PB89 -,.. 52 39

NATIONAL CENTER FOR EAKl'HQUAXE ENGINEERING RESEAIlCH

State University of New )brk at Buffalo

SEISMIC RESPONSE OF PILE FOUNDMIONS

by

s. M. Mamoon, P. K. Banerjee and S. Ahmad Department of Civil Engineering

State University of New York at Buffalo BuffaJo, New York 14260

AEPROOUCED BV u.s. DEPARTMENT OF COMMERCE NATIONAl.. TECHNICAL INFORMATION SERVICE SPRINGAELD. VA. 22161

Technical Report NCEER-88-0034

November 1, 1988

NanCE This report was prepared by the State UnMntity ci New 'bk at Buffalo ... result of iC!8eI1d. spanIOIed .". the National Center for fArthquMe Eftsineerin8 ReIaId\ (NCEER) aut the NIIIional Scierk."l! ftNndation. Neither NCEER. ""O'd"ee 01 NCEER. iIs .,..,.., sa. Uniwnity of New "ibrk .. Buffalo, nor .., perIOIl am.,. 01\ their beNif:

L mMes.., W8M!'ty. eICpNU or impHed. wiIh -.pec.t 10 the ... of MfY infcIrmItioft. ..,.,....., IIIethoIt 01' proceta dIIdoeed 1ft tNt mpart or thIt such -INIJ not iI\fmp upon ,...a,GWNd .... or

b. _ .......... 01 wlt.Dlle •• 1dnd wIh.-pec:t ID'" ... ,arthecta.p ............... _of, ,.", ...... .... ...,........1Idhod Of pnaII ~ in thiI NpOIt.

SEISMIC RESPONSE OF PILE FOUNDATIONS

by

S.M. Mamoon], P.K. Banerjee2 and S. Ahmad3

November 1, 1988

Technical Repon NCEER-88-0034

NCEER Contract Number 87-1013

NSF Master Contract Number ECE 86-07591

1 Graduate Student, Depanment of Civil Engineering, State Universi~y of New York at Buffalo 2 Professor, Department of Civil Engineering, State University of New York at Buffalo 3 Assistant Professor, Department of Civil Engineering, State University of New York at

Buffalo

NATIONAL CENTER FOR EARTIIQUAKE ENGINEERING RESEARCH State University of New York at Buffalo Red Jacket Quadrangle, Buffalo, NY 14261

50272-101 --------.-------__ r---~--------___,

I--R_EPO_RT_DOC.:..:PA.:::.G=~_M_EHT __ A._TI_ON---LI_I_._R~_~:....:.~.:..:E:....:.';.:....~....:...8.:.--8-_0-,-0.:.--34 _______ ~Z~ _____ ~ _ __IPB8·9nt\\C4·5 '2 3 9. 4. Title end Subtitle 5. Report Det.

Seismic Respo.)nse of Pile Foundations November " 1988 6.

7. Authori.) I. Perform ina: Or,.nization Rept. No.

S.M. Mamoon, P. K. Banerjee and S. Ahmad ----- -.-

ID. Proj.,.. fT .. kfWorlt Unit No.

National Center for Earthquake Engineering Research Stat~ University of New York at Buffalo 11. Contrec:I(C) or GtI!llUi) 1'0.

87-1013, cCc 86-07591 IC) Red Jacket Quadrangle

Buffalo, NY 14261

~-------------------~------------------------------.---------~~~~~~~------------n. SponHrinc Oraenlzetion Neme end Address U. TrIM of Jleport & Period Cowre.t

Technical Report

~ ______________________________ 1-i __________ . __ _

15. Supplementery Hate.

This research was conducted at the State University of New York at Buffalo and was partially supported by the National Science Foundation under Grant No. ECE 86-07591.

I----------------------------------.--------------j 16. Abltred (Umlt: zoo wants)

.: A boundary element formulation is presented to analyze the seismic response of a single pile. The pil ,are modeled by compressible beam-column elements and the soil as a hysteretic elastic half-space. A new Grecn's f'.Jnction corresponding to dynamic loads in the interior of a semi-infinite solid is developed for this study. The governing dif­ferential equations of motion for the pile domain have be~n solved exactly for distribut­ed periodic loading intensities. These solutions were then coupled with a numerical solution for the motion of the soil domain by satisfying equilibrium Clnd comoatibility at the pile-soil interface. The responses were evaluated over wide ranges of the para­meters involved to vertically and obliquely incident P, SV and -SH-waves. Results are presented as dimensionless graphs and the pile-soil interaction is studied. It is ob­served that the existence of the pile produces a filtering of the waves, reducing the amplitude of motion as a function of frequency. Also the filtering effects, the pile motions and the amplitudes are found to be dependent on the slenderness ratio, the stiffness ratio and the angle of incidence. Finally, an actual transient analysis is per­formed to study the response of pil~s to seismic waves in the time domain. /

,

1--------------------------------------------- --.. -----------------17. Document An.'rsls •. Oesc:rtplO'"

b. _llIen'0gen-E_cI T.' .... EARTHQUAKE ENGINEERING SEISMIC RESPONSE ANALYSIS DYNAMIC LOADS SOIL STRUCTURE

Release Unlimited

PILE FOUNDATIONS BOUNDARY ELEMENT METHODS PILE-SOIL INTERACTION GREEN'S FUNCTION

I.. Security Class (This RePlKt)

Unclassified ZO. Security CI ... (ThIs P_)

Unclassified 0f'TI0HA1. FOfltII zn (4-77) If ....... <ty NTIS-351 ...... - ..... - ........ _. ,... .............. "r ..

PREFACE

The National Center for Earthquake Engineering Research (NCEER) is devot~d to the expansion and dissemination of knowledge about eanhquakes, the improvement of earthquake-resistant design, and the implementation of seismic hazard mitigation procedures to minimize loss of lives and propeny. The emphasis is on strllctures and lifelines that are found in zones of moderate to high seismiCity throughout l.he United States.

NCEER's research is being carried out in an integrated :-.nd coordinated manncr foI:owing a structured program. The clIrrent research program comprises four main areas:

• Existing ~nd New Structures • Secondary and Protective Systems • Lifeline Systems • Disaster Research and Planning

This technical report pertains to Program 1, Existing and New Structures, and more specifically to geotechnical studies, soils and so~l-structure interaction_

The long term goal of resear;;h in Existing and New Structures is to develop seismic hazard mitigation procedures through rational probabilistic risk assessment for damage or collapse of structures, mainly ex.isting buildings, in regions of moderate to high seismicity. ·1 hL work relies on improved definitions of seismicity and site response, experimental and analytical evaluations of systems response, and more accurate assessment of risk factors. This technology will be incorporated in expen systems tools and improved code formats for existing and new structures. Methods of retrofit will also be developed. When this work is completed, it should be possible to characterize and quantify societal impact of seismic risk in various geographical regions and large municipalities. Toward this goal, the program has been divided into five components. as shown in the figure below:

Program Elements:

I Seismicity. Ground Motions and Seismic Hazards Eslimates

• I GeoleClmica1 Studies. Soils and Soil-Structure Interaction

+ I System Response: I

Testing and Analysis I

• I Reliability Analysis IIId Risk Asseasmenl

I I

, -~

I I

Expert Systems J iii

Tasks: EanhquoIte Haunll Eotimalel. o.-.d Mo«ion~. Ne.o.-.d Mou", .... _lion. Eonhquakc .a am...d Malim Do .. Due.

Site Rmponoc Ellimalel. LaIJC <lrauncI Doformotion Eolirnllfs. SClil-Su .... "", Inll:n~.

Typical s.n-u- and Cribcal S-.I eom,.-. .. , T 1IIIin, and Anal,.;.; Modem Analytical Toola .

v ...... bility Analyoia. Rebability Anal,.;., Itiot.u-I, COl' UJIIrodin&-

~ ..... SttuctwaJ 0.. •. Evahulim tI F.u..u., Buillinp.

Geotechnical studies. soil and soil-structure interaction constitute one of the important areas of research in Existing and New Structures. Current research activities include the following:

1. Development of linear and nonlinear site response estimates. 2. Development of liquefaction and larre ground deformation estim~tes. 3. Investigation of soil-structure interactioli phenomena. 4. Development of computational methods. 5. Incorporation of local soil effects and soil-structure interaction into existing codes.

The ultimate goal of projects in this area is to develop methods of engineering estimation of large soil deformations. site response, and the effect that the interaction of structures and soils have on the resistance of structures against earthquakes.

