ABSTRACT: The 2D scattering and diffraction of plane shear horizontal waves by a surface semicircular canyon on top of an underground circular unlined tunnel ...
ANTIPLANE DIFFRACTION FROM CANYON UNLINED TUNNEL
ABOVE
SUBSURFACE
By V. W. Lee,1 S. Chen,2 and I. R. Hsu3 ABSTRACT: The 2D scattering and diffraction of plane shear horizontal waves by a surface semicircular canyon on top of an underground circular unlined tunnel (cavity) in a homogeneous elastic half-space has been analyzed. Using an exact analytic series solution of the problem for a general angle of wave incidence, the ground motions of the half-space surface on and near the canyon and that of the underground cavity were evaluated. These surface motions depend on the following parameters: (1) The angle of incidence of the plane shear horizontal waves; (2) the dimensionless frequency or wave number ka; (3) a0/a, the ratio of the radius of the surface canyon to that of the underground cavity; and (4) D/a, the ratio of the depth of the cavity to its radius.
INTRODUCTION
MODEL
The wave-propagation problem of the 2D scattering and diffraction of plane shear horizontal (SH) waves by a surface canyon on top of an underground cavity is being investigated here. The same diffraction problems with only surface topographies (canyons and alluvial valleys) or subsurface topographies (underground cavities and tunnels) has separately been studied by many investigators. This paper will put both types of topographies together, the simplest of which would be the case of a semicircular canyon on top of an underground circular cavity. Much of the previous work related to diffraction of elastic waves by surface and subsurface topographies in the elastic half-space will first be summarized here. Some of these diffraction problems are dated as far back as the late 1970s. The diffraction of elastic SH waves by an underground circular cavity and tunnel (Lee 1977) has long been solved for over 20 years. It was later extended to longitudinal (P) and shear vertical (SV) waves (Lee and Karl 1993a,b). Diffraction of elastic waves by shallow circular to semicircular surface canyons was also studied (Cao and Lee 1989, 1990; Todorovska and Lee 1990, 1991a,b). The cases for diffraction by arbitraryshaped surface and subsurface topographies are more recent (Lee and Wu 1994a,b; Manoogian and Lee 1995, 1996). Diffraction of SH waves by circular canyons and a valley on a sloping (wedge-shaped) half-space has also been studied (Lee and Sherif 1996; Sherif and Lee 1996). The 2D problems solved above have also been extended to those in three-dimensions. Lee (1981) solved the 3D diffraction of elastic waves by a surface hemispherial canyon and, later (Lee 1984), the same problem on a hemispherical valley. Later, using the integral representation of Hankel functions, Lee (1988) studied diffraction by 3D underground special inclusion. Deformations near a 3D long surface cylindrical inclusion were presented (Lee et al. 1995), together with a 3D underground cylindrical inclusion (Ghosh and Lee 1995). All of these 2D and 3D diffraction problems studied the surface and subsurface topographies separately, when in fact in the real world they exist together.
The 2D model of the problem to be studied is shown in Fig. 1. It represents a half-space in which a semicircular cylinder of radius a0 is removed to form a canyon, and a circular cylindrical cavity of radius a is situated directly below. The center of the cavity C is at a distance D vertically below the center of the canyon, on the half-space surface. The half-space is assumed to be elastic, isotropic, and homogeneous, with its material properties characterized by its rigidity and shear wave velocity c. Two sets of coordinate axes will be put on each of the centers: the rectangular coordinate system (x, y) with origin at C, and the center of the cavity directly below. On the same axes are the corresponding polar coordinate systems, (r0, 0) and (r, ), respectively (Fig. 1). The various coordinate systems are related by the following identities:
1 Assoc. Prof., Civ. Engrg. Dept., Univ. of Southern California, Los Angeles, CA 90089-2531. 2 Res. Assoc., Civ. Engrg. Dept., Univ. of Southern California, Los Angeles, CA. 3 Chair and Assoc. Prof., Dept. of Build. Sci. and Arch., China Inst. of Technol., Taiwan, R.O.C. Note. Associate Editor: Apostolos Papageorgiou. Discussion open until November 1, 1999. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on September 2, 1998. This paper is part of the Journal of Engineering Mechanics, Vol. 125, No. 6, June, 1999. 䉷ASCE, ISSN 0733-9399/99/00060668–0675/$8.00 ⫹ $.50 per page. Paper No. 19179.
