Topographic Effects on the Seismic Response of Steep Slopes741

7/27/2019 Topographic Effects on the Seismic Response of Steep Slopes741

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B u l l e t i n o f t h e Se i smo l o g i ca l So c i e t y o f Amer i ca , Vo l . 8 7 , No . 3 , p p . 7 0 1 -7 0 9 , Ju n e 1 9 9 7

Topographic Ef fect s on the Sei smic Response of Steep Slopes

by Scott A. Ashford, Nicholas Sitar , John Lysmer, and Nan Deng

Abs t r a c tA frequency-domain parametric study using generalized consistent

transmitting boundaries has been performed to evaluate the significance of topo-

graphic effects on the seismic response of steep slopes. The results show that the

peak amplification of motion at the crest of a slope occurs at a normalized frequency

1t / 2 = 0.2, where H is the slope height and 2 is the wavelengt h of the motion. The

importance of the natural site frequency is illustrated by the analysis of a stepped

layer over a half-space. It was found that the natural freq uency of the region behind

the crest can dominate the response, relative to the topographic effect, for the con-

ditions studied. Moreover, the effect of topography can be handled separately from

the amplification due to the natural freq uency of the deposit behind the crest of the

slope. This concept of separating the amplification caused by topograph y from that

caused by the natural frequency is advantageous to the deve lopment of a simplified

method to estimate topographic effects.

Introduction

"... The e f f ect o f t he v i bra t i on on t he hard pr i mary s l a te ,

whi ch compo ses t he f ound at i on o f t he i sl and , was s t i l l more

cur i ous : t he super f ic i a l par t s o f some n arrow r i dges were

as com pl e t e l y sh i vered as i f t hey had been b l as t ed by gun-

powder . T h i s ef fec t, wh i ch w as rendered consp i cuous by t he

f resh f rac t ures and d i sp l aced so il , mus t be con f i ned t o t he

near sur f ace , f or o t herw i se t here wou l d no t ex i s t a b l ock o f

so l i d rock t hrough out Chi le ; nor i s t h is i mprobabl e , as i t isknow n t ha t sur face o f a v i bra t i ng b ody i s a ff ec t ed d i ff e ren tl y

f ro m t he cen t ra l par t . I t i s, perhaps , ow i ng t o t h i s same rea-

son t ha t ear t hquakes do no t cause such t e rr if ic havoc w i t h i n

d e e p m in e s a s w o u ld b e e x p e c t e d . . . " (Barlow, 1933).

This quote by Charles Darwin describes the effects of

the 20 February 1835 Chilean earthquake and suggests that

topographic amplification of seismic motions is a phenom-

enon that has been well recognized for some time. Certainly,

in the recent past, there have been numerous cases of re-

corded motions and observed earthquake damage pointing

toward topographic amplification as an important effect. Ex-amples include observations from the 1971 San Fernando

earthquake (Boore, 1972), the 1983 Coalinga earthquake

(Celebi, 1991), the 1985 Chile earthquake (Celebi, 1987),

the 1987 Superstition Hills earthquake (Celebi, 1991), and

the 1994 Northridge earthquake (Ashford and Sitar, 1994).

As a result of such observations, a considerable amount of

work has been done in an attempt to model, quantify, and

predict these effects.

One of the first numerical studies of the effect of simple

topography on seismic response was carried out by Boore

(1972) using the finite-difference method. Subsequent stud-

ies on the effect of topography were conducted using finite

elements (e.g., Smith, 1975), boundary methods (e.g., San-

chez-Sesma et aL , 1982), and discrete wavenumber methods

(e.g., Bard, 1982). Geli et aL (1988) reviewed these studies

and others, and they found that all of them in essence con-

sidered the analysis of an isolated two-dimensional ridge on

the surface of a homogeneous half-space and that all yielded

consistent results: (1) the amplification of acceleration of nomore than 2 at the crest, peaking when the wavelength is

about equal to the ridge width, and (2) varying amounts of

amplification and attenuation along the surface of the slope

from the crest to the base. However, these results consider-

ably underestimate amplifications observed in the field,

which mostly range from 2 to 10, and up to as much as 30.

Geli et al . then analyzed a more detailed model configuration

using a layered profile and introduced nearby ridge effects,

and they arrived at conclusions similar to those o f the pre-

vious researchers. In addition, Geli et al . found that neigh-

boring ridges may have greater effect on site response than

layering and concluded that future models should be able toanalyze S V and surface waves, and three-dimensional geo-

logic configurations.