Increasing numbers of structures and buildings are built on poorly performing soils which require pilings for sfability. Thus 'he understanding of the performance of piling in soft soils under dynamic bads is of increasing importance. This paper addresses fundamental methods to analytically describe the belulvior of a single element of a pile foundation during seismk loads. It addresses but one element in a complex problem which nee±; to be solved step by step 10

dev;se methods to increase the eart1u{uake resistance of new and existing structures, many of which require pile fOUndations. Thus. this study must be seen as one step in a sequen('e of investigations carried out by NCEER in the field of soil-structure interaction.

iv

A boundary element formulation is presented to analyze the seismic

response of a single pile. The piles are modeled by compressible beam­

column elements and the soil as a hysteretic elastic half-space. A new

Green's function corresponding to dyMlllic loads in the interior of a semi­

infinite solid is developed for this study. The governing differential

equations of motion for the pile domain have been solved exactly for

distributed periodic loading intensities. These solutions were then

coupled with a numerical solution for the motion of the soil domain by

satisfying equilibrium aOO canpatibility at the pile-soil interface. '!be

responses were evaluated over wide ranges of the pa:::aneters involved to

vertically and cbliquely incident P, SV and SH-waves. Resu:ts are

presented as dimensionless graphs and the pile-soil interaction is studieCL

It is observed that the existence of the pile produces a filtering of the

waves, reruclng the .snplitude of motion as a function of frequency. Also

the filtering effects. the pile rrotions and the amplitudes are found to be

dependent on the slenderness ratio, the stiffness rat.io and the angle of

incidence. Finally, an actual transient analysis is perfooned to study the

resp:>nse of piles to seismic waves in the time cbnain.

v

'!he work presented in this report was made p:>ssible by NCEER Grant No.

87-1013 fran the National Center for EarthqUake Engineerirg Research, State

University of New York at Buffalo. This support is gratefully

acknowledged. The computations were performed on an HP 9000 series

computer that belongs to the Computational Mechanics Division, Department

of Civil En3ineering, State University of New York at Buffalo.

vii

2. MthU> CP MM..YSIS

2.1 Governing Equations 2.2 Numerical SOlution 2.3 In::ident waves 2.4 Transient Analysis

3.1 Introduction 3.2 Periodic Force Normal to the Boundary 3. 3 'Periodic Force Parallel to the Boundary

4. PRBSaI'fATlDH MD ~SIS c:I RPSJLTS

4.1 Foan of Presentation 4.2 ReSp:)nses Due to 58-Waves 4.3 Response Due to P and SV-Waves 4.4 Transient Results

7. RI'.P!RIRPS

ix

1-1

2-1

2-1 2-3 2-7 2-9

3-1

3-1 3-2 3-7

4·1

4-1 4-1 4-7 4-11

5-1

6-1

7-1

2-1 Model for Pile an<.. System of ~rdinates 2-8

3-1 Periodic Force in the Interior of a Semi-infinite 3-3 SOlid: (a) Normal to the Boundary. (b) Parallel to the Boundary

4-1 TranS\'erse Displacement Ratios for Vertically Incident 4-3 SH-waves: Wluence of ~/Es Patios (ao

a O.5)

4-2 Transverse Displacement Ratios for Cbliquely In::i~nt 4-4 SH-<.Iaves: Influeoce of ~fEs Patios (ao,.,O.5, 9 .. 60 )

4-3 Transverse Displacement Ratios for SH-waves in a Short 4-5 Pile (LId-10): Influence of Angle of Incidence

4-4 Transverse Displacement Ratios for SH-waves in a Long 4-6 Pile (L/d .. 30): Influence of Angle of Incidence

4-5 Axial Displacement Ratios for Obliquely Incident 4-8 P and S\1-wav:s: Influence of ~/Es Ratios (ao",o.5, 9 .. 15 )

4-6 Transverse Displacement Patios for Cl>liquely Irv::ident 4-9 P and SV-waveg: Influence of ~/Es Ratios (ao",o.5. 9-75 )

4-7 Transverse Displacement Ratios for P and SV-waves: 4-10 Influence of Angle of Incidence (~/Es-I000)

4-8 Transverse Displacements at Pile Mid-point for 4-12 Vertically Incident Wave: InfAuence of ~/Es Ratio (co .. 0.5 rad/sec. e .. 90 ) P

4-9 Transverse Displacement at Pile Mid-point for Cl:lliquely 4-13 Incident Wave: Infl~ of Ep/Es Ratio (_ • 0.5 rad/sec. 8 - 60 )

xi

In recent years mcch progress has been made in develoPing procedures

for calculating the responses 'f single piles and pile groups to both

static [4,5.7,19} and dynamic' oads [6.7.13,20,21}. However, realistic

dynamic analyses of pile foundations have received co!tlparatively less

attentiom JOOst of the work done is essentially restricted to foundations

subjected to periodic excitations [6,7.13,20,21}. It iswell known that

the excitation in an actua~_artiGuake is a canplex tine dependent function

and theretore. in order to predict the seismic response of pile

fouroation.:;, it is necessary to carry out a transient analysis [12].

The effect of vertically and Obliquely incident seismic waves 0n

surface and embedded shallow foundations has been investigated in some

detail in recent years [9.22]. Simplified rules have been suggeste<:! [9.22 ]

to estimate both the translational am rotational resplnse fran the free­

field motion as a function of the embedment depth. It was found that the

existence of the foundation produces a filtering of the waves. reducing the

amplitude of the ootion as a function of frequeocy.

However. relatively few reports exist in the published literature with

regard to the behavior of deep foundations (piles. piers and caissons) to

traveling waves. The resPJnse of piles to vertically propsgatiD:) S-waves

has been studied by several authors [11.12]. Gazetas [12] has presented

results of a numerical study of the dynamic response of end-bearing piles

etDedded in a n.mtJer of idealized so11 deposits. StilL several questions

remain unanswered. especially with r'!gard to the practical assesanent of

the influence of piles on the seismic excitation of a structure.

Furthermore. only a very limited number of results are available in the

form of dimensionless graphs [16J.

1-1

It is the objective of this report to present the results of a study

on t:re behavior of a single pile subjected to incident P. SV and ~d-waves.

Such cesul ts would be useful not only for developing an improved

understanding of the mechanics of the problem. but. also for deriving

siITi'le preliminary design rules for foundation engineering practice. glCh

a study is well worth undertaking. since the majority of important

structures. such as tall buildings. major highway bridges. offshore oil

platforms. etc.. are supported on piled foundations and the existing state

of knowledge is FOOr. Also. the results would be of great interest to nany

engineers who have built heavy structures on soft soils in seismic prone

areas.

In the present work. the modeling of this problem has been done by

means of Cl hybrid boundary element formulation [3]. ~ numerical scheme

is based on the discretization of the pile and the soil domain around the

pile into elements throughout which the displacement and tractions are

assumed to be constant (or can be interpolated between nodal values in a

higher order f011llulation). The boundary element method offers considerable

advantage over the other numerical methods. for this problem. primari ly

because of its ability to tak~ into account the three dimensional effects

of soil continuity and boundaries at infinity. In this work, for 5011-

domain a new fundamental solution corresponding to a periodic dynamic point

force in the interior of an elastic half-space is dPveloped and then used

in the rnnnerical scheme. '!his solution represents the dynamic equivalent

of Mindlin's [18] static half-space ~int force solution.

1-2

2. 1 <D\7ERmI;; EQlM'I(lqS

The general strategy used for the analysis of the pile foundations

under static and dynamic loading by Banerjee and Driscoll [51. Sen et al

[20,21 J is followe<l here. This involves the construction of an integral

representation for the soil domain and the equations of motion for the

piles, represented by linear strur.:tural ceJrnIXlnents.

For dynamic loading of a hOmogeneous, isotropic. elastic solid. the

governing differential equation in terms of the displacements ui

(neglecting body forces) is [3]:

(),+II)U ••• + flU· i' - pUle = 0 ),)1 ),.,1

(2.1)

where

A, I.l are the lame' s constant

p is the ness density of the solid

2 2 ui = a Ui '8t are the accelerations

If the displacE!lents are periodic (time harrronic), i.e.,

(2.1)

where ui (X,Cd) is the amplitude of the displacement and 411 is the circular

frequency, then, equation <l.1) reduces to Helmholtz's equation {3]

2 {).+l1)U. ". + flUi .. + pili Ui • 0 J ,J1 ,)J (2.3)

For a homogeneous solid of finite extent the solution of this

differential equation for arbitrary boundary conditions can be expressed as

an integral equation in which the full-space Green's function takes the

roles of the kernel function (3]. However. in the present application, in

1-1

which the solid is of semi-infinite extent, it is advantageous to cast tl.e

integral equation in terms of the Green's function for the half-space,

since this reduces the domain of integration to the soil-pile interface

only. The integral equation describing the motion of the soil can be

written as:

(2.4)

where

U j are the displacements of the soil

Gij is the Green's function fe-I an elastic. homogeneous half­

space

'i are the tractions at the pile soil interface

Equation (2.04) must be coupled with equations describiD3 the res!X)nse

of the piles <idealized as beam-collmll'l structures) to axial and flexural

haoronic excitations. '!he equations of motion (20) are:

(2.5)

for the axial dynamic res!X)nse and

(2.6)

for the lateral dynamic res!X)nse, where

Ep the Young's trodulus of the pile material

~ the second moment of the area of pile cross-section

~ the cross sectional area of the pile shaft

2-2

d the diameter of the pile

Ux and Uz are the lateral and axial displacements. respectively

m the mass per unit length of the pile

.x and .z are the lateral and axial tractions on tle pile

Equations (2.4). (2. S) and (2.6) represent the complete set of

equations for the solution of the problem.

