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Soil-Structure Interaction on a ShallowRigid Circular Foundation: SH WaveSource from Near-Field ExcitationsHao Luo a & Vincent W. Lee ba HNTB Corporation , Santa Ana , California , USAb Civil & Environmental Engineering , University of SouthernCalifornia , Los Angeles , California , USAAccepted author version posted online: 12 Sep 2013.Publishedonline: 04 Dec 2013.

To cite this article: Hao Luo & Vincent W. Lee (2014) Soil-Structure Interaction on a Shallow RigidCircular Foundation: SH Wave Source from Near-Field Excitations, Journal of Earthquake Engineering,18:1, 67-89, DOI: 10.1080/13632469.2013.841602

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Journal of Earthquake Engineering, 18:67–89, 2014Copyright © A. S. ElnashaiISSN: 1363-2469 print / 1559-808X onlineDOI: 10.1080/13632469.2013.841602

Soil-Structure Interaction on a Shallow RigidCircular Foundation: SH Wave Source from

Near-Field Excitations

HAO LUO1 and VINCENT W. LEE2

1HNTB Corporation, Santa Ana, California2Civil & Environmental Engineering, University of Southern California, LosAngeles, California

A closed-form wave function analytic solution is presented in this article regarding thetwo-dimensional scattering and diffraction of a flexible wall sitting on a rigid shallow circular foun-dation embedded in an elastic half-space that is activated by a nearby anti-plane line source such as ablast caused by underground construction or mineral exploration or a near-field fault rupture, usingsimilar methodology as the other paper in a series [Lee and Luo, 2013]. These wave propagationinfluences, although often treated as a transient process, may be simulated as linear combinationsof steady-simple harmonic responses as studied in this article. Ground surface displacements spec-tra for wide-band of incident wave frequencies are calculated. Based on the spectra obtained, thedependence of near-field ground displacements are shown with respect to the rise-to-span ratio offoundation profile, frequency of incident waves, distance of source from the foundation, and massratios of various media (foundation-structure-soil). The screening effect of rigid foundation uponground motions behind grazing incident waves is also presented.

Keywords SH Waves; Line (Point) Source; Cylindrical Wave; Soil-Structure Interaction; Closed-Form Analytic Solution; Fourier-Bessel Series

1. Introduction

Most soil-structure dynamic interaction publications concentrate on incident plane waves,e.g., SH, SV, or P-waves, a widely acceptable approximation of far-field seismic waves.However, cylindrical waves radiated from a line source are more suitable to simulate near-field seismic waves or explosions. Same as in Lee and Luo [2013], the Fourier-Bessel wavefunction expansion method is used in this article as well to acquire analytical solutions, butfor incident cylindrical waves.

Other than in Lee and Luo [2013], in which the incident waves are plane waves, theexcitation of the soil-foundation-structure system is altered to a two-dimensional point SH-wave source. Alternatively, from a three-dimensional viewpoint, the source consists of aninfinite number of point sources which are aligned perpendicularly to the two-dimensionalplane. It is essentially a plane strain problem. The displacement and stress responses adja-cent to the soil-foundation-structure system must differ with the variation of the distancebetween the source and system. This is the opposite effect of the plane incident SH-wavecases which are often used to simulate far-field incident waves caused by seismic activitiesfar away from the local area of concern. Emitted-source approximations are often applied

Received 6 April 2013; accepted 21 August 2013.Address correspondence to Hao Luo, HNTB Corporation, Santa Ana, CA 92707 USA. E-mail: hluo@

hntb.com

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68 H. Luo and V. W. Lee

to represent near-field vibration sources in close proximity of seismic faults or nearby(nuclear) major explosions. For these cases, the wavefront is in the form of spherical wavesin a semi-infinite three-dimensional elastic half-space medium. For a two-dimensionalplane strain analysis presented below, the spherical wave is simplified to cylindrical wave.

