain kinematic pile bending at the interface of two-layer soil. In contrast, the current method uses the displacement-influence coefficients Is to consider the pile-soil interaction (resembling the spring constant kx in the Winkler model) and may incorporate the interaction of soil along the pile to improve the accuracy.  Nevertheless, when the ratio of the shear wave velocities Vb/Va of two soil layers exceeds 3, a larger than 15% error (compared with the FE method) in maximum kinematic bending moment may be seen using the simplified method. This is discussed next concerning  Case 12 for kinematic bending at the two-layer interface and at the pile head (at the natural frequency of soil deposit).


### 3.2.1. Kinematic Pile Bending

Figure5 shows the amplitude distributions of kinematic pile bending and shear force from the simplified method and the BDWF solutions. The kinematic pile bending and shear force profiles are slightly sensitive to the value δ. At δ=2.5, the distribution profiles of pile moment and shear force agree with each other between the simplified results and the BDWF solution, although the maximum moment of the simplified method is 17.2% larger than the BDWF solution.Comparison of amplitude of (a) bending moment profile and (b) shear force profile for two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).
(a)
(b)Figure6 provides the corresponding amplitude spectrum of maximum kinematic pile bending moment in the two-layer soil, owing to variation in the frequency ratio ω/ω1, among the FE, the BDWF, and the simplified approaches. It shows a consistent trend of variation in bending moment among various approaches.Figure 6
Comparison of maximum kinematic pile bending amplitude between FE, BDWF, and simplified approaches in a two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).


### 3.2.2. Kinematic Response of Pile Head

Kinematic responses are obtained in form of the ratio of the amplitudes of pile-head displacementup(0) over the excitation motion ug or the ratio of the head displacement up(0) over the free field surface displacement ue(0) [17, 26, 39]. The responses for the free-head pile are plotted in Figure 7 for a spectrum of the frequency ratio ω/ω1. The good agreement of the factors up(0)/ug and up(0)/ue(0) among the simplified approach, the FE method, and the BDWF solution has been attributed to the predominant effect of the free field soil displacement.Normalized kinematic response of pile head to excitation among FE, BDWF, and simplified methods in a two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).
(a)
Amplitude of pile-top deflection
(b)
Pile-head to ground-surface displacementsAs for fixed-head piles, equally successful prediction is seen in Figure8, concerning the kinematic response of pile to soil displacement up(0)/ue(0) among the simplified method, the rigorous boundary integral method (Fan et al. [18]), and the BDWF (Makris and Gazetas [26]). Nevertheless, the current approach offers slightly larger ratio for Ep/Es=1000 than the other two solutions at high frequencies.Figure 8
Normalized response of a fixed-head pile among simplified approach, BDWF [26], and rigorous results [18] (L/d=20, ρs/ρp=0.7, βs=0.05, and μs=0.4).


## 3.2.1. Kinematic Pile Bending

Figure5 shows the amplitude distributions of kinematic pile bending and shear force from the simplified method and the BDWF solutions. The kinematic pile bending and shear force profiles are slightly sensitive to the value δ. At δ=2.5, the distribution profiles of pile moment and shear force agree with each other between the simplified results and the BDWF solution, although the maximum moment of the simplified method is 17.2% larger than the BDWF solution.Comparison of amplitude of (a) bending moment profile and (b) shear force profile for two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).
(a)
(b)Figure6 provides the corresponding amplitude spectrum of maximum kinematic pile bending moment in the two-layer soil, owing to variation in the frequency ratio ω/ω1, among the FE, the BDWF, and the simplified approaches. It shows a consistent trend of variation in bending moment among various approaches.Figure 6
Comparison of maximum kinematic pile bending amplitude between FE, BDWF, and simplified approaches in a two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).


## 3.2.2. Kinematic Response of Pile Head

Kinematic responses are obtained in form of the ratio of the amplitudes of pile-head displacementup(0) over the excitation motion ug or the ratio of the head displacement up(0) over the free field surface displacement ue(0) [17, 26, 39]. The responses for the free-head pile are plotted in Figure 7 for a spectrum of the frequency ratio ω/ω1. The good agreement of the factors up(0)/ug and up(0)/ue(0) among the simplified approach, the FE method, and the BDWF solution has been attributed to the predominant effect of the free field soil displacement.Normalized kinematic response of pile head to excitation among FE, BDWF, and simplified methods in a two-layer soil (Case 12:Vb/Va=1.73, Ep/Ea=500, Ha/Hb=1, and L/d=20).
(a)
Amplitude of pile-top deflection
(b)
Pile-head to ground-surface displacementsAs for fixed-head piles, equally successful prediction is seen in Figure8, concerning the kinematic response of pile to soil displacement up(0)/ue(0) among the simplified method, the rigorous boundary integral method (Fan et al. [18]), and the BDWF (Makris and Gazetas [26]). Nevertheless, the current approach offers slightly larger ratio for Ep/Es=1000 than the other two solutions at high frequencies.Figure 8
Normalized response of a fixed-head pile among simplified approach, BDWF [26], and rigorous results [18] (L/d=20, ρs/ρp=0.7, βs=0.05, and μs=0.4).