As indicated above, most research on topographic ef-

fects has thus far focused on ridges. Some o f the procedures

and concepts developed for the analysis of the ridges may

also be extended to steep slopes. However, there are also

significant differences between the response of steep soil

slopes and the response of rock ridges simulated as homo-

geneous half-spaces; foremost are the semi-infinite nature of

material in the horizontal direction behind the slope crest

and the potential of soil amplification of the motions. With

701



70 2 S.A . Ashford, N. Sitar, J. Lysmer, and N. Deng

t h i s i n mi nd , i t i s i mpor t an t t o r ev i ew s t ud i es unde r t aken

speci f ical ly for soi l s lopes .

One o f t he f i rs t s t ud ie s t o spec i f i ca ll y cons i de r t he se i s -

mi c r e spon se o f so i l s l opes was cond uc t ed by Id r i s s and S eed

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

l ands l i des gene ra t ed dur i ng the 1964 Al askan ea r t hquake

a n d c o n d u c t e d a p a r a m e t r ic s t u d y o f t h e r e s p o n s e o f 2 7 ° c la y

slopes , and la ter of 45 ° s lopes ( Idr i ss , 1968) , us in g t r iangu lar

v i scoe l a s t i c f i n i t e e l ement s t o mode l t he s l opes on r i g i d

f o u n d a t i o n s . W h e n c o n s i d e r i n g t h e N / S c o m p o n e n t o f t h e

18 M ay 1940 E1 Cen t ro se i sm ogram, t he au t hor s found t ha t

mag ni t ude o f t he peak su r f ace acce l e r a t i on was i n a l l ca ses

grea t e r a t t he c r e s t o f t he s l ope t han a t po i n t s l ower on t he

s lo p e . H o w e v e r , w h e n c o m p a r i n g t h e p e a k s u r f a ce a c c e l e r-

a t i on a t t he c r e s t t o t ha t a t some d i s t ance beh i nd t he c r e s t ,

t hey found t ha t whi l e i n some cases t he acce l e r a t i on a t t he

c r es t was much g rea t e r , i n o t he r cases t he r e was l i t t l e d i f -

f e r ence be t ween t he r e sponse a t t he c r e s t and t he r e sponse

a t some d i s t ance beh i nd t he c r e s t . Ver t i ca l mot i ons gen e ra t ed

b y t h e h o r i z o n ta l c o m p o n e n t o f t h e b a s e m o t i o n w e r e g r e a t-e s t nea r t he c r e s t o f t he s l ope ; however , t he ve r t i ca l com-

pone n t o f t he base m ot i on had l i t t le e f f ec t on t he hor i zon t a l

shea r s t r e sses w i t h i n t he embank ment . T he i r r e su l t s sugges t

t ha t the na t u r a l pe r i od o f t he so i l co l umn b eh i nd t he c r e s t o f

a s l ope was r e spon s i b l e fo r mu ch m ore am pl i f ica t i on o f t he

i npu t mot i on t han t he s l ope geom et ry i ts e l f.

K o v a c s e t a l . (1971) pe r fo rm ed l abora t o ry shak ing t ab l e

exper i m ent s on c l ay bank s , i n pa r t t o fu r t he r va l ida t e t he use

of t he f i n i t e - e l ement me t hod ( e .g . , Id r i s s and S eed , 1967) i n

t h i s t ype o f ana l ys i s . T he phy s i ca l mode l r e su l t s ag reed f a -

vorab l y wi t h t he f i n i t e - e l ement ana l yses , and Kov acs e t a l .

conc l ud ed t ha t t he t h i ckness o f t he so i l depos i t was t he p r e -

dom i nan t f ac t o r i n de t e rmi n i ng t he s i t e re sponse .

M ay (1980) s t ud i ed t he e f f ec t o f hor i zon t a l l y p ropaga t -

in g S H and L ove waves on ve r t i ca l s ca rps i n a ha l f - space

and a l aye r ove r a ha l f - space us i ng t he f i n i t e - e l ement

me t hod . H e foun d t ha t r e f lec t i on o f f t he sca rp f ace p l ayed a

l a rge ro l e i n t he r e sponse a nd t ha t t he e f f e c t o f t he sca rp

c o u l d b e r e l a t e d t o t h e r a ti o o f s l o p e h e i g h t / 4 a n d t h e w a v e -

l e n g th o f t h e m o t i o n u n d e r c o n s i d e r a ti o n . H e a l so p e r f o r m e d

t es ts us i ng an i ns t rum ent ed g r an i t e b l ock t o va l i da t e h is nu-

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

t w o m o d e l s .

