CUREe - KAJIMA RESEARCH PROJECT

SEISMIC RESPONSE OF UNDERGROUND

STRUCTURES IN SOFT SOILS

J. Enrique Luco

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California

January 15, 1992 - January 14, 1993

CUREe - KAJIMA RESEARCH PROJECT

SEISMIC RESPONSE OF UNDERGROUND

STRUCTURES IN SOFT SOILS

J. Enrique Luco

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California

January 15, 1992 - January 14, 1993

TABLE OF CONTENTS

1. Introduction

2. Spatial Variation of the Free-Field Ground Motion

3. Response of Pipelines and Tunnels to Obliquely Incident Waves

3.1 Response of an Unlined Tunnel

3.2 Response of a Cylindrical Pipeline

3.3 Response of a Lined Tunnel of Arbitrary Cross-Section

4. Response of Underground Tanks and Vertical Shafts to Spatially

Varying Ground Motions

4.1 Response of a Buried Shell with a Vertical Axis of Symmetry

4.2 Response of Underground Structures to Spatially Random Ground Motion

5. Conclusions

6. List of Publications Resulting from Project

7. Acknowledgements

Appendix A. On the Appropriate Depth Dependence for Plane Waves Reflected in a

Viscoelastic Half-Space (Luco/Barros).

Appendix B. Dynamic Displacements and Stresses in the Vicinity of a Cylindrical

Cavity Embedded in a Half-Space (Luco/Barros).

Appendix C. Diffraction of Obliquely Incident Waves by a Cylindrical Cavity

Embedded in a Layered Half-Space (Barros/Luco ).

Appendix D. Seismic Response of a Cylindrical Shell Embedded in a Layered

Viscoelastic Half-Space. I: Formulation (Luco/Barros).

Appendix E. Seismic Response of a Cylindrical Shell Embedded in a Layered

Viscoelastic Half-Space. II: Validation and Numerical Results (Barros/Luco).

SEISMIC RESPONSE OF UNDERGROUND

STRUCTURES IN SOFT SOILS

J. Enrique Luco

University of California, San Diego

SUMMARY

As part of a research project on the seismic response of underground structures

embedded in soft soils we have completed the following tasks:

(i) Derivation of correct 'radiation' conditions to determine the reflected wave field in

viscoelastic media. This is a critical step in the calculation of the deterministic

free-field ground motion.

(ii) Development and validation of a method to calculate the two- and three­

dimensional response of unlined tunnels (cylindrical cavities) embedded in layered

viscoelastic media and subjected to obliquely incident waves.

(iii) Development and validation of two independent methods to obtain the two- and

three-dimensional response of cylindrical pipelines or lined tunnels buried in

layered viscoelastic media and subjected to obliquely incident waves. The first

method is based on an analytic Donnell model for the shell and applies to tunnels or

pipelines of circular cross-section. The second method is based on a finite element

model for the shell and applies to tunnels of arbitrary cross-section.

(iv) Development of a hybrid formulation and of basic subprograms to calculate the

response of a flexible axisymmetric tank and a lined vertical shaft subjected to

obliquely incident seismic waves.

1. INTRODUCTION

The research effort was concentrated on the development and validation of

methods to calculate the seismic response of underground structures subjected to spatially

varying ground motions. For the purposes of the study the soils were represented as

horizontally layered viscoelastic half-spaces. Two classes of structures were considered.

The flrst class included buried cylindrical pipelines or tunnels which can be considered to

be infinitely long. In this case, excitations in the form of incident waves impinging in a

direction normal to the axis of the pipeline or tunnel lead to two-dimensional plane-strain

problems for P, SV and Rayleigh waves and to two-dimensional anti-plane shear

problems for SH-waves. In the more general case of excitations at an oblique angle with

respect to the axis of the pipeline or tunnel, the response is fully three-dimensional.

The second class of structures considered in the project includes flexible

axisymmetric shells with a vertical axis of symmetry. The structures are buried in a

layered viscoelastic medium representing the soil. The shells represent flexible

underground tanks and lined vertical shafts. The shells are subjected to spatially

varying ground motions in the form of non-vertically incident waves or spatially random

ground motions.

As usual in research, we have encountered some unexpected results which by

providing some new opportunities, have led us to change the emphasis of the work

initially planned. For example, in the process of validating a new procedure to calculate

the response of a buried pipeline we found that our results did not agree with some of the

previous results in the literature while agreeing with others. We proceeded then to

remove the pipeline and consider the simpler problem of an unlined tunnel or cavity

buried in a half-space. Again, our results did not agree with some of the results in the

literature. This lead us to review the process of calculating the free-fleld ground motion

in a viscoelastic medium when no inclusion or cavity is present. As a result of this study

we found that the 'standard' radiation condition used for purely elastic media may not

apply to viscoelastic media and that large errors can be obtained if no attention is paid to

this point. After deriving the correct radiation conditions for the free-field ground motion

we found that we could still not explain the differences between our results and some of

the previous results in the literature for unlined cavities and for pipelines. Only after

developing a second independent approach to calculate the response of buried pipelines

and confirming our initial results could we decide that some of the previous results in the

literature are actually in error. This process has taken much longer than initially

scheduled for this part of the project but it has paid off in that some new results on the

propagation of waves in viscoelastic media were obtained and in that two independent

methods to calculate the seismic response of pipelines and lined and unlined tunnels were

developed.

The project represents an effort involving personnel at UCSD (Luco, Barros) and

USC (Wong, Chou). The work on tunnels and pipelines has been conducted at UCSD.

The work on tanks and shafts is a joint UCSDIUSC effort in which the formulation and

some subprograms have been developed at UCSD while the master program is to be

developed at USC. The following sections describe the work done during the 12 months

of the project.

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Figure 1. Schematic Representation of Axisymmetric Structures (Lined Shaft, Underground

Tank) and Infinitely Long Pipelines and Tunnels Embedded in a Multilayered Half-Space.

2. SPATIAL VARIATION OF THE FREE-FIELD GROUND MOTION

As explained in the previous section we found it necessary to investigate the

procedures used to calculate the variation of the free-field ground motion with depth.

Both deterministic and random models of the spatial variation of ground motion assume

that the variation of ground motion with depth can be derived from deterministic models

involving plane P-, SV- and SH- (homogeneous and inhomogeneous) waves. A critical

step in calculating the reflected waves arising from an incident plane wave is the selection

of the correct depth dependence for the reflected waves. This is usually done by invoking

"radiation" conditions at infinity. The standard choice is to select a depth dependence

given by exp ( -vz) subject to the requirement Re v ~ 0. This insures an exponential

decrease with depth. We have found that when attenuation in the medium is included, the

standard condition Re v ~ 0 is not always correct. In particular, for incident SV -waves

within a range of angles of incidence, the correct choice is Re v < 0 instead of Re v ~ 0.

This apparently implausible choice, which implies an exponentially increasing amplitude

with depth for the reflected wave, can be justified by considering plane wave excitation

as the limiting case of the ground motion resulting from a buried point source as this

source recedes to infinity. We have conducted a detailed and definitive derivation of the

correct depth dependence for reflected waves in an anelastic half-space. We have also

studied the errors introduced by use of the incorrect 'standard' radiation condition on the

free-field ground motion and on the response of embedded foundations and cavities. The

results of this work are described in the paper (Appendix A) :

• J.E. Luco and F.C.P. de Barros (1993). "On the Appropriate Depth Dependence for

Plane Waves Reflected in a Viscoelastic Half-Space" (submitted for publication).

The importance of the findings are illustrated in Fig. 2 which shows the horizontal

luy /u8vl, vertical luz /usv I and rocking response lbex/Usv I of a massless rigid rectangular

foundation of width 2b embedded to a depth h (h!b = 1.0) in a uniform viscoelastic

medium (n = 2.0 ~. ~a= 0.005, ~f3 = 0.01) and subjected to a nonvertically incident SV

wave of amplitude Usv and angle of incidence 95 = 25° (w/r to the vertical z-axis). The

results for Case 1 correspond to those for a nonhomogeneous incident plane wave with

real apparent velocity. Those for Cases 2 ana 3 correspond to a homogeneous incident

plane wave and to the choices Rev> 0 andRe v < 0, respectively. In this case, the

results for Case 2 which correspond to the 'standard' radiation condition Rev > 0 are in

error and the error amplifies the response by a factor larger than two.

-

- 0 as =25 5-..... .... ,_ .. .. .. .. ... ... .. 4

o.o .5

.. ...

----easel

········-····-··· Case 2 -·-·-·-·-·· Case 3

.... ...... .... ....... ..... .......... .... .. ...__ __

2.5

3r-~~--.--r-,.-.--.--.--r-.

- 0 as= 25 ----easel ··-··············· Case 2

2 -·-·-·-·-.. Case 3

·····---··-··················-···-··········-··············-··-

z

- 0 as= 25 ----easel ................... Case 2

-·-·-·-·-·· Case 3

1 1 ....... ··

_,_ ..... ~~-------~~-·--

0.0 .5 2.5

oob/~

... .. , ...

0 ....... .0 .5

.. .. .. ..

1.0

oob/~

Figure 2. Normalized Amplitudes of the Response of a Massless Rigid Strip Foundation Embed­

ded in a Viscoelastic Half-Space and Subjected to a Harmonic Plane SV-Wave with Displacement

Amplitude IUsvl and Angle of Incidence Ba = 25°.

2.5

3. RESPONSE OF PIPELINES AND TUNNELS TO

OBLIQUELY INCIDENT WAVES

3.1 Response of an Unlined Tunnel

As a first step we have considered the response of a cylindrical unlined tunnel

embedded in a layered half-space and subjected toP, SV, SHand Rayleigh waves with

different angles of incidence. An indirect boundary integral method based on the use of

moving Green's functions (Barros and Luco, 1992) in a layered half-space was developed

to solve the problem of an infinitely long cavity subjected to obliquely incident waves.

When the excitation is normal to the axis of the unlined tunnel the problem

becomes two-dimensional. A number of solutions for this case have been presented in

the literature. To validate our approach we have conducted an extensive set of

comparisons and a critical evaluation of the previous results for 2-D cases. These results

are contained in the paper (Appendix B) :

• J. E. Luco and F.C.P. de Barros (1993). "Dynamic Displacements and Stresses in

the Vicinity of a Cylindrical Cavity Embedded in a Half-Space" (submitted for

publication).

Our 2-D results for SH-wave excitation agree reasonably well with results

obtained by Lee (1977) and Datta and Shah (1982) by other methods. For Rayleigh-wave

excitation we found that our results for the displacements on the free-surface above the

cavity or on the wall of the cavity agree in shape with the results of Datta and El-Akily

( 1978) and Kontoni, Beskos and Manolis (1987) but differed in amplitude. The earlier

results appear to include erroneous normalization factors. Our results for Rayleigh waves

also differed from results presented by Wong, Shah and Datta (1985).

For P- and SV -waves our 2-D results again differ from the results presented by

Wong et al. (1985). A typical comparison is shown in Figs. 3a and 3b. These figures

show the amplitudes of the horizontal and vertical displacements on the ground surface

above a cavity {H/a = 1.53) subjected to a vertically incident P-wave. The present results

are shown with a solid line while results obtained by Dravinski (personal communication)

and Motosaka (personal communication) are shown with dash-dot lines and open circles,

respectively. It is apparent that the present results agree very closely with those obtained

by Motosaka and are similar to those obtained by Dravinski. The results obtained by

Wong et al. (1985) are shown with segmented lines and appear to be in error. In this

case, the independent calculation by Motosaka permitted us to confirm the accuracy of

our results.

We found no results in the literature for the fully three-dimensional case of a

cavity or unlined tunnel subjected to obliquely incident waves. We conducted an

extensive study of this case, including different types of waves, embedment depths,

angles of incidence and layering conditions. The results of this study are included in the

paper (Appendix C) :

• F.C.P. de Barros and J.E. Luco (1993). "Diffraction of Obliquely Incident Waves

by a Cylindrical Cavity Embedded in a Layered Viscoelastic Half-Space"

( submitted for publication).

3.2 Response of a Cylindrical Pipeline or Tunnel

We have developed a method to calculate the response of a flexible cylindrical

shell of circular cross-section and infinite length embedded in a layered viscoelastic half­

space. The interaction between the shell and the exterior soil are fully accounted for in

the approach. Excitations in the form of plane P, SV and SH waves impinging at

arbitrary oblique angles with respect to the axis of the shell are considered.

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0_3 3 y/a y/a

Figure 3. Amplitudes of the Normalized Total Horizontal (a) and Vertical (b) Displacements on

the Ground Surface for a Vertically Incident P-Wave. Solid lines correspond to the present results,

open circles to the results of Motosaka (1992), dash-dot lines show the results of Dravinski (1992)

and segmented lines show the results of Wong et al (1985).

y

The formulation used is based on a hybrid method in which the exterior (soil)

domain is represented by means of three-dimensional moving Green's functions (Barros

and Luco, 1992) while the shell is modelled by Donnell's shell theory. The coupling

between the shell and the soil is introduced through conditions of continuity of

displacements and tractions at the soil-shell interface.

The DT/IBF (Donnell Theory/Indirect Boundary Formulation) methodology, an

extensive set of comparisons with previous results, new results for embedded shells and

an extensive bibliography on the seismic response of pipelines are included in the papers

(Appendices D and E) : ,

• J. E. Luco and F.C.P. de Barros (1993). "Seismic Response of a Cylindrical Shell

Embedded in a Layered Half-Space. I : Formulation," ( submitted for publication).

• F. C. P. de Barros and J.E. Luco (1993). "Seismic Response of a Cylindrical Shell

Embedded in a Layered Viscoelastic Half-Space. II : Validation and Numerical

Results," (submitted for publication).

In the second paper we present a very complete validation of our results by

comparisons with previous solutions. In the two-dimensional case of waves impinging

normal to the axis of the shell we have found good agreement with previous results of

Lee and Trifunac (1979), Balendra et al. (1985) and Liu et al. (1991) for SH-waves and

with results of Datta et al. (1983) and Liu et al. (1991) for P and SV-waves.

Figure 4 shows a comparison of our results (solid lines) with results obtained by

Balendra et al. (1985) (open circles) for a concrete circular shell (r0 =3m, ri = 0.27m, t =

0.3m, G1 = 8.4 GPa, ~~ = 1,870 rn/sec, PI= 2,410 kg!m3) buried to a depth h = 2.5 r0 =

7.5m in a uniform half-space (G0 = 0.111 GPA, ~0 = 260 rn/sec, Po= 1640 hg!m3). The

medium is subjected to a nonvertically incident SH-wave (Sv = 30°) propagating normal

to the axis of the shell (Sh = 90°) with a frequency of 10.61 Hz. The figure shows

90 180 270 e (degrees)

3

2

1

(b)

75

90 180 270 90 180 270 360 e (degrees) 9 (degrees)

Figure 4. Two-Dimensional Response of a Concrete Pipeline Embedded in Half-Space and

Subjected to a Non-Vertically Incident SH-Wave (iht = 90°, Bv = 30°). Pre~ent results are

shown with solid lines, those of Balendra et al (1984) are shown with open circles.

comparisons for the normalized displacements Ux, the longitudinal traction I.rx at the

soil/pipeline interface and the shear stress I.9x within the pipeline. The results are

clearly in close agreement.

In the three-dimensional case of waves impinging at an oblique angle with respect

to the axis of the shell we have found that our results do agree with the earlier results of

Wong, Shah and Datta (1986) but not with the results of Liu et al. (1991). However, in

the case considered by Liu et al. (1991) our 3-D results based on the use of the simplified

Donnell's shell theory agree with those obtained by a second approach in which the shell

is represented by a finite element model (refer to 3.3 below). A typical comparison is

shown in Fig. 5 showing the longitudinal and radial displacements for a concrete circular

shell buried in a uniform half-space and subjected to a non vertically incident SV -wave

(8v = 30°) acting in the vertical plane of the shell (Sh = 0°). The solid lines show our

results using a Donnell shell theory (DT/IBF). The solid dots show our results using a

combined FE and indirect boundary integral formulation (FE!IBF). The open circles

correspond to the results obtained by Liu et al. (1991). The earlier 3-D results shown in

Fig. 5 appear to be in error. Some new results for the 3-D response of a pipeline or

tunnel, embedded in a layered medium when subjected to obliquely incident waves are

presented in the second paper referred to above (Appendix E).

3.3 Response of a Lined Tunnel of Arbitrary Cross-Section

We have developed a method to calculate the seismic response of a flexible, lined

tunnel of arbitrary cross-section and infinite length embedded in a multilayered half­

space. The tunnel may be excited by P, SV or SH waves with arbitrary horizontal and

vertical angles of incidence. The procedure relies on a special finite element

representation of the. tunnel cross-section coupled with an indirect boundary integral

1

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0 0

II

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0 00

"•

0 00

0 0

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90 180 270 360 °o 9 (degrees)

90 180 270 360 °o 9 (degrees)

90 180 270 360 9 (degrees)

Figure 5. Three-Dimensional Response of a Concrete Pipeline embedded in Half-Space and

Subjected to a Non-Vertically Incident SV-Wave (Oh = 00, Ov = 30°). Present results obtained

by the DT/IBF and FE!IBF approaches are shown with solid lines and dark dots, respectively.

The results of Liu et al (1991) are shown with open circles.

method based on moving Green's functions for the exterior soil or rock. Although the

response and the excitation are three-dimensional the calculations are of the same order

of numerical difficulty as those in two-dimensional problems~

The hybrid FE/IBF (Finite Element/Indirect Boundary Formulation) has been

carefully validated in the particular case of a tunnel of circular cross-section by

comparisons with the solution obtained by the DT/IBF (Donnell Theory/Indirect

Boundary Formulation) approach described in the previous section. In these comparisons

the cylindrical tunnel lining was represented by 4 annular layers of triangular finite

elements including 80 elements in each layer. The results obtained by both methods are

practically identical (Fig. 5). These comparisons validate both approaches and confirm

the accuracy of our results.

Additional tests of the procedure for non-circular cross-sections are being

conducted. The results of this part of the project will be reported in the paper:

• Luco, J.E. and F.C.P. de Barros (1993). "Three-Dimensional Response of a

Cylindrical Tunnel Embedded in a Layered Medium," (in preparation).

4. RESPONSE OF UNDERGROUND TANKS AND VERTICAL SHAFTS

TO SPATIALLY VARYING GROUND MOTIONS

The work in this area concentrates on flexible buried structures such as

underground tanks and deep ~hafts which can be assumed to be axisymmetric. The

structures are embedded in a multilayered viscoelastic half-space and are subjected to

obliquely incident waves or to a spatially random ground motion.

4.1 Response of a Buried Shell With a vertical Axis of Symmetry

A hybrid procedure that combines that combines a finite element model to

represent a finite axisymmetric region (including the structure) with an indirect boundary

integral approach based on ring-load Green's functions for the exterior layered half-space

has been formulated.

Work on the formulation (Luco), computer program to calculate ring-load Green's

functions (Apsel and Luco, 1983), programs tocalculate free-field ground motion in a

layered medium (Barros and Luco) and programs for axisymmetric finite elements of

arbitrary order (Chou, Wong, Luco) has been completed. Professor H.L. Wong at USC is

responsible for assembling all of these elements into a combined computer program.

Once this program is completed, the validation will start by comparisons with some of

our earlier results for rigid axisymmetric structures embedded in layered media (Luco and

Wong, 1976). Results will then be calculated for a particular underground tank and a

particular shaft of interest to our counterparts at Kajima Corporation.

4.2 Response of Underground Structures to Spatially Random Ground Motion

Initially, we had planned to extend the approach used to calculate the response of

rigid surface foundations to a spatially random ground motion (Luco and Wong, 1986;

Luco and Mita, 1987; Luco et al., 1988) to the case of flexible underground structures.

Unless very drastic assumptions are made, such extension would require an amount of

information as to the properties of the random process which goes beyond what is

currently available. At this point, we are suggesting the use of a simplified approach in

which the free-field is represented as resulting from elastic waves with a distribution of

angles of incidence. The response, in this case, would be obtained by a combination of

results obtained by the approach described in Section 4.1.

5. CONCLUSIONS

Twelve months after the initiation of this ambitious project we have completed

most of the work on the seismic response of pipelines and tunnels buried in layered media

and subjected to obliquely incident seismic waves of different types. Two independent

approaches have been developed and carefully tested. These approaches allow us to

calculate the fully three-dimensional response of cylindrical tunnels or pipelines

including displacements and stresses within the pipeline or tunnel and the contact

tractions between the soil or rock and the buried structure.

Work on the formulation and all subprograms required to calculate the seismic

response of flexible underground tanks and deep shafts has been completed. These

elements will be combined into a master computer program by Professor H. L. Wong.

After validation the resulting program will be used to calculate the seismic response of

underground tanks and vertical shafts subjected to obliquely incident waves.

6. PUBLICATIONS RESULTING FROM PROJECT

Luco, J. E. and F.C.P. de Barros (1993). "On the Appropriate Depth Dependence for

Plane Waves reflected in a Viscoelastic Half-Space," (submitted for publication).

Luco, J. E. and F.C.P. de Barros (1993). "Dynamic Displacements and Stresses in the

Vicinity of a Cylindrical Cavity embedded in a Half-Space," (submitted for

publication).

Barros, F.C.P. de and J. E. Luco (1993). "Diffraction of Obliquely Incident Waves by a

Cylindrical Cavity Embedded in a Layered Viscoelastic Half-Space," (submitted for

publication).

Luco, J. E. and F.C.P. de Barros (1993). "Seismic Response of a Cylindrical Shell

Embedded in a Layered Viscoelastic Half-Space. I: Formulation" (submitted for

publication).

Barros, F.C.P. de and J. E. Luco (1993). "Seismic Response of a Cylindrical Shell

Embedded in a Layered Viscoelastic Half-Space. II : Validation and Numerical

Results," (submitted for publication).

Luco, J.E. and F.C.P. de Barros (1992). "Three-Dimensional Response of a Cylindrical

Tunnel Embedded in a Layered Medium," (in preparation).

7. ACKNOWLEDGEMENTS

The work described here was supported by a grant from Caiifornia Universities

for Research in Earthquake Engineering (CUREe) as part of a CUREe-Kajima

Project. We are most grateful to our counterparts at Kajima Corporation and

particularly to Dr. M. Motosaka for his independent numerical results which allowed

us to settle a number of issues.

Appendix A. On the Appropriate Depth Dependence for Plane Waves

Reflected in a Viscoelastic Half-Space (Luco/Barros).

ON THE APPROPRIATE DEPTH DEPENDENCE FOR

PLANE WAVES REFLECTED IN A VISCOELASTIC HALF-SPACE

J. E. Luco and F. C. P. de Barros

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California 92093-0411.

ABSTRACT

The correct depth dependence for plane waves reflected from the free surface of a vis­

coelastic half-space when subjected to plane incident waves is obtained by a limiting process

from the high-frequency response of the viscoelastic half-space to a harmonic buried line source.

By allowing the source to recede to infinity, a local plane wave representation of the dis­

placement field close to the free surface is obtained. In particular, it is found that for a ho­

mogeneous plane incident SV-wave, the standard "radiation" condition Rev ~ 0 for the re-o

fleeted P-wave applies only if the angle of incidence Bs with respect to the vertical is such that

0::::; B8 < 81 =arcsin [J~o/~p (,Bjc:t)] or Tr/2 > B8 > 82 = axcsin(,B/a) where a, ,8 are the

velocities of the P- and S-waves and eo. ep are the corresponding damping ratios (eo ::::; ep). For 01 < Bs < 02 , the appropriate condition is Im v > 0 which in this case leads to Rev < 0.

The effects of. erroneously using the standard "radiation" condition in the process of

calculating the seismic response of layered soil deposits, rigid embedded foundations, and to­

pographical canyons or valleys embbedded in a viscoelastic half-space are illustrated by several

examples.

INTRODUCTION

Studies of the seismic response of layered soil deposits, of surface and embedded foun­

dations and of topographic or geologic features such as canyons and valleys (Fig. 1) are usually

based on the assumption of plane (homogeneous or inhomogeneous) incident waves. This as­

sumption permits a description of the incident motion in terms of a few parameters (amplitude,

direction of propagation, type of wave) and also uncouples the problem of local site effects from

those related to the source and source-to-site propagation. Also, to improve the representation

of the soil and rock media, attenuation in the fonn of hysteretic damping for P- and S-waves

is introduced in many cases. This combination of plane waves reflecting at the free-surface of

an anelastic medium or reflecting and reffracting at the interface between two anelastic media

leads to some peculiar behavior in which some of the reflected and transmitted waves may have

amplitudes increasing exponentially away from the interface or phases which increase toward

the interface. These behaviors which at first sight appear unplausible have been described by

Buchen (1971b), Borchert (1977, 1982, 1985), Krebes (1983) and Richards (1984).

The problem of determining the free-field ground motion for a uniform or layered medium

subjected to a plane incident wave hinges upon the selection of the appropriate depth dependence

for the reflected waves in the underlying half-space. The choice of depth dependence is usually

made on the basis of a "radiation" condition. The results of Buchen (1971 a, b), Borchert (1973a,

1973b, 1977, 1982, 1985), Krebes and Hron (1980), Krebes (1983) and Richards (1984) and

the earlier studies by Lockett (1962), Cooper and Reiss (1966), Cooper (1967), Shaw and Bugl

(1969) and Schoenberg (1971) suggest or indicate that the standard "radiation" condition used in

the analysis of reflection/reffraction of plane elastic waves may not be applicable in the anelastic

case. Richards (1984) has proposed that the appropriate depth dependence close to the free

surface could be obtained from consideration of the field created by a point or line source buried

in the half-space. Due to the curvature of the wavefront the situation near the free boundary

may be different from that at depth and, hence, a different condition may need to be applied

within the plane wave approximation near the boundary.

The objective of this paper is to obtain in full detail the correct depth dependence for

the reflected P-wave resulting from a homogeneous plane SV -wave incident upon the surface

of a viscoelastic half-space. The approach used is based on the suggestion by Richards (1984)

and relies on the derivation of the high-frequency response of a viscoelastic half-space to a

harmonic buried line source. By taking the limit as the source recedes to infinity the appropriate

depth dependence for plane waves near the surface of the half-space is obtained as a function

of the angle of incidence of the SV -wave. The study is limited to SV incident waves since no

difficulties are apparent for P or SH incident waves. Also, the study is mostly concerned with a

uniform half-space since the difficulties in the case of a layered half-space are only encountered

in the underlying medium. Within the layers, the existence of upgoing and downgoing waves

automatically covers the two possible choices of depth dependence.

The errors resulting from the use of the wrong depth dependence are also examined by

considering the response of a layered soil deposit, a rigid embedded foundation, a topographical

canyon and a valley embedded in a viscoelastic half-space to nonvertically incident SV -waves.

FREE-FIELD GROUND MOTION

To illustrate the situation consider a unifonn viscoelastic half-space ( z > 0) subjected to a

plane incident SV-wave with harmonic time dependence eiwt where w is the frequency (w > 0).

The viscoelastic half-space is characterized by the density p and by the complex velocities for

P- and S-waves

a o=---

1- iea {3 = p

1- ie13

(1a)

(1b)

where ea = Q1 ~ 1 and ep = Q

1 ~ 1 are the damping ratios for P- and S-waves, 2 0 2 /3

respectively and a and p are (to the first order of ea. ep) the real parts of the corresponding

wave velocities. Throughout the discussion it will be assumed that ( 4/3)(~ fa)2 ep < eo ~ e{J·

The incident displacement field for a general plane SV-wave can be represented by

( uinc uinc uinc) = __ z U (v' 0 ik) e"' z-ikx X l y l Z kp SV l l

(2)

where the factor eiwt has been omitted and where U sv is the amplitude of the incident displace­

ment at the origin (0,0,0), k is the (in general complex) horizontal wavenumber, kp = w/{3

and

(3)

The Riemann sheet Re v' ~ 0 is selected so that the incident field decays in amplitude as the

free surface z = 0 is approached from below. This in turn implies that the amplitude of the

incident field tends to infinity as z _... oo.

The non-zero components of the total displacement field including the incident field and

the reflected P- and S-waves are given by

· U SV [ t v' z + · k R -vz 'R -v' z] -ikx Ux = -z ""k';" v e z SP e - v ss e e (4a)

Uz = -i U~v [i k e"'z + vRsp e-vz + i k Rss e-v' z] e-ikx (4b)

where Rsp and Rss are the P- and S-wave reflection coefficients and

(5)

in which k01 = w /a. The stress components of interest in imposing boundary conditions are

u zz/ J.L = i U~v [ -2ikv' e"' z + (2k2 - k~)Rsp e-vz + 2ikv' Rss e-"' z] e-ikx (6a)

Uzz/J.L = i U~v [-(2k2 - k~)e"'z + 2ikvRspe-vz- (2k2 - k~)Rsse-v'z] e-ikx (6b)

where J.L = pf32 is the (complex) shear modulus.