2.2 tomRICAL!DIJl'I(E

By discretizing the pile-shaft-soil interfaces into n cylindrical

segments, the bases of the piles being circular discs. we can use equation

(2.4) to determine the displacements at the pile-soil interface as

n

= 2 [S Gij(X'~m,w)dSpJdl(X'~) p=l ASp

(2.7)

where surface tractions d1 have been assumed to be constant over each

segment of the pile-shaft soil and pile-base soil interfaces p(~l ..... n) •

By taking ~ successively as the centroidal FOint of each elenent on

the boundary, we can obtain the following matrix equations for the

interfaces:

(2.8)

... here superscript's' represents the scattered values and the subscript's'

indicates that the stress and displacement values are obtained from a

consideration of soil domain alone.

The differential equation (2.3), the Helmholtz's equation. is that

governing time harmonic wave scattering. scattering problems dealing with

2-3

infinite and semi-infinite regiol'.5 are usually formulated decorrq:x>sing the

total displacement and stress fields (ut.dt ) into two parts [31: (1) a

known free or undisturbed field (uf.d f ). and (2) the scattered field

(2-9)

The free-field or the incident wave is simply the wave motion that

would be present in the absence of scattering surfaces and the scatterea

part is the wave diverging from the scattering region. uS satisfies the

radiation condition at infinity. which guarantees the absence of reflected

radiation.

Writing (2.8) in terms of eq. (2.9).

or

{u;) s [GJ{~;} - (bsl (2.10)

where.

(2.11)

The set of equations represented by eq. (2.10~ has to be coupled wi th

an equivalent set for the pile danain by sol ving the governing differential

equations (2.S) and (2.6) deSCribing the responses of the piles to axial

and flexural harmonic excitation. The general solutions of these equations

[i.e.. (2. S) and (2.6) J are of the form [20).

uz K ~sin(yz) + ~oos(yz) (2.12)

(2.13)

where

2-4

y c (mw2/E~]1/2

'1\ [1t\(.,2 IEp~] 1/4

(2.13a)

(2.13b)

and the arbitrary constants can be determined frOOl the specified boundary

conditions at the ends of the pile. Since the distribution of tr;octions

along the pile-soil interface is a complex function of nodal values of

tractior., direct determination of the particular solutions of equations

(2.S) and (2.6) is scarcely possible. For arbitrary pile hec:.d

displacements it is necessary to examine the behavior of individual piles

when subjected to axial and lateral dynamic leeds. This can be best done

by considering the effects of applied pile head displacements as the

algebraic sum of the motion of an unsuPIX>rted pile (i.e., no soil reaction)

under arbitrary pile head boundary conditions and those of a fixed head

pile. SuperlXlsition of these solutions leads to the linear set

(2.14)

where (bpJ is the displacenent vector derived fran unit pil~head boundary

conditions, the matrix [OJ comprises constants of equations (2.12) and

(l.U), ;md u1

is the arbitrary pile head displacements or rotation. Here

't' indicates the total field values and subscript 'p' indicates that the

pild-soil interface displacements and tractions are obtained from a

consideration of pile domein alone. Equation (2.14) may be rearranged

into.

(2.15)

where [B] is an identity matrix with {bpJ vector subtracted out from the

first column for the axial case. The corresponding {bpJ vector is

similarly incorporated in the first column for the lateral case.

2-S

Now premultiplying both sides of equation (2.10) with [B] matrix we

have

(2.16)

where, {El =' {El {G] (2.17)

and, (2.18)

By satisfying the equilibrium and compatibility conditions at the

pile-soil interface. the total tractions acting on the piles may be

determined from (2.1S) and (2.16) as

(2.19)

The total displacements are then obtained from (2.10).

The axial loads, shear forces and bending m:::ments in the piles at any

depth due to the seismic forces can then be easily recovered from the

values of ~~ ~ the inertial forces:

(2.20al

(2.2Ob)

M(z) <2.2Oc)

where m is the difference of mass/unit length between the pile and the soil

filled pile domain.

In equation (2.11) information is needed about the free-field stress

and displacement values {el;!. {u~}. which is discussed in the next section.

2-6

2.3 IICIIBft' MVES

Consider the half space z 1 0 (Fig. 2-1), and a train of plane waves.

propagating parallel to the x-z plane. The motions are therefore

independent of the coordinate y and the overall problem can be studied in

two uncoupled pirts, one corresponding to SH-waves with a motion in the '1-

direction and the other to SV and P waves with coupled motion in the x and

z-directions. The solutions adopted for SR. SV and P waves is the one

obtained by direct integration of the differential equations of rotion in

terms of amplitudes used in reference [171, as opposed to solutions in

terms of IX:tentials proposed by some researchers [9i.

SB-Maves: To be consistent with the previous fOD1lUlation, we may consider

the SH-waves to have a motion in the x-direction as a result of waves

propagating parallel to the y-z plane. The motion for an elastic mediun

with one-dimensional geanetry in the case of SH-waves has the form [91:

u ... x

where,

i~ i~ [~xp(C nz) + ASHexp<- C nz)] • f{x,t)

s s

Col is the angular frequency

Ass and ASH are amplitudes of the incident and reflected waves

Cs is the shear wave velocity of the soil

1 and n are direction cosines of direction of propagation

(2.21)

For the half-space, ASH = ASH and for a unit amplitudr: of the motion

on the surface, eq. (2.21) reduces to [9J:

Ux - cos (~nz) • f(x,t) Cs

2-7

(1.21)

y

Incident Waves

Ground Surface

~~----------+-________ ~x

Pile

z

FIGURE 2-1. Model for Pile and System of Coordinates

2-8

where

iwlx Ix f(x.tl "" exp(- --) • exp (iwtl = exp[iw(t - -)] Cs Cs

(2.23)

The shear stresses are obtained from the displacements by

differentiation.

't XZ

where G is the shear modulus.

(2.24)

Equations (2.22) and (2.24) directly furnish {U;} and {d~} variations

along the depth in eq. (2.11) fot SH-wavcs.

SV and p...-tfaves: In the case of SV and P-waves the horizontal displacement

Ux and the vertical displacement Uz depend on both waves. but the

formulation can be simplified assuming the same variation in the x­

direction of all the displacement canponents. Equations similar to (2.22)

and (2.24) can lE written for axial and transverse stress and displacement

values for this case. details of which are given in Ref. [9].

2.4 'l'RMS1EN'l' AlW.YSIS

The transient analysis has been done in the Laplace transform dorrain

followed by a numerical inverse transformation to obtain the responses in

the real time domain. a procedure which was previously used by Doyle [10].

Cruse a:ic Rizzo [8], Ahmad and Manolis [1], and others.

For ~ransient analysis. the actual input is qiven as

u c cos (!- nz) sin(wt _ wlx) x Cs Cs

(2.2!)

w ILl wlx 't - - G (- n) sin (- n) sin(wt - -) xz Cs Cs Cs

(2.26)

2-9

For transient disturbances use is made of the Laplace transform of the

field eqs. (2.2S) and (2.26) with respect to time of the form [1,8]

lID

f(x,sl = Lff(x,tl} = J f(x.t)e-stdt (2.27) o

where 's' is the Laplace transformed parameter. The transformed domain

integral equation representation does not require time convol utions that

are necessary in a time domain formulation. Instead. the formulation is

parametric in the transformed variable and the resulting algori thIn involves

the solution of a ~r of 'quasi-static' t:yFe equations.

Arrj intr~rnal viscous dissipation of energy can easily be accounted for

by replacing the lane constants>.. and 11 t7y their complex counterparts and

leaving Poissons's ratio unaltered. n.e system matrices [G] and [0] in eq.

(2.19) are dependent on the transform parameter s. and are thus complex.

Equation (2.19) is solved for unknowns {u~) in terms of the prescribed

boundary quantitjes {ds } and for a spectrum of values of the transform

parameter. What remains to be done is to invert the solution back to the

real time cbnain. 'n1e inverse Laplace transformation is defined as [1.8):

1 f+iCD

f(x.t} = ---2 . f(x.s) estdS. 7fl ~-illD

i = .r-1 (2.28)

where ~ > 0 is arbitrary. but greater than the real part of all the

singularities of f(s), and s is a complex number with Re(s) > O. Details

of the numerical inversion algorithm can be found in Ref. [1). However. it

has to be noted that the Laplace transform solution is essentially a

superposition of a series of steady-state solutions and is therefore

applicable only to linear problems. Also. since the Laplace transform

casts the entire problem in the complex domain. the storage and computer

ttme requirements are considerably increa~

2-10

3.1 nmoxx::rl~

IITplenentation of the algorithm is de~roent on the availability of

the Green's function [3] for a dynamiC point force buried in the interior

of an elastic half-space. A boundary element formulation of the pile

problem was recently reported by Kaynia and Kausel [13] who obtained

(numerically) dynamic ruried ring load (axial and lateral) sol utions fran

the work of Apse] {21. The major problem here is. of course. the accuracy

of the numerically constructed dynamic solutions. since the convergence of

the semi-infinite integrals in Apsel's work is heavily dependent on the

frequency parameter.

