Calculation of Nonlinear Ground Response in Earthquakes

7/27/2019 Calculation of Nonlinear Ground Response in Earthquakes

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B u l l e ti n o f th e S e i s m o l o g i c a l S o c i e t y o f A m e r i c a . V o l . 6 5 , N o . 5 , p p . 1 3 1 5 - 1 3 3 6 . O c t o b e r 1 9 7 5

C A L C U L A T I O N O F N O N L I N E A R G R O U N D R E S P O N S E I N E A R T H Q U A K E S

BY W ILLIAM B. JOYNER AND ALBERT T. F. CHEN

A B S T R A C T

A me t h o d i s p r e s e n t e d f o r c a l c u l a t i n g t h e s e i s mi c r e s p o n s e o f a s y s t e m o f

h o r i z o n t a l so i l l a y e r s . T h e e s s e n t i al e l e m e n t o f t h e m e t h o d i s a r h e o l o g i c a l mo d e l

s u g g e s te d b y I w a n w h i c h t a k e s a c c o u n t o f t h e n o n l i n e a r h y s t e r e t i c b e h a v i o r o f

s o il s a n d h a s c o n s i d e r ab l e f l e x ib i l it y f o r i n c o r p o r a t i n g l a b o r a t o r y r e s u l t s o n t h e

d y n a mi c b e h a v i o r o f s o i l s . F i n i t e r i g i d i t y i s a l l o w e d i n t h e u n d e r l y i n g e l a s t i c

me d i u m, p e r mi t t i n g e n e r g y t o b e r a d i a t e d b a c k i n t o t h e u n d e r l y i n g me d i u m.

T h r e e a l t e r n a t e w a y s o f i n t e g r a t i n g t h e e q u a t i o n s o f mo t i o n a r e c o m p a r e d , a n i m -

p l i c i t t e c h n i q u e , a n e x p l i c i t t e c h n i q u e , a n d i n t e g r a t i o n a l o n g c h a r a c t e r i s t i c s . A ne x a m p l e is s e t u p f o r c o m p a r i n g t h e d i f f er e n t me t h o d s o f i n t e g r a t i o n a n d f o r

compar ing the non l inea r so lu t ion wi th a so lu t ion based on the wide ly used

e q u i v a le n t l i n e a r a s s u mp t i o n. T h e e x a m p l e c o n s is t s o f a 2 0 0 - m s e c t io n o f fi r m

a l lu v i u m e x c i t e d a t i ts b a s e b y th e N 2 1 E c o m p o n e n t o f t h e T a f t a c c e l e r o g r a m

mu l t i p l ie d b y f o u r t o p r o d u c e a p e a k a c c e l e r a t i o n o f 0 .7 g a n d a p e a k v e l o c i t y

o f 6 7 c m/ s e c . T h e t h r e e t e c h n i q u e s o f i n t e g r a t i o n g i v e v e r y s i m i l a r r e s u l ts , b u t

i n t e g r a t i o n a l o n g c h a r a c t e r i s t i c s h a s t h e a d v a n t a g e o f a v o i d i n g s p u r i o u s h i g h -

f r e q u e n c y o s c i ll a t io n s i n t h e a c c e l e r a t i o n t i me h i s t o r y a t t h e s u r f a c e . F o r t h e c h o s e n

e x a m p l e , w h i c h h a s a t h i c k s o il c o l u mn a n d a s t r o n g i n p u t m o t i o n , th e e q u i v a l e n t

l i n e a r s o l u ti o n u n d e r e s t i ma t e s t h e i n t e n s i t y o f s u r f a c e m o t i o n f o r p e r i o d s b e t w e e n0 .1 a n d 0 . 6 s e c b y f a c t o r s e x c e e d i n g t w o . T h e d i s c r e p a n c i es , h o w e v e r , w o n l d

p r o b a b l y b e l e s s f o r i n p u t mo t i o n o f l o w e r i n t e n s i t y . A t l o n g e r p e r i o d s t h e

equ iva len t l inea r so lu t ion is in e ssen t ia l ag reem ent wi th the non l inea r so lu t ion . Fo r

t h e s a me e x a mp l e b o t h s o l u t i o n s s h o w t h a t , c o mp a r e d t o a s i t e w i t h r o c k a t t h e

s u r f a c e , mo t i o n a t t h e s u r f a c e o f t h e s o i l i s a mp l i f i e d f o r p e r i o d s l o n g e r t h a n 1 . 5

s e c b y a s mu c h a s a f a c t o r o f tw o . A t s h o r t e r p e r i o d s t h e a mp l i t u d e i s re d u c e d .

I N T R O D U C T I O N

T h e p r o b l e m w e a r e c o n c e r n e d w i t h is b a si c a ll y a v e r y s im p l e o n e . W e p o s t u l a t e as y s t e m o f h o r i z o n t a l s o i l l a y e r s b o u n d e d a b o v e b y t h e f re e s u rf a c e a n d b e l o w b y a s e m i -

i n f i n i t e e l a s t i c m e d i u m r e p r e s e n t i n g t h e b e d r o c k . W e f u r t h e r p o s t u l a t e a v e r t i c a l l y

i n c i d e n t s h e a r w a v e i n t h e u n d e r l y i n g m e d i u m , a n d w e a s k t h e q u e s t i o n , " H o w w i l l t h e

o v e r l y i n g l a y e rs r e s p o n d a n d i n p a r t i c u l a r w h a t w i ll b e t h e m o t i o n o f a p o i n t o n t h e f r e e

s u r f a c e ? "

T h i s i s a c l a s s i c a l p r o b l e m i n e n g i n e e r i n g s e i s m o l o g y . T h e r e i s s o m e d i s a g r e e m e n t

c o n c e r n i n g t h e r a n g e o f a p p l ic a b il it y o f th e s o l u t io n ( H u d s o n , 1 97 2; N e w m a r k e t a l .

1 9 7 2 ) , b u t n o o n e w o u l d d e n y t h e i m p o r t a n c e o f s o l v i n g t h i s r e l a t i v e l y s i m p l e p r o b l e m

c o r r e c t l y .

T h e p r o b l e m w a s s o l v e d b y K a n a i ( 19 5 2) s o m e y e a r s a g o f o r t h e c a s e o f l a y e rs w i t hl i n e a r v i s c o e la s t ic i ty o f t h e V o i g t ty p e . W h e n w e a r e d e a l i n g w i t h i n p u t m o t i o n s u f-

f ic i e n tl y i n t e n s e t o c a u s e s e v e re d a m a g e t o s t r u c t u r e s , h o w e v e r , w e c a n n o t a s s u m e l i n e a r

b e h a v i o r o v e r t h e e n t i re r a n g e o f s t ra i n . T o d o s o w o u l d i m p l y s t re s se s m a n y t i m e s

g r e a t e r t h a n t h e s t r e n g t h o f t y p i c a l m a t e r i a l s a s m e a s u r e d i n t h e l a b o r a t o r y .

T o c i r c u m v e n t th i s d i ff i cu l ty t h e m e t h o d i n c o m m o n u s e c u r r e n t l y i s w h a t w e s h a l l

1 3 1 5

1316 WILLIAM B. JOYNER AND ALBERT T. F. CHEN

r e f e r t o a s t h e " e q u i v a l e n t l i n e a r m e t h o d " ( I d r i s s a n d S e e d , 1 9 6 8 ; S c h n a b e l e t a l . ,

1 9 7 2 ) . I t i s b a s e d o n t h e a s s u m p t i o n t h a t t h e r e s p o n s e c a n b e a p p r o x i m a t e d b y t h e r e -

s p o n s e o f a l i n e a r m o d e l w h o s e p r o p e r ti e s a r e c h o s e n i n a c c o r d w i t h t h e a v e r a g e s t r a i n

t h a t o c c u r s a t e a c h d e p t h i n t h e m o d e l d u r i n g e x c i t a t i o n . T h e c a l c u l a t i o n i s i t e r a t i v e .

F i r s t , t r i a l v a lu es fo r av e rag e s t r a in a re ch o sen , th en so i l p ro p er t i e s a re d e te rmin edi n a c c o r d a n c e w i t h t h e t r i a l v a l u e s o f s t r a in , a n d f i n a l l y t h e r e s p o n s e o f t h e m o d e l i s

~ ca lculat ed . I f t h e c a lcu la t ed s t r a in s d i f f e r b y to o m u ch f ro m th e t r i a l v a lu es , t h e cy c le is

r e p e a t e d .

Id r i s s an d S eed (1 96 8) t e s t ed th e v a l id i ty o f th e eq u iv a len t l in ea r a s su m p t io n b y

c o m p a r i n g t h e e q u i v a l e n t l i n e a r s o l u t i o n f o r a t e s t c as e w i t h t h e s o l u t i o n o b t a i n e d u s i n g

a b i l in ea r s t r e s s - st r a in re l a t io n sh ip . R e cen t ly , a n u m b e r o f w o rk e r s h av e d esc r ib ed o th e r

m e t h o d s t h a t u t i l iz e n o n l i n e a r s t r e s s- s tr a i n r e la t i o n s h ip s . T h e R a m b e r g - O s g o o d r e l a t io n -

sh ip w as u sed b y S t ree te r e t a l . (19 74 ), b y C o n s tan to p o u lo s (1 97 3) , an d b y F acc io l i e t a l .

( 1 9 7 3 ) . A n e l a s t o - p l a s t i c r e l a t i o n s h i p w a s u s e d b y P a p a s t a m a t i o u ( w r i t t e n c o m m u n i -

ca t io n ) .A d i f f e r e n t m e t h o d , d e s c r i b e d b y C h e n a n d J o y n e r (1 97 4) a n d e l a b o r a t e d h e r e , i s

b a s e d o n a r h e o l o g i c a l m o d e l p r o p o s e d b y I w a n ( 1 9 6 7 ) . T h i s m o d e l l e a d s t o a v e r y

s imp le a n d e f f ic i en t m e th o d o f ca l cu la t io n an d o f fe rs co n s id e rab le f l ex ib i li t y fo r in co r -

p o r a t i n g l a b o r a t o r y d a t a o n s o il b e h a v i o r .

T h e f o l l o w i n g s e c t io n o f t h is r e p o r t d e s c ri b e s t h e I w a n m o d e l a n d i t s im p l e m e n t a t i o n .

A f t e r t h a t , a b o u n d a r y c o n d i t i o n ( P a p a s t a m a t i o u , w r i t t e n c o m m u n i c a t i o n ) is p r e s e n te d

w h i c h t a k e s a c c o u n t o f f i ni te r ig i d i ty in t h e e l a st ic s u b s t r a t u m . N e x t , t h r e e a l t e r n a t e w a y s

a r e d e s c r ib e d f o r i n t e g r a t in g th e e q u a t i o n s o f m o t i o n . A n e x a m p l e is t h e n d e v e l o p e d w i t h

m a t e r i a l p r o p e r t i e s a s s i g n e d a s a f u n c t i o n o f d e p t h i n a c c o r d a n c e w i t h t h e l a b o r a t o r y

resu l t s o f H ard in a n d D rn ev ich (1 97 2a , 1 97 2b ). T h e e x am p le is f i r s t u sed to co mp are th ed i f f e re n t t e c h n i q u e s o f in t e g r a t i o n . T h e n , t h e e x a m p l e is u s e d t o c o m p a r e r e s u lt s f r o m

t h e e q u i v a le n t li n ea r m e t h o d w i t h t h o s e f r o m t h e n o n l i n e a r m e t h o d .

I w a n ( 19 67 ) e x t e n d e d h is m o d e l t o t h r e e d i m e n s i o n s . A p p l i c a t i o n o f th e e x t e n d e d

I w a n m o d e l t o g r o u n d r e s p o n s e c a l c u l a ti o n s i n t w o d i m e n s i o n s is d e s c ri b e d i n a s e p a r a t e

p a p e r n o w i n p r e p a r a t i o n .