S i t a r and C l oug h ( 1983 ) used an eq u i va l en t l i nea r , t wo -

d i mens i ona l f i n i t e - e lemen t mod e l t o ana l yze t he se i smi c r e -sponse o f s t eep s l opes i n weak l y cement ed sands and found

t ha t acce l e r a t i ons t ended t o be ampl i f i ed i n t he v i c i n i t y o f

t he s l ope f ace . T he i r r e su l ts show up t o a 70% ampl i f i ca t i on

a t t he c r e s t o f t he s l ope a s com pared t o t he f r ee f i e l d beh i nd

t he c r e s t . However , i n con t r a s t t o Ge l i e t a l . (1988) , t hey

no t ed t ha t t hese t opo graph i c e f f ec t s t ended t o be sma l l r e la -

t i ve t o t he am pl i f ica t i on t ha t occur s i n t he f r ee f i e l d due t o

the s i te per iod.

Our s t udy was mot i va t ed by an i n t e r e s t i n eva l ua t i ng

t he r e sponse o f s t eep coas t a l b l u f fs , such a s a r e found a l ong

m u c h o f t h e C a l i f o rn i a c o a st . W e e x p l o r e d a c o m p l e t e r an g e

o f s l o p e a n g l e s b e t w e e n 3 0° and 90 ° t o p rod uce a s e t o f

gene ra l i zed r e su l t s app l i cab l e t o s t eep s l opes . Bo t h ve r t i ca l l y

propaga t i ng S H a n d S V wave s we re cons i de red ; i n add it i on ,

wave sp l it t ing du e t o ve r t i ca l i nc i dence on an i nc l i ned s l ope

was i ncorpora t ed i n t o t he ana l ys i s . F i na l ly , t he r e sponses o f

a s t epped ha l f - space and a s t epped l aye r ove r a ha l f - space

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

f r eque ncy o f t he ma t e r i a l beh i nd t he s l ope c r e s t .

C o m p u t a t i o n a l M o d e l

T he comput a t i ona l mode l used i n t he s t udy i s t he gen-

e r a l i zed cons i s t en t t ransmi t t i ng boun dary (GCT B) deve l oped

and va l i da t ed by Deng (1991) fo r t wo-d i mens i ona l s e i smi c

s i t e - r e sponse ana l ys i s . Kause l and T assou l as (1981) com-

pared seve ra l t ypes o f t r ansmi t t i ng boundar i e s and found

exce l l en t ag reement be t ween t he cons i s t en t t r ansmi t t i ng

b o u n d a r y a n d t h e c l o s e d - f o r m s o l u t io n f o r t h e p r o b l e m o f a

t i me-ha rmoni c an t i p l ane l i ne l oad ac t i ng on a s t r a t um. T he

genera l i zed cons i s t en t t ransmi t t i ng boun dary i s an ex t ens i onof t he cons i s t en t t r ansmi t ti ng bound ary , deve l oped by L ys -

me t and W aas (1972) , t o a ll ow fo r a boun dary o f a rb i t r a ry

shape.

O n e o f t h e k e y a d v a n c e ' s b y D e n g i s th e f o r m u l a t i o n o f

t he so l u t i on t o t he equa t i on o f mot i on a l ong an a rb i t r a r i l y

shaped bo und ary i n a l aye red sys t em, spec i f i ca l ly a l ong a r ec -

t i li nea r curve , and t h i s r epresen t a t i on i s t he bas i s f o r t he fo r -

mul a t i on o f t he GCT B. T he p r e f i x "ge ne ra l i zed" r e f e r s to t he

ab i l i t y o f t hese e l ement s t o conform t o a rb i t r a r i l y shaped

bounda r i e s . T he f r equenc y-do mai n mode l i s li nea r v i scoe l a s -

t i c and u t i l i ze s t he compl ex r e sponse me t hod . Onl y t he key

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

T he GCT B is f o rm ul a t ed by us i ng t he exa c t ana l y t i ca l

so l u t i on i n t he hor i zon t a l d i r ec t i on and a d i sc r e t i zed fi rs t - o r

