Earthquake Englneering. Tenth Wor/d Conference ･e 1 992 Balkema, Rotterdam･ /SON 90 54 10 060 5

The extended quasi‑three‑dimensional ground model
TSuzuki
TeChm'(･aI R(,･wW･(･h & Dcl･CIolm7Cnl /nsTt'TLI((,I Kumagai GILmL'Co･ ･ Ltd･ TbarakLl･ Japan

ABSTRACT: The extended quasi‑three‑dimensional ground皿Odel (EXQ3D nodel) is presented in
thispaper,

forthe

analysisofseisnic

responsesof

irregularlybounded31D

surfaceground･ The quasi‑three‑dimensional ground nodel (Q3D nodel) proposed by Ta皿ura'and Suzuki
is extended to EXQ3D model so that the new model can take fundamental througharbitraryn‑th node of shear vibrations into consideration. The formulation of the model is

described first･ Then, earthquake response analyses of a 3‑D model ground are carried out
using both 3‑D finite element method and the proposed model. The validity of the model isverified through the comparison of the results of analyses obtained using both nodels･ It
is pointed out, finally. that EXQ3Ⅰ) model is effective in evaluating the effect of surface

geology on seismic motions and seismic zonation.

1 ⅠⅣrRODUCTIOⅣ

thanfunda丑ental

one

does

not

affectgrounddisplaceJAent response SO nuCh especially at

the shalloy part of the layer･ This nodel
was applied to sinulate the earthqude re‑

Recent large earthquakes. for example, theMexico Earthquake of 1985 (SuZuki 1986).

the Lena Prieta Earthquake of 1989 (Seed etsponse of a shield‑driven tumel in softal. 1990) and the Luzon Earthquake of 1990

ground at the site tlhere the earthquake(TaJnura et al･ 1990), posed us itBPOrtantlessons suggesting that there exists aobservation vas conducted (Suzuki 1988).

close relationship between severe

血ages

irregular boundaries of 3‑D surface ground.

and

lot

its extention to the model. Which can deal
With ftmdamental through arbitrary order ofsurfacegeology･

There

are

a

Since Q3D zDOdel treats the effect ofofattenuation equations and methods toestimate response spectra to evaluate theshear vibration nodes vhich are predozLinant

effects

at the site, enables us to simulate not

of

ユocal

soil

and

geologicalcondition (Trifunac 1989). Based on these

only ground displaceJnent response but alsomeasures.皿uCh efforts have been taken forseisznic zonation･ In case that a specificground acceleration and velocity responseof 3‑D ground during earthquakes･ Thus. thesuperstructure or an uderground tunnel isextended model. EXQ3D JBOdeL vilュ play anplanned,theprediction

of

ttle

important role in seismic zonation or
microZOnation. This paper describes theSeisnic

ground motions due to large earthquakes isnecessary to evaluate the earthquake
resistance

of

the

structure.

In

forbulation of EXQ3D nodel first. Then, thethisvalidityoftheproposed

皿Odel

is

shown

aspect. much attention has been concentrat‑

based on the comparison of the results ofed on how to evaluate the effect of surface

earthquake response analyses using thismodel and 3‑D finite element approach.

geology on seismic motions and vaious
approaches have been proposed to JnOdel
surface ground･

2 FO肌ATIOⅣ OF EXQ3D MODEL

The quas卜three‑dimensional ground znodel.
Q3D model. was proposed (TaJnura&

Suzuki1987) to analyze horizontal ground dis‑
3‑D surface ground

TIle formulation for a linear analysis uslng
EXQ3D nodel is described in this chapter.

as the input data for the earthquake re‑

The surface ground is divided into a muber

sponse analysis of an underground tunnel.

of soil columns in the first step. Eachcolumn is replaced by a set of N coupledplacement responses Of

Q3D model only takes ftmdamental shear vi‑
bration of surface ground into consider‑ation. which is considered to be valid on

dom Systems, each of which corresponds withthe ground where the fundamental shear

each of the shear vibration node of thespring一mass

systemsortwodegreeoffree‑vibration mode of the surface layer iscolumn. Fig･1 illustrates concretely how to

predominant. because the vibration higher

model

953

the

soil

colu皿.

