Antiplane response of a dike with ¯exible soil-structure interface toincident SH waves

Abdul Hayira,b, Maria I. Todorovskab,*, Mihailo D. Trifunacb

aTechnical University of Istanbul, Civil Engineering Department, 80626 Maslak, Istanbul , TurkeybUniversity of Southern California, Civil Engineering Department, Los Angeles, CA 90089-2531, USA

Accepted 6 May 2001

Abstract

Studies of the effects of differential ground motions on structural response generally do not consider the effects of the soil-structure

interaction. On the other end, studies of soil-structure interaction commonly assume that the foundation of the structure (surface or

embedded) is rigid. The former ignore the scattering of waves from the foundation and radiation of energy from the structure back to the

soil, while the latter ignore quasi-static forces in the foundations and lower part of the structure deforming due to the wave passage. This

paper studies a simple model of a dike but considers both the soil-structure interaction and the ¯exibility of the foundation. The structure is

represented by a wedge resting on a half-space and excited by incident plane SH-waves. The structural `foundation' is a ¯exible surface that

can deform during the passage of seismic waves. The wave function expansion method is used to solve for the motions in the half-pace and in

the structure. The displacements and stresses in the structure are compared with those for a ®xed-base model shaken by the free-®eld motion.

The results show large displacements near the base of the structure due to the differential motion of the base caused by the wave passage.

q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Soil-structure interaction; Wave passage effects; Differential motion; Dynamic interaction; Flexible surface foundation

1. Introduction

Soil-structure interaction is a complex phenomenon,

involving scattering and diffraction of the incident waves

from the structure foundation (wave passage effects or kine-

matic interaction), and propagation of wave energy into the

structure and radiation back into the soil (dynamic interac-

tion). In this process, the soil, the foundation and the struc-

ture all deform. To reduce the complexity of the problem,

various simplifying assumptions are made, starting from

ignoring completely the soil-structure interaction and the

wave nature of the excitation (rigid base assumption; the

excitation is simpli®ed to horizontal driving motion).

Another simplifying assumption is to ignore the wave nature

of the excitation, but account for the ¯exibility of the soil

and the radiation of energy back into the soil by adding

springs and dash-pots between the structure and the rigid

base. The spring and dash-pot constants may be pre-calcu-

lated using wave propagation methods, but assuming that

the foundation is rigid and using linear theory.

The rigid foundation assumption reduces to a minimum

the additional degrees-of-freedom in the formulation of the

system equilibrium equations, at most to three translations

and three rotations of the rigid foundation. The simplest

model is a two-dimensional (2D) anti-plane model, that is

a structure on elastic half-space and excited by plane SH

waves [3,6,15], in which there is only one additional degree-

of-freedom that is translation of the foundation relative to

the soil. The next more complicated model is the same 2D

model but excited by in-plane motion, that is plane P and

SV-waves or Rayleigh waves, which requires three addi-

tional degrees-of-freedom, that are horizontal and vertical

translation and rocking of the foundation relative to the soil

[12]. In all of these models, perfect bond is assumed

between the foundation and the soil, and the soil and struc-

ture are usually assumed to be linear. These assumptions are

violated to various degrees depending on the design of the

foundation and the material properties of the soil.

Numerical simulations of seismic response of soil-structure

systems with ¯exible foundations are rare [1,5] and dif®cult to

evaluate in absence of strong motion records. As far as we

know, there exists no strong motion program to document

distortions and warping of foundations of structures during

the passage of strong seismic waves. An example of experi-

mental evidence of ¯exibility of a building foundation is

Soil Dynamics and Earthquake Engineering 21 (2001) 603±613

0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0267-7261(01)00035-5

www.elsevier.com/locate/soildyn

* Corresponding author. Tel.: 11-213-740-0616; fax: 11-213-744-1426.

E-mail address: mtodonov@usc.edu (M.I. Todorovska).

provided by detailed microtremor measurements of a seven-

storey building in Van Nuys conducted by Trifunac et al. [17].

They show contours of relative amplitude and phase of the

motion on the ground level in the building and in the area

surrounding the building. The wave propagation is seen from

the direction of the primary sources of the ambient noiseÐa

busy freeway to the west and a busy street to the north of the

building. The shape of these contours may change depending

on the time of the day, but the evidence of wave propagation

will be there. There are no such data from strong shaking, but

it is reasonable to expect that the same would occur during

strong motion.