second-orde r d i sp l acement shape func t i on a l ong t he a rb i -

t r a r i l y shaped boundary (Deng , 1991) . T he boundar i e s o f

t hese e l ement s t r ansmi t ene rgy accura t e l y i n t he hor i zon t a l

d i r ec t i on and r epresen t t he pe r f ec t " i n f i n i t e " boundary con-

d i t ion . Nod a l po i n t s ex i s t on l y a t the bounda r i e s be t ween t he

r eg i ons , and on l y t he mot i ons a t t he noda l po i n t s need t o be

so l ved i n t he g l oba l equa t i ons o f mot i on . Once t he noda l

po i n t mot i ons a r e ob t a i ned , t he mot i ons w i t h i n each r eg i on

c a n b e r e c o v e r e d t h ro u g h a n o d a l e x p a n s i o n p r o c e ss .

A s l ope mode l us i ng t he GCT B met hod i s p r esen t ed i n

F i gure 1 . T he s i te i s d i v i ded i n t o t wo semi - i n f i n i te r eg i onson t he l e f t and r i gh t s i des , r e spec t i ve l y . E ach r eg i on i s d i -

v i ded i n t o a g ro up o f pe r f ec t l y hor i zon t a l l aye r s , w i t h ma-

t e r i a l p rope r t i e s va ry i ng f rom l aye r t o l aye r , and wi t h t he

en t i r e mo de l r e s t i ng on a s i mu l a t ed v i scoe l a s ti c ha l f -space .

M at e r i a l P rope r t i e s

T he m a t e r i a l p rope r t i e s used t o mode l each l aye r cons i s t

o f t he mass dens i t y ( p ) , t h e S - w a v e v e l o c i t y ( V s ) , t h e P - w a v e

ve l oc i t y ( V e ) , and t he r e spec t i ve va l ues o f t he f r ac t i on o f

c r i ti ca l damp i ng fo r each wav e t ype . T he m a t e r i a l p rope r t i e s

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



T o p o g r ap h ic E f f e c t s o n t h e S e i s m i c R e s p o n s e o f S t e e p S l o p e s 70 3

- - L e f t G TE

Z --

R i g h tG T E _

i i i i ! i i i i i ! i i i ! i i i ! i i - ] C o n t l ; , - ¢ d d t i o , - , n o u t i o t o n

Fig ure 1. Generalized consistent transmittingboundary representation of a steep slope.

( G ) i s o b t a i n e d f r o m t h e s h e a r - w a v e v e l o c i t y a n d m a s s d e n -

s i t y u s in g t h e r e l a t i o n sh ip G = ( V s ) 2 p , an d th e co n s t r a in ed

m o d u l u s i s o b t a in e d i n a s i m i l a r m a n n e r . M a t e r i a l d a m p i n g

i s a c c o u n t e d f o r t h ro u g h t h e u s e o f a c o m p l e x m o d u l u s . I n

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

d e f in ed a s

G ' = G (1 - 2 f l2 + i 2 f l s 1 , / ~ s )

M" = Mc(1 - 2 f l2 + i 2 f l e S f l 2 )

(1 )

(2 )

w h e r e f l s a n d f i e a r e t h e f r ac t i o n s o f c r i t i ca l d am p in g fo r S

an d P w av es , r e sp ec t iv e ly . T h o u g h i t i s p o ss ib l e t o u se d i f -

f e r e n t v a l u e s f o r f l s an d t i p , b o th w ere a s su m ed to b e t h e

s a m e i n t h e a n a l y s e s p r e s e n t e d h e r e in . A s n o t e d a b o v e , t h e

m o d e l i s l i n ea r v i sco e l a s t i c . I f t h e e f f ec t o f so i l n o n l in ea r i t y

i s t o b e t ak en in to ac co u n t , t h en s t r a in - co m p a t ib l e so i l p ro p -

e r t i e s m u s t b e d e v e l o p e d e l s e w h e r e .

