Proceedings of Ninth World Conference on Earthquake Engineering

August 2-9, 1988, Tokyo-Kyoto, JAPAN (Vol･ⅠⅠ)

3-5-8

PROPOSAL OF A MATHEMATICAL MODELFOR EARTHQUAKE RESPONSE ANALYSIS

OF IRREGUI.ARLY BOUNDED SURFACE LAYER

choshiro TAHURAl and Takevasu SUZUK工2

1Insti亡u亡e Of lndustrial Science, University of Tokyo

Mina亡0-ku, Tokyo, Japan

2Ins亡itute Of Construction Technology} Kumaga1-gumi Co･ナL亡d･

Shinjuku-ku, Tokyo, Japan

StlmY

For earthquake response analysis of a sof亡　surface layer Which is not con-

sidered　亡o be unifom in a three-dimensional expanse, a quasi-three dimensional

ground model is proposed for practical use. Vibration model　亡ests are carried

ou亡　亡O eXamine　亡he verification of　亡he model. The new model is a kind of cotn-

posite system of a lumped mass-spring system and　亡vo-diTnenSional FEN.

INTRODUCTION

SEudies of in亡erac亡ions between underground s亡ruc亡ures and surrounding ground

have included those by Dr･ Okamoto on tnountain tunnels (1948, 1963), I)I. Housner

on BART亡unnels, Dr･ Sakurai on ground surface observa亡ions of pipelines (1967),

Dr･ Tajimi on dynamic analyses of foundations (1969), and Tamura proposing an ana-

lysis model for亡unnels (1975)･ Subsequently,there have been many studies car-

ried ou亡}　and differing from s亡TuCtures in亡he open air, i亡has come to be consi-

dered that, as a measure of亡he seismic force acting on an underground st-cture,

the displacement during earthquake of亡he surrounding ground has been considered

appropriate for evaluation of the earthquake resistance. This has been backed

up by earthquake observa亡ions made on actual structures.

Ⅰt has been recognized that in case there is a large difference in wave impe-

dances be亡veen the surface layer ground and its basement, the influence of　亡be

profile of the surface layer on earthquake notion is substantial, and iTl aSeis一

mic analysis it is itnportant for the behavior of the surface layer during earth-

quake to be investigated. The mathetna亡ical tnodel previously proposed by Tamura

de亡emines earthquake response of the surface layer ground grasplng the varia-

tion in亡he surface layer two-dilnenSionally (length in horizontal direction and

depth). However, i亡is itnaginable that the behavior of the surface layer ground

will be extremelly cotnplex in case the ground condition changed sharply in a

three-ditnensional expanse. In this case, FEN is nonnally used for the three-di-

mensional study of the dynamic behavior of the surface layer ground in a broad

area and in such case the relationship to be solved contains a very large nuTnber

ofunknoms so　亡hat the numerical calculations are actually close to impossible,

and are no亡　practical.

Therefore, the mathetnatical model previously proposed Was extended and a

quasi一亡hree-ditnenSional tnathenatical model that could be applied to three ditnen-

sions Was made up.工t consists of the following main points:

A･　The surface layer ground is divided into vertical soil-colutnn eletnents.

ⅠⅠ-665

B･　The soil-columelements are replaced by one-lumped-nassISPring systems

亡ha亡　show shear vibrations.

C.　The soil-column elements are transfomed into plate elements, and spring

constants for relative displacenents between mass points are computed from these

plate eletnents･

EmployTnent Of FElt will be convenient in computation of spring constants bet-

ween mass points. me new model can be said to be a hybrid system of a lumpedmass-spring systeTn and two-dimensional FEN. Shear vibrations of the ground can

be expressed With this tnodel along with which wave TnOtions propagated through the

ground in planar fom can be expressed･　Comparlsons be亡ween　亡he model and experi一

mental results are also described below.

HATHEHATICAL MODEL

A soil column having a unit cross-sectional area as shown in Fig･ i is corl-

sldered･ The equivalent Young-s modulus (EF) when亡his soil columns deform in亡he

shape of a fundaTnen亡al shearing vibration mode is deterTnined by亡he following

equation.

(EF) I j･oHE(Z)F(Z)dZ

ど(Z) =f(Z)

/oil m(Z)f(Z)dZ/′ou n(Z)dZ

where, Z is depth, E(Z) is Youngls modulus of ground a亡　depth of Z, m(Z) is mass

at depth of Z, f(Z) is fundamental shearing vibration mode, and ti is thickness of

surface layer.

Eq. (2) indica亡eS亡he displacemen亡normalized by average dlsplacemen亡.

Similarly,the average PoissonIs ratio for this soil column is calculated by

亡he equation below.

Ⅴ　=

/oH f(Z)V(Z)dZ

(3)

where, Ⅴ(Z) is Poissonls ratio of ground a亡　dep亡h of Z.

