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Kajima - CUREe Research Project

Building-Foundation-Soil Interaction Effects

By

Dr. Kenji Miura

Pof. G. R. Maftii

Dr. Yuji Mi.yarnoto

Prof. Ronaldo I. Borfa

Ivir. Ariyoshi Yamacia

Prof. H.Allison Smith

Mr. lviasayuki Nagano

Dr. i.P. Bardet

Mr. Yuji Sako

Dr. S.W.Chi

Mr. Yó Hyodo Dr. Q. 1-luang

Mr. Kiyoshi Masuda

lVii. Tatsuya Maeda

Mr. Eiji Kitamura

Mr. Kiichi Suzuki

Mr. YasutsugL Suzuki

Mr. AtsunobuFukiioka

1992.8

Kajima Corporation

CUREe

Kajima Corporation

CUREe;

California Universities for Research

in Earthquake Engineering

* Kajima Institute of Construction Technology

* Information Processing Center

* Structual Department, Architectual Design Division

* Civil Engineering Design Division

* Kobori Research Complex

* The California Institute of Technology

* Stanford University

* The University of California, Berkeley

* The University of California, Davis

* The University of California, Irvine

* The University of California, Los Angeles

* The University of California, San Diego

* The University of Southern California

SUMMARY REPORT

CUREe-Kajima Research Project

Nonlinear Soil-Structure Interaction

Geoffrey R. Martin, Team Leader

PROJECT OBJECTIVES

Many high rise buildings are under construction or being planned on the reclaimed coastal area

around Tokyo and San Francisco bays. These buildings usually have a pile group foundation or a

I) composite foundation, consisting of an exterior wall foundation and internal pile group. Because

of bearing or lateral capacity concerns, the behavior of structures built on such foundations is

greatly influenced by nonlinear soil-foundation-structure interaction during strong earthquakes. In

cases where liquefiable sand deposits occur in foundation soils, further complications arise during

soil-foundation interaction due to pore pressure increases in the saturated sands. An improved

understanding of the effects of nonlinear soil-foundation interaction is needed in order to develop

improved design methods for these types of foundations.

The objectives of this project were to evaluate the ability of nonlinear numerical models to simulate

the complex soil-foundation-structure interaction behavior during strong earthquake shaking. The

collaborative research program was formulated as having three main task areas:

1. A Kajima team project examining the earthquake response of buildings on a pile or

composite foundation system using two-dimensional finite element methods or beam/Winkler

spring models. A specific building (the K-Building) was used for several of the analyses. This

building has 34 stories and a basement supported on an exterior wall foundation and internal piles.

Fckmdation soils comprised liquefiable sand layers and soft clay deposits to depths of about 23

rneters"overlying dense sands. The exterior wall foundation and piles were founded in the

1

underlying dense sand stratum. An earthquake time history with a maximum acceleration of about

0.4g was used to simulate strong earthquake motion.

The second task, undertaken by the University of Southern California (USC) team, also

focused on the use of nonlinear finite element analyses of the K-Building. In these studies, the

numerical simulation of nonlinear soil behavior was accomplished through the use of a simplified

plasticity-based bounding surface theory incorporating a Drucker-type failure surface (BDP

model). The effective stress formulation used simu1ated earthquake-induced pore pressure

increases and allowed simultaneous diffusion.

The third task, undertaken by the Stanford team, focused on the development of a

methodology for evaluating nonlinear soil-structure interaction effects in the time domain for rigid

foundations on soft cohesive soils. In this study, the dynamic response of rigid foundations on an

elasto-viscoplastic-space was investigated using a linear combination of isotropic and kinematic

hardening to model the soil constitutive response.

I) (I) &VLS1t1I1

1. Kajima Team Project

The studies undertaken by the Kajima research team (1) included the following:

A verification study of a simplified numerical model for a soil-pile-foundation system using

beam-Winkler spring models. In this study, centrifuge test results for a 1/50 scale model

consisting of a rigid pile cap and a four pile group embedded in dry or saturated sand was

used to verify the model. The centrifuge test results conducted at the California Institute of

Technology were reported in a previous Kajima/CUREe project report (2). The mod4used

2

to simulate pore pressure increases in the saturated sand under earthquake shaking was that

used in the computer program DESRA.

A numerical simulation of the K-Building response to strong earthquake ground motion

utilizing a simplified stick model to simulate the superstructure and pile foundations, and a

base rotational spring to simulate the rotational stiffness of the assumed rigid exterior wall

foundation. The foundation system was driven by free field earthquake ground motions

(including the effects of pore pressure increases) computed using the DESRA program.

Displacement time histories from free field response were applied to nonlinear Winkler

springs coupled to the simplified foundation model.

A study using two-dimensional nonlinear finite element models to simulate the results of the

centrifuge model tests comprising the rigid pile cap and four pile group in dry sands.

2. USC Team Project

The studies undertaken by the USC research team (3) included the following:

Calibration of the BDP constitutive model by comparison with the centrifuge test results (2)

previously described above. These tests modeled a foundation supported by four piles in

saturated Nevada sand. Pore pressures and pile cap accelerations were monitored during

earthquake excitation in the centrifuge tests. The BDP model soil constants were obtained

from existing laboratory test data on the Nevada sand. Results of analyses were compared to

both free field and pile cap response as measured during centrifuge tests in a similar manner

to the calibration tests conducted by the Kajima team.

The application of a two-dimensional nonlinear finite element analysis to model the

earthquake response of the K-Building. In these analyses, the foundation system comprising

3

the wall/pile basement system was modeled using the foundation parameters supplied by the

Kajima team. The building structure was modeled as a stick model, again using the data

supplied by the Kajima team. The soil pore pressure response during strong ground shaking

was assessed using the BDP model.

3. Stanford Research Team

The studies undertaken by the Stanford team included the following:

Analyses of the nonlinear response of vertically oscillating circular, square and rectangular

foundations to apply harmonic loads using large scale nonlinear finite element analyses. The

constitutive model used for the supporting soil was a deviatoric elasto-viscoplastic model capable

of replicating closed ioop steady state cyclic stress-strain response. The supporting soil medium

was assumed to be a half space that could undergo plastic deformation in the vicinity of the

foundation. The large finite element mesh used was sufficient to simulate radiation damping

without the use of an artificial boundary operator.

Additional studies were performed of the horizontal rocking and torsional vibration modes

of strip and square foundations.

SUMMARY OF RESULTS AND CONCLUSIONS

The results of team research studies may be summarized as follows:

Kajima Team

1. The numerical model comprising beam elements and nonlinear lateral Winkler springs

(taking into account the changing effective stress) successfully predicted the centrifuge test results

4

of the pile foundation model in dry or saturated sand. The model effectively represented the

response characteristics of a pile foundation in nonlinear liquefied soil.

Analyses of the earthquake response of the K-Building supported on composite foundation

showed that response was strongly affected by nonlinear liquefied soil-foundation interaction

during strong earthquake shaking. Response of the superstructure changed depending on the

magnitude of input motion and nonlinearity of surrounding soil. The composite foundation

response was significantly changed by large deformation and degraded resistance of the soil due to

liquefaction.

The proposed two-dimensional finite element model incorporating soil nonlinearity, where

the soil domain is divided into multi-regions according to the degree of soil-pile interaction effects,

was successfully verified by correlation analyses using the centrifuge test results of a pile

foundation in dry sand.

USC Team

The two-dimensional finite element simulation of the centrifuge pile model, utilizing the

BDP model to describe the saturated sand response, was reasonably successful in the case of the

dry sand experiments. However, the simulation of the pile group response in saturated sands

showed lesser agreement, as computed pore pressures diffused less rapidly than measured ones

since the two-dimensional drainage simulation was more constrained than the experimental three-

dimensional drainage. Consequently, computed model response was characterized by stiffer soil-

pile interaction than observed experimentally.

The application of the two-dimensional finite element model to predict the dynamic

response of the K-Building produced results in general agreement with those obtained by the

Kajima team, who used different numerical approaches. However, the limitations of two-

dimensional simulation of pore pressure response were less apparent than in the centrifuge test

simulations as the structural response of the K-Building is strongly controlled by the assumption

that the external foundation wall and piles are elastic and are founded on elastic soil layers.

5

Stanford Team

Analyses of the nonlinear response of vertically oscillating circular, square and rectangular

foundations to harmonic loads showed that for low frequency excitations, resonance amplifies the

motions of the foundation to levels above those obtained at the zero frequency level. Horizontal

rocking and torsional vibration modes of strip and square foundations are characterized by

increased vibrational amplitudes due to material stiffness degradation.

Nonlinear soil effects are shown to be dominant over a wide range of excitation frequencies

for foundations vibrating in torsional and horizontal modes, but over a much narrower range of

frequencies for vertical and rocking modes.

RESEARCH NEEDS

The following research needs were identified as a result of team discussions following the

conclusion of the research program:

The need for improved modeling of local nonlinearity around piles (Winkler spring

simulation) during earthquake induced pore pressure increases, including the effects of pore

pressure diffusion.

The need for improved one-dimensional free field response models to include simulation of

post-liquefaction site response.

The need for improved constitutive models for three-dimensional saturated cohesionless

soil behavior capable of simulating pre-liquefaction and post-liquefaction behavior for use in finite

element models.

11

4. The need for improved two-dimensional finite element hybrid models to more correctly

simulate three-dimensional behavior of pile foundations.

REFERENCES

Miura, Kenji, et. al. "Building-Foundation-Soil Interaction Effects." Final Project Report,

Kajima-CUREe Project, February, 1992.

Scott, R.F. and Hushmand, B. "Soil-Pile Response in the Santa Cruz (Loma Prieta)

Earthquake, October 17, 1989." Kajima-CUREe Report, California Institute of Technology,

University of California, Davis, 1991.

Bardet, J.P., et. al. "A Study of S oil-Pile- Structure Interaction in Liquefiable Soils During

Strong Earthquakes." Kajima-CUREe Project Report, University of Southern California, August,

1992.

Borja, R.I. and Allison-Smith, H. "A Methodology for Nonlinear Soil-Structure

Interaction Effects Using Time Domain Analysis Techniques." CUREe-Kajima Research Project

Report, Stanford University, 1992.

Kajima-CUREe Project

Building-Foundation-S oil Interaction Effects

Topic3-1 ;A Study of Nonlinear Soil-Pile-Structure Interaction in Soft Soils

during Strong Earthquakes

Topic3-2;A Methodology for Nonlinear Soil-Structure Interaction Effects

Using Time-Domain Analysis Techniques

Final Project Report

Kenjl Miura Kiyoshi Masuda

Yuji Miyamoto Tatsuya Maeda

Ariyoshi Yamada Eiji Kitamura

Masayuki Nagano Kiichi Suzuki

Yuji Sako Yasutsugu Suzuki

Yo Hyodo Atsunobu Fukuoka

(Kaj ima Corporation)

February 1992

Contents

Introduction . 1

Simplified numerical model of a pile foundation and verification study ..................2

2.1 Numerical model of a pile foundation in dry or saturated soil .....................2

2.2 Results of correlation analysis .....................................................................12

Earthquake response of a building on composite foundation ..................................23

3.1 Description of the building ...........................................................................23

3.2 Input ground motion for earthquake response analysis ................................23.

3.3 Free ground response by the effective stress analysis ..................................28

3.4 Earthquake response of K-Building .............................................................41

2-Dimensional finite element model and verification study .....................................91

4.1 Numerical model of a pile foundation in dry soil ........................................91

4.2 Results of correlation analysis .....................................................................98

Conclusions .................................................................................................................106

References ......................................................................................................................107

1. Introduction

Many high rise buildings are under construction or being planned on the reclaimed

coastal area around Tokyo and San Francisco Bays. These buildings usually have a pile

group foundation or a composite foundation which consists of wall foundation and pile

group because of increasing bearing capacity and stability of a foundation. The behavior of

structures built on such foundations is greatly affected by nonlinear soil-foundation

interaction during strong earthquakes. Particularly in liquefiable sand deposits, further

complicated interaction between foundation and the surrounding saturated soil will occur

because of pore water pressure generation and large displacement. An improved aseismic

design method is then required to take into account nonlinear soil-foundations interaction

on earthquake response.

The earthquake response of buildings on a pile/composite foundation during strong

earthquakes is usually studied using a 2-dimensional finite element method or a beam-

Winkler spring model. In this study, two analytical methods are employed to investigate

earthquake responses supported on a pile/composite foundation. In Chapter 2, a practical

simplified numerical model of building supported on a pile/composite foundation in non-

linear liquefiable soil deposit is proposed and verified by comparison with centrifuge tests.

In Chapter 3, earthquake response analyses for representative building on a composite

foundation are performed and discussed to develop improved understandings of the

behavior during earthquake. Further, in Chapter 4, 2-dimensional finite element technique

incorpotating soil nonlinearity is proposed and verified by comparison with centrifuge

tests.

- 1 -

2. Simplified numerical model of a pile foundation and verification study 41 Numerical model for calculating earthquake response of a building supported on pile

group/composite foundation indicates in Fig.2.1. Correlation analyses for the centrifugal

test results of a pile foundation model as shown in Fig.2.2 were conducted to verify

the proposed numerical model. The centrifuge tests for verification studies have been

conducted on a 1/50 scale model consisting of a rigid pile cap and a four-pile group

embedded in dry or saturated fine sand in the previous Kajima/CUREe project1).

2.1 Numerical model of a pile foundation in dry or saturated soil

In this analysis procedure, the nonlinear effective stress method, computer program

DESRA proposed by Firm et a12), is utilized for the free ground response. The obtained free

ground responses, including displacements and excess pore water pressures, are applied to

the soil-pile foundation system as earthquake input. Superstructure and pile foundation are

idealized by a one-stick model with lumped masses and bending-shear elements. The

lumped masses are connected to free ground through nonlinear Winkler springs modified

at each step in accordance with the generation and dissipation of excess pore water

pressure. The linear rotational spring, related to the axial stiffness of the piles, is also

incorporated at the pile head.

Analysis of free ground

The computer program DESRA is based on the shear stress-strain relationship of the

Hardin-Dmevich model with the Masing rule expressed as

= G y/(1+G0 (2.1)

in which 'c = shear stress, tmaX = shear strength, y = shear strain, and G0 = initial shear

modulus. The initial shear modulus G0 and shear strength tmax are modified progressively

for the changing vertical effective stress of saturated sand subjected to an earthquake.

Physical constants employed in correlation analyses for the saturated sand are indicated in

— 2 —

Table 2.1. The initial shear modulus G0 employed in the Hardin-Drnevich model is

estimated from

G0 =A ((2.17-e)2/(1+e)) ('m0)1a (2.2)

in which e = void ratio, and 'mO = initial mean effective stress3 . The coefficient A,

obtained by resonant-column tests, is modified to correspond with the resonance frequency

of the free ground in the low input test. The shear strength 'CmaX is obtained by the Mohr-

Coulomb yield condition.

The pore water pressure generation Au is defined by

Au = Er AF,d (2.3)

S where

AF-,d = C1(y - C2EVd) + C3E2Vd / (Y+C4EVd) (2.4)

Er = ((Y')m1'm1C2 (')fl.m (2.5)

in which Er = one-dimensional rebound modulus of sand at an effective stress cy', AEVd =

volumetric strain increment under simple shear condition. Parameters C11 C21 C3 and C41

and K2, m and n are estimated from cyclic simple shear tests and one-dimensional loading

and unloading tests. These parameters are also modified to simulate liquefaction

resistances obtained by the cyclic simple shear test results. The values employed in the

correlation analyses are C1=0.114, C2=1.2, C3=0.17 and C4=1.36, and K2=0.001(tJm2),

m=O.1 and n=O.19. Fig.2.3 indicates the liquefaction resistance curve calculated using

these parameters in comparison with the cyclic simple shear test results of undrained

saturated sand.

Shear modulus values for evaluating initial soil-pile foundation interaction springs are

estimated by the secant modulus of the backbone curve of each layer using Eq.(2.1) at the

effective shear Straifl(O.657m ), which is obtained by the nonlinear free ground response.

M,J Super -Structure

Wink ler Spring Rotational Spring

>i R

Pile -Foundation

Fig.2.1 Lumped masses and bending-shear elements model with lateral Winkler springs for soil-pile foundation system

17.73 3.025

AH4l Laminar Box

1.58

1.44 SG7 - 0.0- - SG6— - -0.0

0.75- - I

ISG3

- -1_i -0.75 AH3 - 3.45 3.65

J

SG4 - LT3 - - - 5.30 £ I SQpp4 G2 -

S 1 h° 10.7 - 4.8-1 -.

0.48 OPP2 SG1 10 4 4. _18 ___

LT2 AH2 1.19 (Dry sand)

- (Saturated sand) - - - -fT1-8.75 - LT1

I

AH11t Base - (Unit: m) AH:Horizontal Accelerometer

PP:Pore Pressure Transducer SG:Strain Gauge LT:Displacement Transducer

Fig.2.2 Centrifuge model of 4-pile group foundation and transducer arrangements (The dimensions in this figure are described in prototype scale)

Table 2.1 Physical soil constants for correlation analyses

Soil : Nevada sand #120

Internal friction angle: ' 350

Void ratio: e 0.69

Density: y 1.98 tIm3

E Permeability: k 5.2x10 3 cm/sec

Coef. of earth pressure at rest: K0 0.45

Shear modulus: G0 570(GLOm) - 3240(GL-1 0.4m) tim2

1'

0.1

S

Gm = 6.20x1 03 tim2 m =1.97t1rn2 CY ,ax

= 5.63 tIm2

Analysis

est

I 0 50 'l'(!)c) 2(10

Number of cycles

Fig.2.3 Liquefaction resistance curve by analysis and cyclic simple shear tests of undrained saturated sand

EIJ

- 7 -

Evaluation of nonlinear lateral Winkler spring EA

The nonlinear lateral load-displacement relationship of a pile is also based on the

Hardin-Drnevich model with the Masing rule, as shown in Fig.2.4. Initial lateral Winkler

springs K 0 at each depth are evaluated by the inversion of soil flexibiities by ring loads at

the nodes, as shown in Fig.2.5. The position of the node corresponds to that of the local

lumped mass of the pile. The soil displacements can be expressed as

(u) = [d1 ][p)

(2.6)

where (u) and (p) are the vectors of lateral displacement and load at the nodes, d jj is

the flexibility matrix of the soil. The soil displacements (u) caused by lateral ring loads

(p) in a layered stratum4 are calculated at low frequency(f=O.2SHz) to form the S flexibility matrix. The soil flexibility at the i-th node is obtained by the superposition of

soil displacements at the i-th node caused by ring loads at all nodes. Then, the lateral

Winkler spring at the i-th node is approximately obtained by the inversion of the soil

flexibility expressed as

K,0= dii (2.7)

where K.Oi is the initial lateral Winkler spring at the i-th node.

The initial ultimate lateral soil resistance Pmax0i at the i-th node is assumed by Broms5)

to be three times the Raukine passive pressure expressed as

maxO = 37'0Kdl

(2.8)

where cF'O is the initial effective stress, K = (l+sin')/(l-sin'), 4' is the internal friction

angle, d is pile diameter and 1 is pile length equivalent to the i-th node.

The initial lateral Winkler spring and the ultimate lateral soil resistance accounting for

the pile group effect is obtained using the pile group efficiency CXH defined by

aH = KHI/(NKHS)

(2.9)

where N, K H N and KHS are the number of piles, the value of horizontal static impedance at

pile head of N-piles, and its value for a single pile respectively. The value of ct is obtained

as 0.56 by the three-dimensional thin layered element method6) using the degraded shear

S modulus. The equivalent lateral Winkler spring K and the ultimate lateral soil resistance

Pmax0C which take into account the pile group effect are approximately defined by

= cXHNKSO (2.10)

maxO = aHNPmaxO (2.11)

where cXHN is the equivalent number of piles. The values of K and P.,x at each time step

in the time domain are also modified according to the change of the effective stress defined

by

= K 0e(a/a 0)la (2.12)

P. = Pmaxoe(/o) (2.13)

S where c' is the effective stress at each time step, and is the initial effective stress.

The rotational spring is also evaluated under a pinned condition at the pile head by the

three-dimensional thin layered element method. The degraded shear modulus of soil is

employed taking into account the effective shear strain obtained from the ground response.

The viscous damping employed in the pile foundation, the lateral springs and the

rotational spring are 1 per cent, 5 per cent and 30 per cent respectively for the first mode

by the eigenvalue analysis.

— 9 —

Lateral load P maxO

P = Ky/(l +KoY/Pmaxo)

K50

Ks = Kso(0'/(y'o)1"2

Displacement y ,

max maxo(''o) ,

- Effective stress at each time step

Initial effective stress

{maxO

Fig.2.4 Non-linear lateral load-displacement relationship of Winkler spring

—10—

S

--- i s

- -H —'

I I 5

node I Ring load

Pi u } = { dJ { p

Ks0' = [ d J-1

Inodej I

K50 Initial lateral Winkler spring

Displacement

Fig.2.5 Evaluation of initial Winkler spring by ring load in layered stratum

—11—

2.2 Results of correlation analysis

I

Centrifuge tests for verification study were conducted with different input acceleration

levels for dry or saturated sand. The input acceleration had a spectrum similar to that of the

1940 El Centro earthquake strong motion record NS component. The maximum input

acceleration measured at the soil container base(AH1) for dry sand were about 85

gal(TEST-Di) and 203 gal(TEST-D2), and those for saturated sand were about 100

gal(TEST-Si) and 250 gal(TEST-S2).

Fig.2.6 indicates predicted acceleration time histories at GL-0.75m and the pile cap as

compared with TEST-D1,D2, and Fig.2.7 also indicates those in TEST-S i,S2.

Acceleration response spectra for 5 per cent damping for predicted and measured time

histories are indicated in Fig.2.8 for TEST-Dl,D2 and in Fig.2.9 for TEST-S i,S2. Fig.2.10

and Fig.2.1 1 indicate comparisons of predicted relative ground displacements and excess

pore water pressures for TEST-Si, S2. Fig.2.12 indicates comparisons of predicted pile

bending moments at SG3 for TEST-Di,D2 and TEST-S 1,S2.

It can be seen that predicted acceleration time histories and response spectra at the

ground and the pile cap in TEST-Di are in good agreement with the measured results, so

that the dry soil-pile foundation system for the low acceleration input is modeled properly.

In TEST-Si, predicted acceleration at GL-0.75m is somewhat smaller than the measured

result. The acceleration response spectrum at GL-4.8m is smaller than measured results

because the generation of excess pore water pressure is overestimated in the analysis

model. The predicted response at the pile cap in TEST-Si, however, shows good

agreement with measured one, and it can be seen that the generation of excess pore water

pressure is too small to effect the response at the pile cap. The predicted responses in

TEST-D2 approximately correspond to the measured results, and the frequency properties

of the measured results are represented well. However, discrepancies of the peak

amplitudes at the pile cap can be seen between the predicted and measured results. These

discrepancies result from the difference of the free ground response. In TEST-S2, the

—12—

predicted responses, except for the peak amplitudes, also agree well with the measured

results, including the relative ground displacements and the generation and dissipation

processes of excess pore water pressure. It is found that the shift in dominant frequency of

the soil-pile foundation system by soil liquefaction is represented well by the proposed

model. This is evidenced by comparing the predicted and measured acceleration response

spectra at the pile cap. The predicted pile bending moments in TEST-Dl,D2 and TEST-Si

also correspond to the measured results, but the peak amplitudes in TEST-S2 are

underestimated. The underestimated peak amplitudes in accelerations and bending

moments by analysis are mainly attributed to the discrepancies in the free ground response.

It is considered that these discrepancies are due to the dilatancy of soil during large strain,

and an advanced effective stress method for predicting ground response in liquefied soil

should be developed.

Fig.2.13 and Fig.2.14 indicate the calculated lateral load-displacement relationships of

the pile at GL-0.6m and GL-3.Om for TEST-Di, D2 and TEST-Si, S2. It can be seen that

the value of lateral springs are degraded by the pile displacement and the decrease in

effective stress due to excess pore water pressure generation in TEST-S2. The degradations

at GL-0.6m are much larger than at GL-3.Om due to large displacement and excess pore

water pressure buildup near the ground surface.

-13-

AL-jo. r-i r\7r-

;ec

-300

Gal 400 AH4: Pile cap - ANALYSIS TEST

30sec

-400 J

-200

ec

S 400 Gal AH4: Pile cap

A.

(TEST-Di)

(TEST-D2)

Fig2.6 Comparisons of predicted acceleration time histories at GL-0.75m and pile cap with TEST-Di ,D2

—14—

-300 Gal

200 AH3: GL-0.75m

Gal 300 * I I A ITh • I - - - - ANAT y:uc

ec

30sec

-200

Gal 450 AH4: Pile cap

-450

Gal 250 1 AH3: GL-0.75m

V ' • V

-250

(TEST-S2)

Fig.2.7 Comparisons of predicted acceleration time histories at GL-0.75m and pile cap with TEST-Si ,S2

30sec

-15-

OF

Gal Gal 2500 2500

AH4 Pile cap h=O.05

4 Pile cap h=O.05

Test Analysis

Test

oJ..T0 0.1 0.5 1.0 5.0 10

Hz 0.1 0.5 1.0 5.0 Hz1°

Gal Gal 2000 2500

AH3: GL-0.75m h=O. 05 AH3: GL-0.75m h=0.05

AnalyAnalysis

Sis Test f' Test

Ol 0.1 0.5 1.0

01 I I b.i 0.5 1.0 50 H 10

z 5.0 io Hz

Gal Gal 1500 1500

h=O.05 h=0.05 AH2: GL-4.8m AH2: GL-4.8m

Analysis A Test Analysis Test

01 o 0.1 0.5 1.0 5.0Hz1° 0.1 0.5 1.0 5.0 Hz

(TEST-Di) (TEST-D2)

Fig.2.8 Comparisons of predicted acceleration response spectra at GL-4.8m, GL-0.75m and pile cap with TEST-Di D2

—16—

Gal Gal 2000 2000

h=0.05 AH4: Pile cap AH4: Pile cap

h=0.05

'SIS Test Anal ,'i Test alysis ( \/

l v I '

0.1 Oji 111111

0.1 0.5 1.0 0.5 1.0 5.0 Hz1° 5.0 Hz1°

Gal Gal

1000

1000 f

AH3: GL-0.75m h0.05 AH3: GL-0.75m h=0.05

Test Test A

Analysis Analysis IV, \ Ij s

DL 0.1 0.5 1.0 5.0 Hz 10 0.1 0.5 1.0 5.0 Hz1°

Gal Gal 1500 1500

AH h=0.05

2: GL-4.8m AH2: GL-4.8m h=0.05

C'

Test Test - Analysis /\,. Analysis W1

o_ _ 0.1 0.5 1.0 5.0 Hz1° 0. 0.5 1 1.0 1 1 5.011 10

Hz

(TEST-Si) (TEST-S2)

Fig.2.9 Comparisons of predicted acceleration response spectra at GL-4.8m, GL-0.75m and pile cap with TEST-Si S2

-17-

1.5 cm

-1.5 J 1.0 -cm

-1.0

10.0 -cm

LT3: GL-0.75m

[12: GL-4.8m

- (TEST-Si)

ANALYSIS TEST -------------

3Osec

3Osec

-10.0 J 5.0 -1cm

L13: GL-0.75m , 1\: I -. "'' ...............,.-- .................................

3osec

-5.0 ----..---sec

Fig.2.10 Comparisons of predicted relative displacement time histories of ground with TEST-Si ,S2

-18-

0.2 1 Kg/cm2 FF4: GL-3.45m

ANALYSIS

L

TEST

30sec -0.2 - 0.2 - Kg/cm2

FF2: GL-5.3m

-0.2 - (TEST-Si)

30sec

0.5 .1Kg/cm2 FF4: GL-3.45m

30sec -0.5 - 0.5 1Kg/cm2

FF2: GL-5.3m

-0.5 (TEST-S2)

30sec

Fig.2.11 Comparisons of predicted excess pore water pressure time histories with TEST-S1,S2

-19-

xiO5Kg.cm SG3: Bending moment - ANALYSIS 101 TEST

-10 1

20

(TEST- Dl) xiO5Kg.cm

- SG3: Bending moment

-20 (TEST-D2)

xlO5Kg.cm SG3: Bending moment : xiO5Kg.cm

20 SG3: Bending moment

ec 20

(TEST-S2)

Fig.2.12 Comparisons of predicted pi'e bendimg moment time histories with TEST-Di ,D2 and TEST-Si $2

-20-

U

Winkler spring at GL-3.0 m r')

(TEST-Di) (I hi I-U2)

Fig.2.13 Calculated non-linear lateral load-displacement relationships of Winkler springs in TEST-Di ,TEST-D2 analyses

Winkler spring at GL-0.6 m Winkler spring at GL-0.6 rn

Winkler spring at GL-3.0 rn

Winkler spring at GL-3.0 m

Fig.2.14 Calculated non-linear lateral load-displacement relationships of Winkler springs in TEST-Si TEST-S2 analyses

3. Earthquake response of a building on composite foundation

3.1 Description of the building

The building (named K-Building) which is used for numerical prediction has thirty-four

stories and a basement supported on wall foundation and piles. The outline of the K-Building

is shown in Fig.3.1. The K-building is 104.5 meters high and has a basement of 6.0

meters deep. The pile length is 18.3 meters and 1.0 meter of the whole length is driven into

the supporting layer of 23.3 meters deep. This building has been planned to construct at

the reclaimed site of Tokyo Bay area which consists of liquefiable sand and clay deposits.

The foundation consists of wall foundation and twenty piles as shown in Fig.3.2. The

dimensions of the basemat and wall foundation are approximately 30 meters square. The

thickness of wall foundation is 1.2 meters. The piles are made of reinforced concrete, and

the diameter is 1.8 meters along the pile shaft and 3.0 meters at the pile tip. The soil

consists of sandy and clay strata, which is 23.3 meters in thickness as shown in Fig.3.3.

The supporting soil below the pile tip is the dense sand stratum.