S i m u l a t i o n o f a S e m i - I n f i n i te B a s e

In o rd e r t o b e t t e r m o d e l r ea l i s t i c co n d it i o n s , a s im u la t ed

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

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

t e c h n i q ue s a r e u s e d i n o r d e r t o m i n i m i z e t h e e n e r g y r e f l e c t e d

b a c k i n t o t h e m e s h t o a c c o u n t f o r r a d i a t io n d a m p i n g ( D e n g ,

1 9 9 1 ) . T h i s i s acco m p l i sh ed b y ad d in g ad d i t i o n a l l ay e r s t o

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

w a s o r i g i n a ll y v a l i d a t e d b y C h e n e t a l . (1 9 8 1 ) ag a in s t ex ac t

so lu t i o n s fo r a u n i fo rm h a l f - sp ace , a s i n g l e l ay e r o v e r a h a l f -

s p a c e ( M o o n e y a n d B o l t , 1 9 6 6 ), a n d t w o l a y e r s o v e r a h a l f -

s p a c e ( S t o n e l e y , 1 9 5 7 ). E x c e l l e n t a g r e e m e n t w a s f o u n d i n

a l l c as e s . O n e o r a c o m b i n a t i o n o f b o th t e c h n i q u e s a r e c o m -

m o n l y u s e d i n f i n i t e - e l e m e n t a n a l y s e s o f s e i s m i c s i t e r e -

sp o n se ( e . g . , H u d so n e t a l . , 1 9 9 4 ; L y s m e r e t a l . , 1981) .

T h e f i r s t t ech n iq u e i s t o ad d so m e ad d i t i o n a l l ay e r s t o

th e o r ig in a l m o d e l o f t h e s i te . T h e t o t a l t h i ck n ess o f t h e

ad d i t i o n a l l ay e r s v a r i e s w i th f r eq u en cy an d i s s e t t o 1 . 5

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

q u e n c y u n d e r c o n s i d e r a ti o n , a s r e c o m m e n d e d b y C h e n e t a l .

(1 9 8 1 ) . T h e ch o ice o f t h i s th i ck n ess i s b ase d o n t h e o b se r -

v a t i o n t h a t f u n d a m e n t a l - m o d e R a y l e i g h w a v e s i n a h a l f -

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

p l i t u d es a t a d ep th co r r e sp o n d in g to 1 . 5 w av e l en g th s , an d

b e c o m e i n s ig n i f ic a n t a t a d e p t h o f t w o w a v e l e n g t h s ( O s t a -

d an , 1 9 8 3 ). I n o u r an a ly s i s , t h e t o t a l t h i ck n ess o f t h e l ay e r s

i s ad ju s t ed acco rd in g t o t h e f r eq u en cy u n d e r co n s id e ra t i o n ,

a n d a l l l a y e r s i n t h e e x t e n d e d r e g i o n a r e o f u n i f o r m t h i c k -

n ess .

T h e s e c o n d t e c h n i q u e , a l s o r e c o m m e n d e d b y C h e n e t

a l. ( 1 9 81 ) , i s t o a t ta c h L y s m e r - K u h l e m e y e r (1 9 6 9 ) v i s c o u s

d ash p o t s a t t h e b ase o f t h e s im u la t ed h a l f - sp ace , t h u s t h e

b a s e b e c o m e s a v i s c o u s b o u n d a r y i n s t e a d o f a r i gi d b o u n d -

a r y . S i n c e t h e d a s h p o t r e p r e s e n t a t i o n o f t h e h a l f - s p a c e i s

o n l y e x a c t f o r t h e v e r t i c a l ly p r o p a g a t i n g P a n d S w a v e s , a n d

s in ce t h e d i r ec t i o n s o f t h e sca t t e r ed m o t io n s a r e u su a l ly u n -

k n o w n , t h is t e c h n i q u e i s a p p r o x i m a t e i n t h e s e n s e t h a t s o m e

o f t h e s c a t t e r e d e n e r g y m a y s t il l b e a b l e t o b o u n c e b a c k i n t ot h e s y s t e m . H o w e v e r , t h e u s e o f b o t h t e c h n iq u e s g i v e s v e r y

sa t i s f ac to ry r e su l ts i n m o s t p r ac t i ca l p ro b l em s (D en g , 1 9 9 1 ).

A n a l y s i s o f a S t e p p e d H a l f - s p a c e

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

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

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

u s e f u l f o r th e d e v e l o p m e n t o f a n u n d e r s t a n d in g o f t h e p a -

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

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

h e ig h t an d th e w av e l en g th . T h i s a l l o w s th e an a ly s i s t o fo cu s

o n t h e r e l a t i o n s h i p b e t w e e n t h e s e t w o p a r a m e t e r s w i t h o u t

h av in g to i n co rp o ra t e t h e n a tu ra l f r eq u en cy o f t h e s it e . O n ce

th i s r e l a t i o n sh ip i s ex am in ed , t h en t h e i n f lu en ce o f o th e r

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

asses sed .