Imposing the boundary conditions u zz = 0 and u zz = 0 at z = 0 leads to the following

expressions for the reflection coefficients

4ikv'(2k2 - k~) Rsp = (2k2 - k~)2 - 4k2vv'

Rss= (2k2 - k~)2 + 4k2 vv'

(2k2 - k~)2 - 4k2vv1

(7a)

(7b)

By use of Eqs. (4a,b) and (7a,b) the total free-field ground motion can be calculated

throughout the medium. In particular, the total free-field ground motion on the ground surface

is given by

2iv'kp(2k2- k~) -ikx

u.x(x,O,O) = (2k2- k~)2- 4k2vv' e Usv (8a)

( ) 4kkpvv' -ikx U

Uz x, 0, 0 =- (2k2 - k~)2- 4k2vv' e sv (8b)

To calculate the free-field ground motion it is necessary to make two decisions: (i)

to select an appropriate value for the complex wavenumber k characterizing the directions of

propagation and attenuation for the incident field, and (ii) to select the appropriate Riemann

sheet for v = ( k2 - k01 ) 112

• It would be desirable to impose the condition Rev > 0 so that the

reflected P-wave decays exponentially from the free boundary but this may not be appropriate

in some cases.

Of the many possible choices for k we consider only two cases: (a) k = ( w I i3) sin 8 8

where 88 is a real angle (0 :5 88 :5 1r 12) corresponding to a plane inhomogeneous incident wave,

and (b) k = k p sin 8 8 = ( w I i3)( 1 - iep) sin 8 8 where again 0 :5 8 8 :5 1r 12 which corresponds to

a homogeneous incident wave. In the following these two cases are considered separately.

(a) Inhomogeneous Incident Wave: k = ( w I P) sin 8 8'

The choice k = ( w I P) sin 8 8 leads to

(9a)

(9b)

and

(10)

In this case, since k is real the choice of the sheet Re v ~ 0, Re v' > 0 leads to Im v > 0,

Im v' > 0. In this case, the incident wave propagates and attenuates toward the boundary while

the reflected P- and S-waves propagate and attenuate away from the boundary. In particular,

from Eq. (9b) we find

v' z- ikx ~ i (~) (z cos 88- X sin 88) + ( w;) ( coe:88) (11)

which indicates that the incident wave propagates with incident angle 88 (with respect to the

vertical axis) and attenuates in the direction of -z.

(b) Homogeneous Incident Wave: k = kp sin88 •

The choice k = kpsin88 = (wiP)(1- iep)sin88 with the condition Rev'> 0 leads to

(12)

and

(13)

which indicates that the incident SV -wave propagates and attenuates upward in the direction

defined by the direction cosines (sin 8s, 0, - cos 8, ).

With respect to the reflected P- wave we find that

(14)

The imaginary part of v2 changes sign at the first characteristic angle

81 =arcsin [ ( ~; )"' ( !) ] (15)

while the real part of v2 changes sign at the second characteristic angle

62 =arcsin(!) (16)

which corresponds to the usual critical angle for purely elastic media. Notice that 61 < 62 for

The locus of k = kp sin 8, in the complex k-plane as 8s varies from 0 to 1r /2 is shown

/ in Fig. 2 by the line OC. The origin k = 0 corresponds to 8s = 0. The point A where OC

intersects the line Rev = 0 corresponds to 88 = 61. When 88 = 62 , the real part of k is

equal to that at the branch point v = 0 at B (Re k = w / &). Finally, as 8s approaches goo, k

approaches the branch point v' = 0 at C. This discussion indicates a change in behavior at the

first characteristic angle 88 = 61. If 0 :5 88 :5 61, then Rev ~ 0 and Im v ~ 0 and the reflected

P-wave propagates and attenuates away from the free boundary z = 0. For 61 < 8 s < goo two

choices are possible: (i) we maintain the requirement Re v > 0 in which case Re v > 0 and

Im v :5 0 resulting in a reflected wave which decays away from the free boundary but propagates

toward it, or (ii) we require that Im v ~ 0 in which case Re v :::; 0 which imply that the reflected

P-wave propagates away from the boundaries but also increases exponentially in amplitude away

from the boundary.

The resulting values of v for k = kp sin Us and for the choices Rev ;?: 0 and Im v ;?: 0

are shown in Fig. 3 for the case eCl' = 0.05, ep = 0.10 and 0:1~ = 2 for which 81 = 20.7°

and 82 = 30° (large values of eCl' and e/J have been used to better illustrate the behavior of v).

For the choice Re v > 0 (Case 2), the locus of v consist of two segments from A (Us = 0)

-- - + - - 0 • to B(8s = 81 ) and from C(8s = 81 ) to D(8s = 82 ) and E(8s = 90 ). In th1s case, there

is a discontinuity at Us = 81 • The choice lm v ;?: 0 (Case 3) leads to the arc passing through

A(Us = 0), B(Us = 81), F(Us = 82 ) and G(Us = 90°). In this case there is no discontinuity

in the behavior of v. Also shown in Fig. 3 are the values of v for k = ( w I~) sin 0 s (Case 1 ).

For this inhomogeneous wave the values of v are in the first quadrant and the reflected P-wave

propagates and decays away from the boundary.

The effects of the choice of k (incident wave) and v (Riemann sheet) on the amplitude

of the free-field ground motion in a uniform viscoelastic half-space (iii~ = 2.0, eCl' = 0.005,

ep = 0.01) subjected to plane incident SV-waves are illustrated in Figs. 4 and 5. The results

in Fig. 4 include the amplitudes of uz:(O, 0, z )IU sv and u .. (O, 0, z )IU sv versus the angle of

incidence Bs for three values of the normalized depth z = wz I fi = 0, 0.5 and 1.0. The three

curves shown in each frame of Fig. 4 correspond to: Case 1 defined by k = ( w I fi) sin 0 s,

Rev > 0 (solid line); Case 2 corresponding to k = kp sin Us and Rev ;?: 0 (dotted line) and

Case 3 defined by k = kp sin Us, Im v > 0 (dash-dot line). The results for the homogeneous

incident wave (k = kp sin Os) with the standard radiation condition Rev 2: 0 (Case 2) show

a marked discontinuity at Os = 81 = 20.7°. Beyond this angle, the free-field for Case 2 is

significantly different from the free-field for Case 3 (k = kp sin Us, Im v ;?: 0) and this difference

increases with depth. For Os < 82 = 30°, the results for Case 1 [k = ( w I 'fi) sin Os, Rev ;?: Ol

and Case 3 [k = kp sin Os, Im v > 0] are similar. For Us > 82 = 30° the values for Case 1

[k = (w I 'fi) sin Os, Re > 0] are similar to those for Case 2 [k = k13 sin Os, Rev > 0].

Figure 5 shows the variation of the normalized free-field amplitudes luz:(O, 0, z )IU sv I and luz(O, 0, z)IU svl as function of the dimensionless depth z = wzl ~for angles of incidence

08 = 25°, 30° and 60°. Comparisons of the results for Case 1, k = (wl~)sinOs, Rev> 0

(solid line); Case 2, k = kp sin 06 , Rev ~ 0 (dotted line) and Case 3, k = kp sin Bs, Im v > 0

(dash-dot line) show significant differences in the distribution of free-field ground motion with

depth depending on the selection of k and v.

S, SS AND SP-WAVES FOR A LINE SOURCE BURIED IN A

VISCOELASTIC HALF-SPACE

Richards (1984) has suggested that appropriate values for v and a full understanding of

the local "radiation" conditions can be obtained from consideration of the field created by a

buried point or line source. Due to the curvature of the wavefront the situation near the free

boundary may be different from that deep below the source and hence a different "radiation"

condition may need to be applied within the plane wave approximation near the free boundary.

Richards suggested to use the location of the saddle point in an integral representation of the

solution of Lamb's problem as a way to determine the appropriate Riemann sheet for v. In

this section we delve into the details of such a procedure. For this purpose we consider the

field created by a uniform vertical load distributed over an infinite horizontal line (x = -x8 ,

-oo < y < oo, z = z8 ) buried at a depth zs below the surface of a uniform viscoelastic half­

space. We consider separately the high-frequency plane-wave representation close to the surface

of the half-space of the direct SV -wave, the reflected SS-wave and the reflected SP-wave. The

general form of the solution for the plane-strain case of a line source buried in a half-space is

similar to that found by Buchen (197lb) for aSH line source close to the interface between two

welded half-spaces.

Direct S-Wave.

The non-zero displacement components associated with the direct SV-wave are given by

the integral

(17)

where

9I(k) = v'lz- zsl + i k(x + Xs) . (18)

The harmonic time dependence factor eiwt has been omitted from Eq. (17). The radiation

condition below the source is satisfied by requiring Re v' ~ 0.

A high-frequency approximation to the direct SV-wave can be obtained by the method

of steepest descent The saddle point k1 of 91 ( k) satisfies

9~(k1) = k~ lz- z,l + i(x + x,) = 0 (19) v1

where v{ is the value of v' at the saddle point Introducing the change of variables

z,- z = S1 cos811 (20a)

X+ x, = s1 sin811 (20b)

where S1 is the length of the path travelled by the S-wave and 811 is the (real) angle of incidence c

(with respect to the vertical), it is found that

v~ = i kp cos 811

and

"(k ) . s1 91 1 = -z k 2 ll

p COS U11

(21)

(22)

(23)

(24)

for z < z, (0 < 8,1 < 1r /2). The path of steepest descent defined by Im [91 ( k) - 91 ( k1 )] = 0

is asymptotic to the lines (Im k /Re k) = ±tan 8,1 in the third and fourth quadrants of the

k-plane, respectively. In the vicinity of the saddle point, the path of steepest descent is inclined

with respect to the real axis by the angle 1r /4 - t:1 /2 where t: 1 ~ ~P is the phase angle of

(Stfkp cos2 8,1). As shown in Fig. 7(a) it is possible to deform the initial contour of integration

into the steepest descent path. The asymptotic approximation of the resulting integral is given

by

{us} 1 u~ = 47rJLk~ (25)

where to the first order in ~P

l9~'(ki)I ~ StP (26) w cos2 881

Eqs. (21), (23) and (25) show that at high-frequencies the direct SV-wave can be represented

locally by a plane homogeneous wave propagating and attenuating in the direction defined by

881.

Reflected SS-Wave.

We consider next the reflected SV-wave resulting from the incident SV-waves. For the

line source considered the corresponding displacement components are given by

{u;

5(x,O,z)} _ 1 100

{ ik } R e-g2 (k)dk u55(x 0 z) - 47r11P k2 fv' 55

z ' ' r fJ -oo (27)

where

92(k) = v'(z + Z8 ) + i k(x + Xs) . (28)

and Rss is the reflection coefficient given by Eq. (7b). It should be noted that Rss depends

on both v and v'. The radiation conditions employed to derive Eq. (27) require that Re v ~ 0,

Rev'> 0.

In this case, the saddle point k2 of g2 ( k) satisfies

g~(k2) = k; (z + Zs) + i(x + Xs) = 0 v . 2

(29)

where v~ is the value of v' for k = k2• At this point it is convenient to introduce the change of

variables

z = s~ cosBs2

(30a)

(30b)

(30c)

where s2 and s~ are the lengths of paths travelled by the incident and reflected waves, respec­

tively, and Bs2 is the angle of incidence with respect to the vertical axis (Fig. 6). We find from

Eqs. (29) and (30a, b, c) that

I . -v2 = z k fJ cos 8 s2 ,

(31)

(32)

(33)

and

"(k ) _ . ( s2 + s~ ) 92 2 - -t kp cos2 88 2

(34)

The path of steepest descent defined in this case by Im [g2 ( k) - 9 2 ( k2 )] = 0 is asymptotic to

the lines (Im k/Re k) =±tan 082 in the third and fourth quadrants of the k-plane, respec~vely.

In the vicinity of k2 the steepest descent plane is inclined with respect to the real axis by the

angle 7r/4- t:2 /2 where t:2 :::::: ep is the phase angle of (S2 + S~)/kp cos2 Bs2 • Formally, the

asymptotic approximation for the reflected SV-wave is given by

27r (k2) {iv~} R (k) -g2 (k 2 )+i(i-.!f)

I "(k )I 1 k ss 2 e 92 2 V2 2 (35)

where

(36)

and Rss(k2 ) is the value of Rss for k = k2• Eqs. (33) and (35) indicate that the reflected

SV -wave can be represented locally as a homogeneous plane wave propagating and attenuating

in the direction defined by the direction cosines (sin 082 , 0, cos 082).

To completely define Rss(k2 ) it is necessary to consider the details of the steepest descent

paths. Three separate cases arise depending on the value of 082 :

(a) 0:::; Bs2 < 81 =arcsin ( JeO//ep (P/a)). In this case Eq. (14) indicates that the saddle

point is to the left of the branch cut Rev = 0 and consequentely the value of v = v2 at

the saddle point k = k2 is such that Re v2 ~ 0 and Im v2 ~ 0. The steepest descent path

is as shown in Fig. 7(b).

(b) 81 < B82 < 82 = arcsin(P/ii). In this case the saddle point is to the right of the branch

cut Re v = 0 and the steepest descent path which starts and ends in the Re v ~ 0,

Re v' ~ 0 Riemann sheet goes through the saddle point k2 on the sheet Re v < 0,

Rev' > 0 as shown in Fig. 7(c). In this case the appropriate value for v = v2 at the

saddle point is such that Re v2 < 0 and Im v2 > 0.

(c) 82 < 082 < 1r /2. In this case the steepest descent path starts in the third quadrant on

the sheet Re v < 0, Re v' > 0, crosses the branch cuts Re v = 0 and Re v' = 0 on the

fourth quadrant, travels on the sheet Rev > 0, Re v' < 0, crosses again the branch cut

Re v' = 0 into the sheet Re v ;::: 0, Re v' 2: 0 near the saddle point and then goes to

the asymptotic line (lmk/Rek) = -tanB82 on the same sheet as shown in Fig. 7(d).

In this case the appropriate value for v = v2 at the saddle point is such that Re v2 ;::: 0,

Rev~> 0.

In the case 82 < B 82 < 1r /2. the original contour is deformed into a contour which

includes a loop around the branch cut Rev = 0 in addition to the steepest descent contour. The

contribution from the loop corresponds to the sPs wave.

Reflected SP Wave.

The displacement components associated with P-waves reflected at the free surface as a

result of SV-waves incident from the source are given by

{ u~=(x,O,z)} = 1 21oo { (k2jv'),} Rspe-g3(k)dk Uz (x, 0, z) 47rJ.tk11 _

00 - (kvfv) (37)

where

(38)

and Rsp is the reflection coefficient given by Eq. (7a). Again, the radiation condition below

the source is satisfied by requiring Re v 2:: 0, Re v' 2:: 0.

We consider next high-frequency saddle-point approximations to the waves given by

Eq. (37). The saddle point k3 of g3 ( k) satisfies

(39)

where v~ and v3 are the values of v' and v at k = k3 • For cases in which the resulting waves

can be represented by rays we introduce the notation

(40a)

z = Pa cosBpa (40b)

(40c)

where Sa and Pa are the paths of the incident SV -wave and of the reflected P-wave, respectively.

The angles 083 and Bp3 shown in Fig. 6 are real. We also write

(41)

where 883 and 8p3 are complex angles. Eq. (41) is a statement of Snell's law in complex form.

From Eq. (41) we find that

v~ = ikp cos Bsa (42a)

(42b)

Substitution from Eqs. (40a,b,c), (41) and (42a,b) into Eq. (39) leads to

(43)

which together with Eq. (41) define 8 8 3 and 8p3·

To obtain the solution we write

(44a)

(44b)

where 8~3 and 8~3 are assumed to be small. Substitution from Eqs. (44a,b) into Eqs. (41) and __/'

(43) leads to

- ("P) -sin 8s3 = 0

sin 8p3

and

(45) .

(46a)

(46b)

where only terms up to first order in eO' and efJ have been kept. Solving for 8~3 and 8~3 results

in

(47a)

(47b)

The resulting expression for the saddle point k3 is

(48)

where

(49)

Fig. 8 for Re (Pk/w) < 0.5 shows the loci of the saddle point ka in the nonnalized

complex k-plane as the incident angle 9 ,a varies from 0 to 82 = arcsin(P /a) for different

values of Pa/ Sa. The results in Fig. 8 correspond to the case a/ "P = 2.0 and ep/ecx = 2. For

P / S > 1 the saddle point ka is always to the left of the branch cut Rev = 0. For P / S < 1

the saddle point may be to the left or to the right of the branch cut Re v = 0 depending on the

value of the incidence angle 9,a. If 0 < 9,a < 8a where

8a =arcsin(!) eQ(Sa + (a/"P)Pa]

e{Js3 + eQ(f3/a)P3 (50)

then the saddle point is to the left of the branch cut. The saddle point is to the right of the

branch cut if 83 < 9,3 < 1r /2. Notice that as P3 / Sa --+ 0 then 83 --+ 81 = arcsin .J[:J[i and

that (--+ ep. In this limiting case the reflected P-wave corresponds to an homogeneo?s incident

wave with k = kp sin 9,.

The expressions for va and v~ are

w ii . c (ep- eQ )S3/3 sm 8p3 [

-.2- ]}

(a) P { P3acos2 8,a+Sa/3cos2 8pa V3 = - COS !7 3 t + 1, Q - (51a)

v~ = (~) cos 9, 3 {i + [efJ + _(..:....;.e.:....p_-:-e=-Q...:....)P_3_a_s-=in_2_9_,a-=--]}

/3 Paa cos2 8,a + Sa/3 cos2 8pa (51b)

which indicate that Im Va 2:: 0, Im v~ 2:: 0 and that Rev~ 2:: 0 if e(3 2:: eQ• The real part of Va

is positive if 9,3 < 83 and negative if 9,3 > 83 • For 83 < 9,3 < 82 the saddle point given by

Eq. (48) is then on the sheet Im v > 0, Rev< 0.

We now investigate whether the path of integration can be defonned into the steepest

descent path through the saddle point. The values of ga ( k) and g~ ( k) at k3 are given by

(52)

and

(53)

The path of steepest descent is defined by Im [93 ( k) - 93 ( k3)] = 0. For I k I large, the steepest

descent path in the sheet Re v ~ 0 Re v' ~ 0 tends to the line

in the third quadrant and to

Im k = ( s3 sin Ba3 + p3 sin Bp3 ) Re k s3 cos 8s3 + P3 cos 8p3

Im k =- ( s3 sinBs3 + p3 sinBp3) Re k s3 cos 8s3 + p3 cos 8p3

(54a)

(54b)

in the fourth quadrant of k-plane. In the vicinity of the saddle point the steepest descent path is

inclined with respect to the real axis by the angle 1r /4 - e3 /2 where e3 is the (small) phase of

the quantity in brackets on the right-hand-side of Eq. (53).

It is possible to show that the initial contour can be deformed into the steepest descent

path. In this case the approximate expressions for u;P and u;P are

(55)

where Rsp(k3) is the value of Rsp for k = k3 and

(56)

For 0 ~ 083 < 83 and for P3fS3 ~ 1, the location of the steepest descent path is similar to that

shown in Fig. 7(b) and the appropriate value for vat the saddle point is such that Rev > 0. For

83 < 083 < 82 and for P3 / S3 ~ l, the location of the saddle point is such that Rev< 0.

Fig. 9 displays the lines of constant amplitudes (solid lines) defined by Re [93 (k3 )] =

(wS3/ P)e/3 + (wP3ja)e01 = ry1 (wz8 / P) and the lines of constant phase (segmented lines)

defined by Im[93(k3)] = (wS3/P) + (wP3jii) = "72(wz8 /P). The results shown corre­

spond to the case ii/ p = 2.0, e(:J/eo = 2.0 and are shown for values of "71 = "72 = 2.125, 2.25, 2.375, 2.5, 2.625, 2. 75, 2.875 and 3.0. The constant phase lines are not cir­

cular indicating that the phase velocity depends on direction. The constant amplitude lines show

that over most of the half-space the amplitude of the reflected SP-waves decays with depth.

However there is a shallow region between (x +xa)lxa = [(ol P) 2(e 13 leo:) -1] tan3 81 = 0.378

and (X + X a) I X 8 = [ ( Q I Pl ( e/3 I eo:) - 1] tan 3 82 = 1.34 7 corresponding to angles of incidence

between 81 = 20.7° and 82 = 30° where the amplitude increases with depth.

We now turn our attention to the wide-angle (Baa > 82) reflected sP-wave. To obtain this

wave we write

from where

v~ = i kp cos8aa

Substitution from Eqs. (57), (58a) and (58b) leads to the condition for the saddle point

(57)

(58a)

(58b)

Za sin 8a3 sinh 8p3 + iz cos 8a3 cosh8p3 + i(x +X a) cos 8a3 sinh Bpa = 0 . (59)

To obtain 8a3 and 8p3 we write 8a3 and 8p3 as in Eqs. (44a) and (44b). Substitution from

Eqs. (44a) and (44b) into Eqs. (56) and (58) leads to

- (P) -sin8aa = 0

cosh8p3 (60a)

(x+xs) -

Zs = tanBs3 (60b)

8:3 = - ( :s) coth Bpa cos2

Baa (60c)

o~, = _ [ (p) (:,) ::~!;: + <~r ~.)] (;) ;~:e;:, (60d)

where only terms up to the first order in eo:. e/3 and z I Zs have been kept The saddle point ka

is again given by Eq. (48) but in this case

(61)

The expressions for v3 and v~ are

w 'nh li . w { c 'nh li l/3 = -::- 51 llp3 -1 -::- I.,Q 51 llp3 + a a

(62a)

v~ = i ~ co5B83 + ~ [ep- ~ (:8

) cothBp3 5in2B83] co5B83 (62b)

which indicate that Re v3 > 0, lm v3 < 0 and that Im v~ > 0 and Re v~ > 0 for z / z 8 ~ 1. The

values of 93 ( k3) and 9~ ( k3) are given by

and

(k ) _ . kpz8

93 3 -1 COS 88 3

"(k ) . [ Z 8 1Z ] 9 3 = -1 -3 kp co53 Bs3 kQ 5inh3 Bp3

(63)

(64)

The loci of the saddle point k3 in the nonnalized k-plane as the incident angle 8 83 varies

from 82 to goo for different values of z/z8 are shown in Fig. 8 in the range 0.5 < Re (Pk/w) <

1.0 . The saddle point k3 for fh < 083 < goo is in the Riemann sheet Rev > 0, Im v < 0.

In this case (82 < B83 < 1r /2), the defonnation of the original contour into the steepest

descent path is similar to that shown in Fig. 7(d). Again, the modified contour includes a loop

around the branch cut Re v = 0. The contribution from the vicinity of the saddle point is given

by Eq. (55) with k3 given by Eqs. (48) and (61), v3 and v~ by Eqs. (62a) and (62b), 93 and 9~

by Eqs. (63) and (64) and where e3 is the phase of i 9~' ( k3 ). It is important to note that in this

case Re v3 > 0 and Im v3 < 0. Thus the reflected P-wave attenuates away from the free-surface

but locally the phase decreases away from the boundary.

Total Field Close to the Free-Surface and Away from the Source.

At this point we allow the source to recede to infinity (x 8 = oo, z8 = oo) in such a way

that x 8 /z8 = tanB8 • In the limit, for xjx 8 ~ 1 and z/zs ~ 1, we find

(65a)

(65b)

(65c)

and

(65d)

where S = Jx; + z;.

Combining the results from Eqs. (25), (35) and (55) we find that the total field associated

with incident SV -waves can be written in the form

where 1

Usv = 4 k 'lrJ.l {3

(~,) (67)

Eq. (66) has exactly the same form as the free-field ground motion for an incident SV-wave

given by Eqs. (4a) and (4b), but now we know that the appropriate values of k, v', and v are

such that

(i) k = k13 sin B8

(ii) Rev' ~ 0, Im v' ~ 0 for 0 :5 B8 < 11:/2

(iv) Rev< 0, lmv ~ 0 if 81 < B8 < 82 = arcsin(P/0:)

(v) Rev ~ 0, Im v :5 0 if 82 < B8 < 11:/2.

EFFECT OF DEPTH DEPENDENCE ON SEISMIC RESPONSE

Next we evaluate the effect of using the wrong "radiation" condition in the calculation

of the seismic response of a layered soil deposit, a rigid embedded foundation, a topographical

canyon and a valley embedded in a viscoelastic half-space when subjected to a nonvertically

incident plane SV-wave.

As a first example we consider the response of a viscoelastic layer of tickness H resting

on a viscoelastic half-space excited by a plane SV-wave with incidence angle B8 = 25°. The

layer (medium 1) and the underlying half-space (medium 2) are characterized by the properties

Oto Pto ealo epto Pl and 02, P2. ea2• eP2• P2· respectively. In particular we consider the case

ad P2 = otf Pl = 2.0, otf P2 = 1.0, ptf P2 = 1.0, eal = 0.02, e{jl = 0.04, ea2 = 0.005 and

ep2 = 0.01. The normalized amplitudes of the horizontal lux(O, 0, O)l and vertical luz(O, 0, O)l

displacement components on the free-surface of the layered half-space are shown in Fig. 10

versus the dimensionless frequency wH / fi1 • The amplitudes are normalized by the amplitude

IUsvl of the displacement for the incident SV-wave. The incident displacement field in the

underlying half-space (i 2:: H) is given by Eq. (2) with kp replaced by kf12 = wj /32. Fig. 10

includes results for the following three cases:

(b) Case 2 defined by k = k{j2 sin Bs = (w/ P2)(l- i6) sin B8 , Re v2 2:: 0 and

(c) Case 3 corresponding to k = kp2 sinB8 , Im v2 2:: 0.

The assumed angle of incidence B s = 25 o falls between the characteristic angles

81 = 20.7° and 82 = 30° for the underlying half-space. In this case, for the homogeneous

incident wave k = kp2 sin B8 the appropriate condition for v2 is Im v2 2:: 0. Thus, Case 3

corresponds to the correct choice while Case 2 is wrong. The results for Case 1 corresponding

to a nonhomogeneous incident wave such that k is real are offered for· comparison. It is apparent

from Fig. 10 that significant differences exist between the results for Cases 2 and 3 and that the

results for Case 1 are not too different from those for Case 3.

Next we consider the response of a cylindrical semi-circular canyon of radius a cut into

a viscoelastic half-space (0:/ p = 2.0, ea = 0.005, ep = 0.01) and subjected to a harmonic plane

SV -wave of displacement amplitude IU sv I and angle of incidence 25 o. The amplitudes of the

horizontal and vertical response on the surface of the canyon and on the adjacent surface of the

half-space are shown in Fig. 11 for the three cases discussed in the previous example and for a

dimensionless frequency waf p = 0.5. Again, the erroneous results for Case 2 are significantly

different to those for Case 3.

As a third example we consider the response of a cylindrical semi-circular valley of radius

a and properties O:I I PI = 2.0, eai = 0.02, ef3I = 0.04 embedded in a half-space of properties

0:2/ P2 = 2.0, ea2 = 0.005, e{32 = 0.01 for the case O:I = P2 and PI = P2· The half-space

is subjected to a harmonic plane SV-wave of displacement amplitude U sv. angle of incidence

88 = 25° and frequency w such that wa/ PI = 1r. The resulting amplitudes of the horizontal and

vertical displacements on the ground surface normalized by U sv are shown in Fig. 12 for the

three cases discussed previously. Again, we find significant differences between the results for

Case 2 and 3.

As a final example we consider the response of a massless rigid strip foundation of

retangular crossection and width 2b embedded to a depth h = bin a uniform viscoelastic half­

space characterized by 0:/ p = 2.0, ea = 0.005 and ep = 0.01. The foundation is subjected to the

free-field ground motion resulting from a harmonic plane SV-wave of displacement amplitude

U sv impinging on the half-space with angle of incidence 88 = 25°. The results shown in

Fig. 13 include the amplitudes of the horizontal luy 1. vertical lu z I and normalized rocking motion

I bB x I at the center of the bottom of the foundation normalized by I U sv I and shown versus the

dimensionless frequency wbjp. Results for the three cases: (a) Case 1, k = (w/P)sin88 ,

Rev > 0, (b) Case 2, k = kp sin 08 , Rev 2:: 0, and (c) Case 3, k = kp sin 08 , Im v > 0 are

shown in Fig. 13.

As in the previous examples, Case 2 corresponds to the wrong choice of v for this

particular angle of incidence. The results in Fig. 13 indicate that the motion of the foundation

in Case 3 is considerably different from the motion in Case 2 but similar to that in Case I.