In general terms, the soil-structure interaction will

lengthen the apparent period of the system, will increase

the relative contribution of rocking excitation of ground

motion to the total response, and will usually reduce the

maximum base shear [13]. The bene®ts of including soil-

structure interaction in the design of structural systems

result from the scattering of incident wave energy from

the foundation, and from additional radiation of structural

vibration energy into the soil. When the soil surrounding the

foundation experiences small to modest levels of non-linear

response, the soil-structure interaction will lead to further

signi®cant loss of the available input energy. The degree to

which the soil-structure interaction modi®es the foundation

and structural response depends mainly on the ¯exibility of

the soil relative to the foundation and to the structure. In

predictions by analytical and numerical models, it also

depends signi®cantly on the modeling assumptions. The

rigid foundation assumption, for example, exaggerates the

effects of scattering and radiation of incident waves, and

may overestimate the damping effects in the soil. It may

also overestimate the rocking and torsional response of

the foundation. Large foundation rocking, coupled with

the gravity forces and with the vertical accelerations, may

be of concern for the stability of the structure, and may be a

possible system failure mode, due to dynamic instability.

This paper presents a model and solution for the response

of a wedge shaped structure on a half-space to incident SH

waves, where the interface between the structure and the

half-space is a ¯exible surface and both kinematic and

dynamic soil-structure interaction effects are considered. In

a way, this is a generalization of the work of Todorovska and

Trifunac [9±11], Todorovska and Lee [8], and Kojic and

Trifunac [2±4] who considered a 2D and 3D structure with

¯exible surface structure-soil interface and the wave passage

effects for incident SH-waves, but neglected the soil-structure

interaction. They showed that nonharmonic `modes' of vibra-

tion also contribute to the response, and their participation is

signi®cant when the horizontal velocity of the wave at the

base corresponds to incidence beyond critical angle. The

deformation of the structure is such that the top moves

very little, but the ®rst story sustains signi®cant deformations

to follow the motion at the soil. In this case, the base ¯ex-

ibility is bene®cial for the upper stories but may lead to

critical conditions for the ®rst story columns [16]. This

paper presents the mathematical model and a method of solu-

tion by the wave function expansion method, discuses

convergence of the Fourier±Bessel series, examines agree-

ment with other published results that are similar or special

cases of this model (the circular valley problem and ridge

problem), and presents results for the displacements on the

free surface and inside the structure, all for different para-

meters of the soil-structure system and for different angles of

incidence of SH-waves. The results for structural response

are compared with those for a rigid base model.

2. Model

The 2D model consists of a circular wedge welded to an

elastic homogeneous half-space. Both media are linear,

elastic, homogeneous and isotropic, with shear wave velo-

cities and shear moduli bD, mD, b s and m s. Fig. 1 shows a

vertical cross-section of the model. The wedge has radius b,

angle 2u 0 and a vertical axis of symmetry through points O

and O1 which are the origins of the two Cartesian coordinate

systems, x-y-z and x1-y1-z1, used to describe the motion of

the system. Axes y and y1 project into points on the plane

shown in Fig. 1 and point towards the reader. Associated

with these two Cartesian coordinate systems are cylindrical

coordinate systems R-u -y and R1-u 1-y1 as shown in Fig. 1.

The distance between the origins of these two coordinate

systems is d� bcosu 0. The half-width of the base of the dike

is a� bsinu 0, which we will use as characteristic length of

the model. The excitation is a plane SH-wave with

frequency v , angle of incidence g , amplitude u0, displace-

ment in the y-direction, and described by

U i� � � u0exp�ks�xsing 2 zcosg�2 ivt �1�where t is time and ks � v=bs is the wave number in the

half-space.

2.1. Equations of motion and boundary and continuity

conditions

The structure and half-space will respond by anti-plane

motions, U(D) and U(s), such that they both satisfy the wave

equation

22U

2x21

22U

2z2� 1

b2

22U

2 t2�2�

subject to the following zero-stress boundary conditions

s �S�uy � 0 at R . a and u � ^p

2�3�

s�D�u1y � 0 at R1 , b and u1 � ^u0 �4�where

suy � m

R

2U

2u�5�

is the shear stress on a surface perpendicular to u .