CONSTITUTIVE RELATION

T h e b a s i c r e q u i r e m e n t f o r a s o l u t i o n t o t h e p r o b l e m i s a c o n s t i tu t i v e r e l a t i o n - - i n

s i m p le t e r m s w e n e e d a r u le t h a t w i l l t e ll e a c h s o i l e l e m e n t h o w t o f i n d i ts w a y a r o u n d t h es t r e s s - s t r a i n p l a n e . F o r t h i s p u r p o s e w e a d o p t e d a m o d e l ( F i g u r e 1 ) p r o p o s e d b y I w a n

( 19 67 ). I t i s c o m p o s e d o f s im p l e l in e a r s p r in g s a n d C o u l o m b f r i c t io n e l e m e n t s a r r a n g e d

a s s h o w n . T h e f r i c t io n e l e m e n t s r e m a i n l o c k e d u n t i l t h e s tr e ss o n t h e m e x c e ed s t h e y i e ld

s t re ss Y v T h e n , th ey y ie ld , an d th e s t r es s ac ro ss th em d u r in g y ie ld in g i s eq u a l to th e

y ie ld s tr e s s . G en era l ly , t h e y ie ld s t re s s o f th e f i r s t e l em en t Y 1 i s se t t o ze ro . B y ap p ro p r i a t e

sp ec i f i ca t io n o f th e sp r in g co n s tan t s G~ an d th e y ie ld s tr e s ses Y ~, w e ca n m o d e l a v e ry

b r o a d r a n g e o f m a t e r i a l b e h a v i o r a s d i c t a t e d b y l a b o r a t o r y e x p e r i m e n t s . T h e a c c u r a c y

o f th e m o d e l i n g d e p e n d s u p o n t h e n u m b e r N o f e l e m e n t s u s e d. W e h a v e f o u n d i t p o s s ib l e

t o u s e l a r g e n u m b e r s e c o n o m i c a l l y . F o r o u r t y p i c a l p r o b l e m N i s 5 0 ; w e h a v e g o n e a s

h i g h a s 1 0 0 w i t h o u t u n r e a s o n a b l e i n c r e a s e s i n c o m p u t i n g t i m e . T h e r e i s o n e m o d e l o f

t h e k i n d d i a g r a m m e d i n F i g u r e 1 f o r e a c h s o il l a y e r i n t h e s y s t e m .

I n p r a c t i c e , t h e m o d e l i s n o t d e s c r i b e d b y s p e c i f y in g t h e G i d i re c t ly . I t t u r n s o u t t h a t

i t is m o r e e f fi c ie n t t o w o r k i n t e r m s o f th e t a n g e n t m o d u l u s o f t h e w h o l e m o d e l . W e

in d ex th e e l em en t s in o r d e r o f in c reas in g y ie ld s tr e s s . A t an y g iv en t ime , a l l t h e e l eme n t s

u p t o a c e r t a i n i n d e x n u m b e r w i ll be y i e l d i n g a n d a l l t h o s e a b o v e w i ll n o t . ( A p r o o f o f

CALCULATION O F NONLINEAR GR OU ND RESPONSE IN EARTHQUAKES 1317

t h is p r o p o s i t i o n i s gi v e n i n A p p e n d i x A . ) W e d e n o t e b y t h e l e t t e r m t h e i n d e x o f t h e

e l e m e n t w i t h t h e l a r g e s t y ie l d s tr e ss o f a ll t h e e l e m e n t s t h a t a r e y i e ld i n g . T h e n t h e t a n g e n t

m o d u l u s Sm o f t h e m o d e l d e p e n d s o n l y u p o n m . I t i s r e l a t e d t o t h e G i b y t h e e q u a t i o n

W e e v a l u a t e t h e Sm f r o m t h e i n i t i a l l o a d i n g c u r v e i n s i mp l e s h e a r . A l t e r n a t i v e l y , w e

c o u l d u s e t h e l o c u s o f th e e n d p o i n t s o f h y s t er e s is l o o p s f o r m e d i n c y c li c l o a d i n g t e s t s a t

d i f fe r e n t p e a k s t r a in l e v e ls . T h e t w o c u r v e s a r e t h e s a m e i n s o f a r a s t h e m o d e l i s c o n c e r n e d .

T h e s e c o n d c u r v e is u s e d b y H a r d i n a n d D r n e v i c h ( 1 97 2 a , 1 9 7 2b ) i n d e s c r ib i n g t h e i r

e x p e r i m e n t a l r e s u l t s o n t h e s t r e s s - s t ra i n b e h a v i o r o f so i ls .

G 2 G i G N

.... .... 7 2

' (2 Y i Y N

F r o . 1 . M o d e l u s e d f o r c o n s t i t u t iv e r e l a ti o n s h i p . M o d e l c o n s i s t s o f s i m p l e e l a st i c s p r i n g s w i t h s p r i n gc o n s t a n t s G ~ a n d C o u l o m b f r i c t i o n e l e m e n t s w i t h y i el d s t r e s s e s Y~.

G i v e n t h e i n i ti a l l o a d i n g c u r v e , w e p r o c e e d b y s e l e c t in g a s e t o f y i e ld st r es s e s Y ,, ( m = 1 ,

N + 1 ). T h e Y i a r e c h o s e n t o c o v e r t h e r a n g e o f st re s s t h e s y s t e m is e x p e c t e d to e n c o u n t e r

a n d a r e d i s t r i b u t e d s o t h a t t h e i n i ti a l l o a d i n g c u r v e c a n b e f a i t h f u ll y r e p r e s e n t e d . F r o m

t h e i n i t i a l l o a d i n g c u r v e a s e t o f s h e a r s t r a i n v a l u e s e m ( m = 1 , N + 1 ) is o b t a i n e d c o r r e s -

p o n d i n g t o t h e s t re s s v a lu e s Ym. T h e t a n g e n t m o d u l i S ,, a r e t h e n g i v e n s i m p l y b y

Y m + l - ] Z m

em+l --ern

T o s i m p l i f y t h e c o m p u t a t i o n s , t h e s t r e s s a n d s t r a i n a r e n o r m a l i z e d i n t h e m a n n e r

u s e d b y H a r d i n a n d D r n e v i c h ( 1 9 7 2b ) . S tr e ss is n o r m a l i z e d b y m u l t i p l y i n g b y 1 /T . . .

1318 W I L L I A M B . J O Y N E R A N D A L B E R T T . F . C H E N

where Zmax is t h e s h e a r s t re s s a t f a i lu r e , a n d s t r a i n i s n o r m a l i z e d b y m u l t i p l y i n g b y

Gm,x / z . . . w h e r e G i n ,x i s t h e l o w- s t r a i n m o d u l u s . T h e s t r e s s - s t r a in c u r v e n o r m a l i z e d

i n t h is w a y h a s a s t re s s l i mi t o f 1 .0 f o r h i g h s t r a i n a n d a s l o p e a t t h e o r i g i n o f 1 . 0.

F o r t h e e x a m p l e s s h o w n l a t e r i n t h is r e p o r t , t h e s a m e n o r m a l i z e d i n it ia l l o a d i n g c u r v e

a n d t h e r e f o r e t h e s a me s e t o f n o r m a l i z e d S , , a n d Y ,, a r e u s e d f o r a ll o f t h e s o i l l a y e r s i nt h e m o d e l . T h e d i f f e r en c e s i n s o il b e h a v i o r f r o m o n e l a y e r t o a n o t h e r r e s u l t f r o m d i f-

f e r e n c e s i n t h e v a l u e s o f Z m,x a n d G~ , x a s s i g n e d t o t h e d i f f e r e n t l ay e r s . T h e f o l l o w i n g

s e t o f v a l u e s w a s a d o p t e d f o r Y ~

Ym - - - - 0.025(0"5) 6 - 'n

Y ~ = 0 . 0 2 5 ( m - 5 )

I . O -

- 1 . 0

2 < m < 6

7 < m < 4 4

Y r n = 1 " 0 - 0 ' 0 2 5 ( 0 " 5 ) m - 4 4 45 < rn < 51

-4.o -2'o ' o'o 2'.o ' 41o

REDUCED STRAIN

FIG. 2. Hysteresis loops for the m odel shown in Figure l. Stress and strain are scaled so that the m axi-mu m stress and the low -strain mo dulus are unity.

A h y p e r b o l i c in i ti a l l o a d i n g c u r v e ( H a r d i n a n d D r n e v i c h , 1 9 72 b ) w a s u s e d a n d n o r m a l i z e d

s t ra i n w a s e x p r e s s e d i n t e rm s o f n o r m a l i z e d s t r es s b y t h e e q u a t i o n

Yme m - - 1 - Y , , "

T h e b a s i c m e t h o d , h o w e v e r , d o e s n o t d e p e n d o n t h e h y p e r b o l i c r e l a t i o n s h i p ; a p u r e l y

e m p i r i c a l s tr e s s- s tr a in c u r v e d e r i v e d f r o m l a b o r a t o r y m e a s u r e m e n t s c o u l d b e u s e d j u s t

as wel l .

T h e t y p e o f h y s t e r es i s l o o p s t h a t t h is m o d e l p r o d u c e s i s s h o w n i n F i g u r e 2 , w h i c h

i l l u s tr a t e s t h e b e h a v i o r o f a s i n g le s o i l l a y e r s u b j e c t t o c y c l i c l o a d i n g o f i n c r e a s i n g a m p l i -

t u d e . T h e " r e d u c e d s t r e s s " a n d " r e d u c e d s t r a i n " p l o t t e d i n F i g u r e 2 a r e t h e s t r e s s a n d

s t ra i n n o r m a l i z e d in t h e w a y u s e d b y H a r d i n a n d D r n e v i c h ( 1 9 72 b ).

D e n o t i n g t h e u p p e r b r a n c h o f a c l o s e d h y s t e re s i s l o o p a s t h e " r e l o a d i n g c u r v e , " i t c a n

r e a d i l y b e s h o w n t h a t f o r a n y h y s t er e s is l o o p s g e n e r a t e d b y t h e I w a n m o d e l t h e r e l o a d i n g

c u r v e i s s i mp l y t h e i n i t i a l l o a d i n g c u r v e wi t h i t s o r i g i n t r a n s l a t e d a n d i t s s c a l e e x p a n d e d

b y a f a c t o r o f t w o b o t h v e r t ic a l l y a n d h o r i z o n t a l l y . T h i s r u l e f o r d e s c ri b i n g h y s te r e s i s

l o o p s is c a l l e d M a s i n g ' s c r i t e r i o n ( Ma s i n g , 1 9 2 6 ; f o r a f u r t h e r d i s c u s s io n o f M a s i n g -

t y p e s y s te m s s ee N e w m a r k a n d R o s e n b l u e t h , 1 97 1, p . 1 6 3, 34 5 - 3 46 ) . T h e I w a n m o d e l

c a n b e u s e d t o r e p r e s e n t , t o a n y d e s i r e d a c c u r a c y , t h e b e h a v i o r o f a n y m a t e r i a l w h o s e

CALCULATION OF NONLINEAR GROUND RESPONSE IN EARTHQUAKES 1319

hysteresis loops satisfy the Masing criterion and do not depend on the number of cycles

of loading. There is some evidence that soils conform at least approximately to the

Masing criterion. Hardin and Drnevich (1972b), commenting on experimentally recorded

hysteresis loops for a wide variety of soils, stated that the slope of the stress-strain

curves immediately after load reversal was approximately equal to the low-strain modulus,independent of the strain amplitude of the loop. Even if a soil does not exactly meet the

Masing criterion, its behavior might still be approximately represented by an Iwan model.

The best approach would probably be to select the hysteresis loop corresponding to the

strain amplitude for which faithful representation was most desired. The reloading curve

from that loop could then be used to evaluate the constants of the model. Even if the

shape of the loop changed with the number of loading cycles, the Iwan model might

still be used, provided the changes were not too drastic. In evaluating the constants of

the model, one would simply use the loop generated after a certain number of cycles.

The number would be chosen to be representative of the number expected during the

postulated earthquake.It should be noted that the rheological model described here has no viscous damping,

and as a result the stress depends on the strain (and strain history) but not on the strain

rate. The energy dissipation per cycle, therefore, does not depend upon the frequency.

There would be no difficulty in adding a dashpot in parallel with the model. As a matter

of fact, that option was allowed in one of our computer programs (Chen and Joyner,

1974). We note, however, that the experimental data of Hardin and Drnevich (1972a,

p. 620) indicate very little or no effect of frequency on damping in soils for frequencies

between about 0.1 and 25 Hz, which covers the range of interest in engineering seismology.