replacingitbythe nodal point i in consideration of n‑th
荒ngn‑.芸…s s.‑fdTiPberra t ;言: tesn;; t eTnheM肝si sOfg 1 :eh:shear vibration mode. and can be defined byby the following equation:(LH̀･i(Z)fi.n(Z)dz)2

芸……t!h:e;三ilnfago,fen:hoefat冨:o:;ic:i oo,vf.eirnEtiizei'.蔓憲

Jif【) =

(1)I"･.i(Z)f!. n(Z)dz
in which:of soilparameter. Which is necessary to form twodimensional elasticity matrix represented
effective mass

as D. is Poisson■s ratio of plate elementc蕊dlen.OnteSn̲tthhe shear vibration∫, yJ.n･ defined as follows:mode; m̀(Z) mass of soil column i at depth
Z; f.

1

I,.n‑

t妄)‑
､･l
normalized modal vector on nA‑th

nV

shea〜

+.!.yi.n(8)

bration mode: H, thickness Of soilcolu‑ i;.and z depth'below ground surfac占･Yi.m=

The sprlng COnStantS and the constants ofviscous dampers of n‑th shear vibrationI"‑yi(≡)I lFi.A(Z) ldz(9)

I"'TFi.n(Z) ldz

system are. then, determined using thecircular frequency of n‑th mode of the soil

三:t iW.hi.cfh : thVei 'lZま,errepartesenn.3sal tAein…OiiS:S̀慧

column aS:

Ⅴ.

is Poisson‑s ratio at nodal point i im

n主tR mode of shear vibration system･ and.is
Ki;:i=Xi;:i‑W!.｡Y;【'

given as the weighted mean obtained uslng(2)

the same manner as that used in equation

ci;:i= Ci;:i=一弘弘=莞Ki;:i (3,ui.n

昌…喜三; TfehiluneSinleenttheel :e;eenT買,;ri.BcfeTl:ufrcneaensfS.brnfat rglix;:≡
element with unit thickness under plane‑shear;岩r 1 :≡ i cch.:nstKa鼠i c.annndec瑞i madsesnoMtlBn,

withstress condition:

盈plll2]';'=

the base ground in the

direction. respectively;/..nBdy(10)

dh::5eaxtps: t connecti鳩a l嘉≡竺≡ta巴俸) o̲f̲. ▲LVisICLO̲uS L̲

mass Mi"I with the base ground in thein which. B and D denote strain matrix andhorizontal

andtwo‑dimensional elasticity natirix respec‑tively under plane‑stress condition. The

D千･n da皿plng factor of soil colu皿i on n‑thda皿ping matrix of plate element ∫ in n‑thmadき｡f shear vibratュ.n.

mode of shear vibration system is, there‑D.

fore. glVen by the following equation:

W.

∫and

circular′directiom. respectively;

frequency

of

n‑th

mode;

瑚iigs tfhaect:erie.hftee.T;anc.claicnulla,tedh.by theand the displacement function. Fi:nni:;:lcl]･;･ ‑空辿【K:]･;･
uJ.n

given aS follows:

4

uJ.n‑

I"'hi.∩(Z)･lFi,n(Z)ldz
D.

‑

1■n

I‑llFi.n(Z)⊥∑ui,n
El卜一1

ldz

DJ,n‑

⊥呈Di.n4Fi.n(7･)=

1

I

ei.nfi.n(Z)

言置(,nl':I:eet he: :ettn芸itnatleslde;:pTltnn̲gtShtnlif:f:nf:S･S.lf買記eA≡elements in n‑th mode ofshear
vibration system is given aS follows:denotes the participation
呈:ctw.hric.hi雪.iIP col.m i ｡n n̲th mode.f

shear vibration.
Each two degree of freedom system com‑posed of a mass, 2springs and[K2】'n'‑ J!.[K2]{;'