CONCLUSIONS

The correct depth dependence for plane waves reflected from the free-surface of a vis­

coelastic half-space excited by a harmonic plane SV incident wave has been derived by a limiting

process from the high-frequency response of the viscoelastic half-space to a harmonic buried

line source. By allowing the source to recede to infinity, a local plane wave representation of

the displacement field close to the free surface is obtained. In particular, it has been found that

for a homogeneous plane incident SV -wave, the standard radiation condition Re v 2: 0 for the

reflected P-wave applies only if the angle of incidence 88 with respect to the vertical is such that

0 < 88 < 81 or 82 < 88 < Tr/2 where 81 =arcsin [.J~o/~.s (,8/a)] and 82 = arcsin(,B/a). If

81 < 88 < 82 the appropriate condition for v is 1m v > 0 which leads to Rev < 0. Thus, the

situation is more complex than previously envisioned. For 0 < 8 8 < 81 , Rev 2: 0 and lm v 2: 0

leading to a reflected P-wave which decreases in amplitude and increases in phase away from

the boundary. For 81 < 88 < 82 (~0 < ~.s), Rev:::; 0 and Im v > 0 corresponding to a reflected

P-wave that propagates away from the boundary but increases in amplitude away from the free

surface. For 82 < 88 < 1r /2, Rev > 0 and Im v < 0 corresponding to a reflected P-wave with

amplitude decreasing with depth but propagating locally toward the free surface.

It has been shown by means of a number of examples that the use of the "standard"

radiation condition Re v 2:: 0 in the region 81 < 8 8 < 82 where it does not apply leads to

significant errors in the calculation of the seismic response of layered soil deposits, embedded

foundations, topographical canyons and valleys embedded in a viscoelastic half-space when

subjected to homogeneous plane SV -waves.

Finally, in an Appendix it is shown that the use of an alternative set of branch cuts may

simplify the process of selecting the appropriate values for v.

ACKNOWLEDGMENT

The work described here was supported by a Grant from California Universities for

Research in Earthquake Engineering (CUREe) as part of a CUREe-Kajirna project.

REFERENCES

1. Borchert, R.D. (1973a). "Energy and Plane Waves in Linear Viscoelastic Media", J.

Geophys. Res. 78, 2442-2453.

2. Borchert, R.D. (1973b)."Rayleigh-Type Surface Wave on a Linear Viscoelastic Half­

Space", J. Acoust. Soc. Amer., 54, 1651-1653.

3. Borchert, R.D. (1977). "Reflection and Refraction of Type-IT S Waves in Elastic and

Anelastic Media," Bull. Seism. Soc. Amer., 67, 43-67.

4. Borchert, R.D. (1982). "Reflection-Refraction of General P- and Type-! S Waves in Elastic

and Anelastic Solids," Geophys. J. R. Astr. Soc., 70, 621-638.

5. Borchert, R.D. (1985). "On Anelastic Earth Structure and Seismic Waves," in Strong

Ground Motion Simulation and Earthquake Engineering Applications., R.E. Scholl and

J.L. King, Eds., Earthquake Engineering Research Institute, Publication No. 85-02, Nov.

1985 ( 27-1 to 27-18).

6. Buchen, P.W. (1971a). "Plane Waves in Linear Viscoelastic Media", Geophys. J. R. Astr.

Soc., 23, 531-542.

7. Buchen, P.W. (1971b). "Reflection, Transmission and Diffraction of SH-Waves in Linear

Viscoelastic Solids", Geophys. J. R. Astr. Soc., 25, 97-113.

8. Cooper, H.F., Jr. and E.L. Reiss (1966). "Reflection of Plane Viscoelastic Wave from

Plane Boundaries," J. Acoust. Soc. Amer., 39, 1133-1138.

9. Cooper, H.F., Jr. (1967). "Reflection and Transmission of Oblique Plane Waves at a Plane

Interface between Viscoelastic Media," J. Acoust. Soc. Amer., 42, 1064-1069.

10. Krebes, E.S. and F. Hron (1980). "Synthetic Seismograms for SH Waves in Layered

Anelastic Medium by Asymptotic Ray Theory," Bull. Seism. Soc. Amer., 70, 2005-2020.

11. Krebes, E.S. (1983). "The Viscoelastic Reflection/fransmission Problem: Two Special

Cases," Bull. Seism. Soc. Amer., 73, 1673-1683.

12. Lockett, F.J. (1962). "The Reflection and Refraction of Waves at an Interface between

Viscoelastic Media," J. Mech. Phys. Solids, 10, 53-64.

13. Richards, P.O. (1984). "On Wave Fronts and Interfaces in Anelastic Media," Bull. Seism.

Soc. Amer., 74, 2157-2165.

14. Schoenberg, M. (1971). "Transmission and Reflection of Plane Waves at an Elastic­

Viscoelastic Interface," Geophys. J. R. Astr. Soc., 25, 35-47.

15. Shaw, R.P. and P. Bugl (1969). "Transmission of Plane Waves through Layered Linear

Viscoelastic Media," J. Acoust. Soc. Amer., 46, 649-654.

APPENDIX

In here we present a simpler derivation of the appropriate values for v obtained by

assuming from the outset that lxfx,l ~ 1 and zfz, ~ 1 and by using an alternative set of

branch cuts for v and v'. We start by writing the total field associated with incident SV-waves

for a line source embedded in a viscoelastic half-space in the form

{uz(x,O,z)} = 1

2100

!._ {U(k)} e-g(k)dk Uz(x,O,z) . 47rp,kp _00 v' V(k)

(A1)

where for z < z,,

{ U( k) } _ ({ -iv' } v' z + { iv' } R -v' z + { k } R -vz) -ikz V(k) - k e k sse -iv SP e e (A2)

and

g(k)=v'z,+ikx, . (A3)

The radiation condition is satisfied by requiring that Re v ~ 0 and Re v' ~ 0 for k real. To

extend the definition of the integrands into the complex k-plane we introduce the branch cuts

for v defined by Re v = - Im v which correspond approximately to the lines

(A4a)

(A4b)

located in the second and fourth quadrants of the k = k R + i k 1 plane. The corresponding branch

cuts for v' defined by Re v' = - lm v' are given by the lines

(A5a)

(A5b)

also in the second and fourth quadrants of the k-plane. The resulting branch cuts are illustrated

in Fig. Al.

At this point we introduce the conditions that lx/xsl ~ 1 and z/z8 ~ 1. Thus, to obtain

the asymptotic expansion of the integral in Eq. (Al) for large values of w [x; + z;] 112 it is

assumed that all terms in the integrand are slowly varying with the exception of those included

in g( k ). The saddle point k0 of g( k) must satisfy

g'(ko) = k~ Z 8 + ixs = 0 Vo

where v~ is the value of v' at k = k0 • Introducing the change of variables

X 8 = SsinB8

it is found that

g(k0 ) = ikpS

and

"(k ) . s g ·o = -z kp cos2 Bs

(A5)

(A6a)

(A6b)

(A7a)

(A7b)

(A7c)

(A7d)

Assuming that the initial contour of integration can be deformed into the steepest descent path

through the saddle point k0 , we find that the asymptotic approximation to ( ux, uz) can be written

in the form

where

{ Ux} = Usv { U(ko)} Uz kp V(ko)

1 Usv = 4 k

7r J.L (3

27r (kvoo'·) ei('Tr/4-E/2)-g(ko) lg"(ko)l

(AS)

(A9)

in which e ~ ~s is the phase of i g" ( k0 ). Eq. (A8) gives the free-field ground motion, including

reflected P and S waves, for an incident SV-wave in the same form as Eq. (4a) and (4b).

To complete the description of the free-field motion it is only necessary to determine the

appropriate value for v0 = [ k5 - k~] 112• This can be accomplished by considering the position

of the saddle point and of the steepest descent path with respect to the branch cuts. The locus of

the saddle point k0 = kp sin 88 as 88 varies from 0 to 1r /2 is shown in Fig. Al by the segment

OD. For 0 < 88 < 81 =arcsin [ Jea/~13 (.8/o)] the saddle point is in the segment OB and

Rev > 0, Im v > 0. For 81 < 8 8 < 82 = arcsin (,8 / o ), the saddle point is in the segment

BC to the left of the first branch cut and Rev~ 0, Im v ~ 0. Finally, for 82 < 88 < 1r /2 the

saddle point is on the segment CD to the right of the first branch cut and Re v > 0, Im v < 0.

It only remains to show that the initial contour of integration can be deformed into the path of

steepest descent defined by lm [g(k)- g(k0 )] = 0. This is done in Figs. A2 (a, b, c) for the

cases 0 < 88 < 82, 82 < 88 < 45° and 45° < 88 < 90°.

FIGURE CAPTIONS

Figure 1. Schematic representation of a unifonn half-space, a layered half-space, a rigid embed­

ded foundation, a buried structure, a canyon, and a layered valley subjected to a nonvertically

incident plane wave.

Figure 2. Nonnalized complex k-plane showing branch points and branch cuts for v and v', and

locus OC of k = kp sin 08 (0 < Bs < 1r /2). Notice that the imaginary part has been selectively

amplified by nonnalizing it by ep. the damping ratio for S-waves.

Figure 3. Values of v in the nonnalized complex v-plane for Case 1: k = (w/fi) sin08 (solid

lines); case 2: k = kp sin 08 , Rev > o (dotted line) and Case 3: k = kp sin 08 , Im v > o (dash-dot line). The results shown are for the case fi 1 a = o.5, ea = o.o5 and eP = 0.1.

Figure 4. N onnalized amplitudes of the horizontal I u x ( 0, 0, z) / U sv I and vertical

luz(O,O,z)/Usvl displacement components at the nonnalized depths z = wzffi = 0., 0.5

and 1.0 in a unifonn viscoelastic half-space (ii/ p = 2.0, ea = 0.005, ep = 0.01) subjected to

plane SV incident waves of amplitude U sv and angle of incidence 08 • Case 1 corresponds to

an inhomogeneous incident SV-wave defined by k = (w/fi)sinB8 • Case 2 and 3 correspond to

a homogeneous incident SV-wave k = kp sin Bs with the conditions Rev~ 0 and Im v ~ 0 on

the reflected P-wave, respectively.

Figure 5. Profiles versus nonnalized depth z wz / fi of the horizontal lux ( 0, 0, z) /U sv I

and vertical luz(O, 0, z )/U svl displacement components in a unifonn half-space (a/ fi = 2.0,

ea = 0.005, ett = 0.01) subjected to incident SV-waves with amplitudes U sv and angles of

incidence 8 8 of 25, 30 and 60°. Cases 1, 2 and 3 are defined in the caption to Figure 4.

Figure 6. Schematic representation of the directS wave, the reflected S wave (SS) and of the

reflected P wave (SP).

Figure 7. Schematic representation of the path of steepest descent in the complex k-plane for:

(a) direct S-wave (0 < Bs < goo)

(b) reflected SS or SP-waves (0 < B8 < 81)

(c) reflected SS or SP-waves (81 < Bs < 82)

(d) reflected SS or sP-waves (82 < 88 <goo).

Figure 8. Loci of the saddle point k3 in the normalized complex k-plane for the reflected SF­

wave. The results shown correspond to the case ii/ p = 2.0 and ep/eOt = 2.0. Notice that the

amplitudes of the imaginary parts have been amplified by dividing by e/3· For Re (Pk/w) < 0.5

and along a line for a given value of P3 / S3 the value of B83 varies from 0 to 1r /2. For

Re (P k / w) < 0.5 the saddle points to the left of the branch cut Rev = 0 are on the Riemann

sheet Re v > 0, Im v > 0 while those to the right of the branch cut are on the sheet Re v < 0,

Imv > 0. For 0.5 < Re (Pk/w) < 1.0, the saddle points are on the Riemann sheet Rev> 0,

Imv < 0.

Figure 9. Lines of constant amplitude (solid lines) and constant phase (segmented lines) for the

reflected SP wave in a half-space characterized by ii/ p = 2.0 and ep/eOt = 2.0.

Figure 10. Amplitude of the horizontal lux I and vertical luz I response on the surface of a

viscoelastic layer of tickness H resting on a viscoelastic half-space and subjected to a harmonic

plane incident SV waves of amplitude I U sv I and angle of incidence (with respect to the vertical)

Bs = 25°.

Figure 11. Normalized amplitud~s of the horizontal and vertical response on the surface of a

semicircular canyon of radius a cut into a uniform viscoelastic half-space (ii/ p = 2.0, eOt =

0.005, e/3 = 0.01). The canyon is subjected to a plane sv incident wave of displacement

amplitude IU sv 1. angle of incidence Bs = 25° and frequency w such that wa/ p = 0.5.

Figure 12. Normalized amplitudes of the horizontal and vertical response on the surface of

a semicircular valley of radius a embedded in a uniform viscoelastic half-space and subjected

a plane SV-wave of amplitude IUsvl and angle of incidence Bs = 25°. The dimensionless

frequency has the value wa/ PI = 7!" and the properties of the media are: GI I PI = 2.0, eo I =

0.02, ef3I = 0.04, G-2/ P2 = 2.0, eo2 = 0.005, e/32 = 0.01, ad P2 = 1.0 and Pd P2 = 1.0.

Figure 13. Normalized amplitudes of the response of a massless rigid strip foundation embedded

in a viscoelastic half-space and subjected to a harmonic plane SV -wave with displacement

amplitude IUsvl and angle of incidence Bs = 25°. (h/b = 1.0, a/P = 2.0, eo = 0.005,

e/3 = o.ol).

Figure Al. Alternative branch cuts Rev= -1m v andRe v' =-1m v' in the complex k-plane

(solid lines). The standard branch cuts Rev = 0 are also shown with segmented lines. The

locus of the saddle point k0 = k13 sin Bs as Bs varies from 0 to 7r /2 corresponds to the segment

OD.

Figure A2. Integration path including the steepest descent path through the saddle point for (a)

0 < Bs < 82, (b) 82 < B8 < 45° and (c) 45° < B8 < 90°.

Figure 1

1.0 ./

./

.5 co.

UJ' -........ a --._ .0 ~

I CO. ~

8 - -.5

-1.0

-1.~.5

/ /

------Rev, v' > 0 Im v, v' > 0

.0

Re [ ~ k I ro]

~/a= 0.5

~a!~~= 0.5

1.5

Figure 2

1.0

.5

8 -.... > .0

I CO..

e --.5

.....

f.-

f.-

1-

1-

1-

-1.~.0

I I I I I I

I' A B

I

G p// 0-·-·-·-·-·-·-·-·-·--·.,JJ H

..------------------------~

/.oD E I .

Case 1 c ··----- Case2 -·-·-·-·-·· Case3

I I j 1 j_ j_

-.5 .0

Re[~v/ro]

.5

-

-

-

-

-

-

1.0

Figure 3

6 6

z=O Case 1 z=O Case 1 Case 2 Case2

4 -·-·-·- Case 3 4 -·-·-·- Case3 > > ~ ~ - -K N :::s :::s

2 2

0

6 6

z=0.5 z=O.s l\ I \

4 : :

4 I I I i

> ' I > ! \ ~ I \ ~ . ' I • - I -K N - :::s - 2 ' 2 I

I /' r• .. ' ....-·-· .......... • \ I '· /...... . ..... , i H . : ..

' :

' ' 0

6 6

z= 1.0 z= 1.0

4 4 > > ~ ~ - ,.._ -K

/" .,_ N

:::s _,· \ :::s 2 / \ 2 / \ .I \

\ \ \

00 90

es (degrees) 98 (degrees)

Figure 4

0

as= 25°

IN 1 //

.,.,../ Case 1 Case 2

-·-·-·- Case 3

2

0

es = 30°

IN 1

2~--~--~----~--~--~--~

o~~-r----r---~----~--~--~

IN 1

6

I Ux/ Usv I

0

es =25°

IN 1

Case 1 Case2

-·-·-·- Case3

2

0

es = 30°

IN 1

2~~~--~----~--~----~--~

0~---n~--~--~----~--~----~

IN 1 ·,.

'· .......... '· '· '· ......

'· ...... ..... .....

I Uz/ Usv I

..... ..... .........

Figure 5

6

t----- X 5 ----+--- X ---1

(x, z)

observer

z

_I

Figure 6

(a)

Rek

Rev'= 0

(b)

Imk

Rek

Rev'= 0

Figure 7a,b

(c)

(d)

I I I I L-

Imk

Imk

;): •• ·r.. ...... , ·. . ... .....

.......... kf3

Re v > 0 , Re v' > 0

--------· Re v < 0, Rev' > 0

· · · · · · · · · · · · · · · · · Re v > 0 , Re v' < 0

Rek

Rek

Figure 7c,d

.......... 1""""'1

a ..........

0 ~

I c::::l.

-.25

Re [ ~ ko I ro ]

~a= 0.05

~~ = 0.10

Figure 8

0.5

1.5 ~----2-. 7-5

.,.. _ .... :;::;..---2.875

2.01--3-.0-

2.5.0

/ .,.. /

/ /

/ /

/ /

I I

I I

I /

/

I I

I I I I

I I

2.5

Figure 9

4

1

10

95 = 25° Case 1 ·--·- Case2 -·-·-·-·-·· Case 3

10

Figure 10

> Cl)

6

~ 4

2

---easel -Case 2

_ _.....---------\ \

\ -·-·-·-·-·· Case 3

' .... ' .-·-·----­........... ~

x/a

---easel

3 -·-·---- Case 2 :-·-·-·-·-·· Case 3

1

3 q_3 3 x/a

Figure 11

16~--~--~--~--~--~--~

14 88 = 25° Case 1 10 ----Case 1 12 ----- Case 2 ----- Case2

-·-·-·-·-·· Case 3

-;10 /"'· ::)g I \ "'- ,I \ ::::~ 6 i .,

----·- / '· ·---... J ... , ··""-. ...........

-·-·-·-·-·· Case 3 8 >

::)6 .......

4

3 3 x/a x/a

Figure 12

-

- 0 as= 25 5 ......... ... , .. .. .. ... ... ... .. 4

2

1

o_o .5

----easel

··················· Case 2 -·-·-·-·-·· Case 3

... ... ... , ... ............

1.0

'··-.. ............ ······-.....

1.5 2.0

---

2.5

3~~--~~--~~--~~--~----~~

- 0 as= 25 ----Case!

······-············ Case 2

2 -·-·-·-·-·· Case 3

·······················-·-··-··--·-······---·-·--·--······-·--·-

1

o_o 1.5 2.5

rob/~

y

z

- 0 as= 25 ----easel

··················· Case 2

2 -·-·-·-·-·· Case 3

I .---···------·-·-·-----··-

···'* .. .. .,,.,.··

.. .. 0 ........

.0 .5 1.0 1.5 2.0 2.5

rob/~

Figure 13

Rev>O

Imv>O

Imk

I I I

Rev>O

Imv>O

I Rev<O

I Im v> 0

Imv=O

I \ Rev=lmv

Rev=O

Figure.Al

, , ,

, , , ,

, ,

, , ,

-----

. . . . . . . . . . .

I ,•' ~ .. ·

.~ -···

. . . . . . .

Figure A2

Appendix B. Dynamic Displacements and Stresses in the Vicinity of a

Cylindrical Cavity Embedded in a Half-Space (Luco/Barros).

DYNAMIC DISPLACEMENTS AND STRESSES IN THE VICINITY

OF A CYLINDRICAL CAVITY EMBEDDED IN A HALF-SPACE

J. E. Loco and F. C. P. de Barros

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California 92093-0411.

ABSTRACT

The two-dimensional response of a viscoelastic half-space containing a buried, unlined,

infinitely-long cylindrical cavity of circular cross-section subjected to harmonic plane SH, P, SV

and Rayleigh waves is obtained by an indirect boundary method based on the two-dimensional

Green's functions for a viscoelastic half-space. An extensive critical review of existing numerical

results obtained by other techniques is presented together with some new numerical results

describing the motion on the ground surface and the motion and stresses on the wall of the

cavity for P, SV, SH and Rayleigh waves.

INTRODUCTION

In this paper we study the diffraction of h~onic plane waves by an infinitely long

cylindrical cavity of circular cross-section embedded in a viscoelastic half-space. The unlined

cavity and the half-space are excited by plane SH, SV, P and Rayleigh waves impinging normal

to the axis of the cavity. This two-dimensional problem is of interest in connection with the

seismic response of unlined tunnels and with the possible modification of the surface ground

motion during earthquakes as a result of the presence of underground cavities. The problem is

also of interest in the development of nondestructive testing techniques.

Although the two-dimensional problem of diffraction of elastic waves by cavities in an

elastic half-space has been considered by several authors there is still the need for a critical review

of the existing numerical results. In particular, while several methods of solution have been

proposed, relatively few detailed across-method comparisons have been made. Also, additional

numerical results for the total motion on the surface of the half-space and for the displacement

and stress fields on the wall of the cavity are needed.

Previous studies of the diffraction of elastic waves by cavities consider two-dimensional

anti-plane or plane strain models of a cavity embedded in a uniform elastic half-space. Lee

(1977), Datta and El-Akily (1978a), Datta and Shah (1982) and Shah, Wong and Datta (1982)

have studied the two-dimensional diffraction of plane SH-waves by a cylindrical cavity buried in a

uniform elastic half-space. The two-dimensional cases of P- and SV -waves have been considered

by Datta and El-Akily (1978a) and Wong, Shah and Datta (1985). The two-dimensional response

to Rayleigh waves has been studied by Datta and El-Akily (1978b), Wong, Shah and Datta

(1985) and Kontoni, Beskos and Manolis (1987). The problem of diffraction of plane waves by

a cylindrical cavity buried in an elastic half-space is _closely related to the problem of determining

the seismic response of a canyon cut into a half-space for which extensive references can be

found in articles by Sanchez-Sesma (1987), Aki (1988) and Luco, Wong and Barros (1990).

A variety of methods of solution have been used to solve the cavity problem. Lee

(1977) solved numerically a truncated infinite system of equations for the infinite number of

coefficients in a series expansion of the scattered field. The method of matched asymptotic

expansions has been used by Datta and El-Akily (1978a, b) and Datta and Shah (1982). Datta

and Shah (1982), Shah, Wong and Datta (1982) and Wong, . Shah and Datta (1985) have also

used a hybrid approach in which an interior region surrounding the cavity is represented by finite

elements while a wave-function expansion or a representation in terms of Green's functions is

used to account for the exterior region. Finally, boundary element methods based on the use of

two-dimensional full-space Green's functions have been used by Kontoni et al (1987).

In the present study, an indirect boundary method based on the two-dimensional Green's

functions for a viscoelastic half-space (Barros and Luco, 1992) is used to obtain the harmonic

motion on the surface of the half-space and the motion and stresses on the wall of the cavity.

The two-dimensional Green's functions used here are obtained as a limiting case of moving

Green's functions for a layered half-space which have been throughly tested (Barros and Luco,

1992). The indirect boundary formulation employed here is similar to that used by the authors

to calculate the response of canyons in a half-space (Ltico, Wong and Barros, 1990). In the case

of canyons, the technique has been carefully tested against other solutions.

The indirect boundary method formulation used to solve the problem is presented next

followed by a number of comparisons with previous two-dimensional solutions for a uniform

elastic half-space. Additional numerical results describing the response of cylindrical cavities

embedded in a viscoelastic half.:·space and subjected to P and SV and Rayleigh waves are then

presented and discussed.

FORMULATION AND METHOD OF SOLUTION

The geometry of the model is illustrated in Fig. Ia. The infinitely-long unlined cavity of

circular cross-section of radius a is parallel to the free surface of the half-space and its centerline

is located at a depth H below the free surface. The soil is represented by a viscoelastic half­

space (z > 0) characterized, for harmonic vibrations, by complex P- and S-wave velocities

a= a[l + 2isgn(w)eo]ll2 and {3 = ,8[1 + 2isgn(w)ep]ll2, and by the density p. The terms

a and ,8 represent (approximately) the real parts of the P- and S-wave velocities, and eo and

ep represent the small hysteretic damping ratios for P- and S-waves, respectively. To avoid

some problems related to the choice of appropriate radiation conditions for a viscoelastic half­

space (Luco and Barros, 1992), it will be assumed that ~o = ~P = e. The seismic excitation

is represented by homogeneous plane P, SV or SH-waves, such that the normal to the wave

front fonns an angle Bv with the vertical axis (Bv = 0 for vertical incidence). We also consider

excitation in the fonn of a surface Rayleigh wave propagating in the direction y > 0.

In what follows, the excitation and the response will have harmonic time dependence of

the type eiwt where w is the frequency. For simplicity, the factor eiwt will be dropped from all

expressions.

Free-Field Ground Motion

As a first step in the formulation, it is necessary to determine the ground motion and the

stress components for free-field conditions, i.e., in the absence of the cavity. In the case of SH,

P and SV-waves, the total free-field ground motion { Uff(Y, z)} can be written in the form

{ Uff} = { Uinc} + { Ureft } (1)

where { Uinc} and { Ureft} represent the incident and reflected displacement fields in the absence of

the cavity. The total free-field ground motion satisties the equations of motion and the conditions

of vanishing tractions on the ground surface (z = 0) while the reflected displacement field { ureo}

satisfies the equations of motion and the radiation conditions at infinity.

Since a major source of confusion in comparing results by different authors results from

differences in the form of the incident or total free-field ground motions we list in detail the

expressions used here.

SH-Wave Excitation. The total free-field ground displacement for a homogeneous incident plane

SH-wave with angle of incidence Bv and amplitude Us is given by

{:i}=Us({~} e"''+{~} e-•'•) e-"• (2)

where k = k13 sin Bv. v' = ik13 cos Bv and k13 = w I (3. The first term on the right-hand-side of

Eq. (2) corresponds to the incident displacement field while the second term corresponds to the

reflected field. The term Us represents the amplitude of the incident displacement (at the origen)

on the ground surface, i.e. u~c(O,O) =Us. The total free-field ground motion at the origen on

the ground surface is given by

{ u~(O, 0)} { 2} u~(O, 0) =Us 0 u~(O,O) 0

(3)

P-Wave Excitation. The total free-field ground displacement for a homogeneous incident plane

P-wave of amplitude Up and angle of incidence Bv is given by

(4)

wherek=k01 sinBv, v=ik01 cosBv, k01 =wla, v'=(k2 -k~)1 12 , Rev'~O, ~f3=wl/3

and

in which

Rpp = - [(2k2 - k~)2 + 4k2 vv'] I l:l.(k)

Rps = -4i kv( 2k2 - k~) I fl.( k)

(5a)

(5b)

(6)

The first, second and third terms on the right-hand-side of Eq. (4) represent the incident P­

wave, the reflected P-wave and the reflected SV-wave, respectively. The term Up represents the

amplitude of the incident displacement vector Up(O, sin Bv,- cos Bv) at the origen on the ground

surface. The total free-field ground motion at the origen on the ground surface is given by

{ u~(O, 0) } { 0 } u~(O, 0) =Up -4kk~vv' /kaA(k) u~(O, 0) -2ik~v(2k2 - k~)/kaA(k)

(7)

SV-Wave Excitation. The total free-field ground displacement for a plane homogeneous incident

SV -wave of amplitude Us and angle of incidence Bv is given by

(8)

where k = k/3 sin Bv' v' = ik/3 cos Bv' k/3 = w I /3' v = ( k2 - k!)112

' Rev 2:: 0 (for ea = ef3).

ka = wjo: and

Rsp = 4ikv'(2k2- k~)/A(k)

Rss =- [(2k2- k~)2 + 4k2 vv'] /A(k)

(9a)

(9b)

in which A( k) is given by Eq. (6). The first, second and third terms on the right-hand-side of

Eq. (8) correspond to the incident SV-wave, the reflected P-wave and the reflected SV-wave.

The term Us represents the amplitude of the incident displacement vector Us ( 0, cos Bv, sin Bv)

at the origen on the ground surface. It should be noted that if ea =f. e/3 the condition Rev > 0

should be replaced by a different condition that depends on 8v (Luco and Barros, 1992). The

total free-field ground motion (at the origen) on the ground surface is given by

{ u~(O,O)} { 0 }. u~(O, 0) =Us 2ik13 v'(2k2 - k~)/ A(k) u~(O,O) ' -4kkt3vv'/A(k)

(10)

Rayleigh-Wave Excitation. The total free-field ground motion for a surface Rayleigh wave in a

uniform half-space can be written in the form

where k = w/CR, v = (k2 - k!)112 , v' = (k2 - k~) 1 12 , k0 = wfo, kp = w/ /3 in which

k is the root of the Rayleigh characteristic equation

(12)

The factor URH corresponds to the amplitude of the horizontal component of the free-field ground

motion (at the origen) on the ground surface. In this case, the total free-field ground motion on

the ground surface is given by

u~(O, 0) = URH 1 { u~(O,O)} { 0 }

u~(O,O) i(2k2- k~)/2kv'

(13)

Numerical values for CR./ /3 and u~(O, 0)/URH for the case ~a = ~fJ are listed in Table 1.