An approximate solution [161 for the displacement field due to a FOint

force ej acting at a depth 'c' below the surface of the half-space can be

constructed by superposing the displacement fields due to two dynamiC

forces acting within an infinite solid. one acts at a depth 'c;' belOlrl the

'surface' and the other acts at a distance 'c' above the surface (i.e., the

mirror image of the buried force). thus

(3.1)

where

(3.2)

The reSUlting stress field arising from G~j does not completely

satisfy the traction free-boundary condi tions at the ground surface. In

the aforenentioned fOD'llUlation, other fundamental solutions [15] such as

double forces, line of forces, doublt'!ts. centers of compression, lines of

3-1

doublets. etc. are added on the rEgative side of the half-space to cancel

the residual tractions. 'iberefore. the solution for the half-space can be

expressed as

(3.3)

where the comp:>rents of Rij (x. ~.w) are contriroted by the other fundamental

solutions mentioned above.

'!his fundamental solution is derived by extending to the dynamic case

of the super~sition techr.ique devised by Mindlin U 8] for the half-space

fundamental solution under static conditions. Mindlin's solution

essentially consists of superposition of a caroinati'''10 of nuclei of strain

[151 derived by synthesis from Kelvin's solution [141. such that the

stresses vanish on the plane boundary of the half-space. Here the same

procedure is employed by starting with the ;Jarmonic oounterf8rt of Kel vin's

solution in the Laplace'S transformed danain (8]. In the follCMing, only

the various ocmp:ments of the residual functions are given. 'n1e problen is

divided into two puts: (1) periodic force normal to tie bOlUldary. and (2)

periodic force tErallel to the boundary.

3.2 PBRImIC PlECE tIllIIU. m 'mE BaJYl\R!

The semi-infinite solid is considered to be bounded by the plane Z .. O.

the positive z-axis penetrating into the body. A perioc1ic dynamic force

peht is applied at point (O,O.+c) and acts in the positive z-direction

(Fig- 3-1(a», where CIJ is the circular frequency and i the imaginary unit.

Since for this case, the displacement and stresses are synmetrical about

the z-axis. the formulations have teen derived in cylindrical coordinate

systems. The solution for this case is campposed of six individual

catqXlnents, which in an unlimited solid, represent: (1) a single periodic

dynamic force at (O.O,+ch (1) a Single periodic dynamic force at (O.O,-ch

3-1

W

I W

IO,O

,-c1

IO

,O,-

c1

-t).

\ \

• x

f ,I

\ ..

x

y

\

Z

I '

\

~t

'R ~

Pe

j,o

t ,~

\

, \

r-,

r ,\

I ~--" ,

I

-~i

~

(x,y

,Z)

z

(a)

Nor

mal

to

th

e B

ound

a ry

FIG

UR

E 3

-1.

y

(tl,o

,+<

:)i

-Wz I'

, \

I "Il

l \

Rl

I ",

\ _

, r

,\

I ----',

\ I

-"

~

(x,y

,Z)

z

(b)

Par

alle

l to

th

e B

ound

ary

Per

iodi

c F

orce

in

the

Inte

rio

r of

a S

emi-

infi

nit

e S

oli

d

(3) a dynamic double force at (O.O.-ch (4) a dynamic center of compression

at (O.O.-ch (5) a line of dynanic centers of canpression extending from z

= -c to z = _CII and (6) a dynamic doublet at (o.O.-c).

1. Single periodic force at (0,0, +c)

'l1le displacements in the r and z directions due to this are {8]:

where Rl is the distance between x and~. In addition,

C3.4d}

O.4e}

where

11 is the shear modulus.

C 1 and <=2 are the pressure and shear wave velocities. respectively. and

s is the Laplace tranf"~ormed ~rameter which is equal to -ifII for the

present case.

:1. Single periodic force at (0.0. ~)

The expressions (U2 and u2 ) due to this are eXactly the same as rz zz

(3.4a) to (3.4e) above. except that Rl now has to be replaced by R2' and

3-4

(3.5)

3. DyncInic dodlle forces at (0.0. -1::)

The displacement components are:

(3.6a)

(3.6b)

where

K = - -..£.. 3 2nll

(3.6c)

(3.6d)

2 e-s~/C2 c2 2 e-s~/ct 3C2 3c2 2 3C t 3Ct B3 = (-- + - + 1) (-- + - + 1) (3.6e)

s2~ s~ ~ 2 s2~ s~ ~ Ct

2 e-s Rz/c2 9<:2 9c2 s~ C3 '" (-- + -- + - + 4)

~ s2~ s~ c 2

3c2 2 e-s~/Ct 2 3c t 3c t -- (-- + -- + 1) (3.6£)

c 2 s2~ s~ ~ 1

3-5

4. Dyn2Inic center of ~essiClll at (O.O,-C)

The displacement components are

where

K _c(I-2\J) 4 - 7111(3-4 \!)

(3.7a)

(3.7b)

(3.7c)

(3.7d)

s. Line of ayn.ic centers of a.pression extending fraa z ... ~ to z .. -

These are obtained by integrating expressions (3.7a) and (3.7b) above

(3.8a)

-.. S J aR U = K. [B"(az' )d~ zz ~ t=-c ~

C3.8b)

where

(3.8c)

BS is the same as B ... above, except that R2 now has to be replaced by

arbitrary R .. (r2+(z+~,2) 1/2. This integral can be only evaluated

nmnerically.

3-6

,. DyJBIic doublet at (O.O.~)

The displacement components are:

(3.9a)

(3.9b)

where

c2

K6 '" - 2l"nd3-4 v ) (3.9c)

(3.9d)

The addition of all the six components produce the complete solution

Un: and Uzz' When s .... O. the expressions reduce identically to Mindlin's

solution (18], of a (static) point force in the interior of a seni-infinite

solid.

In this case there is no axial syrrmetry aJX3 it is therefore convenient

to employ rectangular cartesian ooordinates. (x.y.z). '!he pericxlic force

3-7

Peiwt is applied at (O.O.+C) and acts in the positive x-direction (Fig.

3-lCb). For this case. the solution is composed of six individual

cOlnlX>nents, which in an unlimited solid represent: (1) a single periodic

dynamic force at (O.O.+C). (2) a single dynamic force at (O.O.-c). (3) a

dynamic double force with moment at (D,O.-c). (4) a semi-infinite line of

dynamic double forces with moment extending from z '" -c to Z :. - .... (S) a

dynamic doublet at (O,O,-c) and (6) a semi-infinite line of dynamic

doublets extending from 2 = -c to z = _m with strength proportional to the

distance from z = -c.

1. SiDJle periodic force at (0.0" -fC)

'!he displacement catllXlnents in x. y and z di rections are (8):

u1 xx = Ll

aRl 2 (P1-~(ax ) ) C3.10a)

u1 aRl aRl = Ll (-Q (-)(-»)

yx 1 ay ax C3.1Ob)

1 Uzx = Ll aRl aRl

(-~(az )(ax)] C3.1Oc)

L1 , P1 and Q1 are the same as K1" A1 and B1 in (3.4c). (3.4d) and

(3.4e). respectively.

2. SiDJ!e periodic force at (0.0. -c)

The expressions (U2 "u2 and u2 ) are the same as (3.10a) to (3.10c) xx yx zx

above with R1 replaced by ~. and

3. o.rn-1c dolmle force with &:III!!nt at (O.O.-c)

The displacement components are:

3-8

(3.11)

(3.12a)

(3.12b)

(3.12e)

where

(3.12d)

(3.12e)

(3.12£)

(3.12g)

.t. Line of dynamie double forces with JICIIIeJlt extending frc. z - -c to z ... -These are obtained by integrating expressions (3.12a), (3.12b) and

O.l2e) above

(3.13a)

(3.13b)

(l.Ue)

where.

3-9

(3.13d)

P4' Q4 and T4 are the same as P3, ~ and T3 above. \tiith ~ replaced by

arbitrary R = {x2+y2+(z+e)2}1/2.

5. Dynamic doublets at (O.O.-C)

The displacement COIl1!X'nents are:

(3.14a)

(3.14b)

(3.14cl

where.

(3.14<3)

(3.14e)

(3.Uf)

6. Line of cl.Yn-ic cbmlets extending fraa z ,., -c to z • - with strength proportional. to the distance fran z .. -c

'n1e displacement cxxnponents are:

-. O!x = L6 S~=-c [P6~(:=)2](~-C)d~ <3.15a)

(3.1Sb)

3-10

(3.Be)

where.

L .. _ (1- v)( 1-2v ) 6 "11 (3-4 v)

(J.15d)

£'1 6 and Q6 are the same as P,5 and Q S respectively with R2 replaced by

arbitrary R = {x2+J2+(z+~)2}1/2.

The summation of all the six components furnish the complete solution

Uxx' U and U • The limits -0. reduces to Mindlin's static solution yx zx

[18]. Com];x>nents Uxy. Uyy and Uzy can be similarly obtained.

3-11

This report presents the results of an extensive .~tudy on the seismic

behavior of single piles in a non-dimensionless form. Arrong the group; of

problem parameters influencing the response, most important are: the

stiffness ratio Ep/Es of the pile Young's modulus over a characteristic

Young's modulus of the soil deposit~ the slenderness ratio Lid of the

length over the diameter of the pile frequency, am angle of incidence of

the seismic waves. The dimensionless frequency parmneter is defined as:

wd a = - (4.1) a Cs

where

w circular frequency of excitation

d pile diameter

Cs shear wave velocity

The Young's modulus of soil is assumed to be constant with depth -

typical of stiff overconsolidated clay deposits. '!be Poisson's ratio of

the soil deposit was assumed to be 0.4. The ratio of the density of pile

material to soil was taken as 1.6, typical of concrete piles. Material

damping on was set equal to 0.05.