668 / JOURNAL OF ENGINEERING MECHANICS / JUNE 1999
x = x0;
y ⫹ y0 = D;
x0 = r0 sin 0;
r0 = (x 20 ⫹ y 20)1/2;
x = r sin ;
r = (x 2 ⫹ y 2)1/2;
y0 = r0 cos 0;
y = r cos (1)
It should be noted that the angles and 0 of both systems are measured relative to the vertical y-axis, which is more convenient for the transformation formulas to be used below. EXCITATION The excitation of the half-space with surface and subsurface topographies w i consists of an infinite train of plane SH waves with frequency , particle motions in the z-direction (antiplane) in the half-space ( y0 > 0) and angle of incidence with respect to the vertical axis. It is represented by w i = w i(x0, y0) = exp i(x0 /cx ⫺ y0 /cy ⫺ t)
(2)
where cx and cy = phase velocities of the waves along the xand y-directions, respectively, and are given by
FIG. 1.
Half-Space Model
cx = c/sin ;
cy = c/cos
(3a,b)
w s = w s(r, ) =
冘
H n(1)(kr)(An cos n ⫹ Bn sin n)
(12)
n
Eq. (2) represents an incident wave of unit amplitude. The time factor, exp(⫺it), representing steady harmonic motion with frequency , is present in all the equations that follow. It will be understood and omitted from now on. In terms of the shear wave number k = /c, (2) and (3) give w i = exp ik(x0 sin ⫺ y0 cos ) = exp ⫺ikr0 cos( ⫹ 0)
(4)
In the absence of any surface canyon or subsurface cavity, the incident SH wave would be reflected from the plane free surface ( y0 = 0) as reflected plane SH wave w r with the same angle of reflection and is given by w r = exp ik(x0 sin ⫹ y0 cos ) = exp ikr0 cos( ⫺ 0)
(5)
Using the following expansion theorem (Abramowitz and Stegan 1972): e
冘
=
yz =
εn(⫾i) Jn(kr)cos n n
(6)
n =0
where ε0 = 1, εn = 2 for n > 0, (4) and (5) can be expanded as
冘 冘 ⬁
wi =
ε(⫺i)nJn(kr0)cos n( ⫹ 0);
(7a)
n =0 ⬁
wr =
εn(⫹i)nJn(kr0)cos n( ⫺ 0)
(7b)
n =0
The resultant ‘‘free-field’’ motion in the half-space w i⫹r then becomes
冘
w i⫹r = w i ⫹ w r = 2J0(k0) ⫹ 4 ⫹ 4i
冘 m
(8)
At the surface of the half-space, the free-field motion has a resultant amplitude of |w0| = 2. It corresponds to the input wave motion in the presence of both the surface canyon and subsurface cavity, which, in general, will generate scattered and diffracted waves. The resultant motions will be significantly different from that of free-field in their vicinity. In the presence of the surface canyon, the scattered waves diffracted from the canyon can be represented by w s0 = w s0(r0, 0) =
冘
rz =
冏
⭸w ⭸y
=0
at half-space surface
(13a)
at canyon surface
(13b)
y0 =0
⭸w ⭸r0
冏
=0
r0 =a0
=0
at underground cavity surface
(13c)
r=a
SOLUTION OF PROBLEM To solve this boundary-valued problem, we use the approach of Lee (1979) and Lee and Manoogian (1995) to create additional scattered waves so that all three traction-free boundary conditions [(13a)–(13c)] can be satisfied. The boundary conditions in (13a)–(13c) will be applied one-by-one as follows. Application of Boundary Condition in (13a)
(⫺1)mJ2m(kr0)cos 2m cos 2m0
m
(⫺1)mJ2m⫹1(kr)sin(2m ⫹ 1) sin(2m ⫹ 1)0
冏
⭸w ⭸y
rz =
⬁
⫾ikr cos
which is the same form as that in (9). Both the scattered waves from the surface canyon w s0 and subsurface inclusion w s are represented by wave functions that are outgoing toward infinity, satisfying Sommerfeld’s radiation conditions (Eringen and Suhubi 1975). They will further interact and create additional scattered and diffracted waves. The resultant total displacement w is then a superposition of all incident and reflected plane waves, and all the scattered and diffracted waves from the canyon, cavity, and half-space surface. The total waves must satisfy the wave differential equation [(11)] and the three traction free boundary conditions
