Seismic response for alluvial valleys subjected to SH, P and SV waves

E L S E V I E R 0 2 6 7 - 7 2 6 1 ( 9 4 ) 0 0 0 4 9 - 2

Soil Dynamics and Earthquake Engineering 14 (1995) 249-258 © 1995 Elsevier Science Limited

Printed in Great Britain. All fights reserved 0267-7261/95/$09.50

Seismic response for alluvial valleys subjected to SH, P and SV waves

K. L. Fishman & S. Ahmad Department o f Civil Engineering, State University o f N. Y. at Buffalo, USA

(Received 1 September 1994; revised 26 September 1994; accepted 4 October 1994)

Results of an extensive numerical study on two-dimensional wave scattering by valleys of semi-elliptical cross-section due to incident SH, SV and P waves are presented. The investigation has been conducted using a rigorous boundary element algorithm. The influence of key parameters, such as, valley depth, iml~edance ratio, frequency, and angle of incidence on surface ground motion are studied in detail. Furthermore, the case of a valley within a layered half-space is analyzed and results compared with those obtained for a valley within a homogeneous half-space.

INTRODUCTION

Field observations after large earthquakes show that local site has an important impact on amplification of ground motion. Although numerous factors may have contributed significantly to such observations, it has been suggested that one of the principal causes for the observed distributions of damage may be attributed to the amplification of incident waves by local geological and soil conditions. For example, during the Michoacon, Mexico earthquake of 1985, great damage occurred to the structures atop the alluvial of the Mexico City sedimentary basin while the damage to the buildings atop the base rocks at the outskirts of the basin was considerably smaller (Anderson et al.4). If one takes into account that most of the damage occurred in Mexico City, which is more than 350 km away from the earthquake epicenter, the role of the site effect in the amplification of strong ground motion appears to be of considerable importance. During the Lima, Peru earthquake of 9 November 1974, accelerograms recorded at two sites with nearly the same epicenter distance dis- played pronounced dissimilarities (Zarhardnik & Hron27). The first site in central Lima, which was located atop weak horizontally varying subsurface structure, experienced only minor shaking. The second site, atop the La Molina sedimentary filled valley was subjected to severe shaking during the same earthquake. Similar phenom- enon was also observed during the Niigata, Japan earthquake of 1964. From all these case studies it is evident that there is a need to explain in detail how local site conditions infiuenc~, ~ ground motions during an earthquake.

249

Since the initial work of Trifunac 23 on the two- dimensional response of a semi-circular canyon subjected to SH-wave excitation, a considerable amount of work on wave scattering by canyons and valleys has been reported in geotechnical and seismology literature. A number of numerical methods have been used to solve wave propagation problems, such as the finite difference method, finite element method, and boundary element method. Among these methods, in recent years the boundary element method has become very popular for analysis of site response problems.

The Boundary Element Method (BEM) is well suited to deal with wave propagation problems because it avoids the introduction of fictitious boundaries, and reduces the dimensionality of the problem by one. The two-dimensional scattering and diffraction of SH waves of arbitrary angle of incident from irregular canyon- shaped topography has been formulated in terms of integral equations by Wong & Jennings. 26 BEM has also been coupled with the finite element method to solve wave propagation problems (Ayala & Gomez, 3 Shah et al.21). Another approach which combines the boundary integral equation method with the finite difference method has been presented by Cole et al. 6 for elastodynamics problems.

Direct Boundary Element formulations have been applied to solve two-dimensional scattering of harmonic elastic waves by canyons (Sanchez-Sesma, 14'17 Sabina et al.13), alluvial deposits (Sanchez-Sesma & Esquviel, 13 Dravinski7'8), and ridges (Sanchez-Sesma & Esquviel, 16 Sanchez-Sesma et al.18), for different types of waves and shapes of the scatterers. Sanchez-Sesma 19 and Sanchez- Sesma et al. 2° also used the direct BEM to solve a few

250 K. L. Fishman, S. A h m a d

three-dimensional problems of wave scattering. The frequency domain indirect-BEM has been used by Vogt et al. 24 to calculate generalized scattered motion in the region of a canyon of arbitrary shape in a horizontally layered half-space. Also, Luco et al. 12a has used an indirect BEM to obtain the three-dimensional response of an infinitely long canyon of uniform but arbitrary cross-section cut in a layered viscoelastic half-space.