".2 ~ mE 'lO SlHi\VBS

The motion of the pile under the p.ffects of SH-waves is considered

first. Waves were assumed to produce a unit displacement on the free-

field. The pile was modeled with eleven cylindrical elements. The

displacement and tractions were assumed to be constant throughout each

element.

4-1

The influence of Ep/Es ratio is portrayed in Figs. 4-1(a) and 4-1(b)

for a pile having LId '" 10 and 30 respectively and for a particular

frequency (ao '" O.!). The responses are shown for a vertically incident

SH-wave (angle of incidence '" 90°). '!he figures show the variation along

the depth of the ratio of t.lte total transverse displacement.,t{ to the free­

field transverse displacement, u1 at the ground surface. It is seen that

Ep/Es ratio has a profound e[fect on the responses. at all depths. We

observe high oscillatory 'lariation for a very flexible pile (~/Es '" 100)

and alIOOst low. uniform rigid body IOOtiOr. for a vel.Y rigid pile. Also. for

short piles (LId = 10) bending moment is severe around the bottom mid-half

of flexible piles - region particularly susceptible to fracture and/or

yielding. A comparison between Figs. 4-1(a) and (b) reveals a profound

difference between the behavior of short and long piles. Long piles are

more suscet--tible to oscillatory resp:>nses even for stiffer piles.

Figures 4-2(a) and (b) show similar responses for a short and long

pile respectively. but now for an obliquely incident wave <e .. 60) It is

observed tiut ior both the pile types the lOOtions an: higher as compared to

a vertically incident wave. But stilL the pile movements folIo« identical

pattern.

Figures 4-3(a) and 4-3(b) show the ratio of the total OOrizontal pile

motion. u~ to the free field motion. Ut at the top of a short pile (LId ..

10) to varying angles of incidence (& .. 45°. 60°, 75° and 90°). for a

flexible and rigid pile. respectively. Figures 4-4(a) and 4-4(b) sho.1 the

same variation for a long pile (Lid" 30). It can be seen that the angle

of incidence and the stiffness ratio have a profound influence on the

fil tering effects. It is observed that in the low frequency range. the

filtering effects increase with the angle of incidence. but this is not

true for all frequencies due to oscillations of H.e results. Resonant

4-2

0'1~ =' "-..,n. ='

\.2

(a) Lid 10

1.0

• 8

EriEs = 100 ----- ~ES 1000 ...... ~ ........... piEs 10000

.S

---------• <1 .................................................................. ;;.:.-.~ .. "::':.::: ...................................... -.................... -:.:::.:':':. ... --------- ---.2 \ .e~ __________ ~ __________ J-__________ -L ___________ ~ __________ ~

.90

.<10

• 20

.0 .2 ... .5 .9 1.0

----- ....

Dimensionless Depth, z/L

,- .... " .... " .... " "

---.". ----,.

Dimensionless Depth, z/L

Figure 4-1. Transverse Displacement Ratios for Vertically Incident SH-Waves

4-3

O'~ ::J

...,'0. ::J

1.2~---------------------------------------------------.

1.9

.9

.4

.2

.9

1.6

1.2

.9

. 4

••• ~ ••••• ~ h .... ••••

------------

• Ii! .2

-----................... -_.-

(a) Lid = 10

100 1000 10000

-'

.4

"

-' " ,

'" . •••.•••• •• u ••• :;;~········

" " '"

-' .-

.S

........ ...... .......... ....

.8

Dimensionless Depth, z/L

(b) Lid = 30

~ES = 100

i/:Es = 1000

E~ES = 10000 Ei/Es = 100000 ,

".- ... " , " , .... ~,- .... , , , ..

,,,." ~"",,,/ ' .... "'" ........

1.9

--.:== .. -.:::::::::::::-.~.=--=,=~ .... : .::::::::::: •. ;:::::::::.,:;::.> -. S I I

.S .2 ... .6 .9 1.0

Di~nsionless Depth, z/L

Figure 4-2. Transverse Displgcement Ratios for Obliquely Incident SH-Waves (9 = 60 )

4-4

0''1-4 ::I

.u'o.. ::I

1.29 ---------------------------------------------------------,

1.00

.Be

. 20

,:..

~~, \.. ' ~' \:'\< .. "

'. , •... , :\ ........ ',

". , ~'\ ...... "'I. ". , .... "-

:-" , ,,:: ... ,.. .... ;::.:

9 = e 9 = e =

(a) EIEs

45° 60° 75° 90°

= 1000

.~ .~.~ ...... .

'~ .

. 00 '-~----....L-__ -----'------..l.------<.-----.-J .00 .2B .40 .69 .BB 1.00

Dimensionless Frequency, aa

1.20r---------------------------------------------------------,

1.00

.99

.63

.49

.213

.00

-----.................. ---'-

"­" ,

8 9 8 S

, ,

(u) =' /-:' -'_) J.....os 100000

= 4:;° 6Uo

= 750

90°

\ ..... ''I. \. ... :.~.,~;.~~.--~ .. ~ .. ~

.00 .29 .4e .60 .89

Dimensionless Frequency, ao

Figure 4-3. Transverse Displacement RatiOS for SH-~aves in a Short Pile

4-5

Lee

0"'1<-< :J "-J-Jo. :J

0"'1<-< :J

.u"n.. :J

1.29

1.00

.Be

.S0

.40

.20

". , •.... ,

,:" ",'" :,"'<.', .:\ ..... "

.'. \ .... , :~ ....... ',

.'. , "";-.. "

'::':'::". ' .... ~'"~

(a) EIEs = IGuu

9 = 45° 8 60° 8 = 75° 8 90°

- .. -....-......,'·~I.~ - - - --t'~'I""'"

.eeL-________ ~ __________ L_ ________ _L __________ L_ ________ ~

.00 .20 .4e .S0 .ge 1.00

Dimensionless Frequency, ao

1.20r-----------------------------------------------------~

(b) t:'JE ... , s = 100000

1.00

9 45° ----- 9 = 60°

.se ........... _ ...... 8 75° -_o- S = 90°

.S0

.40

.2e .~ .... , > .... " ~.,~.-

-=:::::. :.:.:.:.:..~.,.~.:..::: ... -= ....... --:::::::=---.ee~------__ ~~------~~------~~------~--------~ .00 .20 .40 .S0 .80 1.00

Dimensionless Frequenc:.' ~ ao

Figure 4-4. Transverse Displacement Ratios for SH-Waves in a Long Pile

4-6

peaks OCCI]r at a lower frequency for a vertically incident wave than

obliquely incident ones. Fil tering effects increase more rapidly for a

very rigid pile than for a flexible one. This is particularly true for

longer piles where the motions rapidly reduce to negligible values at

higher frequencies.

The rotien of sil'X31e piles I.D'lder the influence of a canbination of P

and SV-waves are considered next. As in the case of SH-waves the resul ts

will be presented as a function of the normalized depth. z/L and the

dimensionless frequency parameter. ao' Incident waves are assumed to

produce a unit displacement on the free-field. For vertically incident

waves the pile variations will be equal to that obtained when only an S­

wave is considered (SIJ and SH-waves are the same for e '" 900).

Figures 4-S(a) and 4-S(b) show the variation of axial displacement

t ratio. up versus the free field axial ground motion. ur for an obliquely

incident catbinacion of P and SV-waves (e .. 75°) for a particular frequency

(ao = O.S). for a short and long pile. respectively. Corresponding

transverse displacet!'ent ratios are portrayed in Figs. 4-6(a) and ,,-6(b)'

As in the case of SH-waves. the stiffness ratio and the embedment length

have Significant infl uence on the resp::mses. Whereas. a rigid pile shows

almost a uniform rigid body motion. flexible pGes exhibit oscillatory

movements. resulting in unacceptable arrounts of manents that could cause

yielding and/or fracture of the pile.

The tranaverse displacement ratios u~/Ut at the top of the pile is

depicted in Figs. "-7(a) and 4-7(b) for a short and long pile respectively

with Ep/Es equal to 1000. Here again we observe the tncrease in the

filtering effects with the angle of incidence. For a vertically incident

wave (e - 90°) the results are identical to those in Fig. 4-3(a) and 4-4(a)

4-7

.S0rr------------------------------------------------------~ (a) Lid = 10

100 1000 10000

0>41 =' ....,"-0. .30 ................................................. ~.-.-.-.-.-.--- ___ ._._._._ .......... "".rft.:'T1.~.~.rr..rr.

::l

• IS

• 00 I-:=-------L---.~ .2 .04 .S .8

Dimensionless Depth, z:~

.se r------------------------ - .--(b) Lid = 30

.69

. 20 -----------.-.-.. ~ .. ":'!.~.:::.::.:::.:: ............................................ .