## 4. Application under Seismic Excitation

### 4.1. Seismic Motion and Case Model

The harmonic steady state is rarely seen in a practical engineering design. Kinematic seismic response of a pile should be tailored to cater for the transient excitation [19] owing to earthquake shaking, as is noted in dynamic analysis involving nonlinear pile-soil interaction. A large soil resistance in certain depth may render the nonlinear kinematic pile bending (e.g., caused by SH wave) insignificant compared to pile-head inertial excitation. In addition, previous study [13] does not allow either a detrimental or a beneficial effect on kinematic pile bending to be concluded due to non-linear site response. Consequently, the validity of the simplified method is examined herein for elastic pile and soil.The performance of the simplified approach for the transient response is examined for six typical seismic accelerograms (see Table2), which include 4 actual records selected from ground motion database of the Pacific Earthquake Engineering Research Center (PEER) [41] and 2 artificial motions used in seismic design of a typical site in Shanghai (China). The acceleration time histories for seismic events are plotted in Figure 9 and the associated acceleration response spectra are provided in Figure 10.Table 2
Ground motions employed in the parametric analysis.
Earthquake
Record label
M
w
PGA (g)
T
p (sec)
T
m (sec)
Whittier Narrows 1987/10/01
A-TOR180
6.0
0.05
0.24
0.42
Kobe 1995/01/16
KJM000
6.9
0.82
0.34
0.64
Imperial Valley 1940/05/19
I-ELC180
7.0
0.31
0.46
0.53
ShangHai3-Elcentrol (IV Site)
Artificial3
0.35
0.58
0.82
Friuli 1976/05/06
A-TMZ270
6.5
0.32
0.64
0.50
ShangHai2 (IV Site)
Artificial2
0.35
0.72
0.75
T
p: predominant period and Tm: mean period.Acceleration time history of the input motions at the bedrock roof.
(a)
(b)Figure 10
Acceleration response spectra (5% damping) of the input motions.


### 4.2. Parametric Investigation

The parametric analysis is again conducted for the pile-soil system shown in Figure1, with the following profile parameters: bedrock located at H=30 m, density of either soil layer = 1900 kg/m3, Poisson's ratio = 0.4, soil damping = 5%, pile density = 2500 kg/m3, and pile-to-soil stiffness ratio Ep/Ea=1000. The shear wave velocity is 100 m/s for the upper soil layer, 150 m/s for the lower layer, and 1000 m/s for the bedrock, respectively. The input signals by Sica et al. [31] were scaled in amplitude to a peak acceleration of 0.35 g [31]. Figure 11 shows the comparison of kinematic pile bending between the simplified proposed approach and the BDWF formulation [31]. A good agreement in the predicted bending moment is evident between the simplified method and the BDWF solution, but for the large difference in the peak bending moment around the layer interface and at a shear wave velocity ratio Va/Vb of 1/3.Figure 11
Validation of the proposed procedure under the input motion A-TMZ270 (L/d=33.3, Ha/Hb=1, Ep/Ea=470, ρs/ρp=0.76, and μa=μb=0.4).Figure12 shows the pile-diameter contrast on kinematic bending moment under the six input motions. The ratio of soil layer thickness Ha/Hb is 1, the pile is 20 m in length and 0.6, 0.9, or 1.2 m in diameter, and the head is constrained against rotation (fixed head). The figure indicates that small-diameter piles accommodate more easily to seismically induced soil deformations than lager-diameter piles. Kinematic bending moments at the pile head and the interface of soil layers are nearly proportional to the diameters. This will reduce the safety of pile head, although it does not necessarily increase or reduce the seismic safety of the pile body (depending on the circumstances), as is discussed previously [10, 14].Pile diameters contrast on kinematic pile bending moment under different input motions.
(a)
(b)
(c)Figure13 presents the bending moment profiles owing to variation in the depths of the interface of the two layers under the six input motions. They were obtained for a fixed head pile (20 m in length, 0.6 m in diameter) embedded in a soil with a layer interface located at a depth of 5, 10, or 15 m, respectively. An increase in depth of the interface renders increase in the peak value of kinematic pile bending but has a negligible impact on the kinematic bending moment at the pile head. This preliminary analysis suggests the simplified approach has the potential in modeling kinematic seismic response of piles during the earthquake.Depths of the interface between the two layers contrast on kinematic pile bending moment under different input motions.
(a)
(b)
(c)