3.2 Input ground motion for earthquake response analysis

The incident ground niotion(E0) for earthquake response of aseismic design is

determined at GL-53.5 meters by the semi-empirical method7). Acceleration response

spectra and time history for 2E0 are shown in Fig.3.4. The maximum acceleration value is

407 Gal.

—23—

S

i1lllI I UlIlF't\

- Thi

30-

-

- -__

II II

- i—fj'=j —ç----- (Basemat) (Underground wall and piles)

25— I

11 F-1 F

j I f11 11 11 111 0

1 Li 20 ____ ____

1 0 0

L ___ ___ 0 0

15- 0 0

i:Ji:::it —rtn i

ii i•r—=fl __ __ 0 0 I d=i .8

io'— PFnI:T I _

- 30 0

- I II II II fl Ill i A

(Wall-Pile foundation plan)

- II

!L11 H liirTfl( Liquefiable sand

II II 1 Il II GL 0.Om 4

I I I T'-"•I

S

Wall foundation

Clay

30.0

Gal 500 r

Unit: m -500 lOOsec

(Elevation)

Fig.3.1 Elevation and composite foundation plan of K-Building

-24-

)OO

UNIT:mm

angements)

MATERIAL PROPERTY

Concrete Fc=300kg/cm

E0=2. 57x 106kg/cm AXE OF EXCITEMENT

G0=l. lox lO 6 kg/cm 2 ( v =1/6)

p =2. 3t/i

Fig.3.2 Basemat and wall-pile foundation plan of K-Building

—25—

GLOm ----------------------------------

Vs(m/s) r (t/m 3) 1) GL Urn S:SAND,C:CLAY

BASE MAT OF THE STRUCTURE WB-10657t

GL-6.Om JB-8.65X1OtCI11

Vs=155, y1.7, P =0.48(S) -3.9m

Vs=80, Tl.G. v =0.48 (C) -6.9m

-7. 6m Vs105, r =1. 7, P =0. 48(5)

Vs=130, y=1.6, P =0.48(C) -9.5rn

-10.8m Vs=175, y=1.8, v =0.48(S)

Vs=325, 1=1.9, P =O. 48 (S) -15.

Vs=230, - =1. 7. p =0. 48(C)

L

L-24.3in PILES DIAPHRAGM WALL Vs450, r2.0, p0.40 (ELASTIC)

-29. 85m

Vs450, =1.9, p0.40 (ELASTIC)

-53. 5m

WAVE INPUT POINT

Vs600, y2.06, v =0.40 (ELASTIC)

<PROPERTY OF SOIL LAYER>

Fig.3.3 Soil profile and elevation of composite foundation

-26-

500

-500

Gal

Hz

Acceleration Response Spectra

100

SEC

MRX = Gal

Time History

Fig.3.4 Acceleration response spectra and time history of input motion

-27-

S 3.3 Free ground response by the effective stress analysis

Earthquake response analyses for the free ground were performed by using the effective

stress analysis DESRA to investigate response characteristics during earthquakes.

Physical soil constants employed for analytical studies are indicated in Table 3.1.

Parameters for excess pore water pressure generation in DESRA analyses are determined

by the simple shear tests converted from the triaxial tests for each sand layer. Fig.3.5(a)

and Fig.3.5(b) show the converted test results and the simulated liquefaction curve. The

liquefaction curve for sand-2 is estimated using the simple shear test results of sand-i

because the triaxial test for sand-2 was not conducted. 5 The earthquake response analyses for the free ground were conducted for the following

soil properties with different input acceleration levels.

The analytical conditions are as follows.

Case 1-1: Maximum input acceleration - 100 gal

Soil property - Linear


Soil property - Nonlinear & Liquefaction


Soil property - Nonlinear & Liquefaction

3.3.1 Results of earthquake response for free ground

Acceleration and displacement time histories for the free ground are shown in Fig.3.6

and Fig.3.7 for Case 1-1, Fig.3.8 and Fig.3.9 for Case 1-2 and Fig.3.10 and Fig.3.11 for

Case 1-3 respectively. Acceleration response spectra at the free ground surface are also

shown in Fig.3.12. Fig.3.13 shows the distributions of maximum acceleration, shear stress

and shear strain in the free ground for Case i-i, Case 1-2 and Case 1-3.

It is observed that acceleration .responses in liquefied soil for Case 1-2 are amplified in

the low frequency content compared with those in linear soil for Case i-i. The acceleration

—28—

time histories for Case 1-3 are significantly filtered in the high frequency range. The

displacement time histories for Case 1-3 are also amplified in the low frequency content.

The acceleration response spectra at the free ground surface indicate that the peak

amplitude at 1.5Hz are remarkably reduced by soil liquefaction, but the amplitude at the

low frequency range become larger in Case 1-3.

Amplification of maximum acceleration values cannot be seen in the ground for Case 1-

2 ,which is the low level acceleration input, compared with those of Case 1-1. Maximum

acceleration values for Case 1-3 ,which is the high level acceleration input, are remarkably

reduced in the liquefied sand layer. Maximum shear stress distributions indicate that

degradation of soil shear stress due to soil liquefaction occurs much larger than soil

nonlinearity. It is comfirmed taht larger soil shear strain also occurs in liquefied soil layer.

—29—

Table 3.1 Soil constants and parameters for earthquake response analysis

GL Om

-3. 9m

-6. 9

-7.6

5

8

-15.95

-23.3

-29. 85

-53.5

iui.t WILL

weight

r (tim3)

,wave

velocity

Vs(m/sec)

NdA. iiiu

modulus

Go(t/ n)

r.arui-

press.at

rest K0

rernieaui- lity

Kv(in/sec)

internai

friction

angle q5

5near

strength

v rd)

Sand-i 1.7 155 4170 0.5 6. 9x10 6 28.2 0. 04-U. 64

Clay 1.6 80 1045 1.0 0x108 -- 2.7

Sand-2 1.7 105 1910 0.5 6. 9x10 6 21.6 0.54-0. 58

Clay 1.6 130 2760 1.0 1.0x108 -- 3.4

Sand-3 1.8 175 5625 0.5 6. 9x10 6 30.3 1. 82-i. 97

Sand-4 1.9 325 20480 0.5 6x10 5 44.3 58-5. 18

Clay 1.7 230 9180 1.0 1.0x108 -- 13.3

(Elastic) 2.0 450 41330 -- -- -- --

(Elastic) 2.0 450 41330 -- -- -- --

9 ac rnn '77fl -- --

[Parameters for DESRA] Sand-i, Sand-2

c=1.0, c2=0.4, c'3=c3/c=0. 1612, c4=0.41 k20.09(psi), m0.43, n=0.62

Sand-3

c1.0, c20.4, c'3c3/c0.1612, c4=0.41 k 2=0.017(psi), m=0.43, n=0.62

Sand-4

: c=1.0, c2 0.4, c'3c3/cir0.1612, c4=0.41 k 2=0.021(psi), n=0.43, n=0.62

-30-

lilt I I I I I till I

(Sand-i)

G1IAX = 530000.0

TMRX = 272.00

SIGVO = 1080.0

Cl = 1.000

C2 = 0.400

C3 = 0.161

• 04 = 0.410

K2 = 0.09000

Il = 0.430

2=0.620 •

Analysis

lilt

Test

11111

w

CD

CD

ME

LIQUEFACTION CURVE

D 0

a: (fl t

U) 0 LU a: 1—U)

a: LU

(1)

2 5 10 2 5 100 2

NUMBER OF CYCLES

LIQUEFACTION CURVE

(Sand-2)

G)IPX = 387000.0

TIIRX = 220.00

SIGVO = 1880.0

Cl = 1.000

02 = 0.400

C3 = 0.161

C4 = 0.410

K2 = 0.09000

11 = 0.430

N = 0.620

El ej

Analysis Test

-.--

U•)

C

0

5

2 5 10 2 5 100 2

5

NUMBER OF CYCLES

Fig.3.5(a) Liquefaction curves for sand-i, sand-2 —31-

(Sand-3)

LIQUEFACTION CURVE a

GIIAX = 1030000.0

TIIRX = 652.00

SJGVO = 2300.0

Cl = 1.000

C2 = 0.400

C3 = 0.161

C4 = 0.430

K2 = 0.01700

II = 0.430

0.620

15 Analysis

N =

Test C)

4 5 IU 2 5

100 2 5

NUMBER OF CYCLES

LIQUEFACTION CURVE In

cc Cr

Cr) C) LU a:

U)

a:

LU

U)

C)

im

I I I I I I I

(Sand-4)

Test

GMRX = 49000 Analysis

ThRX = 1300.00

SIGVO = 2820.0

Cl = 1.000

C2 = 0.400

C3 = 0.161

C4 = 0.410

K2 = 0.02100

11 = 0.430

N = 0.620

I III bi 2 5 10 2 5 100 2 5

NUMBER OF CYCLES

Fig.3.5(b) Liquefaction curves for sand-3, sand-4 —32—

S

Gal 260 GLOm

.1 , i A I Lhk A L I h.AALAWa .4"; A. kh1w

-260 0.0

Y v

HRX 2.528E+2 oo-sec 260 GL -6.Om

-260 0.0 MAX 1.872E+2 60.

260 GL -6.9m

-260 0.0 1RX= 1.629E+2 60.

260 GL -7.6m

-260 0.0 MAX=-1.531E+2 60.

260 GL -8.55m

-260

-

0-0 HRX=-1.465E+2 60.

260 CL -9.5m

-260 0.0 HRX= 1.400E+2 60.

260 GL-1 0.8m

-260 0.0 HRX= 1.368E+2 60.

260 GL-13.4m

- W)W"Yetwo1w, 10

-260 0.0 MAX= 1.333E+2 60.

260 CL -1 5.95m

-- --.._

-260 0.0 MAX I .27qE+2 60.

260 GL -29.85m

-----

-260 0.0 MRX= 6.7314E+1 60-

Fig.3.6 Acceleration time histories calculated by DESRA (Linear, Input acc.=1 00 Gal)

-33-

30cm GLOm

3.0 0.0 tix= 2.567E+0 60. sec

3.0 GL -6.Om

-3.0 0.0 MRX= 2.169E+O 60.

3.0 GL-6.9m

-3.0

-..... -

0.0 MRX= 2.02E+0 60.

3.0 GL-7.6m

-3.0 0.0 MRX= 1.953E±0 60.

3.0 GL-8.55m

-3.0 0.0

-

MRX= 1.863E+0 60.

3.0 GL-9.5m

YU h A A ~n'

-3.0 - 0.0

44 -44TAvAV VA 1vv AV .0--v'~w ~"V , V- IV

v 0 'V AV e v̂AhA

MRX= 1.776E+0 60.

3.0 GL-1O.8m

0.0

- --.. -

MRX= 1.715E+0 60.

3.0 GL-13.4m

-3.0 0.0 MRX= I.676E+0 60.

3.0 GL-15.95m

-3.0- 0.0 MRX= I.630E+0 60.

3.0 GL -29.85m

v

-3.0 0.0 1x= 1.285E+0 60.

Fig.3.7 Displacement time histories calculated by DESRA (Linear, Input acc.=100 Gal)

-34-.

I

I

AN

S

120GaI GLOm

-120 0.0 MRXI .112E+2 60. sec

120 GL -6.Om

-120 0.0 MPX=-1.004E+2 60.

120 GL -6.9m

-120 0.0 rIRX=_9.302E+1 60.

120 GL-7.6m

-120 0.0

rvvr1YvTY jvqrTIPVR11M IVI VYV'Y IT I- IV JYWY I -V Y' YIY' I I V ~ IV

HRX=-1.079E+2 60.

120 GL-8.55m

-120 0.0 nX=-1.046E+2 60.

120 GL-9.5m

-120 0.0 tRX=1.013E+2 60.

120 GL-1O.8m

-120 0.0 FIAX=-1.096E+2 60.

120 GL-13.4m

-120 0.0 t1lX=-1.079E+2 60.

120 GL-15.95m

-120 0.0 11X=-1.048E+2 60.

120 GL -29.85m

-120- 0.0 ftX=-8.521E+1 60.

Fig.3.8 Acceleration time histories calculated by DESRA (Nonlinear & Liquefaction, Input acc.=100 Gal)

-35----

0cm GLOm

-14 . 0.0

11.0 GL-6.Om

-11.0 - 0.0

4.0 GL-6.9m

-.w-

-14.0 0.0

4.0 GL-7.6m

-'4.0 0.0

14.0 1 GL-8.55m

hAX 3.371E+0 o. sec

1RX= 2.827E+0 60.

MRX= 2.651E+0 60.

MRX= 1.951E+0 60.

I

0.0

4.0 GL-9.5m

-11.0 0.0

14.0 GL-10.8m

'-,

-14.0 0.0

14.0 GL-13.4m

-'4.0 0.0

GL-15.95m

-14.0 0.0

14.0 GL -29.85m

-14.0 0.0 MRX= 1.057E+0 60.

Fig.3.9 Displacement time histories calculated by DESRA (Nonlinear & Liquefaction, Input acc.=1 00 Gal)

-36-

MRX= 1.871E+U 60.

MRX= 1 .775E+Q 60.

MRX= 1 .529E+O 60.

4IRX= 1.1470Ee-C 60.

MRX= 1.402E+0 60.

MRX=-3.892E+2 60. -'400

0.0

00 Gal GLUm

-u-Do 0.0

GL -6.Om

-oa 0.0

:: H69 uao - GL -7.6m

-1400 0.0

100 GL -8.55m

-oo 0.0

1400 GL -9.5m

1yr

-1403 0.0

1400 ' " el

-'400 0.0

1400 (I .1QAm

-1400 0.0

(l -19m

ilAX=2.260E+2 oo. sec

MRX=-2.1SOE+2 60.

1RX=-2.187E+2 60.

h4X=-2.659E+2 60.

MRX=-2.702E+2 60.

1flX=-2.831E+2 50

MRX=-3.11814E+2 50.

MRX=-3.663E+2 60.

400 GL -29.85m

}-1400

V11111 I fTTI TRYIPY-TIr I I vIV.F-F-Iv - pv'ry.-U, V V-VIV -Y

0.0 HP.X=-3.835E+2 60.

Fig.3.10 Acceleration time histories calculated by DESRA (Nonlinear & Liquefaction, Input acc.=407 Gal)

-37-

Cm GLOm

0.0 IIRX= 1 .32'IE+l 60. sec GL-6.Om

0.0 t1AX= 1.076E+1 60. is. GL-6.9m

0.0 IIRX= 9.775E+0 60. 15. GL-7.6m

---.

0.0 MRX= 9.065E+0 60. 15. GL -8.55m

k

0.0 'IRX= 8.692E+0 60. 15. GL-9.5m

-15.

-. ..-

0.0 lIRX= 8.177E+0

15. GL-1O.8m

0.0 HPX= 7.t453E+O 60. 15. GL-13.4m

0.0 MAX= 6.943E+o 60. 15. GL-15.95m

-

0.0 HRX= 6.153E4-0 60.

GL-29.85rn

-15- - 0.0 MAX= 3.762E+O 60.

Fig.3.11 Displacement time histories calculated by DESRA (Nonlinear & Liquefaction, Input acc.=407 Gal)

-38-

Nonlinear & Liquefaction, Input acc.=407Gal

Nonlinear & Liquefaction, Inout acc.=lOOGal

0 L_. 0.1

0.5 1.0

50' iHz

Gal

200 h=O.02

Linear,Input acc.=1 OOGal

Fig.3.12 Comparisons of acceleration response spectra at the free ground surface

-39-

-20.Om

-23.3m

(Max. Acceleration)

(Max. Shear stress)

(Max. Shear strain)

-10.0 -10.0

20.0 20.Or

23.3r 0.0 100 200 300 400 500

Gal

00Gi

LNonlinear Liquefaction, Input acc.=lOOGaI


I'

Linoar,Input acc.=looGal

I1- .II•• Nonlinear & Liquefaction, Input acc.100GaI /

/

I /

\ ' Nonlinear & Liquefaction, Input acc.=407GaI

.-..

'I

0.5

/ Liriear,Input acc.10OGal

> Nonlinear & Liquefaction, ,.'

.Input acc.=lOOGal

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


GL Orr GLO GL(

1.0 Kg/cm2 1.5 - 0.0 1.0 2.0 3.0 4:0

Fig.3.1 3 Comparisons of free ground responses calculated by DESRA

3.4 Earthquake response of K-Building

3.4.1 Numerical model of superstructure-composite foundation system8)

Superstructure on composite foundation is idealized by a one-stick model with lumped

masses and bending-shear elements as shown in Fig.3.14. In modeling of composite

foundation, shear rigidity(GA) is evaluated using the sectional area of the web part of wall

foundation. Bending rigidity(EI) is evaluated using the whole area of wall foundation.

Lumped mass is evaluated using the whole area of wall foundation and piles. Physical

constants of superstructure and composite foundation employed in this analytical study are

shown in Table 3.2.

Evaluation of nonlinear lateral Winkler spring

Initial lateral Winkler springs K0 are evaluated at each depth which corresponds to

the positions of lumped masses of wall foundation. In the evaluation of initial spring

value, degraded shear modulus is employed which is obtained at shear strain(O.65ymax) by

the nonlinear response of free ground.

The initial lateral Winkler spring values are calculated as follows.

The soil displacements at the discretized nodes corresponds to the position of wall

foundation, as shown in Fig.3. 16(a), can be expressed as

(u) =[D](p)

(3.1)

where (u) and (p) are the vectors of lateral displacement and load at the nodes, [ D ] is the

flexibility matrix of the soil. The soil displacements (u) caused by lateral point loads (p)

in a layered stratum are calculated at low frequency(f=O.25Hz) by the three-dimensional

thin layered element method. The stiffness matrix[ K] of soil-wall foundation system is

obtained by the inversion of soil flexibility matrix[ D ].

Next, assuming that soil displacements at the same depth are equal as shown in

Fig.3.16(b), the displacement vectors[ U] of wall foundation for each depth is expressed as

—41—

a (u)=[R](U) (3.2)

where [ R ] is a constraint matrix which relates between soil displacement and wall

foundation. Stiffness matrix[ S ] for displacement vectors[ U], which corresponds to the

position of lumped masses, is expressed as

[S]=[R]T{K][R] (3.3)

Stiffness matrix[ S] in Eq.(3.3) is obtained as a full matrix, the soil springs at each depth

are coupled. The coupled springs, however, are not appropriate for nonlinear analysis

with liquefaction. In this study, for simplicity, the following procedures are adopted to

convert the coupled soil springs into Winkler type springs(Ku).

Assuming that displacement mode of underground wall is rigid, Winkler type spring at

i-th node are approximately obtained using a matrix relation as follows:

(3.4)

(dij)=[D](l) (3.5)

Kuil/di (3.6)

Assuming that soil reaction along underground wall is uniform:

(ldj)[S](1) (3.7)

Kui=ki (3.8)

Inplace of rigid mode in Eq.(3.5), using displacement mode of underground wall by

loading at the wall foundation head:

(dij)=[D](u) (3.9)

where (u) : Displacement mode of underground wall

Kui= lId (3.10)

The rotational spring(K) at wall foundation tip is also evaluated by three-dimensional

thin layered element method, assuming that the bottom of underground wall is rigid in the

rotational direction. The rotational spring is assumed to be linear because the supporting

layer of wall foundation is enough hard.

The ultimate soil resistance for underground wall is estimated by Eq.(2.8). However,

—42-----

there are few studies for rational evaluation of these soil resistance. Further studies should

be performed to determine the ultimate soil resistance for underground wall.

The nonlinear lateral load-displacement relationship of Winkler spring is also based on

the Hardin-Drnevich model with the Masing rule, and the initial spring values and ultimate

lateral soil resistances at each depth are modified in accordance with the generation and

dissipation of excess pore water pressure.

Table 3.2 shows the physical constants of superstructure and composite foundation, and

the lateral Winkler springs and rotational spring which are evaluated assuming that

underground wall is rigid as shown in Eq.(3.4) to Eq.(3.6). Table 3.3 shows eigenvalue

analysis results of superstructure-composite foundation system with the initial Winkler

springs. Fig.3.17 also shows the natural modes of superstructure -comp o site foundation

system.

—43—

1-str

ucture

A Super

.

34• Ku Lateral Winkler spring

35 GL

36 Wall foundation

37 and Piles

44

45

46

Rotational spring KR

Fig.3.1 4 Numerical model of superstructure-composite foundation

-44 -

Table 3.2 Physical constants of superstructure-composite foundation ,and interaction spring values

NO. HEIGHT (cm)

YElGHT

(t) INERTIA (tcni 2 )

GA (t)

El (tcm 2 )

KH (t/cni)

KR (tcm/rad)

1 1. 045E+04 1. 470E+03 1. 078E+06 4. 509E+13 2 9. 985E+03 1. 096E+03 1. 585E+06 5. 908E+13 3 9. 695E+03 9. 980E+02 1. 713E+06 6. 758E+13 4 9. 405E+03 9. 980E+02 1. 754E+06 7. 131E+13 5 9. 115E+03 9. 980E+02 1. 769E+06 7. 383E+13 6 825E+03 980E+02 1. 798E+06 7. 569E+13 7 8. 535E+03 1. 056E+03 1. 848E+06 7. 822E+13 8 8. 245E+03 1. 002E+03 1. 862E+06 8. 086E+13 9 7. 9 5 5 E + 0 3 1. 002E+03 1. 867E+06 8. 147E+13

10 7. 665E+03 1. 002E+03 1. 877E+06 8. 193E+13 11 7. 375E+03 1. 002E+03 1. 894E+06 8. 251E+13 12 7. 085E+03 1. 056E+03 1. 948E+06 8. 137E+13 13 795E+03 1. 0 0 2 E + 0 3 2. 087E+06 8. 651E+13 14 6. 505E+03 1. 048E+03 2. 361E+06 9. 581E+13 15 6. 210E+03 1. 061E+03 2. 398E+06 9. 589E+13 16 5. 915E+03 1. 061E+03 2. 415E+06 9. 603E+13 17 5. 620E+03 1. 116E+03 2. 435E+06 9. 484E+13 18 5. 3 2 5 E + 0 3 1. 061E+03 2. 464E+06 9. 662E+13 19 5. 030E+03 1. 0 6 1 E + 0 3 2. 511E+06 1. 002E+14 20 4. 7 3 5 E + 0 3 1. 061E+03 2. 521E+06 1. 007E+14 21 4. 440E+03 1. 061E+03 2. 525E+06 1. 007E+11 22 4. 145E+03 1. 116E+03 2. 533E+06 9. 920E+13 23 3. 850E+03 1. 0 6 1 E + 0 3 2. 554E+06 1. 044E+14 24 3. 5 5 5 E + 0 3 1. 061E+03 2. 568E+06 1. 044E+14 25 3. 260E+03 1. 061E+03 2. 611E+06 1. 044E+14 26 2. 965E+03 1. 0 6 1 E + 0 3 2. 694E+06 1. 091E+14 27 2. 6 7 0 E + 0 3 1. 133E+03 2. 747E+06 1. 093E+14 28 2. 375E+03 1. 084E+03 2. 753E+06 1. 102E+14 29 2. 080E+03 1. 084E+03 2. 7 6 0 E + 0 6 1. 104E+14 30 1. 7 8 5 E + 0 3 1. 084E+03 2. 787E+06 1. 102E+14 31 1. 490E+03 1. 084E+03 2. 933E+06 1. 093E+14 32 1. 195E+03 1. 149E+03 3. 762E+06 1. 077E+14 33 9. 000E+02 2. 486E+03 5. 659E+06 1. 199E+14 34 4. 250E+02 3. 797E+03 3. 082E+07 1. 482E+14 35 0. 000E+00 5. 9 6 7 E + 0 3 4. 840E+09 1. 000E+10 1. 000E+17 1. 538E+03 36 -6. 000E+02 5. 520E+03 4. 541E+09 4. 330E+06 5. 560E+14 1. 239E+03 37 -6. 900E+02 3. 740E+02 4. 154E+08 4. 3 3 0 E + 0 6 5. 560E+14 1. 310E+03 38 -7. 600E+02 3. 860E+02 4. 283E+08 4. 330E+06 5. 560E+14 1. 454E+03 39 -8. 550E+02 4. 440E+02 4. 932E+08 4. 330E+06 5. 560E+14 1. 683E+03 40 -9. 5 0 0 E + 0 2 5. 260E+02 5. 841E+08 4. 3306+06 5. 560E+14 2. 107E+03 41 -1. 080E+03 9. 060E+02 1. 006E+09 4. 330E+06 5. 560E+14 2. 641E+03 42 -1. 337E+03 1. 204E+03 1. 337E+09 4. 330E+06 5. 560E+14 2. 916E+03 43 -1. 5 9 5 E + 0 3 1. 175E+03 1. 304E+09 4. 330E+06 5. 560E+14 3. 134E+03 44 -1. 840E+03 1. 146E+03 1. 272E+09 4. 330E+06 5. 560E+14 3. 608E+03 45 -2. 085E+03 1. 146E+03 1. 272E+09 4. 330E+06 5. 560E+14 4. 389E+03 46 -2. 3 3 0 E + 0 3 1 1. 041E+03 1. 1556+09 1 878E+03 1. I0IE+11

-'15-

foundation /

ug

le

Free ground respons€

CL

w,J

KR

Fig.3.15 Numerical modeling of composite foundation

x

(a) Discretized soil corresponding to underground wall foundation

Y

TT4_Ui

z (b) Contraction of soil flexibilities

x

Fig.3.1 6 Evaluation of wall foundation-soil spring

-46-

Table 3.3 Results of eigenvalue analysis

Natural

period

(sec)

Natural

frequency

(Hz)

Participation

coefficient

1st 2.09107 0.47822 1.49125

2nd 0.62435 1.60166 0.77741

3rd 0.35800 2.79331 0.65210

(1 st) (2 nd) (3 rd)

Fig.3.1 7 Natural modes of superstructure-composite foundation system with interaction springs

—47—

3.4.2 Results of earthquake response analysis for K-Building

Earthquake response analyses were performed to investigate the followings:

Effects of initial Winkler springs which are evaluated using the aforementioned

procedures on earthquake response of superstructure-composite foundation.

Differences of earthquake response by the different input acceleration levels.

Differences of earthquake response by equivalent linear analysis and nonlinear analysis

with liquefaction.

Effects of Winkler spring values evaluated using initial shear modulus of free ground and

degraded shear modulus for free ground response.

These analytical results are indicated and discussed below.

(1) Earthquake responses with initial Winkler springs evaluated by different procedures

As described before, coupled lateral soil springs connected to wall foundation are

obtained by Eq.(3.3). Simplified Winkler springs are also evaluated by using Eq.(3.6),

Eq.(3.8) and Eq.(3.10). In this study, effects of soil spring evaluation procedures on

earthquake response are investigated by comparison of the following linear analyses.

Case.2-1 : Analysis with lateral coupled soil springs obtained by Eq.(3.3)

Case.2-2 : Analysis with lateral Winkler springs obtained by Eq.(3.6)

Case.2-3 : Analysis with lateral Winkler springs obtained by Eq.(3.8)

Case.2-4 : Analysis with lateral Winkler springs obtained by Eq.(3. 10)

The conditions of these analyses are as follows:

Maximum input acceleration - 100 gal

Soil property for free ground response - Linear

Soil spring - Linear

Fig.3.18 and Fig.3.19 show acceleration and displacement time histories at the roof and

the first floor. Fig.3.20 shows acceleration response spectra at the roof and the first floor.

—48—

Fig.3.21 and Fig.3.22 also show distributions of maximum accelerations and displacements,

and those of maximum shear forces and overturning moments.

Fig.3.18 and Fig.3.19 indicate that acceleration and displacement responses in Case.2-2

to Case.2-4 give similar results to Case.2-1. Acceleration response spectra of Case.2-2 to

Case.2-4 in Fig.3.20 are also good agreement with Case.2-1. Two peaks in acceleration

response spectra at the roof coincide with the first and second natural frequency of

superstructure-composite foundation system, it can be seen that the second mode of

superstructure at 1.6Hz is amplified because the dominant frequency of input earthquake

correspond to the second frequency of superstructure. Fig.3.21 and Fig.3.22 indicate that

responses of Case.2-2 to Case.2-4 are underestimated in comparison with Case.2-1, but It

can be said that the simplified analysis models with Winkler springs give reasonable

results in acceleration and displacement responses of superstructure on composite

foundation.

—49—

(Gal) 500 1

RF Case.2-1

MPX=4 .215E+2

-500

0.0

60. 1100 - RF Case.2-2

MAX=3.841E+2 (sec)

-1400 J

0.0

60.

I00 1 RF Case.2-3

MPX=3 .5611E+2

-1100 0.0

4 00 1 RF Case.2-4 60. S

MPX=3 .989E+2

-1100 0.0

(Gal) 200 1 1FCase.2-1 MAX=1 .832E+2

iI

-200 J

0.0

60.

200 1 1FCase.2-2

MRX=1.701E+2 (sec)

-200 J

0.0

60

200 1 1FCase.2-3

MPX=1 .613E+2

-200

0.0 60.

200 1FCase.2-4 MPX=1.739E+2

..

-200

0.0 1 60.

Fig.3.18 Acceleration time histories (Case.2) -50-

(cm) is. 1 RFCase.2-1 MAX1.288E+i

-15.J

0.0 60.

15. RFCase.2-2 MAX=1 .267E+1 (sec)

-15.

0.0 60.

is. RFCase.2-3 MPX1 .261E+1

- - - -'IV 4 n v

A v A v

A '. V

vAvpf\\-Jov A

V 6 Q n V A V A V A V v v A

v

-15. 1

. 0.

is. 1 RFCase.2-4 MPX1.272E+1

-15.

0.0 60.

(cm) 2.5 1 1FCase.2-1 MPX2.1124E+0

-2.5

0.0 60.

2.5 1 1FCase.2-2 MAX=2 .321E+0 (sec)

-2.5

0.0 60.

2.5 1 1FCase.2-3 I MPX=2.371E+0

-2.5

0.0 60.

2.5 1FCase.2-4 MFRX=2.361E+0

-2

Fig.3.19 Displacement time histories (Case.2) -51-

Gal RF q o c

Hz

Gal iF 1 SC

Hz

Fig.3.20 Comparisons of acceleration response spectra (Case.2)

—52----

I 4GL

I I I -23.3m 300.0 400.0 500.0 0.0

(Gal)

GL+1 04.5m RF

GL iF 0.Om

30F

25F

1OF

15F

20F

5F

GL -23.3 m

0.0 100.0 200.0

GL+1 04.5m RFI

251

all

1 SF

1OF

5F

GL iF 0.Om

/Cgse.2-4

se.2-2

5.0 10.0 15.0 (cm)

(a)Max. acceleration (b)Max. displacement

Fig.3.21 Comparisons of maximum acceleration and displacement (Case.2)

-53-

GL+1 04.5m

GL+l 0 RF

RF

30F

25F

20 F

30F

25F

20F Case4

4.

15F

1OF

5F

GL iF 0.Om

GL -23.3 m

GL -23.3 m

15F

1OF

5F

GL iF 0.Om

\\ Case.2-1

II

Ca-se.

