seismic strain wedge model to analyze single piles under ... · seismic strain wedge model to...

SEISMIC STRAIN WEDGE MODEL TO ANALYZE SINGLE PILES

UNDER LATERAL SEISMIC LOADING

Aslan SADEGHI HOKMABADI 1, Ali FAKHER

2, Behzad FATAHI

3

ABSTRACT

One of the most effective methods of analyzing a single pile and pile groups under lateral loading is Strain

Wedge Model (SWM). SWM has a number of advantages in comparison with traditional p-y curves, but

this method traditionally could be used to analyze piles under monotonic loads. In the present paper,

SWM has been modified to consider dynamic lateral loading. Based on this new method called Seismic

Strain Wedge Model (SSWM), a Lateral Analysis of Piles (LAP) computer code has been developed.

Using LAP, some case studies have been analyzed and the results show good agreement with test data.

This paper introduces SSWM as a simple and powerful solution to analyze piles under lateral seismic

loading.

Keywords: Strain Wedge Model, Seismic loading, lateral behavior, Single pile

INTRODUCTION

Lateral vibration of piles is an important consideration in the design of piled structures subjected to

dynamic excitations due to earthquake, wind, operation of machines and waves in offshore environment

(Das & Sargand, 1999). The seismic response of pile foundations is a complex process involving inertial

interaction between structure and pile foundation, kinematics interaction between piles and soil, and the

non-linear response of soil to strong earthquake motions (Finn, 2005). Several methods have been

developed for dynamic response analysis of pile foundations listed in Table 1. Depending on how piles

and soils are modeled, these methods may be broadly classified into Continuum-base approaches and

Winkler (or subgrade reaction) approaches.

In the Continuum-base approaches, soil has been modeled as continuum media several soil properties are

required as input for the analysis. Due to complexity and unavailability of soil properties, the application

of methods in the second category, subgrade reaction approaches, is more common. As an example, the

load-transfer method models the pile as an elastic member and the soil is viewed as distributed springs and

dashpots with constant or frequency dependent concentrated at a finite number of nodes. The spring

constants are obtained from analytical calculation or experimental data. The major advantage of this

approach lies in its ability to simulate nonlinearity, heterogeneity, and hysteretic degradation of the soil

surrounding the pile by simply changing the spring and dashpot constants (Das & Sargand, 1999). In

contrast, the main drawback of load-transfer approach is idealization of the soil continuum with discrete

soil reactions (Allotey & El Naggar, 2008). In other words, the shear transfer between the springs is one of

1 PhD Student, School of Civil and Environmental Engineering, University of Technology Sydney (UTS),

Sydney, Australia, e-mail: [email protected] 2 Associate Professor, School of Civil Engineering, College of Engineering, University of Tehran, Iran. 3 Assistant Professor, School of Civil and Environmental Engineering, University of Technology Sydney

(UTS), Sydney, Australia.

Gustavo

Paper No. UHSSA

5th International Conference on Earthquake Geotechnical Engineering January 2011, 10-13

Santiago, Chile the obviously missing fundamental mechanisms in Winkler models (Finn, 2005). Strain Wedge Model

(SWM) overcomes this drawback of Winkler methods by creating relationship between the three-

dimensional responses of soil-pile to the Beam on Elastic Foundation parameters (Ashour et al., 1998).

However, the current SWM analysis is limited to piles under monotonic loads. In the presented research

SWM is modified in order to include dynamic loads.

Table 1. Summary of the lateral seismic analysis of piles methods

Methods Reference Remarks

Equivalent Cantilever

Davisson & Robinson (1965);

Nair et al. (1969)

Pile is reduced to an equivalent cantilever; No information can be obtained on the moment, stresses, and displacements along the length of the pile; Only good for initial analysis.

Beam on Elastic Foundation

Briaud & Tucker (1984)

Presents the soil inertial force on the Winkler springs stiffness, and the pile inertial force and soil damping as an additional force at each node. A new governing equation of Beam on Elastic Foundation (Including inertial force and damping) is solved with the Finite Deference Method.

Novak’s Analysis Method

Novak (1974) This approach derives stiffness and damping constants for piles, and thus enables the use of the simple “lumped-parameter” approach.

