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
sideredsurface layer Which is not con‑

亡o be unifom in a three‑dimensional expanse, a quasi‑three dimensional

ground model is proposed for practical use. Vibration modelou亡亡O eXamine

亡he verification of亡ests are carried

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 groundhave 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‑
lysisried

modelou亡}for亡unnels
anddiffering(1975)･
from

Subsequently,there

s亡TuCturesin亡heopen

have
air,

been

many

i亡has

studiescometobecar‑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 consideredappropriate 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beprofile 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 Tamurade亡emines earthquake response of the surface layer ground grasplng the varia‑
tion in亡he surface layer two‑dilnenSionally (length in horizontal direction anddepth). However, i亡is itnaginable that the behavior of the surface layer groundwill be extremelly cotnplex in case the ground condition changed sharply in athree‑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 soand are no亡

亡hat the numerical calculations are actually close to impossible,
practical.

Therefore, the mathetnatical model previously proposed Was extended and aquasi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･
亡ha亡

The

soil‑columelements

arereplacedby

one‑lumped‑nassISPringsystemsshow shear vibrations.

C.

The soil‑column elements are transfomed into plate elements, and spring

constants for relative displacenents between mass points are computed from theseplate 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 lumped
mass‑spring systeTn and two‑dimensional FEN. Shear vibrations of the ground canbe expressed With this tnodel along with which wave TnOtions propagated through theground 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heshape of a fundaTnen亡al shearing vibration mode is deterTnined by亡he followingequation.

(EF) I j･oHE(Z)F(Z)dZf(Z)
ど(Z) =/oil m(Z)f(Z)dZ/′ou n(Z)dZ

where,Zisdepth,E(Z)isYoungls

modulusofgrounda亡

depthofZ,

m(Z)

is

mass

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

/oH f(Z)V(Z)dZⅤ=(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 theplate 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

figure vibrate a亡土n the reglOn demarcated by

亡he do亡亡ed line in

亡he亡he fundamental shearing frequency at nodal poin亡1, this issubsti亡u亡ed by a one‑lumped‑mass‑spring system which expresses
Equivalent spring Kei}

亡his condition.

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, andfundanen亡al circular frequency as a)i:

KeiI

MiUi2

(4)

(/oF･i ni(Z)fi(Z)dZ)2
Mei = (AREA)i(5)/oHini(≡)fi2(Z)dZⅠト666

yi = /(AREA)dAIoH王mi(Z)dZ

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

uta亡ion of Plate Elements
soil‑column elements in Fig. =王or亡he土ndividual

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sis 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 bysince Eqs. (6) and (7) compu亡e average Values, in applying亡hem i亡is necessary 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･●●●(8)lM｡il ･ lc｡誉). 【K]tYX) ‑ ‑lMe｡封て,,here, [トi】: mass ma亡rix vi亡h l壬i aS COmPOnen亡【C]: damping matrixlK]: sum of lKe] and lKp] in Tnatrix KlMe】: effective mass matrix
x,Y:displacemen亡SOf

masspoints

inxandydirectionsu, W: accelerations of basement in x and y directions
Since the displacetnents of the various mass points) in effect}

displacements of亡he ground a亡

the average

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 canbe 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 issmall compared With the horizon亡al direc亡ionl and that it Was aimed tO reduce亡hescale of calcula亡ions in consideration of practicality. tlowever, in cotnpu亡ationof 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 theappropriateness Of

亡he mathematical model described in the preceding section andresonant vibration experiments were conducted.
especially severe conditions in ordermodel would adequately express

The model ground was made under

亡o investigate wbe亡her this mathematical

亡he dynamic behavior of the ground.

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

亡he basemen亡in units Of cen亡imeters. The basement Was made of plaster, Whileacrylic amide geュ was used forprovlded

亡he alluvial ground, and since contour lines Were

亡o 0,the thickness of the alluvial layer was as much as

upper valley part. Ⅰ亡can be seen thatat

亡he slopes of

亡he left and lower valley sides. The boundaries at

ⅠⅠ‑66720 m a亡亡he亡he valley are very steep

亡he upper valley part and亡he lower valley righ亡
The

shear

side are free boundaries.

wave

velocity

Of亡healluvial

ex亡remely low compared wi亡h亡hat Of亡he basement SOlayer

is240

cm/see,whichis

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
strings buried in the surface layer portion>

are thin rubber

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 15tine. Figs. 5
tudes

andtophases

7ofvibrations

at亡ions are seen very distinctly, a亡thesurface.

higheran easy ma亡ter

亡o es亡ablish

in par亡because of being influenced by

Although

亡han亡heobtain a vibration condi亡lon wl亡h a cons亡an亡is no亡deg a亡ashov low‑Order predominant vibrations detemined from ampli‑
thefundamental

vibra一亡hird order i亡is diHicul亡

phase oveTto亡he entire model, and it

亡he frequency and mode一 This is

亡he damplng ratio of

亡hought

亡O be

亡he ma亡erlal of ap‑

proxlma亡ely 1 percent. The fundamental vibra亡ion mode occurs in a very stablemanner, and亡his appears if

axial direc亡土on of

亡here is the sllgh亡eS亡COmpOnent Of excita亡ion in

亡he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

scribed under HMa亡hema亡ical Model,‑‑ andFigs. 8

亡he method de‑亡he natural vibration modes are shown in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 theanalyses was no亡recognized in亡he experiments･ Thereupon事COrrelating thenatural (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 fromExperiments and AnalysesPredoninant Frequency Natural FrequeTICy
些de̲ No･ froth E甲eriments (Hz) froth A空中yses̲ (甲軍｢)
1

3.64

2

4.40

3.72
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 thatlow一〇rder amplitudes are relatively large indicate how slgnifican亡ⅠⅠ‑668

亡he亡his model is.

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きO alnコTISul aqlol qST凸SユOqユne aqユ̀3uTSOTつuIFrequency : 4.40 lizFrequency : 3.64 Hz

IY Lx

XしFig. 5

Fig. 6

Lovest PredomiTlant Modeof

亡he Experimen亡EIl

Second Predominan亡Itodeof
亡he ExperimentFrequency : 3･72659 Hz
participation Factor : 0･13998

Frequency .･ 4.94 H2:/

ヽi

f‑ i

途

｢Y Lx

YL 剪X

Fig･ 7 Third Predominan亡Modeof

Fig. 8 Fundamen亡al Na亡ural Mode

亡he Experimen亡Frequency : 4.24109

of Ehe如Ialysis

Frequency : 4･30932 HzHz

participation Factor : 0･11880

Par亡icipa【ion Factor : 0.35030

ll I

梯Ⅰ一丁汀 冤 剪＼lf 劔∫｢ 剩＼八HlYL､｣

1Y Lx 劔僮 兀 儉}Lq､｣ 辻･D､/

劔秒rflFigl 9 Second Natural Mode of theAmalysis Fig･ 10 Third Natural Mode of theAnalysis
Frequency : 4.37482 tlz
Frequency : 4.59969 H2:participatiotl Factor : 0･01807

Participation Factor : 0.21212/IY劔 剪Y｢ ユ̲⊥⊥｣X 芳未Fig. ll Fourth‑Natural Mode of剪ｲ

亡heAnalysis Fig. 12 Fifth Natural Mode of theAmalysis. /‑ I.■ ｣ⅠⅠ‑670