2 dampers isN亡connected with each other by finite plateelements with unit thickness as illustratedlc2]'n'=言.lC2]';Iin Fig･2･ Fig･2(b) is the schematic repre‑

in which, N denotes the total nuJnber ofsentation of the n‑th mode of shear vibra‑
groundplate eleme㌔s used in the model. Defining
ass

illustrated as Fig･2(a) is replaced by

and

tion

system

when

the

3‑D

surface

EXq3D modeL The elastic modulus of platerュx

is glVen by the following

element ∫equation.
wEdd nn
the element connects
masses with a rectangular shape

童襲; ;S;t: 1:foffn;e:tsi:S̲:: ̲p:03?nd;̲;rixn:ce苦ntos!rtneyli,tvtau?hstidBPc等蔓草
I

4

tt:tt崇dSiliflfnngesn?aJt:nlPxt.r i缶JrLyI Ln.7f,J三豊m霊of shear vibration system of EXQ3D model
can be written as:

4

EJ.n‑

the

⊥.写.Ri.n4lK]fn)‑ ll2](n) + lX3](n)Ri.∩‑ I"‑Ei(Z)･JFi.n(Z)Idz
lc]…‑ 【C2】l【) + lc3](A)詰:h.liclh :C.El霊'ldeantotdeeS,tthhez eiea;.t三c s:;言霊:冒Thus, the equation of motion for n‑th modeand Ri.n denotes the equivalent rigidity ofof shear vibration system of EXq3D model

954

illustrated in Fig･2(b) is written as:

model

groundisa

3‑Dvalleyboundedbysharp slopes of base ground･ The stratigra‑･珊■二[C,･甘'十〔X,･甘'‑ ‑【y】脚18,

phy of surface layer of the model is shownin Fig･4･ The surface layer is consisted ofin which: lM](n) denotes the effective massproperties･ As shown in the figure, lower

4

different types of soil with different

three modes of shear vibration are predomi一
maれt

in

such

a

layer･In

theanalysesusing……憲Jlf: i be:th19.ipogdynesg意r l:Kt,e airM:‑:t:h: :…………

EXQ3D model. therefore. fundamental throughmatrix of n‑th mode of shear vibration

third modes of shear vibration are taken

system; X

into account.

and Y horizontal displaceJnent inx and y direction. respectively; and 'u','ti
input horizontal acceleration in

∫ and

′

Fig.5

shows the discriti2:ation of themodel ground for EXq3D model. The number of

direction, respectively

nodes used is

A set of N coupled above equations areis 487･ Fig･6 shows a schematic representa‑

315, vわile that of freeedom

induced when fundamental through N‑th modetion of the discritization for FEM3. The

of shear vibrations are taken into accounthorizontal

mesh

for

FEM3

includes121

in EXQ3D model. The earthquake responses ofnodes. which is considerably roughconpared

nodal point i or soil column i at arbitrary

vith that for EXq3D. Suface layer is divi‑

depth z and time t are glVen by the incre‑

ded into

4

sublayers vertically correspon‑

mental mode‑superposition technique adopted

ding

4

soil

in some earthquake response analyses ofnumber of nodes used for FEM3 ismuユti

degreeoffreedomsystemthatregardingtoof

freedomtypes･

isConsequenltly,

972.

the

605, while

Which

is

twice

as

one‑dimensional shear vibration problems

much as that used for EXQ3D.

(Elgamal 1991. Zlohraz 1991).

Ear thquake response ana lyses wereconducted using NS and EW components of

〜 ..

(19)

Hachinohe waves, Tokachトoki Earthquake ofFi.n(Z)(20)

horizontal y and x direction, respectively･Fi.∩(Z)(21)

Fi.∩(Z)+ii(t)(22)

Fi.n(Z)(23)

丈i(Z･t)‑..写.Xi.∩(t) Fi.n(Z)+ui(t)
H

1968

l

ii(Z.t)≡"写.Xi.∩(t)∑.Xi.n(t)〜ジi(Z･t)‑motionsinthefixed to O･1 in both models of analysis.