Column of Soil Parmelee et al.(1964); Idress & Seed (1968)

Soil mass is lumped at discrete points along the depth of the each layer and these masses linked by springs and Dashpots connected in parallel; Earthquake effect is presented as a horizontal acceleration at the base; It is assumed that soil and pile have the same displacement in each node (the pile is moved essentially with the soil) which is not acceptable in layered soils.

P-y Curves Matlock (1970); Reese et al. (1974)

Pile is modelled as an elastic member and the soil is modelled as a series of nonlinear springs (p-y curves); This is a semi-empirical method that accounts for dynamic loads by modifying the static p-y curves based on empirical data. (different researchers suggests different curves)

Springs and Dashpots to model soil

El-Naggar & Bently (2000); Tavaraj(2001);

Gerolymos & Gazetas(2006)

Pile is modelled as an elastic member, and soil is modelled as a series of springs and dashpots (different models are suggested); These methods, based on the determination of the near- and far-field responses, can be grouped into coupled and uncoupled models.

Continuum-Base Approaches

Kuhlemeyer(1979); Banerjee et al.(1987); Brown & Shie(1990)

Soil is modelled as a continuum media and different numerical approaches such as the Finite Element Method or Boundary Element Method are used to solve the governing equations.

METHOD OF ANALYSIS

To develope general equations of SSWM, three methods including Horizontal Slice Method, Pseudo-

Dynamic Method, and Strain Wedge Model are used and combined. Horizontal Slice Method presents the

idea of dividing passive soil into horizontal slices; Pseudo-Dynamic Method is used to calculate the

additional earthquake force applied at each slice or each depth along the pile; and Strain Wedge Model is

employed to determine the soil resistance against the lateral loading by the three-dimensional passive


Santiago, Chile wedge of the soil in front of the pile. In this section, each of these three methods are described, and then

the formulation of SSWM is presented.

Horizontal Slice Method Horizontal Slice Method (HSM) was initially developed for seismic stability analysis of reinforced slopes

and walls (Shahgholi et al., 2001). HSM is a limit equilibrium method assuming a failure surface with a

failure wedge divided into a number of horizontal slices (Figure1). Because of horizontal slices, the failure

surface does not intersect reinforcements; accordingly the reinforcements have no direct influence on inter-

slice forces. According to Nouri et al. (2006), force and moment equilibrium equations for each slice of the

whole sliding wedge can be satisfied.

Figure 1. Horizontal Slice Method: (a) dividing soil in to horizontal slices, (b) Forces acting on a

single horizontal slice containing reinforcement (after Shahgholi et al., 2001)

Pseudo-Dynamic Method

Pseudo-static methods are commonly used for solving various design problems associated with pile

foundations subjected to earthquake motions (Finn, 2005). However, the pseudo-static methods consider

the dynamic nature of earthquake loading very approximately (Kramer, 1996), and do not consider the

effects of time and body waves travelling through the soil during the earthquake. The phase difference due

to finite shear wave propagation can be considered using a new method, called Pseudo-dynamic method

(Steedman & Zeng, 1990). In the Pseudo-dynamic method it is assumed that the shear modulus is constant

with depth along the pile. Therefore, the following equations can be written:

2/1)/( GVs ! (1)

2/1)]21(/)22([ " # $$! GVp (2)

sVLT /4/2 !! %& (3)

where, Vs is shear wave velocity (m/s), G is shear modulus of soil (N/m2), is density of soil (kg/m3), Vp

is the primary wave velocity (m/s), ! is Poisson’s ratio, T is period of lateral shaking (s), and " is angular

frequency of base shaking (rad/s). In Pseudo-dynamic method, only the phase and not the amplitude of the

acceleration is varying. In addition, in this method it is assumed that both the horizontal and vertical

vibrations, with accelerations ah and av, respectively, start at exactly the same time, and there is no phase

shift between these two vibrations resulting in the critical condition for design (Nimbalkar et al., 2006).

However, the seismic acceleration is considered as harmonic sinusoidal acceleration, which is one of the

limitations of the original pseudo-dynamic method proposed by Steedman & Zeng (1990).