- - Case.2-4

- ( t\

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

\"\ - - \

0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 0.5 1.0 1.5 2.0 (ton) x107 (ton cm)

(a)Max. shear force (b)Max. overturning moment

Fig.3.22 Comparisons of maximum shear force and overturning moment (Case.2)

-54-

(2) Earthquake responses by different input acceleration levels

In this study, earthquake response properties of superstructure and composite

foundation are investigated by comparison of different input acceleration levels.

Earthquake response analyses are conducted with the following cases.

Case.3-1 Maximum input acceleration - 100 gal

Soil property for free ground response - Linear

Soil spring - Linear

Case.3-2: Maximum input acceleration - 100 gal

Soil property for free ground response - Nonlinear & Liquefaction

Soil spring - Nonlinear & Liquefaction

Case.3-3 : Maximum input acceleration is 407 gal



In Case.3-2 and Case.3-3, initial Winkler spring values are evaluated using degraded shear

modulus at the effective shear Strain(O.65ymax) which is obtained by the nonlinear response

of free ground for 100 gal and 407 gal input.

Fig.3.23 and Fig.3.24 show acceleration and displacement time histories at the roof

and the first floor. Fig.3.25 shows acceleration response spectra at the roof and the first

floor. Fig.3.26(a) and Fig.3.26(b) show distributions of maximum accelerations and

displacements including normalized results for 100 gal input. Fig.3.27(a) and Fig.3.27(b)

also show maximum shear forces and overturning moments including normalized results

for 100 gal input. Fig.3.28 shows distributions of maximum deformation of Winkler spring

at each depth.

Fig.3.23 and Fig.3.26(a) indicate that acceleration responses of superstructure in

Case.3-2 and Case.3-3 are reduced due to soil nonlinearity and liquefaction in comparison

with Case.3-1. Acceleration response spectra of Case.3-2 in Fig.3.25 also indicate the

—55—

lower peak amplitude at 1.6Hz compared with Case.3-1, which result from the difference

of free ground response. As for composite foundation response, it can be seen that larger

shear force and moment occur in liquefied soil of Case.3-2 and Case.3-3 compared with

Case.3-1. Superstructure responses in Case.3-3 indicate that acceleration response in

nonlinear liquefiable soil is not amplified in proportion to the input acceleration level, but

displacement response with low frequency content is amplified. It is found that

amplification of acceleration response of superstructure in nonlinear liquefiable soil during

strong earthquake is much smaller than those in linear soil, but shear force of wall

founadtion in liquefied soil become larger in comparison with that in linear soil. This is

caused by degradation of soil springs and large soil deformation as shown in Fig.3.28.

n

—56—

60 (sec)

MflY2 .21 2P+2

-1400 J 0.0

250 RF Case.3-2

-250 0.0

1000 1 RFCase.3-3

S -1000

0.0

MRX=8 .1425E-t-2

sI

(Gal) 1400 1 RFCase.3-1 MRX=3 .8Lj1E+2

1 F Case.3-1

MRX=1 .701E+2

-200 0.0

60. (sec)

120 1 1FCase.3-2

MAX=1 .109E+2

-120 0.0

500 1 1FCase.3-3

-500 0.0

MAX=LI .856E+2

sI

uI

Fig.3.23 Acceleration time histories (Case.3)

-57 -

0.0

50 . RFC2s P- MYd 22P-.i

(cm)

1 5. RF Case.3-1 MPX=1 .267E-i-1

-15. 0.0

60. (sec)

15. 1 RE Case.3-2

MAX=1 .213E+1

(cm) 2.5 1

lFCase.3-1 MRX=2.321E+0

-2.5

0.0 60. (sec)

2.5 1 1FCase.3-2 MRX=2.170E+0

-2.5

0.0 60.

15.1 1FCase.3-3 MAX=1.078E+1

-15.

0.0 50.

Fig.3.24 Displacement time histories (Case.3)

—58—

S

RF Gal

6000- h=0.02

0.1 0.5 1 1 1 1.0 5.0 10Hz

iF Gal

3000- h=0.02

Case.3-1

0 lTiñ

0.1 0.5 1.0 5.0 10 Hz

Fig.3.25 Comparisons of acceleration response spectra (Case.3)

- 59—

G L+ 104.5 m RF

30F

25F

20 F

15F

1OF

5F

GL iF 0.Om

GL -23.3 m

GL+1 04.5m RF

30F

25F

20F

15F

1 OF

5F

3L iF ).Om

23.3m

LI

0.0 250.0 500.0 750.0 1000.0 0.0 100.0 200.0 300.0 400.0 (Gal)

(Gal)

Normalized by input acceleration 100 gal

Fig.3.26(a) Comparisons of maximum acceleration (Case.3)

-60-

GL+1 04.5m RF

30F

25F

20 F

1 5F

1OF

5F

GL iF 0.Om

GL+i 04.5m RFI

/

/ ------------------- /

III

/I / "

Case.3-1 7

//

20F

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

1 5F

1 OF

5F

30F

25F

GL -23.3m

0.0 20.0 40.0 60.0 (cm)

23.3m 0.0

I,)

/

5.0 10.0 15.0 (cm)


Fig.3.26(b) Comparisons of maximum displacement (Case.3)

-61-

-62----

G L+ 1 RF

GL+1 04.5m RFIITT

0 5000 (ton)

0 1000 2000 3000

30F

25F

20F

15F

1OF

5F

GL iF 0.Om

30F

25F

20F

15F

1OF

5F

3L iF ).Om

3L 23.3m 0 0

-

HE - .0 5000.0 100000 i.'nUC

(ton)

UT -

U Case.3-1

1J:I: Case.3-3

GL -23.3m

0

S

NormaUzed by input acceleration 100 gal

Fig.3.27(a) Comparisons of maximum shear force (Case.3)

C L+ 1 RF

301'

25F

20 F

15F

lOf

51

CL iF D.Om

CL -23.3 m

GL+1 04.5m RF

30F

25F

20F

15F

1OF

5F

• 0.Om

GL -23.3m

0.0 2.5 5.0 7.5 0.0 0.5 1.0 1.5 2.0

X10 (ton cm) xl 07(ton cm)


Fig.3.27(b) Comparisons of maximum overturning moment (Case.3)

-63-

GLO.Om

GL 0.Om

I

G L-23 .3 m G L-23 .3 m 0.0 3.0 6.0

Fig.3.28

Case.3-1

LPs /

1-

/s II:

/

/

/ /

9.0 12.0

0.0 1.0 2.0 3.0 (cm)

(cm) Normalized by input acceleration 100 gal

Comparisons of maximum Winkler spring deformations (Case.3)

O

(3) Earthquake responses by equivalent linear analysis and nonlinear analysis with

liquefaction

In this study, equivalent linear analysis method is investigated by comparison with

analysis taking into account soil nonlinearity and liquefaction. In equivalent linear

analysis, linear Winkler spring values are employed which are obtained at the effective

shear strain(0.65ymax) by the nonlinear response of free ground.

Earthquake response analyses for 100 gal and 407 gal input are conducted with the

following cases.

Case.4.1-1 : Maximum input acceleration - 100 gal

Soil property for free ground response - Nonlinear

Soil spring - Linear degraded Winkler spring




As for Case.4.1-1 and Case.4.1-2, Fig.3.29 and Fig.3.30 show acceleration and

displacement time histories at the roof and the first floor. Fig.3.31 shows acceleration

response spectra at the roof and the first floor. Fig.3.32 and Fig.3.33 also show

distributions of maximum accelerations and displacements, and those of maximum shear

Is forces and overturning moments.


Soil property for free ground response - Nonlinear

Soil spring - Linear degraded Winkler spring

Case.4.2-2: Maximum input acceleration - 407 gal



—65—




distributions of maximum accelerations and displacements, and maximum shear forces and

overturning moments.

Response results in Fig.3.29 to Fig.3.33 indicate that equivalent linear analysis for 100

gal input gives almost same responses in superstructure and composite foundation as

nonlinear effective stress analysis. Thus equivalent linear analysis method is confirmed to

be useful for the low level input. However, as shown in Fig.3.34 to Fig.3.38, responses for

407 gal input indicate larger displacement of superstructure, smaller shear force and larger

moment of composite foundation in the equivalent linear analysis. It is confirmed that

response of structure during strong earthquake is strongly affected by nonlinear soil around

foundation and the changing of effective stress.

—66-----

(Gal) 250 RFCase4.1-1 MPY=22flflP+2

0.0 60. (sec)

RFCase.4.1-2 MOY- 250

60

120 1 1FCase.4.1-2 MAX=1 .109E+2

iF(p41-1 Mqx=1.nqnF±2 (Gal) 120

60 (sec)

-120 0.0

Fig.3.29 Acceleration time histories (Case.4.1)

-67-

(cm) is. RFCase.4.1-1

-15. 0.0

is. RFCase.4.1-2

0.0

MAX=1 .25L4E+l

MPX=1 .213E+l

0 (sec)

r

(cm) 2.5 1 1FCase.4.1-1 MAX=2 . 18LIE+0

-2.5 0.0

2.5 1FCase.4.1-2

-2.5 0.0

MPX=2 . 170E+0

80 (sec)

80.

Fig.3.30 Displacement time histories (Case.4.1)

Ga I

am

200

Hz

iF Ga

Hz

Fig.3.31 Comparisons of acceleration response spectra (Case.4.1)

—69—

GL+1 04.5m RF

30F

25F

20F

15F

1OF

5F

GL iF 0.Om

GL+1 04.5m RF

30F

25F

20F

15F

1 OF

5F

iF

GLL 1 JGL L/ -23.3m1 I i I-23.3m1 I I

0.0 50.0 100.0 150.0 200.0 250.0 0.0 5.0 10.0 15.0 (Gal) (cm)

(a)Max. acceleration (b) Max. displacement

Fig.3.32 Comparisons of maximum acceleration and displacement (Case.4.1)

-70-

GL+1 04.5m

GL±1 04.5m RFP1

RF

30F - -- - - - - - - - - - - - - - - - - - - - - - - I 30F

25F

25F

20F - ----------- 20F

I

Case.4.1-1 Case.4.1-2

1 5F

1 OF

1 5F

1 OF

\ _Case.4.1-1

Case.4.1-2\

5F I- ------------------ - - - - - - - - - - - - - - - I

5F

GL iF

GL iF O.Om

O.Om

GL r d-23.3m GL r

-23.3mr :1 I I t I

0.0 1000.0 2000.0 3000.0 4000.0 0.0 0.5 1.0 1.5 2.0 (ton) X10 (ton cm)


Fig.3.33 Comparisons of maximum shear force and overturning moment (Case.4.1)

-71-

(Gal) 1 000 1

RF Case.4.2-1 MAX=8 .271E+2

-1000 J 0.0

60 (sec)

1 000 1 RF Case.4.2-2

MAX=8 .425E+2

-1000 0.0

(Gal) 1 F C 500 ase.4.2-1

..4vJVV'v #

-500 0.0

500 1 F Case.42-2

-500 - 0.0

60

MPX=4 .72E+2

MPX=4 .856E+2

60 (sec)

S


-72-

Pt12-1 MPX=4L552E+1 (cm) 50.

60 (sec)

RFCas42-2 M11X=L .AF+1 50.

60

1 F Case.4.2-1

-15. 0.0

i • 1 F Case.4.2-2

w

0.0

MRX=1 .00LIE+1

MAX=1 .078E+1

60 (sec)


-73--.

Ga

Hz

iF

Gal

300

Hz

S


—74—

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

1 OF 1OF

5F

------------------- 5F

GL+1 04.5m

GL+1 04.5m RF

RF

30F

30F

Case.4.2-1

25F

4 ase..- 25F

/ Case.4.2-1

Case.4.?-2

20 F

20F

1 5F

15F

• GL iF

O.Om

0.Om

GL L JGL -23.3m1 I I I I-23.3m

0.0 250.0 500.0 750.01000.0 0.0 (Gal)

(a)Max. acceleration

20.0 40.0 60.0 (cm)

(b)Max. displacement


-75-

GL+1 C RF

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

I'

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

\ Case.4.2-1

Case.4.2-2\

)F

5F I

)F

5F I

25F

20 F

15F

F

\\ \

GLI-J-23.3m

GL -23.3m1 I I!i I

1OF

5F

GL iF 0.Om

1 04.5m F I

0.0 5000.0 10000.0 15000.0 20000.0 0.0 2.5 5.0 7.5

(ton) x107 (ton cm)



-76-

(4) Earthquake responses with different initial Winkler spring values

In the aforementioned analyses, degraded shear modulus by nonlinear response of free

ground is employed in the evaluation of initial Winkler spring values. This procedure

incorporates soil nonlinearity around composite foundation by earthquake input. However it

can be considered that improved soil around composite foundation is not degraded as much

as free ground during earthquake. In this study, earthquake response results are investigated

using initial Winkler spring values which are evaluated in consideration of these

conditions.

Earthquake response analyses for 100 gal and 407 gal input are conducted with the

following cases.

Case.5.1-1 :Maximum input acceleration - 100 gal


Initial Winkler spring - Evaluated using shear modulus of free ground



Initial Winkler spring - Evaluated using degraded shear modulus by nonlinear free

ground analysis





forces and overturning moments. Fig.3.44 show distributions of maximum deformation of

Winkler spring at each depth.



Initial Winkler spring - Evaluated using shear modulus of free ground

—77—

Case.5.2-2 Maximum input acceleration - 407 gal


Initial Winkler spring - Evaluated using degraded shear modulus by nonlinear free

ground analysis





forces and overturning moments. Fig.3.50 show distributions of maximum deformation of

Winider spring at each depth.

Fig.3.39 to Fig.3.43 for 100 gal input indicate similar acceleration and displacement

responses of superstructure for Case.5.1-1 and Case.5.1-2, while shear force of wall

foundation at near ground surface for Case.5.1-1 is much larger than Case.5.1-2. Fig.3.45

to Fig.3.49 for 407 gal input indicate significant difference responses of superstructure and

wall foundation between Case.5.2-1 and Cse.5.2-2. It is found that the degree of soil

nonlinearity around foundation affects the response properties of superstructure and

composite foundation. This is due to degradation of soil resistance depending on the

deformation of Winider springs as shown in Fig.3.44 and Fig.3.50.

S

—78—

RFCasc51-1 MPY=2.17RF+2 (Gal) 250

60. (sec)

RFCase5l-2 MPX=2.212E-i-2 250

uI

(Gal) 120 1

1 F Case.5.1-1 MPX=1 .067E+2

-120 -'

0.0

60 (sec)

120 1 1FCase.5.1-2

MPX=1 .109E+2

-120

0.0

60


—79---

(cm) I RFCase.5.1-1 MAX1 .229E+1

-15. 0.0

is 1 RFCase.5.1-2 MRX=1 .213E±1

-15. J

0.0

(cm) 2.51 1FCase.5.1-1 MAX=2.311E+0

-2.5 0.0

2.5 1 1FCase.5.1-2 MAX=2.170E+0

-2.5 0.0

60 (sec)

sI

sI

S

60 (sec)


Gal

RF

200

Hz

iF

Gal

I

AN

Hz


—81—

GL+1 04.5m

G L+ 1 J4.bm RF

RI

30F

25F

20 F

15F

1OF

5F

- " Case.5.1-1

-

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

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

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

-------------------- ri II I , I , II I ,

7 ,

301

251

201

15F

1 OF

5F

6L iF O.Om

:1. ------------------------ -----------

:i_ -------------------- ------------- I

GL iF O.Om

GL

GL -23.3m •23 .3 rr

0.0 50.0 100.0 150.0 200.0 250.0 0.0 5.0 10.0 15.0 (Gal) (cm)

(a)Max. acceleration (b)Max. displacement


-82-

GL -23.3m

15F 15F

1 OF

5F

GL iF D.Om

3L 23.3m

1OF

5F

GL iF O.Om

GL+1 04.5m

G L+ 10 RFI I

RF

30F

30F

25F

25F

I

20F ----------------------- 20F

Case.

0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 0.5 1.0 1.5 2.0 (ton) X10 (ton cm)



-83-

GL 0.Om

I

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

GL-23.3m L

I

0.0

0.5 1.0 1.5 2.0 (cm)

Fig.3.44 Comparisons of maximum Winkler spring deformations (Case.5.1)

-84----

(Gal) 800 RFCase52-1 McY7 R 1 2F--2

60 (sec)

1 000 RF Case.5.2-2

MAX=8 .'425E+2

_i000

0.0

ip

(Gal)

oo 1FCase.5.2-1 MAX3.308E+2

-L00 J

0.0

60. (sec)

soo iF Case.5.2-2

MRX=4 .856E+2

-500

0.0

MIE


—85—

RF ('.cc c 2_I M (1 V - II 1 fl I E (cm) 50.

60. (sec)

RFCsF2-2 Mr-iv_Tl OCO 50.

(cm) 1 5 .-1 iF Case.5.2-1 MAX=9 .635E+0

-15. J 0.0

60. (sec)

15. 1 1 F Case.5.2-2

MAX=1 .078E+1

-15. J

0.0


-86--

Ga.

RF

iTi

Hz

iF

Ga I


—87 -

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

250.0 500.0 750.0 1000.0 0.0 (Gal)

I

20.0 40.0 60.0 (cm)

GL±1 04.5m RF

30F

25F

20F

15F

1OF

5F

GL iF 0.Om

GL -23.3m

0.0

15F

1OF

5F

31- iF ).Om

23.3 m

GL+1 04.5m RF

25F

30F

20 F

(a)Max. acceleration

(b)Max. displacement


-88-

GL -23.3m

5000.0 10000.0 15000.0 20000.0 0.0 (ton)

(a)Max. shear force

0_u (.0 X10 (ton cm)

(b)Max. overturning moment

GL+104.5m

GL+1 04.5m RF[T

RF

30F

25F

20F

I

1 5F

1OF

5F

GL1F 0.Om

GL -23.3 m

0.0

30F

25F

20F

15F

1OF

5F

GL iF 0.Om


-89-

GL O.Om

Case.5.2-1

G L-23 .3 m 0.0 3.0 6.0 9.0 12.0

(cm)

Fig.3.50 Comparisons of maximum Winkler spring deformations (Case.5.2)

S

-90----

4. 2-Dimensional finite element model and verification study

This chapter describes the correlation analyses for the centrifugal test results of a

pile foundation model which consists of a rigid pile cap and a four-pile group embedded in

dry sana. This analysis is conducted to verify the numerical modeling technique in

considering with the nonlinearity of soil. The detail of the test is shown in the previous

report of KajimaJCUREe project.').

4.1 Numerical model of a pile foundation in nonlinear soil

In this nonlinear correlation analysis, the employed numerical procedure are

described below,

The 2-dimensional Finite Element Method(FEM) for spatial discretization.

Step by step direct time integration by Newmark P.method.

The modified R-O model and plastic theory is applied to the soil element.

Finite element model

The finite element model investigated in this analysis is shown in Fig.4. 1. The soil

is divided to three parts, the inner soil is the part in which the pile-foundation-soil

interaction effects may be induced strongly. On the contrary, the outer and side soil are the

region where can be treated as free field soil deposit. And the both of vertical and

horizontal direction of inner and outer soil is connected by the dashpot element expressed

as C = pVA, in which p = density, V = shear wave velocity, A = governed area of nodal

points. This model is developed to be able to represent 3-dimensional effect by 2-

dimensional FEM solution.

The piles are modeled by flexural beam element, and the soil is modeled by four-

nodes 2-dimensional plane -strain element. Physical constants for the dry sand are

indicated in Table 4.1. The initial shear modulus G0 listed in this table are estimated from

equation (2.2).

-91-

Side Soil r Outer Soil - Inner Soil

Pile2 / Pile,,,,/" -i

CD 00

I Pilel Pile4 00

475 828 475

1778 (Unit ; cm)

Dashpot

Outer S

tl

Je Soil

Fig.4. 1 Analysis Model

-92-

I

Table 4.1 Physical Constants

Level (cm)

Unit Weight (ton/cm3)

Shear Velocity (cm/s)

Poisson's Ratio

1045.0 8.205E+03 985.0 1.161E+04 867.0 1.378E+04 749.0 1.520E+04

Inner 637.0 1.616E+04 Soil 548.0 1.470E-06 1.691E-i-04 0.33

448.0 1 .766E-i-04 342.0 1.834E+04 236.0 1.899E+04 118.0 1 .957E+04

Thickness: t= 119.0 (cm ; Inner) 771.0 (Outer) 890.0 (Side)

-93-

a Step by step time integration

The Newmark 3 method (a = 1/2, = 1/4) and Cholesky factorization by skyline

matrices formulation are employed to step by step time integration. This conservative

method became to be enough efficient for nonlinear dynamic response analysis, because of

recent progress of computer performance.

Nonlinear model

The Drucker-Prager's yield criterion and modified Prager's hardening rule is

applied to the judgment for yield and unloading. The yield function is expressed as

f=aI1 +i2 (4.1) a in which I,, is n-th stress invariant, and the plastic stress-strain relation matrix is

represented by partial differentiation off with respect to stress cr, elastic stiffness matrix,

and hardening coefficient Ct The modified Rumberg-Osgood model is employed to

determine the hardening coefficient. According to the associated flow rule, the equivalent

stress 0e is expressed as the same form with yield functionf (4.1), and the hardening

coefficient (tangent stiffness in this case) Cf 15 decided as the function of equivalent stress

Cf=bI(YeO+b) (4.2)

in which the constant a,b is determined by the expression below,

a=2ith/2(1-y)-h (4.3.a) I

b = z (yG0'y) / (1 - y)(a + 1) (4.3.b)

where, h, yc, ,yc is reference value of damping factor, stiffness degrading ratio, and

shear strain respectively. The shear modulus and damping factor versus shear strain

relation assumed in this correlation analysis is shown in Fig.4.2.

—94—

The correlation analyses for three case of dry sand test are conducted. Table 4.2

summarizes the maximum acceleration measured in each tests. The case of Dry06 is

excited by the smallest acceleration level. The measured acceleration observed at the base

level is subject to FEM model described above, and investigate the correlation of the test

results and analyses, according to the bending moment of piles and acceleration response

spectra at the points of L15, L9, and Pile cap.

—95—

1.0

0.0 1.0x10 6 1.0x10 5 1.0x10 4 1.0x1Q 3 0

1.0x10 2

PIC

(%)

40

On

Refference value

h= 0.12 = 1.0x1O

Fig.4.2 Nonlinear Model

-96-

L15

L9

Table 4.2 Measured Acceleration (gafl

DRY04 DRY05 DRY06

L15 255.4 342.1 189.4

L9 190.9 270.0 137.3

Pile cap 331.9 416.1 304.4

Base 203.1 339.4 85.2

-97-

4.2 Result of correlation analysis

The acceleration response spectra are shown in Fig.4.3 - 5. At the points of side

soil(L9 and L15), the results of test and analyses agree very well but the analysis shows

slightly smaller spectrum value in the case of Dry06. The damping effect in lower strain

range may be overestimated for the nonlinear characteristic assumed in this analyses.

Otherwise, at the point of pile cap, the predominant vibration mode of soil-pile-foundation

system is found at about 0.4 sec. The analysis of Dry06 shows smaller value at also this

point, and the other case show about 1.5 times lager than the test results. The additional

damping effects to material nonlinearity, such as slide and separation between the pile and

soil, are expected especially higher exciting case.

The transition of hardening coefficient are shown in Fig.4.6 - 8. It is found that the

change of yielding stage happens more frequently at the inner soil region.

The bending moment of pile is shown in Fig.4.9. The analysis of Dry06 shows a

little smaller than test results, but they represent good agreement for the response value as

well as the mode shape which the node point is produced at near surface.

As the 2-dimensional F.E.modeling technique for soil-pile-foundation system, the

method which the soil domain is divided to multi-region according to the degree of soil-

pile interaction effects is proposed. And the proposed method is verified according to the

correlation analysis for the centrifuge test of pile model in dry sand.

—98—

-600.0 1 0 0 20 30 (SEC)

L15 OF BOX

-600.0 L._._._,,_._.._. -coo.o[ to zo 30 scc I 0 10 20 30 tOEd 0

PILE CAP L9 OF BOX

Fig.4.3 Response Spectra and Time Histories (DRY04)

1 ouuu

- - 2000

- - 1000

0 0.10 0.20 0.50 1.00 1.40 2.00 0.

I GIlL) 600.0 1 600.0

0 0.20 0.50 1.00 1.40 2.00

[GIlL)

CAP

TEZT - PNflLYSIS 3000

2000

1000

0 0.

600.0

0.20 0.50 LOU 1.

3000

2000

1000

0

L15 OF BOX L9OFBOX

TEST -------------------- INELYSIS TEST --------------------ANIlLYSIS

3000

2000

CAP

PNcLYSIS 3000

2000

L15 OF BOX L9OFBOX NIN

TEST ---- .--------------- flNnLYSIS tsi .................... flNflLYSIS

2000

- 1000 C C

0 0.

600.0

10 0.20 0.50 1.00 1.40 2.1

(CclL)

0 10 0

600.0

10 0.20 0.50 1.00 1.40 2.1

(COL]

(000

0 0 0.

600.0

10 0.20 0.50 1.00 1.40 2.00

(Ct1L)

-600.0 1 0

-600.0 -600.0 .,. 0 20 30 (500) .

0 0 20 30 (SOd 1 0 10 20 30 (SECJ

PILE CAP L9 OF BOX L15 OF BOX

Fig.4.4 Response Spectra and Time Histories (DRYO5)

L15 OF BOX

ANYSIS

L9 OF BOX

TEST --------------------FNLYSIS TEST

3000

2000

1000

3000

2000

- 1000 Q

0 0.

600.0

CAP

N$LYSI6

3000

2000

1000

II 0 0.20 0.50 1.00 1.40 2.00 0.

600.0

0.20 0.50 1.00 1.40 2.00 0.10 0.20 0.50 1.00 1.40 2.00 (CPU

600.0 .

-600.0 0 ;o 30 (SEC) 0 ID 20 30 (SECI 0 0 20 30 (5CC)

PILE CAP L9 OF BOX

L15 OF BOX

Fig.4.5 Response Spectra and Time Histories (DRYO6)

0.0 os 1.0 0.0 0.5 1.0 0.0 0.5 1.0 II ______________________ I

lOsec

20s ec 3 Os ec

Fig.4.6 Hardening Coefficient ( DRYO4)

0 ___________

S

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0

I 1 I I I I

lOsec

2Osec

3Osec

Fig.4.7 Hardening Coefficient ( DRY05)

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5

mmmRw=k I _____________

lOsec

2Osec

3 Os ec

Fig.4.8 Hardening Coefficient ( DRYO6

(cm) 0 TEST

1395 LEVEL

1045

0

/ /

/ o

/ \ \ \

o \

o \

ID I,

1000.0 2000.0 3000.0 4000.0 (ton cm)

ANA. (cm) 0 TEST

M. LEVEL

1045

1i I.

\

0

/ \ \

O\ \.

\\

/0

/

/ /

1000.0 2000.0 3000.0 4000,0 (ton 'cm)

Nfl. RNR.

/ /

0/

0

\ 0

I C

/ 0

I

\

1000.0 2000.0 3000.0 4000.0 (ton 'cm)

DRYO4 DRYO6

Fig.4.9 Pile Moment

(cm) 0 TEST

1395 LEVEL

1045

5. Conclusions

The results in this study can be summarized as follows:

The proposed numerical model, which consists of beam elements and nonlinear lateral

Winkler springs, taking into account the changing effective stress, can effectively predict

the centrifuge test of pile foundation model in dry or saturated sand. It can be confirmed

that the proposed model represents the response characteristics of a pile foundation in

nonlinear liquefied soil.

Earthquake responses of the building supporting on composite foundation are strongly

affected by nonlinear liquefied soil-foundation interaction during strong earthquake.

Response properties of superstructure change depending on the magnitude of input motion

and soil nonlinearity of surrounding soil. Particularly, Composite foundation responses

significantly change by large deformation and degraded resistance of soil due to

liquefaction.

2-dimensional finite element model incorporating soil nonlinearity, which the soil

domain is divided into multi-regions according to the degree of soil-pile interaction effects

is proposed. And the proposed method is verified by the correlation analysis for the 41 centrifuge test of pile foundation in dry sand.

—106—

References

Kajima-CUREe Research Project Report: Dynamic Interaction between Pile Group and Non-linear Soil, 1991. 2

Finn, W.D.L., Lee, K.W. and Martin, G.R. :An Effective Stress Model for Liquefaction, Journal of the Geotechnical Eng. Division, ASCE, Vol.103, pp.517-533, 1977

Richart, F.E., Hall, J.R. and Woods, R.D. :Vibrations of Solids and Foundations, Prentice-Hall Inc., 1970

Kausel, E. and Peek, R. :Dynamic Loads in the Interior of a Layered Stratum-An Explicit Solution-, Bulletin of the Seismological Society of America, Vol.72, pp.1459-1481, 1982

Broms, B.B. :Design of Laterally Loaded Piles, Journal of the Soil Mechanics and Foundations Division, ASCE, Vol.91, pp.79-99, 1965

Masuda, K., Sasaki, F., Urao, K., Ueno, K. and Miyamoto, Y. :Simulation Analysis of Forced Vibration Test for Actual Pile Foundation by Thin Layer Method, Proc. of Reliability and Robustness of Eng. Software Conf., 1987

Takemura, M. and Ikeura, T. :A Semi-Empirical Method using a Hybrid of Stochastic and Deterministic Fault Model-Simulation of Strong Ground Motions during Large Earthquakes-, J. Phys. Earth, 36, 1986

Masuda, K., Kitamura, E., Miura, K. and Miyamoto, Y. :Dynamic Soil-Structure Interaction Analysis Considering Diaphragm Wall and Pile Group, Tnt. Conf. on Computational Eng. Science, 1991

—107—

CUREe-KAJIMA RESEARCH PROJECT

A METHODOLOGY FOR NONLINEAR SOIL-STRUCTURE INTERACTION

EFFECTS USING TIME-DOMAIN ANALYSIS TECHNIQUES

Ronaldo I. Borja and H. Allison Smith

Dept. of Civil Engineering

Stanford University

February 15, 1991 - May 14, 1992

A METHODOLOGY FOR NONLINEAR SOIL-STRUCTURE INTERACTION EFFECTS USING TIME-DOMAIN ANALYSIS

TECHNIQUES

Ronaldo I. Borja and H. Allison Smith Department of Civil Engineering, Stanford University

June 15, 1992

PROJECT SUMMARY

The dynamic response of rigid foundations on an elasto-viscoplastic half-space is inves-

tigated in the context of nonlinear finite element (FE) analysis. A deviatoric viscoplastic

theory with a linear combination of isotropic and kinematic hardening is used to model the

soil constitutive response. Large-scale nonlinear FE computations are made feasible by the

use of a composite Newton-PCG iteration technique, which requires the factorization of the

consistent tangent operator no more than once during the solution process. Time-domain

analyses are used to investigate the nonlinear responses of vertically oscillating circular,

square, and rectangular foundations to harmonic loads, using.two- and three-dimensional

FE modeling. It is shown that for low frequency excitations, resonance is created which

amplifies the motion of the foundation at amplitudes well above those obtained at the

zero-frequency level.