.OO~ ______ ~~ __ ----J-~------~-------L--------~ .9 .2 .4 .6 .s 1.0

Dimensionless Depth, zlL

Figure 4-5. AXial Displacement Ratbos for Obliquely Incident P and $V-Waves (9 • 75 )

4-8

1.2ar---------------------------------------------------------~

1.00

.sa

.S9

------- ---

100 iOOO 10000

/

(a) Lid = 10

................................................. ---------....................... ...................... ...... ..... . . ............................... .

• 49~ __________ ~ __________ ~ __________ ~ __________ ~ _________ ~

.a .2 .4 .6 .8

Dimensionless Depth, z/L

1 .20 .----------------------------------------------- -- - - -

1.00

.8e

100 1000 10000 100000

(b) Lid = 30

" " .'

" .69 -_ "I ... /

" , .-, .-"' , , ............ / ............................ ~.""C....... / ' .... - - - ", ......... 41'!1 ,.......... /

..... "' .. ····· .. ··· .. 1 ....... // . . -.--~ /' ............. .

'<>'------. ;.,.;:;/ '-.--/'

.29

I.e

.OO~ __________ ~ __________ L-________ ~~ ________ ~ __________ ~

.9 .2 .4 .S .9 1.9

Dimensionless Depth, z/L

Figure 4-6. Transverse Displacemen~ Ratios for Obliquely Incident P and SV-Waves (9 = 75 )

4-9

01"-" ::l

->-1'0.. ::l

1.22 r----------------------------- --I (a) Lid = 10

.... '"

. 8a ...•...

.60

• 48

............

...

.......... -', .....

8 ;:; 60° 8 = 75°

8 = 85°

...

'-'.

'"

.ee~ ________ -L~ ______ ~~--------~=_------~~------~ .00 ,20 .4111 .60 .sa 1.00

Dimensionless Frequency, ao

1.20r-----------------------------------, (b) Lid = 30

l.ee 8 60° --- -- 9 = 75°

...... ~.~,

. ................. 8 85° '" ... .....

"" .813 " , "

" ... , .,.. .... ... ....

• S0

.49

... ....

----- ---\\ .....• \ .. \ .. "

.....

• 20 ........

. 00~ ________ -L __________ ~ ________ ~~ ________ ~ ________ __J

.00 .20 . 40 . S0 .80 1. 00

Dimensionless Frequency, ao

Figure 4-7. Transverse Displacement Ratios for P and $V-Waves (yE:.: = 1000)

4-10

when only an SH-wave was considered.

The practical significance of all such curves are apparent: by

mul tiplying a given free-field design response spectrum with the

appropriate interaction curve, one may obtain the deSign response spectrum

that must be input at the base of a structure on pile foundations.

".4 'mANSIENT RESllLTS

For the transient analysis, equations (2.25) and (2.26) represent the

input free-field excitation. '!be excitation frequency, III is set equal to

0.5 rad/sec and the time axis is nonnalized with respect to the she~r wave

velocity (Cs ' and a characteristic le"lgth of the pile as

{ •• l)

Figures 4-S(a) and (b) show the actual transverse displacement. at the mid

length of the pile. for a short and long pile respectively due to

vertically i~ident wave. It is observed that. prese~ of the pile alters

the free-field displacement values substantially. Whereas. rigid piles

show small or negligible movement. flexible piles appear to follow the

ground motions. This observation is more pronounced for longer piles.

Figures 4-9(a) and (b) show the corresIX>nding values for an obliquely

incident wave <e ~ 60°). Hera also rigid piles exhibit negligible

displacement and flexible piles experience higher motion. This phenomena

is more pronounced for longer piles. but still. the magnitudes are much

lower than the corresponding values for a vertically incident wave •

• -11

, .5~ (a) Lid = 10 tREE FIELD

- ---- Ep/Es - 1000 .................. Ep/Es -leeoo -_.- Ep/Es - laoooo

.75

/

.00

/ ,. ... ---~ \: ,.,. " ., .... ,,,,,"" ,"' ...... , .,," ." ... :~.:~ ............ :.~ ..• ~:.::.:.:.:...... -~:.~...... "'--" ... ~:

" ." " ;; "-

',----' \"---'/ ~

-.75

-1.50 L-___________ L-___________ ~------------~~--------~ .00 .25 .50 .75 1. 00

Dimensionless Time, t

4-12

1.00 r-------------------'---------------------------------------,

.sa

. 00

-.5a

,. ,. '-

'" \ \.

FREE FIELD Ep/Es - 1000 Ep/Es - 10000

E"p/Es - 10a000

(a) Lie iu

'" ... / '-

I '" \

\

\ " .:.\ ....• :.~:.~ ....• ··I .•.. :.~-~ \

\

'" " ,. ..... - -""

/ /

/

'I.

/. -------.,...."' .. /

/,

-1.00 ~ ____________ ~ ______________ _L ______________ J_ ____________ ~

.00 .25 .50 .75 1.00

Dimensionless Time, t

1.00 r---------------------------------------------------------~

.5111 -.... /.~ ........ ~,

I..... "'.,," 1" '.\.

i ' .. \ i ..... ,

.I ._. __ ,~ .00

'." ' .. ~ ",

-.5111

FREE FIELD

EptEs - 1000 Ep-'E s - 10000 Ep"E s - 1 eooeB

(b) Lid = 30

\~ .. "" l "', ......... ,.... 1 " ,. .... _,

,. '­

/ " / ., .......... ,.. " I..... "', \

/-." ..... \

/ .... \ i \\

.' '-'--."~ ._.---'.1 \. .... ,

\. /1 "" ,.,' I

'~' .. , ............ './ " .. '-'"

-1.00 ~ ____________ ~ ____________ ~ ____________ ~~~----------~ .00 .~ .~ .~ 1.00

Dimensionless 7ime, t

Figure 4-9. Transverse Displacements for Obliquely Incident Waves (8 = 60 )

4-13

Responses of piles to vertically and obliq'Jely incident ~. SV and P

waves have been analyzed. It is found that the stiffness ratio. angle of

incidence ar'1 the excitation frequency have a significant influence on the

pile rf.sp:mse. While the nurOOer of cases studied is not sufficiently large

to dpcive approximate formulae or general conclusions. it appears that the

exi.;tence of the pile foundation produces a filtering of the waves.

reducing the amplitude of the motion as a function of frequency. Longer

piles are more susceptible to oscillatory response even with higher

stiffness.

For short flexible piles. bending manent is found to be severe around

the bottan half of the pile. PH tering effects increase more rapidly for a

rigid pile than for a flexible one. Resonant peaks occur at a lower

frequency for a vertically incic.ent wave than obliquely incident ones.

Finally. the pilot transient analysis also indicates that the presence of

the pile modifies the ground motion substantially.

The interaction curves presented in this report have significant

practical utility. A deSign response spectrlml for a structure resting on a

pile foundation may be readily obtained by multiplying a given free-field

design response spectrum with the appropriate interaction curve.

5-1

ao dimensionless frequency parameter C'i!fined ty equation (4.1)

Al to A6 expressionz defined in section 3.)

~ cross sectional area of pile !'lase

~ cross sectional area of pile

ASH amplitudes of inCident SH waves

, AsH amplitudes of reflected SH waves

teJ matrix defined in equation (2.1S)

BI to B6 expressior~ defined ir Section 3.1

displacement vector for unit pile head h.c.

known vector defined by equation (2.11)

c depth of source fran free surface of half space

pressure wave velocity

shear wave velocity

d pile diameter

matrix defined in equation (2.18)

In] pile compressibility and flexibility matrix

(El matrix defined in equation (2.17)

6-1

Ep Young'S modulus of pile material

ES Young's modulus of soil material

11 parameter defined by equation (2 .13b)

(~i) vector of pile-soil interface tractions

(~) axial pile stress at base

{dx} vector of lateral pile traction

{dz } vector of axial pile traction

{~~} vector of total field pile tractions

{d~} vector of free field soil tractions

CdS} vector of scattered field soil tractions s

(d;) vector of total field soil tractions

y parameter defined by equation (2.13a)

i imaginary unit. /-1

Ip moment of inertia of pile cross section

Kl to K6 constants defined in section 3.1

1 direction cosine of propagation of 58 wave

L length of pile

Ll to L6 constants defined in section 3.1

6-1

Lame's oonstant

m mass per unit length of pile

r·l( z) bending manent at depth z of pile

Lame's oonstant

n direction oosine of propagation of SH wave

Poisson's ratio

P 1 to p 6 expressions def ined in Section 3.2

Px(Z) '..ransverse load at depth z of pile

Pz(z) axial load at depth of z of pile

n 3.141592654

~ to Q6 expressions defined in Section 3.2

r canpment in polar axes system

R1 distance between field plint and real source point

~ distance between field point and image source point

[R •. ] residual terms in Green's function 1J

p density of pile naterial

s Laplace's transfocn parameter

t time

t dimensionless time defined t¥ equation (".2)

6-3

. ,

inclination angle of seismic waves

U1 pile head displacement

Uij displacements in ith direction due to forc~ in jth direction

Ui acceleration c~lponent in ith direction

{uxl lateral pile displacements

{uzl axial pile displacements

(u~) vector of total field pile displacements

{uf] vector of free field soil displacements s

{U:} vector of scattered soil displacements

{U~} vector of total soil displacements

~ circular frequency

integration parameter

6-4

1. Ahmad. S. and Manolis. G.D .. 'Dynamic Analysis of 3-D structures by a Transforoed BOundary Element Method.' Cc::Irputational Mechanics. Vol. 2, 1987. pp. 185-196.