Free-Field Stresses and Tractions. The free-field stress field [o-ff] can be easily calculated by use

of the stress-displacement relations

(14)

where A = (o2 - 2/32 )p and J.L = (Pp are the Lame constants. The free-field traction vector

" { itr( x)} on the surface that will coincide with the cavity is given by

(15)

in which { v} = ( 0, vy, v z) T is the unit normal at a point x0 on the boundary of the cavity

(pointing into the cavity) or on the free boundary of the half-space (pointing out).

Boundary-Value Problem

In the presence of the cavity, the total displacement vector { u ( x)} and the total traction II

vector { t( x)} on the boundary of the cavity and on the free surface are written in the form

{ u ( x)} = { utr( x)} + {us ( x)} II II II

{ t( x)} = { ttr( x)} + { ts( x)}

II

(16)

(17)

where {us} and {ts(x)} are the scattered displacement and traction vectors, respectively. The

total and scattered fields satisfy the condition of vanishing tractions on the free surface (z = 0).

The scattered field must also satisfy the radiation condition at infinity. The condition of vanishing

total traction on the boundary So of the cavity leads to

II II

{ts(x)} = -{ttr(x)} (18)

To solve the boundary-value problem, the scattered field is represented as resulting from

the action of a distribution of concentrated line loads. These line loads act in the half-space

(without the cavity) on the surface S1 (Fig. lb) located within the region to be occupied by the

cavity. The SCattered displacement field {Us ( x)} iS, then, written in the form

{ Us(xo)} = r [G(xo,tio)] {F(!io)} dlt (!io) ' }Ll

(19)

in which [G(x0 , !io)] is the 3 x 3 matrix of two-dimensional half-space Green's functions and L1

is the intersection of the surface S 1 with the plane x = 0. The first, second and third columns of

the matrix G correspond to the displacement vector at Xo = (0, Yo, zo) for a unit line load passing

through the point !io = (0, y1, z1 ) on L1 and acting in the x, y and z-directions, respectively.

The 3 x 1 vector { F} represents the unknown amplitudes of the line loads. The corresponding II

traction vector { ts ( x)} for the scattered field on the surface S can be written in the form

(20)

where 0 O"~x z

(7 XX

X 0 0 O"xy

[Hex., Yol] = [ ~ lly liz 0 0

~] u;z 0 0 0 0 lly liz (21)

0 uty z 0 0 0 lly O"yy

0 O"~z z 0" yz

0 u:z z 0" zz

In Eq. (21) (lly(xo), llz(xo)) are the direction cosines of the normal to the surface S of the

cavity, and u;x(x0 , fio), u~x(x0 , fio), ... , are the stresses at X'o = (0, Yo, zo) induced by the unit

line loads acting in the x, y and z-directions.

Substitution from Eq. (20) into the boundary condition given by Eq. (18) leads to

(22)

which corresponds to an integral equation for { F( fio)}.

Numerical Solution

To obtain a numerical solution to Eq. (22) the distribution of line sources {F(fio)} over

S 1 is replaced by N s line sources of amplitude { Fsj} (j = 1, N 5 ). The corresponding scattered

displacement and traction vectors are given by

Ns

{us(X'o)} = L[G(xo,thj)]{Fsj} (23) j=l

Ns

{i:(xo)} = L [mxo, ihi)] {Fsj} (24) j=l

where lhi = ( 0, Ysj, Zsj) are a set of N 5 points on Lt. Imposing the boundary condition (22) at

a set of No points x0 ; = (0, y0 ;, z0;), (i = 1, No) on Lo, leads to

f [He Xoi, lhj)] { Fsj} = - { trr( Xoi)} X'o; E Lo ( i = 1, No) (25) j=l

which corresponds to a set of linear algebric equations for { Fsj }. Eq. (25) can be written in the

form

[.H]{l"s} = -{Trr} (26)

where the 3 x 3 ( i, j) block of the ( 3N0 x 3N s) matrix corresponds to [ H( x0 , Ysi) J , { .F\} =

(. - T - T )T - ( 11

... T 11

... T ) T {Fsd , {Fs2} , . . . and {Trr} = {trr(XoJ)} , {trr(Xo2)} , . . . .

The 3No x 3N s system of equations given by Eq. (26) can be solved in the least-square

sense leading to

(27)

in which the asterik denotes the complex conjugate. An alternative, reciprocity-based approach

is to pre-multiply Eq. (26) by (G]T and then solve leading to

(28)

where (G] has for blocks the matrices [G(xoi, Ysi)J. On the basis of reciprocity theorems it can

be shown that as the number of No of observation points increases the matrix [G]T[.tf] tends to

become symmetric (Apsel and Luco, 1987).

Once the forces { 1\} have been obtained from Eq. (27) or (28), then, the scattered

displacement and stress fields, and, also, the total displacement and stress fields can be easily

obtained. It should be noted that the procedure described above is also valid for a cavity of

arbitrary cross-section.

Transformation of Stresses

For the purpose of describing the stresses on the wall of the cavity it is convenient to refer

to the cylindrical coordinates (r,B,x) shown in Fig. lc. The stress components in cylindrical

coordinates are related to the stress components in cartesian coordinates by the relations

. • 2 Ll • 2Ll 2 Ll U99=UyySin u+UyzSlD u+UzzCOS 17 (29a)

Urr = Uyy cos2 8- Uyz sin 28 + Uzz sin2 8

CJ r8 = ~ ( CJ zz - CJ yy) sin 28 - CJ yz cos 28

Uxx=Uxx

(29b)

(29c)

(29d)

(29e)

(29/)

For SH-excitation the problem corresponds to a case of anti-plane shear and u 99 = u rr =

Ur8 = uxx = 0. For P, SV and Rayleigh-wave excitation the problem corresponds to a case of

plane-Strain and Uxz = Urx = CJ9x = Q.

CONVERGENCE OF THE NUMERICAL APPROACH

The numerical results obtained by the procedure described in the previous section depend

in principle on the location and number of source points (Ns) and on the number of observation

points (N0 ) used. The first step is to test the convergence of this approach. For this purpose we

consider a cavity of radius a buried to a depth H = 1.5a in a uniform half-space characterized

by 0 = 2i3 (v = 1/3), and eo = e/J = 0.001. The half-space is subjected to vertically incident

(Bv = 0°) P- and SV-waves such that the incident motion on the ground surface has amplitudes

Up and Us, respectively. The frequency of the excitation is such that 7J = waj1ri3 = 0.5.

Numerical results for the total displacement at a few points on the ground surface (z = 0)

and for the total displacement and for some selected stress components at a few points on the

wall of the cavity (r = a) are presented in Table 2 for different numbers of sources and observers

(N5 , N0 ). In all cases, the sources are equally spaced on a circle of radius a'= a- t(27ra/No)

(No > 27r')') where t = 3. Thus, as the number of observation points increases, the sources

move closer to the actual boundary r = a.

The results listed in Table 2 have been calculated by both the reciprocity-based approach

defined by Eq. (28) and by the least squares approach defined by Eq. (27). The numerical

results presented for source/observer combinations (N 5 , N 0 ) of (20, 40), ( 40, 80) and (80,

160) show that the results are very stable as the number of source and observation points

increases. The displacements on the free-surface (z = 0) are the least sensitive to the number of

sources/observers while the stresses on the cavity are the most sensitive. In general, 40 source

points and 80 observation points are sufficient for most applications. For N s > 40 and No :=:: 80,

the differences between the reciprocity-based and leasi squares results are extremely small. For

Ns = 20, No = 40 the reciprocity-based approach seems to give slightly more accurate results.

COMPARISONS WITH PREVIOUS RESULTS

The objective of this section is to assess the validity and accuracy of the present results

and of those presented by earlier researchers by means of a detailed and critical evaluation of

comparisons of results obtained by different methods. For this purpose we use the numerical

results for SH-waves obtained by Lee (1977) and Datta and Shah (1982), the results for Rayleigh

waves presented by Datta and El-Akily (1978b), Wong, Shah and Datta (1985) and Kontoni,

Beskos and Manoli (1987), and the results for P- and SV-waves obtained by Wong, Shah

and Datta (1985). In judging the comparisons it must be kept in mind that the earlier results

correspond to a purely elastic medium while the present results include a small amount of

attenuation ea = efJ = 0.001. All of the present results were calculated by the reciprocity-based

approach (Eq. 28).

SH-Wave Excitation

Lee (1977) has presented extensive numerical results for a cylindrical cavity buried in

a uniform half-space and subjected to SH-waves with various angles of incidence. The results

presented include the total displacement on the ground surface for two embedment ratios HI a =

1.5 and 5.0, four values of the dimensionless frequency 7J = wal1r~ = 0.5, 1.0, 1.5 and 2.0 and

four angles of incidence Bv = 0°, 30°, 60° and goo. Lee (1977) also presented the amplitudes

of motion at three points on the surface (y I a = -2, 0, 2) and at four points on the cavity wall

((} = 0°, goo, 180° and 270°) as a function of the dimensionless frequency 7J for HI a = 1.5

and for Bv = 0°, 30°, 60° and goo. The solution was obtained by truncating an infinite system

of linear equations on the infinite number of coefficients on a wave-function expansion of the

solution.

A comparison of our numerical results with those obtained by Lee (1977) for a vertically

incident SH-wave (Bv = 0°) with particle motion along the axis of the cavity is shown in Fig. 2.

Results for the amplitude of the total motion on soil surface I U xI = I u xI Us I normalized by

the amplitude of the incident motion also on the ground surface are shown in Figs. 2a and

2b versus the normalized horizontal coordinate y /a for values of the dimensionless frequency

7J = wa/1rfi = 0.5 and 1.0 and for H fa = 1.5 and 5.0. The agreement between the two set of

results is good.

The amplitude of the total displacement along the cavity's wall lUx!= lux/Us! normal­

ized by the amplitude of the incident motion on the ground surface is shown versus (} in Figs. 2c

and 2d. The open circles in Figs. 2c and 2d at (} = 0°, (} = goo and (} = 180° correspond to the

amplitudes obtained by Lee (1977) for H/a = 1.5. The results obtained by Lee for(}= -goo

or (} = 270° are close to the present results but could not be read with sufficient accuracy from

the published figures. The amplitude of the normalized total shear stress !&ex!= luex/wpfiUsl

along the wall of the cavity is shown versus (} in Figs. 2e and 2f.

In 1982, Datta and Shah (1982) presented numerical results obtained by a hybrid approach

combining a finite element solution for an interior region with a wave-function series expansion

for the exterior region. The results include the amplitudes of the scattered displacement field

on the ground surface (7] = 1/Tr, H/a = 1.5, Bv = 0°, 30°, 45°, 60° and goo; Hfa = 1.5,

Bv = 45°, 7J = 1/2Tr and 1/Tr) and the total displacement field on the cavity's wall (HI a = 1.83,

7J = 1/ 1r, Bv = goo). A comparison of the present results with those of Datta and Shah (1982)

for the scattered displacement field on the ground surface IU~s)l = !u~s) /Us! normalized by

the amplitude Us of the incident SH motion on the ground surface is shown in Fig. 3a versus

yfa for 7J = 1/Tr. The results shown correspond to H/a = 1.5, Bv = 0° and Bv = goo, and

HI a = 1.83, Bv = goo. Both sets of results follow the same trends but differences of the order

of 10 percent can be observed just above the cavity.

A comparison for the normalized total displacement lUx! = lux/Us! along the cavity

wall is shown versus (} in Fig. 3b for H /a = 1.83, 77 = 1/ 1r and Bv = goo (horizontal incidence

from the left). It is apparent from Fig. 3b that the two sets of results are similar. Fig. 3b also

shows the normalized free-field displacement along the cavity wall. Comparison of the total and

free-field displacements show amplification on the illuminated side and deamplification on the

shadow side.

Rayleigh-Wave Excitation

Datta and El-Akily (1978b) have presented some numerical results for the scattered field

on the free surface of the half-space for the case of a cylindrical cavity subjected to Rayleigh

waves. The results were obtained by the method of matched asymptotic expansions and were

calculated for a Poisson's ratio of 114 and for values of wHIP= 0.5 and 1.0. The authors do

not list the value of HI a used in the calculations presumably because their normalized values

for the scattered displacement may be independent of HI a. In other words, the dependence of

the scattered displacement field on HI a may be accounted for by the normalization factors used

by Datta and El-Akily.

The real and imaginary parts ofthe normalized horizontal U~s) = u~s) IURH and vertical

U~s) = u~s) IURH components of the scattered displacement field on the free surface are shown

(solid lines) in Fig. 4 versus yla for the case v = 114, a= v'3 fi, ~a = ~fj = 0.001, wH I fi = 0.5

and HI a = 5. The displacements are normalized by the amplitude U RH of the horizontal

component of the free-field ground motion on the ground surface. Also shown in Fig. 4 are the

results obtained by Datta and El-Akily (1978b). These results were renormalized by the procedure

described in Appendix I to account for the use of a different normalization of the original results.

The renormalized results of Datta and El-Akily (1978b) for the real and imaginary parts of the

scattered horizontal displacement (segmented lines, Figs. 4a, 4b) are a factor of 2 smaller than

our results while those for the real and imaginary parts of the scattered vertical displacement

(segmented lines, Figs. 4c, 4d) are a factor 1.69 lower than the present results. When the

horizontal and vertical results of Datta and El-Akily (1978b) are multiplied by factors of 2 and

1.69 (open circles), respectively, then very close agreement is obtained between the two sets of

results. Additional comparisons for HI a= 2.5 and HI a= 10 with wHIP= 0.5lead to similar

conclusions.

It is interesting to point out that comparisons made by Wong, Shah and Datta (1985)

between the results obtained by the matched asymptotic expansion (MAE) of Datta and El-Akily

(1978b) for the case of wH I fi = 1.56 and HI a = 2.6 and those obtained by a hybrid method

show that the MAE results for the scattered vertical displacement on the free surface exceeded

the other results by a factor of two.

A second comparison for the response to Rayleigh waves results from the work of Kon­

toni, Beskos and Manolis (1978). These authors applied a Boundary Element Method based on

the use of the two-dimensional Green's function for the full plane. Consequentely, they had to

discretize the circular boundary of the cavity and also part of the free boundary of the half-space.

Kontoni et al (1987) have presented numerical results for the scattered displacement field on the

free-surface of the half-space and on the cavity wall for the same case ii = 114, wHIP= 0.5

and HI a = 5 considered by Datta and El-Akily (1978). They also provide some numerical

results for the scattered displacement and tangential stress at two points on the cavity wall as a

function of the dimensionless frequency wa I fi for ii = 1 I 4 and HI a = 2.

For the case ii = 1 I 4, wH I fi = 0.5 and HI a = 5, Kontoni et al (1987) used the

same normalization of the scattered displacement field used by Datta and El-Akily (1978b) but

apparentely used a different form for the free-field ground motion and consequently ended up

with normalized amplitudes ofthe order of 106 while the normalized results of Datta and El-Akily

have amplitudes < 6. They state that an exact comparison of their results with those of Datta

and El-Akily was not possible in view of the insufficient data provided by the earlier authors.

Unfortunately, Kontori et al (1987) fail, in turn, to list the expressions that they used for the

free-field Rayleigh wave and thus make it impossible for an exact comparison of our results with

theirs. To get around this problem we assumed that their free-field motion corresponded to that

listed by Datta and El-Akily (1978b) multiplied by the factor 1 l[k(2k2 - k~- 2vv')] in which 1

is an unknown contant and k = wiCR· We then proceeded to find the constant of proportionality

1 that would lead to a resonable match with our results after the differences in normalization

where accounted for. The value of 1 that gave the best fit for the case ii = 114, wH I fi = 0.5,

HI a = 5 was 1 = 1.0877 which happens to coincide with the value of fi I CR. for v = 1 I 4. A

second problem with the results of Kontoni et al (1987) is that they actually correspond to a

time dependence e-iwt contrary to the statement by the authors that they considered the time

dependence e iwt.

Comparisons of the normalized horizontal U~s) = u~s) IURH and vertical U~s) =

u~s) IURH components of the scattered displacement fields on the free-surface of the half-space

and on the cavity wall are presented in Fig. S. The results are normalized by the amplitude U RH

of the horizontal component of the free-field ground motion on the ground surface. The factors

required to renormalize the results presented by Kontoni et al are presented in Appendix I and

are based on 1 = 1.0877. Inspection of Figs. Sa, Sb, Sc and Sd indicates very good agreement

between the two sets of results for the scattered displacements on the free surface for points

immediately above the cavity. As the horizontal distance to the cavity increases the agreement

deteriorates as the results of Kontoni et al become less accurate due to the coarser discretization

of the free surface away from the cavity. The comparisons in Figs. Se and Sf for the scattered

displacement on the cavity wall show excellent agreement between the two sets of results when

1 is set equal to 1.0877.

As a result of the comparisons with the results of Datta and El-Akily (1978b) and Kontoni

et al (1987) we can conclude: (i) The earlier results of Datta and El-Akily match exactly the

shape of the present results for the scattered field on the surface of the half-space but appear to

contain factors affecting differently the amplitudes of the horizontal and vertical components of

motion. The scattered horizontal and vertical components obtained by Datta and El-Akily appear

to be too small by factors 2 and 1.69 [note that 2(CR.I P? = 1.6906], respectively. (ii) The results

of Kontoni et al (1987) suffer from an unknown normalization factor and from a mistatement

about the time dependence. The shape of the results of Kontoni et al (1987) for v = 114,

HI a = 5 and wHIp = 0.5 is fairly accurate for points on the cavity or close to the cavity but

less accurate at increasing distances from the cavity. Most of the differences between the present

results and those of Kontoni et al (1987) for H/a = 5, wHIP= 0.5 can be accounted for by a

common normalization factor of 1 = 1.0877.

From the physical point of view two results are apparent. First, for the case v = 114,

wHIP = 0.5 and HI a = 5 the amplitude of the scattered displacement field is extremely

small compared with the free-field amplitude. On the cavity wall the amplitude of the scattered

displacement is less than 10 percent of the amplitude of the horizontal component of the free­

field ground motion on the ground surface. On the ground surface, the amplitude of the scattered

displacement is less than 5 percent of the horizontal free-field displacement on the ground surface.

Second, the amplitudes of the scattered displacements on the ground surface are almost exactly

proportional to (aiH? for a fixed y, Hand wHIP and for HI a> 2.5 and (wH/P) small. This

is implicit in the matched asymptotic expansion of Datta and El-Akily (1978b) and was verified

by numerical experimentation for the case wHIP= 0.5, v = 1/4 by considering H/a = 2.5,

5.0 and 10.

Kontoni et al (1987) have also presented some numerical results for the scattered dis­

placements and tangential stresses on the cavity wall for the case Hla = 2 and v = 114 for

different values of the dimensionless frequency waf p in the range from 0.1 to 2.0. These results

are normalized in a different way than those described previously for HI a = 5 and v = 1/4.

Since the expression for the free-field motion used by Kontoni et al (1987) is unknown we as­

sumed again that their free-field ground motion corresponded to that used by Datta and El-Akily

(1978b) multiplied by the factor 1 l[k(2k2 - k~- 2vv')] in which 1 is an unkwown constant to

be determined by the comparisons.

Fig. 6a shows the normalized amplitudes of the scattered horizontal and vertical displace­

ments on the cavity wall u~s) /URR and u~s) IURH for Hfa = 2, v = 1/4 and wa/ P = 0.5238 as

calculated by the present approach. Fig. 6b shows the real and imaginary parts of the normalized

scattered tangential stress u~~) fwpPU RH on the cavity wall for the same case. The circles in

Figs. 6a and 6b at 8 = 90° and 8 = 180° correspond to the results obtained by Kontoni et al

(1987) renonnalized as indicated in Appendix I for 1 = 1. The agreement is very good when

1 is set equal to one. It should be noted that different values of the constant 1 are required to

match the results in Fig. 5 for Hla = 5 and waiP = 0.1 (I= 1.0877) and those in Fig. 6 for

HI a = 2 and wal P = 0.5238 (/ = 1.0).

As a final comparison for Rayleigh wave excitation we consider the results obtained by

Wong, Shah and Datta (1985) by use of a hybrid method in which a finite element representation

of a bounded scattering region is matched with a multipolar representation of the scattered field

in the exterior region. The results include the amplitudes of the total and scattered vertical

displacements on the free-surface of the half-space for v = 0.3456 (a = 2.0587 p), HI a = 1.53

and for different values of wal p (1.544, 3.109, 4.632, 6.176). The authors indicate that the

numerical results presented are nonnalized by the amplitude of the incident-field displacement

vector on the ground surface. In the case of Rayleigh waves the meaning of this nonnalization

factor is not clear. In the first place, incident field in the case of Rayleigh waves must be

interpreted as the total free-field. Secondly, the free-field displacement vector on the ground

surface has the complex fonn urr = U RH ( e y + iKe z) where U RH is the amplitude of the

horizontal component in the free-field and "' = 1.5814 for v = 0.3456.

Fig. 7a and 7b show a comparison of the ratios lu~s)(y,O)IURHI and luz(y,O)IURHI

as calculated by the present approach for v = 0.3456, HI a = 1.53 and wa I P = 1.544 with

the results presented by Wong et al (1985) multiplied by appropriate factors so that the peak

amplitudes at y = 0 would match. The factors for the scattered and total vertical components

are 2.43 and 2.15, respectively. It appears that the two sets of scattered vertical displacements

on the ground surface (Fig. 7a) are similar except for the factor of 2.43. The shapes for the total

vertical displacements are quite different particularly for y I a < -1 (Fig. 7b ).

P- and SV-Wave Excitation

Wong, Shah and Datta (1985) have presented a set of numerical results for the amplitudes

of the scattered and total vertical displacement on the free surface above a cavity subjected to

vertically incident plane P- and SV-waves for the case ii = 0.3456, Hla = 1.53 and for

wal fi = 1.544, 3.109, 4.632 and 6.176. Additional results for the amplitudes of the scattered

vertical displacements for the case Bv = 0°, wal fi = 5.66, HI a = 1.53 and 2.5 and some results

for incidence at 45 o are also presented.

For the purpose of comparing the present results with those presented by Wong et al

(1985) we consider the case of a circular cavity embedded to a depth HI a = 1.53 in a medium

characterized by ii = 0.3456 (a = 2.0587 'fi) and subjected to vertically incident P- and SV­

waves with a dimensionless frequency wal fi = 1.544 (7] = 0.4915). The results of Wong et al

(1985) are for a purely elastic half-space while our results include a small amount of dissipation

( ea = e13 = 0.001). Comparisons for the amplitudes lu~s)(y,O)I of the scattered vertical

displacement on the free surface of the half-space normalized by the amplitudes of the incident

displacement field Up or Us also on the ground surface are shown in Fig. 8a and 8b for SV­

and P-excitation, respectively. Significant differences exist between the present results and those

of Wong et al (1985) which supposedly are also normalized by the amplitude of the incident

displacement vector on the ground surface. If the results of Wong et al (1985) are amplified by

a factor of 3 then a reasonable match is obtained as shown in Fig. 8.

Figs. 9a and 9b show a comparison of the normalized amplitudes for the total horizontal

luy(y, O)IUPI and vertical luz(Y, O)IUPI displacements on the ground surface (z = 0) for a

vertically incident P-wave for the same case ii = 0.3456, HI a -:- 1.53 and wal fi = 1.544

considered by Wong et al (1985). In addition to the results obtained by the present approach

Figs. 9a and 9b show results obtained by Motosaka(1992) by a direct boundary element method

(open circles) and by Dravinski (1992) by an indirect boundary method (dash-dot line). It is

apparent that the present results agree very closely with those obtained by Motosaka (1992) and

are similar to those obtained by Dravinski. Fig. 9b also shows the results presented by Wong

et al (1985) (segmented line) which differ in amplitude and shape from the results obtained

by the other authors. The comparisons of the present results with those of Motosaka (1992)

and Dravinski (1992) reinforce the validity of our results and suggest that the numerical results

presented by Wong et al (1985) are in error.

SOME NEW NUMERICAL RESULTS

The comparisons presented in the previous section suggest that the numerical results

available for SH-wave excitation [Lee (1987), Datta and Shah (1982)] are reasonably accurate

while those for P and SV-waves [Wong et al (1985)] and for Rayleigh-waves [Datta and El-Akily

(1978b), Wong et al (1985) and Kontoni et al (1987)] include unspecified normalization factors

and other problems.

In Figs. 10, 11 and 12 we present some new numerical results for a cylindrical cavity of

radius a embedded to a depth H in a uniform viscoelastic half-space characterized by a= 2P (ii = 113) and ~a = ~/3 = 0.001 and subjected to SV-, P- and Rayleigh-waves. Figs. lOa and lOb

show the amplitudes of total horizontal and vertical displacements on the ground surface (z = 0)

normalized by the amplitude Us of the incident ground motion also on the ground surface for

a vertically incident SV-wave. Results for Hla = 1.5 (solid lines) and 5.0 (segmented lines)

are shown for ry = wal1rP = 0.5. For a vertically incident SV-wave, the presence of a shallow

cavity (HI a = 1.5) reduces the horizontal component of motion on the ground surface above

the cavity but increases significantly the vertical component. For a deeply embedded cavity

(HI a = 5.0), the effect of the cavity on the surface ground motion is small. The corresponding

normalized amplitudes of the total displacements on the cavity wall (r = a) are presented versus

()in Figs. lOc and lOd. Finally, the normalized amplitudes lueelwpPU s I and lu xxlwpPU s I of

the total tangential and axial stresses on the cavity wall are shown versus () in Figs. lOe and

lOd, respectively. These results verify that D'xx = iiuee (ii = 113) on the cavity wall.

Numerical results for a vertically incident P-wave normalized by the amplitude Up of

the incident displacement on the ground surface are shown in Fig. 11. These results show a

screening effect on the vertical component of the motion on the ground surface and an increase

in the horizontal component. Finally, results for a Rayleigh wave moving towards y > 0 and

normalized by the amplitude URH of the horizontal component of the free-field ground motion

on the ground surface are presented in Fig. 12. The results in Figs. 12a and 12b show that the

presence of a cavity at a shallow depth (HI a = 1.5) can have a significant screening effect on

the ground motion for y I a > 1.

CONCLUSIONS

An indirect boundary method has been used to obtain the two-dimensional response of

a cylindrical cavity embedded in a uniform viscoelastic half-space and subjected to SH, P, SV

and Rayleigh waves.' A detailed critical review of existing numerical results shows that previous

results for SH-wave excitation are reasonably accurate while those for P, SV and Rayleigh waves

suffer from unknown normalization factors and other problems. Some new, and hopefully more

reliable, numerical results for the total displacement field on the surface of the half-space and

for the total displacement and stress fields on the wall of the cavity are presented for vertically

incident P- and SV-waves and for Rayleigh waves.

ACKNOWLEDGMENTS

--

The work conducted here was supported by a Grant from California Universities for

Research in Earthquake Engineering (CUREe) as part of a CUREe-Kajima Project. The second

author wiskes also to thank the partial support from the CNPq of the Secretariat for Science

and Technology of Brazil. The authors are most grateful to Prof. M. Dravinski and to Dr. M.

Motosaka for provinding some of their unpublished numerical results.

REFERENCES

1. Aki, K. (1988). Local Site Effects on Strong Ground Motion, Earthquake Engineering

and Soil Dynamics II- Recent Advances in Ground Motion Evaluation, (J. Lawrence Von

Thun, Ed.), ASCE Geotechnical Special Publication No.2, New York, N.Y. pp. 103-155.

2. Apsel, R. J. and J. E. Luco (1987). Impedance Functions for Foundations Embedded

in a Layered Medium: An Integral Equation Approach, Earthquake Engineering and

Structural Dynamics, ,li, 213-231.

3. Barros, F. C. P. de and J. E. Luco (1992). Moving Green's Functions for a Layered Vis­

coelastic Half-Space, Report, Dept. of Appl. Mech. & Engng. Sci., University of California,

San Diego, La Jolla, California, 210 pp.

4. Datta, S. K. and N. El-Akily (1978a). Diffraction of Elastic Waves in a Half-Space. I.

Integral Representation and Matched Asymptotic Expansions, Modern Problems in Elastic

Wave Propagation, J. Miklowitz and D. Achenbach Eds., Wiley-lnterscience Publication,

John Wiley & Sons, New York, 197-218.

5. Datta, S. K. and N. El-Akily (1978b). Diffraction of Elastic Waves by Cylindrical Cavity

in a Half-space, Journal of Acoustic Society of America, 64, 1692-1699.

6. Datta, S. K. and A. H. Shah (1982). Scattering of SH-Waves by Embedded Cavities,

Wave Motion,~. 265-283.

7. Dravinski, M. (1992). (personal communication)

8. Gregory, R. D. (1970). The Propagation of Waves in Elastic Half-Space Containing a

Cylindrical Cavity, Proc. Camb. Phil. Soc., 67, 689-710.

9. Kontoni, D-P. N., D. E. Beskos and G. D. Manolis (1987). Uniform Half-Plane Elasto­

dynamic Problems by an Approximate Boundary Element Method, Soil Dynamics and

Earthquake Engineering, §.(4), 227-238.