In addition, horizontal, rocking, and torsional vibration modes of strip and square

foundations are considered using the same methodology developed for vertically oscillating

foundations. The foundation responses for horizontal, rocking, and torsional modes are

characterized by increased vibrational amplitudes due to material stiffness degradation.

Furthermore, one or more resonance frequencies are created which resemble those observed

for vertically oscillating finite-size foundations. Nonlinear soil effects are shown to be

dominant over a wide range of excitation frequencies for foundations vibrating in torsional

and horizontal modes. In contrast, nonlinear soil effects are shown to be dominant over a

much narrower range of excitation frequencies for the vertical and rocking modes.

1

Contents

1 Introduction 3

2 Model Description 5

2.1 Soil-Structure Model ............................... 5

2.2 Solution Algorithm ............................... 6

2.3 Constitutive Model ................................. 7

2.4 Forcing Function and Radiation Boundary Condition ............ 10

3 Foundations Subjected to Vertical Vibrations 11

3.1 Vertical Vibration of Circular Foundations ..................11

3.2 Vertical Vibration of Square Foundations ...................15

3.3 Discussion of Results ..............................16

3.4 Vertical Vibration of Rectangular and Strip Foundations ..........18

3.5 Summary and Conclusions ...........................21

4 Foundations Subjected to Horizontal, Rocking, and Torsional Vi-

brations 23

4.1 Symmetry/Anti-Symmetry and Forcing Functions ..............24

4.2 Horizontal and Rocking Vibrations of Strip Foundations ..........25

4.3 "Critical Frequency" and "Optimal" FE Grid ................28

4.4 Horizontal, Rocking, and Torsional Vibrations of Square Foundations . 30

4.5 Summary and Conclusions ...........................32

5 Boundary Element Model for Plane Strain 34

5.1 Time-Domain BE Formulation .........................34

5.2 Boundary Integrals for Plane Strain .......................36

6 List of Publications 38

7 Acknowledgements 39

P 1 Introduction

Accurate representation of soil-structure interaction effects is a crucial part of earthquake

engineering analyses. A large number of soil-structure models exist of varying degrees of

complexity. The choice of the most appropriate model for a particular analysis is based on

the building-foundation system of interest, the desired level of accuracy, and experienced

"engineering judgment." In many cases, multiple models must be analyzed and experi-

mentally verified before one can select the most effective model for a particular system.

It is generally recognized that the dynamic response of soils is nonlinear, hysteretic, and

irreversible. At sufficiently high strain levels, soil deformations also are known to exhibit

pronounced rate-dependent effects. While nonlinear effects and other unusual behavioral

features dominate soil responses in all but a few exceptional cases, very little work has

been done to incorporate them into soil-structure interaction (SSI) models (Lysmer et al.

1974; Kausel et al. 1976; Jakub and Roesset 1977).

Part of the difficulty associated with nonlinear SSI analyses stems from the lack of ef-

ficient mathematical tools with which one can transform a hyperbolic structural dynamics

problem into a more tractable elliptic problem as in linear analyses. Nonlinearities usually

force the analyst to abandon frequency-domain analyses in favor of more computationally

intensive time-domain analyses. Without numerical enhancements, "conventional" nonlin-

ear time-domain codes usually break down from the rigors of heavy computation associated

with time-stepping and iterations. Thus, analysts typically are forced to neglect nonlin-

ear effects due to the lack of effective SSI models which can accurately model the soil

complexities in a computationally feasible manner.

This report presents a finite element (FE) methodology for modeling nonlinear SSI

effects and applies the model to the problem of machine foundations. While the FE formu-

lation presented here is generalized to be applicable to any nonlinear structural dynamics

problem, the cornerstone of the present model is a state-of-the-art algorithm that empha-

sizes computational efficiency. Thus, the model avoids repeated triangular factorization of

3

the effective mass matrix in the nonlinear analysis while preserving the optimal order of

convergence of an iterative method (Ortega and Rheinboldt 1970).

Hysteresis, rate-dependence, and irreversible deformation are dominant features of

cyclic soil behavior (Perzyna 1971; Prevost 1977; Mróz et al. 1979; Dafalias and Herrmann

1982). This suggests that elasto-plastic and elasto-viscoplastic models may be appropriate

for modeling the soil constitutive behavior. In particular, the presence of hysteretic damp-

ing in machine foundation problems demands that the soil constitutive model be of the

"kinematically-hardening" type (Prager 1956; Mróz 1967). While it can always be argued

that one constitutive model may be more appropriate than another, it is not the objective

of this paper to present such an argument. It is the aim of this paper to demonstrate,

with the simplest possible constitutive model, the effect of soil stiffness degradation, rate

of loading, and hysteresis on the dynamic response of machine foundations. For this pur-

pose, the elasto-viscoplastic theory of Duvaut and Lions (1976) is used to model the soil

behavior.

The effect of repeated loading on machine foundations is to create a zone of intense

yielding and inelastic deformation beneath the foundation. This results in "dynamic non-

homogeneity" with respect to soil elasticities, which perturbs the overall response of an

otherwise homogeneous soil-structure system. This report shows that plastic deformation

(and soil stiffness degradation) for vibrating rigid foundations results in an overall increase

in vibrational displacement amplitudes and the creation of resonance frequencies where 41 motion is amplified above those at zero-frequency level. The resOnance phenomenon is

explained in the context of a single-degree-of-freedom (SDOF) oscillator analog which has

been used successfully by previous investigators to model prototype continuum SSI prob-

lems (Hsieh 1962; Lysmer 1965; Richart and Whitman 1967; Richart et al. 1970; Veletsos

and Wei 1971; Kausel 1974; Luco 1974; Gazetas 1983).

4

2 Model Description

This section contains a description of the soil-structure interaction model investigated in

this report. The equations of motion are formulated for a nonlinear soil-structure sys-

tem, and the constitutive model representing the nonlinear soil properties is presented.

In addition, the composite Newton-PCG iterative solution algorithm is formulated and

discussed.

2.1 Soil-Structure Model

Consider a massless, rigid foundation of arbitrary shape shown in Fig. 2.1, and let the

forcing function be denoted by the nodal vector FEXT(t). The forcing function FEXT

may arise from an externally applied dynamic load (as in machine foundation) or from

propagating incident waves (such as seismic excitations). The FE dynamic equation of

motion then reads

Mx + FINT = FEXT (1)

where M = mass matrix; x = nodal acceleration vector; and FINT = internal nodal force vector, which takes the form

FINT = if BTcrdfI

(2)

where BT = s train- displacement transformation matrix; o = Cauchy stress vector; and S Q = problem domain. The stress vector o, may depend in general on the nodal velocity

vector * and on the nodal displacement vector x either in a linear manner or through a

more complex relationship.

Assuming that FEXT is given over a certain time domain of interest, then Eq. (2) may

be time-integrated in the form

Ma+j + (1 + c)(FINT)fl+l - a(FINT) = FEXT(tn+l+a) (3)

5

where a+1 is an approximation to (+1) and a is a parameter which takes on a parallel

meaning to the parameter introduced by Hilber et al. (1977) for linear structural dynamics.

Defining the predictor values for displacements and velocities as

2(1 - 23) = d + LtV + 2

a

Vn+i=Vn+Lt(17)an (5)

where /3 and are integration parameters, then the finite difference (FD) equations for

displacements and velocities take the form

d 1 = + Lt2/9a+1 (6)

n+l = Vn+l + Ltya+i (7)

Note that when a = 0, Eq. (3) degenerates to Newmark's method. The purpose of in-

troducing the parameter a in Eq. (3) is to achieve an optimal balance between effective

numerical dissipation and loss of accuracy compared with the trapezoidal rule (Hughes

1987).

2.2 Solution Algorithm

Eq. (3) can be written in the following residual form:

r(a 1) = FEXT(tn+1+a) - Ma1 - (1 + a)(FINT ) +l + a(FINT ) 0 (8) 41 The vector a+1 can then be determined iteratively for the nonlinear case by Newton's

method. The iteration takes the form

= r(a +1); a+1 .' a +1 + ak k+1k

__ (9)

where k is an iteration counter. For a fully implicit solution, —r'(a +1) is the consistent

tangent operator derived from

-r'(a 1) M + (1 + a)LiyCT + (1 + a)1t2/3KT (10)

-- 6

where CT = ôFINT/ôv' +i = algorithm material damping matrix, and KT =

= algorithmic material stiffness matrix. Iterations commence by allowing d+1 and v+i

to initially take on their respective predictor values.

A fully implicit solution to Eq. (8) requires simultaneous equation solving for the lin-

earized problem at each iteration. This is done by a sequence of computations involving

triangular factorization of the effective mass matrix (which also serves as the tangent

operator) followed by a backsubstitution. In general, triangular - factorization is a very

costly operation for large systems and should be avoided whenever possible. To cut down

on computing costs associated with triangular factorization, Eq. (9) is herein solved itera-

tively using a linear equation solving technique based on preconditioned conjugate gradients

(PCG), in which the preconditioner is the elastic component of the effective mass matrix.

Thus, for the nonlinear case, equation solving is done by the use of so-called composite

Newton-PCG iteration (Borja 1991a). The use of a composite iteration guarantees that

with a constant time step, no additional subiterations will be required for linear analysis

since the preconditioner is identical to the tangent operator itself. Furthermore, since the

tangent operator is factored only once throughout the entire analysis, the savings engen-

dered by the Newton-PCG iterations are significant particularly for nonlinear analysis of

large systems (Borja 1991a).

2.3 Constitutjve Model

The constitutive model used in the present study is the deviatoric elasto-viscoplastic model

of Duvaut and Lions (1976). The inviscid counterpart of this model is the Von-Mises elasto-

plastic model with a linear combination of isotropic and kinematic hardening (Borja 1991b).

This model can replicate the following features considered typical in soils subjected to

dynamic loads: path-dependence, plasticity, hardening/softening, the Bauschinger effect,

and viscous damping. Though this constitutive model may seem simplistic compared to the

complexities of real soil behavior, its well-behaved nature facilitates a clear interpretation

of the significance of each of the model features. Furthermore, this simple model provides a

backdrop against which the performance of more robust constitutive models (e.g., Prevost

7

1977; Mróz et al. 1979; Dafalias and Herrmann 1982) may be assessed for calibration and

response predictions.

For the inviscid case the evolution equations for the proposed constitutive theory are

summarized in Hughes (1984). Specifically, the radius R of the yield surface is assumed to

vary linearly with the effective plastic strain, while the backstress a is assumed to evolve

according to Prager's translation rule (Prager 1956). An accurate and stable integration

algorithm for this model is embodied in the generalized radial return concept of Krieg

and Key (1976), which is also summarized in Hughes (1984). This algorithm produces the

time-discrete evolution equations for the inviscid solutions ã,+1, R, 4, and

For the viscoplastic case the general rate-dependent constitutive model takes the form

= tr —g()( —a); g(ij) = 1

where & = inviscid solution of the rate-constitutive equation, btl' is the usual trial rate-of-stress, and ri = viscosity coefficient. Other forms for g(i) also may be considered, such

as the exponential and rational laws (Simo et al. 1988), but the simpler form given by

Eq. (11) is chosen here. Employing the one-step backward difference scheme on Eq. (11)

yields

tr — o. + ( t/)a1 12 — 1 + ( t/)

where zt/77 step size-to-damping ratio. Note that the elastic solution is recovered from • Eq. (12) as —* 0, while the inviscid solution is recovered as --* x.

We also postulate the following time-discrete evolution equation for radius:

R1 — (13) — 1 + (t/77)

and time-discrete evolution equation for backstress:

= a + ( t/)a +1 1 + (i.t/77) (14)

It then follows that the algorithmic material and damping stiffness matrices specialize to

CT = O (15)

KT = 1 K + 1 + ( 1 + (t/) J BTa+l(Efl+l)Bd (16)

where K = elastic stiffness matrix and consistent algorithmic moduli of

Simo and Taylor (1985). Note that for the present constitutive model, viscous damping

is embedded in the KTterm and not in the CTterm. The use of the above viscoplastic

model allows modeling of material stiffness degradation via a plastic-modulus parameter H', modeling of the Bauschinger effect via a kinematic hardening parameter 0, and modeling

of nonproportional damping via the viscosity coefficient i.

' The significances of the model parameters are shown in Fig. 2.2. Note in Fig. 2.2

that there is no upper bound on the value of the plastic hardening parameter H'; as H' —poo, the inviscid elasto-plastic solution approaches the elastic solution. Also note

that the model does not preclude softening plasticity, i.e., H' < 0; the only restriction is that H' > -3i, where y = elastic shear modulus. The kinematic hardening parameter

6 allows the yield surface to translate in the deviatoric stress space: 0 = 0 implies pure

kinematic translation, 0 = 1 implies pure isotropic hardening/softening, and 0 < 0 < 1

results in a linear combination of isotropic and kinematic hardening features. However, this

constitutive model does not have the capability to replicate the shear-volume relationship

that could be important in predicting the onset of liquefaction in saturated sands.

' Not all nonlinear constitutive models may be used for steady-state forced-vibration

analysis of rigid foundations. It is important to note that, in order to be useful for

time-domain vibration analysis, a constitutive model must first exhibit a steady-state

(closed-loop) hysteretic stress-strain response in the limit as the number of load cycles

approaches infinity. This requirement rules out any elasto-plastic model possessing some

degree of isotropic hardening since these models have an elastic stress-strain limiting re-

sponse. Therefore, even though the model can replicate the isotropic hardening effect,

but this feature is not as meaningful as the kinematic hardening effect in the analysis of

steady-state response, and is thus suppressed altogether. Furthermore, from a numerical

-- 9

analysis standpoint, a constitutive model must also exhibit a finite elastic region—bounding

surface-type models (e.g., Dafalias and Herrmann 1982) and constitutive theories which

use the notion of a vanishing elastic region (e.g., Mroz et al. 1979) are unacceptable—in

order that the elastic infinite-domain solutions may be exploited in the far-field region.

The latter requirement thus guarantees that in the limiting condition the extent of the

yield zone formed in the supporting soil medium is finite. The above requirements are

satisfied by the elasto-viscoplastic constitutive model used in this work.

2.4 Forcing Function and Radiation Boundary Condition

In this report, the nonlinear responses of vibrating rigid foundations to harmonic forcing

functions are investigated. Vibrations in vertical, horizontal, rocking and torsional modes

are considered. Harmonic forcing functions are used so that the nonlinear responses can be

interpreted against the corresponding linear responses where closed-form or semi-analytical

solutions are available for most foundation shapes (Cazetas 1991).

Long-duration analyses require the use of an accurate radiation boundary operator

to allow an effective transmission of the outgoing waves (Lysmer and Kuhlemeyer 1969;

Manolis 1983; Givoli 1988; Pinsky and Abboud 1991). Radiation boundary conditions are a

separate computational issue which can introduce additional complexities and uncertainties

into the solution. Fortunately, with a sufficiently large FE -mesh, steady-state responses

can be established from an initially quiescent condition before the waves reflecting from

the mesh boundaries reach the model foundation. Thus, with a judicious choice of mesh S size, time step, and duration of analysis, the model responses can be made completely

free from the inaccuracies brought about by the use of the various artificial boundary

operators. These computational issues will be elaborated further in Chapter 4. For long-

duration analyses, it may be necessary to introduce artificial boundary operators to make

computations economically feasible.

10

3 Foundations Subjected to Vertical Vibrations

Here we compare the responses of vertically oscillating rigid foundations on a visco-plastic

half-space to the corresponding linear elastic responses. The FE code used in the present

study is an enhanced nonlinear version of DLEARN (Hughes 1987) and runs in 64-bit-

per-word single precision operation on a Cray Y-MP supercomputer at San Diego Super-

computer Center. This code has a dynamic storage allocation feature that automatically

calls the composite Newton-PCG routine when the problem size exceeds 1K degrees of

freedom; otherwise, for smaller problems the solution is carried out by the conventional

triangular factorization technique. Two error tolerances are prescribed in the code for

execution: a global Newton iteration control which determines the overall accuracy of the

solution, TOLl = iO, and a local iteration control which determines the accuracy of

the PCG linear equation solver, TOL2 = 10-2 (Borja 1991a). In all of the cases studied

the elastic constants are assumed to be as follows: Young's modulus E3 = 240,000 kPa, Poisson's ratio v = 1/3 and shear velocity v = 200 rn/sec for the soil; E f = 109E3 for the

"rigid" foundation. Material responses are sampled at the Gauss points via a four-point

integration for two-dimensional quadrilateral elements and an eight-point integration for

three-dimensional brick elements, both considered "standard" for these elements. Finally,

the time-integration parameters of Hilber et al. (1977) are set to /3 = 0.3025, y 0.60, and o = —0.10, and the time step is taken to be /t = T/16 (where T = period of the exciting force).

3.1 Vertical Vibration of Circular Foundations

Consider the well-studied problem of a vertically oscillating circular foundation bonded to

the surface of an elastic half-space (Lysmer 1965; Luco and Westman 1971; Veletsos and

Wei 1971; Veletsos and Verbic 1973). Under a harmonic vertical forcing function 1(t), the response 5(t) of the system is also harmonic and is related to f(t) by the so-called dynamic impedance function

Kd = = K5(ã + iao ) (17)

11

where K3 = static vertical impedance of the foundation, a and /3 are frequency- depende nt

coefficients, i = unit complex number, and

wr a0 = -

V3 (18)

is a dimensionless frequency parameter which accounts for the combined effects of the

excitation frequency w, the foundation radius r, and the shear wave velocity v3. The ratio between the dynamic displacement amplitude i and the static displacement L is called the amplitude ratio, and is given by

LK 3 1 -JIkJ - a2 + aj2 (19)

In practice, the dynamic responses of rigid foundations to harmonic forcing functions are

usually described in terms of the variation of the amplitude ratio /L with the dimensionless frequency a0.

The FE mesh investigated is composed of 936 nodes and 875 bilinear axisymmetrjc

elements shown in Fig. 3.1, which gives rise to a total of 1,751 unknown degrees of freedom.

The rigid foundation is represented by five ring elements subjected to a sinusoidal vertical

forcing function f(t) is applied at the center of the foundation.

Fig. 3.2 shows how the mesh of Fig. 3.1 was studied to determine if the spatial dis-

cretization pattern adequately represents the dynamic behavior of the SSI model in the

linear regime. For each value of a0, the foundation was subjected to a harmonic forcing I function f(t) starting from an initially quiescent condition. The steady-state displacement amplitude for each a0 was then normalized with respect to the corresponding displacement

amplitude at zero frequency and plotted in Fig. 3.2a to construct an amplitude-frequency plot for this mesh. The good agreement between the time-domain FE solution and the

frequencydomajn amplitude-frequency curve, determined from Eq. (19) together with the values of a and reported by Veletsos and Tang (1987), suggests that the mesh of Fig. 3.1

is adequate for this study.

12

It is crucial that the steady-state condition be reached in the solution prior to the arrival

of the reflected waves since no artificial boundary operators are used in the present analysis.

In general, the time required for the arrival of the reflected waves can be determined from

the mesh dimensions and the P-wave velocity for the soil material, and should be long

enough to obtain a meaningful physical interpretation of the results. Fig. 3.2b further

affirms the adequacy of the mesh of Fig. 3.1 as the harmonic foundation response lags the

forcing function by a phase distance 0 and continues to manifest after only the first cycle

of loading from an initially quiescent condition. The fact that the steady-state condition

can be established so soon for the half-space problem is hardly surprising considering the

significant influence of radiation damping on the system response, a phenomenon which

will be discussed further in the next section.

Having established the accuracy of the mesh of Fig. 3.1, the half-space is now assumed

to be elasto-viscoplastic in which the yield surface is assumed to translate (kinematic

hardening), but not expand (isotropic hardening), in the stress space. It is important to

note that for harmonically vibrating foundations, isotropic hardening does not influence

the foundation response at steady state; expansion of the yield surface contributes only

to permanent deformation during the transient period of building up the dynamic motion

from an initially quiescent condition. Thus, in all of the nonlinear test problems, a value

of 0 = 0 was assumed.

Results of nonlinear time-domain analyses are shown in Figs. 3.3 and 3.4. In Fig. 3.3, S the soil was assumed to be a strain-hardening material with H' = 0.20E3 and uniaxial

yield stress cr, = 0.001E3. An implication of these values of model parameters is a 50%

increase in static displacement due to plastic deformation.

Amplitude-frequency plots and time-history responses (normalized with respect to the

half-space elastic zero-frequency displacement are portrayed in Figs. 3.3a and 3.3b, re-

spectively, showing the influence of material damping coefficent 77 on the system response.

The following observations can be made from Fig. 3.3: (1) plasticity results in increased

vibrational displacement amplitudes, and hence, in smaller effective dynamic impedance

13

compared to the elastic impedance; (2) resonance frequencies are created at sufficiently low

values of 77 and a0; (3) increasing the material damping coefficient 77 decreases the vibra-

tional displacement amplitudes (however, note that the material damping in the present

model is nonproportional); (4) resonant peaks disappear at sufficiently large values of

i; (5) the elasto-viscoplastic zero-frequency displacements for different values of material

damping coefficient are the same; and (6) the elasto-viscoplastic response approaches the

elastic response as a0 becomes large (i.e., a0 > 2).

The last two observations directly result from the deviatoric elasto-viscoplastic model

proposed in this study. At the zero-frequency level (ao = 0), the increment of time step At

approaches infinity and consequently, Lt/77 - oo for any finite value of i. As indicated

previously, this corresponds to the inviscid solution. On the other hand, as a0 becomes large, At decreases to very small values causing Lt/71 -+ 0 for any finite, nonzero value of

i, implying an elastic solution. Furthermore, it is well known that the amplification factor

(amplitude ratio) is independent of damping at high frequencies. Hence, the solution for

= 0 also approaches the elastic curve as a0 becomes large.

In Fig. 3.4, the soil was assumed to be elasto-plastic (i 0) with the same uniaxial

yield stress as that used in Fig. 3.3. Amplitude-frequency plots then are constructed for

different values of H' using the same FE methodology to show the influence of material

stiffness degradation on the system response. The following additional observations can

be made from Fig. 3.4: (1) as H' approaches zero, the resonant peak increases and, al-ternately, as H' becomes large the resonant peak disappears; and (2) H' <0 results in

an unstable time-history response, i.e., no steady-state solution appears to exist for rigid

foundations oscillating on a strain-softening elasto-plastic soil. The second observation is

understandable considering that strain-softening causes the yield surface to collapse with

harmonic vibration.

A more illustrative picture of the nonlinear SSI phenomenon is shown in Fig. 3.5 where

snapshots of deformed meshes for the circular foundation problem are portrayed. This

figure also shows the zones of plastification at the indicated time instants, where "full

14

plastification" implies that all of the four Gauss points in an element yielded while "par-

tial plastification" implies yielding of at least one, but not all four, Gauss point. Note

that the effect of repeated loading on machine foundations resting on an elasto-viscoplastic

half-space is to create a zone of intense yielding and inelastic deformation beneath the

foundation, which in turn results in "dynamic nonhomogeneity" with respect to soil elas-

ticities. The yield zone thus created is a finite zone where the soil moduli have degraded to

their elasto-plastic or elasto-viscoplastic values; however, it is not a fixed region in space.

Instead, at steady-state the yield zone oscillates in such a way that it defines the same

region in space it occupied one period of time earlier. The effect of radiation damping, in

which outgoing waves quickly die out, is also portrayed in Fig. 3.5 to illustrate the ease

with which the steady-state waves are established in a half-space.

3.2 Vertical Vibration of Square Foundations

The three-dimensional FE mesh for this problem is shown in Fig. 3.6. The mesh is com-

posed of 3,856 nodes and 3,159 trilinear brick elements including the foundation elements,

giving rise to a total of 9,896 unknown degrees of freedom. In this problem, the forcing

function f(t) and temporal discretization are the same as those used in Example 1.

A mesh convergence study similar to that discussed in Example 1 was performed to

assess the adequacy of the spatial and temporal discretizations. Results of this study are

summarized in Fig. 3.7, which shows that the time-domain FE elastic solution compares

10

very well with the frequency-domain amplitude-frequency curve for a square foundation on

an elastic half-space presented by Wu (1991). It must be noted that for square foundations,

the expression for a0 takes the form

a0 = vs (20)

where b = half-width of the foundation.

Fig. 3.7 also shows the nonlinear responses derived from running three-dimensional FE

analyses where the half-space is represented by an elasto-plastic (ij = 0), strain-hardening

15

(H' = 0.20E3) material. In this study, the free variable is the uniaxial yield stress

which controls the size of the yield zone. The following observations can be made from

Fig. 3.7: (1) as o, decreases the dynamic displacement amplitude increases; (2) for the

range of values of o, shown in Fig. 3.7, resonance is created in all three nonlinear cases;

(3) as a, decreases, the resonant peak increases; and (4) the displacement amplitudes for

the elasto-plastic case approach those for the elastic case as a0 becomes large (i.e., a0 > 2).

Finally, the deformed meshes and yield zones at time instants I = 1.5T (corresponding to "deepest push") and t = 2.OT (corresponding to "highest pull") are shown in Figs. 3.8

and 3.9, respectively. As seen previously, he effect of radiation damping for the three-

dimensional case is to enhance the formation of steady-state waves over a short period of

time.

3.3 Discussion of Results

The response of a foundation on a soil stratum containing a local yield zone is similar

to the response of a foundation on a nonhomogeneous half-space. It is known that the

response of a foundation on a nonhomogeneous half-space can be substantially different

from the response of an identical foundation resting on a uniform half-space (Kausel 1974;

Kausel and Ushijima 1979). For example, the presence of a bedrock at a relatively shallow

depth causes the amplitude-frequency curves for vertically oscillating circular foundations

to exhibit resonant peaks (Gazetas 1983). Since the effect of local yielding is to create a

finite zone that is generally softer than the surrounding medium, resonant peaks can also S be expected to occur under this condition.

The creation of resonant peaks can be explained better with the aid of a simple

single-degree-of-freedom (SDOF) oscillator analog. The idea here is to replace the proto-

type foundation-soil system with an SDOF mass-spring-dashpot oscillator with frequency-

dependent stiffness and damping coefficients (Hsieh 1962; Lysmer 1965). Under a harmonic

exciting force the dynamic impedance function for an oscillator with a natural frequency

w is given by Eq. (19), with a = 1 and 9 = cv3/r, where c = dashpot coefficient.

16

The dashpot serves to represent the effect of radiation damping which arises when stress

waves propagate outward from the contact surface as the foundation moves against the

soil (Reissner 1936). Thus, the amplitude ratio, i = / (a2 + a2)_h/2, can be plotted

against a0 for each of the equivalent oscillators.

Fig. 3.10 shows the amplitude-frequency plots for vertically oscillating circular foun-

dations on (1) a homogeneous half-space, (2) a stratum of thickness H = 2r underlain

by a firm bedrock, and (3) a cylindrical medium of radius R = 2r and thickness H = 2r

surrounded by a firm bedrock (a physical meaning for the third case is that of a soil in a

valley). The rationale for presenting these curves is that as the ratio between the soil-to-

rock moduli approaches unity, the problem approaches that of a uniform half-space. On

the other hand, as the moduli ratio approaches zero, the problem degenerates to any of the

remaining two specific cases mentioned above. For values of the moduli ratio between zero

and unity, the problem reduces to a composite soil system in which the foundation soil is

either underlain by or is embedded in a stiffer medium, the latter resembling the case of a

half-space containing a local yield zone.

For the half-space problem, radiation damping is so significant that resonance is not

possible. The amplitude-frequency plot for the stratum-over-bedrock problem was con-

structed from known variations of the oscillator parameters a and 3 with a0, as developed

by Kausel (1974), Kausel and Ushijima (1979), and Tassoulas (1981). Here, the dimen-

sionless parameter 9 for the equivalent oscillator is an irregular, nonnegative function of

a0, which has a value nearly equal to zero when a0 < 1. Since the damping coefficient

fi is small for low-frequency excitations, the foundation response exhibits resonant peaks.

Finally, the foundation response for the "valley" problem was constructed directly by the

authors by running extended time-domain analyses to achieve steady-state conditions. The

response curve portrayed in Fig. 3.10 shows more pronounced resonant peaks and multiple

resonance frequencies. For convenience in presentation, the curves plotted in Fig. 3.10 all

have been normalized with respect to their corresponding elasto-static displacements so

that they all converge to A*/A = 1 at a0 = 0.

17

It is evident from the examples presented in this section that local yielding in an other-

wise homogeneous elastic half-space tends to reduce the effective dynamic impedance of the

half-space and create resonance frequencies where the foundation motions are amplified to

values well above those at zero-frequency levels. There was no attempt on the part of the

writers to develop nonlinear "design curves" similar to those available for linear systems

because the system response is generally a function of the specific constitutive model cho-

sen to describe the soil behavior. Furthermore, the response curves are nonlinear functions

of the constitutive model parameters. However, with a well-validated and calibrated con-

stitutive model, a FE methodology for generating nonlinear response curves was presented

herein so that more complex soil behavioral features can be incorporated into the solution

for routine engineering applications.

3.4 Vertical Vibration of Rectangular and Strip Foundations

In this section the responses of vertically oscillating rectangular and strip foundations on

elasto-plastic half-space are compared to the corresponding linear elastic responses. The

FE mesh investigated, shown in Fig. 3.11, is composed of about 4K nodes and over 3K

trilinear brick elements, resulting in a total of about 10K unknown degrees of freedom.

Note that this mesh represents one-fourth of the total 3-D mesh, and can thus be used for

analysis of vibrational response in the vertical direction only.

The foundation dimensions are represented by the fixed half-width b and the varying half-length 1, with i/b > 1.0 representing the various foundation aspect ratios. In the limit I as i/b -* oc which corresponds to a strip foundation, a plane strain condition results, and

the mesh of Fig. 3.12 will degenerate to an equivalent two-dimensional mesh. No artificial

absorbing boundary was used in the numerical model.