Equations (1), (2) and (3) are used in compu亡a亡ion of the.stiffness of the

plate elements.

Subs亡i亡u亡ion of Soil Column Elemen亡　b One-Lum ed-Mass-S stem Soil column

elements are made by dividing the ground into the mesh shown in Fig. 2. The

cross-sectional shape of a soil column was made rectangular in this case, bu亡i亡

would be亡he same with a亡riangular cross section.

Le亡ting a soil column　土n the reglOn demarcated by　亡he do亡亡ed line in　亡he

figure vibrate a亡　亡he fundamental shearing frequency at nodal poin亡1, this is

substi亡u亡ed by a one-lumped-mass-spring system which expresses　亡his condition.

Equivalent spring Kei}　and effective mass Mei are Obtained by亡he following equa-

tions With total mass of soil column as Mi, fundanent.al vibration node as fi, and

fundanen亡al circular frequency as a)i:

Kei I MiUi2　　　　　　　　　　　　　　　　　　　　　　　　　(4)

Mei = (AREA)i

(/oF･i ni(Z)fi(Z)dZ)2

/oHini(≡)fi2(Z)dZ

Ⅰト666

(5)

yi = /(AREA)dAIoH王mi(Z)dZ

Stiffness matrix lKe] Connecting basement and mass point can be made from Eq. (4).

uta亡ion of Plate Elements The respective plate elements　王or　亡he　土ndividual

soil-column elements in Fig. =　are cotnputed. At soil-column element I, the four

nodal points are numbered j, j+1, i+1, i, and substitution into plate elemen亡s

is done determining equivalent Youngls modulus and PoissonIs ratio by亡he equa-

tions.below.

EJ -壬il=jEF)見　　　　　　　　(6)

vJ -壬EIL=iVL　　　　　　　　　(7)

冒….tii豊V;:de｡Ⅴ言La:ti禁…三d;:ti.r:Xl皇K富ま…:n慧i:gf冒;;三e霊…sss?an be computed by

since Eqs. (6) and (7) compu亡e average Values, in applying亡hem i亡is neces~

sary for appropria亡e divisions tO be made in order亡hat亡here will no亡be exces-

sive differences be亡ween亡he values of the four poin亡S･

ua亡ion of Ho亡ion The equation of motion is as glVen below･

● ●●

lM｡il ･ lc｡誉). 【K]tYX) - -lMe｡封 (8)

て,,here, [トi】: mass ma亡rix vi亡h l壬i aS COmPOnen亡

【C]: damping matrix

lK]: sum of lKe] and lKp] in Tnatrix K

lMe】: effective mass matrix

x, Y: displacemen亡S Of mass points in x and y directions

u, W: accelerations of basement in x and y directions

Since the displacetnents of the various mass points) in effect}　the average

displacements of亡he ground a亡　the individual nodal points are determined by solv-

ing亡his equa亡ionI亡he distribution of displacemen亡s in the direction of dep亡h can

be computed by Eq. I(2). The reasons earthquake response in the vertical direction

Was not included Were that the response displacement in the vertical direction is

small compared With the horizon亡al direc亡ionl and that it Was aimed tO reduce亡he

scale of calcula亡ions in consideration of practicality. tlowever, in cotnpu亡ation

of the matrix lKp] obtained froth plate elements･ since a plane stress state is

assumed, vertical displacement accOmpanylng earthquake response in the horizontal

direction is computed from these calculations.

MODEL VIBRATION EXPERIMENTS

A soft surface layer ground model Was made on a shaking table to verify the

appropriateness Of　亡he mathematical model described in the preceding section and

resonant vibration experiments were conducted.　The model ground was made under

especially severe conditions in order　亡o investigate wbe亡her this mathematical

model would adequately express　亡he dynamic behavior of the ground.

Fig･ 4 is a plan of a model alluvial ground,the contour lines being亡hose

of　亡he basemen亡in units Of cen亡imeters. The basement Was made of plaster, While

acrylic amide geュ was used for　亡he alluvial ground, and since contour lines Were

provlded　亡o 0,the thickness of the alluvial layer was as much as　20 m a亡　亡he

upper valley part. Ⅰ亡can be seen that　亡he slopes of　亡he valley are very steep

at　亡he left and lower valley sides. The boundaries at　亡he upper valley part and

ⅠⅠ-667

亡he lower valley righ亡　side are free boundaries.

The shear wave velocity Of亡he alluvial layer is 240 cm/see, which is

ex亡remely low compared wi亡h亡hat Of亡he basement SO　亡ha亡na亡ural frequency is am-

plified at the surface layer. The PoissonTs ratio is estimated to be approximate-

ly 0.499, and since the propagation velocity of compression waves is extremely

high compared wi亡h亡he velocity of shear waves, a dis亡inc亡　Separa亡ion between

shear waves and compression waves is made, and in亡he frequency range of　亡he ex-

periments, i亡is practically only shear waves tha亡are predominant.