2. Apsel. R.. 'Dynamic Green's Functions for Layered Media and Applications to Bouneary Value Problems,' Ph.D. '!besis. Dept. of ~l. Mech. and Eng. Sci •• univeristy of california. San Diego. 1979.

3. Banerjee, P.K. and Butterfield. R., Boundary Element Methods in Engineering Sciences. Mc-Graw Hill. LOndon and New York. 1981.

4. Banerjee, P.K., 'Analysis of Axially and Laterally Loaded Pile Groups.' in C.R. Scott (ed.) Developments in Soil Mechanics. Applied Science Publishers, u>ndon. Olapter 9. 1978. pp. 317-346.

5. Banerjee. P.K. and Driscoll. R. ..... 'Three-dimensional Analysis of Raked Pile Groups,' Proc. lnstn. of Civil Engrs., Part 2. Vol. 61. 1976. pp.653-671.

6. Banerjee. P.K. and Sen. Ro. Dynamic Behavior of Axially and Laterally Loaded Piles and Pile Groups. Chapter 3 in Dynamic Behavior of Foundations and Buried Structures (Developments in Soil Mech. Foundation Engineering, Vol. 3). Ed. P.R. Banerjee and R. Butterfield. Elsevier A(:plied SCience, wOOon. Woo 95-133, 1987.

7. Banerjee. P.K., sen, R. and Davies, T.G .. static and Dynamic Analyses of Axially and Laterally Loaded Piles and Pile Groups, Chapter 7 in Geotechical Modeling and Applications (Dean Alexander Vesie Memorial Volume), Ed. S.M. Sayed, Gulf Publishing Co .. Houston. pp. 322-35". 1987.

8. Cruse, T.A. and Rizzo, F.J., 'A Direct Formulation and Numerical SOlution of the General Transient Elastodynamlc Problem, I.' Journal of Anal. and Appl. Math., Vol. 12. 1968. pp. 2""-259.

9. Dominguez, J and Roesset, .J.M., 'Response of Embedded Foundations to Travelling Waves.' Report No. R78-H, Dept. of Civil Engr .. MIT. cambridge, MA, 1978.

10. Doyle,J.M., 'Integntion of the Laplace Transformed Equations of Classical Kinetics, ' Journal of Awl. Math., Vol. 13, 1966.

11. Flores-Berrones. R. and Whitman. R.V .. 'Seismic Responses of End Bearing Piles,' Journal of Geotech. Eng., ASCE, Vol. 108, GT4, 1982. pp. 554-569.

12. Gazetas, G., 'Seismic Responses of End Bearing Single Piles.' Soil DyMnics and Farthq. Eng., Vol. 3, No.2, 1984, pp. 82-93.

13. Kaynia, A.M. and Kausel, E .. 'DynamiC Behavior of Pile Groups,' 2nd Int. Conf. on Num. ~thods in Offshore Piling. Austin. '!'X. 1982.

7-1

14. ReI vin. (Lord). 'Displacements Due to a Point Load in an Indefinitely Extended Solid.' Math. and Phys. Papers. 1.On<bn. VoL 1. 1848. pp. 97.

B. LOVe. A.E.H.. A Treatise on the Mathematical 'lbeory of Elasticity. 4th Ed., Carrbridge University Press. London. 1952.

16. Mamoon, S.M •• Al"Inad, S •• and Banerjee, P.R.. 'seismic Behavior of Pile Foundations.' status Report. National Center for Earthquake Engineering Research. State University of New York at Buffalo. May 1988.

17. Michalopoulos. E.. 'Anl'lification of Generalized Surface waves.' Mo8. '!besis, DePlrtment of Civil Engineering. MIT. Carttlridge. MA, 1976.

18. Mindlin. R.D •• 'Force at a Point in the Interior of a Semi-infinite SOlid.' Journal cf Applied Physics. Vol. 7. 1936, pp. 195-102.

19. Poulos. B.G. and Davis. E.H .. Pile Foundation Analysis and Design, John Wiley and Sons. Inc •• New York. New York. 1980.

20. Sen, It.. Davies. T.G.. and Banerjee. P.K.. 'Dynamic Analysis of Axially and Laterally Loaded Piles and Pile Groups Embedded in Hanogeneous Soils.' Journal of Earthq. Eng. and struct. ~. Vol. 13. 1985. pp. 53-65.

21. Sen. R., Kausel. E .. and Banerjee, P.R •• 'Dynamic Analysis of Piles and Pile Groups in Non-homogeneous Soils.' Int. Journal of Num. and Anal. Meth. in Geanech •• Vol. 9. 1985. 1'1'. 507-524.

22. Wong. H.L. and Luco. J.E.. 'Dynamic Response of Rectangular Foundations to Cl>liquely Incident Seismic Waves.' Journal of Earthq. fng. and Struct. I>jn.snics. Vol. 6. 1978.

7-2

NATIONAL CEl'oTER FOR EARTHQUAKE ENGISEERING RESEARCH LIST OF PUBLISHED TECHmCAL REPORTS

The National Cenler for Earthquake Engineering Research (NCEER) publishes technical reports on a variety of subjects related to earthquake engineering written by authors funded through NCEER. These reports are available nom both NCEER's Publications Department and the National Technical Information Service (NTIS). Requests for reports should be directed 10 the Publications Department, National Center for Earthquake Engineering RCiearch. State University of New York at Buffalo, Red Jacket Quadrangle. Buffalo, New York 14261. Repons can also be requested through NTIS. 5285 Port Royal koad. Springfield. Virginia 22161. N11S accession numbers are shown in parenlhesis. if available.

NCEER-87-0001

NCEER-87-0002

NCEER87·0003

NCEER·87-0004

NCEER-87-0005

NCEER-87 -0006

NCEER -R7 ·0007

NCEER-R7-0008

NCEER -87-0009

NCEER-87·0010

NCEER-87·0011

NCEER-87-0012

NCEER-87-OO13

NCEER-87 -0014

N('E~R-87-OO1.5

NCEER-87-OO16

"First- Year Program in Research. Education and Technology Transfer,'· 3/5/87, (PB88-134275IAS).

"Experimental Evaluation of Jrut.antaneous Optimal Algorithms for SlrUctural Control," by R.C. Lin. T.T. Soong and A.M. Reinhorn, 4/20/87, (PB88-134341/AS).

"Experimentation Using the Earthquake Simulation Facilities at Univenity al Buffalo," by A.M. Reinhorn and R.L. Ketter, to be published.

·'The System Characteristics and Performance of a Shaking Table,'· by J.S. Hwang. K.C. Chang and G.C. Lee, 6/1(81, (PB88-134259IAS).

·'A Finite Element Formulation fOT Nonlinear Viscoplastic Material Using a Q Model,'· by o. Gyebi and G. Dasgupta, 11/2187, (PB88-213764/AS).

"Symbolic Manipulalion ProgTam (SMP) - Algebraic Codes fOf Two and Three Dimmsional Finite Element Formulations," by X. Lu ar.d G. Dasgupta. IlfJt87, (PB88-219522/AS).

"[rul811taneous Optimal Control Laws for TaU Buildings Under Seismic ExcitatioN,'· by IN. Yang. A. Alc.b.Irpour and P. Gha.emmaghami, 6/10187, (PB88·134333/AS).

"lDARC: Inelastic Damage Analysis of Reinforced Concrete Frame - Shear-Wall SUUCURS,'· by Y.J. Park, A.M. Reinhom and S.K. KlIIUlath. 7/20/87, (PB88-134325/AS).

"Liquefaction Potential for New York State: A Preliminary Report on Sites in Manhattan and Buffalo," by M. Budhu. V. Vijayakumar, R.F. Giese and L. Baurngns, 8/31187, (pB88-1637041AS). This report is Available only Ihrough NTIS (see address given above).

"Vertical and Torsional Vibration of Foundations in Jnhomogeneous Media," by A_S. Veletsos and K.W. Dotson, 6/1/87, (PB88-1342911AS).

"Seismic Pr<>babilistic Risk Assessment and Seismic Margins Studies fOT Nuclear Power Planu:' t:' Howard H.M. H .... ang. 6/15187, (PB88-134267(AS). This report is available only through NTIS (see address given above).

"Parametric Studies of Frequency Respon~ of Secondary Systems Under Ground-Acceleration Excitations," by Y. Yong and Y.K. Lin, 6/10/87, (PB88-134309/AS).

··Frequency Response of Secondary Systems Under Seismic Excitation. .. by J.A_ Hol.un&. 1. Cai and Y.K. Lin, 7f3l187, (PB88-134317/AS).

"M.xIelling Ear1hquake Ground Motions in Seismically Active Regions Using Parametric Time Series Methods," byG.W. Ellis and A.S. Cakmlk. 8'2.5/87, (PB88-134283/AS).

"Detection and Assessment of Seismic SlrUctural Damage." by E. DiPasquale and A.S. Cakmak. 8/25/87, (pB88-1637121AS).