T h e m o d e l u s e d i n t h e a n a l y s e s i s s i m i l a r t o t h a t s h o w n

in F ig u re 1 , w h ere t h e l e f t an d r i g h t G C T B an d th e v i sco -

e l a s ti c h a l f - s p a c e h a v e u n i f o r m p r o p e rt i e s : a s h e a r - w a v e v e -

lo c i t y o f 3 0 0 m / sec an d a P o i s s o n ' s r a t i o o f 0 .3 , w i th a s l o p e

h e ig h t H o f 3 0 m . T h e co n t ro l p o in t fo r t h e i n p u t m o t io n i s

l o c a t e d 9 0 m d i r e c t ly b e l o w t h e b a s e o f t h e s l op e . T h e f r a c -

t i o n o f c r i t ica l d am p in g ( f l) w as v a r i ed f r o m 1 t o 2 0 % , a n d

i t w a s f o u n d t h a t d a m p i n g h a d v e r y l i tt l e e f f e c t o n a m p l i f i-ca t i o n a t t h e c r e s t o f t h e s lo p e . T h u s , t h e r e su l t s fo r 1 %

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

su l t s fo r 5 % a re p resen t ed fo r t h e f i r s t ca se fo r co m p ar i so n .

T h e r e su l t s o f t h e an a ly ses a r e p resen t ed a s a fu n c t io n

o f 1-1/2, i .e . , t h e r a ti o o f t h e s lo p e h e ig h t an d th e w av e l en g th

o f t h e m o t io n u n d e r co n s id e ra t i o n . T h i s d e f in i t i o n o f t h e n o r -

m a l i z e d w a v e l e n g t h d i f f er s f r o m e a r l i e r s tu d i e s o f r i d g e e f -

fects (e .g . , Boore, 1972 ; Gel i e t a L , 1 9 8 8 ) an d d am s ( e . g . ,

G aze t a s an d D ak o u las , 1 9 9 2 ) , i n w h ich t h e co r r e l a t i o n w as

m a d e b e t w e e n t h e w a v e l e n g t h a n d t h e w i d t h o f t h e t o p o -

g rap h ic f ea tu re b u t i s s im i l a r t o t h e "d im en s io n l e s s f r e -





Topograph ic E f fec ts on the Se ismic Response o f S teep S lopes 7 0 5

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

t h o u g h in an y case , t h e am p l i f i ca t i o n i s o n t h e o rd e r o f 1 5

to 2 0 % o f t h e f r ee - f i e ld m o t io n . In ad d i t i o n , a t t en u a t io n o c -

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

c res t . A t l o w v a lu es o f H / 2 , a t w h ich t h e t o p o g rap h ic s t ep i s

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

o n t h e r e sp o n se .

E f f e c t o f S V ( I n - P l a n e ) W a v e s o n a V e r t i c a l S l o p e

F o r t h e a n a l y s i s o f t h e r e s p o n s e t o S V w a v e s , b o t h a

h o r i z o n t a l a n d v e r t i c a l c o m p o n e n t n e e d t o b e c o n s i d e r e d .

S i n c e t h e i n p u t m o t i o n o n l y c o n s i s t s o f h o r i z o n ta l m o t i o n ,

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

t i v e t o t h e h o r i zo n ta l i n p u t m o t io n , an d th e v e r t i ca l am p l i -

f i ca t i o n i s r e l a t i v e t o t h e f r ee - f i e ld h o r i zo n ta l r e sp o n se .

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

S V w a v e a r e s h o w n i n F i g ur e 4 f o r f r e q u e n c ie s r a n g i n g f r o m

0 . 1 t o 1 0 H z . T h e r e su l t s a r e v e ry s im i l a r t o t h o se o b t a in ed

fo r t h e S H w a v e s . T h e f i r s t p e a k a m p l i f i c a t i o n o c c u r s a t

H I 2 = 0 .2 , a n d t he s e c o n d p e a k o c c u r s a t H / 2 = 1.0. Th em a g n i t u d e s o f b o t h a m p l i f i c a t io n p e a k s a r e o n t h e o r d e r o f

5 0 % , w h i c h a r e h i g h e r t h an t h o s e o b s e r v e d f o r S H w a v e s ,

w i th t h e seco n d p eak s ig n i f i can t ly so . T h e p a t t e rn o f a t t en -

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

t h e s l o p e i s a l s o s i m i l a r t o t h e S H - w a v e c a s e , t h o u g h t h e

m a g n i t u d e s a r e g r e a t e r f o r t he S V case .