Harmonic vibrations were established by applying a sinusoidal vertical excitation force,

f (t), in the time domain. For convenience, and for consistency with results for founda-

tions having different aspect ratios, the harmonic force was computed for each rectangular

foundation in such a way that the mean pressure, p = f(t)/bl, is unchanged (in the plane

IV

strain limit, 1(t) was computed on the basis of a unit i). More specifically, the ampli-

tude of f(t) is proportional to the foundation area. It is important to note that since no

artificial absorbing boundaries were used in the analyses, the only meaningful results are

the computed foundation responses prior to the arrival of the reflected waves. The longest

possible duration of analysis can be estimated as the total time it takes for a travelling

P-wave to radiate from the short edge of the foundation (in the i-direction), reflect on the

boundary, and arrive back to the same point as a reflected wave, see Fig. 3.11.

Fig. 3.12 shows the results of linear elastic FE analyses in the form of amplitude-

frequency curves. For ease in presentation, the steady-state amplitudes were normalized

with respect to the zero-frequency elastic amplitude corresponding to a square foundation.

The excitation frequencies were nondimensionalized through the use of the parameter a0 =

wb/v8, where w is the excitation frequency and v is the shear wave velocity.

The following observations can be made from Fig. 3.12: (1) the higher the aspect ratio,

the larger the steady-state amplitude; (2) as i/b -p oo, the zero-frequency amplitude ap-

proaches infinity; (3) at sufficiently low frequencies (e.g., a0 < 1.0) steady-state amplitudes

are a strong function of the foundation aspect ratio; (4) at the zero-frequency level, the

static amplitudes are nearly proportional to the square root of the foundation areas; and

(5) at sufficiently high frequencies (e.g., a'0 > 2.0), the amplitudes are nearly indepen-

dent of the foundation aspect ratio. In general, these behaviors are consistent with those

presented in Wu (1991).

Some explanations of the foregoing observations are given in the following. It has

been proven that the vertical static stiffness of a typical rectangular foundation can be

approximated with good accuracy by the corresponding value of a circular foundation

having the same area (Gazetas 1983). Since the static stiffness of a circular foundation

has an analytical expression and is proportional to its radius (or the square root of its

area), observations (3) and (4) mentioned above are expected. Furthermore, since a strip

foundation does not have a characteristic length scale, the static foundation impedance

degenerates to zero in the limit as i/b -i 00, i.e., the amplitude becomes unbounded. On

19

the other hand, at high-frequency excitations radiation damping dominates the system

response. Since the radiation energy flux travels predominantly in the vertical direction as

the excitation frequency increases (Gazetas 1987), for the same force per unit foundation

area the steady-state amplitude remains nearly constant regardless of the foundation aspect

ratio. Conclusively, there is an important point to make from the above observatons. As

a0 -* 0, the magnitudes of vertical impedance functions for rectangular foundations are

approximately proportional to the square root of their areas, whereas the corresponding

magnitudes are proportional to their areas at high frequencies.

Fig. 3.13 illustrates results from the nonlinear elasto-plastic analyses as well as ob-

servations noted from the elastic case. Additional observations from Fig. 3.13 can be

summarized as follows: (1) local soil yielding does amplify the motion of the foundation;

(2) resonance is created when the aspect ratio becomes reasonably close to unity; and (3)

at sufficiently high frequencies (e.g. a0 > 1.0), the steady-state amplitudes are nearly the

same regardless of the foundation aspect ratio. This last point also is true for the elastic

case, but at higher values of a0.

Of particular interest in this example is the creation of resonance frequencies when the

foundation aspect ratio becomes reasonably close to unity, i.e., when foundation becomes

nearly square. Recall that the effect of local soil yielding is to create a finite yield zone

below the foundation where the soil moduli have degraded to their elasto-plastic values.

Furthermore, the effect of having a soft zone embedded in a stiff medium is to reduce the

effective radiation damping of the system. Consequently, resonance is created for square

and nearly square foundations. Alternately, it was noted previously that the magnitude of

the impendance function approaches zero in the static limit as the aspect ratio approaches

infinity; consequently, the radiation damping is large enough to suppress the resonant

peak in the limit as i/b - oo. As illustrated in Fig. 3.13, the radiation damping per unit

foundation area increases with the increasing aspect ratio such that the resonant peak does

not occur in slender foundations (e.g., i/b = 3.2). Observation (3) mentioned above can

be explained by the degraded soil moduli which cause the "high frequency" level (relative

- 20

to the shear velocity of the soil) to be reached under lower excitation frequencies.

Figs. 3.14 through 3.17 illustrate the extent of oscillating yield zones for the rectangular

(b/i = 2) and strip foundation problems under an excitation frequency of a0 = 2.0 (consid-

ered high-frequency excitation, see Fig. 3.13). The deformations for the strip foundation are

shown in perspective view so that they may be compared directly with those for rectangular

foundation, but the former problem was actually analyzed under a two-dimensional plane

strain condition. Although the yield zones for the two foundations clearly differ in extent

and geometry, note that they produced nearly the same steady-state amplitudes, a stark

contrast to low frequency behavior where amplitudes increse noticeably with increasing

aspect ratio. This points to the previously mentioned fact that at high frequency excita-

tions, the extent and geometry of yield zones balance in such a way that the magnitude of

impedance function per unit foundation area is the same regardless of the foundation as-

pect ratio. Furthermore, note that surface waves dissipate rapidly with increasing distance

from the foundation. This implies that "steady-state" waves can be established reasonably

rapidly in time before the reflected waves reach the foundation. In general, the required

time to reach a "steady-state" condition is longer for elasto-plastic problems than it is for

the elastic half-space problem since the presence of a local yield zone embedded in a stiffer

medium does tend to lessen the radiation damping effects. Time-domain models designed

to handle material nonhomogeneity generally require artificial absorbing boundaries so that

extended time-stepping computations may be made feasible.

S 3.5 Summary and Conclusions

The dynamic response of vertically-excited rigid foundations on an elasto-viscoplastic half-

space has been investigated in the context of nonlinear FE analysis. The effect of repeated

loading on machine foundations is to create a zone of intense yielding and inelastic defor-

mation beneath the foundation, resulting in "dynamic nonhomogeneity" with respect to

soil elasticities. In general, radiation damping is smaller for a half-space problem contain-

ing a local yield zone than it is for the same half-space problem without a yield zone. Local

yielding thus creates resonant peaks on the amplitude-frequency curves which increase as

21

the ratio between the elastic to elasto-plastic (or elasto-viscoplastic) moduli increases. This

phenomenon is consistent with known results for the problem of finite-size foundations on a

uniform layer over bedrock and for the problem of finite-size foundations on a finite region

embedded in a half-space.

Furthermore, local yielding influences the foundation response at low frequency excita-

tions, but its influence in high frequency regime is generally insignificant. Another factor

which is of special interest in this report is the effect of the foundation aspect ratio on the

amplitude-frequency response of vertically vibrating rectangular foundations. Although

low frequency amplitudes depend significantly on the foundation length 1 for a given fixed foundation width b, it has been shown that high-frequency amplitudes do not.

A notable contribution of this report is the advance in nonlinear soil-structure inter-

action analysis. A nonlinear model such as the one presented in this report provides a

means for understanding the influence of other factors that could dominate the SSI re-

sponse, such as the hysteretic soil behavior and soil stiffness degradation. When validated

and calibrated, nonlinear models allow the extrapolation of numerical results well beyond

the limits or capabilities of those that employ traditional linear models. With the advent

of supercomputers and with more efficient numerical tools, it is shown that large-scale

nonlinear SSI analyses can be carried out in a computationally feasible manner despite the

costly time-stepping and iterations. Specifically, three-dimensional nonlinear SSI analyses

of the order of 10K degrees of freedom and requiring a total of about 100 iterations can

be run on the Cray supercomputer in about two hours of CPU time. With efficient par-

allel computation techniques coming of age, the feasibility of performing these complex,

computationally intensive nonlinear analyses will be enhanced.

22

4 Foundations Subjected to Horizontal, Rocking, and Torsional Vibrations

It is shown in Chapter 3 that the effect of local yielding on the dynamic response of

vertically excited circular and rectangular foundations (including strip foundations) does

induce the creation of resonant peaks on the amplitude-frequency curves. Furthermore,

parametric studies indicate that as the excitation frequency increases, the nonlinear effect

decreases (i.e., the elastic behavior dominates the foundation response in the high-frequency

range). The next phase of this project involves extending this investigation to include

horizontal, rocking, and torsional modes of vibration.

While the results presented for vertically oscillating foundations are helpful in demon-

strating the significance of plastic deformation on the dynamic response of vibrating foun-

dations, they are not directly applicable to problems involving seismically induced excita-

tions where typically the foundation motion is dominated by lateral and rocking modes.

In this chapter we employ the same soil constitutive model and methodology for nonlin-

ear analysis of vertically oscillating foundations to study the nonlinear responses of finite

size foundations in lateral, rocking, and torsional modes. The methodology is generalized

further by considering both two- and three-dimensional problems with implicit integration

in the time domain. Whenever possible, symmetry and anti-symmetry are exploited in

the analyses so that the size of the problem is reduced. For example, symmetry cuts the

size of the problem by one-half as does anti-symmetry; thus, a three-dimensional analysis

of rectangular foundations for either lateral, rocking or torsional vibration modes can be

performed successfully with the use of a quarter-size mesh.

The enhanced nonlinear version of DLEARN is again used and run on a CRAY Y-

MP 8/864 supercomputer to construct time-history responses and steady-state amplitude-

frequency curves for various modes of oscillation. The amplitude-frequency curves are used

to assess the impact of soil nonlinearities on the foundation behavior as well as determine

the range of excitation frequencies over which nonlinear effects may be dominant. Two

23

error tolerances, TOLl = iO and TOL2 = 10-2, were used in the code for control of global

Newton and local PCG iterations, respectively (Borja 1991). The elastic parameters for the

foundation-soil model are: Young's modulus E3 = 213 MPa, Poisson's ratio ii = 1/3, and elastic shear wave velocity v3 = 200 rn/sec for the soil; and Young's modulus E f = 109E3 for the "rigid" foundation. Material responses were sampled at the Gauss points using

a full four-point Gaussian integration for 2D analysis and an 8-point integration for 3D

analysis. Time-stepping was done via the a-method of Hilber et al. (1977) using time-

integration parameters of /9 = 0.3025, -y = 0.60, a = —0.10, and a time step of At = T/16, where T = period of excitation.

4.1 Symmetry/Anti...Symmetry and Forcing Functions

Except for the axisymmetric case of vertically vibrating circular foundations, all problems

associated with oscillating finite-size foundations are three-dimensional which induces im-

mense increases in the bandwidths of the storage matrices as compared to two-dimensional

analysis. Therefore, whenever possible, nonlinear time-domain analyses of the 3D foun-

dation vibration problems requires utilization of symmetry and anti-symmetry conditions

where each reduces the size of the problem by one-half to make the computation feasible.

Fig. 4.1 illustrates how these conditions can be exploited to reduce the overall size of the

matrix problem.

Consider a rigid square foundation resting on a half-space. Under vertical excitation this

foundation will oscillate in such a way as to form two vertical planes of symmetry passing S through the foundation axes. Vertical planes of symmetry can be prescribed numerically

using horizontal roller supports in the manner shown in Fig. 4.1. Thus, this problem can

be analyzed by modeling only one-quarter of the half-space, as we did in Chapter 3.

Consider next a square foundation subjected to torsional excitation. In this case, only

horizontal shear waves propagate; therefore, the two vertical planes cutting the foundation

axes represent planes of anti-symmetry. Planes of anti-symmetry can be prescribed numer-

ically by releasing the constraints normal to these planes in a manner shown in Fig. 4.1.

24

Thus, the solution also requires only one-quarter of the half-space to be modeled.

Finally, consider a square foundation subjected to either lateral or rocking excitation.

In this case, one vertical plane passing through a foundation axis represents a plane of

symmetry while the other represents a plane of anti-symmetry. Thus, the size of the

matrix problem can be reduced analogously. The appropriate boundary conditions for FE

vibration analysis of rectangular foundations are shown schematically in Fig. 4.1.

Forcing functions for forced vibration analysis are sinusoidal nodal forces acting on the

foundation. These forcing functions used in this study are shown in Fig. 4.1 for various

modes of vibration. Note that the amplitudes of these forces (or moments) are only 25%

of their total values since only one-fourth of the total half-space is modeled.

Vertical and lateral forcing functions are concentrated forces applied on the foundation

in a manner shown in Fig. 4.1. For a rocking foundation the forcing function is a couple

represented by a pair of eccentric and oppositely directed vertical nodal forces. However,

due to the fact that the center of the foundation lies on a plane of anti-symmetry where

vertical motion is prohibited, only one of the two forces need be prescribed. Finally,

torsional forcing functions are represented by a pair of orthogonal oscillating forces acting

in phase on the foundation sides. Because of the small strain formulation adopted in the

numerical model and since the finite elements have no rotational degrees of freedom, it

is not possible to simulate torsional excitations with a single force applied only on one

foundation side. For output interpretation purposes, the "responses" of the foundation

to lateral, rocking, and torsional excitations are the horizontal displacement L, angular rotation e, and angular twist , respectively.

4.2 Horizontal and Rocking Vibrations of Strip Foundations

As discussed earlier, nonlinear analysis of 3D meshes are prohibitively expensive to perform

unless special techniques are utilized to reduce the model size. An exception is the strip

foundation analysis which can be reduced to plane strain conditions with uniform loading

along the longitudinal direction. Hence, a two-dimensional FE mesh is sufficient in this

- 25

case, and the choice of strip foundations makes it feasible to investigate the horizontal and

rocking vibrations of foundations by the FE methodology without intensive computation.

The two-dimensional FE mesh used here is similar to the one shown in Fig. 3.1 where

the vertical vibration is considered. To make this mesh also suitable for investigating

horizontal and rocking vibrations, utilization of anti-symmetry conditions mentioned earlier

is required. Harmonic vibrations are established by applying a sinusoidal horizontal force

or rocking moment along the longitudinal direction of the foundation.

Classical theory of elasticity predicts that the static vertical and horizontal impedances

of strip foundations are. zero, which implies that the zero-frequency foundation displace-

ments are infinite. To generalize the presentation of steady-state amplitude-frequency I response curves for horizontally vibrating strip foundations, the horizontal displacement

amplitudes L are herein normalized with respect to their elastic value, at a nondimensionalized frequency of a0 = wblv, = 1, where w = excitation frequency and b =

foundation half-width. No such singularity arises for rocking strip foundations, however,

since their rocking impedances are finite even at the zero-frequency level. Thus, for rocking

foundations the rotational amplitudes E can simply be normalized with respect to their

elastic value at the zero-frequency level, W. The normalized nonlinear response curves are

then plotted versus a0 in Figs. 4.2 and 4.3 for strip foundations vibrating in the lateral and rocking modes.

Fig. 4.2 shows the influence of plastic modulus H' and viscosity coefficient 77 on the I steady-state amplitude-frequency responses of horizontally vibrating strip foundations. For

the first case, an elasto-plastic soil (77 = 0) with an initial uniaxial yield stress of o, = 0.0013E3 was assumed, and H' was allowed to take on different values. For the second case, H' was set to 0.40E3, o, to 0.0013E3, and i, was allowed to vary. It is clear from

the response curves of Fig. 4.2 that (a) local soil yielding results in increased vibrational

amplitudes; (b) increasing H' and/or 77 decreases the vibrational amplitudes; (c) minor

resonant peaks are created at sufficiently low values of H' and i; and (d) nonlinear soil effects are dominant over a wide range of excitation frequencies (ao = 0 .- 5). The first two

RIi

observations are similar to those observed in the vertical vibration of rigid foundations and

can be explained by the stiffness degradation of the soil. The third observation is a peculiar

feature and may have resulted from the participation of higher natural frequencies of the

system. The last observation differs remarkably from the results in Chapter 3, which show

that for vertically oscillating strip foundations the nonlinear amplitude-frequency response

curves approach the elastic solution at a much lower excitation frequency (e.g., a0 < 2).

Thus, this example shows that while for a vertically vibrating foundation nonlinear soil

effects may be relatively unimportant in the high-frequency range, they could be significant

over a wide range of excitation frequencies for a horizontally vibrating foundation.

A similar parametric study was conducted for rocking strip foundations and the results

are shown in Fig. 4.3. Note that for rocking strip foundations nonlinear soil effects are

dominant only in the low-frequency range (e.g., a0 <3). However, Fig. 4.3 shows that at

or near the resonance frequencies the angular displacements can be amplified dramatically

by nonlinear soil effects. One can explain the similarity in responses between a vertically

vibrating foundation and a rocking foundation from the fact that both systems generate

predominantly P-waves. On the other hand, a horizontally vibrating foundation generates

mostly S-waves, which radiate at a slower speed than P-waves. Thus, the effects from

radiational damping are reduced, resulting in the nonlinear model for horizontal modes

converging to the elastic model at larger frequency values than those observed for vertical

and rocking modes.

Finally, Figs. 4.4 and 4.5 show snapshots of the deformed meshes and yield zones at

a0 = 2.0 for strip foundations vibrating in the lateral and rocking modes, respectively.

For presentation purposes, a full mesh was employed to show the implications of the anti-

symmetric feature exploited in the analyses. Fig. 4.4 shows the predominantly S-waves

generated by the horizontally vibrating strip foundation, while Fig. 4.5 suggests that a

rocking strip foundation generates mostly P-waves. The similarity of the yield zones even

at the early pairs of time instants (e.g., t = 1.75T and t = 2.25T) suggests that "steady-

state conditions" can be achieved over a reasonably short period of time, thus affirming

- 27

the potential of a time-domain solution methodology for nonlinear FE analysis of vibrating

foundations.

4.3 "Critical Frequency" and "Optimal" FE Grid

In dynamic analysis the accuracy of a FE grid is a function of the excitation frequency.

Because of spatial discretization error, accurate representation of higher-order vibration

modes requires increasingly finer meshes. However, the cost of a three-dimensional FE

analysis grows at an enormous rate as the resolution of the mesh is increased.

Fig. 4.6 shows two FE grids consisting of eight-node trilinear brick elements, which

are considered "optimal" for three-dimensional forced vibration analysis in the low- and

high-frequency ranges. These grids have the same total number of degrees of freedom

(approximately 10K) but have different mesh dimensions. The coarse mesh (Fig. 4.6(a))

is optimal for low-frequency excitations where high-frequency modes are not significant.

The long dimensions of this mesh allow a sufficient number of long-period load cycles to

be applied at the foundation prior to the arrival of the reflected waves. Alternately, the

mesh of Fig. 4.6(b) is appropriate for high-frequency excitations where the capturing of

higher-order modes requires finer resolution. Although the dimensions of this mesh were

reduced for reasons of economy, this representation still provides an optimal modeling since

it allows a sufficient number of short-period load cycles to be applied at the foundation

prior to the arrival of the reflected waves. Since the total number of degrees of freedom is

the same for both grids, the cost of running a full three-dimensional FE dynamic analysis

is also approximately the same for either mesh.

Fig. 4.6 also shows two FE half-meshes (Figs. 4.6(c) and 4.6(d)) for two-dimensional

analysis of forced vibration of strip foundations in both lateral and rocking modes (note

that the centerline for each mesh is a plane of anti-symmetry). Since the increase in

computing costs with mesh size is not as dramatic for 2D as it is for 3D analyses, the high-

resolution mesh of Fig. 4.6(d) may be used both for low- and high-frequency excitations,

as we did in the previous section on investigating the horizontal and rocking vibrations of

NEZ

strip foundations. The low-resolution mesh of Fig. 4.6(c) was used in this study only to

verify the adequacy of the fine mesh.

In interpreting the FE results, one should be careful in evaluating the adequacy of a

FE mesh since its accuracy is generally a function of the mode order of the system and

the excitation frequency. Mesh convergence studies are thus an important part of the nu-

merical analysis since the predicted dynamic responses could vary dramatically depending

upon the quality of the FE mesh. There is no a priori rule for cho6sing an "optimal" mesh

particularly for nonlinear applications, but in general, for low-frequency excitations, the

coarser the mesh, the stiffer the predicted response. For a mesh dominated by nonhomo-

geneous material response arising from local plastification, an opposite trend is possible

depending upon the magnitude of the excitation frequency.

Figs. 4.7-4.10 show the predicted time-history responses of strip foundations vibrating

in the horizontal and rocking modes using the two plane-strain FE grids of Fig. 4.6(c) and

an extended version of Fig. 4.6(d). For a horizontally vibrating foundation, Fig. 4.7 shows

that the resolution of the coarse mesh in the elastic regime is adequate for low-frequency

excitation (ao = 1.0) but not for high-frequency excitation (ao = 5.0). For this mode of

vibration, the "critical frequency" below which the coarse mesh is acceptable but above

which it is unacceptable is approximately a0 = 3.0. On the other hand, for a rocking

foundation Figs. 4.9 and 4.10 show that the coarse mesh is adequate for both low- and

high-frequency excitations (ao = 1.0 and a0 = 5.0, respectively). The adequacy of the fine

mesh was established from a separate mesh convergence study employing a yet finer mesh.

In general, the accuracy of the mesh degrades with increasing values of the ratio h/k, where h = element dimension and ,\ = wavelength, which equals v/w, where v = prop-agating wave velocity and w = excitation frequency. For an elastic soil with a Poisson's ratio of v = 1/3, the P-wave velocity v, is two times greater than the S-wave velocity v

(or, equivalently, the wavelength ratio in a homogeneous soil is ),/) = 2 under the same

excitation frequency), which explains why the coarse mesh may not be sufficient at high

frequencies for the S-wave dominated horizontal motion even though it is adequate for the

- 29

P-wave dominated rocking motion.

Much can be learned from the time-history responses of Figs. 4.7 and 4.8 for a hor-

izontally vibrating strip foundation. For an excitation frequency of a0 = 5.0, Fig. 4.7

indicates that, for the coarse mesh, reflected waves arrive at the foundation much sooner

than expected (the theoretical arrival time of reflected S-waves is t = 9.9T). The early

arrival time might be due to the fact that while the elements near the foundation are

fine enough to effectively transmit outgoing S-waves, the exterior elements could be too

coarse to transmit the same waves on to the reflecting mesh boundaries. Thus, the coarse

elements might have acted as wave reflectors, reducing the effectiveness of the mesh.

Contrasting observations are indicated in Fig. 4.8 which shows that nonlinear soil effects

tend to 'improve' the performance of the non-uniform coarse mesh of Fig. 4.6(c). With local

yielding the velocity of propagation inside the plastic zone decreases to v1 < v3; however, it is possible that h/v is still below the critical value inside the yield zone allowing the waves

to continue propagating effectively, beneath the foundation. As the shear waves reach

the iinyielded zone, they regain their initial elastic propagation speeds. Thus, it takes

longer for the reflecting waves in the nonlinear case to reach the foundation, resulting in

the performance of the coarse mesh being improved by the introduction of the nonlinear effects.

Since local plastification near the foundation could retard the propagation of body

waves, we define an "optimal" FE grid as one where the elements inside the probable yield I zone are small enough to effectively transmit waves traveling at reduced speeds, and with

exterior elements that are effective enough to transmit elastic waves. An "optimal" mesh

thus provides a balance between solution accuracy and computational cost.

4.4 Horizontal, Rocking, and Torsional VibraCions of Square Foundations

Classical theory of elasticity predicts that the static displacements of finite-size founda-

tions under vertical, lateral, rocking, and torsional loads are finite, and can thus be used

30

to normalize the corresponding amplitude-frequency response curves. For the horizontal,

rocking, and torsional modes, the normalized responses shown in this paper are the horizon-

tal displacements */e, angular rotation e*/ee, and angular twist */e, respectively.

Figs. 4.11-4.13 show the predicted amplitude-frequency plots for a vibrating square

foundation subjected to lateral, rocking, and torsional excitations. Each plot demonstrates

the respective influence of the plastic hardening modulus H', viscosity coefficient i, and

initial yield stress o, on the foundation's dynamic response. All the response curves of the

elastic cases are compared with the frequency domain solutions in Wu (1991) to show the

accuracy of the time domain analyses. It is clear from the response curves of Figs. 4.11-4.13

that (a) local soil yielding increases the predicted vibrational amplitudes of the foundation

motion particularly at decreasing values of H', 77, or o; (b) primary resonance peaks

are created at sufficiently low values of a0 in all modes; (c) compared to the rocking

mode, nonlinear soil effects are more dominant over a wider range of excitation frequencies

(ao = 0 5) for the lateral and torsional modes. The latter observation is consistent with

those observed for a vibrating strip foundation and can be explained from the type of body

waves that each mode of vibration generates.

In generating the above response curves, the results from the mesh convergence studies

of the previous section were utilized to determine which FE grid is appropriate for each

analysis. Thus, for the lateral and torsional vibration modes which produce mostly S-

waves, the coarse mesh of Fig. 4.6(a) was used for low-frequency excitations (ao < 3.0), and the fine mesh of Fig. 4.6(b) for high-frequency excitations (ao > 3.0). On the other

hand, for the rocking mode which generates mostly P-waves, the coarse mesh of Fig. 4.6(a)

was used for all values of excitation frequencies. Results of an additional mesh convergence

study for the S-wave dominated torsional vibration are shown in Fig. 4.14. Nonlinear soil

and high-frequency excitations are chosen to show the inadequate performance of the coarse

mesh in this case. This phenomenon is similar to that of the S-wave dominated horizontal

vibration of strip foundations and reveals that the methodology used for the 2D strip

foundation problem also applies to the 3D square foundation problem.

31

Finally, Figs. 4.15-4.17 show snapshots of the deformed meshes and yield zones at

a0 = 2.0 for square foundations vibrating in the lateral, rocking, and torsional modes.

A low-frequency excitation was chosen to plot the relevant yield zones since it is at low

frequencies where nonlinear soil effects tend to be more dominant. Again, the similarity of

the yield zones at the early pairs of time instants suggests that "steady-state" conditions

had been achieved over a short period of time, and affirms the potential of a time-domain

solution methodology for nonlinear FE vibration analysis.

4.5 Summary and Conclusions

The dynamic response of harmonically excited rigid foundations has been investigated

in the context of two- and three-dimensional nonlinear FE analysis. Lateral, rocking, and

torsional vibration modes were considered. Localized soil yielding beneath the foundation is

shown to result in increased vibrational amplitudes of the system response and the creation

of resonance particularly at low-frequency excitations. At high-frequency excitations (ao '

5.0), amplification of the harmonic response due to nonlinear soil effects is also possible

particularly when the motion is dominated by shearing modes. The latter result is contrary

to common understanding that nonlinear soil effects are generally unimportant in the

high-frequency range. Results presented in this paper apply only to soil conditions where

the deviatoric elasto-viscoplastic constitutive model is appropriate (e.g., soft soils and

undrained conditions).

Some computational issues which impact the quality of the numerical results have been I addressed. Mesh convergence studies are extremely important in assessing the adequacy of

a FE grid. The time-domain solution methodology is shown to be effective in the analysis

of nonlinear machine foundation response in all possible modes of vibration. Furthermore,

with a supercomputer and an enhanced FE code, it is shown that implicit time-stepping

schemes can be performed in a feasible manner even for large-scale nonlinear 3D systems.

Studies are now underway to investigate the potential of the methodology for nonlinear

analysis of seismically excited structures and for developing nonlinear transfer functions

that are critical in nonlinear soil-structure interaction analysis.

32

5 Boundary Element Model for Plane Strain

The objective of this chapter is to describe an on-going research at Stanford University on

coupled finite element (FE)-boundary element (BE) modeling for nonlinear soil-structure

interaction analysis. New implementational ideas that pertain to the BE formulation for

plane strain are also described. We are now testing segments of the BE code for eventual

integration into the nonlinear FE code. Note that all analyses are done in the time domain.

5.1 Time-Domain BE Formulation

We consider an exterior domain çE bounded by a regular surface 49ç1E in the sense of

Kellogg (1929). Let u and v be two single-valued, twice continuously differentiable vector

fields. We can look at u as the real displacement field and v as the virtual field. The

boundary integral equation for linear elastodynamics with quiescent initial condition and

zero gravity forces then reads (Kobayashi 1987)

c.u()=j(u*v_u*v)dS (21)

where c is a 'jump' tensor, V = n cc : V is a gradient mapping operator which maps a

given displacement field onto a traction field n is the unit normal to 811E, is

the source point, Ce is the elasticity tensor, V is the gradient operator, and the symbol *

denotes a time convolution.

An important element of a boundary integral formulation is the fundamental solutibn

Here, we employ Green's function for a point pulse t at source point for our fundamental

solution to determine the response of point x in an infinite domain. The fundamental solu-

tion for the response v(x, ; 8; t) has the explicit form (Eringen and Suhubi 1975; Manolis

and Beskos 1988)

1 1r1r3 ,' r v2(x,;63;t) = - - -) + -( - r

4 [ c1 c2 r r3 -

3rr 6 c' ' '

IV

8(t - r) d] (22) r

33

while for Vv(x, ; ; t) the fundamental solution is

=

--- { [( - )7jfl - c (i2 '1rirmnm - 2rjn1 + r1n + Sii'mflm)]ö(t

- -f-) 4ir r3 c 12 r5 r3

+(l2 i'm flm - 3rn3 + 6iirmnm) -

_64(5r - - r,rz1 + ran, + öljrmnm - fc;~

ir3 b(t

+1 2 i'i'ji'mm 1 -

2 rjrjrmnm 1 rn, + Siirmnm)( - (23) C2 r4 c2 r2 c2

I where r2 = - x2 J, r = - xli, c1 is the P-wave velocity, and c2 is the S-wave velocity.

Substituting Eqs. 22 and 23 in Eq. 21 results in the exact boundary integral equation

cu() 4ir p (24)

where

1, = I [ -i'd-

u( r 1 - ri x,t - -) + t - - -) c r3 Cl c2 r r3 c2

3rr - r3 r IC u(t - r) d] dS (25) 1

and

[I

JE { [(24_l)rL nj

C (i'n - 2rn + rn + Sjjrmnm x, t -

r5 r3 )]u1(

Cl -) — 2 rjn2 - 3r1n + 6ijrmnm

)u1(x,t - .1r5

r3 r3)

rn2 + r1n, +

r5 r3

{ )rni

IC-1 u

-

(x, t - \r)\ dA l

+ 2 C rrjrmnm

+ - r)c

(4 2 ,c 4 r 1 c Cl 2 rjrjrmnm - 1 rn + Siji'mflm r 2

)üi(x,i--)}dS (26)

34

5.2 Boundary Integrals for Plane Strain

Consider a plane strain problem on the x, y-plane. The appropriate boundary integral

equation for this type of problem can be obtained by viewing &E as the surface of an

infinite cylinder parallel to the z-axis. Let us take only the region z > 0 (and then multiply

the results- by two) so that 81E E (x, y) x [0, oo). Equation (4) then reads

c .u(e) = 2lrfL.L 0 dzdL

(27)

where L is the projection of 91E on the x, y-plane.

