Modeling of Pile Group Pile Caps Lam Pf1267

7/31/2019 Modeling of Pile Group Pile Caps Lam Pf1267

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EVALUATION OF CURRENT PRACTICE FOR ANALYSIS OF

PILE GROUPS UNDER SEISMIC LOADING

W.D. Liam Finn1 and Thuraisamy Thavaraj2

ABSTRACT

Semi-empirical methods are widely used in the seismic response analysis of pile foundations because the

complexity of the problem precludes 3-D dynamic finite element analysis. The most common approach for

the analysis of pile foundations is the use of nonlinear Winkler springs and dashpots to simulate the

interaction between piles and soil. The properties of these springs are specified by p-y curves. The most

widely used curves are those recommended by the American Petroleum Institute. In order to include the

effects of inertial interaction with the superstructure, a very simplified foundation-superstructure model is

employed in the analysis. This paper evaluates the effectiveness of p-y curves and the simplified foundation-superstructure model in simulating the response of pile foundations. The p-y curve approach is shown to be

potentially unreliable. The simplified model is shown to work very well provided the pile foundations

undergo very little rotation of the pile cap and the pile foundation is analyzed using a simplified nonlinear

continuum model of the soil-foundation system.

INTRODUCTION

Seismic soil-structure interaction analysis involving pile foundations is one of the more complex

problems in geotechnical earthquake engineering. The analysis involves modelling pile-soil-pile interaction,

the effects of the pile cap, nonlinear soil response and inertial interaction with the superstructure.

Commercial structural analysis programs can not include the pile foundations directly. Therefore in the

seismic analysis of bridges and buildings on pile foundations, various semi-empirical procedures are widelyused.

Dynamic nonlinear finite element analysis in the time domain using the full 3-dimensional wave

equations is not feasible for engineering practice at present because of the time needed for the computations.

However, by relaxing some of the boundary conditions associated with a full 3D analysis, it is possible to get

reliable solutions for nonlinear response of pile foundations with greatly reduced computational effort. The

results are very accurate for excitation due to horizontally polarized shear waves propagating vertically (Finn

and Wu, 1994). A full description of this method, including numerous validation studies, has been presented

by Wu and Finn (1997a,b). The method is incorporated in the computer program PILE-3D.

The most common approach for the analysis of pile foundations is to use nonlinear Winkler springs with

dashpots to simulate soil stiffness and damping. Some organizations such as the American Petroleum

Institute (API, 1993) gives specific guidance for the development of nonlinear pressure-deflection (p-y)

curves with depth as a function of soil properties. These recommendations are based on static or slow cyclicloading field tests. The API p-y curves are most widely used in engineering practice.

The California Department of Transportation (CALTRANS) has adopted a simplified model of a bridge-

foundation system that facilitates taking nonlinear soil behaviour and inertial interaction between foundation

soils and superstructure into account (Abghari and Chai, 1995). The foundation pile group is represented by

a single pile that supports a concentrated mass corresponding to its proportion of the total static force carried

by the group. The mass is supported in a single degree of freedom (SDOF) system with a period equal to the

first mode period of the superstructure assuming fixed supports. The function of the SDOF system is to

model approximately the inertial contribution of the superstructure to the response of the pile foundation.

The interaction between the soil and the pile is modelled using Winkler springs and dashpots with properties

1Anabuki Chair of Foundation Geodynamics, Kagawa University, Takamatsu, Japan

2University of British Columbia, Vancouver, B.C.


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equivalent to p-y curves. The pile head is maintained fixed against rotation. This model is designated

simplified-pile-superstructure model (SPSM).

SIMPLIFIED 3D SEISMIC ANALYSIS OF PILE FOUNDATIONS

The basic assumptions of the simplified 3D analysis

are illustrated in Figure 1. Under vertically propagating

shear waves the soil undergoes primarily shearing

deformations in xOy plane except in the area near the

pile where extensive compressional deformations

develop in the direction of shaking. The compressional

deformations also generate shearing deformations in

yOz plane. Therefore, the assumptions are made that

dynamic response is governed by the shear waves in the

xOy and yOz planes and compressional waves in the

direction of shaking, Y. Deformations in the vertical

direction and normal to the direction of shaking are

neglected. Comparisons with full 3D elastic solutionsconfirm that these deformations are relatively

unimportant for horizontal shaking. Applying dynamic

equilibrium in Y-direction, the dynamic governing

equation of the soil continuum in free vibration is

written as

2

2*

2

2*

2

2*

2

2

sz

vG

y

vG

x

vG

t

v

+

+

=

(1) Figure 1 : Quasi-3D model of pile-soilresponse.