The majority of the studies reported to date have considered two-dimensional (plane strain) wave scattering by idealized semi-circular or elliptical shaped canyons or valleys within a homogeneous half-space, subjected to incoming SH waves. Few results have been obtained which consider incident P and SV waves, or a layered half-space. Kawase l~ considered wave scattering of semi-circular canyons in response to incoming SH, SV, P and Rayleigh waves. Two angles of incidence, 0 ° and 30 ° were considered, but for all analyses the dimensionless frequency ratio was kept constant. Toborovska & Lee 22 investigated scattering of SH waves by shallow circular valleys. The valleys were shallow since only a portion of the circular shape was embedded in the half- space such that valley side walls need not be perpendi- cular to the half-space surface. Zhao & Valliappan 2s considered V shaped, rectangular, and trapezoidal shaped canyons. Harmonic SV and P waves with vertical incidence as Well as real earthquake input motions were applied to the half-space. The effect of weathered canyon walls was also studied. Bielak et al. 5 considered wave scattering by nonhomogeneous semi-circular valleys within an isotropic, homogeneous, linear, elastic half- space subjected to SH waves with an angle of incidence of 0 ° or 60 °. Hadley et al. l° constructed an admittance function over a wide range of frequencies to describe wave scattering by inclusions, peaks, and valleys within a layered half-space subjected to verticaUy incidence SH

12 waves. Liu et al. studied wave scattering by semi- elliptical shaped valleys within a layered half-space subjected to incident P, SV and SH waves. The effect of angle of incidence as well as the plane of wave propagation was investigated. A limited parametric study was performed covering a range of frequencies and valley depth to width ratios.

In this paper a rigorous algorithm of the direct BEM for steady-state dynamics is used to conduct an extensive parametric study of two-dimensional wave scattering by elliptical valleys within an elastic half-space (or layered soil) due to incident SH, SV and P waves. The influence of key parameters, such as depth and shape of the valley, ratio of shear wave velocity of the valley to that of the underlying soil, excitation frequency, incidence angle, and layered soil profile are studied and discussed. The study of these key parameters on surface ground motion for all types of body waves presented herein is more comprehensive and systematic compared to the earlier work cited. The purpose is to provide the practitioner with a range of possible ground responses, emphasizing

the most significant, and complex interrelationships between attributes affecting wave scattering.

BOUNDARY INTEGRAL FORMULATION

The boundary integral equation of steady-state elastodynamics for an isotropic body occupying a volume V and bounded by a surface S can be expressed as (Cruse & Rizzo 6a)

oJ) t/(x, o))

- F i j ( x , ~ , w ) a i ( x , w ) ] d S ( x ) (1)

where x is source point, ~ is the field point; w is the circular frequency of excitation; cij is discontinuity term; and ui and t~ are the ampfitudes of the i-th component of displacement and traction vectors respectively.

In eqn (1), Gij and Fij are the fundamental solution tensors representing the displacements and tractions at x in direction i due to a unit harmonic force applied at

in the j-direction. For two-dimensional steady-state dynamics, these fundamental solutions can be found in Ahmad & Banerjee. 2

Analytical integration of (1) is almost impossible, except for a few idealized cases. Hence, (1) has to be numerically integrated. For this purpose, the boundary of the domain is discretized into number of elements, and quadratic variation of the field quantities is assumed over each element.

After carrying out the necessary numerical integra- tions and applying the known boundary conditions, the system of boundary element equations can be put into the following matrix form (Abroad & Banerjee2),

AX = BY (2a)

AX = b (2b)

where Y and X are the vectors of known and unknown field quantities, respectively; and A and B are the coefficient matrices related to X and Y, respectively.

Finally, the vector X of unknown displacements and tractions on the boundary are obtained by solving this linear system matrix equation. Knowing the boundary values, displacements at selected interior points can be determined by eqn (1) with ciy as unit diagonal matrix.

This BEM formulation solves the layered media problem by considering each layer as a separate homogeneous region, forming the BEM equations indepen- dently, then assembling and solving these equations together by satisfying equilibrium and compatibility across common interfaces (Ahmadl). Material damping (linear hysteretic type) has also been incorporated in the present BEM formulation by a simple transformation of the Lam6 constants into a complex form (Ahmad & Banerjee2).

Alluvial valley seismic response 251

/ / f lai l space /7

w a v 6

z

Fig. 1. Semi-elliptical valley and surrounding half-space.

PROBLEM DEFINITION

Figure 1 represents a half-space containing a valley with an elliptical cross-section. Soil is assumed to be isotropic, homogeneous and linear elastic. The contact between valley and half-space is assumed to be perfectly bonded. For studies involving a layered half-space a soft upper layer having a thickness equal to the width of the alluvial valley is considered. The incident SH, SV and P waves are assumed to have an incident angle 0, oscillation frequency ~v, and a unit amplitude of displacement.

COMPARISON WITH PUBLISHED RESULTS

To establish the accuracy and reliability of the present analysis, results are compared with those from a limited number of cases available in the literature.