The Masing criterion is satisfied by the elasto-plastic stress-strain relationship used

by Papastamatiou (written communication), by the bilinear relationship used by Idrissand Seed (1968), and by the Ramberg-Osgood relationship used by Streeter e t a l . (1974),

Constantopoulous (1973), and Faccioli e t a l . (1973). All of these relationships could

therefore be represented by a model of the Iwan type.

BOUNDARY CONDITION

In solving soil response problems of this kind it is common to prescribe the motion

at the base of the soil column. This is an approximation that allows no energy to be

radiated back into the underlying medium. The accuracy of the approximation depends

upon the contrast in rigidity across the boundary and upon the energy dissipation withinthe soil column. If the energy dissipation is small, multiple reflection within the soil

column will give rise to resonances whose strength will depend upon the rigidity con-

trast. Prescribing the motion at the base of the soil column is the equivalent of assuming

infinite rigidity in the underlying medium.

To take account of the finite rigidity of the underlying medium we use a method

suggested by Papastamatiou (written communication). That method allows us to satisfy

the boundary conditions exactly for a wave vertically incident on the boundary from

below. The approach is similar to that of Lysmer and Kuhlemeyer (1969).

We have assumed a vertically incident plane shear wave in the underlying elastic

medium. That assumption allows us to obtain an expression for the shear stress in theunderlying medium at the boundary in terms of the particle velocity of the incident

wave and the particle velocity on the boundary. Particle displacement for the incident

wave is a function of depth x and time t given by

u~ = v , (x+ v , t )

132 0 WILLIAM B. JOYNER AND ALBERT T. F. CHEN

w h e r e v s i s t h e s h e a r v e l o c i t y i n t h e u n d e r l y i n g m e d i u m . F o r t h e r e f le c t e d w a v e , p a r t i c l e

d i s p l a c e m e n t i s g iv e n b y

U R = g . ( x - v / ) .

T h e s h e a r s tr e ss a t t h e b o u n d a r y i s

( a v , + e v .

w h e r e / @ is t h e r i g id i ty o f t h e u n d e r l y i n g m e d i u m .

c~Ux 1- v ,

w h e r e V~ i s t h e p a r t i c l e v e l o c i t y f o r t h e i n c i d e n t w a v e a n d

- VR.OX V

T h e p a r t i c le v e l o c it y o n t h e b o u n d a r y i s t h e s u m o f t h e v e l o c i ty o f t h e i n c i d e n t a n d

r e f l e c t e d w a v e s ,

v , , = v R + v , .

S o l v i n g f o r VR a n d s u b s t i t u t i n g i n t h e e q u a t i o n f o r s h e a r s t r e s s g i v es

o r , e q u i v a l e n t l y ,

Z B = / ~ e ( 2 I 1 1 - - V B )Us

z ~ = p ~ G ( 2 V I - V B) (1)

w h e r e p e i s th e d e n s i t y o f t h e u n d e r l y i n g m e d i u m .

TECHNIQUES OF INTEGRATION

G i v e n a c o n s t i t u ti v e r e l a t i o n a n d a b o u n d a r y c o n d i t i o n a t t h e b a s e o f th e s y s t e m ,

w e n e e d o n l y in t e g r a t e th e e q u a t i o n s o f m o t i o n s t ep b y s t e p in t i m e t o o b t a i n t h e m o t i o n s

a t t h e s u r fa c e . W e h a v e e x p e r i m e n t e d w i t h t h r e e d i f f e r e n t t e c h n i q u e s f o r i n t e g r a t i n g t h e

e q u a t i o n s o f m o t i o n , a n i m p l i c i t t e c h n i q u e , a n e x p l i c it t e c h n i q u e , a n d i n t e g r a t i o n a l o n g

c h a r a c t e r i s t i c s .

T h e i mp l i c i t t e c h n i q u e i s d e s c r i b e d i n d e t a i l e l s e w h e r e ( C h e n a n d J o y n e r , 1 9 7 4 ) . I t i s

b a s e d o n t h e a p p r o a c h u s e d b y I d r is s a n d S e e d (1 9 67 ) f o r a n a l y z i n g a s y s t e m o f so i l l a y e rs

w i t h b i l i n e a r s t r e s s - s t r a in r e l a t i o n s h i p s . B r i e f ly , t h e s o i l p r o f i l e is d i v i d e d i n t o l a y e r s , a n d

t h e m a s s o f e a c h la y e r is l u m p e d e q u a ll y a t th e t o p a n d b o t t o m . T h e e q u a t i o n s o f m o t i o n

a r e w r i t t e n i n th e f o r m o f a s e t o f s i m u l t a n e o u s e q u a t i o n s w h i c h is s o l v e d o n c e f o r e a c h

t i m e s t e p . T h e r h e o l o g i c a l m o d e l u s e d w i t h t h e i m p l i c i t t e c h n i q u e i s a n I w a n m o d e l o f

t h e t y p e s h o w n i n F i g u r e 1 i n p a r a ll e l w i t h a d a s h p o t t o g i ve v i s c o u s d a m p i n g . T h e d a s h -

p o t w a s i n c l u d e d t o p e r m i t c o m p a r i s o n w i t h t h e e a r li e r w o r k o f Id r i ss a n d S e e d ( 19 6 7) .

C A L C U L A T I O N O F N O N L I N E A R G R O U N D R E SP O NS E IN E A R T H Q U A K E S 1321

F o r t h e e x a m p l e p r e s e n t e d i n t h i s r e p o r t , w e u s e a d a s h p o t c o e f f ic i e n t s m a l l e n o u g h s o t h a t

th e e f f ec t o f t h e d as h p o t i s neg l igib l e. We d o th i s b ecau se o f t h e ev id en ce c i t ed p re -

v i o u s l y i n d i c a t i n g t h a t d a m p i n g i n s o i ls is n o t s i g n if i c an t ly d e p e n d e n t o n f r e q u e n c y .

T h e b o u n d a r y c o n d i t i o n a l l o w i n g f in i te r i g id i t y i n t h e s u b s t r a t u m w a s n o t i m p l e m e n t e d

i n t h e c o m p u t e r p r o g r a m f o r t h e i m p l i c i t t e c h n i q u e a l t h o u g h i t c o u l d p r o b a b l y b e d o n ew i t h o u t d i f fi cu l ty .

I n t h e e x p l ic i t t e c h n i q u e , a l so , t h e m a s s o f t h e s o i l l a y e rs is l u m p e d a t t h e t o p a n d

b o t t o m o f t h e l a y e rs . T h e s t a t e o f t h e s y s t e m is d e s c r i b e d i n t e r m s o f t h e p a r t i c le v e l o c i t y

V i a t t h e to p o f each l ay e r i an d th e n o rm al i zed sh ea r s t r e s s s i i n each l ay e r i. T h e l ay e r s

a r e n u m b e r e d f r o m t h e t o p d o w n . T h e c o m p u t a t i o n s p r o c e e d s t e p b y s te p i n s p a c e f r o m

t h e b o t t o m u p a n d s t e p b y s t ep i n t i m e . A s i m p li f ie d o u t li n e o f th e s e q u e n c e o f c o m -

p u t a t i o n s i s a s f o ll o w s :

1 . A t e a c h l a y e r b o u n d a r y t h e p a r t i c le v e l o c i t y V i i s k n o w n a t t i m e t , a n d i n e a c h

l a y e r th e n o r m a l i z e d s h e a r s t r e ss s~ i s k n o w n a t t i m e ( t - A t /2 ) , w h e r e A t is t h e t i m e i n c r e -

m e n t .

2 . T h e c h a n g e i n n o r m a l i z e d s h e a r s t r a i n A e~ i n l a y e r i i s c o m p u t e d f o r a t i m e i n t e r v a l

A t c e n te r e d a b o u t t , u s i n g t h e f o l lo w i n g f o r m u l a

A e i = ( V i + l - V i ) k i A t / A x i

w h ere A x~ i s t h e th i ck n e ss o f lay e r i an d k~ i s a n o rm al i z in g co n s t a n t w h ich i s eq u a l t o

t h e r a t i o Gmax/Zmax o r th e l ay e r .

3 . T h e v a l u e s o f A e~ a r e u s e d i n c o n j u n c t i o n w i t h t h e r h e o l o g i c a l m o d e l , i n a m a n n e r

d e s c r ib e d s u b s e q u e n t l y , t o u p d a t e t h e n o r m a l i z e d s h e a r s t re s s t o t i m e ( t ÷ A t /2 ) .

4 . T h e v a l u e s o f n o r m a l i z e d s t r e ss i n t h e la y e r s a b o v e a n d b e l o w a l a y e r b o u n d a r y

a t t i m e ( t + A t / 2 ) a re u s e d t o u p d a t e t h e p a r t ic l e v e l o c it y a t t h e b o u n d a r y t o t i m e ( t + A t )

a c c o r d i n g t o t h e fo l l o w i n g f o r m u l a , b a s e d o n N e w t o n ' s s e c o n d l a w

V i + l ( t + A t) = Vi+a(t ) + [('~max)i+lSi+l - (Tmax) iSi] At /m i+ 1 .

w h e r e m i + 1 i s th e t o t a l m a s s p e r u n i t a r e a l u m p e d a t t h e t o p o f l a y e r ( i + 1). T h e f a c t o r s

o f Zmaxco n v er t n o rm al i zed s t r e s s t o ac tu a l s t re s s .

I n s t e p 4 s p ec i a l t r e a t m e n t i s r e q u i r e d f o r t h e l a y e r b o u n d a r i e s a t t h e t o p a n d b o t t o m

o f th e s y s t e m . A t t h e t o p t h e s t re s s a b o v e t h e b o u n d a r y i s z e ro a n d t h e p a r t ic l e v e l o c i t y i su p d a t e d b y

V l ( t + A t) = V l ( t ) + (Zm,x)lS At/m ~.

I f th e r e i s a t o t a l o f N l a y e r s i n t h e s y s t e m , th e i n d e x o f t h e b o t t o m b o u n d a r y i s

( N + 1). W e c a n w r i te

V u + l ( t + A t) = V u + l ( t) + A u + l ( t + A t / 2 ) A t (2 )

w h e r e A u + a is t h e a c c e l e r a t i o n a t t h e b o u n d a r y . N e w t o n ' s s e c o n d l a w g i ve s

A N + ( t + At/2) = [z . ( t + At /Z) - - ('Cmax)NS.~(t÷ A t /2 ) ] /m N+1 (3 )

where -cn i s t h e s h e a r s t r es s in t h e u n d e r l y i n g m e d i u m a t t h e b o u n d a r y . U s i n g e q u a t i o n

(1 ) w e m a k e t h e a p p r o x i m a t i o n

zB ( t+ A t /2 ) ~ p E v s [ 2 V i ( t + A t ) - V N + I ( t + At)]. (4)

S u b s t i tu t i n g f r o m e q u a t i o n ( 4) i n t o e q u a t i o n ( 3 ) a n d f r o m e q u a t i o n ( 3) i n t o e q u a t i o n ( 2 )

and so lv ing f o r VN + 1 t + At ) g ives

VN+I(t+A t ) = [ 1 . 0 - 1 . O / ( p E v s A t / r n N + 1 + 1.0)] .

{2 V , ( t + A t ) - [ ( Z m J N S N ( t + At / 2 )

- V N + I ( t ) / ( A t / m N + I ) ] / ( p E v ~ ) } . ( 5 )

T h e a p p r o x i m a t i o n m a d e i n e q u a t i o n ( 4 ) l e ad s t o t h e f o r m o f e q u a t i o n ( 5) su c h t h a t t h e

l i mi t o f VN+a a s ( p E v s ) g o e s t o i n f i n i t y i s e q u a l t o 2 V I , a s i t s h o u l d b e . O t h e r a p p r o x i -

m a t i o n s , a l t h o u g h t h e y m a y a p p e a r r e a s o n a b l e , d o n o t l e a d t o t h a t l i m i t i n g c o n d i t i o n

a n d r e s u l t i n i n s ta b i l i t y f o r l a r g e ( p ~ v s ) , i n wh i c h t h e e r r o r s a r e ma g n i f i e d a t e a c h s t e p i n

t h e c o m p u t a t i o n .