10. Lee, V. W. (1977). On the Deformations near Circular Underground Cavity Subjected to

Incident Plane SH-Waves, Proc. of the Application of Computer Methods in Engineering

Conference, University of South California, Los Angeles, 951-962.

11. Luco, J. E., H. L. Wong and F. C. P. de Barros (1990). Three-Dimensional Response of

a Cylindrical Canyon in a Layered Half-Space, Earthquake Engineering and Structural

Dynamics, 19(6), 799-817.

12. Luco, J. E. and F. C. P. de Barros (1992). On the Appropriate Depth Dependence for

Plane Waves Reflected in a Viscoelastic Half-Space, (to be submitted for publication).

13. Motosaka, M. (1992). (personal communication)

14. Sanchez-Sesma, F. J. (1987). Site Effects on Strong Ground Motion, Soil Dynamics and

Earthquake Engineering, 7_(2), 124-132.

15. Shah, A. H., K. C. Wong and S. K. Datta (1982). Diffraction of Plane SH-Waves in a

Half-Space, Earthquake Engineering and Structural Dynamics, 10, 519-528.

16. Thirunrenkatachar, V. R. and K. Viswanathan (1965). Dynamic Response of an Elas­

tic Half-Space with Cylindrical Cavity to Time Dependence Surface Tractions on the

Boundary of the Cavity, J. Math. Mech., 14, 541-571.

17. Wong, K. C., A. H. Shah and S. K. Datta (1985). Diffraction of Elastic Waves in a

Half-space. II. Analytical and Numerical Solutions, Bull. Seism. Soc. Am., 75, 69-92.

APPENDIX I

Relation between Normalizations used by

Datta and EI-Akily (1978b), Kontoni et al (1987) and Present Study

To compare the results presented by Datta and El-Akily (1978b) and Kontoni et al (1987)

with the present results for Rayleigh wave excitation it is necessary to account for the differences

in the scaling of the free-field ground motion and for the different normalizations used. The

expressions used for the horizontal and vertical components of the free-field ground motion on

the ground surface are compared in Table A.l. Since Kontoni et al (1987) did not report the

expressions used for the free-field Rayleigh wave we assume here that their free-field corre-

sponded to that used by Datta and El-Akily multiplied by the factor -r/[(2k2 - k~)k~/2k] in

which 1' is an unknown constant of proportionality . The terms k = w/CR. and ken kp, v, v'

fork= w/CR. have been defined previously and € = wafa = (P/a)(a/H)(wH/fi). Table A.1

also includes a comparison ofthe expressions used for the horizontal and vertical scattered dis­

placement components in which the dimensionless factors (U~s)' o~s>), (Uyv' UZD) and (UyK'

UzK) represent the normalized scattered displacements in the present study, the work of Datta

and El-Akily (1978b), and Kontoni et al (1987), respectively. The asterisks affecting U D and

UK represent the complex conjugate and need to be introduced to compensate for the time

dependence e-iwt used by Datta and El-Akily (1978b) and Kontoni et al (1987) while eiwt is

use here [Contrary to the statement by Kontoni et al (1987) that they used a time dependence

eiwt we had to assume that they actually used e-iwt in order to match the sign of the real and

imaginary parts]. Finally, Kontoni et al (1987) used two different normalizations designated by

(a) and (b) in Table A.l. In interpreting Table A.l it should be kept in mind that for a Rayleigh

wave k(2k2 - k~- 2vv') = (2k2 - k~)k~/2k.

By equating the ratios of response to free-field we find that the relations between our

results and those of Datta and El-Akily (1987b) is given by

u<s> = i €2 (!!._) 2 u· Y CR. YD

(A.l)

(J(s) = i e? p u· (

2k2- k2) Z 2kv' ZD

(A.2)

For fi = 1/4 (a= .;3p, CR/!3 = 0.9194) and H/a = 5, wHIP= 0.5 (e2 = 1/300) the factors

in Eqs. (A.1) and (A.2) correspond to 3.943 x 10-3 and 4.892 x 10-3 , respectively.

The relation between our results and those of Kontoni et al (1987) for the normalization

(a) is given by

(A.3)

2 (J(s) = i :__ (vk2f3) u·

Z f ZK (AA)

For fi = 1/4, H/a = 5, wa/P = 0.1, a= Sin and "Y = 1.0877 the values of the two constants

appearing in Eqs. (A.3) and (A.4) are 4.446 x 10-6 and 5.516 x 10-6 , respectively. These

constants give the best fit between our results and those of Kontoni et al (1987). It is interesting

to note that "Y = 1.0877 happens to coincide with the value of !3/CR = (0.9194)-1 = 1.0877 for

fi = 1/4.

For the second normalization (b) the relation between our results and those of Kontoni

et al (1987) corresponds to

(J(s) = i ~ (!!_) 2 u· Y f CR. YK

(A.5)

o~s) = i ~ (!!_) 2 u·

f CR. ZK (A.6)

For fi = 1/4, CR./ {3 = 0.9194 and for "Y = 1, the factor (2/"Y )({3 / CR.)2 = 2.366 which corresponds

exactly to the ratio between our numerical results and those of Kontoni et al ( 1987) for H /a = 2.

Kontoni et al (1987) have also presented some numerical results for the scattered tan­

gential stress :E88K normalized as indicated in Table A. I. The relation between the results of

Kontoni et al (1987) and the present normalized result u~~) = u~~ J(wpPURH) is giving by

U(s) = 2i (!!_)3 :E* (A.7) "Y CR. K

For i/ = 1/4 and 1 = l, the factor (2/l)(f3/CR)3 is equal to 2.573 which corresponds to the

actual ratio of the present results to those presented by Kontoni et al (1987) for HI a = 2.0.

\ Table A.l. Free-Field Ground Motions, and Normalization of Scattered Displacements and

Stresses used in Present Study and by Datta and El-Akily (1978b) and Kontoni et al (1987).

Present Study Datta and El-Akily (1978b) Kontoni et al (1987)

Free-Field

u~(O, 0, 0) URn -k~v'i 2kv'i

-~ (2k2- k~)

u~(O,O,O) [2k2- k~ l (2k2

- k~)k~ i 2kv' URH 2k I

Scattered

(s) uRHu~s> f 2 k2v'U* (a) €2 k2v'U* Uy YD YK

(2k2 - k2)k (b) f3 U*

Pv YK (3

(s) uRHu~s> (2k2 - k2 )k2 (2k2 - k2 )k2

Uz €2 f3 f3 U* (a) f 2 13 13 U* 2k ZD 2k ZK

(b) (2k2

- k~)k * k2v Uzx

(3

Scattered Stress

O'(s) wp~URHC;(s) -2 (2k2

- k~)k2 * - p(3 k2 ~K

f3v

FIGURE CAPTIONS

Figure 1. (a) Cylindrical Cavity Embedded in a Half-Space, (b) Location of Sources and

Observers and (c) Stresses in Cylindrical Coordinates.

Figure 2. Normalized Total Horizontal Displacement on the Ground Surface (a, b) and on

the Cavity Wall (c, d) and Normalized Total Shear Stress uex on the Cavity Wall (e, f) for a

Vertically Incident SH-wave. Present results are shown with solid (HI a = 1.5) and segmented

(Hia = 5.0) lines, the results of Lee (1978) are shown with open circles (17 = wal1rfi = 0.5

and 1.0, HI a = 1.5 and 5.0, ep = 0.001).

Figure 3. (a) Normalized Scattered Horizontal Displacement on the Ground Surface for Hori­

zontally [Bv = goo, HI a = 1.5 (segmented line) and 1.83 (solid line)] and Vertically [Bv = 0°,

HI a = 1.5 (dash-dot line)] Incident SH-waves. (b) Normalized Total Displacement on Cav­

ity Wall for a Horizontally Incident SH-waves [Bv = goo, Hla = 1.83 (solid line)]. Solid,

segmented and dash-dot lines correspond to the present results, open symbols correspond to

the results of Datta and Shah (1982) and dotted line in Fig. 3b represent free-field quantities

(7J = wa17rfi = 117r, ep = 0.001).

Figure 4. Real and Imaginary Parts of the Normalized Scattered Horizontal (a, b) and Vertical

(c, d) Displacements on the Ground Surface for a Rayleigh Wave Moving towards y > 0.

(H/a = 5, walfi = 0.1, ii = 1/4, a= ..j3fi, ea = ep = 0.001). Present results are shown

with solid lines, the original results of Datta and El-Akily (1978b) are shown with segmented

lines and those modified by factors of 2.0 and 1.6g for the horizontal and vertical components

are shown with open circles.

Figure 5. Real and Imaginary Parts of the Normalized Scattered Horizontal (a, b) and Vertical

(c, d) Displacements on the Ground Surface and on the Cavity Wall (e, f) for a Rayleigh Wave

Propagating towards y > 0. (H fa = 5, waf P = 0.1, iJ = 1/4, a = v'3 p, ~a = ~13 = 0.001).

The solid and dashed lines represent the current results, open circles represent the results of

Kontoni et al (1987) scaled by an appropriate common factor.

Figure 6. (a) Amplitudes of the Normalized Scattered Horizontal and Vertical Displacements on

the Cavity Wall and (b) Real and Imaginary Parts of the Normalized Scattered Tangential Stress

on the Cavity Wall for a Rayleigh Wave moving towards y > 0. (H /a = 5, waf P = 0.5238,

iJ = 1/4, a = v'3 p, ~a = ~/3 = 0.001). Solid and dashed lines represent the current results,

open circles correspond to the results of Kontoni et al (1987) scaled by an appropriate factor.

Figure 7. Amplitudes of the Normalized Scattered (a) and Total (b) Vertical Displacements

on the Ground Surface for a Rayleigh Wave moving towards y > 0. Solid lines show present

results, segmented lines show results of Wong et al (1985) multiplied by factors of 2.43 and 2.15

for the scattered and total vertical displacements, respectively. (H fa = 1.53, waf P = 1.544,

iJ = 0.3456, ~0' = ~/3 = 0.001).

Figure 8. Amplitudes of the Normalized Scattered Vertical Displacement on the Ground Surface

for Vertically Incident SV- (a) and P-waves (b). (Hja = 1.53, wa/P = 1.544, iJ = 0.3456,

a = 2.0587 p, ~a = ~/3 = 0.00!). The present results are shown with solid lines, the original

results of Wong et al (1985) are shown with segmented lines, and the results of Wong et al

multiplied by a factor of 3 are shown with dot-dash lines.

Figure 9. Amplitudes of the Normalized Total Horizontal (a) and Vertical (b) Displacements on

the Ground Surface for a Vertically Incident P-Wave (H/a = 1.53, wa/P = 1.544, iJ = 0.3456,

a = 2.0587 p, ~a = ~13 = 0.001). Solid lines correspond to the present results, open circles to the

results of Motosaka (1992), dash-dot lines show the results of Dravinski (1992) and segmented

lines show the results of Wong et al (1985).

Figure 10. Normalized Amplitudes of the Total Horizontal (a) and Venical (b) Displacements

on the Ground Surface, Total Horizontal (c) and Venical (d) Displacements on the Cavity Wall

and Total Stresses u88 (e) and CTzx (f) on the Cavity Wall for a Venically Incident SV-Wave

[H/a = 1.5 (solid line) and 5.0 (segmented line), waf'rr~ = 0.5, ii = 1/3, 0: = 2~. ~a = ~!3 = 0.001].

Figure 11. Normalized Amplitudes of the Total Horizontal (a) and Vertical (b) Displacements on

the Ground Surface, Total Horizontal (c) and Venical (d) Displacements on the Cavity Wall and

Total Stresses u 88 (e) and CTxx (f) on the Cavity Wall for a Vertically Incident P-Wave [H/a = 1.5

(solid lines) and 5:0 (segmented lines), wa/7r~ = 0.5, ii = 1/3. 0: = 2~. ea = e/3 = 0.001].

Figure 12. Normalized Amplitudes of the Total Horizontal (a) and Venical (b) Displacements

on the Ground Surface, Total Horizontal (c) and Vertical (d) Displacements on the Cavity Wall

and Total Stresses u 88 (e) and CTzx (f) on the Cavity Wall for a Rayleigh Wave Moving towards

y > 0 [H fa = 1.5 (solid lines) and 5.0 (segmented lines), waj1r~ = 0.5, ii = 1/3, 0: = 2~,

ea = e/3 = 0.001].

Table 1. Values of CR./{3 and u~(O,O)/u:(O,O) as a Function of Poisson's Ratio ii for ~c.r = ~/J·

ii 0 1/4 1/3 0.3456 0.5

Cp_/ {3 0.8740 0.9194 0.9325 0.9344 0.9553

u~(O, 0)/u~(O, 0) i1.2720 i1.4679 i1.5652 i1.5814 i1.8393

Table 2. Convergence of the Total Displacements and Stresses for Vertically Incident P- and

SV-Waves as a Function of the Number of Source and Observation Points (ii = 1/3, 0: = 2/3,

ea = e~ = 0.001, H/a = 1.5, TJ = waj1r'j = 0.5).

P..,Wave

Ns/No

Location 20/40 40/80 80/160 Location

luy/UPI, z = 0

yfa = ±1 1.4982(l) 1.4965 1.4965 yja = 0 1.5012(:1) 1.4965 1.4965

yfa = ±3 0.7863 0.7871 0.7871 yfa = ±3 0.7814 0.7871 0.7871

lu:/UPI, z = 0

yfa = 0 1.8411 1.8464 1.8464 yfa = ±1 1.8034 1.8462 1.8464

yfa = ±3 1.3938 1.3931 1.3931 yfa = ±3 1.4005 1.3931 1.3931

luy/UPI, r = a

B = oo. 1.2725 1.2728 1.2728 B = oo 1.26g6 1.2728 1.2728

B = 45° 0.9056 0.9061 o.g062 ()=goo 0.8g4o o.go60 0.9061

lu:/UPI, r=a

B = 0° 0.7801 0.7824 0.7825 () = oo 0.7775 0.7824 0.7825

B =goo 2.22g4 2.2311 2.2308 () = 45° 2.1852 2.2308 2.2308

lo-ee/wpj3Upl, r = a

B = oo 4.1278 4.13g4 4.1400 () = oo 4.1242 4.1392 4.13g7

B = 90° 2.5085 2.5885 2.5g27 () = 45° 2.4494 2.5875 2.5g32

<1 > Obtained by reciprocity-based approach. <2> Obtained by least squares method.

SV-Wave

Ns/No

20/40 40/80 80/160

luy/Usl, z=O

1.6369 1.6423 1.6424 1.6176 1.6422 1.6424

1.7507 1.7511 1. 7512 1.7350 1. 7510 1.7512

lu:/Usl, z = 0

2.4327 2.4303 2.4302 2.4007 2.4301 2.4302

0.3474 0.3463 0.3463 0.3411 0.3463 0.3463

luy/Usl, r=a

1.2046 1.2019 1.201g 1.2174 1.2020 1.201g

1.5765 1.5g26 1.5g21 1.5228 1.5g2Q 1.5g21

lu:/Usl, r =a

0.3g87 0.4005 0.4004 0.3862 0.4003 0.4004

2.4335 2.41g2 2.41g1 2.3928 2.418g 2.41g1

io-ee/wp,BU s I, r = a

4.2744 4.3472 4.3475 4.266g 4.3471 4.3476

1.7236 1.6g5Q 1.6952 1.6g53 1.6g5g 1.6g51

I I

Kpl

SH

SV

(a)

I I

I

(c)

H

q X%

u XT

y y

L •• s •. s~~ _;~~~en I I .__ ' ' ' ' ' \ \ ~ . \ \ I I - ' • .' I ' ...... .,.." -' / ....... .."' - ---

(b)

Figure 1

4 4

(a) 11 = 0.5 (b) 11 = 1.0 3 3

o, ... o. ... Q_Q_Q_Q_!) ... ~~

>< >< ~ ~

1 1

4 4

(c) 11 = 0.5 (d) 11 = 1.0 3 3

IZl IZl

~ 2 ~ 2 >< >< ~ ~

1 1

270 270

4 4

(e) 11 = 0.5 (t) 11 = 1.0 3 3

IZl IZl . 1\

:::::> :::::> I \ I \

I CO.. I CO.. I I

a.2 a.2 I I I I a a I I I I

~ ~ I

>< >< I I

~ oO I

1 1 I I

Figure 2

4 4

(a) (b)

3 3

r/.l Cl)

2 2 2 2 - . >< V,) / '-' ::s ><

.~ ::I

1 .~_,/' 1

3 270 y/a e (degrees)

Figure 3

,......., ::I: 0:::

(a)

--0~100

(c)

(b)

100 --0~100

(d)

::::J • 0 l-*-" ~:r-1'if-~r----l~---=:JIIC--~b=--,J... ..........

--0~100 100 --0~100

100

100

Figure 4

:I: ~

-.0~100

(b)

100 --0~100

(d)

~ .o~~~~~~~~~~~ ....._

-.0~100

(e)

-.~90

--Real ------------ Imag. ~,e,e;e-~&

_c;1

100 --0~100

(f)

100

100

--Real -------------· Imag.

270

Figure 5

3 8 I u/s) /URH I :t (a) (b)

~ ::::> -----1 Uz(s) / URH I 4 ........ - 2 :t til ~ '-'

N ::::> ::s I CO. a. 8 \

~ ........ \ \

1 - \

::::> til

~ '-'

........ ~ -4 ~ ' - '

, til

,_, '-' >. ::s ...,

Qgo 270 -8-90 270 e (degrees) e (degrees)

Figure 6

4 4

(a) (b)

3 3

:I: :I: Cl:: Cl:: ~ 2 ~ 2 ..._ ..._ ,-...

N en '-"' ::s N ::s

1 1

3 y/a y/a

Figure 7

5 (a)

4 (b)

::3" 3 .......... -V,)

2 -N ::s

1

Q3

y/a y/a

Figure 8

4 4

(a) (b)

3

c. :::::> 2 ....._

>. N ::s ::s

1 1

y/a y/a

Figure 9

4 4

(a) (b)

3 3

~ ~

2 2 2 2 >. N

= = 1 1

4

4 4

(c) (d)

3 3

~ ~

2 2 2 2 >. N = =

1 1

270

9 3

(f)

~ 6 ~ 2 ::::> ::::>

lc:::L lc:::L a. a. 8 8

......... ......... CD 3 ~ 1 CD t5 \::)

Figure 10

4

3

Cl..

~ 2 >-. ::s

1

4

3

Cl..

::: 2 >-. ::s

1

9

Cl.. 6 ;:J I CO.. a. 8 ..._

CD 3 ~

(a)

(c)

(e)

\ \ \ \ \ I ' ........ -- .... __ ,

270

4

3

Cl..

~ 2 N ::s

1

4

3

0..

::: 2 N ::s

1

Cl.. 2 ;:J I CO.. a. 8

J 1

(b)

(d)

(f)

\ \ \ \ \ I ,...,_ ... ______ ,

4

270

270

Figure 11

4 4

(a) (b)

3 3

::c ::c ~ ~

~ 2 ~ 2 ..._ ..._ >. N

::s ::s

1

4 4

(c) (d)

3 3

~ ~ ~ 2 ~ 2 ..._ ..._

>. N ::s ::s

1 1

__ .... -270

9 3 (e) (f)

::c 6 :I:: 2 ~ ~

:::> :::> I ca.. ICQ. a. a. 8 8 ';3

..._ 1 ><

CI) >< t) t)

270 -90

Figure 12

Appendix C. Diffraction of Obliquely Incident Waves by a Cylindrical

Cavity Embedded in a Layered Half-Space (Barros/Luco ).

DIFFRACTION OF OBLIQUELY INCIDENT WAVES

BY A CYLINDRICAL CAVITY

EMBEDDED IN A LAYERED VISCOELASTIC HALF-SPACE

F. C. P. de Barros and J. E. Luco

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California 92093-0411.

ABSTRACT

The three-dimensional harmonic response in the vicinity of an infinitely-long, cylindrical

cavity of circular cross-section buried in a layered, viscoelastic half-space is obtained when the

half-space is subjected to homogeneous plane waves and surface waves impinging at an oblique

angle with respect to the axis of the cavity. The solution is obtained by an indirect boundary

integral method based on the use of moving Green's functions for the viscoelastic half-space.

Numerical results describing the motion on the ground surface and the motion and stresses on

the wall of the cavity are presented for obliquely incident P, SV, SH and Rayleigh-waves with

different horizontal angles of incidence.

INTRODUCTION

The diffraction of hannonic waves by an infinitely-long, cylindrical cavity of circular

cross-section embedded in a horizontally layered viscoelastic half-space (Fig. 1 a, b) is considered

in this paper. The cavity and the half-space are excited by homogeneous plane SH, SV and P­

waves and by Raleigh-waves impinging at an oblique angle with respect to the axis of the cavity

(Fig. 1 b, c). Although the geometry of this model can be considered to be two-dimensional,

the response is fully three-dimensional due to the variation of the incident motion along the axis

of the cavity. The problem under consideration is of interest in connection with the seismic

response of unlined tunnels and with the possible modification of the surface ground motion

during earthquakes as a result of the presence of underground cavities.

Previous studies on the diffraction of waves by cavities have been limited to two­

dimensional anti-plane or plane-strain models of a cavity buried in a uniform half-space. Lee

(1977), Datta and El-Akily (1978a), Datta and Shah (1982) and Shah, Wong and Datta (1982)

have studied the two-dimensional diffraction of plane SH-waves by a cylindrical cavity buried

in a uniform elastic half-space. The cases P- and SV-waves have been considered by Datta

and El-Akily (1978a) and Wong, Shah and Datta (1985). The response to Rayleigh waves has

been studied by Datta and El-Akily (1978b), Wong, Shah and Datta (1985) and Kontoni, Beskos

and Manolis (1987). Recently, the authors (Luco and Barros, 1993) have presented additional

results for SH, P, SV and Rayleigh waves and have conducted a detailed comparative study

with previous two-dimensional solutions. A variety of methods of solution have been used in

these studies. Lee (1977) solved numerically a truncated infinite system of equations for the

infinite number of coefficients in a wave series expansion of the scattered field. The method of

matched asymptotic expansion has been used by Datta and El-Akily (1978a, b) and Datta and

Shah (1982). Datta and Shah (1982), Shah, Wong and Datta (1982) and Wong, Shah and Datta

(1985) have used a hybrid approach in which an interior region is represented by finite elements

while a series solution is used to account for the exterior region. A direct Boundary Element

Method has been used by Kontoni et al (1987) while the authors (Luco and Barros, 1993) have

used an indirect boundary integral method.

The problem of diffraction of plane waves by a cylindrical cavity buried in an elastic

half-space is closely related to the problem of determining the seismic response of a canyon

cut into a half-space for which extensive references can be found in articles by Sanchez-Sesma

(1987), Aki (1988) and Luco, Wong and Barros (1990).

The present work on the three-dimensional response of an infinitely-long cylindrical cavity

buried in a layered viscoelastic half-space is directed at a model which is more general than purely

two-dimensional models but is easier to analyze than fully three-dimensional problems involving

a cavity of finite length. In the present study, an indirect boundary integral method based on

moving Green's functions for a layered viscoelastic half-space (Barros and Luco, 1992) is used

to obtain the harmonic motion on the surface of the half-space and the motion and stresses on the

wall of the cavity. The indirect boundary integral formulation employed here is similar to that

used by the authors to calculate the three-dimensional response of cylindrical canyons embedded

in a half-space (Luco, Wong and Barros, 1990). In the case of canyons, the technique has been

carefully tested against other solutions (Luco et al, 1990; Zhang and Chopra, 1991). For buried

cavities, the method of solution has also been tested in the limiting two-dimensional case when

the waves impinge normal to the axis of the cavity (Luco and Barros, 1993).

The proposed indirect boundary integral method is presented first followed by a set of

numerical results describing the response of cylindrical cavities embedded in layered media when

subjected to obliquely incident P, SV, SHand Raleigh-waves.

FORMULATION AND METHOD OF SOLUTION

The geometry of the model is illustrated in Fig. 1. The infinitely-long, unlined cavity

of circular cross-section of radius a is parallel to the free surface of the half-space and its

centerline is located at a depth H below the free surface. The soil is represented by (N - 1)

horizontal viscoelastic layers overlying a viscoelastic half-space. Each of the media in the half­

space is characterized, for harmonic vibrations, by complex P- and S-wave velocities Oj =

a;[1+2isgn(w)eo:;Jll2 and /3; =.8;[1 +2isgn(w)(a;Jll2, and by the density P; (j = l,N).

The terms a; and ,83 represent (approximately) the real parts of the P- and S-wave velocities,

and eo:; and e/3; represent the small hysteretic damping ratios for P- and S-waves, respectively.

In what follows, the excitation and the response will have harmonic time dependence of

the type eiwt where w is the frequency. For simplicity, the factor eiwt will be dropped from all

expressions.

Free-Field Ground Motion

As a first step in the formulation, it is necessary to determine the ground motion and

the stress components for free-field conditions, i.e., in the absence of the cavity. The seismic

excitation is represented by homogeneous plane P-, SV- or SH-waves, such that the normal to

the wave front in the underlying half-space forms an angle Ov with the vertical axis (Ov = 0 for

vertical incidence) and the angle Oh with the axis of the cavity (x-axis).

To calculate the free-field ground motion it is convenient to consider· the coordinate

system x', y', z' (z' = z) shown in Figs. la and lb. Referred to this coordinate system, the

incident motion within the underlying exterior half-space is represented by

{u/v }inc= A {U'}e-ik'x'+v.Nz'

where A is the amplitude of the incident . displacement, k' = ( w //3 N) sin Ov,

(1)

.,1 -VN -

i(w/f3N)cos8v for S-waves and k' = (wfo:N)sinOv, v~ = i(w/o:N)cosOv for P-waves, and

{U'} is the vector

(O,l,O)T for SH-wave excitation

{U'} = (cosOv, 0, sinOv)T for SV-wave excitation (2)

(sin Ov, 0,- cos Ov )T for P-wave excitation

The resulting total free-field ground motion satisfying all the continuity, free-surface and

radiation conditions for the layered geometry shown in Fig. lc can be calculated by the approach

described by Luco and Wong (1987). The resulting free-field displacement and stress fields in

the x' y' z' - coordinate system are denoted here by

{ u' tr} = {U' tr(z')}e-ik' x'

[u' tr] = [E' tr(z')]e-ik'x'

(3)

(4)

where the elements of U' tr(z') and E' ff(z') are independent of y' and depend only on z' =

z. In the case of a uniform half-space we also consider an incident surface Rayleigh wave.

The expressions for the corresponding displacements and stresses in the free-field are listed in

Appendix I.

To impose boundary conditions on the surface of the cavity wall it is necessary to intro­

duce the rotation of coordinates

sin Oh 0] { x } { x } co~Ob ~ ~ = [C]T ~ (5)

For future reference it is also convenient to point out that the incident ground motion in the

underlying half-space, referred to the (x, y, z) coordinate system, is represented by a plane wave

described by

{u}inc = A{U} exp { -i~ [x sin Ov cos 8b + y sin Ov sin Oh] + vjyz} (6)

where A is the amplitude of the incident displacement and v = f3N, vjy = i(w/f3N) cosBv for

S-waves and v = O:N, vjy = i(w/o:N)cosBv for P-waves. The vector {U} is given by

(-sin~' cos~' O)T for SH-waves,

for SV-waves and (7)

It should be noted that if ~ = 90° or if Bv = 0°, then, the incident displacement becomes

independent of the coordinate x and the problem becomes two-dimensional.

The free-field displacement and stress fields in the xyz-coordinate can be written in the

form

{ Uff} = {U tr(Y, z) }e-ikx

[atr] = [Etr(y, z)]e-ikx

(8)

(9)

where k = (w/f3N)sinBvcosBh for S-wave excitation and k = (w/a:N)sinBvcosBh for P-wave

excitation. In Eqs. (8) and (9),

{Utr(y,z)} = [C] {U'tr(z)}e-ik'ysin8h

[Etr(y, z)] = [C] [E' tr(z)] [C]T e-ik'y sinBh

in which the y-dependence enters only in the last exponential factor.

Boundary-Value Problem

(10)

(11)

In the presence of the cavity, the total displacement vector { u(X)} and the total traction v

vector { t(X)} on the boundary of the cavity can be written in the form

{u(X)} = {utr(x)} + {Us(x)} v v v

{t(X)} = {ttr(X)} + {ts(X)}

(12)

(13)

v v where {Us} is the scattered displacement field, and { tcr( X)} and { ts (X)} are the traction vectors

associated with the free-field and the scattered field, respectively. The vector { ucr(X)} is given v

by Eqs. (8) and (10) and the vector {tcr(X)} can be written in the form

(14)

where X0 = (0, Yo, Z0 ) and

{ Tcr(Xo)} = [Err(xo)] {v(xo)} (15)

in which {v} = (0, vy, Vz)T is the unit normal to the cavity's boundary at X0 •

The condition of vanishing traction on the boundary S of the cavity leads to

v { v } "k {ts(X)} =- Tcr(X) e-t x , xES . (16)

Numerical Solution

To solve the boundary-value problem, the scattered field is represented as resulting from

the action of a number (N 5) of moving concentrated loads moving in the direction of the x-axis

with the (complex) velocity

{

f3N

c = w I k = sin Bv cos oh O'N

sin8v cos~

for S-wave excitation, (17)

for P-wave excitation.