Valleys within a homogew~ous half-space

Vertically incident SH waves, corresponding to the two-dimensional antiplane shear problem were considered by Wong & Trifunac. 2s Using the present BEM two semi-elliptical alluvial valleys have been analyzed: (a) a shallow valley with depth to half width ratio R = ~ = 0.15 and (b) a deep valley with R = 1.0. The value of dimensionless frequency was r /= ~vL/TrVs = 0"5, Comparison in Fig. 2, of the present results (lines) with exact solutions of Wong & Trifunac 25 (points) show good agreement.

There is no exact solut~ion for the response of a valley due to incident P and SV waves. Based on the Boundary Element Method, Mossessian & Dravinski 9 have presented results for the two-dimensional case of a semi- elliptical valley subjected to P and SV waves. Parameters are as follows: dimensionless frequency 7/= 0.5; Pois- son's ratio = 1/3; mass density ratio PR/PE = 2/3; shear wave velocity ratio V~./Vs = 2.0. In this example, cv = circular frequency, a = half-width of valley, p = mass density, V~ = shear wave velocity; and subscripts R, E represent valley and elastic half=space respectively.

(a)

E ,<

E

Q.

12

10

8

6

4

2

0

R,,0.15, ETA,, 0.5,vsl/vs2-0.5

/ t ,

(b)

-3 -2 -1 0 1 2 X/L

12

10 " 0

ZL S E

s Q

E

V i~ 2

o

//~ R,,1.0, ETA,, 0.S

-3 -2 -1 0 1 2 3 X/L

Fig. 2. Comparison with results of Wong & Trifunac, 2s (a) shallow valley, (b) deep valley.

The total displacement amplitude at the surface is plotted against the distance X / L . Comparisons are made for 0 = 0 ° and 30 ° in Fig. 3 and Fig. 4 for P and SV waves, respectively. The solid and dashed lines represent results obtained by the present BEM. Solid lines are horizontal surface displacement and dashed lines represent vertical surface displacement. The symbols represent results obtained by Mossessian & Drivinski. 9 It is evident from Fig. 3 and Fig. 4 that the present results are in complete agreement with those obtained by Mossessian & Drivinski. 9

Valley within a layered half-space

The BEM is also used to study scattering of incident waves by an elliptical valley in a layered half-space. Results are compared with those obtained by Lui et al. 12 The elliptical valley has width of 400 m, and depth of 200 m. The layered half-space consists of a soft layer with H = 1000 m overlying bedrock. Material properties are listed in Table 1. The material properties of the valley and the underlaying bedrock are normalized with respect to those of the upper layer. The spatial coordinates are normalized with respect to the layer thickness H. In Figs 5 and 6, comparisons of normalized displacement amplitudes on the surface are presented. The symbols represent results obtained by Liu et al l 2 Both results agree well.

252 K. L. Fishman, S. Ahmad

(a) 8

"O ~ e E <

E ¢B g ~ 2

Surface Amplitude Displacement

P Wave,valley,Theta-O

• "~". R-0.5, ETA- 0.5 ..' •

: - V$1Ns2-0.5

.. / "..

. . . . . . . . . . -e....., . . . . . -e" " ' i - , . - ...,..e..- . . . . . . . . .

-3 -2 -1 0 1 2

X/L

(b) e Surface Amplitude Displacement

~=s Q, E <

~ 4 E

m .~ 2 Q

P Wave,vaUey,Theta-30

, " ' . . . P,-0.5, ETA,. 0.5 ..." '..

.. ~.. VslNs2-0.5

. " " i "

~ .. . . . . . . .

-3 -2 -1 0 1 2

X/L

Fig. 3. Comparison with results of Mossessian & Dravinski 9 for incident P waves, (a) 0 = 0 °, (b) 0 = 30 °.

INFLUENCE OF IMPORTANT PARAMETERS

In what follows, results of a parametric study intended to document the effects of valley shape, and impedance, and wave characteristics such as wave type, frequency and angle o f incidence on the wave scattering by alluvial valleys embedded in an elastic half-space, are described.

The parameters used in this work are defined as follows:

Pl = mass density of the valley; P2 = mass density of the half-space;

\ vs] = shear wave velocity of the valley; vs2 = shear

wave velocity of the half-space or upper layer; v~3 = shear wave velocity of bedrock underlaying the top layer (for layered half-space case)

= 2L/A = ~L/Trvs] = dimensionless frequency; where a; is frequency of the incident wave

L = half-width of the valley; X / L = dimensionless distance from the center of valley P]/P2 = ratio of mass density of the valley to the half-

space (or top layer) %1/%2 = ratio o f shear wave velocity of the valley to

the half-space (or top layer) 0 = Angle of wave incidence.

(a) s Suaace Amp,rude ~ r n e n t

"o ~ e ¢L E <

E o =o

(b)

0 -3

12

SV wave,vaUey,'rhela-O

R-0~, ETA,, 0.5

-2 -1 0 1 2 3

X/L


o 10 SV Wllve,VaUey,Thela,,30 / ,~ .,~ / \ ...o s, ErA- o s

< 6 .4"

/ ~ 4 ...

i5 2 . . . . . . . . . . . ,t . . . . . . . : """" "e- • . . . . . . . . .