We next discretize L into n,1 boundary elements and integrate over the elements Lq

numerically to obtain

1 el g c . u() =

(Jo dz)dLq F9 W9 (28) q=1 g=1

where n9 is the number of Gauss integration points, W9 are the Gauss weights, and

F9 = fo D dz (29)

Because of the time-translation property of Vu and u, Eq. 29 should be convoluted with

z (or, equivalently, with t).

Performing a convolution in the time domain requires a temporal discretization. Due

S to the presence of a velocity term that multiplies the tranction kernel, the temporal dis-

cretization needs to be at least C0 -continuous. We consider the following piecewise-linear

temporal discretization for particle displacement (Manolis and Beskos 1988):

UX,i_ (

t_tn _r/c) nlI ' r'

C n At C _)

[(t_tn_i_rIc) (30)

and the following piecewise constant temporal interpolation for particle velocity:

u(x,t__)=>(u Ltn_un-i

) On (-_r) CI (31)

Also, consider the following piecewise linear temporal discretization for the gradient field:

35

___________ t—i,, —r/c Vux,t - = [(_ — —_r)u - ( )'u] (t - ) (32) C n

where c represents the relevant wave speed, n is a time step counter, At is the time increment, and

q(t) = H(t - (n - 1)zt) - H(t - nLt) (33)

in which H is the Heaviside function. Furthermore, the causality condition allows Eq. 29 to be written in the form

F91fR

Dg dz (34)

where a2 = x 2 + y, which is fixed at a given integration point, while R represents the

relevant distance traveled by the singular-point disturbance. Also, the integrands D 9 have typical forms (a2 + 2)_m/2, where m = 1,... , 4, and are therefore analytically integrable.

Note in this formulation that the convolution is done at the Gauss integrations points,

unlike in most conventional formulations where the convolution is done at the nodes. Con-

sequently, piecewise continuous space-time interpolations need to be assumed. To illustrate

this point, assume that the spatial and temporal interpolations are piecewise linear; then

the space-time shape functions must be bilinear. Studies are underway to investigate the

performance of the BE code for eventual integration into the nonlinear FE code.

36

6 List of Publications

Borja, R. I., Wu, W.H., and Smith H. A., "Nonlinear response of vertically oscillating

rigid foundations," submitted to ASCE Journal of Geotechnical Engineering (in review).

Borja., R. I., Wu, W.H., and Smith H. A., "Nonlinear vertical vibration of rectangular

foundations," to appear in the Proceedings of the Tenth World Conference in Earthquake

Engineering Madrid, Spain, July, 1992.

Borja, R. I., Wu, W.H., Amies, A. P., and Smith H. A., "Nonlinear lateral, rocking,

and torsional vibration of rigid foundations," submitted to ASCE Journal of Geotechnical

Engineering (in review).

"Coupled finite element-boundary element formulation for nonlinear dynamics problem

in plane strain" (in preparation).

"Numerical modeling of kinematic interaction for nonlinear soil-structure interaction

analysis" (in preparation).

S

37

7 Acknowledgements

Financial support for this research was provided by the California Universities for Research

in Earthquake Engineering (CUREe), and by the Kajima Corporation, Japan which is

gratefully acknowledged. Computer time was provided by a grant from the San Diego

Supercomputer Center. The authors also wish to acknowledge the help of Mr. Wen-

Hwa Wu and Mr. Alexander Amies, both Ph.D. students studying under the direction of

Professors Smith and Borja.

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41

Perzyna, P. (1971). "Thermodynamic theory of viscoplasticity." in Advances in Applied Mechanics, Vol. 11, Academic Press, New York, N.Y.

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42

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n

43

MASSLESS RIGID EXT0)

FOUNDATION

a

Figure 2.1. Soil-Structure Model.

Ca

I

71=00

H'=O / tj>0

H'<O

PimEa (a) (b)

Ca

S

Cl

0=

0<0<

0=1

ta

L STRESS

>0

10

:0< 1

(d) a3

(C)

t

Figure 2.2. Geometric Interpretation of ConstitutiveModel Parameters: (a) Hardening, Perfect Plasticity, and Softening; (b) Viscous and Inviscid Plasticity; (c) Isotropic, Kinematic,

and Combined Hardening; (d) Viscous Translation and Expansion of Yield Surface on Pi-Plane.

MESH SCALE: lOm

Figure 3.1. Finite Element Mesh for Analysis of Vertically Oscillating Circular Foundation.

S

1.2

p1.0

60.8 - 0 FREQUENCY-DOMAIN SOLUTION 0 TIME-DOMAIN (FE) SOLUTION

0.2

0.0 L

0

2 4 6

8 FREQUENCY PARAMETER, a0 = ar/v3

(a)

TIME, tIT

Figure 3.2. Mesh Convergence Study. Vertically Oscillating Circular Foundation on Elastic Half-Space: (a) Amplitude Versus

Frequency Plot; (b) Time-History Plot.

I 2 4 6 8

FREQUENCY PARAMETER, a0 = cor/v,

(a)

I

z' u..

0 0.4

I— z 0.0 - TIME, tIT w

\t

() -04

=

0 -j 1 !! -. -. -. 0.0002 a

- 0 . 8 17 0.0005 ESTIC SOLUTION

(b)

Figure 3.3. Vertically Oscillating Circular Foundation on Kinematically Hardening Viscoplastic Half-Space: (a) Amplitude

Versus Frequency Plot; (b) Time-History Plot.

2.5

2 4 6 8 FREQUENCY PARAMETER, a0 =

0.0 0

(a)

'.1 ' I'. ! 0 8

S / •-'% S O-r* , \.

20.4 1 A it

I \\

0.0 TIME, tIT ii i\ '.3

-0.4 /1 Ilk '. / f \\ I _o_H'=._o.oSEs I ------ H'=O.1E5 - 0.8 I l \j - - - H' = 0.2Es

1 - ELASTIC SOLUTION

(b)

Figure 3.4. Vertically Oscillating Circular Foundation on Elasto-Plastic Half-Space. Influence of H' on (a) Amplitude Versus

Frequency Plot; (b) Time-History Plot.

E

C

E

E

C

C

-U----

Iliflunaus IIIIIiiiiiu•

FULL PLASUFICAI1ON PARTiAL PLAS11FICA11ON

Figure 3.5. Establishing Steady-State Waves and Yield Zones for Circular Foundation Problem: Arrows Denote Instantaneous Direction

of Force; E = Extensional Yielding; C = Compressional Yielding.

Figure 3.6. Finite Element Mesh for Analysis of Vertically Oscillating Square Foundation; Mesh Dimensions are 60x60x60 m.

0

2.5

.0 FREQUENCY-DOMAIN SOLUTION

TIME-DOMAIN (FE) SOLUTION:

D a=O.00O9E3

L. c7=O.0011E

a=O.0016E

o ELAS11C SOLUTION

0 1 2 3 4 5

FREQUENCY PARAMETER, a0 = t-ob/v

Figure 3.7. Amplitude-Frequency Plot for Vertically Oscillating Square Foundation on Elasto-Plastic Half-Space.

.5

[s]

I

S

Figure 3.8. Square Foundation Problem: Deformed Mesh and Steady-State Yield Zone at 'Deepest Push".

TS:

Figure 3.9. Square Foundation Problem: Deformed Mesh and Steady-State Yield Zone at "Highest Pull".

14

12

10 0

O T

Uj

0

J1JIH

VALLEY

2 2r

1H NUllIllIll IIIIIIIIII

STRATUM OVER BEDROCK

IJ 4 =O.O5

I

1 2 3 4 5 6

7 8

FREQUENCY PARAMETER, a0 = o.,r/v 3

Figure 3.10. Influence of Boundary Conditions on Amplitude-Frequency Plots and Resonant Peaks for Vertically

Oscillating Circular Foundation (Hir = 2): as Soft Region Becomes More Finite, Resonant Peak Increases.

Figure 3.11. Finite Element Mesh for Analysis of Vertically Oscillating Rectangular Foundations; Mesh

Dimensions are 60x60x60 m.

ASPECT RATIO:

1/b =1.0 G i/b = 2.0 L i/b=3.2

i/b=oo

2 3 4 5 6

FREQUENCY PARAMETER, a0 = tob/v,

Figure 3.12. Amplitude-Frequency Plot, Elastic Case.

.v

.5

.0

.5

5

ELASTIC

® i/b = 1.0 ELASTO-PLASTIC

i/b=1.0 ' i/b =2.0. t i/b =3.2

l/b=oo

0.01 I I I I I 0 1 2 3 4 5

6

FREQUENCY PARAMETER, a0 =

Figure 3.13. Amplitude-Frequency Plot, Elasto-Plastic Case. r

S PARTIAL PLASTIFICATION

FULL PLASTIFICATION

Figure 3.14. Deformed Mesh at t=1.5T for a Rectangular Foundation with 1/b = 2.0.

PARTIAL PLASTIFICATION FULL PLASTIFICATION

Figure 3.15. Deformed Mesh at t=2.OT for a Rectangular Foundation with I/b = 2.0.

a

PARTIAL PLASTIFICATION

FULL PLASTIFICATION

Figure 3.16. Deformed Mesh at t=1.5T for a Strip Foundation.

PARTIAL PLASTIFICATION FULL PLASTIFICATION

Figure 3.17. Deformed Mesh at t=2.OT for a Strip Foundation.

VERTICAL

ROCKING

LATERAL

TORSIONAL

Figure 4.1 Symmetry and Anti-Symmetry for Vertical, Lateral, Rocking, and Torsional Vibration Modes. Arrows Denote Free

DOFs, Circles Denote Fixed DOF's.

3.0

2.5

< 2.0 0

Li H'=O.3E

ô

1

0

2 3 4 5 6

3.0

2.5

< 2.0 0

Li

Ai=O.002

<> 11= 0.005 0 ELAS11C SOLUTION

1. 2 3 4 5 6

FREQUENCY PARAMETER, a0 = o.th/v

Figure 4.2 Amplitude-Frequency Plots for Horizontally Vibrating Strip Foundations Showing Effects of Plastic Modulus and Viscosity Parameter.

3.5

3. go

Ii

H'=O.4E

A H'=O.7E

<C> H'=Es ELASTIC SOLUTION

0.0 0

1 2 3 4 5 6

2.5

0

A i=O.002

a ELASTIC SOLUTION

1 2 3 4 5

FREQUENCY PARAMETER, a0 = ob/v M.

Figure 4.3 Amplitude-Frequency Plots for Strip Foundations Subjected to Rocking Vibration Showing Effects of Plastic Modulus and Viscosity Parameter.

(b)

(a)

MESH (C) (d) SCALE:

10 m

Figure 4.6 Three-Dimensional FE Grids for (a) Low-Frequency Excitation and (b) High-Frequency Excitation. Plane Strain FE Grids: (c) Coarse Mesh and (d) Fine Mesh. Note: Foundation

Dimensions are the Same for all Grids.

a0 = 1.0

1

2

c 08

1 0.4 I- z LU

0.0 LU C.) 4 -J CIL Cl) a0 = 3.0

-u.a 0

4

6

1.0

< 0.5

I- z LU

0.0 LU C-) 4

Cl)

_1 fl

a0 =5.0

0 2 4 6 8 10

TIME,t/T

Figure 4.7 Normalized Displacement-Time History Responses of Horizontally Vibrating Strip Foundation on Elastic Soil; Solid Lines

Denote Fine Mesh, Dashed Lines Denote Coarse Mesh.

1.0

2.0

I— z w 0.0 w C.)

-J -2.0

Cl) 0

a0= 1.0

0 1 2

0 2 4 6

1.0

< 0.5

I- z LU

0.0 LU C.)

-0.5

Cl) 0

_1 fl

a0 =5.0

2 4 6 8 10 TIME,t/T

Figure 4.8 Normalized Displacement-Time History Responses of Horizontally Vibrating Strip Foundation on Elasto-Plastic Soil (H' = O.4E , i = 0.00I3Es); Solid Lines Denote Fine Mesh, Dashed

Lines Denote Coarse Mesh.

S

a0 = 1.0

1.0

O 1 2

a0 3.0

-1.0' 0 2

0.6 1 a0=5.0 C)

0.3

00 0•

-0.3

-0.6 0 2 4 6 8 10

TIME,t/T

Figure 4.9 Normalized Rotation-Time History Responses of Rocking Strip Foundation on Elastic Soil; Solid Lines Denote Fine Mesh, Dashed

Lines Denote Coarse Mesh.

4

6

a0= 1.0

0 1 2

10

z 00.0

0 -1.0

a0= 3.0

0 2 4 6

0.8 a0 =5.0

-0.8 0 2 4 6 8 10

TIME,t/T

Figure 4.10 Normalized Rotation-Time History Responses of Rocking Strip Foundation on Elasto-Plastic Soil (H' = 0.4E, 1 = O,y= 0.0013E);

Solid Lines Denote Fine Mesh, Dashed Lines Denote Coarse Mesh.

2.0

--@- FREQUENCY-DOMAIN SOLUTION

1.5 TIME-DOMAIN (FE) SOLUTION:

El i=O.O i=O.002

1.0

0 ELAS11C SOLUTION

0.5

1 2 3 4 5

FREQUENCY PARAMETER, a0 = wb/v

Figure 4.11 Amplitude-Frequency Plot for Horizontally Vibrating Square Foundation Showing Effects of Viscosity Parameter.

3.0

t) 2.5 1-6

-- FREQUENCY-DOMAIN SOLUTION

2.0 0

TIME-DOMAIN (FE) SOLUTION: H' =O.4E

AH'=E ELASTIC SOLUTION

me 0 1 2 3 4 5

7.

FREQUENCY PARAMETER, a0 = wb/v

Figure 4.12 Amplitude-Frequency Plot for a Horizontally Rocking Square Foundation Showing Effects of Soil Plastic Modulus.

--®-- FREQUENCY-DOMAIN SOLUTION

TIME-DOMAIN (FE) SOLUTION: a=O.00l4E

t a=O.0028E ELASTiC SOLUTION

0

I I I I

0 1 2 3 4 5

FREQUENCY PARAMETER, a0 =

Figure 4.13 Amplitude-Frequency Plot for a Torsionally Vibrating Square Foundation Showing Effects of Initial Yield Stress.

4

KI

S

a0=4.0

-o i 2 3 4 5

4

e2

z 0O I-

0 -2 I

I

V

a0 = 5.0 1

2 4 6 8

TIME,t/T

Figure 4.14 Normalized Torsional Rotation-Time History Responses of Square Foundation on Elasto-Plastic Soil (H' = O.4E5 , 0, = 0.00I3Es); Solid Lines Denote Fine Mesh, Dashed Lines Denote Coarse

Mesh.

t= 1.50T t = 1.75T

t = 2.00T t = 2.25T

— FULL PLASTIFICATION EM PARTIAL PLASTIFICATION

Figure 4.15 Establishing Steady-State Waves and Yield Zones for Horizontally Vibrating Square Foundation: Arrows Denote

Instantaneous Direction of Force.

t= 1.50T t=1.75T

S

t = 2.00T t = 2.25T

FULL PLASTIFICATION

E.x.x.:. PARTIAL PLASTIFICATION

Figure 4.16 Establishing Steady-State Waves and Yield Zones for Rocking Square Foundation: Arrows Denote Instantaneous

Direction of Moment.

t=1.50T

t=1.75T

6M

()

t = 2.00T

t = 2.25T

FULL PLASTIFICATION ::: PARTIAL PLASTIFICATION

Figure 4.17 Establishing Steady-State Waves and Yield Zones for Torsionally Vibrating Square Foundation: Arrows Denote

Instantaneous Direction of Torque.

UNIVERSITY OF SOUTHERN CALIFORNIA

Department of Civil Engineering

Research Center for Computational Geomechanics

A STUDY OF SOIL-PILE-STRUCTURE INTERACTION IN LIQUEFIABLE SOILS

DURING STRONG EARTHQUAKES

by

J.P. Bardet, S.W. Chi, Q. Huang, and G.R. Martin

Kajima-CUREe project report

August 1992

Table of contents

INTRODUCTION................................................................................................................ 1

SIMULATION OF CENTRIFUGE TESTS ON PILE GROUPS ................................................... 1 Summary of experimental results ........................................................................ 1 Definition of finite element model......................................................................2

Geometry................................................................................................. 2 Calibration of material parameters..........................................................2

Equivalent 2D structural properties.............................................2 Linear soil properties................................................................... 3 Nonlinear soil properties............................................................. 3

Soil permeability .......................................................................... 5

Eigenvalueanalysis............................................................................................. 5 Linearanalysis ...................................................................................................... 5

Method.................................................................................................... 5 Results............................................ ......................................................... 6

Nonlinearanalysis............................................................................................... 6 Freefield response .................................................................................. 6

Response of pile group............................................................................ 7 Conclusion........................................................................................................... 8

PREDICTION OF SEISMIC RESPONSE OF K-BUILDING FOUNDATION .............................. 8 Definition of finite element model......................................................................9

Geometry................................................................................................. 9 Calibration of soil parameters................................................................. 9

Linear soil properties................................................................... 9 Nonlinear soil properties............................................................. 9

Eigenvalueanalysis.............................................................................................10 Linearanalysis..................................................................................................... 10 Nonlinearanalysis............................................................................................... 11

Freefield response .................................................................................. 11 Moderate earthquake loading (100 gal) ..................................................11 Strong earthquake loading (407 gal).......................................................12

Conclusion........................................................................................................... 12

REFERENCES................................................................................................................... 13

ACKNOWLEDGMENT....................................................................................................... 13

LIST OF FIGURE CAPTIONS .............................................................................................. 15

LIST OF TABLES............................................................................................................... 86

INTRODUCTION

The interaction between building, foundation and soil during earthquakes is a

challenging problem for researchers which has practical and financial implications in the

design of buildings on liquefiable soils in Japan and the United States of America. Until

now, mostly simplified methods have been employed to describe the nonlinear effects of

soil-pile response in interaction problems. These methods use nonlinear, hysteretic and

degrading springs to represent the soil response in the pile vicinity. However, there is still

very limited experience in using continuum techniques based on nonlinear finite elements

and nonlinear constitutive models. This report summarizes the findings of the University of Southern California

(USC) during the research on building-foundation-soil interaction conducted under the

) sponsorship of Kajima-CUREe. The purpose of this research is (1) to verify a finite

element procedure developed at USC for analyzing liquefaction problems by comparison

of numerical and centrifuge results in the case of a pile group, and (2) to apply the

procedure to predict the response of an existing building founded in liquefiable soils.

The report is organized into two distinct parts. The first part is related to the

verification and calibration of our procedure by comparison to centrifuge results. The

second part is concerned with the application of the finite element procedure to a

particular building (the K-building) in Tokyo.

SIMULATION OF CENTRIFUGE TESTS ON PILE GROUPS

Summary of experimental results

Scott and Hushmand (1991) carried out a series of seven centrifuge tests at the

) California Institute of Technology in the framework of the Kajima-CUREe research

project on building-foundation-soil interaction effects. The objective of their

experimental program was to better understand the interaction between pile-foundations

and liquefiable soil during earthquakes. The prototype foundation was made of a group of

four piles, about 1 im in length and covered with a rigid cap. The pile group model was

placed in a laminar box filled with dry or saturated Nevada sand of various initial

densities. The laminar box was laterally shaken at its base by earthquake input motions of

various magnitudes. The response of the structure was monitored by recording the

bending moment in the piles, the acceleration in soil and pile cap, the horizontal

displacement, and the pore pressure in the case of saturated tests. The test geometry and

position of sensors are shown in Fig.1. Further detail on the description of the test

procedure and results may be found in Scott and Hushmand (1991).

- page 1 -

The objective of the present analysis is to simulate the results of the centrifuge

tests by using nonlinear finite element analysis. As for analyses by Miura et al (1991,

1992), four centrifuge tests, out of seven tests, have been selected for analysis. These tests

are referred to as tests 4,6, 8 and 9 by Scott and Hushmand (1991) and D2, Dl, S2 and

Si by Miura et a! (1992), respectively.

Definition of finite element model

Geometry

Fig.! shows the finite element mesh used in the analysis. The geometry is similar

to that of Miura et al (1992), except that the mesh has only a single layer of elements,

instead of several layers as described by Miura et al (1992). The soil layers, piles and pile

cap are represented by using four noded isoparametric elements.

In the nonlinear and saturated analyses, the pore-pressure is allowed to dissipate

and diffuse, owing to soil permeability. There are three degrees of freedom per node.

Two degrees are for the horizontal and vertical solid displacements, and one for the pore

pressure. This finite element formation is referred to as U-p formulation (Zienkiewicz and

Shiomi, 1984) Nodes 'on the left and right.vertical boundaries at the same e1evation•have

identical horizontal displacement, which prescribes the lateral boundary conditions of the

laminar box. However, the vertical component of the boundary nodes is left free. The

earthquake input motion is applied by specifying the horizontal acceleration at all the

lowest nodes. The earthquake input motion was provided under digitized format by R.F.

Scott and B. Hushmand. The acceleration corresponds to record No.13 of Scott and

Hushmand (1991) data.

Calibration of material parameters

The structural properties are assumed to be linear; they are scaled to account for

three-dimensional effects. The soil which surrounds the pile group is considered in both

linear and nonlinear cases.

Equivalent 2D structural properties

One of the problems in the two-dimensional (2D) modeling of three-dimensional

(3D) soil-structures is to preserve the dynamic characteristics of 3D soil-structures. The

inclusion of nonlinear soil behavior requires that both initial stresses (applied by the static

weight of structures) and dynamic stresses (applied by the dynamic motion of structures)

be calculated with a reasonable degree of accuracy. The dynamic motion of structures

generates transient stresses on the soils, in addition to those directly resulting from the

earthquake waves. Owing to their highly nonlinear nature, soil properties cannot be

simply scaled without altering their characteristic responses. It is therefore important that

- page 2 -

the 2D scaling of 3D structures preserve the fundamental modes and periods in the range

of earthquake ground motion. This scaling requires not only selecting appropriate unit mass for soil and

structure, but also the careful choice of structural stiffnesses. We propose to verify the

assumption on the 2D scaling of structural masses and stiffnesses by performing an

eigenvalue analysis using elastic properties for both soil layers and structures. We will

also verify the assumption on structural unit masses by examining the initial soil stresses

at the end of the gravity loading, before the application of earthquake input motion. The scaled 2D and 3D properties for the pile and pile cap are summarized in table

1 and the corresponding calculation steps are reported in tables 2 and 3. The plane strain

analyses are performed on a 1-rn thick layer of soil and structure. The scaling of the structural properties assumes that the soil interacting with the pile group has the same

thickness as the pile spacing (1.67rn).

Linear soil properties The linear soil properties selected in the analysis are similar to those of Miura et

al (1991, 1992) that were derived from laboratory test results on Nevada sand compiled

by Scott and Hushmand (1990). The following relation is used to calculate the low-strain

shear modulus G0 in kPa:

G0 -- A (1+e) m

(2. 17-e)2 3 0.5

(1)

where e is the void ratio and (Nm the mean effective stress in kPa. The coefficient A is set

equal to 5840. (Miura et al, 1992). The variation of G and Young's modulus E with depth

is reported in tables 5 and 6 in the case of tests 4,6, 8 and 9. The Poisson ratio is set

equal to 0.3.

Nonlinear soil properties

)

The nonlinear soil behavior of Nevada sand is simulated by using the BDP model.

This model which derives from previous work (Bardet, 1990) is currently in publication

and is detailed in Proubet (1991). This section only provides a summary necessary to

understand the calibration of BDP model parameters. The BDP model has 11 material constants that are calibrated, in our analysis, from

experimental tests on the Nevada sand.. Table 7 lists the constant values used in the

analysis. The following procedure has been adopted to define these constants.

The elastic properties in the BDP model depends on mean pressure p. The

Young's modulus E is:

E =E0 po (2)

- page 3 -

where the elastic parameters E0, no, and V are calculated from the initial slope of stress-

strain response during triaxial or simple shear tests. When using the results of low-strain-

amplitude resonant column tests, it is recommended to multiply E0 obtained from Eq. 1

by a factor of 0.6 to account for larger but reversible strain levels.

The failure parameters a and c are defined from the friction angle (1) and the

apparent cohesion of Nevada sand. In the simple shear test, a and 4) are related through:

a=sin4) (3)

The parameter Pa that is the pressure associated with undrained shear strength 'tmax is

calculated by using the following relation:

tmax - C Pa=

(4) a

The parameter 0 controlling the transition between contracting and dilating behavior is

defined by selecting a positive attractive pressure Pb close to zero: I

R 'max ' Pa - Pb

The pressure Pb attracts the effective stress paths during undrained loadings. The dioser

Pb is to zero, the softer is G, the mean-pressure-dependent shear modulus.

The material parameters H and n, that control the variation of plastic deformation

when the stress point approaches the failure surface, are calibrated by using a one-

element program called SOIL (Bardet, 1989). SOIL is based on linearized integration

techniques (Bardet and Choucair, 1991). H and n are adjusted by trial and error so that the

simulated response fits the experimental data points. The tests selected for fitting the

material parameters is the simple shear test at 8 psi (55 kPa) initial pressure reported by

Scott and Hushmand (1990).

The tenth and eleventh parameters of the BDP model, Hu and nu, control only the

evolution of plastic deformation during cyclic loading. Independent of other constants

calculated from monotonic tests, Hu and flu are calibrated from liquefaction stress curves.

The liquefaction is assumed to take place when the pore pressure becomes minimum

during a cyclic loading. The BDP model assumes that the pore pressure remains positive

so that the shear modulus remains defined. Figs.2 and 3 show the simulated evolution of

mean pressure and shear strain during cyclic simple shear tests at prescribed shear stress

amplitude. All the tests are carried out from 55 kPa initial effective stress.

The liquefaction strength curve of Fig.4 is constructed from the simulated results

of Figs.2 and 3. This simulated liquefaction curve is in good agreement with the

experimental liquefaction curve (Scott and Hushmand, 1990).

(5)

- page 4 -

Soil permeability The permeability coefficients used in the analysis were selected as recommended

by Miura et al (1992). The values used in the analysis are reported in table 4. The

selected values are higher than the experimental values (Scott and Hushmand, 1990) to

account for the centrifuge scaling of diffusion coefficient.

Eigenvalue analysis

The eigenvalue analyses were performed to verify not only the distribution of

stiffness and mass in the finite element models but also boundaiy conditions. All the

eigenvalue analyses are carned in the linear case, by using linear properties for soil and

structure.

The analysis results are shown in Figs. 5 to 6 in the case of dry and saturated

Nevada sands. The first four fundamental frequencies are 2.49, 5.05,7.44, and 8.21 Hz in

the dry case and 2.28, 4.57, 6.37, and 7.02 in the saturated case. In the dry case, effective

stresses are calculated by using dry unit mass, whereas, in the saturated case, effective

stresses are calculated by using buoyant unit mass. The natural modes of the model are

similar in the dry and saturated cases. The first mode is a simple shear mode. In the

second mode, the pile group and soil undergo shear modes of opposite direction. In the

third and fourth modes, the soil mass moves vertically while the pile barely moves.

Linear analysis

Before performing nonlinear analyses, it is useful to perform a few linear

analyses. Linear analyses are generally faster to perform than nonlinear analyses, and

allow the assessment of nonlinear effects by comparing the results of linear and nonlinear

analyses.

Method

The elastic properties used in the linear models are assumed constant, i.e.

independent of mean pressure and shear strain amplitude, but they vary with depth as

specified in tables 5 and 6. A Rayleigh damping factor 8 , that creates viscous damping

proportional to the stiffness matrix, is included to simulate the effect of soil structural

damping caused by hysteretic soil behavior and wave radiation. The following

proportionality constants 8 were used in the linear analyses:

test number 4 6 8 9

test condition dry dry saturated saturated

maximum acceleration (g) 0.203 0.085 0.255 0.112

damping factor 8 1 0.2 0.005 0.15 0.03

- page 5 -

The values of 8 were found by trial and error in order to simulate the peak acceleration.

As expected, the damping increases with the level of earthquake ground motion owing to

more extended irreversible soil behavior. The linear analyses in the saturated cases are

carried out without considering pore pressure. There are only two degrees of freedom per

node in the linear analysis instead of three in the saturated nonlinear analysis. The shear

modulus G, which depends on the initial effective stress, is calculated by using buoyant

unit mass in the saturated cases ( 8 and 9) and dry unit mass in the dry cases (4 and 6).

Results

Figs. 7 to 22 show the results of linear analyses for tests 4, 6, 8 and 9. The input

acceleration in test 4, 6, 8 and 9 and corresponding acceleration response spectra are

shown in Figs. 7, 11, 15, and 19. The computed and experimental accelerations at

location L15 and L9 and in the pile cap are also shown.

Linear analyses show that the dry response of the pile group can be realistically

simulated by using a Rayleigh type damping. The dominant period of the pile group can

be simulated by using elastic soil properties. The maximum acceleration amplitude can be

simulated by adjusting the amount of viscousdamping; However, in the saturated cases,

the measured response spectra shift toward lower frequency, owing to the softening of the

elastic properties of liquefied sands. This spectral shift, not taken into account by linear

analyses, requires the use of nonlinear analyses.

Nonlinear analysis

Before analyzing the pile group, a free field analysis is performed to characterize

the soil response, assuming that the soil layers, free of the pile group, are of infinite

extent.

Free field response I

The soil colunm of the free field analyses Consists of a single vertical column Cut

into the fmite element mesh of Fig. 1. The pore pressure, and horizontal and vertical

displacements of the nodes on the leftand right vertical bOundaries are set to coincide.

The stresses and pore pressure are initialized by applying a constant gravity loading. This

gravity initialization ensures that the model is in an equilibrium state before beginning the

earthquake loading.

The results of the free field responses are shown in Figs. 23 to 25 for test 8. Figs.

23 shows the computed accelerations and acceleration response spectra at L15 and L9.