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

i n F i g u r e 5 . T h e v e r t i c a l r e s p o n s e i s m o s t p r o n o u n c e d a t t h e

c res t o f t h e s lo p e , an d a t H / 2 > 0 . 2 , i t i s g r ea t e r t h an th e

f r ee - f i e ld h o r i zo n ta l r e sp o n se . T h o u g h n o t p r e sen t ed h e re in ,

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

d am p in g (A sh fo rd an d S i t a r , 1 9 9 4 ) . T h e am p l i f i ca t i o n o f t h e

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

5 0 % o f t h e f r e e - f ie l d m o t i o n a n d d e c r e a s e s w i t h i n c r e a s e d

d am p in g . F in a l l y , t h e am p l i t u d e o f t h e v e r t i ca l r e sp o n se a t

t h e c r e s t t en d s t o i n c rease w i th i n c reas in g f r eq u en cy an d

s e e m s t o b e i n d e p e n d e n t o f t h e h o r i z on t a l r e s p o n s e a t f r e -

q u e n c i e s a b o v e H / 2 > 0 .2 .

E f f e c t o f S l o p e A n g l e

T h e e f f e c t o f s l o p e a n g l e o n t o p o g r a p h i c a m p l i f ic a t i o n

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

F ig u re 1 . S in ce s t ee p s lo p es a r e t h e su b j ec t o f th i s s t u d y ,

o n ly s lo p es b e tw e en 4 5 ° an d 9 0 ° a r e co n s id e red (3 0 ° an d 9 0 °

fo r S V w a v e s ) . T h e s l o p e - c r e s t a m p l i fi c a t i o n o f t he S H - w a v e

f r ee - f i e ld m o t io n i s sh o w n in F ig u re 6 . W i th d ec rea s in g s lo p e

an g le , t h e m a g n i tu d e o f t h e am p l i f i ca t i o n a t t h e f i r s t p ea k

d e c r e a s e s f r o m a b o u t 2 5 % t o a b o u t 1 5 % , w h i l e t h e re s p o n s e

a t h ig h e r f r eq u en c i e s t en d s t o i n c rease t o ab o u t 5 0 % , w i th

n o a p p a r e n t s e c o n d p e a k . T h e h o r i z o n t a l r e s p o n s e d u e t o S V

w a v e s i s s h o w n i n F i g u r e 7 . I n g e n e r a l, t h e m a g n i t u d e o f t he

am p l i f i ca t i o n d ec reases w i th d ec reas in g s lo p e an g le , f ro m

a b o u t 5 5 % t o a b o u t 1 5 % , f o r H / 2 < 0 . 4 . R esu l t s a t h ig h e r

f r e q u e n c ie s , a b o v e H / 2 = 0 . 4 , i n d i ca t e n o c l ea r t r en d . T h e

v e r t i c a l r e s p o n s e d u e t o S V w a v e s i s p r e s e n t e d i n F i g u r e 8 .

A s w i th t h e h o r i zo n ta l r e sp o n se , t h e v e r t i ca l r e sp o n se d e -

c reases w i th d ec reas in g s lo p e an g le ; h o w ev er , t h e re i s n o

re l a t i o n sh ip b e tw een th e am p l i f i ca t i o n an d th e n a tu ra l f r e -

q u en c y o f t h e l ay e r b eh in d th e s lo p e c r e s t .

A n a l y s i s o f a S t e p p e d L a y e r o v e r a H a l f - s p a c e

T h e p a r a m e t r i c s t u d y o f t h e s t e p p e d h a l f - s p a c e p r o v id e s

a b as i c u n d e r s t an d in g o f t h e in f lu en ce o f t o p o g rap h y o n s i t e

r e sp o n se . T h e n ex t s t ep i s t o ev a lu a t e t h e r e l a t i o n sh ip b e -

2zOm

< 1 . 5(..)L L

,--.I13 -

< 1~. , - I

<I. -zON 0 . 5E :O'1 -

a

~ . ' ~ . /....~. rl .