0 "" "* -3 -2 -1 0 1 2

X/L

Fig. 4. Comparison with results of Mossessian & Dravinski 9 for incident SV waves, (a) 0 = 0 °, (b) 0 = 30 °.

For all figures presented in this study, ratios of shear- wave velocities, "Osl/'Os2 , a r e taken as 0"3, 0.5, 0.75, 1.5, and 2.0. Values of dimensionless frequency, 7, considered are 0.5, 1.0, 1.5, and 2.0. Ratio of depth to half width of the valley (R) considered are 0" 15, 0"5, 1"0, and 1.5 (R = 1-0 actually is the semi-cylindrical valley). The ratio of the densities of the half-space and valley is assumed to be 1.5. The Poisson's ratio for all materials is v = 1/3. Figures present the surface displacement amplitudes plotted against the dimensionless distance, X/L, for various dimensionless frequency, ~7; shear wave velocity ratio vsl/vs2 , and depth to width ratio of valley, R. The value of X / L = - 1 corresponds to the left rim of the valley, and X / L -- 1 to the right rim of the valley.

Table 1. Material properties

~Iaterial p Vp Vs Poisson's properties (kg/m 3) (m/s) (m/s) ratio, v

Valley 1930 1470 600 0'4 Layer 2050 3113 1500 0.35 Half-space 2230 4677 2500 0.3

Notes: p = mass density; Vp -- pressure wave velocity; V s = shear wave velocity.


1 6


1 4

_ 1 2 n E l 0 <

s E o 6

.~ 4

2

SH Wave,Valley,Theta-0

R-0.S, VslNs2,0.5

ETA,,0.5 ETA,,1.0 ETA-I.S

• ""A f \ \ .•'"'.. #" ' • " "4 ..^ - [ ~ .. ..,.

~'~ "\ ". Z---~--4~" / ;~

~ " - ' : - ~ - + f ~,""~" " " " d ~ : " - ' ......

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

X/H

Fig. 5. Comparison with results of Liu e t al. 12 considering a layered half-space and incident SH waves.

Effect o f valley shape

Figure 7 presents displacement amplitudes in the y-direction along the valley surface due to vertically propagating SH waves, for a range of valley shapes. The influence of the vaUey is much less significant for the shallow valley case, when R = 0.15 and R = 0-5, and is much more pronounced for the deep valley case, R = 1.0, and 1.5. The valley does not considerably influence the ground motions unless the depth of the valley is equal to at least 1/4 the wavelength of the incoming wave.

(a) e

0

-0.6

P Wave,vagey,Theta.,0

- - ETP.-0.5 R-0.5, Vsl/Vs2-(L5 . . . . . . ETA,,1.0

- - - - - ETA-I.5

.,"X ./'~',, / \ . / \

V. : '~" . . . . . • "~ / " • . . . . . . ".--~_ ~ " ~ ' ~ ' ~ ' " " " " " • "N--" • ""m - " " ~ ' ~ ' ' ~ ' ~

-0.4 -0.2 0.0 0.2 0.4 0.6

X/H

6

Ux 4

Surface Amplitude Displacement (b) s

P Wave,Valley,Thela-0

ETA-0.5 R-0.S, VslNs2-0.5 . . . . . . ET/~,I.0

- - . - - ETA,,1.S

i.\. t ' ~ /.,~ / ~ / \ i \

i \ / ~ i , \ i ....... ~. / '~ t ...... '

. . . . . . . ~ : - i - - ' - " , . . , . ~ . ~ ' .... :

U z 4

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

X/H

Fig. 6. Comparison with results of Liu et al. 12 considering a layered half-space and incident P waves, (a) Ux, (b) Uz.

1 6

14

12

10

Uy 8

6

4

2

0

S u r f a c e Displacement Amplitude

R-0.15 . . . . . R-0.5 - - - - - I1,,1.0 VI1NIQ,-O.S, ETA,, 0.5 ~ - P,,,1.5 / - - - \

/ \ i _ - . \

i l / \ \ i / \ "~ ." . . . . . . '

-3 -2 -1 0 1 2

X/L

Fig. 7. Effect of valley shape on ground motion considering vertically incident SH waves.

Figures 8 and 9 present similar results for plane P and SV waves, respectively considering both horizontal and vertical components of displacement. These results are different from the case of SH waves in the sense that a shallower valley (R = 0.5) has significant effect on ground motions. Compression waves, inherent to these scattered wave patterns, have a velocity of approxi- mately twice the shear wave velocity. Therefore, with R = 0.5 the depth of the valley is 1/4 the wavelength.