Consider an unbound medium (a full-space) and suppose there is another identical cylindrical cavity of the same radius a with origin at C1, at the same distance of D above the halfspace as the other one is below (Fig. 2). Introduce two additional coordinate systems with origin at C1, the Cartesian (x1, y1) and polar coordinate (r1, 1) systems. Let there be addi-
C2nH (1) 2n (kr0)cos 2n0
n
⫹ C2n⫹1H (1) 2n⫹1(kr0)sin(2n ⫹ 1)0
(9)
It should be noted that both the free-field plane waves w i⫹r and diffracted waves w s0 in the above equations satisfied the traction free-field boundary condition at the half-space surface ( y = 0) yz =
冏
⭸w ⭸y0
y0 =0
=
冏
⭸w ⭸0
=0
(10)
0 =⫾/2
The scattered waves w s0 also satisfy the steady-state elastic wave equation (Helmholtz equation) (Pao and Mow 1973) ⭸2w 1 ⭸w 1 ⭸2w ⫹ 2 ⫹ k 2w = 0 2 ⫹ ⭸r r ⭸r r ⭸2
(11)
and represent outgoing waves propagating toward infinity. The presence of the underground cavity will result in additional scattered waves, which, with respect to the coordinate system (r, ) with origin at C, the center of the cavity, can be represented by
FIG. 2.
Full-Space Model
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tional scattered waves w 1s from the top cavity as that of w s from the underground cavity [(12)]
冘
w ⫹w =2
H (1) n (kr1)(An cos n1 ⫹ Bn sin n1)
relative to the polar coordinate system (r1, 1) at C1 (with B0 = 0). These additional waves w s1 are a ‘‘mirror image’’ of the scattered waves w s in the half-space (Lee 1979). The resultant wave motion in the half-space will now be given by w = w i ⫹ w r ⫹ w 0s ⫹ w s ⫹ w 1s
To apply this boundary condition at the surface of the canyon, the scattered waves w s [(12)] and w s1 [(14)] from the respective cavities will have to be transformed to the polar coordinate systems (r0, 0) with origin at O, the center of the canyon (Fig. 2). The following addition theorem (Abramowitz and Stegun 1972) is used: For the coordinate systems (r, ) and (r0, 0) in Fig. 2, with r0 < D
再 冎 冘 冘 再 ⬁
=
⬁
=
H (1) n⫹m(kD)Jm(kr0)
m=⫺⬁
冎
再
冎
cos m0 sin m0
m=0
(16)
where εm m (1) (H (1) n⫹m(kD) ⫾ (⫺1) H n⫺m(kD)) 2
P⫾ mn(kD) =
so that all m = 0, 1, 2, . . . H (1) 2m⬘(ka0) ⫹ C2m ⫹ 2 A2m = 2εm(⫺1)m cos 2m J⬘2m(ka0) H (1) 2m⬘(ka0) ⫺ C2m⫹1 ⫹ 2B2m⫹1 = 4i(⫺1)m sin(2m ⫹ 1) J⬘2m(ka0)
冘
冘
⫹ P2m,n (kD)An = 2εm(⫺1)m cos 2m
w = w (r0, 0) =
(24a)
n =0
and for m = 1, 2, 3, . . . H (1) ⬘ (ka0) 2m⫹1 C2m⫹1 ⫹ 2 J⬘2m⫹1(ka0)
冘 ⬁
⫺ P2m⫹1,n (kD)Bn = 4i(⫺1)m sin(2m ⫹ 1)
n =1
(24b)
Eqs. (24a) and (24b) each constitute a set of matrix equations of infinite order for the unknown coefficients.
(17)
This last boundary condition on the surface of the underground cavity will now require all waves to be expressed in the polar coordinate system (r, ). Start with the incident and reflected plane waves [(4) and (5)], using the identities in (1) w i = exp ik(x0 sin ⫺ y0 cos ) = exp(⫺ikD cos )exp ik(x sin ⫹ y cos ) = e⫺ikDcoseikrcos(⫺) = e⫺ikDcos
冘
εni nJn(kr)cos n( ⫺ )
n
⫹ m
(23b)
⬁
⬁
s
(23a)
using the orthogonality of the sine and cosine functions. Expressed in terms of the original set of coefficients An, Bn, and Cn, n = 0, 1, 2, . . . , (23) becomes, for m = 0, 1, 2, . . .