I n s t e p 3 w e u s e t h e I w a n m o d e l t o d e t e r m i n e t h e s t r e s s c h a n g e s f r o m t h e s t r a i n

c h a n g e s . E a c h s o il la y e r i s r e p r e s e n t e d b y a m o d e l o f th e t y p e s h o w n i n F i g u r e 1 . T h r o u g h -o u t t h e c o m p u t a t i o n s w e k e e p t r a c k o f t h e s t re s s i n e a c h o f t h e sp r in g s o f e a c h m o d e l

a n d t h e i n d e x m o f t h e e l e m e n t w i t h t h e h i g h e s t y ie l d i n g s t re s s a m o n g a l l e l e m e n t s t h a t

a r e c u r r e n t l y y i el d i n g . W h e n we o b t a i n t h e v a l u e o f Ae~ f o r a l a y e r , we f ir s t c h e c k t o s e e

i f t h e d i r e c t i o n o f d e f o r m a t i o n h a s b e e n r e v e r s e d f r o m t h e p r e v i o u s c y c l e. I f i t h a s , w e

s e t m e q u a l t o o n e . W e t h e n u s e th e t a n g e n t m o d u l u s Sm t o c o m p u t e a t r i a l v a l u e o f

A s i = S m A e i .

W e t h e n c h e c k t o s e e i f t h e n e w s t re s s ( s i + As i) w i ll c a u s e t h e ( m + 1 ) e l e m e n t t o y i e l d .

T h a t w i ll h a p p e n i f

[ s , + A s , - H , . + , . , I - - > Y m + l

w here Hm+ 1 , ~ is the s t r e s s in the (m + 1 ) sp r ing o f l aye r i. I f the (m + 1 ) e lem ent do es y ie ld ,

we s e t t h e s t re s s s l t o t h e v a l u e w h e r e y i e l d i n g b e g i n s a n d c o r r e c t t h e s t re s s i n c r e m e n t .

W e t h e n s u b t r a c t f r o m A e~ t h e s t r a in i n c r e m e n t c o r r e s p o n d i n g t o t h e c o r r e c t e d s t r es s

i n c r e m e n t a n d i n c r ea s e t h e v a l u e o f m b y o n e . T h i s p r o c e s s i s r e p e a t e d u n t i l w e r e a c h a

v a l u e o f m s u c h t h a t t h e ( m + 1) e l e m e n t d o e s n o t y i e ld .

F o r i n t e g r a t i o n a l o n g c h a r a c t e r i s t i c s w e f o l l o w e d t h e a p p r o a c h o f S t r e e t e r e t a l .

(1 97 4). T h e I w a n m o d e l i s u s e d i n t h e m a n n e r ju s t d e s c r i b e d t o k e e p t r a c k o f t h e s t re s s -s t r ai n h i s t o r y o f e a c h l ay e r a n d p r o v i d e a v a l u e o f t h e ta n g e n t m o d u l u s f o r e a c h l a y e r

a t e a c h ti m e s t e p . T h e t a n g e n t m o d u l u s i s u s e d t o o b t a i n t h e s t r a i n - d e p e n d e n t s h e a r - w a v e

v e l o c it y , w h i c h i s re q u i r e d f o r i n t e g r a t i o n a l o n g c h a r a c te r i st i cs . T h e b o u n d a r y c o n d i t i o n

a l l o wi n g f i n i te r ig i d i t y i n t h e s u b s t r a t u m i s i n c l u d e d .

E X A M P L E

F o r o u r e x a m p l e w e c h o s e t h e s o il p r o fi l e i l lu s t r a te d i n F i g u r e 3 , r e p r e s e n t i n g a

2 0 0 - m s e c t i o n o f fi r m a l lu v i u m c o n s i s ti n g p r e d o m i n a n t l y o f c o h e s iv e m a t e r ia l . T h e k e y

m a t e r i a l p a r a m e t e r s a r e t h e m a x i m u m s h e a r s t r e s s Z m aX a n d t h e l o w - s t r a in m o d u l u sG m a x. B o t h p a r a m e t e r s a r e s t r o n g l y d e p e n d e n t u p o n t h e s t a t e o f e ff e ct iv e s tr e ss p r i o r t o

s e i s m i c e x c i t a t i o n , w h i c h i n t u r n d e p e n d s u p o n d e p t h . A n a t t e m p t w a s m a d e t o a s s i g n

r e a li s ti c v a l u e s t o t h e s e p a r a m e t e r s a s f u n c t i o n s o f d e p t h u s i n g t h e m e t h o d s o f H a r d i n a n d

D r n e v i c h ( 1 97 2 a , 1 9 72 b ) w i t h m i n o r m o d i f i c a t i o n s . A d e n s i t y o f 2 . 05 g m / c m 3 w a s

a s s u m e d a n d t h e l o w - s t r a i n s h e a r m o d u l u s w a s c o n v e r t e d t o s h e a r v e l o c i ty f o r p l o t t i n g

C A L C U L A T I O N O F N O N L I N E A R G R O U N D R E S PO N S E I N E A R T H Q U A K E S 1323

i n F i g u r e 3 . A p a s t c o n s o l i d a t i o n v e r t i c a l st r e s s o f 2 .9 4 b a r s w a s a s s u m e d . A s a r e s u l t ,

t h e m a t e r i a l w a s o v e r c o n s o l i d a t e d a b o v e a d e p t h o f 29 m , c a u s i n g k i n k s a t t h a t d e p t h i n

t h e t w o c u r v e s o f F i g u r e 3 . T h e m a t e r i a l w a s a s s i g n e d a p l a s t i c i ty i n d e x o f 20 p e r c e n t ,

a n d c o m p l e t e w a t e r s a t u r a t i o n w a s p r e s u m e d a t a ll d e p t h s .

I n t h e n o r m a l l y c o n s o l i d a t e d p a r t o f t h e s e c t i o n Zmax w a s a s s u m e d p r o p o r t i o n a l t ove r t i ca l e f fec t ive s t re ss ,

Zmax = CsPve

w h e r e Pve i s th e v e r t i c a l e f f e c ti v e s t r es s p r i o r t o s e i s mi c e x c i t a t i o n . P~e c a n r e a d i l y b e

c a l c u l a t e d g i v en t h e d e n s i t y a n d t h e d e p t h . T h e c o e f f ic i e n t C~ is e v a l u a t e d o n t h e a s s u m p -

t i o n t h a t n o p o r e - p r e s s u r e c h a n g e o c c u r s d u r i n g s h e a r u s in g t h e e q u a t i o n

w h e r e ~ is t h e a n g l e o f d r a i n e d s h e a r r e s i s ta n c e a n d K o i s t h e c o e f f ic i e n t o f e a r t h p r e s s u r e

D E P T H

( M E T E R S )

0 5 0 0

2 0 00 I 0 .0

V E L O C I T Y ( M E T E R S PE R S E C O ND )

I 0 0 2 0 0 3 0 0 4 0 0

2 . 0 4 . 0 6 . 0 8 . 0

S T R E S S ( B A R S )

F I o . 3 . D y n a m i c p r o p e r t i e s f o r s o il p ro f i le u s e d i n s a m p l e p r o b l e m .

a t r es t. T h e e q u a t i o n i s o b t a i n e d f r o m a n e q u a t i o n o f H a r d i n a n d D r n e v i c h ( 1 9 72 b ,

p . 6 7 3 ) b y a s s u m i n g t h a t t h e c o h e s i o n c f o r n o r m a l l y c o n s o l i d a t e d m a t e r i a l i s z e r o

( T e r z a g h i a n d P e c k , 1 96 7, p . 1 12 ). E s t i m a t i o n o n t h e b a s i s o f p l a s ti c i t y i n d e x g i v e s a

v a l u e o f 3 1 ° f o r q5 ( T e r z a g h i a n d P e c k , 1 96 7, p . 1 12 ) a n d 0 . 5 5 f o r K o ( B r o o k e r a n d I r e l a n d ,

1 96 5). C a r r y i n g o u t t h e c o m p u t a t i o n s g i v e s a v a l u e o f 0. 3 3 f o r C s.

T h e a s s u m p t i o n o f n o p o r e p r e s s u r e c h a n g e w il l t e n d t o l e a d t o a n o v e r e s t i m a t e o f

t h e s t r e n g t h a n d t h e r e b y o f th e c a p a c i t y o f t h e m a t e r i a l t o t r a n s m i t s h e a r . N o r m a l l y

c o n s o l i d a t e d c l a y s t e n d t o c o n t r a c t w i th s h e a r , i n c r e a s in g t h e p o r e p r e s s u r e a n d r e d u c i n gt h e s t r e n g t h .

F o r t h e o v e r c o n s o l i d a te d p a r t o f th e s e c t io n w e a s s u m e d t h a t w e c o u l d r e p r e s e n t t h e

v a r i a t i o n o f ' C m a x b y a n e q u a t i o n o f t h e f o r m

"Cmax = CsPve(OCR) T

w h e r e O C R i s t h e o v e r c o n s o l i d a t i o n r a t i o i n t e r m s o f v e r t ic a l e f f ec t iv e st r es s e s. A n a l y s i s

o f u n d r a i n e d s t r e n g t h d a t a f r o m L a d d a n d E d g e r s ( 1 9 72 ) g a v e a v a l u e o f 0 .7 5 f o r T .

H a r d i n a n d B l a c k ( 19 69 ) a n d H a r d i n a n d D r n e v i c h ( 1 97 2 b) h a v e s h o w n t h a t f o r a

w i d e v a r i e t y o f u n d i s t u r b e d c o h e s i v e so i ls , a n d a l s o sa n d s , t h e l o w - s t r a i n s h e a r m o d u l u s

G m a x

i s g i v e n b y a n e q u a t i o n w h i c h c a n b e r e w r i t t e n i n th e f o r m

Gmax = C6P,nel/2(OCRrn)

w h e r e P ine i s t h e m e a n e f f e c t iv e s t re s s , OCRm i s t h e o v e r c o n s o l i d a t i o n r a t i o i n t e r m s o f

m e a n e f fe c ti v e s t re s s e s, C G i s a c o n s t a n t d e p e n d i n g o n v o i d r a t i o , a n d K d e p e n d s o n

p l a s t i c it y in d e x . F o r a P I o f 2 0 p e r c e n t K h a s a v a l u e o f 0 .1 8 .

I t i s m o r e c o n v e n i e n t i n t h e p r e s e n t c a s e t o d e a l i n t e r m s o f v e r ti c a l s tr e s se s . U s i n g

K o v a l u e s f r o m B r o o k e r a n d I r e l a n d ( 19 65 ) t o c o n v e r t f r o m m e a n s t re s s t o v e r t ic a l

s t r e s s , i t w a s f o u n d t h a t t h e r e s u l t s f r o m t h e p r e c e d i n g e q u a t i o n c o u l d b e r e p r o d u c e d

w i t h i n a f e w p e r c e n t a g e p o i n t s o v e r t h e r a n g e o f 1 .0 t o 3 2 i n O C R b y t h e f o l l o w i n ge q u a t i o n

Gmax = cmevea /2(OCR) Q

w h e r e Q t a k e s o n a v a lu e o f 0 .2 8 f o r a P I o f 2 0 p e r c e n t . A v a l u e o f 0 .9 × 1 0 6 ( d y n e s /

c ru Z) 1/2 w a s c h o s e n f o r C m b a s e d o n s h e a r v e l o c i t y m e a s u r e m e n t s i n a 1 8 0 - m d r i ll h o l e

i n a l l u v i u m i n t h e S a n F r a n c i s c o B a y A r e a ( W a r r i c k , 1 97 4).