These moving loads act in the layered half-space (without the cavity) and within the region to

be occupied by the cavity (Fig. ld). The scattered displacement field is then written in the form

Ns

{us(X)} = L [G(xo,Xsj)] {Fsj}e-ikx (18) j=l

where [G(x0 ,Xsj)] is the 3 x 3 matrix of moving Green's functions. The first, second and third

columns of the matrix G correspond to the displacement vector at X0 = (0, y0 , z0 ) for a unit

point load acting in the x, y and z-directions, respectively, and moving with velocity c = wjk

along a line parallel to the x-axis passing through the point Xsj = (0, Ysh Zsj). The 3 x 1 vectors

{ Fsj} represent the unknown amplitudes of the j-th moving load.

The traction vector of the scattered field on the surface S of the cavity can be written in

(19)

(20)

in which (vy(x0 ), llz(x0 )) are the direction cosines of the normal to the surface S of the cavity

and a;x(x0 , Xsj). a~x(X0 , Xsj), .. . , are the stresses at X0 = (0, Yo, Zo) induced by the moving unit

point loads acting in the x, y and z-directions.

Substitution from Eq. (19) into the boundary condition given by Eq. (16) leads to

N

L [fflxo,Xsj)] {Fsj} =- { Tff(Xo)} (21) j=l

where f2 is the intersection Of the boundary 8 with the plane X = 0. Imposing the boundary

condition (21) at a set of No points, Xoi. (i = 1, No) on n, leads to

N

L [ fflxo;, Xsj)] {Fsj} =- { Tff(Xo;)} j=l

Xoi E f2 (i = 1,No) . (22)

which can be written in the form

[h]{Fs} = -{Tff} (23)

where the 3 x 3 (i,j) block of the (3No x 3Ns) matrix corresponds to [H(xo,Xsj)], {Fs} =

T T T - ( II ... T II ... T )T ({Fst} ,{Fs2} , ... ) and {Tff} = {Tff(Xot)} ,{Tff(Xo2)} ,... .

The 3N0 x 3N, system of equations given by Eq. (23) can be solved in the least-square

sense leading to

(24)

in which the asterik denotes the complex conjugate. An alternative, reciprocity-based approach

is to pre-multiply Eq. (23) by [G]T and then solve leading to

(25)

where [G] has for blocks the matrices [G(x0 ;, Xsj)]. On the basis of reciprocity theorems it can

be shown that as the number N0 of observation points increases the matrix [G]T[.H] tends to

become symmetric (Apsel and Luco, 1987).

Once the forces {Fs} have been obtained from Eq. (24) or (25), then, the scattered

displacement and stress fields, and, also, the total displacement and stress fields can be easily

obtained. It should be noted that the procedure described above is also valid for a cavity of

arbitrary cross-section.

For the purpose of describing the stresses on the wall of the cavity it is convenient to

refer to the cylindrical coordinates (r, 8, x) shown in Fig. 2. The stress components in cylindrical

. coordinates are related to the stress components in cartesian coordinates by the. relations

aee = ayy Sin2 8 + ayz Sin 28 + azz COS2 8

arr = ayy COS2 8- ayz sin28 + azz sin2 8

are = ~ (azz- ayy) sin 28- ayz COS 28

a XX = a XX

a ex = -a xy Sin 8- axz COS 8

(26a)

(26b)

(26c)

(26d)

(26e)

(26!)

CONVERGENCE OF THE NUMERICAL APPROACH

The numerical results obtained by the procedure described in the previous section depend

in principle on the location and number of source points (N s) and on the number of observation

points (N0 ) used. The first step is to test the convergence of this approach. For this purpose we

consider a cavity of radius a buried to a depth H = 2a in a uniform half-space characterized by

a = 2~ (ii = 1/3), and ea = ef3 = 0.01. The half-space is subjected to a non-vertically incident

(Bv = 45°) SH-wave propagating in the direction of the cavity (Ott = 0°). The frequency of the

excitation is such that TJ = waj1r~ = 0.5.

Numerical results for the total displacement components at a few points on the ground

surface (z = 0) and at a few points on the wall of the cavity (r =a) are presented in Table 1

for different numbers of sources and observers (N s. No). In all cases, the sources are equally

spaced on a circle of radius a' = a- 3(27ra/ N0 ) (No > 20). Thus, as the number of observation

points increases, the sources move closer to the actual boundary r = a.

The results listed in Table 1 have been calculated by the reciprocity-based and by the least

squares approaches defined by Eqs. (25) and (24), respectively. The displacement amplitudes

lui I are normalized by the amplitude A of the incident displacement vector on the ground surface.

The numerical results presented for source/observer combinations (N8 , N 0 ) of (20, 40), (40, 80)

and (80, 160) show that the procedure is very stable as the number of source and observation

points increases. In particular, the displacements on the free-surface (z = 0) are the least

sensitive to the number of sources/observers. In general, 40 source points and 80 observation

points are ·sufficient for most applications.

NUMERICAL RESULTS

Figure 3 through 9 illustrate different aspects of the response of a cylindrical cavity

of radius a embedded to a depth H in a uniform half-space when the half-space is subjected

to SH, P, SV and Rayleigh waves. The halfspace is characterized in all cases by a = 2~

(v = 1/3), ~a = ~!3 = 0.01. Each figure shows the amplitudes of the normalized displacement

components Ux = lux/AI, U11 = lu31 /AI and Uz = luz/AI on the ground surface (a, b, c)

and on the cavity wall (d, e, f) normalized by the amplitude A of the incident displacement

vector at the origin on the ground surface. Also shown are the amplitudes of the non-zero

stress components Eee = laee/aol. Eex = laex/aol and Exx = laxxfaol on the cavity wall

normalized by a0 = wp~A. [In the case of Rayleigh wave excitation (Fig. 6) the displacements

are normalized by the amplitude AH of the horizontal component of the free-field Rayleigh

ground motion and the stresses by (wp~AH )]. The parameters H /a, waj1r~, Oh. Ov and the type

of incident wave vary from Figure to Figure.

Fig. 3 shows the effects of the horizontal angle of incidence (Jh on the response for a

non-vertically incident (Ov = 45°) SH-wave in the case Hfa = 2.0 and waj1r~ = 0.5. The

results show that the horizontal angle of incidence has a significant effect on the response. In

particular, the results for (Jh = goo which correspond to the two-dimensional anti-plane solution

are significantly different from those for other horizontal angles of incidence. To interpret these

results it is convenient to refer to Table 2 listing the amplitudes of the corresponding normalized

displacements on the free-field ground surface (x = y = z = 0).

The effects of (Jh on the response for non-vertically incident P- and SV-waves (Ov = 45°)

and for Rayleigh-waves are presented in Figs. 4, 5 and 6, respectively. It should be noted that

for Rayleigh-wave excitation some numerical difficulties were encountered when Oh = 0°. For

this reason, results for Rayleigh waves are presented for (Jh = 5°, 30°, 60° and goo.

Fig. 7 illustrates the effects of the vertical angle of incidence Ov for an SH-wave propa­

gating in the direction of the axis of the cavity (Oh = 0°) and for Hfa = 2.0 and waj1r~ = 0.5.

It is apparent from Fig. 7 that the vertical angle of incidence Bv has a significant effect on the

displacements and stresses in the vicinity of the cavity.

The effects of the embedment depth H of the cavity on the response to a non-vertically

incident (Bv ::::: 45°) SH-wave propagating in the direction of the axis of the cavity (Bh = 0°) are

shown in Fig. 8 for wa/1rP = 0.5. Clearly, the embedment depth has a significant effect on the

response and, particularly, on the vertical displacement components.

Fig. 9 illustrates the effects of the dimensionless frequency 'fJ = wa/1rP on the response to

a non-vertically incident (Bv = 45°) SH-wave propagating along the axis of the cavity (Oh = 0°)

for H/a = 2.0. It is apparent that frequency is an important factor in determining the response

in the vicinity of the cavity.

The effects of layering for the simple case of a profile consisting of a layer over a half­

space [Fig. 2(a)] are presented in Fig. 10. The excitation corresponds again to a non-vertically

incident (Bv = 45°) SH-wave propagating along the axis of the cavity (Oh = 0°). Numerical

results were calculated for H/a = 2, hi/a= 4, 'fJ = wa/1rP2 = 0.5, i/1 = i/2 = 1/3, Pl = P2·

~o 1 = ~o2 = ~(31 = ~(32 = 0.01 and for ~I/~2 = 1 and ~1/~2 = 0.25. The displacements

are normalized by the amplitude A of the incident displacement vector at an outcrop with the

same properties as the underlying half-space and at the same elevation as the free-surface. The

stresses are normalized by lwp2~2AI. In the two cases considered the amplitudes of the free­

field displacements on the ground surface are Ux = 0, Uy = 2 and Uz = 0 for PI/ P2 = 1 and

Ux = 0, Uy = 1.979 and Uz = 0 for ~d ~2 = 0.25. The results in Fig. 10 show that the effects

of layering on the response are very strong.

CONCLUSIONS

A procedure has been presented to calculate the three-dimensional displacements and

stresses in the vicinity of an infinitely-long cylindrical cavity embedded in a layered viscoelastic

half-space and subjected toP, SV, SHand Rayleigh waves impinging at an oblique angle with

respect to the axis of the cavity. The results obtained show that the horizontal angle of incidence

has a marked effect on the response on the ground surface and on the wall of the cavity. In

particular, the results for oblique incidence are significantly different from the two-dimensional

results that are obtained when the horizontal incidence is normal to the axis of the cavity. The

effects on the results of the vertical angle of incidence, the embedment depth of the cavity, the

excitation frequency as well as those introduced by layering are also documented.

ACKNOWLEDGMENTS

The work conducted here was supported by a Grant from California Universities for

Research in Earthquake Engineering (CUREe) as part of a CUREe-Kajima Project.

REFERENCES

1. Aki, K. (1988). Local Site Effects on Strong Ground Motion, Earthquake Engineering

and Soil Dynamics 11- Recent Advances in Ground Motion Evaluation, (J. Lawrence Von

Thun, Ed.), ASCE Geotechnical Special Publication No.2, New York, N.Y. pp. 103-155.

2. Apsel, R. J. and J. E. Luco (1987). Impedance Functions for Foundations Embedded

in a Layered Medium: An Integral Equation Approach, Earthquake Engineering and

Structural Dynamics, 15, 213-231.

3. Barros, F. C. P. de and J. E. Luco (1992). Moving Green's Functions for a Layered Vis­

coelastic Half-Space, Report, Dept. of Appl. Mech. & Engng. Sci., University of California,

San Diego, La Jolla, California, 210 pp.

4. Datta, S. K. and N. El-Akily (1978a). Diffraction of Elastic Waves in a Half-Space. I.

Integral Representation and Matched Asymptotic Expansions, Modern Problems in Elastic

Wave Propagation, J. Miklowitz and D. Achenbach Eds., Wiley-lnterscience Publication,

John Wiley & Sons, New York, 197-218.

5. Datta, S. K. and N. El-Akily (1978b). Diffraction of Elastic Waves by Cylindrical Cavity

in a Half-space, Journal of Acoustic Society of America, 64, 1692-1699.

6. Datta, S. K. and A. H. Shah (1982). Scattering of SH-Waves by Embedded Cavities,

Wave Motion, .4, 265-283.

7. Kontoni, D-P. N., D. E. Beskos and G. D. Manolis (1987). Uniform Half-Plane Elasto­

dynamic Problems by an Approximate Boundary Element Method, Soil Dynamics and

Earthquake Engineering, §(4), 227-238.

8. Lee, V. W. (1977). On the Deformations near Circular Underground Cavity Subjected to

Incident Plane SH-Waves, Proc. of the Application of Computer Methods in Engineering

Conference, University of South California, Los Angeles, 951-962.

9. Luco, J. E. and H. L. Wong (1987). Seismic Response of Foundations Embedded in a

Layered Half-Space, Earthquake Engineering and Structural Dynamics, 15(2), 233-247.

10. Luco, J. E., H. L. Wong and F. C. P. de Barros (1990). Three-Dimensional Response of

a Cylindrical Canyon in a Layered Half-Space, Earthquake Engineering and Structural

Dynamics, 19(6), 799-817.

11. Luco, J. E. and F. C. P. de Barros (1992a). Dynamic Displacements and Stresses in the

Vicinity of a Cylindrical Cavity Embedded in a Half-Space, (submitted for publication).

12. Luco, J. E. and F. C. P. de Barros (1992b). On the Appropriate Depth Dependence for

Plane Waves Reflected in a Viscoelastic Half-Space, (submitted for publication).

13. Sanchez-Sesma, F. J. (1987). Site Effects on Strong Ground Motion, Soil Dynamics and

Earthquake Engineering, 1(2), 124-132.

14. Shah, A. H., K. C. Wong and S. K. Datta (1982). Diffraction of Plane SH-Waves in a

Half-Space, Earthquake Engineering and Structural Dynamics, 10, 519-528.

15. Wong, K. C., A. H. Shah and S. K. Datta (1985). Diffraction of Elastic Waves in a

Half-space. II. Analytical and Numerical Solutions, Bull. Seism. Soc. Am.,·15, 69-92.

16. Zhang, L. and A. K. Chopra (1991). Three-Dimensional Analysis of Spatially Varying

Ground Motions around a Uniform Canyon in a Homogeneous Half-Space. Earthquake

Engineering and Structural Dynamics, 20, 911-926.

APPENDIX I

Free-Field Ground Motion for

a Rayleigh Wave on a Uniform Half-Space

The free-field displacement and stress fields for a surface Rayleigh wave propagating in

the x 1 -direction in a uniform half-space z > 0 are given by

[ ( -2 2) ] ( -) - 2k - k I 2ik .- I

U1 =A -i ke-vz- f3 e-"' z - e-tkz zl H 2ik . k~

(1.1)

(1.2)

2k - k I 2ik •- I

[ ( -2 2) ] ( -) U~l = AH -ve-vz + 2v1 f3 e-v% k~ e-tk:t (1.3)

(1.4)

I - I ( 2ik ) 'k- I a zlzl = p.AH 2 i kv[e-vz - e-"' z] k~ e-' z (!.5)

u;,,, = JlAn (2k'- k~)[e-••- e-•'•J ( ~:) e-ikz' (1.6)

1 A (2k2 k2) -vz ( 2ik) -ikz1

aylyl = p. H a - f3 e k~ e (1.7)

(!.8)

(1.9)

in which k = w /CR. is the root of the Rayleigh equation

(1.10)

The parameter AH represents the amplitude of the horizontal component of the free-field

ground motion on the ground surface at x' = y' = z' = 0. From (1.1, 1.2, 1.3) we find

u~, (0, 0, 0) = AH (1.11)

u~,(O,O,O) = 0 (1.12)

[2P- k

2] u~, (0, 0, 0) = i 2kv' f3 . AH (1.13)

Some typical values for CR./~ and i(2P- k~)/2kv' are listed in Table 1.1 as a function

of Poisson's ratio for ~a: = ~f3·

Table 1.1 Values of ep_j ~ and Ratio of Vertical to Horizontal Displacement on the Ground Surface