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613604

Displacements U(D) and U(S) also have to satisfy the conti-

nuity of stresses and displacements at the contact surface,

U�D� � U�S� at R1 � b and 2 u0 # u1 # u0 �6�

s�D�R1y1� s�S�R1y1

at R1 � b and 2 u0 # u1 # u0 �7�where

sR1y1� m

2U

2R1

�8�

(8) is shear stress on a surface perpendicular to R1.

2.2. Representation of the motion in the half-space and in

the structure

The motion in the half-space can be represented as a sum

of the free-®eld motion, U( ff ), and perturbation, U(R), repre-

senting waves scattered from the wedge

U�S� � U� ff � 1 U�R� �9�The free-®eld motion is the sum of the incident wave, U(i),

and the wave U(r) re¯ected from the half-space surface if the

structure were not there

U� ff � � U�i� 1 U�r� �10�The re¯ected wave is then

U�r� � u0exp�ks�xsing 1 zcosg�2 ivt� �11�The free-®eld motion U( ff ) satis®es the zero-stress condi-

tion (Eq. (3)). Then, the scattered ®eld must also satisfy the

zero-stress condition (Eq. (3)) as well as the wave equation

(Eq. (2)), and it should be an outgoing wave. Such motion

can be represented as the following Fourier±Bessel series

[14]

U�R��R; u� � u0

X1n�0

�ARn H�1�2n �ksR�cos2nu

1 BRn H�1�2n11�ksR�sin�2n 1 1�u�exp�2ivt� �12�

where ARn and BR

n are unknown coef®cients and H�1�n �:� is the

Hankel function of the ®rst kind with order n.

The displacement in the structure that satis®es automati-

cally the zero±stress condition (Eq. (4)) can be represented

as

U�D��R1; u1� � u0

X1n�0

�AD

n J2n�kDR1�cos

�2n

p

2u0

u1

�

1 BDn J2n11�kDR1�sin

��2n 1 1

�p

2u0

u1

��

£ exp

�2 ivt

��13�

where ADn and BD

n are unknown coef®cients.

2.3. Solution

The representations of the motions in the structure and in

the half-space, i.e. U( ff ), U(R) and U(D), are such that the zero-

stress conditions on the outer boundaries are automatically

satis®ed. The unknown coef®cients are then determined

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613 605

Fig. 1. Vertical cross-section of a two-dimensional circular wedge (`structure') welded to an elastic homogeneous half-space.