## 4.1. Seismic Motion and Case Model

The harmonic steady state is rarely seen in a practical engineering design. Kinematic seismic response of a pile should be tailored to cater for the transient excitation [19] owing to earthquake shaking, as is noted in dynamic analysis involving nonlinear pile-soil interaction. A large soil resistance in certain depth may render the nonlinear kinematic pile bending (e.g., caused by SH wave) insignificant compared to pile-head inertial excitation. In addition, previous study [13] does not allow either a detrimental or a beneficial effect on kinematic pile bending to be concluded due to non-linear site response. Consequently, the validity of the simplified method is examined herein for elastic pile and soil.The performance of the simplified approach for the transient response is examined for six typical seismic accelerograms (see Table2), which include 4 actual records selected from ground motion database of the Pacific Earthquake Engineering Research Center (PEER) [41] and 2 artificial motions used in seismic design of a typical site in Shanghai (China). The acceleration time histories for seismic events are plotted in Figure 9 and the associated acceleration response spectra are provided in Figure 10.Table 2
Ground motions employed in the parametric analysis.
Earthquake
Record label
M
w
PGA (g)
T
p (sec)
T
m (sec)
Whittier Narrows 1987/10/01
A-TOR180
6.0
0.05
0.24
0.42
Kobe 1995/01/16
KJM000
6.9
0.82
0.34
0.64
Imperial Valley 1940/05/19
I-ELC180
7.0
0.31
0.46
0.53
ShangHai3-Elcentrol (IV Site)
Artificial3
0.35
0.58
0.82
Friuli 1976/05/06
A-TMZ270
6.5
0.32
0.64
0.50
ShangHai2 (IV Site)
Artificial2
0.35
0.72
0.75
T
p: predominant period and Tm: mean period.Acceleration time history of the input motions at the bedrock roof.
(a)
(b)Figure 10
Acceleration response spectra (5% damping) of the input motions.


## 4.2. Parametric Investigation

The parametric analysis is again conducted for the pile-soil system shown in Figure1, with the following profile parameters: bedrock located at H=30 m, density of either soil layer = 1900 kg/m3, Poisson's ratio = 0.4, soil damping = 5%, pile density = 2500 kg/m3, and pile-to-soil stiffness ratio Ep/Ea=1000. The shear wave velocity is 100 m/s for the upper soil layer, 150 m/s for the lower layer, and 1000 m/s for the bedrock, respectively. The input signals by Sica et al. [31] were scaled in amplitude to a peak acceleration of 0.35 g [31]. Figure 11 shows the comparison of kinematic pile bending between the simplified proposed approach and the BDWF formulation [31]. A good agreement in the predicted bending moment is evident between the simplified method and the BDWF solution, but for the large difference in the peak bending moment around the layer interface and at a shear wave velocity ratio Va/Vb of 1/3.Figure 11
Validation of the proposed procedure under the input motion A-TMZ270 (L/d=33.3, Ha/Hb=1, Ep/Ea=470, ρs/ρp=0.76, and μa=μb=0.4).Figure12 shows the pile-diameter contrast on kinematic bending moment under the six input motions. The ratio of soil layer thickness Ha/Hb is 1, the pile is 20 m in length and 0.6, 0.9, or 1.2 m in diameter, and the head is constrained against rotation (fixed head). The figure indicates that small-diameter piles accommodate more easily to seismically induced soil deformations than lager-diameter piles. Kinematic bending moments at the pile head and the interface of soil layers are nearly proportional to the diameters. This will reduce the safety of pile head, although it does not necessarily increase or reduce the seismic safety of the pile body (depending on the circumstances), as is discussed previously [10, 14].Pile diameters contrast on kinematic pile bending moment under different input motions.
(a)
(b)
(c)Figure13 presents the bending moment profiles owing to variation in the depths of the interface of the two layers under the six input motions. They were obtained for a fixed head pile (20 m in length, 0.6 m in diameter) embedded in a soil with a layer interface located at a depth of 5, 10, or 15 m, respectively. An increase in depth of the interface renders increase in the peak value of kinematic pile bending but has a negligible impact on the kinematic bending moment at the pile head. This preliminary analysis suggests the simplified approach has the potential in modeling kinematic seismic response of piles during the earthquake.Depths of the interface between the two layers contrast on kinematic pile bending moment under different input motions.
(a)
(b)
(c)


## 5. Conclusion and Discussions

A simple approach is formulated to predict the elastic, kinematic pile bending during harmonic or transient excitation. The approach employs a circular pile (rather than a simplified thin strip). The kinematic response of a pile embedded in two-layer soil is resolved in the frequency domain owing to specified soil displacement field. The simplified approach is generally valid to nonhomogeneous soil problems, in light of the good comparison with the dynamic FE method and BDWF solution. The main conclusions from the study are as follows.The simplified method employs the soil-displacement-influence coefficientsIs to consider the pile-soil interaction (resembling the spring constant kx in the BDWF). It provides conservative estimation of maximum kinematic bending moments at soil-layer interface (with a sharper stiffness contrast) despite an adequate accuracy in general. The accuracy may be improved by incorporating the interaction of soil into the soil-displacement-influence coefficients Is for such cases with Vb/Va<3.The parametric studies during the seismic excitation show the impact of pile diameters and depths of the interface of two layers and demonstrate the simplified approach comparing well with published results in gaining kinematic pile bending during the earthquake.The formulated simple method is intended for elastic soil, elastic pressure on pile-soil interface, and elastic seismic response of the free field soil, for which a judicious choice of elastic modulus of the soil is required. The method may be extended to elastoplastic case though introducing a similar technology as cut-off method [28, 33] in the current procedure.



---

*Source: 241482-2013-09-02.xml*