2. The Mathematical Model: Incident Cylindrical SH Waves

The model studied in this article is identical to Lee and Luo [2013] except for the type ofincident waves. An elastic wall with width B and height H is set on a shallow circular rigidfoundation with radius a embedded on a half-space (Fig. 1). All materials here are treated ashomogeneous, elastic, and isotropic except for the rigid foundation. An antiplane SH-wavepoint source is located at point O′ with radial distance of R from the center of foundation Oand incidence angle γ with respect to the horizontal. The perpendicular distance betweenthe circular center O and ground surface is d. The half width of the foundation is b. Thecentral angle of the circular-arc sector L is δπ . The two Lamé constants of half-space andwall, rigidity and shear wave velocity, are, respectively, denoted by μ,Cβ and μB,CβB. Thetwo interfaces L and L between the soil, foundation, and wall are assumed to be fused withno relative movement.

As in the plane wave case discussed in Lee and Luo [2013], the Helmholtz differentialwave equation as well as the traction-free boundary condition Eq. (2) and kinetic Eq. (3)still must be satisfied:

∂2w

∂r2+ 1

r

∂w

∂r+ 1

r2

∂2w

∂θ2+ k2w = 0 (1)

τθz|L = 0 at r > b; θ = 0, π (2)

w|L = w0eiωt at r = a; φ ∈[−δπ

2,δπ

2

](3)

o'

SH Source

Rr'

a

L L

δπ

hr Ly

dy

xo φ

y r

φ

o x

b b

H

B

FIGURE 1 Mathematical model.

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Soil-Structure Interaction on Shallow Foundation 69

where

τφz = μ

r

∂w

∂φor τθz = μ

r

∂w

∂θ(4)

τrz = μ∂w

∂r(5)

in which w is the total anti-plane wave field in the half space and w0 is the unknownmovement of the rigid foundation.

The incident waves with unit amplitude are expressed as follows:

w(i)o′ (r′) = H(1)

0

(kr′)∣∣∣H(1)

0 (kR)

∣∣∣ . (6)

Here, the denominator∣∣∣H(1)

0 (kR)

∣∣∣ is a scaling factor used to normalize the resultant

displacement amplitude at origin point O by incident waves to a unit. In other words, thetotal resultant displacement amplitude equals two by combining incident waves with theirreflection waves. A harmonic time factor e−iωt is assumed to be understood and omittedhereafter unless noted otherwise. Due to the asymptotic properties of Hankel function,the amplitude of incident waves degenerate along with their respective radial argumentsincrease [Bowman et al., 1969; Pao and Mow, 1973]. This conforms to the behavior ofthe source-emitted waves and Sommerfeld radiation condition. Sherif [1993] solved for theeffects on the diffraction of wedge topography activated by a near SH-wave point sourceusing the same expression of incident waves, as in Eq. (6). The reflected waves could beviewed as divergent from an image point source with respect to the flat ground surfaceoriginated at O′′, as shown in Fig. 2:

w(r)o′′ (r′′) = H(1)

0

(kr′′)∣∣∣H(1)

0 (kR)

∣∣∣ . (7)

φo x

SH Source

o'

r' Ry r

ryy

φ

o x

Reflected Source

r''

o''

D

D

d

γ

γ

FIGURE 2 Free-field response by method of images and coordinate transformation byGraf’s formula.

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70 H. Luo and V. W. Lee

Using Graf addition formula and a set of coordinate transformation are applied, the incidentand reflected wave equations in the coordinate system (r, θ) can be obtained [MacDonald,1902]:

w(i)(r, θ ) =∞∑

n=0

(−1)n H(1)n (kR)∣∣∣H(1)0 (kR)

∣∣∣ Jn(kr)ein(θ+γ ) (8)

w(r)(r, θ ) =∞∑

n=0

(−1)n H(1)n (kR)∣∣∣H(1)0 (kR)

∣∣∣ Jn(kr)ein(θ−γ ) (9)

and the free-field motion, in the absence of any variation of topography, is the sum of thesetwo wave fields:

w(f) (r, θ) = w(i) + w(r) =∞∑

n=0

a(f)n Jn (kr) cos nθ , (10)

where Jn(·) is the Bessel function of the first kind with order n. And for n = 0, 1, 2, . . .

a(f)n = 2εn(−1)n H(1)

n (kR)∣∣∣H(1)0 (kR)