where G*

is the complex modulus, v is the displacement in the direction of shaking, s is the mass density of

soil, and is a coefficient related to Poisson's ratio of the soil. Piles are modelled using ordinary Eulerianbeam theory. Bending of the piles occurs only in the yOz plane. Dynamic soil-pile-structure interaction ismaintained by enforcing displacement compatibility between the pile and soils. Nonlinear soil behaviour is

modelled using a variation of the equivalent linear approach used in the SHAKE program (Schnabel and

Seed, 1972) that includes soil yielding. Potential slip between the pile and the soil is modelled by using

contact elements with friction angles corresponding to the maximum mobilized friction angle between the

pile and the soil. Pile gapping is taken into account by not allowing tension to occur in the contact elements.

As these elements approach gapping, displacement occurs with very low default modulus and compressive

stress.

A finite element code PILE-3D (Finn and Wu, 1994; Wu and Finn, 1997a,1997b) was developed to

incorporate the dynamic soil-pile-structure interaction theory described previously.

SEISMIC RESPONSE ANALYSIS OF A SINGLE PILE

PILE-3D Analysis

PILE-3D was used to analyze the seismic response of a single pile in a centrifuge test conducted at the

California Institute of Technology. Details of the test may be found by Finn and Gohl (1987). Figure 2

shows the soil-pile-structure system used in the test. The system was subjected to a nominal centrifuge

acceleration of 60 g. A horizontal acceleration record with a peak acceleration of 0.158g is input at the base

of the system. The distribution of shear moduli was measured prior to shaking, while the centrifuge was in

flight, using bender elements.

The computed and measured moment distributions along the pile at the instant of peak pile head

deflection are shown in Figure 3. The moments computed by PILE-3D agree quite well with the measured

moments. The peak moment predicted by PILE-3D is 344 kNm compared with a measured peak value of325 kNm.


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Figure 2 : The layout of the centrifuge test for a

single pile.Figure 3 : Comparison of measured and

computed bending moments.

Analysis Using API p-y Curves

A dynamic analysis of the foundation-superstructure system was also conducted using the p-y curves

prescribed by the American Petroleum Institute (API, 1993) to model the soil-pile interaction. These p-y

curves are defined by the equation,

= yp9.0

Hk

tanhp9.0P uu (3)

where pu is the ultimate bearing capacity at depth H, k

is the initial modulus of subgrade reaction, y is the

lateral deflection, and H is the depth. The relative

density of the sand surrounding the pile is Dr = 38%.

This corresponds to a k of approximately 15000 kN/m3

according to the API recommendations. The analysis

shows that the p-y curves were much too stiff under

strong shaking. The distribution of moments for a value

of k = 15000 kN/m3

is shown in Figure 3. A reasonable

approximation to the peak moment in the pile isobtained using k = 2500 kN/m

3, which is only 1/6 of

the value recommended by API (1993). In another test

in the same sand, run at a very low peak acceleration of

0.04 g, the API stiffness k = 15000 kN/m, gives a very

good approximation to the measured bending moments

(Figure 4). The response in this case was almost elastic

and the initial stiffness controls the response. These

results suggest that the initial stiffness of the API p-y

curves is reasonable, but that the curves do not head

away fast enough from the initial tangent at the origin.

Therefore, stiffness at close to the initial value is being

mobilized over too large a displacement range understrong shaking.

Figure 4 : Comparison of measured and computed

pile moments for near elastic responseusing API procedure.

-25 0 25 50 75 100Bend ing Moment (kN.m)

-1 2

-1 0

-8

-6

-4

-2

0

2

Depth

(m

)

soi l surface

Measured

Com puted, p -y

kh=15000 kN/m3

-6 00 -400 -200 0 20 0

Bend i ng Moment (kN . m)

-1 2

-1 0

-8

-6

-4

-2

0

2

Depth

(m

)

soi l surface

Measured

Computed-PILE3D

Computed-API p-y


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In view of the widespread use of p-y curves in engineering practice, it would seem necessary to

investigate their reliability by more centrifuge tests and dynamic field studies. It should be noted that the

API p-y curves have been shown to be fairly unreliable also in predicting the response to static and slow

cyclic loading tests in the field (Murchison and ONeill, 1984; Gazioglu and ONeill, 1984).