The results of free-field simulations may be compared to the experimental results on the

pile group, keeping in mind that the pile group may influence the dynamic response of

- page 6 -

the soil layers. The computed acceleration is similar to the experimental acceleration in

Fig.30. The input acceleration is slightly attenuated at the ground surface, indicating that

the soil layers do not amplify the input ground motion. Fig.24 shows that the computed

excess pore-pressure build-up at PP2 and PP4 is larger than the experimental values (with

the pile group) in Fig.33. Fig. 25 shows that the relative displacement is the same order

of magnitude as the experimental value in Fig. 32.

The results of the free field responses are shown in Figs. 26 to 28 for test 9. The

conclusions drawn for test 8 may be extended to test 9. The peak acceleration at ground

surface is 0.3 g and 0.2 g for test 8 and 9, respectively, which shows that the

amplification of acceleration is larger in test 9 than in test 8. Both spectral peaks of

computed acceleration (1.5 g for test 8 and 0.9 g for test 9) are higher than the

experimental ones (1 .g for test 8 and 0.6 g for test 9). The computed and measured

displacements have the same order of magnitude. The computed pore-pressures in tests

8 and 9 are larger that the measured pore-pressures.

Response of pile group

Figs.29 to 33 show the results of the nonlinear finite element analysis for test 8.

They compare the computed and experininta1 time histories of acceleration at L9 and

L15, and at pile cap, acceleration response spectra, displacement at L3, L9 and L15, and

finally pore pressure at PP1, PP2, PP3 and PP4.

As shown in Figs. 29 and 30, the computed and experimental accelerations are in

agreement. However, as shown in Fig.31, the computed acceleration of the pile cap has a

dominant frequency larger than the experimental one, which indicates that the numerical

model is stiffer than the centrifuge model. As shown in Fig. 32, the analysis predicts

permanent displacement slightly larger than the experiment. As shown in Fig. 33, the

computed pore pressures are larger than the measured ones. The computed and measured

pore pressures increase at the same rate but the computed pore-pressure diffuses less

rapidly than the measured one. This difference is expected since two-dimensional

drainage is more constrained than three-dimensional drainage.

Figs.34 to 38 show the results of the nonlinear finite element analysis for test 9.

As shown in Fig.34 to 36, the agreement between computed and measured accelerations

is better in test 9 than in test 8. The acceleration level in test 9 is lower than in test 8, and

generates less nonlinear effects in the soil layers. As shown in Fig. 37, the computed

displacements are slightly larger than the measured ones. The square erratic lines in

Fig.37 result from a round-off error during the post-processing phase, but this does not

invalidate the computed results. As shown in Fig.38, the computed pore pressure is much

larger than the measured one. This discrepancy may come from faster three-dimensional

drainage. The same problem was noticed in the analysis of Miura et al (1992).

- page 7 -

The permeability coefficient is an important factor that controls the build-up and

dissipation of pore-pressure. 2D models do not easily explain 3D pore-pressure

dissipation mechanisms. Close to the piles, the 3D pore pressure builds up more slowly

and dissipates more rapidly than in 2D cases. This 3D effect could be artificially

simulated in 2D analyses by decreasing the model capacity to generate pore-pressure and

by locally increasing the permeability coefficient. The 2D scaling of 3D permeability and

drainage mechanism needs to be further investigated.

Conclusion

The response of pile groups in dry sands may be simulated by using linear

analysis, provided that an adequate value of Rayleigh damping be selected. However, the

simulation of the response of pile groups in saturated sands requires the use of nonlinear

models having the capability of building up and diffusing pore pressure to account for

the structural softening observed during pore pressure build-up.

in the context of the nonlinear finite element analyses with pore-pressure

diffusion, the BDP model is a useful model to describe the cyclic response of saturated

sands. It is capable of predicting the pore-pressure build-up as well as the softening of the

material properties resulting from pore-pressure build-up. The BDP model is a useful tool

to analyze the experimental results of soil-structure-foundation interaction in saturated

sands.

PREDICTION OF SEISMIC RESPONSE OF K-BUILDING FOUNDATION

The finite element technique that was previously calibrated by comparing

numerical and centrifuge results is now applied to predict the seismic response of a

particular building, referred to as K-building. The foundation and superstructure of the K-

building is defined by Miura et al (1992). Our objective is to illustrate the application of

our procedure in a particular case of practical interest to designers. It is not our intent to

perform a parametric study, as Miura et a! (1992) have reported in order to understand the

influence of various factors on the dynamic response of the K-building.

Following the finite element model formulation of the K-building and its

foundation, an eigenvalue analysis and a linear transient analysis are carried out. These

linear analyses are followed by a free-field response for an earthquake input motion equal

to 100 gal. The nonlinear dynamic response of K-building is then predicted in the case of

moderate (100 gal) and strong (407 gal) earthquake input motion.

- page 8 -

Definition of finite element model

Geometiy

Fig.39 shows the finite element model used in the analysis of the 34-story K-

building. The superstructure model is identical to the one defined by Miura et a! (1992).

However, since our plane-strain analysis applies to a im thick layer, all the properties of

the three-dimensional structure are divided by a depth thit is assumed equal to the width

of the K-building, i.e. 30.m. The superstructure is represented by beam elements, the

properties of which are listed in table 9. The beam elements are assumed weightless. The

superstructure inertial mass is lumped at the floors by using concentrated mass elements,

the mass of which is listed in table 9. The static weight or, the superstructure is applied by

using vertical nodal loads distributed on the first floor. The piles underlying the K-

building are represented by truss elements that are capaNe of transmitting only axial

forces. The properties of the truss elements are listed in table 8. The basemat and

diaphragm wall are represented by using four noded isopzrametric linearly elastic

elements. The soil layers are also represented by four noded isoparametric elements that

have the capability of describing nonlinear soil behavior and pore-pressure diffusion. The

earthquake input motion (see Fig.46) is specified by presribing the horizontal

acceleration at depth 53.5m. This input acceleration was provided by Miüra et al (1992).

Calibration of soil parameters

Linear soil properties The linear soil properties selected in the analysis (see table 10) are similar to those

defined by Miura et a! (1992). The shear modulus G varies with the initial effective

stress, whereas Poisson ratio v is constant. v is assumed ro be equal to 0.3 for sands and

0.48 forclays (Miura et al, 1992).

Nonlinear soil properties

The clay layers, referred to as clay-i, -2, and-3, are modeled by using the elastic-

perfectly plastic von Mises model. The elastic properties.. i.e., Young's modulus and

Poisson ratio, are identical to those used in the linear analyses. The maximum size of the

von Mises failure surface is directly related to clay cohesion. Their values are reported in

table 10. The sand layers, referred to as sand-i, -2, -3, and-4, are modeled by using the

BDP model. The model constants have been calibrated zs outlined in the previous

chapter. The elastic constants E0 were selected to be 60 of the ones calculated from low

amplitude resonant tests. The slope a of the failure surfae was calculated from the

friction angle 0 by using Eq.3. The apparent cohesion was arbitrarily taken equal to zero.

- page 9 -

The slope of the characteristic state was selected to have Pb close to zero. The attractive

pressure Pa , which is ielated to the undrained shear strength through Eq.4, was assumed

to be at least twice the value of the initial effective pressure. This assumption is useful for

controlling the position of the elbow in the effective stress path during simple shear tests.

This assumption was necessary since no laboratory data was available for monotonic

simple shear tests on sand-i, -2, -3 or -4. The parameter H0 and n were adjusted by trial

and error so that the effective stress path has an elbow at the undrained shear strength

given by Miura et al (1992) (see table 9). The parameters Hu and nu were also adjusted to

fit the liquefaction strength curves provided by Miura et a! (1992). The liquefaction

strength curve of FigAO is constructed by simulating the response of sand- 1, -2, -3 and-4

during cyclic simple shear tests at constant shear stress amplitude. In these simulations,

the initial effective stress is equal to the in situ geostatic stress. The soil permeability

selected in our analysis (see table 10) comes form Miura et a! (1992).

Eigenvalue analysis

Fig.41 shows the first six eigenmodes of K-building. In the first mode (0.65 Hz)

deformation is concentrated in the building. The first mode has a higher frequency than

the one calculated (0.48 Hz) by Miura et al (1992). This discrepancy results from a larger

rotational stiffness for the foundation, which Miura et al (1992) softened by adding a

rotational spring. The five other modes have frequencies in the range determined by

Miura et al (1992), who found 1.60 and 2.79 Hz for the natural frequency of second and

third mode, respectively, in the case of a simplified spring model. The soil layers deform

simply in shear at 1.58 Hz, and in more complicated patterns when the frequency

increases. The elastic soil layers have a fundamental frequency higher than the building,

which implies that soil layers will probably interact, not with the fundamental mode of

the building, but with its second and thinl modes. This remark pertains to elastic soil

behavior, it does not rule out that the first structural mode may interact with soil layers if

the soil elastic modulisoften due to pore-pressure build-up.

Linear analysis

The linear analysis was carried out to assess the contribution of nonlinear soil

behavior to the response of the K-building using the 100 gal acceleration input shown in

Fig.42. No Rayleigh damping was introduced in the linear analysis. Fig.43 shows the

time history and acceleration response spectra of the calculated acceleration at the first

floor and top of K-building. The first structural mode (0.65 Hz) is clearly seen in the

response spectra. As shown in Fig.44, the maximum displacement at roof and first floor is

8.5 cm and 2 cm, respectively. Figs. 45. and 46 show the variation of maximum values of

horizontal acceleration, lateral displacement, bending moment and shear force in the

- page 10 -

superstructure. The results of Figs.45 and 46 are in agreement with those of Miura et al

(1992).

Nonlinear analysis

Free field response

Figs.47 to 51 show the free-field response of K-building in the case of nonlinear

soil properties and a 100 gal maximum input acceleration. Fig. 47 shows the acceleration

and acceleration response spectra at various depths. The acceleration is larger at 10.8m

than at the ground surface due to the truncation of acceleration that takes place in the

sand-2 layer. Fig.48 shows that the permanent displacement localizes in the sand-2 layer,

and that layers above and under sand-2 layer behave elastically. Fig.49 shows that pore-

pressure builds up mainly in sand-2. Fig.50 shows the effective stress path and the stress-

strain response in the middle of sand-2 layer. The vertical effective stress is initially about

40 kPa. It decreases toward zero after. 10 sec as shear stress cycles are applied by the

earthquake input motion. The stress-strain response shows that the shear strain reaches

8% in sand-2 layer. Fig. 51 shows that pore-pressure builds up also in sand-3, but to a

lésserextent, generating less-softening of the stress-strain response. '-

Moderate earthquake loading (100 gal)

Figs. 52 to 59 show the response of K-building for a 100 gal input acceleration

earthquake. Fig. 52 shows the time histoiy and acceleration response spectra on the roof

and first floor. This nonlinear response is similar to the linear response of Fig. 43, which

indicates that nonlinear effects do not dominate the building response. This remark is

corroborated by the results of Fig.53, 54 and 55 that show the time history of relative

displacement on the roof and first floor, and the maximum values of horizontal

acceleration, lateral displacement, bending moment, and shear force in the

superstructure. In spite of similarity between linear and nonlinear analyses, soil layers

undergo plastic deformation as shown in Figs. 56, 57, 58 and 59. Fig.56 shows that the

acceleration is truncated in sand-2 layer as in the free-field response. Fig.57 shows that

permanent displacements are observed in the sand-2 layer, in spite of the K-building

foundation. Fig.58 shows that pore-pressure builds up in all sand layers, but -

predominantly in sand-2 which liquefies. Finally, Fig. 59 shows that the vertical

effective stress is reduced in all sand layers, which should result in reducing the

fundamental frequency of soil layers.

The similarity of structural response for linear and nonlinear analyses indicates

that the effects of nonlinear soil behavior are reduced by the elastic diaphragm wall and

- page 11 -

piles, that rest directly on elastic soil layers. The stresses in the diaphragm wall and piles

should however remain tolerable for sustaining this conclusion.

Strong earthquake loading (407 gal)

Figs. 60 to 67 shows the response of K-building for a 407 gal input acceleration

earthquake, which corresponds to a maximum acceleration four times larger than in the

previous analysis. Fig.60 shows that the time histories of the horizontal acceleration on

the roof and first floor are different from the ones obtained in linear analysis. The

acceleration response spectra are wider, which indicates that there are more nonlinear

effects for 407 gal than for 100 gal input acceleration. Fig. 61 shows that the

displacement reaches 20 cm at the top of K-building. Figs. 62 and 63 shows that the

maximum values of horizontal acceleration, lateral displacement, shear force, and

bending moment has a similar distribution, but a larger magnitude, in the 407 gal case

compared to the 100 gal case. The computed values are in agreement with those

calculated by Miura et al (1992). Figs. 64 to 67 show that the soil layers undergo severe

nonlinear behavior. Pore-pressure build-up is more extended than in the previous case.

S and- i and -2 undergo liquefaction for the complete duration of the earthquake while

sand-3 and -4 undergo liquefaction for a shorter time.

As noted in the 100 gal analysis, the elastic diaphragm wall and elastic piles,both

resting on elastic soil, limits the nonlinear effects arising from soil liquefaction.

Additional analysis would be appropriate to examine the assumption of elastic support

and behavior for the piles and diaphragm wall.

Conclusion

The fmite element procedure utilizing the BDP model was calibrated by using

centrifuge test data. Applications of the model were then demonstrated to predict the

dynamic response of the K-building in Tokyo. Although the nonlinear soil constants were

calibrated in an approximate way owing to a lack of laboratory data, the overall

procedure produces results in general agreement with results obtained by Miura et al

(1992) with different methods. The procedure is capable of describing structural

response during earthquake loadings while soil experiences pore pressure increases up to

the point of liquefaction. Our analysis indicates that the structural response of the K-

building is strongly controlled by the assumption that the diaphragm wall and piles are

elastic and rest on elastic soil layers.

-page 12-

REFERENCES

Bardet., J.P., 1989, "SOIL: an interactive program to simulate soil behavior" Civil

Engineering Department, University of Southern California.

Bardet., J.P., 1989, "LINOS: a finite element for geomechanics" Civil

Engineering Department, University of Southern California.

Bardet., J.P., 1990, "An hypoplastic model for sand't ASCE, Engineering

Mechanics, Vol.116, No.9, PP.1973-1994.

Bardet., J.P., and W. Choucair, 1991, "Linearized integration techniques for

incremental constitutive equations," mt. J. Numerical and Analytical Methods in

Geomechanics, Vol.15, pp.1-19.

Miura, K, K. Suzuki, Y. Miyamoto, K. Masuda, and S. Uchiyama, 1991,

"Earthquake response of a structure on pile group in liquefiable sand deposit,"

Kajima-CUREe report, Kajima Corporation, Japan, p.123.

Miura, K, Y. Miyamoto, A. Yamada, M. Nagano, Y. Sago, Y. Hyodo, K.

Masuda, T. Maeda,E.Kitamura, K, Suzuki,-Y. Suzuki, A. Fukuoka, 1991,

"Building-Foundation-Soil Interaction Effects," Kajima-CUREe report, Kajima

Corporation, Japan, p.107.

Proubet, J., 1991, "Application of numerical methods to geomechanics problems,

PhD thesis, University of Southern California,.

Scott, R.F., and B. Hushmand, 1990, "Compilation of laboratory test results on

Nevada Sand," Private communication to K. Miura (Kajima Corporation), p.83.

Scott, R.F., and B. Hushmand, 1991, "Soil-pile response in the Santa Cruz (Loma

Prieta) earthquake, October 17, 1989." Kajima-CUREe report, California Institute

of Technology, Pasadena, California, University of California, Davis, California.

Zienkiewicz, O.0 and T. Shiomi, 1984, "Dynamic behavior of saturated porous

media: the generalized Biot formulation and its numerical solution." mt. J.for

Numerical and Analytical Methods in Geomechanics, Vol.8, pp.71-96.

ACKNOWLEDGMENT

The authors acknowledged the financial support of Kajima-CUREe and are

thankful for the interest, enthusiasm and patience of Kajima Corporation. The authors

thank B. Hushmand and R.F. Scott for communicating their centrifuge data. The finite

element analyses were performed and supervised by J.P. Bardet assisted by S.W. Chi and

- page 13 -

LIST OF FIGURE CAPTIONS

Geometry of finite element model for pile-group.

Shear-stress versus mean-pressure and shear-stress versus shear-strain simulated by

BDP model during cyclic simple shear test on Nevada Sand (At/a0= 0.232 and

0.193, (= 55 kPa)


BDP model during cyclic simple shear test on Nevada Sand (ttt/sY0= 0.154 and

0.128, = 55 kPa)

Experimental and simulated liquefaction strength curves for Nevada sand during

cyclic simple shear tests at c= 55 kPa.

First four natural modes of pile group for test No.6.

First four natural modes of pile group for test No.8.

Time history and acceleration response spectrum of input horizontal acceleration for

test No.6.

Computed and experimental time histories, and acceleration response spectra of

horizontal acceleration at L9 for test No.6.




horizontal acceleration at pile cap for test No.6.

S11. Time history and response spectrum of input horizontal acceleration for test No.4.








test No.8.



-page 15-





Time history and response spectrum of input horizontal acceleration for test No.9.







Computed time histories and acceleration response spectra of horizontal acceleration

at L9 and Li 5 for free field of test No.8

Computed time histories of excess pore pressure at PP2 and PP4 for test No.8

Computed time histories of relative displacement at W. L9, and L15 for test No.8.

Computed time histories and acceleration response spectra of horizontal acceleration

at L9 and Li 5 for free field of test No.9

Computed time histories of excess pore pressure at PP2 and PP4 for test No.9

Computed time histories of relative displacement at L3, L9, and L15 for test No.9.




horizontal acceleration at L15 for test No.8. S Computed and experimental time histories, and acceleration response spectra of


Computed and experimental time histories of relative displacement at L3, L9, and

L15 for test No.8.

Computed and experimental time histories of excess pore-pressure at PP1, PP2, PP3

and PP4 for test No.8.





- page 16 -



Computed and experimental time histories of relative displacement at L3, L9, and

L15 for test No.9.

Computed and experimental time histories of excess pore-pressure at PP1, PP2, PP3

and PP4 for test No.9.

Geometry of finite element model for K-building.

Experimental and simulated liquefaction strength curves for sands 1, 2, 3 and 4

during cyclic simple shear tests.

Fundamental modes of K-building


K-building.

Time histories, and acceleration response spectra of horizontal acceleration on roof

and first floor of K-building computed by linear analysis.

Time histories of relative displacement on roof and first floor of K-building

computed by linear analysis

Distribution of maximum acceleration and relative displacement versus elevation

computed in linear analysis of K-building.

Distribution of maximum bending moment and shear force versus elevation


Time histories, and acceleration response spectra of horizontal acceleration at depth

0, 10.8, and 53.5 m for the free field of K-building.

48. Time histories of relative displacement at depths 0,6.9, 7.6, and 10.8m for free field

of K-building.

Time histories of excess pore-pressure computed at the middle of sand layers 1, 2, 3

and 4 for free field of K-building.


BDP model at the middle of sand-2 layer during free field response of K-building.


BDP model at the middle of sand-3 layer during free field response of K-building.


and first floor of K-building computed for 100 gal input acceleration.

- page 17 -


computed for 100 gal input acceleration.

Computed distribution of maximum acceleration and relative displacement versus

elevation of K-building (100 gal).

Computed distribution of maximum bending moment and shear force versus



6.9 and 7.6 m for K-building (100 gal).

Time histories of relative displacement at depths 0, 6.9, 7.6, and 10.8m for K-

building (100 gal).


and 4 for K-building (100 gal).

Time histories of vertical effective stress computed at the middle of sand layers 1, 2,

3 and 4 for K-building (100 gal).


and first floor of K-building computed for 407 gal input acceleration.



Computed distribution of maximum acceleration and relative displacement versus


Computed distribution of maximum bending moment and shear force versus



6.9 and 7.6 m for K-building (407 gal). I

Time histories of relative displacement at depths 6.9 and 7.6 m for K-building (407

gal).


and 4 for K-building (407 gal).

Time histories of vertical effective stress computed at the middle of sand layers 1, 2,

3 and 4 for K-building (407 gal).

- page 18 -

rr,z##,zs rt,z,,ztI

L3

Figure 1. Geometry of finite element model for pile-group.

- page 19 -

Cf C\1 -

0

0 c:z

mean pressure p (kPa) mean pressure p (kPa)

Figure 2. Shear-stress versus mean-pressure and shear-stress versus shear-strain

simulated by BDP model during cyclic simple shear test on Nevada Sand (Lxt/ 0= 0.232 and 0.193, = 55 kPa).

- page 20-

C)

20 40

mean pressure p (kPa)

strain €12

! I

oressure p (kPa) C)

Sr

V tr

C,

Tr

—iO —5xiO 4 0 5x10 4 10

strain €12


simulated by BDP model during cyclic simple shear test on Nevada Sand

(At/ 0= 0.154 and 0.128, y= 55 kPa).

-page2l -

0

0 0 4-

Cl) 0 (I)

- 4-- Cl)

10 Ca

Cl)

0

Figure 4. Experimental and simulated liquefaction strength curves for Nevada sand

during cyclic simple shear tests at 55 kPa

- page 22 -

Figure 5. First four natural modes of pile group for test No.6.

t'J Centrifuge Test No.08 uode No.4 Fraq ?.025 Oçarmode dot. I Ornox -- L222C-01

x,xx,otIox x.00OC,O ...J 2.IO

Figure 6. First four natural modes of pile group for test No.8.

0 0

0 0

.=' N a

C 0

U

0 N

10 20

time (sec)

0 0 Ln

0 0 0 N

0 o 0 '- 0 C if) 0 -

S U 0

0 0 In

due

frequency (Hz)

Figure 7. Time history and acceleration response spectrum of input horizontal

acceleration for test No.6 (1.g = 1000.gal).

- page 25 -

C 0

- ANALYSIS

C 0 - (N

01

C 0

C) C) U U o 0

0 (N

0

0 0

TEST

0 0

10

0 0 In (N

0 0 0 (N

0 0

0 C In 0 - 0 1.

() 0

0 0 In

onalysis test

10: 20

time (sec)

frequency (Hz)

Figure 8. Computed and experimental time histories, and acceleration response spectra

of horizontal acceleration at L9 for test No.6 (l.g = 1000.gal).

- page 26 -

0

I V

OM

C

(N - analysis

C -- test 0 0 (N

0 0' o

0 C tfl 0'-

0 '- 0

0 0

C C

0 L

0.1

ANALYSIS

0 0

TEST

h, ii

?

4 k~v~

10 20

time (sec)

frequency (Hz)


of horizontal acceleration at L15 for test No.6 (l.g = 1000gaI).

- page 27 -

0 0

0 0 LU N

0 0 0

_ N 0

0 c LU 0- 0 - 0

U 0

0 0 LU

0 0

ANALYSIS 0 0

- N C 01

0

G) a) U

0 N

0 0

TEST

J~ i i1ji!

10 20

time (sec)

onolysis -- test

I'

'I

0- 0.1

frequency (Hz)


of horizontal acceleration at pile cap for test No.6 (l.g = 1000.gal).

- page 28 -

I

0 0

0 0 N 0

cDn

0

CD C-)

0 N

0 0

i0

10 20

time (sec)

41

0 0 0 N

0 C 0

0 C Lfl 0 - 0 '- 0

So 0

0 0LO

frequency (Hz)

Figure 11. Time history and response spectrum of input horizontal acceleration for test

No.4 (1.g = 1000.gal).

- page 29 -

C C

ANALYSIS

0 0

TEST 0 0 (N

:.

C U U C 0

0 IN

fo 20

time (sec)

- analysis I 1

-- test 1

C 0 0

0 c if)

o

0 0 if)

WE F

frequency .z)


of horizontal acceleration at L9 for test NoA (1.g = 1000.gal).

- page 30 -

0 0

10

C C to (N

0 0 0 (N

ANALYSIS

C C

0 0

- (N 0

C 0

0 0

TEST

20

time (sec) 0 0 C!)

C')

0 © 0 (N

0 o ©

0 C LU 0 -

0

U 0

0 0 C!)

- analysis -- test

OR

frequency (Hz)


of horizontal acceleration at L15 for test No.4 (1.g = 1000gal).

-.page3l -

0 0

0 0

TEST

H'

11wtmj

10• 20

0 0 (N

0 0 0 (N

0 0

O In 0W-

0 0

— 0

C-, 0

0 0 tiq

time (sec)

0 '-- 0.1 1 10

frequency (Hz)


of horizontal acceleration at pile cap for test No.4 (1.g = 1000.gal).

- page 32 -

0 0

C 0

0 .10 20

(sec)

C 0 (N

0 0

- U•)

01

00

C o 0 0) ci) 0

So

C tO

0.1

1 10

frequency (Hz)


acceleration for test No.8 (Lg = l000.gal).

- page 33 -

ANALYSIS

LI 0

TEST

0 10 20

time (sec)

0 0 (N

- onaysis test

0 0

-' In 0

C 00

0 00 ci) ci) 0 0 0 0

0 (0

ra

0.1 1 10

frequency (Hz)



- page 34 -

0 0 0 (N

0 0 (0

o - 0'

C 0 0 0

a) a) U 0

0 0 (C)

0 0

-' 0 0

C 0

Q) 0 o (N

0 0

ANALYSIS

0 0

0

0' (N

C 0

a) o (_) 0 o

0 0

TEST

0 10 20

time (sec)

- onolysis test

frequency (Hz)



-page 35-

A

0

-' 0 0 (N

C 0

a) a)

0 (N

0 0

ANALYSIS

TEST

0 10 20

0 time (sec)

0 0 (N

- - anolysis - -- test

1 10

frequenc: (Hz)



page 36 -

0 0

- In o

C 00

0 00 a) — a) 0 0

0 In

me

0 0

0 _ 0 - N 0

C 0

0 0

0 N

0 0 tO

10 20

time (sec)

0 0 0 N

0 0

0

C o 0

*ZI 0

O 00 C- 0 0 0 0 0 0

0 If)

no -

1 10

frequency (Hz)

Figure 19. Time history and response spectrum of input horizontal acceleration for test

No.9 (l.g = 1000.ga1)

- page 37 -

A

0

0 0

0 0

- (N 0 0'

C 0

C,

C) U 0

0 0 (N

0 0

TEST

'U 10 20

0 time (sec)

0 0 (N

- Oflolysis test

0 0

- U) 0

C o 0

0

C)

C) 0 C)

0 0 U)

I

10

frequency (Hz)


of horizontal acceleration at L9 for test No.9 (1.g = 1000.gal).

- page 38 -

C L)

ANALYSIS

0 0

- C') 0 0'

C 0

0) 0) 0

C C')

0 0

I0

10 20

0 0 0 N

- onoysis -- test

time (sec)

S 0 0

- U) 0• 0'

C 0 0 0

4) 0) 0 0 0

0 LI)

TEST

0k-. 0.1

frequency (Hz)


of horizontal acceleration at L15 for test No.9 (Lg = 1000.gal).

- page 39 -

C C It

0 0 - (N 0

C 0

a) 0

0 (N

ANALYSIS

0 0

0 0 ::'

0

C 0

a) a) 0 - 0

0 0 (N

TEST

0 0

0

10 20

.00 0 0 LO

0 o

C 00

0 2° a) —a) 0 0

0 0 LI•)

time (sec)

- analysis test

/

frequency (Hz)


of horizontal acceleration at pile cap for test No9 (Lg = 1000gal).

-page 40-

d

(N

C 0

d

(N

0

0 10 20

time (s)

O 0

0

frequency (Hz)

Figure 23. Computed time histories and acceleration response spectra of horizontal

acceleration at L9 and L15 for free field of test No.8 (1.g = IOO(igal).

- page 41 -

L

p iti:1;\'tc I ,

-

-

~-j

tL 1 -

bottom -- PP2

PP4 .. ground surface

-I .. . .. . . . 0 on

10

20

time (s)

I

Figure 24. Computed time histories of excess pore pressure at PP2 and PP4 for test No.8

- page 42-

L3

L15 -

C a) E 04 0

a) 0 0 U,

D 0 a) > o (N

-i3O '- 0

11

10 20 30

time (s)

Figure 25. Computed time histories of relative displacement at L3, L9, and L15 for test

No.8.

— page 43 —

L9

d

-.' (N

0

2° a) C) U

Q

0

115

E! TTT 5 10 15 20 25

time (s)

Ui

d 1

- input . L9 L15

- s

I

0.1 1 10

frequency (Hz)

Figure 26. Computed time histories and acceleration response spectra of horizontal

acceleration at L9 and L15 for free field of test No.9 (l.g = 1000.gal).

-page 44-

I

_I

boto-..-.. -- FF2 -. -. FF4

ground surface

...-r•... -I.-..--(-•, ....

5 10 15 20 25

S time (s)

Figure 27. Computed time histories of excess pore pressure at PP2 and PP4 for test No.9

-page45 -

L9

0

E (N

E C)

CN

:;

L15

S

5 10 15 20 25

:ime (s)

Figure 28. Computed time histories of relative displacement at L3, L9, and L15 for test

No.9.

- page 46 -

ANALYSIS

0 0

0 =. C o c'J 0'

C 0

C) C)

00 (N

C 0

TEST

5 10 15 LU

time (sec) 0 Ii) (N

0 0 0 (N

0 0 0 '- 0 C 0- 0 '- 0

(•) 0

0 0 In

- dnciyss - - - test

0 - 0.1 El

frequency (Hz)



- page 47 -

C ' 0

(N

0 0

0

0 (N

0 0

0 0 If, (N

0 0 0 (N

0 0.. 0

0 C LI) 0-

0

2o 0. 0 0

0 0 a-,

'IT

ANALYSIS

TEST

0 o (N

0

C 0

Q)

C-, 0 0 (N

0 0

0 0 Nf-

5 10 15 20 25

- analysis test

I

frequency (Hz)



-page 48-

0 0

0 :' o

(N o

C 0

a) a)

0 (N

0 0

0 0

C .= 0

(N o

C 0

a) a)

0 0 (N

0 0

ANALYSIS

TEST

5 10 15 20 25

:tme (sec)

- - onaIyss - - -. test

1 10

frequency (Hz)



- page 49 -

0 0 Ln (N

0 0 0 (N

o 0

C to 0 -0 L0

0 0

0 0 to

10 15

time (sec)

Figure 32. Computed and experimental time histories of relative displacement at L3, 1-9,

and L15 for test No.8.

- page 50-

PP PP2 PP3 Pp4

NAL.YSIS

jA4

5 10 15 20 25

PP2 PP3 • TEST

5 10 15 20 25

time (sec'

Figure 33. Computed and experimental time histories of excess pore-pressure at PPI,

PP2, PP3 and PP4 for test No8.