COMPARISON OF INTEG RATIONTECHNIQUES

F i v e d i f fe r e n t r u n s w e r e m a d e i n c o m p a r i n g t h e t h r e e d i f f e re n t te c h n i q u e s o f i nt e -g r a t i o n . T h e s o i l c o l u m n u s e d w a s t h e o n e i l l u s t r a t e d i n F i g u r e 3 a n d d e s c r i b e d i n t h e

p r e c e d i n g s e c t i o n . I n f i n i t e r i g i d i t y w a s a s s u m e d f o r t h e u n d e r l y i n g m e d i u m , b e c a u s e t h e

f in i te - ri g id i ty b o u n d a r y c o n d i t i o n h a d n o t b e e n i n c o r p o r a t e d i n t o t h e c o m p u t e r p r o g r a m

f o r t h e i m p l i c i t t e c h n i q u e . T h e i n p u t m o t i o n a t t h e b a s e w a s t h e f i rs t 2 0 s e c o f th e N 2 1 E

c o m p o n e n t o f t h e T a f t s t r o n g - m o t i o n r e c o r d m u l t i p l ie d b y a f a c t o r o f f o u r, g i v in g a

p e a k a c c e l e ra t i o n o f 0 .7 g a n d a p e a k v e l o c it y o f 67 c m / se c . ( T h e T a f t a c c e l e r o g r a m w a s

r e c o r d e d d u r i n g t h e 1 952 K e r n C o u n t y , C a l i fo r n i a , e a r th q u a k e . )

L a y e r t h ic k n e s s e s a n d t i m e i n c r e m e n t s f o r t h e fi v e r u n s a r e g i v e n i n T a b l e 1. F o r

i n t e g r a t i o n a l o n g c h a r a c t e r i s t i c s , o n c e t h e t i m e i n c r e m e n t i s c h o s e n , t h e l a y e r t h i c k n e s s e s

a r e d e t e r m i n e d b y t h e r e q u i r e m e n t t h a t t h e t r a v e l t i m e t h r o u g h a l a y e r a t t h e l o w - s t r a i ns h e a r v e l o c i t y b e e q u a l t o t h e t i m e i n c r e m e n t . F o r r u n C 1 a t i m e i n c r e m e n t o f 0.0 1 s e c

w a s c h o s e n i n o r d e r t o o b t a i n g o o d r e s ul ts f o r fr e q u en c i e s b e l o w a b o u t 1 0 H z . T h i s c o rr e s -

p o n d s t o a r e q u i r e m e n t o f 1 0 p o i n t s p e r w a v e l e n g t h f o r g o o d r e s o lu t i o n , a r e q u i r e m e n t

s u g g e st e d b y t h e w o r k o f B o o r e ( 19 72 ). T h e t i m e i n c r e m e n t w a s h a l v e d f o r r u n C 2 i n

TABLE 1

VALUES OF AX AND A t FOR METHODS USED

Symbol Technique Ax (m) At (sec)

I Implicit 2.4-2.5 0.0025E2 Ex plic i t 2 .25 0 .005E1 E xp lic it 1.8-4.5 0.01C1 Ch aracteris tics 1.8-4.5 0.01C2 Characteristics 0.8-2.2 0.005

C A L C U L A T I O N O F N O N L I N E A R G R O U N D R E SP O N SE I N EA R T H Q U A K E S 1325

o r d e r t o s h o w c o n v e r g e n c e . R u n E 1 w a s d o n e b y t h e e x p l i c i t t e c h n i q u e w i t h t h e s a m e

t i m e i n c r e m e n t a n d l a y e r t h i c k n e s s e s as w e r e u s e d fo r C 1 . R u n E 2 w a s d o n e b y t h e e x p l ic i t

t e c h n i q u e u s i n g a c o n s t a n t l a y e r t h i c k n e s s, a n d r u n I w a s d o n e b y t h e i m p l i c i t t e c h n i q u e .

L a y e r th i c k n e s s e s f o r r u n s E 2 a n d I w e r e c h o s e n t o g i v e a f r e q u e n c y r e s o l u t i o n c o m -

p a r a b l e t o t h a t o f C 1 a n d E l . T i m e i n c r e m e n t s f o r E 2 a n d I w e r e c h o s e n t o s a ti s f yth e r eq u i remen t s o f s t ab i l i t y . S tab i l i t y w i th th e imp l i c i t t ech n iq u e i s d i scu ssed b y C h en

an d Jo y n er (1 97 4) . In p red ic t in g s t ab i l i ty fo r ru n s w i th th e ex p l ic i t t ech n iq u e , t h e s t ab i l i t y

c r i t e r io n fo r th e l in ea r e l a s t ic case (R ich tm y er an d M o r to n , 1 9 6 7 , p . 2 63 ) w as ap p l i ed

u s i n g t h e l o w - s t r a i n v a l u e o f t h e v e l o c it y . N o n u m e r i c a l e v i d e n ce o f i n s ta b i l i ty w a s n o t e d

f o r r u n s i n w h i c h t h e t i m e i n c r e m e n t s a t is f ie d t h a t c r i t e r io n .

t 3O_ 3la.t

TIME (SECONDS)

FIG. 4. S urface particle velocity by different techniques of integration.

S u r f a c e p a r ti c l e v e l o c it y f o r t h e f iv e r u n s is c o m p a r e d i n F i g u r e 4 , a n d t h e a g r e e m e n t

i s q u i t e g o o d . S u r face acce le ra t io n i s co mp ared in F ig u re 5 . T h ere th e ag reemen t i s a l so

g o o d e x c e p t f o r t h e h i g h - f r e q u e n c y o s c i ll a t io n s t h a t a p p e a r i n t h e e x p l ic i t a n d i m p l i c i t

ru n s . T h ese o sc i l l a t io n s a re d i scu ssed in g rea te r d e ta i l b y C h en an d Jo y n er (1 9 7 4 ) .

T h e i r f r e q u e n c y d e p e n d s u p o n t h e t h i c k n e s s o f t h e la y e r s in t o w h i c h t h e s o i l c o l u m n i sd i v i d e d i n t h e l u m p e d m a s s p r o c e d u r e . T h e f r e q u e n c y c a n b e m a d e a s h i g h a s d e s i r e d

b y ch o o s in g th e l ay e r th i ck n esses su f f ic i en t ly smal l . T h e o sc i l l a t io n s can th en b e r em o v e d

b y ap p ly in g a su i t ab le d ig i t a l lo w -p ass f i l te r . R em o v a l b y d ig i t a l f il t e r in g i s b e t t e r t h an

th e u se o f v i sco u s d am p in g to su p p ress th e o sc i l l a t io n s , b ecau se i t is eas ie r to av o id

d i s t o r t i o n o f th e s i gn a l w i t h i n t h e f r e q u e n c y b a n d o f i n t er e s t . A s m e n t i o n e d p r e v i o u s ly ,

t h e d a m p i n g c h a r a c t e r i s t ic s o f s o i ls a r e e s s e n ti a l ly i n d e p e n d e n t o f f r e q u e n c y a b o v e

0 .1 H z i n d i c a t i n g t h a t v i s c o u s d a m p i n g is n o t a v a li d r e p r e s e n t a t i o n o f t h e m a t e r i a l

b e h a v i o r .

A s a m a t t e r o f f a c t, f o r m a n y p u r p o s e s t h e r e m a y b e n o n e e d t o r e m o v e t h e o s c i ll a ti o n s.

F i g u r e 6 s h o w s t h e a c c e l e r a t i o n r e s p o n s e s p e c t r a a t f iv e p e r c e n t d a m p i n g f o r t h e f iv er u n s . T h e d i f fe r e n c e s a r e s m a l l f o r p e r i o d s g r e a t e r t h a n a b o u t 0 .1 s ec .

W e c o n c l u d e t h a t a l l t h r e e t e c h n i q u e s g i v e v a l i d re s u l ts . T h e r u n w i t h t h e s h o r t e s t

c o m p u t a t i o n t i m e w a s E l , w h i c h w a s d o n e b y t h e e xp l ic i t t e c h n i q u e w i t h l a y er th i c k -

n e s s es c h o s e n t o g i v e c o n s t a n t t r a v e l t i m e a t t h e l o w - s t r a i n v e lo c i t y . R u n C 1 , h o w e v e r ,

t h e c o m p a r a b l e r u n d o n e b y i n t e g r a t i o n a l o n g c h a r a c t e ri s t ic s , r e q u i r e d o n l y 2½ m i n u t e s

o n a n I B M 3 7 0 -1 5 5 . I n t e g r a t i o n a l o n g c h a r a c t e ri s ti c s m a y b e p r e f e r re d f o r m a n y

a p p l i c a t i o n s b e c a u s e i t a v o i d s t h e p r o b l e m o f t h e h i g h - f r e q u e n c y o s c i ll a t io n s .

,- , E2

F - - o

(..)~. )(12 ~ C1

o ' . o o e ' . o o 4.0o G ' .o o s ' .o o i b . o o i b . o o i b . o o i b . o o i b . o o 2 b . o o

T I M E ( S E C O N D S )

FIG. 5. Surface acceleration by d ifferent techniques of integration.

C O M P A R I S O N W I T H T H E E Q U I V A L E N T L I N E A R M E T H O D

F o r c o m p a r i n g t h e r e s u l ts o f t h e e q u i v a le n t l i n e ar m e t h o d w i t h t h o s e o f t h e n o n l i n e a r

m e t h o d , t h e s a m e s o i l c o l u m n ( F i g u r e 3 ) i s u s e d a s b e f o r e . T h e s u b s t r a t u m i s a s s i g n e d

a s h e a r v e l o c i t y o f 2 .0 k m / s e c a n d a d e n s i t y o f 2 .6 g m / c m 3. T h e i n p u t i s t h e f i r st 5 0 s e c

o f th e N 2 1 E c o m p o n e n t o f t h e T a f t s t r o n g - m o t i o n r e c o r d m u l t i p l i ed b y a f a c t o r o f f o u r ,

a s b e f o r e ; s o t h a t t h e i n c i d e n t w a v e i f i n c i d e n t o n a f r e e s u r f a c e w o u l d p r o d u c e a p e a k

a c c e l e r a t i o n o f 0 .7 g a n d a p e a k v e l o c i t y o f 6 7 c m / s e c . I t s h o u l d b e n o t e d t h a t t h is e x -a m p l e , w i t h a t h i c k s o i l c o l u m n a n d s t r o n g i n p u t m o t i o n , w a s d e l i b e r a t e l y c h o s e n t o

p r o v i d e a s e v e r e t e s t f o r t h e e q u i v a l e n t l i n e a r a s s u m p t i o n .

T h e n o n l i n e a r c a l c u l a t i o n s w e r e d o n e b y i n t e g r a t i o n a l o n g c h a r a c t e r is t i c s. T h e

e q u i v a l e n t l i n e a r c a l c u l a t i o n s w e r e d o n e b y a m e t h o d s i m i l a r t o t h a t d e s c r i b e d b y S c h n a b e l

e t a l . ( 19 7 2) . B e c a u s e o f th e h i g h l e v e l o f th e i n p u t m o t i o n a n d t h e c o n s e q u e n t l a r g e e n e r g y

C A L C U L A T I O N O F N O N L I N E A R G R O U N D R E SP O NS E I N E A R T H Q U A K E S ] 3 2 7

1 . 6 .

z 1 .e, -E 1

¢y t n r i t t

L IJ. _ _ _ - ~ E 2 B AL d

0 . 8 - , ~ . - -

_Jo / ' | . . . .o') f~-.im ~ C 2

\ C 1 i

O ' i0 0 . 2 0 .#

O . 6 0 . 8

1.0 I .e

N R T U R R L PERIOD - SECONDS

F IG . 6 . A c c e l e r a t i o n r e s p o n s e a t f iv e p e r c e n t d a m p i n g f o r d i f f e re n t t e c h n i q u e s o f i n t e g r a t i o n .

"- E O U I V R L E N T L I N E R R

z(23I - . - t

(12rYLLI__1W

" N O N L I E R R

] I N P U T

0 , 0 0 5 , 0 0 I 0 . 0 0 1 5 , 0 0 9 ' 0, 0 0 2 5 , 0 0 3 0 . 0 0 3 5 , 0 0 4 0 , 0 0 q .5 ,0 0 5 0 , 0 0

T I M E ( S E C O N D S )

F IG . 7. C o m p a r i s o n o f s u r fa c e a cc e l e ra t i o n f o r e q u i v a l e n t l in e a r a n d n o n l i n e a r m e t h o d s .

dissipation in the soil column, special attention was required to the relationship between

soil damping and the dissipation constant of the equivalent linear model. The parameters

of the equivalent linear model as functions of strain were calculated from the properties

of the Iwan model used for the nonlinear calculations, so that the only difference between

the two sets of calculations is the equivalent linear assumption. Further descriptionof the equivalent linear procedure is given in Appendix B.