Vi

Hi+1

x

y

Ti

Ni

Si

(1+Kv)WiKhWi

L

Li+1

(b)(a)

Reinforcement

elements

Failure surface

Vi+1


Santiago, Chile As shown in Figure 2, if the base is subjected to harmonic horizontal and vertical seismic accelerations

with amplitudes ah and av, the horizontal and vertical accelerations at depth x below the ground surface at

time t can be express as:

)(sin),(s

hhV

xLtatxa

$$! %

(4)

)(sin),(p

vvV

xLtatxa

$$! %

(5)

In the SSWM, the above accelerations are employed to calculate the additional earthquake force applied at

each depth along the pile.

Figure 2. Pseudo-dynamic method for slopes

Strain Wedge Model

Non-linear springs (p-y) are widely used to analyze single piles under static lateral loads. However, the p-

y curve employed does not consider soil continuity and pile properties such as pile stiffness, pile cross

section, and pile head conditions (Reese, 1983). SWM overcomes these limitations and allows the

assessment of the p-y response of laterally loaded piles based on the three-dimensional soil-pile

interaction and its dependence on both soil and pile properties (Norris & Abdollaholiae, 1985).

In the SWM, the soil resistance against the lateral loading is determined by the three-dimensional passive

wedge of soil that develops in front of the pile. As shown in Figure 3, geometry of the mobilized passive

wedge is characterized by base angle, m; the current passive wedge depth h; and the wedge fanning angle,

!m. All of these parameters are the functions of soil horizontal strain in each step of loading. Therefore,

the geometry of the soil wedge is a function of the strain level in soil (i.e level of loading). The complete

formulation of the SWM is addressed by Ashour et al. (1998 & 2004).

av

av

ah (x,t)

av (x,t)

x


Santiago, Chile

Figure 3. Passive wedge of soil as developed in front of the pile in the SWM (Ashour et al., 1998)

The SWM analysis maintains two types of stability, local stability of the soil sublayer (horizontal slice);

and global stability of the pile and the passive soil wedge. According to Hetenyi (1946) for the global

stability of the system, the governing equation of Beam on Elastic Foundation (Equation 6) should be

satisfied.

0)()()(

2

2

4

4

!'' yEdx

ydP

dx

ydEI sx

(6)

where, EI is pile stiffness (N. m2), y is pile deflection at depth x (m), x is vertical distance between any

point in soil mass and external borders of soil mass (m), Px is applied axial load at the pile head (N), and

Es is modulus of subgrade reaction (N/m2). On the other hand, local stability is performed by dividing the

soil layer into thin sublayers and the equilibrium of forces acting on each sublayer (slice) should be

satisfied. As shown in Figure 4, the existing loads are horizontal stresses in front of the wedge (#$h), pile

side shear stresses ((), soil-pile reaction forces (p), and the wedge side forces (F). It should be noted that

the wedge side forces are in equilibrium in the normal direction.

Figure 4. Geometry and equilibrium of each sublayer in SWM

B C

P

!"h

F F #m


Santiago, Chile

SEISMIC STRAIN WEDGE MODEL (SSWM)

The Seismic Strain Wedge Model (SSWM) is developed by considering the earthquake horizontal

acceleration (ah) and vertical acceleration (av) at the base of the soil-pile system (Figure 5). Using pseudo-

dynamic method, acceleration at each depth along the pile at particular time can be obtained from

Equations (4) and (5). It is assumed that all the assumptions of SWM are valid in SSWM. Therefore, the

soil medium in the passive wedge represents the soil response to the lateral loads. As shown in Figure 5b,

the soil mass of ith sublayer can be calculated as below:

))(tan(3 xhL m $! )

(7)

ii H

DBCLV )

2

)(3

'! (8)

iii Vm *! (9)

ii HDBC

Lm +,

-./

0 '!

23 (10)

where, L3 is Horizontal length of wedge (m), %m is passive wedge angle (rad), h is depth of passive wedge

(m), Vi is the volume of ith sublayer (m3), BC is front width of the soil wedge (m), D is the pile diameter

(m), Mi is the mass of ith sublayer (kg), and Hi is thickness of the soil sublayer.

Figure 5. Development of Seismic Strain Wedge Model (SSWM)

Using the concept of Horizontal Slice Method (HSM), the additional horizontal force that is generated by

the earthquake in ith layer (Pdhi) is,

ih

dhi Wg

txaP *!