∣∣∣ cos nγ (11)

are the coefficients of the free-field waves, with ε0 = 1 and εn = 2 for n >0.The wave field in the half-space, scattered and diffracted in the presence of the

foundation is

w(s)(r, θ ) =∞∑

n=0

AnH(1)n (kr) cos nθ , (12)

where the An are a set of complex unknowns to be determined by boundary conditions.Substituting (12) into (4), then

τ(s)θz (r, θ ) = μ

r

∂w(s)(r, θ )

∂θ= −μ

r

∞∑n=0

nJn(kr)An sin nθ . (13)

Since θ = 0, π on the flat boundary L + L, traction-free boundary condition Eq. (2) is

satisfied automatically. Rewrite (12) into its exponential form in terms of(

r, φ)

as r = r

and φ = π/

2 + θ

w(s)(r, φ) =∞∑

n=−∞AnH(1)

n

(kr

)einφ , (14)

where An = (−i)|n|

ε|n|A|n| for n = 0, ±1, ±2, · · ·

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Soil-Structure Interaction on Shallow Foundation 71

Using the following Graf’s formula [Abramowitz and Stegun, 1972],

Cn(kr

)einφ =

∞∑m=−∞

Jn+m (kd) Cm (kr) eimφ , for r > d (15)

expressions in terms of lower coordinates (r, φ) may be transformed to the upper coordinate

system(r, φ

). Here, Cn(·) can be either Jn(·) or H(1)

n (·). After transformation, the scatteredwave w(s) in Eq. (14) turns to

w(s)(r, φ) =∞∑

m=−∞AmH(1)

m (kr) eimφ (16)

in which

Am =∞∑

n=−∞AnJn+m(kd) = Jm(kd)A0 +

∞∑n=1

An

2(−i)n

[Jm+n(kd) + Jm−n(kd)

]. (17)

Similarly, the free-field wave Eq. (10) in the lower coordinate system is:

w(f)(r, φ) =∞∑

n=−∞a(f)

n Jn(kr

)einφ , (18)

where a(f)n = (−i)|n|

ε|n|a(f)

|n| for n = 0, ±1, ±2, · · · . After applying Graf’s formula, Eq. (18) can

turn into the upper coordinate system:

w(f)(r, φ) =∞∑

m=−∞a(f)

m Jm (kr) eimφ (19)

in which

a(f)m =

∞∑n=−∞

a(f)n Jn+m(kd) = Jm(kd)a(f)

0 +∞∑

n=1

a(f)n

2(−i)n

[Jm+n(kd) + Jm−n(kd)

]. (20)

The coefficients may be further simplified to an explicit formula

a(f)m =

∞∑n=0

εninH(1)

n (kR)∣∣∣H(1)0 (kR)

∣∣∣ cos nγ[Jm+n(kd) + Jm−n(kd)

], (20)*

especially for m = 0, a(f)0 = 2

∞∑n=0

n even

εnin H(1)n (kR)∣∣∣H(1)0 (kR)

∣∣∣ Jn(kd) cos nγ .

3. Boundary Conditions and Solution of Rigid Body Movement

As in Lee and Luo [2013] for plane SH waves, the rigid body movement of foundationw0 can be resolved using the displacement continuity boundary conditions at the interface

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72 H. Luo and V. W. Lee

of foundation and half-space L, traction-free boundary conditions along the free edges ofthe wall, together with kinetic equation of foundation. The formulation of the subsequentequations appears to be the same as to the ones for plane waves [Lee and Luo, 2013], but theconvergence of truncated infinite series tends to be relatively inferior, especially when theincidence angle γ is exactly zero and the source is right located at the free ground surface.This makes the model ill-conditioned. Therefore in the numerical results shown in the nextsection, a small incidence angle γ = 5◦ is used instead of genuine grazing incidence (γ =0◦) to improve the convergence of the solution.