Horizontal displacement amplitudes are nullified in the center of the valley considering vertically incident P waves. Similarly, the vertical component of displacement is nullified in the center of the valley considering incident

(a)

Ux

10

9

8

7

- - R. ,0.1S . . . . . . R-O.S - - - - - R-1.0 - - - - R - I . S

Vs l /Vs2 , ,0 .S , ETA,, 0.5

,,,~.,...- - ~ ' ~ . f ~ . . . . .

/ : ' . . . . ":N.

. . . . . . . . . . . . . . . . . . . . . . . ., , * ' . . l l l l l P . . .. . . . . . . . . . . . .

-3 -2 -1 0 1 2

X/l_

(b)

Uz

10

9

8

7

6

5

4

3

2

1

0

- - P , -0 .15 . . . . . . R-0.S VslNs2,,0.5, ETA,,0.5 - - - - - R-1.0

- - R - I . S ..°...%

. . . . . .~ -~" . . . . . . ~ . . . . ~ . • . -~. . . . . . . . . . ~ : . . . . . . . . .,. . . . . - ~< . < . ~y " ~ , , ~ . : ~ . . "- ..~.

-3 -2 -1 0 1 2

X/L

Fig. 8. Effect of valley shape on ground motion considering vertically incident P waves, (a) [Ix, (b) Uz.

254 K. L. Fishman, S. Ahmad

(o) lo 9

8

7

6

U x 5

4

3

2

1

0

e~0.1s / ' \ ...... , .os / ~ v.w-..os, ET*.0.S - - - - - R-1.0 I •

- - - R-~.S i ~. i ~.

.i ............ .~,• -.7/\ / i I \ \ "

. . . . . : ~ :'°.~,_.

( b ) lO

9

8

7

6

Uz 5

4

3

2

1

o

-3 -2 -1 0 1 2 3

X/L

R-0.15 .... . . R-0~ - - - - - R-1.0 - - - - R-1.5

VslNS2-0.5. ETA-0~

• " " • j • ' • •••.•

• " \ 1 . . . . . _ ~....>.... ~ ¢ x/,--:::~., . . ......-.--.l • - " " ~ - - ~ - ~ ' ~ " ~ A . ~ ' ~ J ' / ~ , v" " ~ ' ~ ~ " ~ "":4

~ ~ . . i

-3 -2 -1 0 1 2 3

X/L

Fig. 9. Effect of valley shape on ground motion considering vertically incident $V waves, (a) Ux, (b) Uz.

SV waves. This effect is most likely due to wave inter- ference at the valley surface.

Effect o f va l ley impedance

Figure 10 depicts the influence of impedence ratio (vsl/vs2) on the perturbation of surface displacements considering vertically propagating SH waves. When the valley material is stiffer than the half-space, %1/%2 > 1, a

slight deamplification of ground motion occurs through- out the width of the valley. In general the effect of a soft valley to amplify ground motions increases as the stiffness of the valley decreases relative to that of the

16

Surlace Disp~cement.~nl~tUde

Uy

14

12

10

8

6

4

2

0

Vs lNe2 , ,0 .3 . . . . . V s l N d , , 0 • 5

-3 -2 -1 0 1 2 3

X/ l

Hg. 10. Effect of impedance ratio on ground motion considering vertically incident SH waves•

(a) 2o

15

Ux lO

(b ) 20

15

Uz l o

5

0

V s l N d , , ~ 3 . . . . . V s l N I 2 - 0 • S - - - - - V s l N d ~ • ~ ~ - Y s l N I 2 , , 1 . S

R-0.5, ETA,, 0.5

r

-3 -2 -1 0 1 2

X/L

V s l / V d = 0 ~ . . . . . V I 1 N d - 0 . $ ----- VslN~,0.TS R-0J$, ETA- 0.5 ~ - Vsl/VI~,I~$

~ -,.= - - = = = a , . ~

-3 -2 -1 0 1 2 3

X/l.

Fig. 11. Effect of impedance ratio on ground motion consider. ing vertically incident P waves, (a) Ux, (b) Uz.

(a) lo

8

6

Ux 4

2

0

V s l N S 2 - 0 . 3 . . . . . V I I N I 2 - 0 . 5 - - - - - V s l N I Q - 0 . 7 5 - - - V s l N s 2 . 1 . 5

R-0.5. ETA- 0.5

• . . . . - . . . . . . .

[ " " • • ' . :

-3

( b ) 10

-2 -1 0 1 2 3

X/L

Uz

V s I N d , , 0 . 3 . . . . . V I 1 N I 2 = 0 . 5

8 - - - - vsl N~..-n.75 ~ - VSlNd,I.5

6

. . - • . 4 /

R-0.5, ETA- 0.5

.,•'-.. • . . . .