I I I

0 . 3

S L O P E H E I G H T / W A V E L E N G T H

/I

I I I I i I I I t I i

0 . 0 1 0 . 0 3 0 . 1 1

Fig ure 4. Horizontal amplifications for vertically incident S V wave on a steppedhalf-space for various distances behind crest, fl = 1%.





Topographic Effects on the Seismic Response of Steep Slopes 7 07

zOiF-<L)L L

, . d

f t .

<

1 .5

0 .5

9 0 d e g r e e s lo p ev

7 5 d e g ~ e s l o p e

60 degr~ .e s lope

4 5 d e r ~ s lo pe

L ~ . .t ~ .. ~ t" s ~ , = q - ~ " ~ - . ~ " ~

i i i i i i i i i i i i i

0 . 0 1 0 . 0 3 0 .1 0 . 3 1

SLOPE HEIGHT/WAVELENGTH

Fig ure 6 . Horizontal amplification at the crest for a vertically incident S H wave onan inclined slope, fl = 1%.

zOF-<

L L

Q .

<

2

1 .5

0.5

I90 degr~ee s lope

v

75 dec l~ .e s lope

60 degr~..e,s lope

45 degrge s lope I30 deg.r~_es lo p e ~ i

i i i i i i i i

0.01 0.03 0.1 0.3

SLOPE HEIGHT/WAVELENGTH

" Y ' % V . . . r ~ i

Fig ure 7 . Horizontal amplification at the crest for a vertically incident S V wave onan inclined slope, fl = 1%.

p o in t s . F i r s t , t h e n a tu ra l f r eq u en cy o f t h e s i t e h as a g rea t e r

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

h ig h l ev e l s o f im p e d an c e r a t io . S eco n d , i t ap p ea r s t h a t t h e

t o p o g r a p h i c a m p l i f i c a t i o n c a n b e a d d e d o n t o t h e a m p l i f i c a -

t i o n cau sed b y th e n a tu ra l f r eq u en cy .

C o n c l u s i o n s

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

o f a s t e p p e d h a l f - s p a c e a n d a s t e p p e d l a y e r o v e r a h a l f- s p a c e ,

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

a s t e e p s l o p e o n t h e s e i s m i c r e s p o n s e o f t h a t s l o p e c a n b e

n o rm al i zed a s a fu n c t io n o f t h e r a ti o o f th e s lo p e h e ig h t (H )

a n d t h e w a v e l e n g t h o f t h e m o t i o n ( 2 ) . T h e r e l a t i o n sh i p b e -

t w e e n s l o p e h e i g h t a n d w a v e l e n g t h w a s a l s o n o t e d b y M a y

(1 9 8 0 ) fo r h o r i zo n ta l ly p ro p ag a t in g S H w a v e s i n c i d e n t o n a

v e r t i ca l s ca rp , an d s im i l a r r e l a t io n sh ip s w e re o b se rv ed b e -

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

B o o re , 1 9 7 2 ; G e l i et aL , 1 9 8 8 ; D ak o u las , 1 9 9 3 ) .

F o r b o t h S H a n d S V w a v e s , t h e m a g n i t u d e o f t h e r e -

sp o n se a t t h e c r e s t o f t h e s lo p e i s s ig n i f i can t ly r ed u ced b y

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





Topographic Effects on the Seismic Response of Steep Slope s 709

arately from amplification due to the natural frequency of

the layer behind the crest of the s lope. T his conc ept of sepa-

rat ing the ampl i f icat ion caused by topography f rom that

caused by the natural f requenc y i s advantageous to the de-

velopment of a s impl i f ied method to est imate topographic

effects .

A c k n o w l e d g m e n t s

This research was supp or ted by the U.S. Geolog ical Survey under USGS

Aw ard Numb er 14-08-0001-G2127. The v iews and conclusions con ta ine d

in th is ar t ic le are those o f the au thors and should no t be in terpre ted as

necessar i ly represen t ing the o f f ic ia l po l ic ies , e i ther expressed or impl ied ,

of the U.S. governm ent . Par t ia l suppor t was a lso p rov ided by the Cal i fo rn ia

Depar tm ent o f Transpor ta t ion . The f inancia l suppor t o f the donors is g rea t ly

appreciated by the authors.

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De p a r tm e n t o f Ap p l i e d M e c h a n ic s a n d E n g in e e r in g Sc i e n c e

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Manu scr ip t received 19 Octobe r 1995 .

Documents

Topographic Effects on the Seismic Response of Steep Slopes741