3.1. Displacement Continuity Condition between the Soil and Foundation

The displacement continuity boundary condition at the circular-arc interface of foundationand half-space is

(w(f) + w(s)

)∣∣r=a = wf = w0e−iωt. (21)

Here, w0 is the assumed displacement amplitude of rigid-body foundation movement andthe harmonic factor e−iωt is not omitted in the right-hand side and included implicitly inboth two terms on the left-hand side. By substituting foregoing wave functions in

(r, φ

)into Eq. (21) with harmonic factor eliminated, the variables of series in wave functionequations can be separated. For n = 0,

A0H(1)0 (ka) + a(f)

0 J0 (ka) = w0 (22)

for n = ±1, ±2, ±3, ...

AnH(1)n (ka) + a(f)

0 Jn (ka) = 0. (23)

Thus, An for n �= 0 are already solved but A0 remains unknown due to the undetermined w0:

An =

⎧⎪⎪⎨⎪⎪⎩

w0 − a(f )0 J0 (ka)

H(1)0 (ka)

, for n = 0

−a(f )n Jn (ka)

H(1)n (ka)

, for n = ± 1, ± 2, ± 3,...

. (24)

3.2. Boundary Conditions of the Wall

Since the foundation is fully rigid and any horizontal cross-section of the wall in parallelwith the half-space surface must have the same motion w0, the motion of the wall is inde-pendent of abscissa x so that can be represented as wB = wB(y)e−iωt. This must satisfy thefollowing ordinary differential equation

d2wB

dy2+ k2

BwB = 0. (25)

The general solution of Eq. (25) is

wB(y) = E cos kBy + F sin kBy, (26)

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Soil-Structure Interaction on Shallow Foundation 73

where E and F are complex constants to be determined. With that, the wall is essentiallyanalogous to a single degree-of-freedom (SDOF) oscillator or a uniform shear beam witha deformation that may be characterized by the displacement at the given height. Theboundary conditions around the upper and vertical sides of the wall are

wB|y=0 = w0e−iωt (27)

τyz

∣∣y=−H = μB

∂wB

∂y

∣∣∣∣y = −H

= 0. (28)

Substitution of Eq. (26) into (27) and (28) leads to the wall motion equation (with E = w0

and F = −w0 tan kBH). The motion equation of the wall is solely dependent upon ordinate y

wB(y) = w0 (cos kBy − tan kBH sin kBy) = w0

cos kBHcos kB(y + H). (29)

The shear stress along the interface of wall and foundation L is

τyz

∣∣y=0 = μB

∂wB(y)

∂y

∣∣∣∣y=0

= −w0μBkB tan kBH. (30)

The resultant shear force along the interface L per unit length is

fB = −w0μBkBB tan kBH. (31)

Equation (31) may be further modified to the following equation:

fB = −ω2MBw0tan kBH

kBH, (32)

where kB = ωCβB

, CβB =√

μB

ρBare the wave number and wave velocity in the wall,

respectively. MB is the mass of the wall per unit length MB = ρBBH.

3.3. The Kinetic Equation of the Foundation

The displacement of the foundation w0 may be determined by the kinetic equation below

Mfwf = − (fs + fB) e−iωt, (33)

where Mf is the mass of the rigid foundation per unit length along the z axis, and fs denotesthe action of half-space soil applied onto the foundation. fB is defined above in Eq. (31). wf

represents the displacement function of rigid foundation in terms of time factor t, as shownbelow:

wf = wf(t) = w0e−iωt (34)

Mf = ρfa2

2(δπ − sin δπ) . (35)

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74 H. Luo and V. W. Lee

The term fs is shown below:

fs = a∫ δπ/2

−δπ/2τrz|r=a dφ = μka

∞∑n=−∞

(a(f)

n J′n (ka) + AnH(1)′

n (ka)) ∫ δπ/2

−δπ/2einφdφ (36)

wf = −w0ω2e−iωt. (37)

–6 –4 –2 0 2 4 60

1

2

3

4

5

6

7

8

9

10

Dis

pla

cem

ent

Am

plit

ud

e

x/b

R/a = 1.25R/a = 1.5R/a = 2.0R/a = 3.0R/a = 5.0

FIGURE 3 Ground surface displacement amplitude for η = 3.0, Mf/Ms = 2.0, MB/Ms =5.0, kBH/kb = 2, δπ = 0◦ (h = b = a), γ = 5◦ , and R/a = 1.25,1.5,2.0,3.0,5.0.