SPSM AND FULL GROUP ANALYSES

The CALTRANS SPSM model, together with soil and pile properties is shown in Figure 5.

Figure 5 : Simplified pile-superstructure model.

The seismic response analysis based on p-y curves

has been shown to be unreliable. However, the

CALTRANS representative pile concept is a very

attractive computational feature. Therefore the concept

is evaluated below by treating the foundation soil as a

nonlinear continuum in both the representative pile and

full pile group analyses using PILE-3D. In this way, the

representative pile concept can be tested against fullgroup analysis under identical foundation conditions.

Figure 6 shows the comparison of bending moment

profiles for the two different pile groups and the single

pile. The bending moment profiles from the (22) and(44) analyses do not deviate much from the momentprofile in the single pile analysis. The difference in the

peak moment is less than 8%. The shear force profiles

show similar behaviour. It is evident from Fig. 6 that

the group effect does not appear to be a significant factor

in the response of the different pile groups analyzed here

even though the piles are spaced at two diameters, centre

to centre. Figure 6 : Comparison of pile moment profiles.

0 1000 2000 3000

Max imum Bend ing Momen t (kNm )

0

2

4

6

8

10

Depth

(m

)

Single Pi le

2x2 Group

4x4 Group

Homogeneous

Clayey Soi l

Pi le

Nonl inear Spr ingsand Dashpots

So i l Prop ert iesUndrainedShear S trength, S

u= 50 kPa

Uni t we ight, s

=18 kN/m 3

Shear Mod ulus, Gm ax

= 50 MPa

Pi le Prope rt iesFlexura l Rig id ity , EI = 1310 kNm3

Uni t we ight, p

= 25 kN/m 3

Diameter , D = 0.5 mLength, L = 9 m

E p/E s=500

p/s =1.4

L/d >15

=0.4

SuperstructureParametersMass, M

s= 5 0 M g

Natura l Frequency ofSDO F sys tem, f = 4Hz

Free F ieldSoi l


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Table 1 shows the lateral stiffnesses of the single pile and the pile groups both under initial elastic

conditions and at the time of maximum straining at which the minimum lateral stiffness is mobilized.

Table 2 shows the group factor, calculated as a ratio of group stiffness over the single pile stiffness

multiplied by the number of piles in the pile group. It was 0.61 for (22) pile group and 0.31 for (44) pilegroup under elastic conditions. The group effect corresponding to the minimum stiffnesses for the (22) pile

group is 0.76 and for the (44) pile group is 0.41. The same spacing was used in both pile groups and thegroup effect increased with the number of piles in the pile group as expected. Also, there is a reduction inthe group effect during the periods of strong shaking at which the pile foundations reached their minimum

stiffnesses. This is in keeping with the general perception that the range of pile to pile interaction is reduced

as the soil behaviour becomes nonlinear. Despite the group interaction effects, the bending moment and

shear force responses were not significantly affected. The reason for this behaviour is explained below.

Table 1 : Lateral stiffnesses of single pile and pile groups

Type of Pile

Foundation

Initial Elastic Lateral

Stiffness (MN/m)

Minimum Lateral

Stiffness (MN/m)

Reduction in Stiffness

(%)

Single Pile 183 35 81

(22) Group Pile 444 106 76(44) Group Pile 921 228 76

Table 2 : Group effect corresponding to initial elastic and minimum lateral stiffnesses

Group Effect, Kgroup/(n*Ksingle)Type of Pile FoundationInitial Elastic Stiffnesses Minimum Stiffnesses

(22) Group Pile, K22 0.61 ~0.76

(44) Group Pile, K44 0.31 ~0.41*n is the total number of piles in the pile group.

The fundamental frequencies of the three pile foundation systems were determined corresponding to theinitial stiffnesses and the minimum stiffnesses that occur during the time of strongest shaking. The

frequencies are shown in Table 3. Although the group effects on foundation stiffnesses of the pile groups are

significant, the differences in the global system frequencies of the pile groups are not significantly different

from the frequency of the single pile system under either elastic or nonlinear response. The maximum

difference is about 15%. This is due to the fact that the global system frequencies result from the combined

stiffnesses of the superstructure and pile foundation rather than the stiffness of the pile foundation alone. In

this study the superstructure stiffness dominates the predominant system frequency. The similarity in

frequencies of the different superstructure-foundation systems is responsible for the similarity in outputs.