3,2 Comparison of results

Iをi(Z.t)‑.≡.Yi.n(t)The results of analyses obtained by EXq3D

PJ

yi(z･t)=ground"n写.Yi.n(t)
N

input

The daJnping factor of surface layer wasNxi(Z･t)≡asn∑,Yi.n(i)Fi.n(z)and FEM3(24)

are compared with respect to both

acceleration and displacement responses.

Table 1 sllmarizes the values of maximum
三Es.wlhuitceh : axcic'ezietr';t iXoli'.Z ･ tr'e laantdi v;i ( zv･etl'o｡言‡;accelerationand displacement, respectively. of nodal

obtained by the two models. In terms ofpoint

maximu皿

i

in

the

horizontal∬direction

at

and

maximl皿

reSpOnSeS,

aS

displacement

ShoⅥl

in

the

table.

results obtained by EXQ3D give good approx‑
imations of finite element solutions except;;冒… : ; iz, a:a:bdSaO:1憲: ca:ciceenyf字rzait:i:ojne･cg.i i'vzretltlia.t ;≡;for cases of

nodal point i in the horizontal rdirection

eration of the rough discritization inat depth z and time t; Ⅹ.

FEM3, 1t can be concluded that the results

X.

in Table 1 validate EXq3D sufficiently.

Ⅳ0.137, G工..‑5.0

m. In consid‑｡鵡enreenlta.t i ;ees,aeccctel,veerlal,t :no'n: ; 'vneXaiiscni'ltti1 n:nnSx
Table 1 Comparison of maximum responses

direction at time t in n‑th mode of shear☆.

Y.
dV還;nnTeenltaStyl:vtfeenm;acscseYiel･rna'lttni'o'n; dVllFrlenocttti主.yn :≡…
time t in n‑th mode of shear vibration

Location 磐

7#､T&ﾅ

137‑Ⅹ‑0

137‑Y‑0

三霊:主aacntdinご恩S.ainld c'wi息lln冒nut xgarnodund,

137‑Ⅹ‑5

direction. respectively, at time t.

137‑Y‑5綯#b

137‑Y‑10
3.Ⅵ;RIFICATION OF EXQ3D MODEL

137‑Ⅹ‑15

3･1 Model ground and conditions of analyses

403.3

縱Sb

2.896

繝迭

1.533

268.5

經c

CR紕

190.4

surface

177.4

途

225.4

c偵2

1.813

sr
コ

0.923
0.386

紊

0.333

163.2

c

0.165

c

141‑Ⅹ‑0

cb繧

355.4

經ビ

141‑Y‑0

R

328.7

繝C

420.5

縱cb

2.990

311.5
393.5

繝釘
經

1.473
2.494

200‑Y‑0
204‑X‑0

3‑D

ｲ

186.7

quake

of

F俯6褪

cr

200‑Ⅹ‑0analyses磐137‑Y‑15ln order to validate EXq3D model, earth‑ground were carried out. using EXQ3D modeland finite element model (FEM3). The modelｲ

EXn3D 播Tﾘ6ｨ

329.3

137‑Ⅹ‑10

responsefv･●● 播b蒄C2

2〔】4‑Y‑∩

#2縒
cR縒

r

.309.9

0.153

縱sr

2.324

1.642

1̲750

137‑Ⅹ‑0: No･ 137,Direction‑X,Depth=0･Om

ground used is illustrated in Fig.3. The

955

sｦﾘ

ｲ

h:.nHh:i寮ナ
m.(Z)=pi(Z)･A.

M(21

十 日知 日,., h

+

･

●

･

●

･

･

+

.t一 ● .I‑ ●

CC

｡l ■l

香

Soil ColuJln i

Fig･l A schematic representation of a set of N coupled spring‑mass‑damper Systems

Plate El印ent ∫K̀こ'. Kt.n:(a) irregularly bounded 3‑D surface ground(b) n‑th mode of shear vibration system

Fig･2 A schematic representation of EXQ3D model

ヽ

ヽ

劍

ﾂ

′

lヽ

ーヽ

勍‑.l.‑.‑.‑.‑.lD...‑.‑.‑..Y

′一

o′

一.∫
ー

15 200 白ﾒ簫簫簫簫簽&ﾒ簫簫
′

ﾒ

I)