from the continuity conditions (Eqs. (6) and (7)). Matching

the continuity conditions requires same representation for

U( ff ), U(R) and U(D). Such a representation is expansion in

Fourier±Bessel series of R1 and u 1 with period 2p . The

expansions of U(i) and U(r) are

U�i� � u0eiksdcosgX1m�0

em�2i�mJm�ksR1�cosm�u1 1 g�exp�2ivt�

�14�

U�r� � u0e2iksdcosgX1m�0

em�i�mJm�ksR1�cosm�u1 2 g�exp�2ivt�

�15�

where

em �1; m � 0

2; m . 0

(�16�

The scattered waves, U�R�, are transformed using the

exterior region form of the Graf's Addition Theorem by

which

H�1�n �ksR�cosnu

sinnu

( )� �21�n

X1m�0

H�1�n �ksR1� em

2{�Jn1m�ksd�

1 �21�nJm2n�ksd��cosmu1

2 �Jn1m�ksd�2 �21�nJm2n�ksd��sinmu1}

�17�

Substitution of Eq. (17) into Eq. (12) gives

U�R��R1; u1� � u0

X1m�0

X1n�0

H�1�m �ksR1��ARn Pm;n�ksd�cosmu1

1 BRn Qm;n�ksd�sinmu1�exp�2ivt�

�18�

where

Pm;n�s� � em

2�Jm12n�s�1 �21�2nJm22n�s��

Qm;n�s� � em

2�Jm12n11�s�2 �21�2n11Jm22n11�s�� �19�

The expression for U�D� is transformed by expanding

cos 2np

2u0

u1

� �and sin �2n 1 1� p

2u0

u1

� �

in Eq. (13) in Fourier series of period 2p . This leads to

U�D��R1; u1� � u0

X1m�0

X1n�0

�CnJ2n�kDR1�Mm;ncosmu1

1 DnJ2n11�kDR1�Nm;nsinmu1�exp�2ivt��20�

where

Mm;n � 1

p

Zp

2pcos 2n

p

2u0

u1

� �cosmu1du1 �21�

Nm;n � 1

p

Zp

2psin �2n 1 1� p

2u0

u1

� �sinmu1du1 �22�

The above Fourier expansion assumes extension of U�D�

beyond the domain of the structure, i.e. on �2p;p�, to mini-

mize the Gibb's effects at u1 � ^u0. After evaluation of the

integrals

Mm;n �

1;

1 1 sin 4npp

2u0

� �= 4np

p

2u0

� �;

�21�msin 2npp

2u0

� �= 2

m2p

4n

p

2u0

� �21

1npp

2u0

" #;

n � m � 0

m � 2np

2u0

m ± 2np

2u0

8>>>>>><>>>>>>:�23�

Nm;n �

0;

1 2 sin 2�2n 1 1�p p

2u0

� �= 2�2n 1 1�p p

2u0

� �;

�21�msin �2n 1 1�p p

2u0

� �= 2

mp

21�2n 1 1�2p

2m

p

2u0

" #;

n � m � 0

m � �2n 1 1� p

2u0

m ± �2n 1 1� p

2u0

8>>>>>><>>>>>>:�24�

The stresses are

s s� �rz �R1; u1� �

s0

kDR1

ms

mD

X1m�0

X1n�0

ARn Tm�3; ksR1�Pm;2n�ksd�cosmu1

1BRn Tm�3; ksR1�Qm;2n11�ksd�sinmu1

1Tm�1; ksR1�em�2i�mexp�iksdcosg�cosm�u1 1 g�1Tm�1; ksR1�em�i�mexp�2iksdcosg�cosm�u1 2 g�

�25�

s �D�rz �R1; u1� � s0

kDR1

X1m�0

X1n�0

ADn T2n�1; kDR1�Mm;ncosmu1

1 BDn T2n11�1; kDR1�Nm;nsinmu1

�26�where

Tm�j; s� � mCm�s�2 sCm11�s�;

Cm�s� �

Jm�s�;Ym�s�;

j � 1

j � 2

H�1�m �s�;H�2�m �s�;

j � 3

j � 4

8>>>>><>>>>>:�27�

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613606

and

s0 � mDkDu0 �28�is the reference stress to be used for normalization if not

speci®ed otherwise.

The corresponding substitutions in the continuity condi-

tion Eqs. (6) and (7) then gives the following two in®nite

systems of equations written in matrix form

: ...

:

¼ �Wsymm;n �2£2

¼

: ...

:

26666643777775

m£n

..

.

ARn

ADn

( )2£1

..

.

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;n£1

�...

{Vsymm }2£1

..

.

8>>>><>>>>:

9>>>>=>>>>;n£1

�29�where

�W symm;n �2£2 �

J2n�kDb�Mmn 2H 1� �m �ksb�

T2n�1; kDb�Mmn 2ms

mD

Tm�3; ksb�

264375

2£2

�30�

{Gm}2£1 �emJm�ksb�cosg��2i�mexp�iksdcosg�1 imexp�2iksdcosg��

ms

mD

emTm�1; ksb�cosg��2i�mexp�iksdcosg�1 imexp�2iksdcosg��

8><>:9>=>;

2£1

�31�and

: ...

:

¼ �Wasymm;n �2£2

¼

: ...

:

26666643777775

m£n

..

.

BRn

BDn

( )2£1

..

.

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;n£1

�...

{Vasymm }2£1

..

.