Table 3 : First mode frequency of the superstructure-pile foundation system

First Mode Frequency (Hz)Type of StructureInitial Minimum

Single Pile- Superstructure 3.66 1.89

(22) Group Pile -Superstructure 3.45 1.82(44) Group Pile -Superstructure 3.07 1.61

CALTRANS adjusts the results of the SPSM for group effects using a group reduction factor. Since

group effects need to be incorporated during the dynamic analysis as they affect the nonlinear response, their

use after the event may not be appropriate.

The results of the evaluation study suggest that the SPSM model concept for the analysis of pile

foundations is only valid if the system frequency of the representative pile-superstructure model and the

frequency of the full foundation-superstructure model are approximately the same. This is likely to occuronly when the stiffness of the support structures of the bridge dominates the system frequency. For this to


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happen, the support structures must be much more flexible than the pile foundation in the degree of freedom

under consideration. Parametric studies using many earthquakes with different frequency contents would be

desirable to explore how full foundation system frequencies compare with representative pile system

frequencies for typical bridges and foundation soils. Such studies would allow the reliability of the

representative pile concept to be evaluated properly.

The problems associated with system frequencies and p-y curves can be avoided if PILE-3D or any other

program is used that can treat the soil as nonlinear continuum and model the whole pile group conveniently

enough for engineering practice.

Effects of Pile Cap Rotation

In line with the CALTRANS procedure, the

evaluation analyses were conducted with the

assumption that the pile cap was fully restrained

against rotation. The analyses were repeated without

preventing rotation of the pile cap. In these latter

analyses, the only restraint on the pile cap is

provided by the moments on the pile cap caused bythe axial forces in the piles.

Profiles of maximum bending moments are

shown in Fig. 7. In this case there is very poor

agreement between results of analyses based on the

representative single pile concept and the full group

analyses, even though the foundation soils are

treated identically for all three systems. It would

seem that the differences in computed rotations may

be primarily responsible for the differences in

moments from the representative pile and full pile

Figure 7 : Comparison of pile moment profiles.

foundation group analyses. Therefore the representative pile concept is probably not appropriate for small

pile groups that can not effectively restrain the pile cap. However for large pile groups, the rotationalrestraint of the pile cap by the axial forces in the pile may be sufficient to approximate a fixed condition.

Parametric studies to establish a limiting pile group size are necessary.

REFERENCES

Abghari, A. and Chai, J.(1995), Modeling of soil-pile-superstructure interaction for bridge foundations,

Performance of Deep Foundations Under Seismic Loading, ASCE Geot. Special Publ. No. 51, pp. 45-59.

API (1993), Recommended practice for planning, designing, and constructing fixed offshore platforms,

American Petroleum Institute, Washington, D.C.

Finn, W.D. Liam and Gohl, W.B. (1987). Centrifuge model studies of piles under simulated earthquake

loading from dynamic response of pile foundations - experiment, analysis and observation. ASCE

Convention, Atlantic City, New Jersey, Geotechnical Special Publication No. 11, pp. 21-38.Finn, W.D. Liam and Wu, G. (1994), A recent development in the static and dynamic analysis of pile

groups, Proc. of the Annual Symp. of the Vancouver Geotechnical Society, Vancouver, BC, May 1994,

pp. 1-24.

Gazioglu, S. M. and O'Neill, M.W. (1984), An evaluation of p-y relationships in cohesive soils, Proc. of

the ASCE Symposium on Analysis and Design of Pile Foundations, ASCE National Convention, San

Francisco, California, Oct. 1-5, 1984, Edited by J.R. Meyer, pp. 192-213.

Murchison, J.M. and O'Neill, M.W. (1984), An evaluation of p-y relationships in cohesionless soils, Proc.

of the ASCE Symposium on Analysis and Design of Pile Foundations, ASCE National Convention, San

Francisco, California, Oct 1-5, 1984, Edited by J. R. Meyer, pp. 174-191.

Wu, G. and Finn, W.D. Liam (1997a), Dynamic elastic analysis of pile foundations using finite element

method in the frequency domain, Canadian Geotechnical Journal, Vol. 34, No. 1, pp. 34-43.

Wu, G. and Finn, W.D. Liam (1997b), Dynamic nonlinear analysis of pile foundations using finite element

method in the time domain, Canadian Geotechnical Journal, Vol. 34, No. 1, pp. 44-52.

0 1000 2000 3000

Max imum Bend ing Mom en t (kNm)

0

2

4

6

8

10

Depth

(m

)

Single Pi le

2x2 Group

4x4 Group

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Modeling of Pile Group Pile Caps Lam Pf1267