ﾒ

卓′

ｲ
I

一

I

I

ヽ一

..F

ヽ

Fig.3 Three‑dimensional surface ground used for earthquake response analyses

ー召FS. EEL

Fundamental

Mode'Frequency‑1 ･ 74Hz

*Eff. Mass Rat.=0.47
uodJ

L

∵1

V●otorF

tH

Second

Mode

Third

*Frequency=4 ･ 34Hz

★Eff. Mass Rat.三0.18

NodJI

V4CtOrF(I)Mode*Frequency=6 ･ 65Hz

*Eff. Mass Rat.‑0.20
LJodAL

L二才｣

V40tOrF

Soil PropertiesYs
tz)(:/tn,)(1/see)ン

1.5

60

0.495

I.6

80

0.495

I,8

210

0.470

I.9

270

0.450

Fig.4 Soil properties and shear vibration modes of surface layer used im analyses956

す一一一一fixedI

7.7. N VH 0 0 0 0

204 20 141 137 米ed

1

yQ‑fixed㌧′r
Fig･5 Discritization of plate elenet meshused for the analysis by EXq3D

EXQつP

H●Ih182

一.O

一.8

t

l

5

rEI

e

0

0J

6.0

(i

O

Fig,6 Discritization of finite element meshused for the analysis by FEM3

FEHコ

IA

7.O

L

I2

l.0

EXOつO

6B

l●CI

9

0

JO

M■t4‑a.

O

0

0

I.0

2.0

1.8

3.

Cl

1

5.0

140

l

6.0

i

o

1lS9

0

t

(】

Lll

rJL

7

0

:I

0

A

C

l

l

NI

0

0M,

Dl

I

tl

A

a

0

0

1

0

2

E)

1

8

JI

L

2.D

⊃

O

1.0

3.0

I

q

i

O

L

a

S,0

l.0

M

†.0

Hl

̀l

16

O

つ2

Ne

1+い

9

O

JO.0

D.0

I.0

CL

0.EIM.

7.0

Dl

1.0

f

rlHり20

O

I

O

1

t

l

l.0

H

14日

9.0

日).0

6.0

101l

7.0

l■

‑

13

S8

日cJ

O

r〇日●llLX

1.0

5.0

L l a)B I s o cJつ13t Hb Q I s o c)

I

rt‑X

Mい･･36tl

0

0

4■

cl

rl■L.‑0.

0FrH3

Nt. J31 CL. ‑LQ.Orl. DH pl HtAIl

m

0

早

t

s

6

0

e

C)

0.

ミO

71

1り

7̲O

I

O

(3.15

9

事●CI

0

)O

?.D

I.0

9.a

JO.0

CI

r14[･‑I.ll

L2

EL

l3.67

1+EI

O

L

Fig･7 Comparison of tiqle histories of acce‑

l爪e

(s8C7

Fig･8 Comparison of tine histories of dis‑lation between the analyses by EXQ3D

placeJnent between the analyses by

and FEM3

EXQ3D and FEM3957

In order to compare time‑dependent

extended quas i ‑three‑dimens ional grotmd

characteristics of responses obtained bymodel (EXQ3D model) for earthquake response

the two models. time histories of acceler‑sinulation of surface ground tinder threeation and displacement were plotted atdimensional geological conditions･

selected locations･ Fig.7

shows comparative

validity of the model vas verified, basedplotting on time histories of acceleration.

on the comparison between the results of

The

The upper tyro histories are acceleration

earthquake response analyses using both

responses at GL.‑10

EXq3D model andm of No.131. where the3‑D finite element model.

effect of second and third modes of shear

Through the analyses. the followings arevibration is relatively large. The lower

su皿mariZed as conclusions :two time histories are. on the other hand,

1. EXq3D model enables us to simulate

acceleration responses on the groundeasilysurface of

surface ground, vhile Q3D model can not

EXQ3D

Ⅳo･141･ As shovn in the figure.

gives

good

approximationofacceleration

responsesof

3‑D

sufficiently siJnulate because it takes only

3‑D

finite element solutions in terms of tine

fundamental node of shear vibration into

‑dependent characteristics of acceleration.
histories of displacement. The selectedconsideration.
2. Using EXQ3D model, the accuracy ofearthquake response analyses of undergroundFig･8

shows comparative plotting on timelocations for plotting are the sa血e thatstructures

are used in Fig･7･ Displaceznent responses

structures are buried deeply ill 3‑D surface

near ground surface are mostly governed byground ･fundamental mode of shear vibrat ion.
However.