The s亡raigh亡11nes intersecting in grid fom in Fig. 4　are thin rubber

strings buried in the surface layer portion>　and are for measuring and observing

the vibration mode of the surface. A multiple number of miniature accelerographs

are installed a亡　亡he shaking亡able and the surface for observa亡ions of frequen-

cies, ampli亡udes, and phases.　Fur亡hermore, the vibrating condi亡iorsof the surface

were recorded by photographs also.

The exper土men亡s were conducted ro亡a亡ing the exciting direction 15　deg a亡　a

tine. Figs. 5　to　7　shov low-Order predominant vibrations detemined from ampli-

tudes and phases of vibrations at the surface. Although the fundamental vibra一

亡ions are seen very distinctly, a亡　higher　亡han亡he　亡hird order i亡is diHicul亡　to

obtain a vibration condi亡lon wl亡h a cons亡an亡　phase oveT　亡he entire model, and it

is no亡　an easy ma亡ter　亡o es亡ablish　亡he frequency and mode一 This is　亡hought　亡O be

in par亡because of being influenced by　亡he damplng ratio of　亡he ma亡erlal of ap-

proxlma亡ely 1 percent. The fundamental vibra亡ion mode occurs in a very stable

manner, and　亡his appears if　亡here is the sllgh亡eS亡　COmpOnent Of excita亡ion in　亡he

axial direc亡土on of　亡he valley.

Next, On dividing into an element mesh like the grid in Fig. 4, the natural

vibra亡ions of　亡be surface layer are calculated by Eq. (9) using　亡he method de-

scribed under HMa亡hema亡ical Model,-- and　亡he natural vibration modes are shown in

Figs. 8　亡hrough 12.

lK]tYXユニu2[畑YX)　　　　　　　　　　　　　(9)

CO肝ARISONS OF EXPERIMENTAL AND ANALYTICAL RESULTS

On comparisons of experimental and analytical results, it can be seen亡ha亡

there is very good agreemen亡between亡he predominant vibra亡ion a亡　亡he lowes亡　fre-

quency and亡he fundamental vibra亡ion･ In Figs･ 9 and lO亡he modes are fairly

similar, the frequencies being close亡ogether at 4･24 ti∑ and 4･31 Hz} respectively･

The second-Order node in the experimental results can be seen to be iTltermediate

among亡hese modes･ The亡hird一〇rder mode in the experimental results correlate

With the fifth-Order mode in the analytical results. The fourth-order mode in the

analyses was no亡recognized in亡he experiments･ Thereupon事　COrrelating the

natural (predominant) vibrations of similar modes in亡he experiments and analyses,

the frequencies will be as　亡abulated below.

Table Na亡ural (Predominant) Frequencies from

Experiments and Analyses

Predoninant Frequency Natural FrequeTICy

些de_ No･ froth E甲eriments (Hz) froth A空中yses_ (甲軍｢)

1　　　　　　　　3.64　　　　　　　　　　　　3.72

2　　　　　　　　4.40　　　　　　　　　　　　4.24

3　　　　　　　　4.94　　　　　　　　　　　　4.60

Although it nay be considered that the Tnethod of division had also affected

the analytical results,亡he foregoing results may be evaluated as glVing good

agreetnen亡betveen analyses and experiments in亡he low-Order predominant vibration

reglOn in spite Of the various factors previously men亡ionedt That low-order pre-

dominant vibrations of　亡he surface layer are generated predominantly, and that　亡he

low一〇rder amplitudes are relatively large indicate how slgnifican亡　亡his model is.

ⅠⅠ-668

699-ⅠⅠ

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I � �� �� ��

Y Lx �� �� �� ��

Fig. 5　Lovest PredomiTlant Mode

of　亡he Experimen亡

Frequency .･ 4.94 H2:

ヽ �� ��

Y Lx ���� ���

Fig･ 7 Third Predominan亡Mode

of　亡he Experimen亡

Frequency : 4.24109　Hz

Par亡icipa【ion Factor : 0.35030

l ����������

l I ������梯���

∫ � � ��

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Fig. 6　Second Predominan亡Itode

of　亡he Experiment

Frequency : 3･72659 Hz

participation Factor : 0･13998

/ ��

f- i i 途�

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X �� �"�� ��

Fig. 8 Fundamen亡al Na亡ural Mode

of Ehe　如Ialysis

Frequency : 4･30932 Hz

participation Factor : 0･11880

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Frequency : 4.37482 tlz

participatiotl Factor : 0･01807/

Y ��劔������剪�

I

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Frequency : 4.59969 H2:

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Fig. ll Fourth-Natural Mode of　亡heAnalysis Fig. 12 Fifth Natural Mode of theAmalysis

ⅠⅠ-670

..../--I.■~｣