8>>>><>>>>:

9>>>>=>>>>;n£1

�32�where

�Wasymm;n �2£2 �

J2n11�kDb�N�m11�;n 2H�1�m11�ksb�T2n11�1; kDb�N�m11�;n 2

ms

mD

Tm11�3; ksb�

264375

2£2

�33�

{Vasymm }2£1

�em11Jm11�ksb�cosg��2i�m11exp�iksdcosg�1 im11exp�2iksdcosg��

ms

mD

em11Tm11�1; ksb�cosg �2i�m11exp�iksdcosg�1 im11exp�2iksdcosg�h i

8><>:9>=>;

�34�The systems of Eqs. (29) and (32) can be solved numeri-

cally for the unknown coef®cients of expansion, after trun-

cation to ®nite systems.

2.4. Solution for a wedge on a rigid base

For the rigid base case, the solution of the problem is

simple. Because the motion at the base of the wedge is

synchronous, the sum in Eq. (13) collapses into one

termÐthe even term for n� 0. If the excitation is a driving

harmonic motion equal to the free-®eld motion at point O,

i.e. 2u0exp�2ivt�, then the displacement in the wedge is

U�D� � 2u0

J0�kDR1�J0�kDb� exp�2ivt� �35�

3. Results and analysis

We present the results in terms of dimensionless para-

meters. We de®ne dimensionless parameter

h � 2a

bT� va

pb�36�

equal to the ratio of the width of the `foundation' and the

wavelength, l , of shear waves in the soil (l � bT ). It can

be interpreted as dimensionless length of the base of the

structure or as dimensionless frequency.

The results we present were calculated by a code written

in Mathematica 4.0 interpreter language. All computations

were done in double precision. In all cases, the amplitude of

the incident wave is u0 � 1.

3.1. Accuracy

While in theory the accuracy of the results should

increase with increasing number of terms, in practice the

Fourier±Bessel series start diverging for number of terms

grater than a critical value, which depends on the wave-

length of the incident waves. The reason for this is that

Bessel functions of very high order are progressively smal-

ler numbers, becoming numerically indistinguishable for

suf®ciently large order, and the system of equations

becomes ill conditioned. The convergence is faster for

longer incident waves (small h ), and for larger angle of

the wedge. It is the fastest for the special case when 2u0 �1808 which is the circular valley problem for which the

matrices of the systems of equations to be solved are

block-diagonal [14]. We show results typically for

2u0 � 1508. The convergence also depends on where the

motion is evaluated.

The convergence is illustrated in Fig. 2, showing normal-

ized displacement amplitude uU�D�u=u0 for N � 5; 6 and 7,

where N is the highest order term in the truncated series, for

g � 458 and for a `soft' structure with mD=mS � 1=6 and

rD=rS � 2=3. uU�D�u=u0 is plotted vs. h for 0 , h , 1 and

2u0 � 1508. Different parts correspond to different points

along the surface of the wedge, at x=a � 0, ^0.5, ^0.75

and ^1. These results show that, as h increases, the conver-

gence is the best in the center (x=a � 0) and worst at the

corners (x=a � 1) of the base. In this paper we analyze

results for h # 1 and for 1508 # 2u0 # 1808.Table 1 shows upper bounds for the normalized displace-

ment and stress residuals along the surface of the wedge for

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613 607

vertical and horizontal incidence and for h � 0:25, 0.5, 0.75

and 1. It is seen that the residuals are small.

3.2. Results for special casesÐcircular valley and two

ridges

When the angle of the wedge is 2u0 � 1808, the wedge

structure becomes a circular valley [14]. Fig. 3 shows

displacement amplitude (normalized by the amplitude of

the incident wave) along the free surface for

23 , x=a , 3, for h � 0:5 and for angles of the wedge

2u0 � 1508, 1658 and 1808. Top and bottom parts corre-

spond to incident angles g � 08 and 908. The ratio of mate-

rial constants is mD=mS � 1=6 and rD=rS � 2=3. It is seen that

inside the structure, the displacements are larger for a shar-

per wedge. The results for 2u0 � 1808 coincide with those

of Trifunac [14].

When the material properties of the wedge and half-space

are the same, the model represents a topographic irregular-

ity±a triangular hill. Fig. 4 shows the amplitudes along the

free surface for 23 , x=a , 3, h � 0:5, angles of the

wedge 2u0 � 106:38, 126.98 and 151.98, corresponding to

d/a ratios 0.75, 0.5 and 0.25, respectively, and incident

angles g � 08, 458 and 908. Our results agree qualitatively

with those of SaÂnchez-Sesma et al. [7] for a ridge with

boundary de®ned by the function

y � d�1 2 �x=a�2�e23�x=a�2 �37�

where d is the height of the symmetrical ridge and 2a is the

width at the base. For d/a� 0.75 and g � 90 our method

does not converge for N , 7 and for this reason our results

for d/a� 0.75 (top right corner) are not shown in Fig. 4.