Poisson's Ratio v 0 1/4 1/3 0.5

~~~ 0.8740 0.9194 0.9325 0.9553

i [ (2k2- k~)/2kv'] i1.2720 i1.4679 i1.5652 i1.8393

Table 1. Normalized Displacement Components for a Non-Vertically Incident (Ov = 45°) SH­

Wave Propagating in the Direction of the Axis of the Cavity (Oh = 0°) for Different Numbers

of Source (N5 ) and Observation (N0 ) Points [h/a = 2.0, waj1r~ = 0.5, v = 1/3, 0: = 2~,

ea = e/3 = 0.01].

{Ns = 20, No= 40) {Ns = 40, No= 80) {Ns = 80, No = 160) Variable Location Eq. (25) Eq. (24) Eq. (25) Eq. (24) Eq. (25) Eq. (24)

U:r: yfa = ±1 1.0232 1.0229 1.0235 1.0235 1.0235 1.0235 (z = 0) yfa = ±2 0.9701 0.9700 0.9701 0.9701 0.9701 0.9701

Uy yfa=O 1.2405 1.2407 1.2406 1.2406 1.2406 1.2406 (z = 0) yfa = ±1 1.3208 1.3203 1.3210 1.3210 1.3210 1.3210

yfa = ±2 1.1269 1.1270 1.1268 1.1268 1.1268 1.1268 .

Uz yfa = ±1 0.7701 0.7694 0.7703 0.7703 0.7703 0.7703 (z = 0) yfa = ±2 0.6003 0.5999 0.6003 0.6003 0.6003 0.6003

U:r: () = oo 1.0678 1.0676 1.0681 1.0681 1.0681 1.0681 (r =a) () = 45° 0.5122 0.5117 0.5121 0.5121 0.5121 0.5121

Uy () = oo 1.6136 1.6131 1.6140 1.6140 1.6140 1.6140 (r =a) () = 45° 0.3360 0.3363 0.3345 0.3344 0.3344 0.3344

()=goo 1.4069 1.4042 1.4080 1.4079 1.4080 1.4079 () = 270° 1.1382 1.1377 1.1379 1.1379 1.1380 1.1380

U:e () = oo 0.3422 0.3410 0.3422 0.3422 0.3421 0.3421 (r =a) () = 45° 0.7976 0.7979 0.7970 0.7970 0.7970 0.7970

Table 2. Amplitudes of the Normalized Displacement Components of the Free-F:ield Ground

Motion on the Ground Surface (z = 0) for Non-Vertically Incident SH, P and SV-Waves

(Ov = 45°) and for Rayleigh Waves with Different Horizontal Angles of Incidence (a = 2P.

eo = e/3 = 0.01).

SH-Wave P-Wave SV-'Wave Rayleigh-Wave

eh Ux Uy Uz Ux Uy Uz Ux Uy Uz Ux Uy uz

oo(50)* 0.000 2.000 0.000 1.285 0.000 1.457 0.000 0.000 1.414 0.996 0.087 1.565

30° 1.000 1. 732 0.000 1.113 0.643 1.457 0.000 0.000 1.414 0.866 0.500 1.565

60° 1. 732 1.000 0.000 0.643 1.113 1.457 0.000 0.000 1.414 0.500 0.866 1.565

goo 2.000 0.000 0.000 0.000 1.285 1.457 0.000 0.000 1.414 0.000 1.000 1.565

* (~=5° for Rayleigh-Wave only)

FIGURE CAPTIONS

Figure 1. Schematic Representation of a Cylindrical Cavity Embedded in a Layered Half­

Space. (a) Top view showing horizontal angle of incidence, (b) Cross section, (c) Free-field

model showing vertical angle of incidence and (d) Location of sources and observers.

Figure 2. (a) Example of Cavity Buried in a Layer Overlying a Half-Space, (b) Cylindrical

Coordinates used to Describe Stresses on Cavity Wall.

Figure 3. Effects of the Horizontal Angle of Incidence Oh on Displacement Components on

the Free Surface z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f)

and Stress Components on the Cavity Wall (g, h, i) for a Non-Vertically Incident SH-Wave

(Ov = 45°. wa/7rP = 0.5, H/a = 2.0, ii = 1/3. a= 2P. ea = ef3 = 0.01) [ (Jh = 0°,

- - - Oh = 30°, -·-·- Oh = 60°, ·········· Oh = goo].

Figure 4. Effects of the Horizontal Angle of Incidence Oh on Displacement Components on

the Free Surface z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f)

and Stress Components on the Cavity Wall (g, h, i) for a Non-Vertically Incident P-Wave

(Ov = 45°, wa/7rP = 0.5, H/a = 2.0, ii = 1/3, a= 2P. ea = ef3 = 0.01) [ (Jh = 0°,

- - - Oh = 30°, -·-·- Oh = 60°, ·········• Oh = goo].

Figure 5. Effects of the Horizontal Angle of Incidence Oh on Displacement Components on

the Free Surface z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f)

and Stress Components on the Cavity Wall (g, h, i) for a Non-Vertically Incident SV-Wave

(Ov = 45°, waf7rP = 0.5, H/a = 2.0, ii = 1/3. a= 2P. ea = ef3 = 0.01) [ -- (Jh = 0°,

--- Oh = 30°, -·-·- Oh = 60°, ·········· Oh = goo].

Figure 6. Effects of the Horizontal Angle of Incidence Oh on Displacement Components on the

Free Surface z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f) and Stress

Components on the Cavity Wall (g, h, i) for a Rayleigh Surface Wave (waj1r~ = 0.5, H/a = 2.0,

i) = 1/3, a= 2~. ~Q = ~{3 = 0.01) [ ~ =5°,--- oh = 30°, -·-·- oh = 60°,

.......... oh = 90°].

Figure 7. Effects of the Vertical Angle of Incidence Ov on Displacement Components on

the Free Surface z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f)

and Stress Components on the Cavity Wall (g, h, i) for a Non-Vertically Incident SH-wave

(~ = 0°, waf'rr~ = 0.5, H/a = 2.0, v = 1/3, a= 2~, ~a: = ~/3 = 0.01) [ Ov = 45°,

--- Ov = 22.5°, ·········· Ov = 0°].

Figure 8. Effects of the Embedment Depth H on Displacement Components on the Free Surface

z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f) and Stress Components

on the Cavity Wall (g, h, i) for a Non-Vertically Incident SH-Wave (Oh = 0°, Ov = 45°,

waj1r~ = 0.5, Hfa = 2.0, v = 1/3, a = 2~, ~a: = ~/3 = 0.01) [ --- H/a = 1.5,

-- H/a = 2.0, ·········· H/a = 5.0].

Figure 9. Effects of the Dimensionless Frequency waj1r~ on Displacement Components on

the Free Surface z = 0 (a, b, c), Displacement Components on the Cavity Wall (d, e, f) and

Stress Components on the Cavity Wall (g, h, i) for a Non-Vertically Incident SH-Wave (Oh = 0°,

Ov = 45°, waj1r~ = 0.5, Hfa = 2.0, v = 1/3, a= 2~, ~a: = ~/3 = 0.01) [ 'fJ = 0.5,

-- - 'fJ = 0.2, .......... 'fJ = 0.1].

Figure 10. Effects of Layering on Displacement Components on the Free Surface z = 0 (a, b, c),

Displacement Components on the Cavity Wall (d, e, f) and Stress Components on the Cavity Wall

(g, h, i) for a Non-Vertically Incident SH-Wave (Oh = 0°, Ov = 45°, waj1r~2 = 0.5, Hfa = 2.0,

hi/a = 4, ii1 = ii2 = 1/3, ~a: 1 = ~a:2 = ~/31 ~ ~/32 = 0.01, P1 = P2) [ ~1/~2 = 1,

.......... ~d ~2 = 0.25].

(a) (c) X

1 x'

2

3

"'y'

z'

(b) (d)

y y

Sources Observers

z z

Figure 1

(]' :Z:%

(b)

Figure 2

2

·-. ' 1 \ \ I . , .......... '

·'·-'l- / I ' . . I \_.

0

2

Uz (c)

1

2

3

1

o~~._._~~~~~

3~----~~~~~~

2

0

2

Uz (f)

0 90 180 9 (degrees)

Lea

4

2

o~~~~~~~._~~

3~~~~~~-T~~

·Lex

1

0

4

Lxx (i)

3

Figure 3

(a)

1

0.

3

Uy

2

1

o~~~~~~-L~~

3~~~~~~~~~

2

1

0

4

3

2

1

-2 0 2 4 ~90 y/a

Uy (e)

0

2

Lex (h)

1

o~~~~~~~~~

3~~~~~~~-,~

2

1

0 90 180 270 ~90 0 90 180 270 e (degrees) e (degrees)

Figure 4

(a)

1

0L....-.J-....L..-....J.-.....L..---L.--L.--L---J

2~~~~~~~~~

(b)

2

4 ~90

(d)

0 90 180 e (degrees)

Lax (h)

1

(i)

2

1

0 90 180 270 a (degrees)

Figure 5

1

0

2

1

3

Uy '~:\ I I).·

.._ I i · \'' 'I o I

\ I /i \\\ \ I !! \ \~ . I :, .

\ v l \\\ 'l \ 11 \ \ \ . . \ \ \ I . ., i . \l ,,\ I . I

Dz

0 2 y/a

(a) (d)

0

2 3

(b) Uy (e)

2

1

1

0 0

2 4

Dz (f)

3

0 90 180 a (degrees)

Figure 6

(a) (d)

1 l

o~~--~~~~~~

3~~r-~~~~~~

o~~--~-=--~--~

3~~~--~~~~~

0

2

.... I \ ..

1

-2 0 2 y) a

(c)

2

1

0

2

Uz (f)

1

4 ~90 0 90 180 e (degrees)

Lex (h)

1

0

3

2

Figure 7

(a) (d)

1

o~.~~~~~~~~

3~~~~~~~--~

0

2

Uz ,, (c) I \ I l I I

1 I

2

2

1

0

2

Uz I' (f) I \ I \

\

1

4 Q90 0 90 180 e (degrees)

2

1

0

4

3

2

1

0 90 180 270 e (degrees)

Figure 8

2 2

Ux (a)

1 1

0 0

4 3

3 -- ,_ 2 , ' / ..........

' 2 ·--~ /--·--/.---·

1 1

0 0

2 2

Dz (c)

1 1

Ux

Uy

Dz

(d)

(f)

0 90 180 e (degrees) .

8

6

0

4

3

1

0

4

3

2

r, I \ I \ I \ I \

Lax

Lxx (i)

f\ I \ I \

270

Figure 9.

1

3

2

1

0

2

1

Dz

0 y/a

(a)

1

0

3

l' 2 .

I • l' I

i'v\J . . 1

0

2

(c)

1

2

(d)

2

, .. ·" ..-\ ~~ d i r\ r. " I '\ r. I A • II • ~I I' v.iV V\ \.., jV V ~\I \; \4 \ . \, '

0

2

Lax (h)

1

0

3

' 1\ .. .. .. .. . . . I . . 2 . I

1

0 90 180 270 e (degrees)

Figure 10

Appendix D. Seismic Response of a Cylindrical Shell Embedded in a

Layered Viscoelastic Half-Space. I : Formulation (Luco/Barros).

SEISMIC RESPONSE OF A CYLINDRICAL SHELL

EMBEDDED IN A LAYERED VISCOELASTIC HALF-SPACE.

I: FORMULATION

J. E. Loco and F. C. P. de Barros

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California 92093-0411.

ABSTRACT

A method to obtain the three-dimensional harmonic response of a infinitely long cylin­

drical shell of circular cross-section embedded in a layered viscoelastic half-space and subjected

to harmonic plane waves impinging at an oblique angle with respect to the axis of the shell

is presented. The procedure combines an indirect integral representation for the field in the

exterior half-space with a model of the pipeline or tunnel based on Donnell shell theory. The

integral representation for the soil is based on the use of moving Green's functions for the

layered viscoelastic half-space. The accuracy of the formulation is tested by comparison of re­

sults obtained by use of different discretizations. Extensive comparisons with previous two- and three-dimensional results for the case of a shell embedded in a uniform half-space and some new

numerical results for a shell embedded in a multilayered half-space are presented in a companion

paper.

INTRODUCTION

In this paper we consider the three-dimensional response of an infinitely long cylindrical

shell of circular cross-section embedded in a horizontally layered half-space (Fig. 1 a, b). The

shell representing a pipeline or a tunnel and the soil are excited by plane waves impinging at an

oblique angle with respect to the axis of the shell (Fig. 1 b, c). Although the geometry of this

model is two-dimensional, the response is fully three-dimensional.

Most of the previous work on the seismic response of pipelines or tunnels has been

concentrated on two-dimensional models. In particular, the anti-plane shear case of a cylindrical

shell of circular cross-section buried in a uniform elastic half-space and subjected to plane SH­

waves normal to the axis of the shell has been considered by Lee and Trifunac (1979) and

Balendra et al (1984). The solution is obtained by expanding the fields in both the exterior

and interior region in series of cylindrical wave-functions. The coefficients of the terms in

these series are obtained by truncating an infinite system of equations in the infinite number

of unknown coefficients. The plane-strain case for P-, SV- and Rayleigh-wave excitation has

been considered by El-Akily and Datta (1980, 1981), Datta et al (1983, 1984), Wong et al

(1985) and Chin et al (1987). El-Akily and Datta (1980, 1981) considered a cylindrical shell

of circular cross-section buried in a uniform half-space. The external field was represented by

a series expansion in cylindrical wave-functions while the shell was modelled using Fliigge's

bending theory. Matched asymptotic expansions and a successive reflection technique were used

to determine the coefficients in the expansion. Datta et al (1983,1984) considered a cylindrical

pipe of circular cross-section lying in a concentric cylindrical region of soft soil buried in a

uniform half-space. The fields within each of the three regions were expanded in series of

wave-functions. The unknown coefficients in the expansion were obtained by truncating an

infinite set of linear equations in the infinite number of unknown coefficients. Wong et al (1985)

considered the two-dimensional response of a lined tunnel of non-circular cross section buried in

a uniform elastic half-space by use of a technique involving wave-function expansion in the half­

space combined with a finite element representation of the tunnel and its immediate cylindrical

vicinity. The same approach was used by Chin et al (1987) to study the response of a pipe of

circular cross-section buried in a back-filled trench embedded in a uniform elastic half-space.

Three-dimensional models of infinitely long pipelines similar to those considered in this

study have been considered previously by Datta and Shah (1982) for a full space and by Wong et

al (1986a, 1986b) and Liu et al (1988) for a half-space. In particular, Wong et al (1986a, 1986b)

have considered the response of a cylindrical pipeline of circular cross section buried in a uniform

elastic half-space and subjected to obliquely incident plane P-, SV -, SH- and Rayleigh-waves.

The solution was obtained by expansion into wave-functions in both the half-space and the shell

and by truncation of the resulting infinite system of equations. Liu et al (1988) have obtained the

three-dimensional response of an infinitely long pipeline buried in a backfilled trench embedded

in a uniform half-space when subjected to obliquely incident P- and SV-waves. These authors

have used a hybrid approach in which an internal region including the pipeline is modelled by

finite elements while the exterior region is modelled by use of a boundary integral representation

in terms of Green's functions for a uniform half-space.

In the present paper, a method of solution which combines an indirect integral repre­

sentation for the exterior soil with a simplified shell theory (Donnell, 1933) for the internal

pipeline or tunnel is presented. The integral representation for the exterior domain is based on

the moving Green's functions for a layered viscoelastic half-space obtained by Barros and Luco

(1992, 1993). In this way, the physical three-dimensional problem is reduced to an essentially

two-dimensional mathematical problem. The integral representation for the exterior domain, the

solution for the shell and a discussion of the accuracy of the formulation is presented next. An

extensive bibliography on the seismic response of pipelines and tunnels is included at the end

of the paper. Detailed comparisons with previous two- and three-dimensional solutions for a

shell in a uniform half-space and new numerical results describing the response of cylindrical

shells embedded in layered media and subjected to obliquely incident P-, SV- and SH-waves are

presented in a companion paper (Barros and Luco, 1993).

FORMULATION

' The geometry of the model is illustrated in Fig. 1. The external region representing the

soil consists of (N- 1) horizontal viscoelastic layers overlying a viscoelastic half-space. The

infinitely long shell of circular cross-section is parallel to the free surface of the half-space and

is located at a depth H. Perfect bonding is assumed to exist between layers and between the

shell and the exterior medium. Each of the media in the exterior half-space is characterized,

for harmonic vibrations, by complex P- and S-wave velocities O:j = Oj(1 + 2i~a;) 112 and /3j =

,Bj(1 + 2i~.a;) 112 , and by the density Pi (j = 1, N). The terms Oj Pi represent (approximately)

the real parts of the P- and S-wave velocities, and ~a; and ~fJ; represent the small hysteretic

damping ratios for P- and S-waves, respectively. The shell is characterized by the centerline

radius a, tickness h, Young's modulus E0 , Poisson's ratio v0 and density Po·

In what follows, the excitation and the response will have harmonic time dependence of

the type eiwt where w is the frequency. For simplicity, the factor eiwt will be dropped from all

expressions.

Free-Field Ground Motion.

As a first step in the formulation, it is necessary to determine the ground motion and

the stress components for free-field conditions, i.e., in the absence of the shell. The seismic

excitation is represented by homogeneous plane P-, SV- or SH-waves, such that the normal to

the wave front in the underlying half-space forms an angle Ov with the vertical axis (Ov = 0 for

vertical incidence). The projection of the normal to the wavefront on the horizontal plane forms

the angle Ott with the axis of the shell (x-axis).

To calculate the free-field ground motion it is convenient to consider the coordinate

system x', y', z' (z' = z) shown in Figs. la and lc. Referred to this coordinate system, the

incident motion within the underlying exterior half-space is represented by

z> ZN (1)

where A is the amplitude of the incident displacement, k' = (w/f3N) sinOv for S-waves and

k' = (w / O:N) sin Ov for P-waves. The term v'N is defined by

1 ( i (~) cos Ov for S-wave excitation

"N = i (:;, ) cos 0, for P-wave excitation {2)

In Eq. {1), ZN is the depth of the last interface with the underlying half-space and {U'} is the

vector (sin Ov, 0,- cos Ov )T for P-wave excitation

for SV-wave excitation (3)

(O,l,O)T for SH-wave excitation

The total free-field ground motion satisfying all the continuity, free-surface and radiation

conditions for the layered geometry shown in Fig. lc can be calculated by the approach described

by Luco and Wong (1987). The resulting free-field displacement and stress fields in the x'y' z' -

coordinate system are denoted here by

{u'ur} = {U'ur(z')}e-ik'x'

[a' ur] = [E' ur(z') ]e-ik' x'

(4)

(5)

where the elements of U'lff(z') and E'ur(z') are independent of y' and depend only on z' = z.

To impose boundary conditions at the interface between the layered half-space and the

shell it is necessary to introduce the rotation of coordinates

leading to the free-field displacement and stress fields in the xyz-coordinate system:

{ uur} = {Utff(Xo)}e-ikx

[<1tff) = [Etff(Xo)]e-ikx

(6)

(7)

(8)

where x0 = (O,y,z), k = (w/f3N)sin8vcos8h for S-wave excitation and k -

(w / O:N) sin Ov cos ()h for P-wave excitation. In Eqs. (7) and (8),

{Uur(xo)} =[C] {U'ur(z)}e-ik'ysinBh

[Eur(xo)] = [C] [E'ur(z)] [C]T e-ik'ysinBh

in which the y-dependence enters only in the last exponential factor.

II

(9)

(10)

The free-field traction vector { tur(x)} on the area that will be in contact with the shell

can be written in the form

(11)

where II

{11ff(xo)} = [Elff(xo)]{v(xo)} (12)

in which {v} = (0, vy, vz)T is the unit normal to the shell's boundary at x0 pointing into the

shell. It is noted that the variable x appears only in the exponential factors exp( -ikx) affecting II

{Utff}, [atff] and {ttff}.

Finally, it is convenient to recall for future reference that the incident ground motion in

the underlying half-space referred to the (x, y, z) coordinate system is represented by the plane

wave

{ U1N }inc= A{U} exp { -i: [x sinOv cos~+ ysinOv sin Oh] + v}vz} (13)

where A is the amplitude of the incident displacement and the vector { U} is given by

for SH-wave excitation,

{U} = (cos~ cos Ov, sin~ cos Ov, sin Ov )T for SV-wave excitation, and (14)

(cos ()h sin Ov, sin Ott sin Ov, -cos Ov )T for P-wave excitation,

in which Ov and ()h are the vertical and horizontal angles of incidence. The velocity v appearing in

Eq. (13) corresponds to the (complex) velocity O:N in the underlying half-space for P-excitation

and to {3 N for SV- and SH-excitation.

It should be noted that if Ott = 90° or if Bv = 0°, then, the incident displacement becomes

independent of the coordinate x and the problem becomes two-dimensional.

Contact Problem.

In the presence of the shell, the total displacement vector { u1 (X)} and the total traction v

vector { t1 (x)} in the exterior region are written in the form

v v v {t1(X)} = {tur(X)} + {t1s(X)}

v

(15)

(16)

where {u1s} and {t1s(X)} are the scattered displacement and traction vectors, respectively. The

exterior field satisfies the conditions of vanishing tractions on the free surface (z = 0) and

the continuity conditions at layer interfaces. The exterior scattered field must also satisfy the

radiation conditions at infinity. At the interface S between the shell and the exterior medium,

the continuity conditions

xES

v

(17)

(18)

apply, in which {u2(x)} and {t2(X)} are the displacement and traction vectors for the shell. In v

here we assume that { u2 (X)} and { t2 (X)} can be related in the form

(19)

where [G22(x,x')) is the 3 x 3 matrix of Green's functions for the shell.

To solve the boundary-value problem, the exterior scattered field is represented as result­

ing from the action of a distribution of concentrated loads moving in the direction of the x-axis

with velocity c = w / k. These moving loads act in the layered exterior half-space (without the

shell) on the surface 81 (Fig. 1d) located within the region to be occupied by the shell. The

scattered displacement field { Uts (X)} is, then, written in the form

(20)

where

(21)

in which [G11 (x0 , Yo)] is the 3 x 3 matrix of moving Green's functions (Barros and Luco, 1992,

1993). The first, second and third columns of the matrix G correspond to the displacement

vector at Xo = (0, y, z) for a unit point load acting in the x, y and z-directions, respectively,

and moving with velocity c = w / k along a line parallel to the x-axis passing through the point

Yo= (O,yt,zt) on St. The 3 x 1 vectors {F} represent the unknown amplitudes of the j-th II

moving loads. The corresponding traction vector {t15(X)} for the scattered field at the interface

S with the shell can be written in the form

(22)

where

{ fts(xo)} = 1 [Hu (xo, :lfo)]{F(yo)} dl(:lfo) ' L1 (23)

in which a;x O'~x CT~x

X O'~y z

[H(Xo,Yol] = [~ 0 0 !] O'xy O'xy

Vy Vz X y z 0 0

O'xz O'xz CTxz Vy Vz

X O'~y z 0 0 0 Vy O'yy O'yy

(24)

X O'yz O'~z z

CTyz X

O'zz ~O'~z z CT.zz

In Eq. (24) (vy(xo), v.z(xo)) are the direction cosines of the normal to the surfaceS of the shell,

and a;x(xo,Yo). a~x(xo,:lfo), ••• , are the stresses at Xo = (O,y,z) induced by the moving unit

point loads acting in the x, y and z-directions.

v

The continuity conditions (17), (18) and Eqs. (7), (11), (20) and (22) indicate that { u. (X)} J

and { tj (X)} can be written in the fonn

{uj(X)} = {Uj(Xo)} e-ikx (j = 1, 2) (25)

{~(X)} = { ~(xo)} e-ikx (i = 1, 2) (26)

The relation (19) is then given by

(27)

where

[a22(xo,x1)] = L: [c22(x,:?)] e-ik(x-x') dx . (28)

By use of Eqs. (15), (16), (21), (23), (25), (26) and (27) the displacement fields {U1 (x0 )}

and{U2(x0 )} at the interfaceS between the shell and the soil can be written in the fonn

{Ut (xo)} = {Uur(xo)} + f [Gn (xo, Yo)]{F(yo)} dh (Yo) (29) jL1

{U2(xo)} =- [ [a22(xo,x1)] { i111(X:,)} dl(x1,)

- f f [ G22 (xo, x1,)] [ Hn (x1,, :lfo) J { F(:lfo)} dh (:lfo)dl(x1,) (30) jLjL1

At this point, we use a weighted version of the displacement continuity condition and

require that

[ [Hn(xo,Y1)]T ({UI(xo)}- {U2(xo)}) dl(xo) = 0

Substitution from Eqs. (29), (30) into Eq. (31) leads to

(31)

(32)

where

[s<V..Y.l] = ( L [Hn(X.,y1f [Gn(X.,!7.)) dl(X.) +

L L [ Hu (X., V.f [ G,,(X., x1) J [ Hn (x1, !7.) J ell (X0 )dl(x1)) (33)

and

{D(y1)} = -{ L [Hn(X.,y1Jf {Uur(X.)} dl(X.) +

L L [ H,., (X., Y1l r [ G, (X., x1) J { i\ff(x1J} dl(X.)dl(x1) } . (34)

Eq. (32) represents an integral equation for the unknown distribution of forces {F(Yc,)}.

The kernel [B] and the right-hand-side {D} of Eq. (32) depend on the moving Green's

function matrix [ G22 (.io, x1) J for the shell. These Green's functions are derived in the next

section.

MOVING GREEN'S FUNCTIONS FOR A CYLINDRICAL SHELL

The equations of motion for harmonic vibrations of the cylindrical shell are given by

(35)

where {u~} = (u, v, w)T represents the midsurface displacement vector in the local coordinates v

shown in Fig. 2, { t2e'} is the effective traction vector also referred to the local coordinates, Po is

the density of the shell, h the tickness, K 0 = Eoh/(1- v;) in which Eo is the Young's modulus

and V0 is the Poisson's ratio. The operator matrix [L] has for elements (Donnell, 1933)

(36)

v v The effective traction vector { t2e'} is related to the actual traction vector { t2'} on the

outside surface (r =To= a+ h/2) of the shell through the relation

To solve Eq. (35) we make explicit the exponential x-dependence and write

{u~(x, 0)} = {OHO)} e-ikx

{t2'(x, 0)} = { n'(O)} e-ikx

(37)

(38a)

(38b)

Applying the Finite Fourier transform

1 r27r . Wn =

211" Jo w(8)e-me dO (39a)

(39b) n=-oo

to Eq. (35) leads to

(40)

where

[A,.]=- ( 1 ~ vJ) [1,]- ( J;:/ p.) 2

[I] (41)

in which [I] is the 3 x 3 unit matrix. The elements of the matrix [Ln] are given by

L11 =- [(ka)2 + ( 1 ~ llo) n2]

L22 =- [ ( 1 ~ Vo) (ka)2 + n2

]

(42)

- - (1 + ll0 ) £12 = £21 =

2 (ka) n

L13 = -£31 = -i llo (ka)

The matrix [Bn] is given by

0:] 1+ 2: 0

[Bn] = [ ~ -i (~) (ka) i (~) n

(43)

Solving Eq. (40) and inverting the Finite Fourier transform leads to

{VHO)} =a t• (a;,(o, 8')] { f.'(O')} dO' (44).

where

(45)

To impose the continuity conditions at the interface between the shell and the external

medium we must consider not the mid surface displacements { V~ ( 8)} but the displacements

{U2(8)} on the outside boundary r =To= a+ h/2. These displacements are related by

[

1 0 _l!~] {t4} = 0 1+ ,';. -~i. {ii2}

0 0 1

(46)

The resulting expression for the outside displacement vector in terms of the tractions is

where 00

[G~2(8,8')] = 2~ ~o (x) L [Dn] [An]-1 [Bn] ein(e-e'> n=-oo

in which i(k;)(~)l

·n (h) 0

1 + 2: -z2 a 0 1

Some of the stresses of interest are given by

([l~ ~!L

Il { U, } K a a9 1 a

2D"x9/~:- Zlo) = ho a. a9 a

a a9 ax

-z [:

~a ( a2 ~ a2 ) -a ae p+~OP ])

1 a (~~ + vog:2) {u~(x,B)} -o:r8iJ 18 2 82

-a8x aax89

(47)

(48)

(49)

. (50)

where z = r- a [z = h/2 on the outside wall ( r = r0 ) and z = -h/2 on the inside wall

(r = r,) ]. By writting

{ Uz } { Ex } - ~ ~b u9 = LJB e

2u:J:9/(1- Zlo) 2Ex9/(1- Vo) it is found that

where

in which

[

-i(ka)

[En] = -iv~(ka) 11

0

] [0 1 + (~) 0 0 . 0 -i(ka)

iv0 n

in

-ika

(ka)2 + v0 n

2]

V0 (ka) 2 + n 2

-2(ka)n zn

(51)

(52)

(54)

The relation between displacements and tractions given by Eq. (47) involves the dis­

placement and traction vectors referred to the local (cylindrical) coordinates. To connect the r

displacements and tractions on the shell with those of the soil it is necessary to refer these vee-II

tors to the global cartesian coordinates (x, y, z). Denoting by {U2} and {12} the vectors {UH II

and {12'} when referred to the global coordinate system we have

(55a)

and

(55b)

where

[

1 0 0 ] [Co] = 0 -sin 0 cos 0

0 - cos 0 - sin 0

(56)

The relation between the displacement and the traction can now be written in the form

{27f {U2(0)} =a Jo (622(8, O')] { n(O')} dO' (57)

where

(58)

NUMERICAL APPROACH

The integral equation (32) is discretized by replacing the unknown distributed forces

{F(y0 )} by a set of unknown concentrated forces {Fsj} (j = 1, N 5 ) acting at N 5 source points

Yo= Ysi on Lt and by imposing Eq. (32) at the same set of discrete points Y'o = Ysi (j = 1, N 5).

In addition, the integrals over L appearing in Eqs. (33) and (34) are discretized by use of

numerical integration formulae involving a set of N0 observation points x0 = X0 ; (i = 1, N0 ) on

L. The resulting set of linear algebric equations can be written in the form

[B] {P} = {b} (59)

where the 3 x 3 blocks of the 3N5 x 3N5 matrix [B] correspond to [B(y5;,Ysj)], {P}T =

( { F(yst}T, { F(Ys2}T, ... ) and { D}T = ( { D(Yst }T, { D(Ys2}T, ... ).

To reduce the possibility of ill-conditioning it is useful to write { P} in the form of a

finite Fourier expansion with respect to the angular coodinates Oj =arctan [(H- Zsj)/Xsj] of the

source points Ysi (j = 1, N 5). In this case,

{ P} = [M] {Fo} ' (60)

where the ith row of 3 x 3 blocks of [.M] is given by ([I], cos Oi[I], sin Oi[I], cos 20i[I],

sin 20i [I], ... , cos ~s Oi [I]) in which [I] is the 3 x 3 identity matrix, Ns is assumed to be even

and 81 = 0. The coefficients {Fo} in the expansion are obtained from Eq. (59) in the form

(61)

Once the forces {P} have been calculated by use of Eqs. (60) and (61), the displacement fields

in the external medium and in the shell can be calculated from Eqs. (29) and (30).

CONVERGENCE OF THE NUMERICAL APPROACH

The numerical results obtained by the procedure described in the previous section depend

in principle on the location and number of source points (N 5 ) and on the number of observation

points (No) used. The first step is to test the convergence of this approach. For this purpose

we consider a concrete circular shell (p0 = 2,240kg/m3 , Eo= 1.6 x 1010 N/m2, Z10 = 0.2,

h = 0.1ri = 0.0909r0 ) buried to a depth H = 5.0ri = 4.545r0 in a uniform half-space

(p1 = 2,664kgjm3 , E1 = 7.567 x 109 N/m2, Z10 = 0.333, ~a= ~{3 = 0.001). The half-space

is subjected to non-vertically incident (8v = 30°) P- and SV -waves propagating in the direction

of the shell <8tt = 0°). The frequency of the excitation is such that ., = wrolrr'Pt = o.1o5.

Numerical results for the normalized displacement components at a few points on the

ground surface (z = 0) and at a few points on the external wall of the shell (r = ro) are presented

in Table 1 for different numbers of sources and observers (N5 , N0 ). Also shown are some values

for the normalized hoop stress on the centerline (r = a) of the shell. In all cases, the sources

are equally spaced on a circle of radius a'= r0 - 3(27rr0 /N0 ) (No > 20). Thus, as the number

of observation points increases, the sources move closer to the actual boundary r = r0 •

The displacement amplitudes Ui = lui/ AI are normalized by the amplitude A of the

incident displacement vector on the ground surface. The normalized hoop stress is given by

'Eee = lcree(a)/wptPtAI . The numerical results presented for source/observer combinations

(N s. N 0 ) of (20, 40) and ( 40, 80) show that the procedure is very stable as the number of source

and observation points increases. It appears that 20 source points and 40 observation points are

sufficient for most applications.

CONCLUSIONS

A procedure has been presented to calculate the three-dimensional response of a cylin­

drical shell of infinite length embedded in a layered viscoelastic half-space and subjected to

obliquely incident waves. The procedure combines an indirect integral representation for the

field in the exterior half-space with a simplified Donnell shell theory for the pipeline or tunnel.

The convergence of the procedure has been successfully tested. Extensive critical comparisons

with previous results for the particular case of a shell buried in a uniform half-space and new

results for shells embedded in a layered media are presented in a companion paper(Barros and

Luco, 1993).

ACKNOWLEDGMENTS

The work conducted here was supported by a Grant from California Universities for

Research in Earthquake Engineering (CUREe) as part of a CUREe-Kajima Project.

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coelastic Half-Space, Report, Dept. of Appl. Mech. & Engng. Sci., University of California,

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to a Moving Point Load, (to be submitted for publication).

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Plane Waves Reflected in a Viscoelastic Half-Space, (submitted for publication).

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Table 1. Normalized Displacement Components for Non-Vertically Incident (Ov = 30°) P- and

SV-Waves Propagating in the Direction of the Axis of a Shell (Oh = 0°) for Different Numbers

of Source (N 5 ) and Observation (N0 ) Points.

P-Wave SV-Wave (No,Ns) (No,Ns)

Variable Location (20,40) (40, 80) (20,40) (40, 80)

Ux y/To = 0 1.0347 1.0347 3.4015 3.4015 (z = 0) y/To = ±1 1.0303 1.0303 3.4013 3.4013

y/To = ±3 1.0036 1.0036 3.4007 3.4007

Uy y/To = ±1 0.0520 0.0520 0.0087 0.0087 (z = 0) Y/To = ±3 1.0216 1.0215 0.0167 0.0167

Uz y/To = 0 2.0071 2.0070 0.0846 0.0846 (z = 0) y/To = ±1 l.g884 l.g884 0.0874 0.0873

y/To = ±3 1.8881 1.8881 0.10g3 0.10g3

Ux () = oo 0.3565 0.3565 2.0488 2.0487 (T =To) () = 45° 0.6606 0.6606 2.4g5g 2.4g5g

()=goo 0.7847 0.7847 2.66gg 2.66gg

Uo () = oo 1.2737 1.2737 0.8ggo 0.8ggo (T =To) () = 45° 1.1403 1.1404 0.5430 0.5430

Ur () = oo 0.023g 0.0238 0.2444 0.2446 (T =To) () = 45° 1.2153 1.2153 0.7654 0.7656

()=goo 1.8732 1.8733 0.8865 0.8865

Eoe () = oo 10.141 10.140 2.g166 2.g204 (T =a) () = 45° 5.61gg 5.61g8 3.428g 3.4314

()=goo 1.6686 1.6684 5.4410 5.4436 () = 270° 2.og2g 2.og21 5.1771 5.1800

FIGURE CAPTIONS

Figure 1. Schematic Representation of Cylindrical Shell Embedded in a Layered Half-Space.

(a) Top view showing horizontal angle of incidence, (b) Cross section, (c) Free-field model

showing vertical angle of incidence and (d) Location of sources and observers.

Figure 2. Schematic Representation of Shell Showing Local Coordinates.

(a) (c) X

x' 1

2

3

"'y'

z'

(b) (d)

y y

Sources Observers

z z

Figure 1

Z=H s

1

z

w

y

u

X

Figure 2

Appendix E. Seismic Response of a Cylindrical Shell Embedded in a

Layered Viscoelastic Half-Space. II: Validation and Numerical

Results (Barros/Luco ).

SEISMIC RESPONSE OF A CYLINDRICAL SHELL