2 ./ '.. " :..

o -3 -2 -1 0 1 2 $

X/L

Fig. 12. Effect of impedance ratio on ground motion considering vertically incident SV waves, (a) Ux, (b) Uz.

10

8

6

t~ 4

2

0

Alluvial valley seismic response

s u m ~ ~ ~ m a , t

ETA-O.S ..... ETC.1.0 V s l ~ O . S , R-O~ - - - -" k'TAml.S m.

li i !

l i i ! i i i ! i ' ' t i A i /

,'-.,, .../ ! :;¢/'-~,~/~...~ .... /--I

-3 -2 -1 0 1 2 3

X/L

Fig. 13. Effect of frequency ratio on ground motion considering vertically incident SH waves.

half-space. When the valley impedance is very low, %l/Vs2 = 0.3, significant amplification of ground motion may occur even for lower frequency ratios (~1 < 1.0) and shallower valleys (R < 0.5). For %]/%2 = 0"3 the effect of the valley on ground motions may be significant even when the depth of the valley is less than 1/4 the wavelength of the incoming wave.

Figures 11 and 12 present similar results for the case of vertically incident P and[ SV waves, respectively. The same observations with respect to SH waves also apply to vertically incident P and SV waves. Although, compared to Vsl/Vs2=0"3, higher amplification with Vsl/V~2 =0.5 was observed for SV waves even for shallow valleys and low excitation frequency. This was not apparent under these conditions for other incoming wave types.

Effect of excitation frequency

The effect of frequency ratio on the response of the valley surface to incident SH, P and SV waves is a complex phenomena whereby the influence of frequency, valley impedance, and shape are coupled. For a given impedance ratio and value of R the frequency affects the

12

10

8

Uz 6

4

2

0


ETA-0.5 I . . . . . ETA,1.0 ----- ETA..1.5

- ETA,,2.0 Vsl/Vs2-0.5, R- 0.5

- • ~ . . ~ 7 . ~ " " " ' . ~ - ~ :

-3 -2 -1 0 1 2 3

X/L

Fig. 14. Effect of frequency ratio on ground motion considering vertically incident P waves.

1 0

Surfa~ Sunl~ude lX~la~ment

255

'1--'1 Erl~l1.0 8 ETk.LS VslNs2,.O.S. R- 0.5 7 ETA-~O

UX 5 • '

4

3

0

-3 -2 -1 0 1 2 3

X/L

Fig. 15. Effect of frequency ratio on ground motion considering vertically incident SV waves.

response in terms of the amount of amplification of ground motion due to the presence of the valley and also the shape of the displacement amplitude (admittance) function. In general the highest amplification occurs at the center of the valley for lower frequency ratios and shallow valleys. For higher frequency ratios and deeper valleys the peak response is closer to the edge of the valley, and the Shape of the admittance function may contain several peaks and troughs. Figure 13 is a plot depicting displacement amplitudes as affected by frequency, along the valley surface for vertically incident SH waves, If one considers a column of soil above a point in the valley basin and applies the one-dimensional theory of wave propagation, features of the valley response including the level and location of maximum displacement amplitude may be qualitatively explained in a rather general sense. However, there are cases when the amplification is higher, and resonance apparently occurs at lower frequencies than those predicted with a one-dimensional idealization. Therefore the existence of a nonrigid basement and the two-dimensional effects of wave scattering caused b y the valley are considered significant.

Figures 14 and 15 show the effect of varying frequency for vertically incident P and SV waves, respectively. The same general relationships observed for SH waves are also evident for incident P and SV waves. Considering shallow valleys the amplification of ground motions due to SV waves and P waves is larger for higher frequencies compared to the response due to SH waves. This effect however, is not as prevalent for valleys with higher impedance ratios.

Effect of incidence angle

Figures 16, 17, and 18 depict the effect of angle of incidence, 0, on wave scattering by alluvial valleys considering incoming SH, P and SV waves. As expected for 0 other than zero the shape of the admittance function is no longer symmetric with respect to the

256

10

K. L. Fishman,

Surlace AmpF~ude Displacement

Uy

9

8

7

6

5

4

3

2

1

0

THE'rA-0 . . . . . THETA-30 R=0.5, ETA~ 0.5 - - - - - THETA-60 - - - ~ . Q o / '~

11"".\, 1 / \ 3

.... ~ " ' : U "'" ..... '~ . . . . . . . . . I -3 -2 -1 0 1 2 3

X/L

Fig. 16. Effect of angle of incidence on ground motion considering incoming SH waves.

centedine of the valley. For incoming SH waves the point of maximum displacement amplitude shifts toward the far confluence as the angle of incidence becomes more oblique and may be twice as high as the maximum amplitude for O = 0 °.