0 1 2η

3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

|w0|

|w0|Envelope of w0

Backbone Curve of w0

FIGURE 4 Envelope and backbone curve of foundation rigid-body movement spectrumfor Mf/Ms = 2, MB/Ms = 5, h/b = 0.451, kB H/kb = 2, R/a = 2, γ = 90◦.

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Soil-Structure Interaction on Shallow Foundation 75

FIGURE 5 3-D ground surface displacement for Mf/Ms = 1, MB/Ms = 1, h/b = 0.6,kBH/kb = 2, R/b = 2, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

The definite integral in the right-hand side of Eq. (36) has explicit solution

∫ δπ/2

−δπ/2einφdφ =

⎧⎪⎨⎪⎩

δπ , n = 0

− ieinφ

n

∣∣∣∣∣δπ/2

−δπ/2= 2 sin nδπ

2

n, n �= 0

. (38)

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76 H. Luo and V. W. Lee

FIGURE 6 3-D ground surface displacement for Mf/Ms = 1, MB/Ms = 4, h/b = 0.6,kBH/kb = 4, R/b = 2, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

So fs can be rewritten as

fs = μka

⎡⎢⎢⎣

∞∑n=−∞

n�=0

2 sin nδπ2

n

(a(f)

n J′n (ka) + AnH(1)′

n (ka))

+δπ(

a(f)0 J′

0 (ka) + A0H(1)′0 (ka)

)

⎤⎥⎥⎦ . (39)

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Soil-Structure Interaction on Shallow Foundation 77

FIGURE 7 3-D ground surface displacement for Mf/Ms = 1, MB/Ms = 1, h/b = 0.6,kBH/kb = 8, R/b = 2, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

Therefore, the displacement of the rigid foundation w0 is obtained as:

w0 =− 2i

πka

∞∑n=−∞

a(f)n ξn

H(1)n (ka)

ka

2

(1 − sin δπ

δπ

)(Mf

Ms+ MB

Ms

tan kBH

kBH

)+ H(1)

1 (ka)

H(1)0 (ka)

(40)

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78 H. Luo and V. W. Lee

FIGURE 8 3-D ground surface displacement for Mf/Ms = 2, MB/Ms = 1, h/b = 0.4,kBH/kb = 2, R/b = 4, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

in which

ξn ={

1, when n = 0sin nδπ

2

/nδπ

2 , when n �= 0. (41)

Equalities

Ms = ρa2

2(δπ − sin δπ) (42)

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Soil-Structure Interaction on Shallow Foundation 79

FIGURE 9 3-D ground surface displacement for Mf/Ms = 2, MB/Ms = 4, h/b = 0.4,kBH/kb = 4, R/b = 4, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

ω2 = μk2a2

2Ms(δπ − sin δπ) . (43)

The following Wronskian formula [Abramowitz and Stegun, 1972] were used to obtain Eq.(40):

W{Jp (z) , H(1)

p (z)} = Jp (z) H(1)′

p (z) − J′p (z) H(1)

p (z) = 2i

πz. (44)

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80 H. Luo and V. W. Lee

FIGURE 10 3-D ground surface displacement for Mf/Ms = 2, MB/Ms = 1, h/b = 0.4,kBH/kb = 8, R/b = 4, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

4. Numerical Computation of Displacement Amplitudes and Case Study

The equations of∣∣we

0

∣∣ and w∗0, namely the envelope and backbone curve of the rigid

foundation motion w0, are listed below, same as [Lee and Luo, 2013]:

∣∣we0

∣∣ =∣∣∣∣∣(J2

0(ka) + Y20(ka)

) ∞∑n=−∞

a(f )n ξn

H(1)n (ka)

∣∣∣∣∣ (45)

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Soil-Structure Interaction on Shallow Foundation 81

FIGURE 11 3-D ground surface displacement for Mf/Ms = 4, MB/Ms = 1, h/b = 0.4,kBH/kb = 2, R/b = 6, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

w∗0 = w0| MB

Ms=0

MfMs

=1

=− 2i

πka

∞∑n=−∞

a(f )n ξn

H(1)n (ka)

ka2

(1 − sin δπ

δπ

) + H(1)1 (ka)

H(1)0 (ka)

. (46)

Note that, although infinite series expansion expressions of incident and reflectedwaves, Eqs. (10) or (19) were applied in the formation of boundary condition equations.