EXQ3D

model

also

gives

3.

good

sitnulation of ground displacement responses

improved.

model

works

for

not

even

only

if

the

eliminates

generating

3‑D

mesh

in

3D‑FEH. but also reduces computation time
due to the reduction of number of freedom
needed.

in the deep part･ as shoym in the figure･
Therefore. EXQ3Ⅰ) model enables us easily tosimulate earthquake response of

EXQ3D

laborious

is

4. In applying EXQ3D model. limitations ofundergroundstructures deeply buried in surface ground.

the model, based on the assuJnPtion that theincremental mode‑superpos i t ion can beapplied. should be kept in mind.

4.LIMITATIONS OF EXQ3D MODEL
Since EXq3D is the simplified model based

REFERENCES

on the incremental mode‑superposition tech‑nique. there are soJne limitations in its
application･ The model is fully applicable

ElgaLmal,to the surface ground consisted of a layer

system･ Earthq. Eng･ Struc･ Dyn･ 20: 371‑

with

uniform

thickness

andwithout

A.W.

1991.

Shear

hystereticelasto‑plastic earthquake response of soil

any

389

local irregularity･ However. a large amountMohraz,of error may generate, if the incremental
mode‑superposition technique is applied tonode‑superposition for non‑linear dynamic

the ground. where the frequency of funda‑

471‑481.

B.

et al. 1991.

Am

incremental

analysis･ Earthq･ Eng･ Struc･ Dpl･ 20:

mental shear vibration mode of a soil

Suz山こi, T. 1986. A summary report On earth‑column is almost identical to that of thesecond or third mode of adjoining columns.
Therefore, the following limitations should
be kept in mind;
1, The discritization of a mesh should not
be excessively rough. The difference inquake damage in Mexico･ Tech･ Report ofthicknessofadjoiningcolu皿S

ShouldRu皿agai Guni 39: 99‑109.
Suヱuki, T. et al. 1988. Earthquake observa‑

tion and response analysis of a shieldtunnel. Proc. 9WCEE

6: 563‑568.

Seed. 氏.B. et al. 1990. Preliminary reportbeonkept at the level less than 50 ㌔;

theprincipal

the October2･ The model can not be applied to thegeotech皿ical

17, 1989

aspectsofLoma Prieta Earth‑quake. Report No.UCB/EERC‑90/05.
Tamura et al. 1990. Preliminary report ofsurface ground with excessively sharp slopeof base rock･ The slope angle should not

JSCE reconnaissance team on July 16. 1990

exceed 45 degrees at naLXinuJn;

Luzon Earthquake: Earthq. Eng. Comitt･

3. The fifth mode will bemode desirable to be

and

JSCE.

the heighest
incremental ly

Tamura.

C.

&

Suzuki.

T.

1987.

A

quasi‑

superposed. Modes higher than the fifth one

throe‑dimensional ground model for earth‑

do not predominate in naturalsoil deposits

quake response analysis ofin general. In such cases.

the extent ofabovementioned error becomescompatible toundergrotlnd

structures. Monthly Jour. Inst. Indust.
Scl‥

University of Tokyo 39‑1: 37‑40.

Trifunac, H.D. 1989. Dependence of Fourier

Or larger than the increase in accuracy due

to the incremental mode‑superposition.

spectrum a皿plitudes of recorded earthquake

accelerations on magnitude. local soil
conditions and on depth of sedimemts.

5. CONCLUSIONS

Earthq. Eng･ Struc･ Dyn. 18: 99911016･

In this paper. the author proposed the958