Figure 7 shows the input acceleration time history and compares the surface accelera-

tion computed by the nonlinear method and the equivalent linear method. There are

definite points of similarity, but it is clear that the equivalent linear approximation does

not reproduce the short-period components of motion present in the nonlinear solution.

Comparing the nonlinear solution with the input shows the effect of the postulated

"q EQUI gR LEN T L.INERR

N O N L I N E R R

> -I - -

o._1LL I>

f 3 . 0 0 5 . 0 0 1 0 , 0 0 1 5 . 0 0 2 0 , 0 0 2 5 . 0 0 3 0 . 0 0 3 5 . 0 0 q.o 00 q .5 ,00 50 .00

TIME (SECONDS)

FIo. 8. Compari son of surface particle velocity for equivalent linear and nonlinear methods.

soil profile on ground motion. The peak acceleration is sharply reduced. The longer

period components are amplified, however, and the overall effect may be a more damag-

ing motion as will be illustrated subsequently.

Figure 8 shows the corresponding velocity time histories for the same example.

Comparing the nonlinear and equivalent linear solutions we see much better agreement,

indicating that the equivalent linear approximation is adequate with respect to the longer

period components that are dominant in the velocity time history. Comparing either of

the solutions with the input shows clearly the amplifying effects of the soil profile for

long-period motions.

To illustrate the consequences of these results for structures we have computed response

C A L C U L A T I O N O F N O N L IN E A R G R O U N D R E SP O NS E IN E A R T H Q U A K E S 1329

s p e c t r a . F i g u r e 9 s h o w s t h e a c c e l e r a t i o n r e s p o n s e a t 5 p e r c e n t d a m p i n g f o r t h e t h r e e

m o t i o n s b e t w e e n 0 - a n d 3 -s ec p e r i o d . T h e l in e r e p re s e n t s t h e i n p u t , h e x a g o n s t h e n o n -

l i n e a r s o l u t i o n , a n d c r o s s e s t h e e q u i v a l e n t l i n e a r s o l u t i o n . I t i s c l e a r t h a t t h e e q u i v a l e n t

l i n e a r m e t h o d s i g ni f ic a n tl y u n d e r e s t i m a t e s t h e i n t e n s i t y o f m o t i o n f o r p e r i o d s b e t w e e n

0 .1 a n d 0 .6 s e c i n t h is c a s e . T h i s p e r i o d r a n g e c o r r e s p o n d s t o b u i l di n g s b e t w e e n o n e a n da b o u t s ix s to r ie s . T h e i m p o r t a n c e i s o b v i o u s .

C o m p a r i n g t h e in p u t r e s p o n s e w i t h th e n o n l i n e a r r e s p o n s e i n F i g u r e 9, o n e m i g h t b e

t e m p t e d t o c o n c l u d e t h a t f o r s h o r t - p e r i o d s t r u c t u r e s , t h e m o t i o n s o n a l l u v i u m w o u l d b e

l e s s d a m a g i n g t h a n o n b e d r o c k . C o n s i d e r a b l e c a u t i o n i s i n d i c a t e d h e r e . F o r o n e t h i n g ,

d i f f e r e n t s o i l m a t e r i a l s , d e n s e s a n d s f o r e x a m p l e , m i g h t g i v e m o r e i n t e n s e m o t i o n s .

ZOHF -El:

LdOC-)

L~I . - -

c~(] :

~ x ~ x,~D~au~ ~n ^ ~

0.5 t .0 t ,5 2,0 2.5 3,0

N R T U R R L P E R I O D - SE C O N D S

Fio . 9. Acceleration response at five per c ent d am ping. The line represents the input, hexagons th enonlinear solution, and crosses the equivalent linear solution.

P o s s i b le l e n g t h e n i n g o f s t r u c t u r a l p e r io d s d u e t o d e f o r m a t i o n b e y o n d t h e l i n e a r r a n g en e e d s t o b e c o n s i d e r e d a s w e l l a s t h e e ff e ct s o f d u r a t i o n n o t a c c o u n t e d f o r in t h e r e s p o n s e

s p e c t r u m . C o n s i d e r a t i o n s h o u l d a l s o b e g i v e n t o t h e e f fe c ts o f g r o u n d d e f o r m a t i o n a n d

g r o u n d f a i l u re .

F i g u r e 1 0 c o m p a r e s t h e r e l a t i v e v e l o c i ty r e s p o n s e s p e c t r a a t 5 p e r c e n t d a m p i n g f o r t h e

r a n g e f r o m 0 - to 6 - s ec p e r io d . T h e r e s u lt s s h o w t h a t t h e e q u i v a l e n t l i n e a r a p p r o x i m a t i o n

i s a d e q u a t e f o r t h e l o n g e r p e r i o d s a n d t h a t t h e s o i l s i t e g i v e s l a r g e a m p l i f i c a t i o n f o r

l o n g e r p e r i o d s .

D u r i n g t h e r u n n i n g o f t h e n o n l i n e a r s o lu t io n , w e m o n i t o r e d t h e p e a k s tr a in f o r e a c h

d e p t h i n t er v a l . T h e m a x i m u m w a s 6 x 1 0 - 3 f o r t h e i n t e r v a l f r o m 3 2 to 3 5 m .

T h e c o m p a r a t i v e c o s t s o f th e t w o m e t h o d s i s d if fi c ul t t o e v a l u a t e i n t h e g e n e r a l c a s eb e c a u s e i t i s p o s s ib l e t o r u n t h e e q u i v a l e n t l i n e a r m e t h o d u s i n g f e w e r la y e r s, d e p e n d i n g

o n t h e d e t a il o n e w i s h es t o r e p r e s e n t i n t h e s o i l p r o f il e . F o r t h e e x a m p l e p r e s e n t e d h e r e ,

h o w e v e r , u s in g t h e s a m e n u m b e r o f la y e rs f o r b o t h m e t h o d s , t h e n o n l i n e a r s o l u t i o n

r e q u i r e d l e s s t h a n h a l f a s m u c h c o m p u t e r t i m e a s t h e e q u i v a l e n t l i n e a r . S o , w e b e l i e v e

t h a t i n g e n e r a l i t w i ll b e c o m p e t i t i v e , a t l e a st .

CONCLUSIONS

T h e I w a n m o d e l l e ad s t o a s i m p l e a n d e f f ic i e n t m e t h o d f o r c a l c u l a ti n g t h e s e i sm i c

r e s p o n s e o f a s y s t e m o f h o r iz o n t a l s o il l ay e r s. T h e m e t h o d t a k e s a c c o u n t o f t h e n o n l i n e a r

h y s t e r e t i c b e h a v i o r o f s oi ls a n d h a s c o n s i d e r a b l e f le x i b il i ty f o r i n c o r p o r a t i n g l a b o r a t o r y

r e s u l ts o n t h e d y n a m i c b e h a v i o r o f s o il s . F i n i t e r i g i d it y is a ll o w e d i n t h e u n d e r l y i n g

e la s ti c m e d i u m , p e r m i t t i n g e n e r g y t o b e r a d i a t e d b a c k i n t o t h e e l a st ic m e d i u m . I n t e g r a -

t i o n b y i m p l i c i t o r e x p l i ci t te c h n i q u e s o r i n t e g r a t i o n a l o n g c h a r a c t e r i s t ic s g i v e s v a l i d

r e s u lt s . I n t e g r a t i o n a l o n g c h a r a c t e r i s t i c s h a s t h e a d v a n t a g e o f a v o i d i n g s p u r i o u s h i g h -

f r e q u e n c y o s c i l l a ti o n s i n th e s o l u t i o n f o r t h e a c c e l e r a t i o n t i m e h i s t o r y a t t h e s u r f a c e .

C o m p a r i s o n w i t h r e s u lt s o b t a i n e d b y t h e w i d e l y u s e d e q u i v a l e n t li n e a r a s s u m p t i o n

i n d i c a t e s t h a t f o r a t h i c k so i l c o l u m n a n d a h i g h le v e l o f i n p u t m o t i o n t h e e q u i v a l e n t l i n e a r

m e t h o d m a y u n d e r e s t i m a t e t h e s h o r t p e r i o d c o m p o n e n t s o f s u r fa c e m o t i o n b y a f a c t o r

o f tw o o r m o r e . A t l o n g e r p e r i o d s ( l o n g e r t h a n 0 .6 s ec in t h e e x a m p l e ) t h e t w o m e t h o d s

a g r e e . B e t t e r a g r e e m e n t a t s h o r t p e r i o d s w o u l d b e e x p e c t e d f o r l o w e r l e v e ls o f i n p u t

m o t i o n . C o m p a r a t i v e c o m p u t e r t i m e r e q u ir e m e n t s d e p e n d o n t h e c i r cu m s t a n ce s o f th e

0- JIllJ

800 o o x x o0 0 0o~

" d. t l0¢ 50

~" 00 0.5 i .0 1.5 8.0 8.5 3.0 3.5 q.O q.5 5.0 5.5 6.0

NATURAL PERIOD - SECONDS

Fla . 10. Relative velocity response a t five per cent damp ing. T he line represents the input, h exagonsthe nonlinear solution, and crosses the equivalent linear solution.

p a r t i c u l a r p r o b l e m . W e b e l ie v e t h a t i n m a n y c a se s n o n l in e a r m e t h o d s w i ll b e c o m p e t i t iv e ,

a t le a s t, i n t h i s r e g a r d . I n g e n e r a l , n o n l i n e a r m e t h o d s a r e p r e f e r a b l e b e c a u s e t h e y h a v e a

m o r e r i g o r o u s b a si s , b u t e q u i v a l e n t li n e a r m e t h o d s m a y b e a d e q u a t e f o r m a n y p u r p o s e s

w h e r e a c c u r a t e e v a l u a t i o n o f r e s p o n s e s p e c t r a f o r s h o r t p e r i o d s i s n o t r e q u i re d .

O u r r e s u l t s g iv e a n in d i c a t i o n o f th e g e n e r a l n a t u r e o f th e e f f e c t o n e m i g h t e x p e c t

f o r a t h i c k s o il c o l u m n a t a h i g h l ev e l o f i n p u t m o t i o n , c o m p a r e d t o a s it e w i t h b e d r o c k

a t t h e s u rf a c e. T h e m o d e l s h o w s a m p l i f i c a ti o n b y a f a c t o r o f t w o o r m o r e a t l o n g p e r i o d s

a n d s o m e r e d u c t i o n i n a m p l i t u d e a t s h o r t p er i o d s .

ACKNOWLEDGMENTS

We are indebted to H arold W . O lsen for his advice on questions of soil dynamics and to R oger D.Borcherdt for assistance in the theory of linear viscoelasticity. David M. Boore and D. J. Andrews m adehelpful suggestions concerning numerical methods. W illiam D. Stuart and Ha rold W . O lsen read themanuscript and made a number of valuable suggestions.

C A L C U L A T IO N O F N O N L IN E A R G R O U N D R E S P ON S E I N E A R T H Q U A K E S 1331

A P P E N D I X A

A T h e o r e m R e g a r d i n g th e R h e o l o g i c a l M o d e l

G i v e n t h e r h e o l o g i c a l m o d e l i l l u st r a t e d in F i g u r e 1 , w e p r o v e t h e f o l l o w i n g t h e o r e m :I f e l emen t j i s no t y ie ld ing , then a l l e lemen ts i fo r wh ich Y~ i s grea ter than Y j are a l so no t

y ie ld ing .

W e d e n o t e t h e s tr e ss i n t h e i s p ri n g b y H i a n d t h e s t r es s i n t h e i f r i c t io n e l e m e n t b y R ~.