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82 H. Luo and V. W. Lee

FIGURE 12 3-D ground surface displacement for Mf/Ms = 4, MB/Ms = 4, h/b = 0.4,kBH/kb = 4, R/b = 6, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

To avoid excessive deviations of free-field responses at locations far away from the foun-dation, the original Eqs. (6) and (7) of free-field wave motion, together with the scatteredwave motion, are used to assemble the total displacement in the half-space.

The wave frequency is defined as the ratio of foundation width 2b to the wavelengthof incident waves λ so that it is normalized to a dimensionless parameter:

η = 2b

λ= kb

π= ωb

πCβ

. (47)

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Soil-Structure Interaction on Shallow Foundation 83

FIGURE 13 3-D ground surface displacement for Mf/Ms = 4, MB/Ms = 1, h/b = 0.4,kBH/kb = 8, R/b = 6, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

To investigate the correlation of displacement responses on the ground surface withrespect to the location of SH-wave source, Fig. 3 combines multiple steady-state displace-ment amplitude curves that corresponds to various distance R’s, the radial distance betweensource focus and midpoint of foundation O. It is obvious that the screening effect of therigid semi-circular foundations are so effective for grazing incidence (γ = 5◦) that the dis-placement amplitudes behind foundation are nearly zero. On the other hand, the peak valuesof each curve are all located right above the source focus, as it is so close to the groundsurface with a small incidence angle γ .

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84 H. Luo and V. W. Lee

FIGURE 14 3-D ground surface displacement for Mf/Ms = 2, MB/Ms = 2, h/b = 0.6,kBH/kb = 4, R/b = 2, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

Figure 4 shows the effectiveness of envelope and backbone curve Eqs. (45) and (46) forcylindrical incident wave cases. Figure 5 and all subsequent figures are three-dimensionalplots that illustrate the correlation of frequency and ground displacement amplitudes in asteady-state manner. The normalized ground surface displacement amplitudes are plottedwith varying parameters that include source distance R

/b, location x/b, dimensionless fre-

quencies η, angle of incidence γ , as well as dimensionless ratios Mf/

Ms, MB/

Ms, h/

b, andkBH

/kb. Differing from plane-wave cases [Lee and Luo, 2013], the smallest incident angle

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Soil-Structure Interaction on Shallow Foundation 85

FIGURE 15 3-D ground surface displacement for Mf/Ms = 4, MB/Ms = 4, h/b = 0.6,kBH/kb = 8, R/b = 4, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

that may be used to get convergent and accurate results cannot be exact zero degree (graz-ing incidence), but a small angle, say 5◦ for the figures below. The displacement peaksat the edge of rigid foundation facing the source side for nearly grazing incident angle,when R/b is small (e.g., <3.0), with maximum amplitudes that exceed 9.0. As the sourceis moving further from the foundation-wall system, as R/b gets bigger, the peak of dis-placement amplitudes is moving further accordingly, reasonably locates right above thesource. Compared with plane-wave cases, the screening effect of rigid foundation is even

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86 H. Luo and V. W. Lee

FIGURE 16 3-D ground surface displacement for Mf/Ms = 2, MB/Ms = 2, h/b = 0.8,kBH/kb = 4, R/b = 4, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

more prominent for horizontal incidence waves. The aspect ratio of foundation shape h/

bhas minor influence on the ground motion behind the foundation when the incident angleis nearly horizontal. When incident angle is oblique (e.g., 30◦ or 60◦), the smaller the h

/b,

the more intense the ground motion behind the foundation. In other words, h/

b is inverselyproportional to the amount of energy can “leak” to the shadow side of the rigid foundation.The three parameters Mf/Ms, MB/Ms, and kBH/kb characterize the mass and geometryratio of foundation/wall/soil. In real applications these ratios should be in a reasonable

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Soil-Structure Interaction on Shallow Foundation 87

FIGURE 17 3-D ground surface displacement for Mf/Ms = 4, MB/Ms = 4, h/b = 0.8,kBH/kb = 8, R/b = 6, η ∈ [0.25,5.0], and γ = 90◦, 60◦, 30◦, 5◦.

scale. In the figures below, these parameters only vary in the range of 1–8. In this range,these parameters do not significantly influence the ground surface motion amplitudes.