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613608

Table 1

Upper bounds for normalized (Normalization factor for displacements is

the amplitude of the incident waves, u0, and for the stresses is

s0 � mDkDu0) displacement and stress residuals along the contact surface,

for unit amplitude incident wave and for a `soft' structure (mD=ms � 1=6 and

rD=rs � 2=3)

h N g � 08 g � 908

eU es eU es

0.25 4 1.2 £ 10211 8.7 £ 10211 1.7 £ 10210 8.4 £ 10210

0.50 5 3.2 £ 1028 2.6 £ 1027 8.8 £ 1028 8.6 £ 1027

0.75 6 8.2 £ 1026 8.1 £ 1025 1.3 £ 1025 1.4 £ 1024

1.00 7 1.1 £ 1024 1.3 £ 1023 7.6 £ 1024 1.2 £ 1022

Fig. 2. Normalized displacement amplitudes |U|/u0 vs. h , for incidence angle g � 458, evaluated at seven points along the free-surface (x/a�21, 20.75,

20.5, 0, 0.5, 0.75 and 1). The index of the highest order term in the series is N� 5, 6, and 7.

Fig. 5 shows surface displacements for a ridge de®ned by

a circular arc studied by Yuan and Liao [18]. In spite of the

fact that this is geometrically different from a triangular

ridge, which we analyze in this work, our results are similar,

for small values of h (0.5 and 1.0) and for the corresponding

d/a ratios.

3.3. Free-surface displacements for a `soft' and a `hard'

structure

Fig. 6(a) and (b) shows the displacements on the free

surface of the half space and along the base of the structure

for a structure `softer' (bottom) and `harder' (top) than the

half-space (mD=mS � 50 and 1/16 and rD=rS � 1=4 and 1),

and for two incident angles (g � 0 and p /4). |U|/u0 is shown

for ±3 , x/a , 3 and for 0.2 # h # 1.0. It is seen that the

displacement pattern is more complex for larger h and for

g � 458 incidence. The displacement amplitudes on the half-

space surface oscillate about the free-®eld amplitude (� 2).

3.4. Displacements inside the structure

In Figs. 7(a), (b) and 8(a), (b), we compare displacement

amplitudes in the structure with those for a ®xed-base

model. The displacement amplitudes are shown along ®ve

radial cross-sections inside the structure (u1 � 2u0;

2p=4; 0;p=4; u0) including the edges.

Fig. 7(a) and (b) shows the displacements for a `soft'

structure (mD=mS � 1=6 and rD=rS � 2=3), for h � 0:5 and

1. Fig. 8(a) and (b) show the same for a `stiff' structure (mD/

mS� 6, rD/rS� 3/2) for h � 0.5 and 1. It can be seen that

the displacements for the ®xed-base model are similar to

those for vertically incident waves, and are the largest at

the top of the structure. The displacements for the ®xed

model are larger than those for vertically incident waves,

more so for the larger value of h, due to the radiation damp-

ing and also due to the smaller effective input (the ®xed-

base model is driven by motion with amplitude equal to the

free-®eld motion on the surface, where it is the largest,

while due to its curved base our model is driven by motion

that has smaller amplitude towards the bottom (more so for

h � 1 than for h � 0:5). It is also seen that the location of

the nodes changes with the incident angle. For inclined

wave incidence, the displacement amplitude is not necessa-

rily the largest on the top. For h � 0:5, along the edge

opposite to the direction of arrival of the incident waves,

the displacements amplitudes for the soil-structure interac-

tion model are larger than those for the ®xed-base model,

contrary to the popular belief that they should be smaller if

there is radiation damping. For h � 1, the displacements for

horizontal wave incidence may be two times larger than

those for the ®xed-base model; these large displacements

occur in the lower part of the structure, near the base.