E q u i l i b r i u m r e q u i r e s t h a t

H i + R i = H j + R j = s ( A 1 )

w h e r e s i s t h e s t r es s f o r t h e m o d e l . T h e f r i c ti o n e l e m e n t i is f ix e d w h e n

IR , I < Y ,

a n d l i k ewi s e f o r j . T h e f r i c t i o n e l e m e n t i i s y i e l d i n g t o i n c r e a s i n g s t r es s wh e n

a n d t o d e c r e a s i n g s tr e ss w h e n

R i = Y i

R i = - ~ i

a n d l i k e w is e f o r j . W e c a n n o w r e s t a t e t h e t h e o r e m a s f o l l o w s

Y i > Y j

t h e n

I R , I < Y iF r om e q u a t i o n ( A 1 ) w e h a v e

R i = R j + ( H j - H i ) .

S i n c e

] R j l < Y jw e h a v e

R i < Y j + ( H j - H , )

R i > - Y j + ( H j - H , ) .

I t f o l lo w s t h e n t h a t

R , < Y j + IH ~ -H ,I

Ri > - 5-IHj=Hi[.

S o l o n g a s [ H i - H ,I < ( Y i - r~ )w e h a v e t h a t

I R , I < r , .

S o , w e h a v e p r o v e d a w e a k f o r m o f t h e th e o r e m w h i c h s t at es t h a t

Y i > Y ,a n d

I g j l <

a n d]Hi - H i l < ( Y , - Y j ) ,

t h e n

IRi] < Yi.

W e n o w p r o c e e d to u s e th e w e a k f o r m o f t h e t h e o r e m t o p r o v e t h a t I H j - H i l c a n n e v e r

ex ceed ( Y i - Y j ) , w h i c h c o m p l e t e s th e p r o o f o f t h e t h e o r e m i n i t s o r ig i n a l f o r m . T h e

in i t i a l v a lu e o f ( H j - H i ) i s z e r o . T h e q u a n t i t y r e m a i n s c o n s t a n t s o l o n g a s b o t h e l e m e n t s

a re f ix ed an d a l so so lo n g as b o th e l emen t s a re y i e ld in g in e i th e r d i r ec t io n . S o lo n g as

t h e a b s o l u t e v a l u e r e m a i n s l e s s t h a n ( Y i - Y j ) , t h e w e a k f o r m o f t h e t h e o r e m t e l l s u s

t h a t e l e m e n t i c a n n o t y i e l d w h i l e j i s f ix e d . T h e o n l y w a y , t h e r e f o r e , t h a t t h e v a l u e o f

( H j - H i) c a n c h a n g e i s f o r e l e m e n t j t o y i e l d w i t h e l e m e n t i f ix e d .I f j i s y ie ld in g u n d e r in c reas in g s t r e s s an d i i s f i x ed , t h e n ( H j - H i) wil l increase . S ince

R~= r j

s u b s t i t u t i o n i n t o e q u a t i o n ( A 1 ) g iv e s

R i = ( H j - H i ) + Y j . (A2)

A s e l e m e n t j y i e ld s , t h e q u a n t i t y ( H i - H i ) w i ll i n c re a s e , b u t w h e n i t r e a c h e s t h e v a l u e

( Y i - Y j) , s u b s t i t u t i o n i n t o e q u a t i o n ( A 2) s h o w s t h a t

R i = Y i

a n d f u r t h e r i n c r e a s e o f s t re s s w i ll c a u s e e l e m e n t i t o y i e l d . W i t h b o t h e l e m e n t s y i e l d in g ,

t h e q u a n ti t y ( H i - H i ) re m a i n s c o n st a n t.

I f e l em en t j i s y i e ld in g u n d e r d ec reas in g s t r e s s an d e l em en t i i s f i x ed , t h e n ( H i - H i )wil l decrease . S ince

R j = - Y j

s u b s t i t u t i o n i n t o e q u a t i o n ( A 1 ) g iv e s

R i = ( H i - H i ) - Y r. (A 3 )

A s e l e m e n t j y i e ld s , th e q u a n t i t y ( H i - H i ) w i l l d e c r e a s e b u t w h e n i t r e a c h e s t h e v a l u e

- ( Y i - Y j) , s u b s t i t u t i o n in t o e q u a t i o n ( A 3 ) s h o w s t h a t

Ri = - - Y i

a n d f u r t h e r d e c r e a s e o f s t r es s w i ll c a u s e e l e m e n t i to y i e ld . A s n o t e d b e f o r e , w h e n b o t h

e l e m e n t s a r e y i e l d i n g , t h e q u a n t i t y ( H j - H i) r e m a i n s c o n s t a n t .

APPENDIX B

Equivalent Linear Procedure

T h e e q u i v a l e n t l i n e a r m e t h o d f o r s o l v i n g s o i l r e s p o n s e p r o b l e m s i s b a s e d o n t h e

a s s u m p t i o n t h a t a l i n e a r m o d e l w i ll gi ve a s a t i s f a c to r y a p p r o x i m a t i o n p r o v i d i n g t h a t t h e

d y n a m i c p r o p e r t ie s a r e c h o s e n t o a c c o r d w i t h t h e a v e r a g e s t r a i n t h a t o c c u r s i n t h e m o d e l

d u r i n g e x c i t a t io n . T h e m e t h o d i s it e r at iv e . A s s u m e d v a l u e s o f a v e r a g e s t r a in a r e u s e d f o r

t h e f i r s t r u n . A f t e r t h e r u n , t h e c o m p u t e d s t r a i n i s c o m p a r e d t o t h e a s s u m e d s t r a i n .

CALCULATION OF NONLINEAR GRO UND RESPONSE IN EARTHQUAKES 1333

I f n e c e s s a ry , a d j u s tm e n t s a r e m a d e a n d t h e p r o c e s s i s r e p e a t e d a s m a n y t i m e s a s n e e d e d

t o o b t a i n s a t i s f a c t o r y a g r e e m e n t .

T h e l i n e a r m o d e l w e u s e i s b a s e d o n K a n a i ' s ( 1 9 5 2 ) s o l u t i o n t o t h e h o r i z o n t a l l a y e r

p r o b l e m , e x t e n d e d t o g e n e r a l l in e a r v is c o e la s ti c m a t e r i a l . W e p o s t u l a t e a s y s t e m o f N

h o r i z o n t a l p l a n e l a y e rs c o m p o s e d o f h o m o g e n e o u s i s o t r o p i c li n e a r v is c o e la s ti c m a t e r i a la ll r e s ti n g o n a n e l as t ic h a l f- s p a c e . W e f u r t h e r p o s t u l a t e a n i n c i d e n t p l a n e S H w a v e i n

t h e h a l f -s p a c e , p r o p a g a t i n g a t a n a n g l e q5 w i t h t h e v e r t ic a l . F o r t h e c o m p u t a t i o n s

p r e s e n t e d i n th i s p a p e r q5 w a s t a k e n t o b e z e r o , b u t t h e m e t h o d w a s o r i g i n a l ly d e v e l o p e d

f o r a r b i t r a r y ~b. W e u s e a C a r t e s i a n c o o r d i n a t e s y s t e m , x , w i t h t h e o r i g i n o n t h e s u r f a c e

o f th e h a l f - s p a c e a n d w i t h a n o r i e n t a t i o n s u c h t h a t x 3 i s v e r t i c a l a n d t h e r a y p a t h f o r

t h e i n c i d e n t S H wa v e l i e s i n t h e x l - x 3 p l a n e . W i t h i n e a c h l a y e r w e l o o k f o r s o l u t i o n s

t o t h e e q u a t i o n o f m o t i o n i n w h i c h d i sp l a c e m e n t ta k e s t h e f o r m

U = u e x p [ i t o t ] (B1)

w h e r e u j is a c o m p l e x f u n c t i o n o f t h e s p a t i a l c o o r d i n a t e s , t o i s a n g u l a r f r e q u e n c y a n d ti s t i m e . T h e d i s p l a c e m e n t U j i s c o m p l e x , b u t w h e n s o l u t i o n s f o r p o s i t i v e a n d n e g a t i v e

to a r e a d d e d t h e r e s u l t w i ll b e r e a l. W i t h t h i s f o r m u l a t i o n t h e e q u a t i o n o f m o t i o n f o r a

h o m o g e n e o u s i s o t r o p i c l i n e a r v is c o e la s t ic m a t e r i a l i s ( B o r c h e r d t , 1 97 3)

w h e r e

2vj(k+# /3) VO+#V 2Uj = (B2)

~ U 1 ~ U 2 ~ U 3

# = (ito/2) r~

k = (ito/3) r k

p i s t h e d e n s i t y a n d t h e f u n c t i o n s r s a n d r k a r e th e F o u r i e r t r a n s f o r m s o f th e r e l a x a t i o n

f u n c t i o n s c h a r a c t e r i s ti c o f t h e s h e a r a n d b u l k b e h a v i o r o f th e m a t e r i a l . S u b s t i tu t i n g

f r o m e q u a t i o n ( B 1 ) i n t o e q u a t i o n ( B 2 ) g i ve s

w h e r e

( k + # / 3 ) V0 + # V 2 I , / j -- ~ - (.O2pb/j

0 : 8u l 8/ ' /2 8~/3

a - i "

T h e o r i e n t a ti o n o f th e c o o r d i n a t e s y s t em w a s c h o s e n s o t h a t t h e d i s p l a c e m e n t f o r

t h e i n c i d e n t w a v e d o e s n o t d e p e n d o n x 2 . S i n c e t h e g e o m e t r i c a n d m a t e r i a l p r o p e r t i e s

o f t h e s y s t e m d o n o t d e p e n d o n x 2 , t h e r e i s n o w a y t h a t o n e v a l u e o f x 2 c o u l d b e d i s-

t i n g u i s h e d f r o m a n o t h e r . W e t h e r e t b r e a s s u m e t h a t ~?uf lc?x2 = 0 a n d c o n s e q u e n t l y

t h a t c~O/c~x2 = 0 . W i t h t h a t a s s u m p t i o n t h e e q u a t i o n f o r t h e j = 2 c o m p o n e n t i n (B 3 )

is u n c o u p l e d f r o m t h e o t h e r t w o a n d b e c o m e s

i J V 2 u = - o o Z p u . ( B 4 )

T h e s u b s c r i p t i s o m i t t e d i n ( B 4 ) a n d i n w h a t f o l l o w s , b e c a u s e w e a r e c o n c e r n e d o n l y

w i t h t h e j = 2 c o m p o n e n t .

D i s p l a c e m e n t f o r t h e i n c i d e n t w a v e i n t h e u n d e r l y i n g m e d i u m i s r e p r e s e n t e d b y

w h e r e

U t = a t ex p ( i c o t - i l x l - i b e x 3 )

l = c o s i n 4 ) / V s

h i = c o c o s 4 / v ~

an d v~ is t h e sh ea r v e lo c i ty in th e e l a s t i c h a l f - sp ace . T h e so lu t io n to eq u a t io n (B 4) in

th e n th l ay e r is

( u ), = e x p ( - i l x l ) [ a . e xp ( - i h n x 3 ) + b , e x p ( i h n x 3 ) ]

w h e r e

h , 2 = c o 2 p , / # , - I z .

T h e f i rs t t e r m i n t h e b r a c k e t s r e p r e s e n ts a n u p g o i n g w a v e a n d t h e s e c o n d t e r m a d o w n -

g o in g w av e . In th e g en era l case , s in ce /~ , i s co m p lex , h , w i l l b e co mp lex . A t each b o u n d a ry

a p p l i c a t i o n o f t h e c o n d i t i o n s r e q u i r in g c o n t i n u i t y o f s tr e s s a n d d i s p l a c e m e n t l e ad s t o t w o

r e l a t io n s h i p s a m o n g t h e c o m p l e x c o ef f ic i en t s a , a n d b , f o r a d j a c e n t l a y e r s. W i t h t h e

ass u m p t io n o f a s t r e s s- f r ee su r face , t h ese r e l a t io n sh ip s a re so lv ed u s in g a m o d i f i ca t io n

o f H a s k e l l ' s ( 19 53 ) m a t r i x m e t h o d t o g i v e t h e c o e f fi c ie n ts a , a n d b , i n t e r m s o f a ~ .

T h e F o u r i e r co e f f i c i en t s a t f o r a p o s t u l a t e d i n p u t m o t i o n a r e c a l c u l a t e d u s i n g t h e

F a s t F o u r i e r T r a n s f o r m p r o g r a m H A R M f r o m t h e I B M S c ie n ti fi c S u b r o u ti n e P a c ka g e .