Result for both vertical, Uz and horizontal, Ux, displacement amplitudes are presented for incident P waves. The displacement amplitude for the vertical

(a)

Uz

10

8

6

4

Sudace Amplitude Displacement

R.0.5,THETA-0 ... . . R'0.5, T M I

I - - ' - - R,O.S,THErA-60 I VslNs2= 0.5, ETA,, 0.5 I ~ - R,I.S.'n-IETA-O I I . . . . R.,I .S,THETA-30 I L - - " R.I.S.THETA-6O .l

.."i ................. - - .." . . . - " . ~

.7...~, . ~ ~ . Z , . . I - I - .-__~ . . . . . . . . . ~..~/ "%/

-3 -2 -1 0 1 2 3

X/L

(b)

Ux

10

R.0.&'n-IETA-O R.0.§,THETA-30 R.0.5,11-1ETA-60 I:I-I.S,'n-IETA-0 R-I.S,THET~.30 R-1.5,1HETA-60

Vsl/Vs2=, 0.5, ETA- 0.5

/ ' ~ ' N .

z: ....... .i x . I ~ . . ~ : ~ - , / ....~----%~.X L .... . < ~ > - . - . - ~ , ~ , : / / / ~ . ~ _ . .. ./~.~...% • , ; . o,,,,,~..~, --,-- .... P- ....................... . . . . . . .

-3 -2 -1 0 1 2

X/t.

Fig. 17. Effect angle of incidence on ground motion considering incoming P waves, (a) U~, (b) Ux.

S. Ahmad

component decreases with increasing angle of incidence. Furthermore, the largest vertical displacement amplitude occurs towards the near confluence or close to the middle of the valley. However, the [Ix component of displacement due to an incoming P wave is significantly amplified and shifted toward the far confluence as 0 is increased.

With incident SV waves both the vertical and horizontal components of the maximum displacement amplitude increase with increasing angle of incidence. The Ux component of displacement amplitude for SV waves may be 4 to 5 times higher with 0 = 60 ° than for 0 = 0 °. The Uz component may also be significantly increased, and may occur shifted toward the far or near confluence depending on the valley depth.

Effect of layered half-space

Wave scattering and diffraction by alluvial valleys within a layered half-space may be studied and compared to results obtained for valleys within a homogeneous half- space. Figures 19 and 20 show results obtained for a ratio of Vs2/Vs3 equal to 0"4 considering vertically incident SH and SV waves. Valley shapes represented by R values ranging from 0" 15 to 1.5, are considered. The dimensionless frequency, ~7, is maintained at 0.5 for all layered half-space analyses presented herein.

(a) 25


Uz

20

15

10

(b ) 25

20

15

Ux 10

R.0.5,THETA-O . . . . . R-0.5,THETA-30 - - - - - R-O.5,THETA-60 - - - R-I.iI,THETA-0 . . . . R.1.5,THETA-30 ~ - R-1.5,THETA-60

. i " ,":7\

. i - " " , \ f . ~ . - - . . , .

-3 °2 -1

I / '~ Vs lNs2- 0.5, ETA,, 0.5

i f", / \ 'k '~, \ i // i i i i

i i i " i i

\ i \,... f - - - - - • . t ~ . . - . ~ / _ . - . - "

0 1 2 3

X/L

I ~ P,.0.S,THETA-O .. P,-0.5 "n4ETA-30 /" \ R.0.5,'I'HETA-60 I I / " ~ R.I.S,THET~-0 I i / "~ ' \ VslN$2= 0.5, ETA,, 0.5 R-I.S,THETA-30 I • : . \ F~.I ST.~rA-SO ] I / ~-

' i i i \ i l ~.\ i l ~',

/ - - ' . ~ i~ ,\ / ...-..~,, i , .. ....... ~,

/ ,," .~.1 I ." " ' :v . . . . . . . . . . . . . . . ,_, . / . . / . / . : .......... __ ' . ~ . ' , ~ . . . . . . ~ .

-3 -2 -1 0 1 2

X/L

Fig. 18. Effect angle of incidence on ground motion considering incoming SV waves, (a) U:, (b) Ux.


Uy

16

14

12

10

Su~ace r)~olacem~ k.plRude

SH W~Fe.valley on L I ~ HS

R-0.15 / ' "~, VsINs2-0.S,Vs2Ns3-0A . . . . . . ~o.s / \ - . - R-~.o / \ m - o ~ - - - R-~.S / X.

i \ i

/ / ~ - X X \ i / . . " ....... \ ' l

........................ i . / - - ~,~k'l .........................

-3 -2 -1 0 1 2 3

X/L

Fig. 19. Wave scattering by elliptical valleys in the soft upper layer of a half-space considering incident SH waves.