Eq. (12) for w s can be rewritten as s
(22)
Application of Boundary Condition in (13c)
P⫹ mn(kD)cos m0 ⫺ Pmn (kD)sin m0
Jm(kr0)
⭸w ⭸ = (w i ⫹ w r ⫹ w 0s ⫹ w ⫹ w 1s ) = 0 ⭸r ⭸r
H(1) 2m⬘(ka0) C2m ⫹ 2 J⬘2m(ka0)
Application of Boundary Condition in (13b)
(21)
The symmetry of the two waves resulted in the odd coeffi⫺ cients A⫹ 2m⫹1 and even coefficients B2m canceling out. With all waves expressed in the (r0, 0) coordinate system, the traction free boundary condition [(13b)] at r0 = a0 can now be applied
(15)
It should be noted that the incident and reflection plane waves, w i⫹r = w i ⫹ w r, in (7), already satisfy the stress-free condition at the surface. The scattered waves from the canyon [(9)] are also in the form that satisfy this boundary condition [see (10)]. The remaining two scattered waves, w s and w s1, are a mirror image of each other with the line of symmetry being the surface of the half-space. This symmetry resulted in the sum of the functions w s ⫹ w s1 having zero slope at the line of symmetry. In other words, the y-derivative of w s ⫹ w s1 is zero at the half-space surface, so that (13a), the traction-free boundary condition, is satisfied by all components of the resultant waves.
cos n sin n
A⫹ 2m J2m(kr0)cos 2m0
⫹ B⫺ 2m⫹1J2m⫹1(kr0)sin(2m ⫹ 1)0
(14)
n =0
H (1) n (kr)
s 1
m=0
⬁
w s1 = w s1(r1, 1) =
冘 ⬁
s
⫺ m
Jm(kr0)(A cos m0 ⫹ B sin m0)
(18)
m=0
(25a)
w = exp ik(x0 sin ⫹ y0 cos ) r
= exp(⫹ikD cos )exp ik(x sin ⫺ y cos )
where
冘 ⬁
A⫹ m =
冘
= e⫹ikDcose⫺ikrcos(⫹) = e⫹ikDcos
⬁
P⫹ mn(kD)An;
n =0
B⫺ m =
P⫺ mn(kD)Bn
冘
εn(⫺i)nJn(kr)cos n( ⫹ )
n
(19a,b)
(25b)
n =0
In a similar way, (14) for w s1 becomes, for r0 < D (Fig. 2)
s 0
Next, the scattered waves from the canyon w can be written as follows, for r < D:
⬁
w s1 = w s1(r0, 0) =
冘
w = w (r, ) = s 0
m=0
⫺ m
⫹ B sin m( ⫺ 0))
s 0
⫺ Jm(kr)(C⫹ m cos m ⫹ Cm sin m)
(26)
m=0
(20)
⫺ with A⫹ m and Bm the same as in (19). It should be noted that the angle ( ⫺ 0) is used in (20) instead of 0 as in (18) for the sine and cosine terms because of the difference in orientation between the pair of coordinates (r0, 0) and (r1, 1) when compared with that between (r0, 0) and (r, ). Combining (18) and (20) for w s and w s1, gives, for r0 < D
670 / JOURNAL OF ENGINEERING MECHANICS / JUNE 1999
冘 ⬁
Jm(kr0)(A⫹ m cos n( ⫺ 0)
using the same transformation formula above in (16), where C⫹ m =
冘
n =0,2,4,...