Fig. 8(a) and (b) shows the displacements for a `stiff'

structure on `soft' soil (mD=mS � 6 and rD=rS � 2=3 and 3/

2), for h � 0:5 and 1. It can be seen that for h � 0:5 the

displacements for the ®xed-base model are larger than those

for the ¯exible-base model practically everywhere. However,

for h � 1 that is not the case. And for horizontal incidence,

the displacement of the ¯exible-base model can be signi®-

cantly larger than for the ®xed-base model (this occurs on the

surface of the structure facing the incident waves).

Fig. 8(b) shows that for this example of a `stiff' struc-

ture displacements U(D)/u0 are larger near the interface

with the soil, and then their amplitudes decay towards

the top of the structure. These displacements are reminis-

cent of exponential decay of displacements, from base

towards top of the structure, described by Todorovska

and Trifunac [9±11], and should be associated with the

analogous `pseudo static' deformations, driven by the soil

underneath, and associated with imaginary phase veloci-

ties inside the structure. Because of the curved soil-struc-

ture interface the critical incidence angles of the waves

striking the soil-structure interface are functions of g and

u 1 (see Fig. 1) and consequently the pseudo static displa-

cements are also dependent on u 1. This results in their

more complex spatial dependence than for the shear

plate model of the structure discussed in Todorovska

and Trifunac [9±11].

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613 609

Fig. 3. Dependence of the surface displacements on wedge angles, 2u 0, and

comparison of the results for 2u 0� 1808 with the motions on the surface

and near an alluvial valley from Trifunac [14].

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613610

Fig. 5. Dependence of surface displacements on the height-to-width ratio (d/a) of a triangular ridge, for g � 08, h � 0.25 and 1.0 with the results of Yuan and

Liao [18] for surface motion of a cylindrical ridge.

Fig. 4. Dependence of the surface displacements on the height-to-width ratio (d/a) of a triangular ridge, for g � 08, 458 and 908, and comparison with the

results of Sanchez-Sesma et al. [7] for a similar shape two-dimensional ridge.

4. Conclusions

The response of a wedge-shaped structure with a ¯exible

structure-soil interface was investigated for incident SH-

waves in a half-space, and was compared with the ®xed-

base response of the same structure. The ¯exible-base

model accounts for the kinematic and dynamic soil-struc-

ture interaction effects. The results show that the ®xed-base

model does not give conservative results (contrary to a

popular belief), both when the structure is `softer' and

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613 611

Fig. 6. (a) Normalized displacements |U|/u0 vs. x/a and h , for a `stiff' structure (mD/mS� 50) and a ªsoftº structure (mD=mS � 1=16), for a ridge with

2u0 � 1508 and for vertically incident waves. (b) Normalized displacements |U|/u0 vs. x/a and h , for a `stiff' structure (mD/mS� 50) and a `soft' structure

(mD/mS� 1/16), for a ridge with 2u 0� 1508 and incident angel g � 458.

when it is `stiffer' relative to the soil. Of concern for design

should be large stresses near the base of the structure due to

quazi-static deformation of the base resulting from the wave

passage.

Acknowledgements

This work was done while the ®rst author was on

leave from the Technical University of Istanbul and was a

A. Hayir et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 603±613612

Fig. 8. Variation of U(D)/u0 along R1=b for u1 � 2u0, 2p=4, 0, p /4 and u 0, for incident SH waves with g � 0, p/4 and p /2, for a `hard' structure (mD/mS� 6)

and for h � 0.5 and 1.0. For comparison, U�D�=u0 is shown for synchronous base excitation ignoring soil-structure interaction (`Fixed base').

Fig. 7. Variation of U(D)/u0 along R1=b for u1 � 2u0, 2p=4, 0, p=4 and u 0, for incident SH waves with g � 0, p /4 and p /2, for a `soft' structure

(mD=mS � 1=16� and for h � 0.5 and 1.0. For comparison, U(D)/u0 is shown for synchronous base excitation ignoring soil-structure interaction (`Fixed base').

Postdoctoral Fellow at the University of Southern California.

The ®nancial support for his stay from TUBITAK-NATO is

gratefully acknowledged. The authors also thank T.Y. Hao

for her help in preparation of the illustrations presented in this

paper.

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