),( (11)

ah (x,t)

av (x,t)

x

av

ah

1

1

3

i

n

H

L3

BC

D


Santiago, Chile

ihidhi WKP *! (12)

where, Khi is seismic acceleration coefficient in horizontal direction, g is acceleration due to gravity, and

Wi is the weight of soil in the wedge at each sublayer (N). As shown in Figure 6, for the equilibrium of the

ith layer, local stability should be satisfied by including the induced earthquake load (Pdhi).

Figure 6. Local equilibrium of a soil-pile sub layer in the presence of seismic loads

The critical condition occurs when Pdhi is in the same direction as the static lateral soil-pile interaction

load (Figure 5) Thus:

dhisii PPP '! (13)

21 2)( DSSBCPihsi (1 '2!

(14)

where, Psi is the static lateral soil-pile interaction load, Pi is the total lateral soil-pile reaction (static plus

seismic), S1 and S2 are pile shape factors, and $hi is the horizontal stress change in passive wedge in front

of pile. Similarly, the additional vertical force induced by the earthquake in the ith sublayer is obtained by:

i

vdvi W

g

txaP *!

),( (15)

ividvi WKP *! (16)

where, Pdv is the vertical seismic force in the ith sublayer (N) and kvi is seismic acceleration coefficient in

vertical directions. As shown in Figure 5, the upward vertical acceleration is the critical direction (this

assumption is explained further in the following section). The upward vertical acceleration, which is

critical, contributes to reduce the soil effective vertical stress as explained below,

3$

!

$!21

1

)(m

i i

dviiimvd

A

PH 41 (17)

3$

! '

'*

$!21

13

3

)2

(

)2

(

)(m

i

iivi

iimvd DBCL

HDBC

LK

H

441 (18)

iivi

m

i

mvd HK 41 )1()(1

1

$!2 3$

!

(19)

where, ($vd)m is the total (static and seismic) effective vertical stress (N/m2). Pseudo-static condition is

occurred when no phase change in the body waves travelling through the soil medium is considered.

Under this condition, accelerations ah(x,t) and av(x,t), in all of the sublayers are assumed to be constant,

thus:

B C

Pdhi

Psi

!"h


Santiago, Chile

ihih

ih

vdhi

vWKW

g

aW

g

txap

ss

!!!5656

)),(

(lim)(lim (20)

33 3$

!

$

!

$

!565656

$!$!$!21

1

1

1

1

1

)1()1(lim)),(

1(lim)(limm

i

iiv

m

i

m

i

iiv

vii

v

vmvd

vHKH

g

aH

g

txa

ppp

4441 (21)

To compare the results of earthquake modelling with pseudo-static and pseudo-dynamic method, a case

study is present. The algorithm of solution with the SSWM and the developed computer code (LAP) is

described in the following section.

LATERAL ANALYSIS OF PILES (LAP) CODE

Lateral Analysis of Piles (LAP) computer code was initially developed by the authors to analyse the

lateral behaviour of single piles and pile groups under static loading (Fakher et al., 2009). LAP allows the

use of different methods such as SWM to analyse piles under lateral static loading (Hokmabadi et al.,

2009). In this study, LAP has been modified to analyse piles under seismic lateral loading using SSWM.

A detailed flowchart of calculation method is presented in Figure 7. The flowchart is a modification of the

original development by Ashour et al. (2004) for SWM.

Figure 7. Flowchart of Seismic Strain Wedge Method for a single pile

Input DATA

Soil static properties (&, ', (50, Su and layer thickness)

Pile Properties (EI, D, L, Targeted deflection, Y0)

Soil dynamic properties (G, !)

Loading (M, L, Px, PT, ah, av)

Dividing soil layers into thin soil sublayers (i)

Assume lateral strain (() in soil in front of the pile,

#t=T/Nt, t= #t

Based on ah, av , and Eq. (4),(5) calculate ah(x,t),

av(x,t) and Pdhi and (#$dv)i at time t

Using the current profile of Es, the laterally loaded

pile is analyzed by Eq. 3 under an arbitrary pile-

head lateral static load (PT). The pile head

deflection (Y0) and h assumed using BEF analysis

are compared to those of the SWM analysis.