Displacement amplitudes due to vertically incident SH waves are increased for valleys included in a soft upper layer. The degree to which the response is affected by half-space layering depends mostly upon the shape of the valley, and the ratio of V~2/V~3. With R = 1.0 and Vs2/~Js3 = 0"4 the amplification of Uy is increased by 30% at the center of the valley compared to the hom- ogenous half-space result,,;. Very little difference in results are realized for vs2/vs3 := 0.8 compared to the homogeneous half-space results.

Displacement amplitudes along the surface of alluvial

(a) 20

la 16

14

12

U x 10

8

6

4

2

0

SV Wm~e.Vgley on Lwemd I-6

- - R-0.15 VslNs2-0.S,Vs2NI3,,0.4 . . . . . . R-0.5 ETA.. 0.,5 - - - - - R-1.0 - - - - R..I.S

, / "\ / "\

. :~ , .,<..-:..~- ,q....

- . . . . . . . : : . . - . . . . .

(b ) 20

-3 -2 -1 0 1

x/t. Surface Displacement Amplitude

Uz

18

16

14

12

10

8

6

4

2

0

SV Wave,Valley on Layered ItS

1:1-0.15 .... . . R-0.5 Vs INs2-0.5,Vs2JVs3,,0.4 - - - - - R-1.0 ETA,, 0.5 - - - - R-1.5

~ .-.. , , , /.,[ "~..: ; . , ' . \ ,.,.,,...,',- . '= -~ - ~:':"'~.L'~,--_/,," '-,. \ / ~.. ",,.~._o .~. . . . . "'~

-3 -2 -1 0 1 2

X/t.

Fig. 20. Wave scattering by elliptical valleys in the soft upper layer of a half-space considering incident SH waves, (a) Ux,

(b) U=.

valleys due to vertically incident SV waves are not much affected by the presence of the softer upper half-space layer. Any increase in amplification realized is small and in some instances a reduction in amplification is observed compared to results obtained with a homogeneous half- space.

CONCLUSIONS

The direct Boundary Element Method has been applied to the problem of ground motions at the site of an alluvial valley. A detailed evaluation of the influence of key parameters ong round motion was undertaken. Parameters were related to the geometry and impedance of the valley, and characteristics of incoming waves including frequency, angle of incidence, and particle motion. The occurrence of SH, SV, and P type body wave forms within homogeneous and layered elastic half- spaces containing an alluvial valley have been analyzed and discussed.

In summary it was demonstrated that in general shallow valleys have little impact on ground motion, and the valley response is similar to free field. The effect of the valley on ground motion becomes significant if the depth of the valley is at least one quarter of the incoming wavelength. Wave scattering from the valley complicates ground motion for deep valleys and high frequency incoming waves. In these cases the two-dimensional nature of the problem becomes important and one may not get a good qualitative or quantitative understanding of the problem on the basis of one-dimensional analysis. Very soft valleys may impact ground motions even for shallow valleys and low frequency incoming waves. Conversely, if the valley impedance is higher than that of the half-space a slight deamplification of ground motion within the valley may be realized compared to free field.

For vertical incidence, ground motions were most dramatically amplified within the valley for incoming SH waves, compared to SV or P waves. For a valley having an impedance at least half that of the surrounding elastic half-space, ground motions within the valley due to incoming SH waves were amplified by a factor as high as six, and when the valley impedance was very low the factor was as high as twelve. However, for non-vertical angles of incidence, the ground motions within the valley were most dramatic for incoming SV waves, with high angle of incidence.

The angle of wave incidence has a dramatic effect on ground motion within the valley. In general as the angle of incidence becomes more oblique the shape of the admittance function is less symmetric and the maximum amplitude of displacement is increased compared to the 0 = 0 ° case.

Compared to results obtained with a homogeneous half-space the effect of a soft layer overlaying the half-

258 K. L. Fishman, S. Abroad

space is to further amplify ground motions within the valley when incoming SH waves are considered. The valley shape and impedance of the upper layer have the most influence on the valley response. The presence of the soft upper layer had little effect on ground motions within the valley for incoming SV waves.

Future work in this area should be directed towards including more complicated soil behavior models. Soil behavior within the valley may be nonlinear, and also, there exists a possibility for slip to occur at the interface between the valley and the half-space. Analysis must be performed in the time domain in order to include such behavior. A possible application is to study the dynamic response of below grade, lined, MSW landfills. The behavior of MSW is nonlinear, often considered similar to that of soft clay, and the possibility for slip exists along the liner at the base of the landfill.

Other extensions to this work may include considera- tion of anisotropic soil behavior, and the possibility of layering within the valley itself.

ACKNOWLEDGEMENT

All of the computations presented in this paper were performed by Mr. Hongbo Xue in partial fulfillment of the requirements for the Degree of Master of Science at the State University of New York at Buffalo.

REFERENCES

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Seismic response for alluvial valleys subjected to SH, P and SV waves

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