5. Conclusions

In this article, an analytical solution is presented for the scattering of anti-plane (SH) cylin-drical waves emitted from a point source nearby a shallow circular-arc rigid foundationsupporting a rectangular structure using the method of Fourier-Bessel wave-function series

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88 H. Luo and V. W. Lee

TABLE 1 List of parameters tested in the 3-dimensional Figs. 5–17

Figure No. R/b Mf/Ms MB/Ms h/b kBH/kb

5 2.0 1.0 1.0 0.6 2.06 2.0 1.0 4.0 0.6 4.07 2.0 1.0 1.0 0.6 8.08 4.0 2.0 1.0 0.4 2.09 4.0 2.0 4.0 0.4 4.0

10 4.0 2.0 1.0 0.4 8.011 6.0 4.0 1.0 0.4 2.012 6.0 4.0 4.0 0.4 4.013 6.0 4.0 1.0 0.4 8.014 2.0 2.0 2.0 0.6 4.015 4.0 4.0 4.0 0.6 8.016 4.0 2.0 2.0 0.8 4.017 6.0 4.0 4.0 0.8 8.0

expansion. Different from the regular plane wave as discussed in Lee and Luo [2013], thepoint source (or the cylindrical waves) introduces an additional variable, i.e., the distancebetween source and scatterer R, into the definition of the mathematical model.

Similarly, for nearly grazing angles of incidence, the rigid foundation acts like a barriershielding the propagation of energy from the source towards the far-end field. However,in comparison with the plane-wave cases, the effectiveness of the foundation, serving asa screen, is even more prominent. Secondly, the pattern of ground surface displacementbecomes more complicated as the wave frequency increases. Thirdly, the distance of sourcerelative to the foundation-wall system also has impact on the location and amplitudes ofnearby ground motion. Lastly, peak amplitudes are about 9.0 at the near edge of the rigidfoundation.

References

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Bowman, J. J., Senior, T. B. A., and Uslenghi, P. L. E. [1969] Electromagnetic and AcousticScattering by Simple Shapes, North-Holland Publishing Company, Amsterdam.

Gicev, V. [2005] “Investigation of Soil-flexible Foundation-structure Interaction for Incident PlaneSH Waves,” Ph.D. thesis, University of Southern California, Los Angeles, California.

Housner, G. W. [1957] “Interaction of buildings and ground during an earthquake,” Bulletin of theSeismological Society of America 47(3), 179–186.

Lee, V. W. and Luo, H. [2013] “Soil–structure interaction on shallow rigid circular foundation:Plane SH waves from far-field earthquakes,” Earthquake Engineering and Engineering Vibration(in press).

Lee, V. W., Manoogian, M. E., and Chen, S. [2002] “Antiplane SH-deformations near a surface rigidfoundation above a subsurface rigid circular tunnel,” Earthquake Engineering and EngineeringVibration 1(1), 27–35.

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Trifunac, M. D. [1972] “Interaction of a shear wall with the soil for incident plane SH waves,” Bulletinof the Seismological Society of America 62(1), 63–83.

Wong, H. L. [1975] “Dynamic soil-structure interaction,” Report No. EERL 75-01, EarthquakeEngineering Research Laboratory, California Institute of Technology, Pasadena, California.

Wong, H. L. and Trifunac, M. D. [1974] “Interaction of a shear wall with the soil for incident planeSH waves: Elliptical rigid foundation,” Bulletin of the Seismological Society of America 64(6),1825–1842.

Wong, H. L. and Trifunac, M. D. [1975]. “Two-dimensional, antiplane, building-soil-building inter-action for two or more buildings and for incident plane SH waves,” Bulletin of the SeismologicalSociety of America 65(6), 1863–1885.

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