IF

(h)SWM = (h)BEF

(Y0)SWM = (Y0) BEF

Final (converged) geometry of the

passive wedge ('m, %m, BC in each soil

sublayer i and h) and the associated (E

and Es profile,Pxt, and Yot) of the pile

NO

STOP

Based on ( (assumed), (50, $vo(Ashour et al. 98)

calculate #$h, SL, 'm, h (assume h in the first trial),

BC, and Es in each sublayer i (i.e. Es profile along

the pile) at #$h = $d and (

IF

Tt 7

TtYY t 77! 0),max( 00

TtPP t 77! 0),max( 00

Yes

NO

Yes

t=t+ t

START

(h)new and (P0i)new


Santiago, Chile

VERIFICATION

In order to verify the SSWM, a centrifuge test results on a single pile carried out by Finn & Gohl (1987)

are used. In the prototype model, a single pile with a diameter of 0.57 meter and a length of 12.9 meter

was subjected to a base motion of the maximum horizontal acceleration of 0.158g, and the pile head

acceleration and the maximum moment along the pile were measured. The pile was embedded in sand

with '=30 degrees, 4 =14.7 kN/m3, and e0=0.78. Details of the test and the prototype model are explained

further by Tabesh (1997).

The above mentioned piles are analysed by LAP including SSWM. Figure 8 presents the results of these

analyses for four set of data including measurement data, Pseudo-Dynamic analysis, Pseudo-Static

analysis, and analysis using SAPAP program performed by Tabesh (1997). As shown in Figure 8, the

results of analysis by LAP using SSWM are in good agreement with the measured maximum moment

along the pile and the location of the maximum point. It should be mentioned that SSWM uses a simple

method to consider the seismic behaviour of the system by including the minimum number of parameters.

Parametric study is performed in order to investigate the affect of base acceleration on the pile head

displacement. Various weights on piles head and different input base acceleration are included in this

parametric study (Figure 9).

Figure 8 Measured and calculated values of maximum moment along the pile


Santiago, Chile

Figure 9. Effect of base acceleration on the pile head displacement using the SSWM

As shown in Figure 9, larger horizontal base acceleration (Kh) causes larger pile head displacement. Based

on Equation (4), when the horizontal base acceleration (Kh) increases, the horizontal seismic acceleration

at each depth at time t increases, resulting in increasing of additional earthquake horizontal force in each

layer (Pdhi) (Equations 11 & 12). Therefore, considering the local equilibrium at each sublayer (Figure 6)

and the governing equation of beam on elastic foundation (Equation 6), increasing of earthquake

horizontal force in each layer (Pdhi) causes larger pile head displacement.

Larger upward vertical base acceleration (Kv) increases the pile head displacement as well. According to

Equation (5), when the vertical base acceleration (Kv) increases, the vertical seismic acceleration at each

depth at the time t increases resulting in the increasing of the additional earthquake horizontal force in

each layer (Pdhi) (Equations 15 & 16). Therefore, according to Equation (19), the soil effective vertical

stresses decrease. Reduction in the soil effective vertical stresses results in decreasing of the springs’

stiffness (soil resistance) against the lateral loading in SWM and increasing of pile head displacement.

However, the influence of the vertical base acceleration on the pile head displacement is less than the

horizontal base acceleration.

CONCLUSION

In this study, a new method called Seismic Strain Wedge Model (SSWM) is proposed for analysing piles

under lateral seismic loads. The main formulation of this method is based on SWM and the additional

Earthquake force is calculated using Pseudo-Dynamic Method. Verification of the model with a case study

shows good agreement between the predictions and laboratory measurements of the maximum moment

along the pile. However, further verifications will be helpful to examine the model to predict the pile head

movement. In addition, according to the parametric study performed to investigate effects of input

earthquake base acceleration on the pile head displacement, larger horizontal base acceleration (Kh) causes

larger pile head displacement. Increasing in the upward vertical base acceleration (Kv) has the similar

influence on the pile head displacement; however, vertical base acceleration has less impact on the pile

head displacement compared to the horizontal base acceleration.

The SSWM approach presented here provides a simple and powerful method for solving the problem of

the seismic analysis of laterally loaded piles. The SSWM overcomes the main drawback of the traditional

p-y approach, which is not considering stress transfer between the springs. In addition, the SSWM can


Santiago, Chile consider the effect of phase difference in seismic travelling through the soil during the earthquake

excitation. It should be noted that input data, required in the SSWM, can be obtained from the basic soil

properties that are typically available to the designers.

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