EMBEDDED IN A LAYERED VISCOELASTIC HALF -SPACE.

II: VALIDATION AND NUMERICAL RESULTS

F. C. P. de Barros and J. E. Luco

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California 92093-0411.

ABSTRACT

A procedure to calculate the three-dimensional harmonic response of a infinitely long

cylindrical shell of circular cross-section embedded in a layered viscoelastic half-space and

subjected to harmonic plane waves impinging at an oblique angle with respect to the axis of the

shell is validated by extensive comparisons with previous two- and three-dimensional results for

the particular case of a shell embedded in a uniform half-space. New numerical results describing

the motion and stresses within a shell embedded in a multilayered half-space and subjected to

obliquely incident P, SV and SH-waves with different horizontal angles of incidence are presented

and discussed.

INTRODUCTION

In a companion paper (Luco and Barros, 1993) the authors have presented a procedure to

calculate the three-dimensional seismic response of an infinitely long cylindrical shell of circular

cross-section embedded in a horizontally layered viscoelastic half-space (Fig. 1 a, b). The half­

space is subjected to P-, SV- and SH-waves impinging at an oblique angle with respect to the

axis of the shell (Fig. 1 b, c). The procedure relies on an indirect integral representation to model

the external region and on a simplified Donnell shell theory to represent the internal pipeline

or tunnel. The integral representation is based on the moving Green's functions for a layered

viscoelastic half-space obtained by Barros and Luco (1992, 1993).

In this paper, the procedure proposed by the authors is verified by extensive comparisons

with previous two- and three-dimensional solutions for the particular case of a shell embedded in

a uniform half-space. These comparisons not only serve as validation for the present approach

but also offer an opportunity for a critical appraisal of numerical results presented by several

authors over the last 15 years. In particular, detailed comparisons with displacements and stresses

calculated by Lee and Trifunac (1979), Datta et al (1983, 1984), Balendra et al (1984), Wong et

al (1986) and Liu et al (1988, 1991) for two-dimensional cases and by Wong et al (1986) and

Liu et al (1988, 1991) for three-dimensional cases are presented.

New numerical results describing the two- and three-dimensional response of cylindrical

shells embedded at different depths in multilayered media are also presented and discussed.

These results include displacements on the external boundary of the shell and stresses on the

midsurface of the shell for excitation in the form of obliquely incident P-, SV- and SH-waves.

VALIDATION AND CRITICAL COMPARISONS

In the comparisons that follow the displacements are normalized by the amplitude A of

the incident displacement field. All of the stress components with the exception of cr :rB = cre:r

are normalized by wp~A = (wr0/~)jl(A/ro) where p, ~ and jl = ~2p are the density, shear

wave velocity and shear modulus of reference and r 0 is a length of reference corresponding to

the external radius of the circular shell. The shear stress cre:r is normalized by (1- vo)wp~A/2

where v0 is the Poisson's ratio of the shell. The reference quantities p, ~ and fl are taken to

correspond to those of the underlying half-space (which correspond to PI· ~I and fli in the case

of a uniform half-space). In judging the comparisons it must be kept in mind that the present

results include a small amount of attenuation in the soil eo = e/3 = 0.001 and no attenuation

in the shell <eoo = e{3o = 0) while the results by other authors typically do not include any

attenuation.

Finally, the present results have been calculated using Ns = 20 source points equally

spaced on a circle of radius rs = r0 - 61rro/No where No = 40 is the number of observation

points equally spaced along the external shell boundary (r = r0 ).

Two-Dimensional Anti-Plane Shear Cases.

A first comparison is made with results presented by Lee and Trifunac (1979) for a

circular cylindrical shell of external radius ro. internal radius ri = 0.9ro. thickness h = O.lro

and embedment depth H = 1.5r0 subjected to a vertically incident SH-wave with particle motion

along the axis of the pipeline (Ott = 90°, Bv = 0°). The shell is characterized by shear modulus

jl0 , shear wave velocity ~o and density Po (flo = ~';p0) and the surrounding uniform half-space

is characterized by flt. ~1 and PI· Lee and Trifunac (1979) present results for flo/ilt = 3 but

do not state the value for Pol Pl or ~o/ ~1· In here we assume (Lee, personal communication)

that Pol PI = 3 and, consequently, ~o/ ~~ = 1. The present results were calculated by assuming

a small amount of attenuation e/31 = 0.001 in the half-space and no attenuation in the shell

~f3o = 0. The results of Lee and Trifunac correspond to purely elastic media. Finally, the

comparisons were made for the dimensionless frequency 1J = WT0 /1rfh = 0.5.

Figs. 2a and 2b show the comparisons for the amplitudes of the normalized displacements

u% = lux/AI on the surface of the half-space (z = O) and on the external boundary of the

shell (T = T0 ). These displacements are normalized by the amplitude A of the incident SH­

wave at z = 0. Fig. 2c shows the comparisons for the amplitude of the normalized stress

Erx = lur%/WPt.BtAI on the external wall of the shell (T = T0 ). The results in Figs. 2a and

2b show excellent agreement between the present results with the results of Lee and Trifunac

(1979) for the surface displacements (z = 0) and for the displacements on the external pipe

wall (T = T0 ). The results of Lee and Trifunac (1979) for the normalized contact stress Er% on

T =To differ in shape and amplitude from the present results (Fig. 2c). The peak value for IErxl

obtained by Lee and Trifunac is about 6 times larger than our result.

As a second comparison we consider the results presented by Balendra et al (1984) for

a concrete circular shell (T0 =3m, Ti = 2.7m, h = 0.3m, flo= 8.4GPa, .Bo = 1,870m/sec,

Po = 2,410kg/m3 , ll0 = 0.2) buried to a depth H = 2.5T0 = 7.5m in a uniform half­

space (jl1 = O.lllGPa, ,81 =260m/sec, p1 = 1,640kgfm3). The medium is subjected

to a non-vertically incident (Bv = 30°) SH-wave propagating normal to the axis of the shell

with a frequency of 10.61 Hz. In this case the dimensionless frequency 1J takes the value

1J = WT0 /7r.Bt = 0.245. Comparisons for the amplitudes of the normalized displacement Ux =

lux(Ti)/AI, and normalized shear stresses Erx = lurx(To)/wpt.Btal and Eex = l2uex(Ti)/[(1-

vo)WPt.Bta]l are presented in Figs. 3a, 3b and 3c, respectively. The figures show excellent

agreement between the present results and those obtained by Balendra et al (1984). The small

differences for Eex can be attributed to the fact that the present calculation is based on a thin

shell theory which in the 2-D case (ka = 0) leads to a shear stress O"fJx which does not vary

across the tickness of the shell.

As a final comparison for the two-dimensional anti-plane shear case we consider the

results presented by Liu et al (1991) for a concrete circular shell (p0 = 2. 24 x 103 kg I m 3 ,

Eo = 1.6 x 1010 Nlm2, v0 = 0.2, h = 0.1ri = 0.0909r0 ) embedded to a depth H = 5n =

4.545ro in a uniform half-space <Pt = 2.664 x 103 kglm3 , Et = 7.567 x 109 Nlm2, v1 =

0.333) and subjected to a vertically incident SH-wave (Fig. 8d in paper in reference). The

corresponding comparison for the amplitude of the normalized displacement Ux = lux (ro) I AI

for a dimensionless frequency TJ = wr0l7r~t = 0.105 is shown in Fig. 4. The agreement

betweeen the two sets of results is excellent.

Two-Dimensional Plane-Strain Cases.

As a first test for the plane strain case we consider the results presented by Datta et al

(1983) [see also Datta et al (1984)] for a concrete circular shell ( p0 = 2.24 x 103 kglm3 , Eo =

1.6 x 1010 Nlm2, V0 = 0.2, h = 0.1ri = 0.0909r0 ) buried to a depth H = 8.33ri = 7.573r0

in a uniform half-space (p1 = 2.665 x 103 kglm3 , Et = 6.9 x 108 Nlm2, Vt = 0.45). The

medium is subjected to vertically incident P- and SV -waves with a dimensionless frequency

'f/ = wrol1r~t = 0.132.

Comparisons for the normalized amplitudes of the radial displacements Ur = lur(ro)IAI

on the external wall of the pipe are presented in Fig. 5a and 5b for vertically incident P- and

SV-waves, respectively. There is a excellent agreement between the present results and those

presented by Datta et al (1983). To make these comparisons it was necessary to renormalize

the results presented by Datta et al (1983) for P- and SV-waves by multiplying these results by

factors of 1.362 and 2.0, respectively. These factors correspond to the ratio of the peak free-field

radial displacements at r = r0 to the amplitude A of the incident waves. These peaks occur at

() = 90° and()= 0° for P- and SV-waves, respectively.

The corresponding comparisons for the normalized hoop stresses Eee(r0 ) for P- and SV­

waves are shown in Figs. 5c and 5d, respectively. The agreement between the two sets of results

is very good. The results of Datta et al (1983) shown in Figs. 5c and 5d were renormalized by

multiplying by factors of 5.434 and 0.629 which correspond to the peak values of the normalized

hoop stresses Eee(T0 ) in the free-field. Also shown in Figs. 5c and 5d are the normalized hoop

stresses E88 (a) on the centerline of the shell (segmented lines). The significant differences

between the stresses at T =To and T =a indicate a significant amount of bending of the shell.

As a second test for the plane strain case we consider the results presented by Wong et

al (1986) for a concrete circular shell (Po= 2,240kgfm3 , Eo= 16 x 109 N/m2 , Zlo = 0.2,

h = 0.1Ti = 0.0909T0 ) embedded to a depth H = 2.0Ti = 1.818T0 in a uniform half-space

<Pt = 2, 665 kgjm3, Et = 0.69 x 109 N jm2, Zit = 0.45). Values for the hoop stress Eee(To)

and for the longitudinal stress Exx(T0 ) at T = To were presented for nonvertically incident

(8v = 10°) P- and SV-waves for 17 = WT0/7r~t = 0.132.

Comparisons for the normalized hoop Eee(To) and longitudinall:xx(To) stresses at T =To

for both P- and SV -waves are shown in Fig. 6. The agreement between the present results and

those of Wong et al (1986) for Eee is very good. Some small differences for L:xx(To) (Figs. 6c

and 6d) can be attributed to the present use of a simplified shell theory in which l:rr is considered

to be much smaller than Eee and Exx· The results of Wong et al (1986) shown in Fig. 6a, 6b,

6c and 6d were renormalized by multiplying by factors of 2.026, 1.856, 1.840 and 0.5764,

respectively. These factors correspond to the peak values of the corresponding normalized

stresses in the free-field.

Also shown in Figs. 6a, b, c, d are the normalized stresses (segmented lines) calculated

by the present approach on the centerline T = a of the shell. It is apparent, particularly for

SV -excitation, that the bending effects are significant.

As a third test for the plane strain case we consider the results presented by Liu et

al (1991) for a concrete circular shell ( p0 = 2.24 x 103 kgjm3 , Eo = 1.6 x 1010 N jm2,

Zl0 = 0.2, h = 0.1Ti = 0.0909T0 ) buried to a depth H = 5.0Ti = 4.545T0 in a uniform half-space

<Pt = 2.664 x 103 kgfm3 , Et = 7.567 x 109 Njm2, Zit = 0.333). The medium is subjected to

vertically incident P- and SV-waves with dimensionless frequency 17 = WTo/1r~t = 0.105 ..

The amplitudes of the nonnalized radial displacements Ur = lur(To)/AI on the external

wall of the pipe are compared in Figs. 7a and 7b for vertically incident P- and SV-waves,

respectively. To compare both sets of results it was necessary to renonnalize the results presented

by Liu et al (1991) by multiplying by the factors 1.667 and 0.424 which correspond to the peak

values of the nonnalized amplitudes of the free-field radial displacements at T = To for P- and

SV-waves, respectively. In this case, the peak values of ur(To, 8) in the free-field occur at

8 = 135° and 8 =goo for P- and SV-wave, respectively.

Comparisons for the nonnalized hoop stresses Eee(a) on the centerline of the shell are

shown in Figs. 7c and 7d. For the purpose of the comparison the results of Liu et al (1991)

for Eee have been renonnalized by multiplying by 2.766 and 1.g74 corresponding to our peak

values for Eee(a) in the free-field. It is appparent from Fig. 7 that excellent agreement exist

between the two sets of results. We note that Liu et al (1991) present two sets of results, one

labeled "analytic" and a second set calculated by a hybrid approach. The comparisons in Fig. 7

refer to the "analytic" results. The agreement with the hybrid results of Liu et al (1991) is also

good but not as close as that shown in Fig. 7.

Three-Dimensional Case.

To test the results in the three-dimensional case of waves impinging on the shell at

angles other than goo we consider first the results presented by Wong et al (1986) for a concrete

circular shell ( p0 = 2,240kgjm3 , Eo= 16 x 109 N/m2, V0 = 0.2, h = 0.1Ti = 0.0909T0 )

embedded to a depth H = 2.0Ti = 1.818T0 in a unifonn half-space (pt = 2, 665 kgjm3 ,

Et = 0.6g x 109 N/m2, Vt = 0.45). Values for the hoop Eee(T0 ) and longitudinal Exx(T0 )

stresses on the pipewall (T = T0 ) were presented for obliquely incident P-, SV- and SH-waves

characterized by 8h = 30° and 8v = 10° for 7} = WT0j1r~1 = 0.132.

Comparisons for the amplitudes of the nonnalized stress Eee (To) for P-, SV- and SH-wave

are presented in Figs. 8a, 8b and 8c, respectively. The corresponding comparison for Exx(To)

are presented in Figs. 8d, 8e and 8f. Clearly, there is good agreement between the two sets of

results. The results of Wong et al (1986) in Figs. 8a, b, c, d, e and f have been renormalized by

factors of 1.941, 1.082, 1.466, 1.834, 1.076 and 0.284, respectively, corresponding to our peak

values for the normalized stresses on r = r 0 in the free-field.

Figs. 8a to 8f also show with segmented lines the normali~ed stresses Eee(a) and Exx(a)

on the centerline r = a of the shell. In is apparent that large differences exist between Eee ( r 0 )

and Eee(a) for SV- and SH-waves indicating the importance of bending of the shell.

As a second test of the results in the three-dimensional case we consider the results

presented by Liu et al (1991) for a concrete circular shell (Po = 2.24 x 103 kgjm3 , Eo = 1.6 x 1010 Njm2 , v0 = 0.2, h = 0.1ri = 0.0909r0 ) buried to a depth H = 5.0ri = 4.545r0 in

a uniform half-space (p1 = 2.664 x 103 kgjm3, E1 = 7.567 x 109 N/m2, v1 = 0.333). The

medium is subjected to non-vertically incident (Ov = 30°) P- and SV -waves impinging in the

direction of the pipeline (Oh = 0°). The dimensionless frequency corresponds to TJ = wr0 /rrflt =

0.105.

The amplitudes of the normalized radial Ur(r0 ) and longitudinal Ux(ro) displacements

at r = r o and of the normalized hoop stress Eee (a) on the centerline r = a are compared in

Fig. 9. The results of Liu et al (1991) for P-waves were renormalized by multiplying by factors

of 1.428, 0.573 and 2.325 which correspond to the peak values of Ur. U:r and Eee on the free­

field. The corresponding results for SV-waves were renormalized by factors 0.937, 2.482 and

1.621, respectively. Significant differences can be seen between the present results and those of

Liu et al (1991). The discrepancies are smaller for the dominant displacement components [Ur

for P-waves, Fig. 9a and U:r for SV-waves, Fig. 9e] than for the secondary displacements [Ux

for P-waves, Fig. 9b and Ur for SV-waves, Fig. 9d]. The discrepancies between the two sets of

hoop stresses Eee(a) for SV-waves (Fig. 9f) are particularly large.

The differences shown in Fig. 9 between our results and those of Liu et al (1991) for

the three-dimensional case are somewhat surprising considering the excellent agreement found

between the two sets of results for two-dimensional cases (Figs. 4 and 7). To confirm our

results we have recalculated our three-dimensional results by use of a hybrid approach which

combines a finite element model for the shell with an indirect boundary formulation for the

external half-space (FE/IBF, Luco and Barros, 1993). In this case the shell is represented by

four concentric layers including 160 triangular elements in each layer. The displacements on the

contact area r = r0 calculated by the hybrid FE!IBF approach coincide almost exactly with the

results obtained by use of the present approach (DT/IBF). The moving Green's functions (Barros

and Luco, 1992, 1993) which have been used in both the DT/IBF and FEJIBF approaches have

been carefully tested. Calculations for the three-dimensional response of a cylindrical canyon

embedded in a uniform half-space and subjected to obliquely incident waves (Luco et al, 1990)

based on the use of the same Green's functions have been validated by subsequent calculations

by Zhang and Chopra (1991). Also, our three-dimensional results appear to agree with the earlier

results of Wong et al (1986) (Fig. 8). These considerations tend to reinforce the validity of our

present three-dimensional results. We note that the comparison with Liu et al (1991) involves

a case in which the three-dimensional effects are much stronger than in the comparison with

Wong et al (1986). The apparent horizontal speed of the excitation along the shell for SV-waves

in the case considered by Liu et al (Oh = 0°, Ov = 30°) is c/ fi1 = 2.0 while the corresponding

apparent speed for the case considered by Wong et al (Oh = 30°, Ov = 10°) is 6.65.

NUMERICAL RESULTS FOR A LAYERED MEDIUM

As an example we consider the response of a cylindrical concrete shell of circular cross

section embedded in a layered viscoelastic half-space. The shell of external radius To = 2.5 m

and thickness h = 0.25m is characterized by Eo = 2.646 x 1010 N/m2, Zlo = 0.167 and

Po = 2, 500 kg jm3• The soil is represented by four viscoelastic layers overlying a viscoelastic

half-space. The properties of the model are listed in Table 1. Two locations of the shell

are considered. In the first and second cases the centerline of the shell is located at depths

H = 11.5 m (H/To = 4.6, first layer) and H = 42 m (H/To = 16.8, third layer), respectively.

Excitations in the form of non-vertically incident P-, SV- and SH-waves (Bv = 30°) impinging

normal (Bh = goo) and along (Bh = 0°) the axis of the shell are considered. All calculations were

performed for a frequency of 10Hz corresponding to a dimensionless frequency TJ = wro/7r~s =

o.og8, The response is normalized by the amplitude A of the incident displacement field at an

outcropping with the same properties as the underlying half-space. The normalized amplitudes

Ux =lux/A!, Uy = luy/AI and Uz = luz/AI on the free-field ground surface (x = y = z = 0)

in absence of the shell are listed in Table 2.

The response in the two-dimensional case of P-, SV- and SH-waves impinging normal to

the axis of the shell (Bh = goo) is illustrated in Fig. 10 for Bv = 30°. The results shown include

the amplitudes of the normalized displacements Ur(r0 ), Uo(r0 ) and Ux(r0 ) on the interface

between the shell and the soil (T = ro) and the amplitudes of the normalized hoop :Eoo(a) and

shear Eox (a) stresses on the centerline r = a.

Results for the three-dimensional case of non-vertically incident (Bv = 30°) P, SV and

SH-waves impinging along the axis of the shell (Bh = 0°) are shown in Figs. 11 and 12. The

results in Fig. 11 include the amplitudes of the normalized longitudinal Ux(r0 ), tangential Uo{r0 )

and radial Ur(To) displacements on the soil-shell interface (T = T0 ). The results in Fig. 12 include

the amplitudes of the normalized logitudinal Exx(a), tangential (hoop) Eoo(a) and shear Eox(a)

stresses on the centerline of the shell (r =a)

CONCLUSIONS

A procedure to calculate the three-dimensional response of a cylindrical shell of infinite

length embedded in a layered viscoelastic half-space and subjected to obliquely incident waves

has been tested by comparison with previous solutions for a shell embedded in a uniform half­

space. The effects of layering have been illustrated by a set of new numerical results for the

two- and three-dimensional response of shells embedded in multilayered media and subjected to

P-, SV- and SH-waves.

Comparisons for the particular two-dimensional case of excitation impinging normal to

the axis of the shell indicate that the present results are consistent with earlier results of Lee

and Trifunac (1979), Balendra et al (1984) and Liu et al (1991) for SH-waves and with those of

. Datta et al (1983, 1984), Wong et al (1986) and Liu et al (1991) for P- and SV-waves. These

comparisons confirm the accuracy of the present approach in the two-dimensional case. In the

three-dimensional case the situation is more controversial. The present results do agree very

closely with three-dimensional results for the stresses within the shell presented by Wong et

al (1986) but do not agree with the three-dimensional results of Liu et al (1991). However,

the present results for the three-dimensional case considered by Liu et al (1991) agree very

closely with a second set of results obtained by the authors by use of an hybrid approach

(Luco and Barros, 1993) in which the shell was represented by a finite element model while

the exterior region was accounted for by means of an indirect boundary formulation based on

moving Greens's functions. The comparisons with the work of Wong et al (1986) and the

confirmatory results obtained by a second method suggest that the present approach is also valid

in the three-dimensional case.

ACKNOWLEDGMENTS

The work conducted here was supported by a Grant from California Universities for

Research in Earthquake Engineering (CUREe) as part of a CUREe-Kajima'Project.

.'

REFERENCES

1. Balendra, T., D.P. Thambiratnam, C. G. Koh and S-L Lee (1984). Dynamic Response of

Twin Circular Tunnels due to Incident SH-Waves, Earthquake Engineering and Structural

Dynamics, 12, 181-201.

2. Barros, F. C. P. de and J. E. Luco (1992). Moving Green's Functions for a Layered Vis­

coelastic Half-Space, Report, Dept. of Appl. Mech. & Engng. Sci., University of California,

San Diego, La Jolla, California, 210 pp. '-

3. Barros, F. C. P. de and J. E. Luco (1993). Response of a Layered Viscoelastic Half-Space

to a Moving Point Load, (to be submitted for publication).

4. Datta, S. K., A. H. Shah and K. C. Wong (1983). Dynamic Amplification of Stresses

and Displacements Induced in a Buried Pipe in a Semi-Infinite Medium, Technical Re­

port CUMER-83-3, Dept. of Mechanical Engineering, University of Colorado, Boulder,

Colorado.

5. Datta, S. K., A. H. Shah and K. C. Wong (1984). Dynamic Stresses and Displacements

in Buried Pipe, Journal of Engineering Mechanics, 110(10), 1451-1466.

6. Lee, V. W .. and M. D. Trifunac (1979). Stresses and Deformations near Circular Un­

derground Tunnels Subjected to Incident SH-Waves, J. of the Engineering Mechanics

Division, ASCE, 105, 643-659.

7. Liu, S. W., K. R. Khair and A. H. Shah (1988). Three Dimensional Dynamics of Pipelines

Buried in Back-Filled Trenches due to Oblique Incidence of Body Waves, Technical

Report CUMER-88-4, Dept. of Mechanical Engineering, University of Colorado, Boulder,

Colorado.

8. Liu, S. W., K. R. Khair and A. H. Shah (1991). Three Dimensional Dynamics of Pipelines

Buried in Backfilled Trenches due to Oblique Incidence of Body Waves, Soil Dynamics

and Earthquake Engineering, 10(4), 182-191 ..

9. Luco, J. E., H. L. Wong and F. C. P. de Barros (1990). Three-Dimensional Response of

a Cylindrical Canyon in a Layered Half-Space, Earthquake Enginnering and Structural

Dynamics, 19(6), 799-817.

10. Luco, J. E. and F. C. P. de Barros (1993). Seismic Response of a Cylindrical Shell Em­

bedded in a Layered Viscoelastic Half-Space. 1: Formulation, (submitted for publication).

11. Luco, J. E. and F. C. P. de Barros (1993). Three-Dimensional Response of a Cylindrical

Tunnel Embedded in a Layered Medium, (to be submitted for publication).

12. Wong, K. C., A. H. Shah and S. K. Datta (1986). Three Dimensional Motion of a Buried

Pipeline II. Numerical Results, Journal of Engineering Mechanics, 112(12), 1338-1345.

13. Zhang, L. and A. K. Chopra (1991). Three-Dimensional Analysis of Spatially Varying

Ground Motion around a Uniform Canyon in a Homogeneous Half-Space, Earthquake

Engineering and Structural Dynamics, 20, 911-926.

Table 1

Properties of the Layered Soil Model

Medium H p a p eo= e/3 m mfsec mfsec kgfm3

1 23 185 827 1,900 0.01

2 11 305 1,304 1,800 0.01

3 16 370 1,428 1,800 0.01

4 28 443 1,478 1,800 0.01

5 00 510 1,634 1,800 0.01

Table 2. Normalized Amplitudes of Free-Field Displacements on the Ground Swface

(x = y = z = 0) for NonVertically Incident P-, SV- and SH-waves (Bv = 30°) for 8h =goo and

lh. = oo.

[Bt. =goo] [Bh = 0°] Wave U:r: Uy u% U:r: Uy u%

p 0 0.473 3.036 0.473 0 3.036

sv 0 3.366 0.073 3.366 0 0.073

SH 2.870 0 0 0 2.870 0

FIGURE CAPTIONS

Figure 1. Schematic Representation of Cylindrical Shell Embedded in a Layered Half-Space.

(a) Top view showing horizontal angle of incidence, (b) Cross section, (c) Free-field model

showing vertical angle of incidence and (d) Local coordinates for shell.

Figure 2. (a) Normalized Longitudinal Displacement Ux on the Ground Surface z = 0, (b)

Normalized Displacement Ux on the Soil-Shell Interface T =To and (c) Normalized Longitudinal

Shear Stress Erx on T =To for a Vertically Incident SH-Wave (Oh =goo. Ov = 0°) Impinging on

a Shell (h = 0.1T0 ) Embedded to a Depth H = 1.5T0 in a Uniform Half-Space. Present results

are shown with solid lines; the results of Lee and Trifunac (1979) are shown with open circles.

In Fig. 3c the results of Lee and Trifunac have been divided by a factor of six (7] = 0.5).

Figure 3. (a) Normalized Longitudinal Displacement Ux at T = T0 , (b) Tangential Stress Erx

at T = T0 and (c) Shear Stress Eex at T = Ti for a Nonvertically Incident SH-Wave (Ov = 30°)

Impinging Normal (Oh = goo) to a Shell (h = 0.1T0 ) Embedded to a Depth H = 2.5T0 in a

Uniform Half-Space. Present results are shown with solid lines, those of Balendra et al (1984)

are shown with open circles (7] = 0.245).

Figure 4. (a) Normalized Longitudinal Displacement Ux at T = T0 , (b) Tangential Stress Erx at

T =To and (c) Shear Stress Eex at T = Ti for a Vertically Incident SH-Wave (Ov = 0°) Impinging

Normal (Oh = goo) to a Shell (h = O.ogogT0 ) Embedded to a Depth H = 4.545T0 in a Uniform

Half-Space. Present results are shown with solid lines, those of Liu et al (1991) are shown with

open circles (17 = 0.105).

Figure 5. (a), (b) Normalized Radial Displacements Ur at T = T0 and (c), (d) Normalized Hoop

Stresses Eee at T = To (solid lines) and T = a (segmented lines) for Vertically Incident (Ov = 0°)

P- and SV-Waves Impinging Normal (Oh =goo) to a Shell (h = O.ogogT0 ) Embedded to a Depth

H = 7.573T0 in a Uniform Half-Space. Present results are shown with solid lines or segmented

lines, results of Datta et al (1983) are shown with open circles (17 = 0.132).

Figure 6. (a), (b) Normalized Hoop Stresses Eoo at T =To (solid lines) and r = a (segmented

lines) and (c), (d) Normalized Longitudinal Stresses 'Exx at T = T0 (solid lines) and T = a

(segmented lines) for Nonvertically Incident (Ov = 10°) P- and SV-Waves Impinging Normal

(~ = goo) to a Shell (h = O.ogogTo) Embedded to a Depth H = 1.818T0 in a Uniform Half­

Space. Present results are shown with solid lines or segmented lines, results of Wong et al (1986)

forT= To are shown with open circles (7] = 0.132).

Figure 7. (a), (b) Normalized Radial Displacements Ur at T =To and (c), (d) Normalized Hoop

Stresses Eoo at T = a for Vertically Incident (Ov = 0°) P- and SV-Waves Impinging Normal

(~ = goo) to a Shell (h = O.ogogT0 ) Embedded to a Depth H = 4.545T0 in a Uniform Half­

Space. Present results are shown with solid lines; the results of Liu et al (1991) are shown with

open circles (7] = 0.105).

Figure 8. (a), (b), (c) Normalized Hoop Stresses and (d), (e), (f) Normalized Axial Stresses Exx

for Nonvertically Incident (Ov = 10°) P- ,SV- and SH-Waves Impinging Obliquely (Oh = 30°)

on a Shell (h = o.ogogT0 ) Embedded to a Depth H = 1.818T0 in a Uniform Half-Space. The

present results at T = T 0 are shown with solid lines while those at T = a are shown with

segmented lines. The results of Wong et al (1986) at T = To are shown with open circles

(7] = 0.132).

Figure 9. (a), (d) Normalized Radial Displacements Ur at T = To , (b), (e) Normalized Axial

Displacement Ux at T = T0 and (c), (f) Normalized Hoop Stresses Eoe at T = a for Nonvertically

Incident (Ov = 30°) P- and SV-Waves Impinging with Angle Oh = 0° on a Shell (h = O.ogogT0 )

Embedded to a Depth H = 4.545To in a Uniform Half-Space. Present results are presented by

solid lines (DT!IBF) and solid dots (FE/IBF). The results of Liu et al (1991) are shown with

open circles (7] = 0.105).

Figure 10. Normalized Radial Ur. Tangential Ue, and Longitudinal Ux Displacements at T . To

and, Hoop Stresses Eee and Shear Stresses Eex at T = a for Nonvertically Incident P-, SV- and

SH-Wave (Bv = 30°) Impinging Normal ((}h = 90°) to a Shell (h = 0.1T0 ) Embedded to Depths

H/To = 4.6 (segmented lines) and 11.8 (solid lines) in a Multilayered Half-Space (TJ = 0.098).

Figure 11. Normalized Longitudinal Ux, Tangential Ue, and Radial Ur Displacements at T = T0

for Nonvertically Incident (Ov = 30°) P-, SV- and SH-Wave Impinging with Angle Ott = 0° on

a Shell (h = 0.1r0 ) Embedded to Depths H/To = 4.6 (segmented lines) and 11.8 (solid lines)

in a Multilayered Half-Space (TJ = 0.098).

Figure 12. Normalized Axial 'Exx• Hoop Eee. and Shear Eex Stresses for Nonvertically Incident

(Ov = 30°) P-, SV- and SH-Wave Impinging with Angle Ott = 0° on a Shell (h = 0.1r0 )

Embedded to Depths H/To = 4.6 (segmented lines) and 11.8 (solid lines) in a Multilayered

Half-Space (TJ = 0.098).

(a) (c)

1 x'

2

3

"y' z'

(b) (d)

y

Z-H ,-

z

Figure 1

(a) (b) (c)

3

2 1

1

0 360 °o 90 180 270 Y fro 8 (degrees)

Figure 2

3

1

·0o 90 180 270 e (degrees)

(b)

90 180 270 e (degrees)

75

360

Figure 3

(a) (b)

1

0o 90 180 270 e (degrees)

Figure 4

P-WAVE SV-WAVE

Lee

90 180 2 0 360 e (degrees)

Figure 5

P-WAVE SV-WAVE 25 25

Lee (a) Lee ,, (b) 20 20 I \

I \ I \ I ' I ' I ' 15 15 I ' I ' I \ I I \ I I \ I I \,

10 10 I I I I

5 5 I I

0 0

5 5

4 Lxx (c)

4 Lxx (d)

3 3

2

1 1 I I

00 360 °o 90 180 270 360 e (degrees)

Figure 6

P-WAVE

(a)

3

2

1

o~~--~~~~~~

20~~~~~~~~

(c)

0o 90 180 270 e (degrees)

SV-WAVE

(b)

(d)

15

10

5

90 180 2 0 e (degrees)

Figure 7

5

5

90 180 270 360 °o 360 e (degrees)

Figure 8

3

2

1

0

2

1

1.5

Ur P-WAVE

1.0

.5

.0

4

Ur SV-WAVE (d)

o0o 3

0 0 2

1

90 180 270 360 °o e (degrees)

15

P-WAVE

10

ocPoo 0 0 5

0

15

Ux SV-WAVE (e)

10

5

90 180 270 360 °o e (degrees)

Lee P-WAVE

Lee sv-wAvE (f)

0 0 00 00

0 0

90 180 270 360 e (degrees)

Figure 9

P-WAVE SV-WAVE

0 0 SH-WAVE

3 4 2

Ue Ue Ux (g)

3 2

2 1

1 1 ,,,----............. ... .. ..... ____________

0 0 0

60 20 60

Lee Lee (f) Lex 15 \

\

40 40 \ \ \ \ \

10 \ , ... -.......... \

\

---------- \ .... , .. ... ... \ ... __ ... \ 20 20 I I

5 \ \

90 180 270 360 °o 90 180 270 360 °o 90 180 270 360 e (degrees) e (degrees) e (degrees)

Figure 10

P-WAVE SV-WAVE SH-WAVE 1.2 1.2 .4

Ux Ux Ux (g)

.8

.2

.4 .4 ,,--,,

, ' , ' , ' , ' ,---- ,' ' , .... ,, .. _,

.0 .0 .0

2.4 1.2 3.2

Ua 2.4

1.6 .8 ' \ \ \ \ 1.6 \

\ \ \

.8 .4 \ \ \ \ .8 \ \ \

.0 .0 .0

3 1.0 3

Ur (f) (i) ,-,

I \

2 I \ 2 I \ I \ I \ I \

.5 I \ I \ I \ I \ I \

1 I \ I \ 1

I I

90 180 270 36o·0o 90 180 270 90 180 270 360 e (degrees) e (degrees) e (degrees)

Figure 11

15

10

5

40

30

20

10

15

10

5

P-WAVE

(a)

30

20

10

Lee (b)

--- --------- ...... , ', .,, .......... ____ ,

(c) 40

30

20

10

90 180 270 360 °o e (degrees)

SV-WAVE

(d)

Lee (e)

Lex (f)

20

10

90 180 270 360 °o e (degrees)

SH-WAVE

(g)

Lee (h)

90 180 270 360 e (degrees)

Figure 12