-page5l -

0 0

0 171

0

C)

0 0 N

0 0 LU N

0 0 0 N

0 o0

0 C LU 0 - 0 '- 0

0 0

0 0 LU

ANALYSIS 0

-0 0

C 0

0 0 N

TEST

Hi 10 20

time (s'

- onolysis test

0 0.1

LIM

frequency (Hz)



- page 52 -

0 0 LN

ANALYSIS

0 0

TEST

10 20

time (s)

analyss

I

-- test

0 0.1 IN

frequency (Hz)


- of horizontal acceleration at L15 for test No.9 (l.g = 1000.gal).

-page 53-

C C CN

0 (C)

0 0 0 04

0' 0

S.-

ctfl 0- 0 '-0

0 0

0 0 LU

0 0 N

0

C 0

C) C) 0 0 0

0 0 N

ANALYSIS

TEST

I

10 20

time (s) 0 0 LO N

0 0 0 N

0 0 '- 0 c ti) 0- 0 '- 0

0 0

0 0

- analysis test

0.1 1 10

frequency (Hz)


of horizontal acceleration at pile cap for test No.9 (l.g = 1000.gal).

- page 54

time (s)

Figure 37. Computed and experimental time histories of relative displacement at 0, L9, and L15 for test No.9.

- page 55 -

S.'

F ppi

NALYSII if

a)

a.

0 'U 20

time (s)

_PP1 -- PP2

P?3 TEST PP4

S.'

a)

a. 0

10 20

time (s)

Figure 38. Computed and experimental time histories of excess pore-pressure at PPI,

PP2, PP3 and PP4 for test No.9..

- page 56 -

levation (m)

) -3.9 -6.9 -7.6 10.8 -9.5 15.95

23.3

29.85

-53-5

sand-i sand-2 sand-3 sand4

earthquake input motion

Figure 39. Geometry of finite element model for K-building.

- page 57 -

0.5

0.4

0.1

0

1 10 100 1000

Number of cycles

—a----- Sand-i (test)

0 Sand-i (analysis)

Sand-2 (Kajima)

0 Sand-2 (analysis)

Sartd-3 (test)

o Sand-3 (analysis)

-A- Sand-4 (test)

£ Sand-4 (analysis) I

S

Figure 40. Experimental and simulated liquefaction strength curves for sands 1, 2, 3 and

4 during cyclic simple shear tests.

- page 58 -

K-BuIdinq elQenoolue on&ysis

K-9uiIdng egenvoIue anolvss 4o4n No.6

- 2.619 62çowoodo 404.4

- 2.665C-02

K-Buildirg elgenvolue CnOIys iOCe NO.4 Feq - 2.t9 U.ço.000dn 404. 0— - 3.4724

Figure 41. Fundamental modes of K-building

- -page59-

0 0

0 10 20 .30 60

TIME ('.cc)

I

0.1 1

RE0U€NC (Hz)


acceleration for K-building (l.g = 1000.gal).

- page 60 -

0 0

0 0 c'4

C 0

C) C) 0 oO

0 ('4

0 0

40 0 20 40

time (sec)

0 20

time (sec)

roof first floor

0 0 0

0 0

—0

C 00

0 00

c'J C) U 0 00

0 0

S '.1 1

frequency (Hz)

- 0— IC 0.1

frequency (Hz)

IsJ

Figure 43. Time histories, and acceleration response spectra of horizontal acceleration

on roof and first floor of K-building computed by linear analysis (1.g

1000. gal)

- page 61 -

roof

20 40

time (sec)

first floor

20 40

time (sec)

Figure 44. Time histories of relative displacement on roof and first floor of K-building

computed by linear analysis.

- page 62 -

I 0(

8C

0

2 4 6 8

Max. displa cement (clu)

-40

0

EPZoX!I

100.0

80.0

60.0

20.0

0.0

-20.0

/ .400

0.00 100.00 200.00 300.00 400.00 500.00

Max. acceleraUon (ga!)

E

40.0

0

Figure 45. Distribution of maximum acceleration and relative displacement versus

- elevation computed in linear analysis of K-building (1.g = 1000.gal).

- page 63 -

120 120

100.0 100

80.0 80

60.0 0 C,

E

Z 60 0

Ron

20.0 20

0.0 01

0 0.0 2000.0 4000.0 6000.0

Max. shear force ((on)

5000000 10000000 15000000

Max. bendtng moment (ton an)

Figure 46. Distribution of maximum bending moment and shear force versus elevation


- page 64 -

6

0 20 40

time. (s)

20 . 40 60

time (s)

6 GL=-10.8 (m) -

0

- GL=-53.5m) - -- GL-15.95rn) -

GL=-10.8m) I ' - CL=0.Om) j \

i I' ......... i % t

o 20 40

L

time (s) 0.1

1 10

frequercy (Hz)

6' U

Figure 47. Time histories, and acceleration response spectra of horizontal acceleration at

depth 0, 10.8, and 53.5 m for the free field of K-building (1 g = 1000.gal).

- page 65 -

GLOm -- GL —6.9m

CL —7.6m CL —10.8m

0 20 40 60

time (s)

Figure 48. Time histories of relative displacement at depths 0, 6.9, 7.6, and 10.8m for

free field of K-building.

- page 66 -

0 10

0 20 40 Awl

time (s)

Figure 49. Time histories of excess pore-pressure computed at the middle of sand layers

1, 2, 3 and 4 for free field of K-building.

- page 67 -

99

'0

10 20 30 40 50

men-pressure (iPo) -.

0 5 10

shear strom


simulated by BDP model at the middle of sand-2 layer during free field

response of K-building.

- page 68 -

0 20 40 60

meon pressure (kPa)

shear strain y


simulated by BDP model at the middle of sand-3 layer during free field

response of K-building.

- page 69 -

0 0

-0 0 C4

C 0

a) a)

0 0 CN

0 0

0 20 40

time (sec) 0 0 - 01

0 0

-S. 0 -at')

0 0

a) 0 0

0 0 0

ii

C C

roof

C C

0

0 0

0 0

60

0 0 0

0 0 C r')

0 0 0 r'J

0 0 0

Ere I

frequency (Hz)

frequency (Hz)


on roof and first floor of K-building computed for 100 gal input acceleration

(1.g = 1000.gal).

- page 70 -

roof

E -0

0 20 40

time (sec)

E n first floor

0 20 40

time (sec)



- page7l -

120.0

100.0

80.0

60.0

E

Z 40.0

0

20.0

0.0

-20.0

0.00 100.00 200.00 300.00 400.00 500.00

Max. acceleration (gal)

itt,.

100.

R0.(

60.0

E

Z 40.0

0

20.0

0.0

-20.0

40.0

0 1 2 3 4 5 6

Max. displacement (Cm)

Figure 54. Computed distribution of maximum acceleration and relative displacement

versus elevation of K-building (100 gal).

- page 72 -

120.0

100.0

80.0

E

60.0

0

40.0

20.0

0.0

120.0

100.0

80.0

40.0

20.0

0.0 0.OE+0 2.0E+6 4.OE+6 6.OE+6 8.0E+6 1.OE+7 1.2E+7

Max. bending moment (ton cm)

0.0 :c0.0 20010 3000.0 4000.0 5000.0

Max. s:%ear force (ton)

E

Z 60.0

0

Figure 55. Computed distribution of maximum bending moment and shear force versus


- page 73 -

0 0 (N

0 0 (N

0 0 (N

20 40

time (sec' 0 0 CO

0 0 aD

C 0

0 0

a) 0 0

0 0 N

- depth=6.9 m - .. depth=7.6

0. Ii

frequency (Hz)


depth 6.9 and 7.6 m for K-building (100 gal) (1.g = 1000.gal).

- page 74 -

F II lL't It

tI' • r

' 1111

IIII

liii It U. jr

%l •'

I It

lf

- depth=6.9 m depth=7.6 m

20

time (sec)

Figure 57. Time histories of relative displacement at depths 6.9 and 7.6 m for K-

building (100 gal).

-page 75-

0 0

1,

V

I

- - -1.-

sand-4 - -- sand-3

sand—i

0)

0 0 0

LI)

20

time (sec)

Figure 58. Time histories of excess pore-pressure computed at the middle of sand layers

1, 2, 3 and 4 for K-building (100 gal).

-page 76-

I X~Vi* P~

I

- sand-4 -- sand-3

sand-2 sand—i

20

time (sec)

Figure 59. Time histories of vertical effective stress computed at the middle of sand

layers 1, 2, 3 and 4 for K-building (100 gal).

- page 77 -

C C C

0 0 0

0 0'

C 0 0 - 0 a,

0

0 0.1

frequency (Hz)

roof

0 0 LU

0 0 LU

0 0 0

20 40

time (sec)

0 0 0

0 0 0

0 . 0.1 1 Ic

frequency (Hz)


on roof and first floor of K-building computed for 407 gal input acceleration

(l.g = 1000.gal).

-page 78-

20 40

(N 0

time (sec)

first floor

0

20 40

time (sec)



- page 79 -

60.0

E

100.0

80.0

Z 40.0

C,

20.0

120.0

-40.0

0.00 200.00 400.00 600.00 800.00 1000.00

Max. acceleration (gal)

120.

100.0

80.0

60.0

E

E 40.0

C,

20.0

0.0

-20.0

en a

0 5 10 15

Max. displacement (Cm)

Figure 62. Computed distribution of maximum acceleration and relative displacement

versus elevation of K-building (407 gal) (l.g = 1000gal).

- page 80 -

80.0

Z 60.0

C

40.0

20.0

5.OE+6 1.OE+7 1.5E+7 2.OE#7 2.5E+7

Max. bending moment (ton-cm)

120.0

100.0

80.0

E

a

60.0

40.0

20.0

0.0

0.OE+0 0.0 2000.0 40000 6000.0 8000.0 10000.0 12000.0

Max. shear force (ton)

1200

100.0

Figure 63. Computed distribution of maximum bending moment and shear force versus


- page 81 -

0 9

0 0 0 N

20 40

- depth=6.9 rn - - depth=7.6 m

0 0

0 0

- N 0 01

GL=-6.9 (m)

0 0

GL-7.6 (m)

0 - 0.1

frequency (Hz)


depth 6.9 and 7.6 rn for K-building (407 gal) (Lg = 1000gal).

- page 82 -

tI -1 -4

(1•'

jj4t1.1

I / ' 1

depth=6.9 m depth=7.6 m

20

time (sec)

Figure 65. Time histories of relative displacement at depths 6.9 and 7.6 m for K-

building (407 gal).

- page 83 -

- sand—i .; - - sand-2 -- SQflcj-3 * ...' ..

I/; i Ih jflhl H l 4 L ' ' k — TI

t I

114

t('IA1/ft':;

20

NO

time (s)

Figure 66. Time histories of excess pore-pressure computed at the middle of sand layers 1, 2, 3 and 4 for K-building (407 gal).

- page 84 -

20

0 0

C') CO a) L0

Ln (I)

a) > I . 00

0 4- a)

0 0 0

Ln

a) >

0 0 (N

::z

Zz

sand—i sand-2 sand-3 sand-4

. 1

we

0 LC

time (s)

Figure 67. Time histories of vertical effective stress computed at the middle of sand

layers 1, 2, 3 and 4 for K-building (407 gal).

- page 85 -

LIST OF TABLES

Summary of properties of three- and two-dimensional pile group.

Three- and two-dimensional properties of pile for analysis of centrifuge tests.

Three-dimensional and two-dimensional properties of pile cap.

Summary of material properties for Nevada sand used in analysis.

Variation versus depth of horizontal and lateral effective stress and elastic

properties in tests No.6 and 4 on dry Nevada sand.

Variation versus depth of horizontal and lateral effective stress and elastic

properties in tests No.9 and 8 on saturated Nevada sand.

BDP model constants for Nevada sand used in analysis.

Two- and three-dimensional properties of piles of K-building.

Two- and three-dimensional properties of basemat and superstructure of K-building.

Material properties for soil layers of K-building.

BDP model constants for soil layers of K-building.

I

- page 86 -

Table 1. Summary of properties of three- and two-dimensional pile grouP.

3D PROPERTIES 2D PROPERTIES(input data)

PART E 6 unit mass Ref E Unit mass Poisso

kPa kPa ton/m3 kPa ton/m3 ratio

pile 2.O0E08 8.30E07 1.91E00 I 1.43E07 1.91E00 0.2

pile cap part A (steel) 2.O0E08 8.30E+07 7.67E00 1 2.00E08 5.90E+00 0.2(

part B (Al) 7.00E+07 2.56E07 2.74E00 I

part C (Al) 7.00E07 2.56E07 2.74E00 1

Aluminium block 700E07 2.56E07 2.74E00 1

Reference: E.P. Popov, Mechanics of Materials, Prentice-Hall. 1978, p.570.

Table 2. Three- and two-dimensional properties of pile for analysis of centrifuge tests.

pile in 3D - - pile in 2D

external diameter (m) 0.48 width (m) 0.48

thickness(m) 0.0127

pile length(m) 10.73 length(m) 10.73

pile cross-section area(m2) 1.86E-02 cross-section area(m2/m) 0A8

inertial (m4) 5.52E-04 inertial (m4/m) 922E-03

Young's modulus E (kPa) 2.00E08 YOuflgS modulus E (kPa) 1-13E07

pile EA(kN) 3.73E+06

pile EI(kN-m2) L10E05

steel unit mass (t/rn3) 7.67E00 unit mass (t/rn3/m) 1.91E00

pile mass (ton) 1.53E00

soil unit mass (t/rn3) 1.97

total depth (m) 1.67 total depth (rn) 1.00E-00

average mass per unit depth (ton/rn) 9.85E00 mass per unit depth (ton/rn) 9.85E00

average El per unit depth((kN-m2/rn) 1.32E05 El per unit deptli((kN-m2/rn) 1.32E05

average EA per unit depth (kN/m) 4.47E06 EA per unit depth (kN/m) 688E06

Note: El and mass per unit length are preserved, but EA is not preserved

total depth 20

- page 87 -

Table 3. Three-dimensional and two-dimensional properties of pile cap.

I 3D pile cap I 2D pile cap I

part A (circular) diameter Cm) 3.840

height Cm) 0.690 unit mass (ton/m3) 7.670

mass (ton) 61.291 part B (circular) diameter(m) 2.100

height.(m) 0.370 unit mass (ton/m3) 1.570

mass (ton) 2.012

part C (square) width (m) 3.270

height Cm) 0.520

unit mass (ton/m3) 2.735

mass (ton) 15.207 Alum Thium block (square) width Cm) 1.260

height Cm) 1.260

unit mass (ton/m3) 2.735

mass (ton) 5.471

total depth Cm) 1.67

vertical center of gravity Cm) 1.109

horizontal center of gravity (m) 0.061 average.mass per unit length (t/m) 50.29

iidth (m) 3.8W-00 eight (rn) 2:22E00 nit mass Ct/m3/m) 5.90E+00 iass per unit length (t/rn) 5.03E+01 ccentricity of center Qf qravity (m)6.06E-02

Alu1inium block 3D pile cap

A

C

3D eccentricity

2D pile cap

2D

- page XX -

Table 4. Summary of material properties for Nevada sand used in analysis.

- test number 8 9 6 4

void ratio 0.697 0.697 0.791 0.831

porosity 0.411 0.4 11 0.442 0.454

relative density (%) 5l.l 77 51.177 22.366 10.345

Poisson ratio 0.330 0.330 0.330 0.330

saturated unit mass (tim3) 1.978 1.978 1.927 1.907

dry unit mass (tim3) 1.568 1.568 1.485 1.453

buoyant unit mass (t1m3) 0.978 0.978 0.927 0.907

permeability (m/s) 1.50E-03 1.50E-03 1.50E-03 1.50E-03

Yours modulus CoflStaflt 1.35E04 135E+04 1-20E+04 1.14E04

CoefficientofearthpreSsUre 0.493 0.493 0.493 0.493

water unit mass (t/m3) soil specific gravity

maximum dry unit weight minimum dry unit weight

minimum void ratio maximum void ratio

MWIIIIII 2.660

16.970 kN/m3 13.980 kN/m3 0.5 36

9:865 .

reference: R.F. Scott, letter to Kajima. Caltech, 30 October 1990 pp.1-5

RE. Scott & B. Huslimand, letter to Kajima. Caltech, 16 November 1990.

Note: Permeability coefficieti is multiplied by 50 to account for centrifuge scaling

reference: Kajima report. February 1991. p.48&69

- page 89 -

Table 5. Variation versus depth of horizontal and lateral effective stress and elastic

properties in tests No.6 and 4 on dry Nevada sand.

tp't f

depth (m)

vertIcal elf stress (kPa)

lateral elf stress (kPa)

Youngs modulus E (kPa)

Shear modulus G (kPa)

0.00 437 2.15 2.50E01 9.40E03

0.60 17.34 8.54 4.98E0'1 1.87E-04

1.78 34.53 17.01 7.03E,04 2.64E04

2.96 51.28 25.26 8.56E+04 3.22E+04

408 65.92 32.47 9.71E04 3.65E+04

4.97 79.69 39.25 1.07E05 4.01E+04

5.97 94.69 46.64 1 . 1 6E05 4.37E+04

7.03 110.14 54.25 1.25E+05 4.72E04

8.09 126.45 62.28 1 .34E05 5.05E04

9.27 143.64 70.75 1.43E+05 5.39E04

10.45 152.24 . . 74.98

i-eference:Kajima report February 1991. pp.51-52

note: stresses are evaluated at middle of layer, except for the last row that is evaluated at bottom

depth(m)

vertical elf stress(kPa)

lateral elf stress(kPa)

Youngs modulus E(kPa)

Shear modulus 6(kPa)

0.00 4.28 2.11 2.35E-04 8.83E03

0.60 16.96 8.35 '1.68E04 1.76E04

1.78 33.78 16.64 6.61E04 2.48E04

2.96 50.17 24.71 8.05E-04 3.03E04

4.08 6450 31.77 9.13E~04 3.43E04

4.97 77.97 38.40 1 .00E+05 3.77E+04

5.97 92.65 45.63 I .09E05 4.11 E+04

7.03 107.76 53.08 1.1 8E05 4.'13E+04

8.09 123.73 60.94 1.26E05 '1.75E04

9.27 140.55 69.22 1.35E05 5.06E04

10.45 148.96 73.37

reference: Kaj ima report, February 199 I, pp.51-52

note: stresses are evaluated at middle of layer, except for the lait row that is cvilj;itcd at bottom

- page 90 -

Table 6. Variation versus depth oi horizontal and lateral elfective stress and elastic

properties in tests No.9 :nd 8 on saturated Nevada sand.

vertical elf lateral ef f Young's modulus Shear modulus

depth (m) stress (kPa) stress (kPa) E (kPa) G (kPa)

0.00 2.88 1.42 2.29E04 8.61E03

0.60 11.42 5.63 4.56E04 1.71E04

1.78 22.75 11.20 6.43E04 2.'12E04

2.96 33.79 16.64 7.84E+04 2.95E04

4.08 43.43 21.39 8.89E+04 3.34E+0-1

4.97 52.51 25.86 9.78E04 3.68E04

5.97 6239 30.73 1 .07E05 4.01 E04

7.03 72.57 35.74 1.1 5E05 4.32E04

8.09 83.32 41.04 1.23E05 4.63E04

9.27 94.64 46.62 1.31E05 4.93E+04

10.45 100.31 49.41

reference: Kajima report. February igi, pp.51-52 note: StressèS are evaluated at middle of layer, except for the last row thai is evaluated at bottom

vertical elf lateral eff Young's modulus Shear modulus

depth (m) stress (kPa) stress (kPa) - E (kPa) 6 (kPa)

0.00 2.88 1.42 2.29E,04 8.61E03

0.60 11A2 5.63 4.56E04 1.71 E04

1.78 22.75 11.20 6.43E,04 2.42E04

2.96 33.79 16.64 7.84E04 2.95E04

4.08 43.43 21.39 8.89E04 3.34E-04

4.97 52.51 25.86 9.78E04 3.68E+04

5.97 6239 30.73 1 .07E05 4.01 E04

7.03 72.57 35.74 1.15E05 4.32E04

8.09 83.32 41.04 1.23E+05 4.63E04

9.27 9464 46.62 1.31E05 4.93E04

10.451 100.31 49.41

reference: Kajima report, February 19l, pp.51-52 note: streseS are evaluated at middle c layer, except for the last row that is evaluated at bottom

- page9l -

Table 7. BDP model constants for Nevada sand used in analysis.

definition notation values unit

constant for elastic Youngs modulus E0 8000. kPa

exponent for elastic Youngs modulus no 0.5

constant Poisson ratio v 0-3

slope of failure surface a 0-6

apparent cohesion c 1. kPa

slope of characteristic line 0 0.63

attractive pressure Pa 70. kPa

constant for mpnotonic plastic modulus .. H0 0.1

exponent for nionotonic plastic modulus n 1.6

constant for cyclic plastic modulus Hu 0.8

exponent for cyclic plastic modulus flu 4.

I

I

I

- page 92 -

Table 8. Two- and three-dimensiorzal properties of piles of K-building.

pile row In 3D truss In 2D

external diameter (m) 1.8 length(m) 21.3

pile length(m) 243 cross-section area(m2) 0.121

pile cross-section area(m2) 2.51E-00 mass per unit length (tim) 0.251

inertial (m4) 5.15E-01 Young's modulus E (kPa) 2.52E07

Youngs modulus E (kPa) 2.52E-07 EA (kN) 1 .O7EO7

pile EA(kN) 6.41E07 total mass(t) 6.18E-00

pile El(kN-rn2) I .3OE07 total mass on foundation (t/m) I .26E,03

concrete unit mass (t/m3) 2.300

pile mass (ton) 1.12E02

average soil unit mass (t/m3) 1.700

total depth (m) 30

numberof piles . 5

total mass on foundtion (tim) I .26EO3

average EA per unit depth (kN/m) i.O7EO7 . .

Notes: EA and mass per unit length are preserved, but bending is neglected Piles are modelled by using truss elements; they transmit only axial forces.

0 0 0 0 0

top view 2DQ

- pace 93

Table 9. Two- and three-dimensional properties of basernat and supersulicture of K-

building.

m2t

structure mass per unit depth (tim) 1319.03

number of point for mass distribution 11.00

distributed nodal load (kNim) 1201.87

width(m) 30

depth(m) . 6

total mas(t). . 10657

mass per unit depth (t/m) 355.23

unit mass (t/m3) l.92J

references: Kajima report, 1-eDruary I'z, rlg....

Notes: The mass of the superstructure is applied by using 11 applied nodal forces The inertia mass elements do not apply any gravity loads.

averaging depth (m) 30

Young's modulus (kPa) 2.52E07

total building mass per unit depth (t/m) 1349.03 number of point for mass distribution 11.00

distributed nodal load (kN/m) 1201.87

references: Kajima report, February 1992, pA5

(Kajima report, Feb 1992, Fig.3.2)

-. page 94 -

9 continued) 20 data

M ass El (t.cm2) shear El corrected average average

(t) correction for shear inertial mass

GA (t) (actor Phi (t.cm2) (m4/m) (t/m)

1.078E+06 1470 4509E+13 2.32E+03 1.942E+10 2.518E-02 49.00

1.585E+06 1096 5.908E+13 5.32E+03 1.11 IE+10 1.440E-02 36.53

1.713E+06 998 6.758E+13 5.63E03 1.200E10 1.557E-02 33.27

1.754E+06 998 7.131E13 5.80E+03 1.229E10 1.594E-02 33.27

1.769E+06 998 7.383E+13 5.96E+03 1.240E+10 1.608E-02 33.27

1.798E+06 998 7.569E+13 6.01E+03 1.260E+10 1.634E-02 33.27

1.848E+06 1056 7.822E+13 6.04E+03 1.295E10 1.680E-02 35.20

1.862E+06 1002 8.086E+13 6.20E+03 1.305E10 1.692E-02 33.40

1.867E+06 1002 8.147E+13 6.23E+03 1.308E+10 1.697E-02 33.40

1.877E+06 1002 8.193E+13 6.23E+03 1.315E10 1.706E-02 33.40

1.894E+06 1002 8.251E+13 6.22E03 1.327E10 1.721E-02 33.40

1.948E+06 1056 8.137E+13 5.96E+03 1.365E10 1.770E-02 35.20

2.087E+06 1002 8.651E+13 5.91E+03 1.462E10 1.897E-02 33.40

2.361E+06 1048 9.581E13 5.60E+03 1.712E10 2.220E-02 34.93

2.398E+06 1061 9.589E+13 5.51E03 1.739E10 2.255E-02 35.37

2.415E06 1061 9.603E+13 5.48E+03 1.751E10 2.271E-02 35.37

2.435E06 1116 9.L184E+13 5.37E03 1.766E10 2.290E-02 37.20

2.464E06 1061 9.662E+13 5.41E+03 1.787E10 2.31.7E-P2 35.37

2.511E+06 1061 1.002E14 5.50E+03 1.821E+10 2.361E-02 35.37

2.521E+06 1061 1.007E14 5.51E+03 1.828E+10 2.371E-02 35.37

2.525E06 1061 1.007E+14 5.50E+03 1.8311+10 2.3751-02 35.37

2.533E06 1116 9.920E+13 5.40E+03 1.837E10 2.382E-02 37.20

2.554E+06 1061 1.044E14 5.64E+03 1.852E+10 2.402E-02 35.37

2.568E+06 1061 1.04-1E14 5.61E+03 1.862E10 2.415E-02 35.37

2.61 1E-06 1061 1.044E+14 5.51E+03 1.893E+10 2.455E-02 35.37

2.694E+06 1061 1.091E+14 5.58E+03 1.953E10 2.534E-02 35.37

2.747E+06 1133 1.093E+14 5.49E+03 1.992E10 2.583E-02 3777

2.753E+06 1054 1.102E+14 5.52E+03 1.996E+10 2.5891-02 36.13

2.760E06 1084 1.10'IE+14 5.52E+03 2.00IE+10 2.596E-02 36.13

2.787E+06 1054 1.102E14 5.45E+03 2.021E+10 2.621E-02 36.13

2.9331+06 1084 1.093E+14 5.14E03 2.1271+10 2.7581-02 36.13

3.762E+06 1149 1.077E+14 3.95E+03 2.728E+10 3.5381-02 38.30

5.659E+06 2486 1.199E+14 1.13E-03 1.0631+11 1.3791-01 82.07

3.082E+07 3797 1.482E+l1 3.191+02 4625E+1 I 998E-01 126.57

1.000E10 5967 i.000E+17 3.33E+02 2.9911+14 3.8791+02 198.90

43301-06 5520 5.5601+14 1.90E+05 2.923E+09 3.7911-03 184.00

43301+06 374 5.560E14 3.14E+05 1.768E+09 2.293E-03 12.47

43301+06 386 5.560E+11 1.71E-05 3.257E+09 422-1E-03 12.87

4.330E+06 -1.14 5.560E14 1.71E~05 3.2571+09 422-11-03 1480

4.330E+06 526 5.560E+11 9 12E04 6.098E+09 7.9091-03 17.53

4330E06 906 5.560E+I-1 2.33E+04 2.383E-10 3.0911-02 30.20

43301+06 1204 5.560E+14 2311+04 2.402E+10 3.1151-02 40.13

43301+06 1175 5.560E+14 257E+04 2.166E10 2.8091-02 39.17

t1.330E06 I I4 5.560E14 2.57E+04 2.166E10 2.8091-02 38.20

4.330E-06 I I- 5.560114 2571+04 2.166E10 2.8091-02 38.20

3D data (1

height (cm)

I .045E-O

9.985E+0

9.695E+0

9.-105E+O

9.1 l5E+0

8.825E+0:3

8.5 35E + 03

8.245E+03

7. 955E +03

7.665E+03

7.3 75E +03

7.085E03

6.795E03

6.505E03

6.210E+03

5.9 1 5E+03

5.620E03

5.32 5E + 0.3

5.030E+03

4.7 35E +03

441OE+03

4.1 45E+03

3.850E+03

3.555E+03

3.260E+03

2.965E+03

2.670E+03

2.375E+03

2.080E +03

I .785E+03

I .190E03

1.195E+03

9.000E02

4250E+02

0.000E+00

-6.000E+02

-6.900E +02

-7.600E 02

-8.550E +02

-9.500E +02

- I .Ot3OE+03

- I .337E+03

-i.595E+03

-1 .840E03

-2.085E +03

-2.330E 03

- pagc 95 -

Table 10. Material properties for soil layers of K-building.

total unit permeability Poisson fficient

- mass (m/s) ratio earth 1pre

material depth (m) (t/m3) _________ ssure

9 sand-I 0 1.7 6.90E-06 0.3 0.43

8 clay-i -3.9 1.6 1.00E-08 0.48 0.92

7 sand-2 -6.9 1.7 690E-06 0.3 0.43

6 clay-2 -7.6 1.6 1.00E-08 0.48 0.92

5 sand-3 -9.5 1.8 6.90E-06 0.3 0.43

4 sand_Li -10.8 1.9 2.60E-05 03 0.43

3 clay-3 -15.95 1.7 LOOE-08 0.48 0.92

2 eiastic-1 -23.3 2.0 0.3

1 -29.85 1.9 0.3 ela

!~j

-53.5

material sand-1 clay-i sand-2 clay-2 sand-3 sand-4 clay-3 elastic-I

eiastic-2

Specific gravity

2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66

void ratio 1.371 1.767 1371 1.767 1.075 0.844 1.371 0.660 0.844

porosity 0.578 0.639 0.578 0.639 0.518 0.458 0.578 0.398 0.4581

mises c (kPa)

15.277

19.237

75.252

shear modulus

(t/m2) 4170 1045 1910 2760 5625

20480 9180

41330

41330

Youngs modulus (kPa)

1.06E05 3.03E04 4.87E04 8.01E04 1.43E+05 5.22E+05 2.66E+05 1.05E06 1.05E06

layer 9 8 7 6 5 4 3 2 1

Notes: All layers are assumed saturated

layer material

vertical eff stress

(kPa)

hcizontal efI stress (Pa)

friction

angie (deg)

shear strength

t/m2)

9 sand-I 13. 5.7 282 0.040.64

8 clay-i 35.6 32.8 2.7

7 sand-2 46.8 20.1 21.6 0.540.58

6 clay-2 54.8 50.6 3.4

5 sand-3 65.5 28.1 30.3 1 .82 1.97

4 sand-4 93.3 40.0 1,1.3 3.585. 18

3 clay-3 141.2 130.3 13.3

2 elastiC 1

I elastic-2

17,

- p3gc 96 -

Table 11. BDP model constants for soil layers of K-building.

page 97 -

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