"Pipeline Experiment a' Parkfield. California," by J. Isenbeq md E. Riclwdlon. 9115/87, (pB88-16371O/AS)_

A-I

NCEER-87-OO11

NCEER87-0018

NCEER-87 -0019

NCEER-81-0020

NCEER-81-0021

NCEER-81-0022

NCEER-87-0023

NCEER-87 -0024

NCEER -81-0025

NCEER -87 -0026

NCEER-87 -0027

NCEER-87 -0028

NCEER-88-0001

NCEER-88-OOO2

NCEER-8:; -0003

NCEER-88-0004

NCEER -88-0005

NCEER-88-0006

NCEER-88-OOO7

"Digital Simulation of Seismic Ground Motion," by M. Shinozuka. G. DcOOatis and T. Harada, 8/31/81, (pB88-ISS197/AS). This report is available only through NTIS (see address given above).

"Practical Considerations for Structural Control: System Uncertainty, System Time Delay and Trunca­tion of Small Control Fon:es," IN. Yang and A. Akbarpour, 8/10/87, (PB88-163138IAS).

"Modal Analysis of Nondassically Damped Structural Systems Using Cllllonical Transformation," by IN. Yang, S. Sarkani and F.x. Long, 9(11/87, (PB88-187851/AS).

"A Nonstationary Solution in Random Vibration Theory," by l.R. Red-Horse and P.O. Spanos, 11[3/87, (PB88-163746/AS).

"Horizontal Impedances for Radially Inhomogeneous Viscoelastic Soil Layers," by A.S. Veletsos and K.W. Dotson, 10/15/87, (PB88-150859/AS).

"Seismic Damage Assessment of Reinforced Concrete Members," by Y.S. ChWlg, C. Meyer and M. Shinoruka, 10/9/87, (PB88-150861/AS). This report is availabl~ only through NTIS (see address given above).

"Active Structural Control in Civil Engineering," by T.T. Soong. 11111/87, (PB88-187118/AS).

Vertical and Torsional Impedances for Radially Inhomogeneous Viscoeiastic Soil Layers, H by K.w. Dotson and A.S. Veletsos, 12/87, (PB88-1877;;6/AS).

"Proceedings from Ihe Symposium on Seismic Hazards, Ground Motions. Soil-Liquefaction and Er:gineering Practice in Eastern North America," October 20-22, 1987, edited by K.H. Jacob, 12/81, (pB88-1881ISIAS).

"Report on Ihe Whittier-Narrows, California, Eanhquake of October 1, 1987," by J. Pantelic and A. Reinhom, 11/87, (PB88-lti1152/AS). nus report is available only through NTIS (see address given above).

"Design of a Modular Program for Transient Nonlinear Analysis of Large 3-D Building Structures," by S. Srivastav and l.F. Abel, 12/30/87, (PB88-187950/AS).

"Second-Year Program in Research. Education and Teclmology Transfer," 318188, (PB88-219480/AS).

''Workshop on Seismic Computer Analysis and Design of Buildings With Interactive Graphics," by W. McGuire, J.F. Abel and C.H. Conley, 1/18/88, (PB88-187760IAS).

"Optimal Control of Nonlinear Aexible Structures," by J.N. Yang, r.x. Long and D. Wong, 1(12/88, (PB88-213n2/AS).

"Substructuring Techniques in the Time Domain for Primary-Secondary Structural Systems," by 0.0. Manolis md G. Juhn, 2/10/88, (PB88-2l3"180/AS).

'1terative Seismic Analysis of Primary-Secondary Systems," by A. Singhal, L.D_ Lutes and P.O. Spmos, 2/23/88, (PB88-213798/AS).

"Stochastic Finite Element Expansion for Random Media," by PoD. Spanos and R_ Ghanem, 3/14188, (pB88-213806/AS).

·CombininB Structural Optimiulion md Structural ConIrol," by F.Y. Chen& and CoP. Panlelides, 1/10188, (pB88-213814/AS)_

"Seismic Performance Assessment of Code-Designed Structures," by H_H-M_ Hwq, '-W. Jaw IIId H-I. Shau, 3f20/88, (PB88-219423/AS)_

A-2

NCEER-88-0008

NCEER-88-0009

NCEER-88-00IO

NCEER-88-OUll

NCEER-88-0012

NCEER-88-0013

NCEER-88-0014

NCEFR-88-0015

NCEER-88-00J6

NCEER-88-0017

NCEER-88-0018

NCEER-88-0019

NCEER-88-0020

NCEER .R8-OO21

NCEER-88-0022

NCEER-88-OO23

NCEER-88-0024

NCEER-88-OO25

NCEER-88-0026

NCEER-88-0027

"Reliability Analysis of Code-Designro Structures Under Natural H82IIJds," by H.H-M. Hwang, H. Ushiba and M. Shinozuka. 2/29/88, (PB88-229471/AS).

"Seismic Fragility Analysis of Shear Wall Structures," by J-W Jaw and H.H-M. Hwang, 4/30/88.

"Base Isolation of a Multi-Story Building Under a Harmonic Gro.md Motion - A Comrarison of Performances of Various Systems," by F-G Fan, G. Ahmadi and l.G. Tadjbakhsh, 5/18/88.

"Seismic Floor Response Spectra fOT a Combined System by Green's Functions." by F.M. Lavelle, L.A. Bergman and P.D. Spanos, 5{1)88.

"A New Sollltion Technique for Randomly Excired Hysteretic SlrUctUTes," by G.Q. Cai and Y.K. Lin. 5/16188.

"A Study of Radiation Damping and Soil-Structure Interaction Effects in the Centrifuge," by K. Weissman, supervised by J.H. Prevost, 5/24/88.

"Parameter Identification and Implementation of a Kinematic Plasticity Model for Frictional Soils,"' by I.H. Prevost and D.V. Griffiths, 10 be published.

"fwo- and Three- Dimensional Dynamic Finite Element Analyses of the Long Valley Dam," by D.V. Griffiths and 1H. Prevost, 6/17/88.

"Damage Assessment of Reinforced Concrete Structures dl Eastern United States," by A.M. Reinhorn. MJ. Seidel, S.K. Kwmalh and Y.J. Patk, 6/15)88.

"Dynamic Compliance of Vertically Loaded Strip Foundations in Multilayered Viscoelastic Soils," by S. Ahmad and A.S.M.1srail, 6/17/88.

"An Experimental Study of Seismic Structural Response With Added Viscoelastic Dampers," by R.C. Lin, Z. Liang, T.T. Soong and R.H. Zhang. 6/30/88.

"Experimental Investigation of Primary - Secondary System InleractiOll," by G.D. Manolis, G. Julm and A.M. Reinhom, 5/27/88.

"A Response Spectrum Approach For Analysis of Nonclassically Damped Structures," by J.N. Yang, S. Sarkmi and F-X. Long, 4122188.

"Seismic InLeraction of Structures and Soils: Stochastic Approa:h," by A.S_ Veletsos and A.M. Prasad. 7121/88.

"Identification of the Serviceability Limit Stale Uld Detection of Seismic SlnIctunl DlITlage," by E. DiPasquale and A.S. Cakrnak, 6/15188.

"Multi·Hazard Risk Analysis: Case of a Simple Offshore Structure," by B.K. Bhartia and E.H. Vanmarc1r.e, 7/21/88.

"Aulomated Seismic Design of Reinforced Concrete Buildings," by Y.S. Chung, C. Meya- IItd M. Shinolltka, 7(5{88.

"uperimental Srudy of Active Control of MDOF Structures Under Seismic Excitations," by L.L. Chung, R.C. Lin. T.T_ Soong Uld A.M. Reinhom, 7110188.

"EarthquUe Simulation Tesll of • Low-Rise Metal SIrUCIUnI," by 1.S. Hwang, K.C. Chilli, G.C. Lee and R.L. KetLer, 811188.

"Systems Study of UrbIII Response md RcoonstruCtion Due to Catastrophic Earthquakes," by F_ Kozin and H.K. Zhou, 9(2.2188,10 be published.

A-3

NCEER-88-OO28

NCEER-88-oo29

f"-':EER-88-oo30

NCEER -88-0031

NCEER-88-oo32

NCEER-88-oo33

NCEER-88-oo34

"Seismic Fragility Analysis of Plane Frame StnlCl\lreS," by H.H-M. Hwang and Y.K. low, 7/31/88.

"Response Analysis of Stochastic SlrUctures," by A. Kardara, C. Bucher and M. Shinozulca, 9f22!88, In

be published.

"Nonnorrnal Accelerations Due to Yielding in a Primary Structure," by D.C.K. Chen and L.D. Lutes, 9/19/88.

"Design Approaches for Soil-Structure Interaction," by A.S. Veletsos, A.M. Prasad and Y. Tang, to be published.

"A Re-evaluation of Design Spectra for Seismic Damage Control," by C.l. Turkstra and A.G. Tallin. Iln/SS.

'The Behavior and Design of Noncontact Lap Splices Subjected to Repeated Inelastic Tensile Loading," by V.E. Sagan, P. Gergely and R.N. While, 10 be published.

"Seismic Response of Pile Foundations," by S.M. Mamoon, P.K. Banerjee and S. Alunad, 11/1/88.

A-4