G i v e n t h e a t , t h e c o e f f ic i e nt s a , a n d b , a r e f o u n d . T h e m o t i o n a t a p o i n t i n t h e l a y e rs ist h e n r e p r e s e n t e d a s a f u n c t i o n o f t im e b y a F o u r i e r s e r ie s , w h i c h i s e v a l u a t e d u s i n g th e

H A R M p ro gr am .

F o r t h e p u r p o s e o f c o m p a r i n g t h e e q u i v a l e n t l i n e a r w i t h t h e n o n l i n e a r m e t h o d , w e

d e t e r m i n e t h e c o m p l e x m o d u l u s # f o r e a c h l a y e r b y r e q u i r i n g t h a t t h e v i s c o e l a s t i c

m a t e r i a l s h o w s t r e ss - s tr a i n b e h a v i o r s i m i l a r t o t h a t o f th e I w a n m o d e l . W e d e f in e th e

d a m p i n g r a ti o D b y

D = w / ( z ~ E ; , ) (B5)

w h ere zp i s t h e p e ak s t r e s s an d W i s t h e en e rg y p e r u n i t v o lu m e d i s s ip a t ed p e r cy c le

a t t h e g iv en p eak s t r a in l ev e l E p . T h i s c o r r e s p o n d s t o t h e d e f i n i t i o n o f d a m p i n g r a t i o

c o m m o n l y u s e d i n s oi ls d y n a m i c s ( e. g ., H a r d i n a n d D r n e v i c h , 1 97 2a ). U s i n g e l e m e n t a r y

p r in c ip l es , W an d "co can b e ca l cu la t ed fo r an I w a n mo d e l g iv en Ep .

T h e h y s t e res i s l o o p s o f v i sco e las ti c m a te r i a l fo r s in u so id a l l o ad in g a re e l li p ses an d ,

o f c o u rs e , a r e n o t i d e n t i c a l t o t h o s e o f a n I w a n m o d e l . W e d e t e r m i n e / ~ t / / ~ , t h e r a t i o

b e t w e e n t h e i m a g i n a r y a n d r e a l p a r ts o f t h e c o m p l e x m o d u l u s , b y r e q u i r in g t h a t t h e

v i s co e l a st ic m a t e r i a l h a v e t h e s a m e v a l u e o f D a s t h e I w a n m o d e l , r e a li z i n g t h a t i n t h e

v i s c oe l a s ti c m a t e r i a l p e a k s tr e ss a n d p e a k s t r a i n w il l n o t o c c u r a t t h e s a m e i n s t a n t .

In a v i sco e las ti c ma te r i a l

w = ~ z ~ d t .c y c l e

F o r s i n u s o id a l lo a d i n g

E = E . c o s c ot

z = Zp co s ( c o t + ( 5 )

CALCULATION OF NO NLINEAR GROUND RESPONSE IN EARTHQUAKES 1335

w h e r e

6 = ar c ta n ( /2~//zR) .

P e r f o r m i n g t h e i n t e g r a t i o n a n d s u b s t i t u t i n g i n t o e q u a t i o n ( B 5) g i v e s

D = ( s in 6)/2.

W i t h s o m e a l g e b r a i c m a n i p u l a t i o n t h is e x p r e s s io n c a n b e s o l ve d t o g iv e

/ q / # R = 2 D / ( 1 - 4D2) a/z.

T h e d e t e r m i n a t i o n o f/~ R i s s t r a i g h t f o r w a r d . W e r e q u i r e t h a t

f r o m w h i c h w e e a s i ly o b t a i n

I~R = (zp/Ep)(1 + [Izr/pR]2) -1 /2 .

I n r u n n i n g t h e p r o b l e m d e s c r i b e d i n t h e te x t , v a l u e s o f E v w e r e a s s u m e d a t d e p t h s o f

1 6, 3 2 , 6 4 , 1 28 , a n d 1 99 m a n d w e r e i n t e r p o l a t e d i n b e t w e e n . D u r i n g e a c h i t e r a t i o n ,

s t r a i n w a s m o n i t o r e d a t t h o s e d e p t h s . A s t r a in v a l u e e q u a l t o 7 0 p e r c e n t o f t h e m a x i m u m

a b s o l u t e v a l u e w a s t a k e n f o r c o m p a r i s o n w i t h t h e Ep a s s u m e d a t t h e b e g i n n i n g o f t h e

i t e r a t i o n . I f n e c e s s a r y , E p w a s a d j u s t e d a n d a n o t h e r i t e r a t i o n m a d e . F o u r i t e r a t i o n s w e r e

r e q u i r e d f o r t h e e x a m p l e g i v e n in t h e t e x t. T h e c h o i c e o f 70 p e r c e n t a s th e c o n v e r s i o n

f a c t o r f r o m m a x i m u m s t ra i n p e a k to a v e r a g e s tr a i n p e a k is a d m i t t e d l y s o m e w h a t

a r b i t r a r y . W e b e l i ev e , h o w e v e r , t h a t i t i s a r e a s o n a b l e c h o i c e a n d w e d o u b t t h a t t h e

r e s u l t s a r e v e r y s e n s i t i v e t o t h e c h o i c e .

REFERENCES

Boore , D. M. (1972). Fin i te d if ference m ethods fo r se ismic-wave p ropa gat io n in he terogeneous m ater ia ls ,

in Methods in Computational Physics, Vol. 11, Seismology, Surface Waves and Earth Oscillations,

Br u ce A . Bo l t , E d i to r , Acad em ic P r e ss , New Yo r k , 1 -3 7 .

Borche rd t , R. D. (1973). Energy and p lane w aves in l inear v iscoelas t ic me dia , J . Geophys. Res. 78 , 2442-

Broo ker , E . W . and H . O . I re land (1965) . Ear th p ressures a t res t re la ted to s t ress h is to ry , Can. Geotech. J.

2~ 1-15 .

Ch en , A . T . F . an d W . B . Jo y n e r ( 1 97 4 ). M u l t iq in ea r an a ly s i s f o r g r o u n d m o t io n s tu d ie s o f l ay e red

systems, Report No. USGS-GD-74-020, NT IS No. PB232-704/AS, Clear inghous e , Spr ingf ie ld , VA22151.

Co nstan top oulos , I . V. (1973) . Am pl i f ica t ion s tud ies fo r a no n l inear hystere t ic so i l mo del , Research Report

R73-46, De par tm ent o f Civ i l Engineer ing , M assachuse t ts Ins t i tu te o f Technology , 204 pp .

Facc io l i , E . , E . San to yo V. , and J . L . Le 6n T. (1973). Micro zonat ion cr i te r ia and se ismic response s tud ies

f o r t h e c i ty o f M an ag u a , Proc. Earthquake Eng. Res. Inst. Conf. Managua, Nicaragua, Earthquake o fDec. 23, 1972 1 ,271-291 .

Har d in , B . O . an d W . L . B lack (1 9 69 ). V ib r a t io n m o d u lu s o f n o r m a l ly co n so l id a t ed c l ay ( di scu ss io n ),

Proc. Am. Soc. Civil Eng.,J. Soil Mech. Found. Div. 95, 1531-1537.

Har d in , B . O . an d V . P . Dr n ev ich ( 19 7 2a ). Sh ea r m o d u lu s an d d am p in g in so i l s : Measu r em en t an d p a r a -me ter effects, Proc. Am. Soc. Civil Eng., J. Soil Mech. Found. Div. 98, 603-624.

Har d in , B . O . a n d V . P . Dr n ev ich ( 1 9 72 b ). Sh ea r m o d u lu s a n d d am p in g in so i l s : Des ig n eq u a tio n s an d

curves , Proc. Am. Soc. Civil Eng., J. Soil Mech. Found. Div. 98, 667-692.Ha skel l , N. A . (1953) . The d ispers ion of su r face waves on mul t i layered media , Bull. Seism. Soc. Am. 43,1 7 - 3 4 .

Hu dson , D. E . (1972). St rong mo t ion se ismology , Proc. Inter. Conf. Microzonation (Seattle, Washington),

1, 29-60.

Id r iss , I . M . and H. B. Seed (1967) . Response of Horizontal Soil Layers During Earthquakes, D e p a r t m e n tof C iv i l Engineer ing , U nivers i ty o f Cal i fo rn ia , Berkeley .

Id r i s s , I . M. an d H . B . Seed (1968). Se ism ic respo nse o f hor iz on ta l so i l l ayers , Proc. Am. Soc . Civ i l Eng . ,

J . So i l Mech . Found . D iv . 94, 1003-1031.

Iw an , W . D . (19 6 7) . O n a c l as s o f m o d e l s fo r t h e y i e l d i n g b eh a v i o r o f co n t i n u o u s an d c o m p o s i t e s y s tem s ,

J . A p p l. Me ch . 34 , 612-617 .

K an a i , K . (19 52 ). R e l a t i o n b e t w een t h e n a t u re o f su r f ace l ay e r an d t h e am p l i t u d es o f e a r t h q u ak e m o t i o n s ,

Bull . Ear thquake R es . Ins t . , Tokyo Univ. 30 , 31-37 .L ad d , C . C . an d L . E d g e r s (19 7 2) . C o n s o l i d a t ed -u n d ra i n ed d i r ec t - s im p l e s h ea r te s t s o n s a t u ra t ed c l ay s ,

R es ea rch R ep o r t R 7 2 -8 2 , D e p a r t m e n t o f C i v i l E n g i n e e r i ng , M a s s a c h u s et ts I n s t it u t e o f T e c h n o l o g y ,

C am b r i d g e , M as s ach u s e t t s .

L y s m er , J . an d R . L . K u h l em ey e r (1 96 9) . F i n i t e d y n am i c m o d e l fo r in f i n i te m ed i a , Proc. Am. Soe . Civ i l

Eng . , J . Eng . Mech . D iv . 95, 859-877.

M as i n g , G . (1 9 26 ). E i g en s p an n u n g en u n d V e r fe s t ig u n g b e i m M es s i n g , Proe. Intern. Congr. Appl . Mech.

332-335.

N e w m a r k , N . M . , A . R . R o b i n s o n , A . H . - S . A n g , L . A . L o p e z , a n d W . J . H a l l (1 97 2) . M e t h o d s f o r d e t e r-

m i n i n g s i te ch a rac t e r i s t ic s , Proc. Intern. C onf . M icrozon ation, (Se att le, Washington) 1, 113-129.

N e w m a r k , N . M . a n d E . R o s e n b l u e t h ( 1 9 7 1 ) . Fundamentals o f Earth quak e Engineering, P ren t i ce -H a l l ,

E n g l ew o o d C l i f f s, N . J .

R i ch t m y e r , R . D . an d K . W . M o r t o n (1 9 6 7 ) . Difference M etho ds fo r Ini t ial Value Prob lems, 2r id ed i t ion ,I n te r s ci e n c e, N e w Y o r k .

S ch n ab e l , P . , H . B . S eed , an d J . L y s m e r (19 72 ). M o d i f i ca t i o n o f s e i s m o g rap h r e co rd s fo r e f f ec t s o f l o ca l

s o i l co n d i t i o n s , Bull . Se i sm . Soc . Am . 62, 1649-1664.

S t r ee t e r, V . L . , E . B . W y l i e , an d F . E . R i ch a r t ( 1 97 4 ). S o i l m o t i o n co m p u t a t i o n s b y ch a rac t e r i s ti c s m e t h o d ,

Proe. Am. Soc. Civi l Eng., J. Geotech. Eng. Div. 100, 247-263.

T e rzag h i , K . an d R .B . P eck (19 67 ). SoilMechanics in EngineeringPraetice, 2 n d e d i t io n , W i l e y , N e w Y o r k .

W ar r i ck , R . E . (1 97 4 ). S e i s m i c i n v es t i g a t i o n o f a S an F ran c i s co B ay m u d s i t e , Bul l . Se i sm . Soc . Am. 64,

375-385.

U.S. GEOLOGICAL SURVEY

345 MIDDLEFIELD ROAD

MENLO PARK, CALIFORNIA 94025

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Calculation of Nonlinear Ground Response in Earthquakes

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