P⫹ mn(kD)Cn;
Cm⫺ =
冘
⫺ Pmn (kD)Cn
(27a,b)
n =1,3,5
It should be noted that C⫹ m uses only the even coefficients C0, C2, C4, . . . and C⫺ m uses only the odd coefficients C1, C2, C3, . . . . Finally the scattered waves w s1 from the ‘‘image’’
cavity above the half-space in (14) can be written as follows, for r < D:
冘 ⬁
w s1 = w s1(r, ) =
Jm(k)(A* m cos m ⫹ B* m cos m)
(28)
m=0
where
冘 ⬁
A* m =
冘 ⬁
P⫹ mn(2kD)An;
B* m =
n =0
P⫺ mn(2kD)Bn
(29a,b)
n =0
Combining w i and w r in (25a) and (25b) gives
冘 ⬁
wi ⫹ wr =
εn Jn(kr)(e⫺ikDcosein/2 cos n( ⫺ )
n =0
冘 ⬁
ikDcos ⫺in/2
⫹e
cos n( ⫹ ) = 2
e
εnJn(kr)(cos n cos(n/2
n =0
⫺ kD cos )cos n ⫹ i sin n sin(n/2 ⫺ kD cos )sin n) (30)
With all the waves expressed in the (r, ) coordinate system, the traction free boundary condition at r = a can now be applied, giving, for m = 0, 1, 2, . . . H (1) m ⬘(ka) ⫹ Am ⫹ A* m ⫹ Cm = ⫺2εm cos m cos(m/2 ⫺ kD cos ) J⬘m(ka) (31a)
and for m = 1, 2, 3 H (1) m ⬘(ka) ⫺ Bm ⫹ B* m ⫹ Cm = ⫺4i sin m sin(m/2 ⫺ kD cos ) J⬘m(ka) (31b)
again using the orthogonality of the sine and cosine functions. Expressed in terms of the original set of coefficients An, Bn, and C⬘s, (31) becomes, for m = 0, 1, 2 n H (1) m ⬘(ka) Am ⫹ J⬘m(ka)
冘
冘
⬁
⫹ Pmn (2kD)An ⫹
n =0
⫹ Pmn (kD)Cn
n =0,2,4...
= ⫺2εm cos m cos(m/2 ⫺ kD cos )
and for m = 1, 2, 3 H (1)⬘ m (ka) Bm ⫹ J⬘m(ka)
冘 ⬁
n =0
⫺ Pmn (2kD)Bn ⫹
冘
(32a)
兩w兩 = (Re(w)2 ⫹ Im(w)2)1/2;
⌽ = tan⫺1(Im(w)/Re(w))
(33a,b)
where Re(⭈) and Im(⭈) = real and imaginary parts of a complex number, respectively. The following dimensionless parameter will be used: ka =
a 2a = c
(34)
where ka = dimensionless wave number. It also represents a dimensionless frequency a/c and is also 2 times the ratio of the radius of the cavity to the wavelength of the waves. Fig. 3 displays two 3D plots of surface displacement amplitudes 兩w兩 versus the dimensionless distance x/a and dimensionless frequency or wave number ka of the incident plane waves w i at horizontal ( = 90⬚), and 45⬚ angles of incidence. The dimensionless parameters used in this and all subsequent figures are as follows: • ka = a /c, the dimensionless frequency or wave number as given in (34). • x/a, the ratio of the horizontal distance x and a radius of the canyon. The horizontal distance is measured from O, the center of the canyon, along the half-space surface. • a0 /a, the ratio of the radius of the canyon to that of the underground cavity. In Fig. 3, a0 /a = 1. • D/a, the ratio of the depth of the cavity’s center below the surface to its radius. Here in Fig. 3, D/a = 2.5.
⫺ Pmn (kD)Cn
n =1,3,5...
= ⫺4i sin m sin(m/2 ⫺ kD cos )
above analysis is to describe and analyze the displacement amplitudes and the relative phases of motions at points on the surface on or close to the canyon and at points on the subsurface cavity. The detailed description of the amplitudes and phases of the surface ground and subsurface cavity motions will give the space-dependent transfer functions of displacements at points around the canyon and cavity, from which the strains and stresses can be calculated. This information can then be used to understand and interpret the effects of the surface and subsurface topographic features similar to the model studied here. The resulting motion will be characterized by the displacement amplitudes of the total motion w [(15)], 兩w兩, and relative phases , where
(32b)
Eqs. (32a) and (32b) each constitute a set of infinite-order matrix equations for the unknown coefficients. The sets of infinite-order matrix equations in (24) and (32) are now ready to be solved. As the forms of both equations suggested, the first term involving the quotient H (1) m ⬘/J⬘ m in each equation is the dominant term. This means that the matrix equations will be well-banded along the diagonal and hence they are well-conditioned. These equations may thus be easily solved by truncating the matrices to finite orders, large enough so that the last few terms in the series for the scattered waves w s0, w s, and w s1 will all contribute to less than the desired percentage of the error. The higher the wave number of the input waves, the higher will be the order of the matrix. For ka = 15, as much as n = 35 terms of each of the scattered wave functions in (9) and (12) are used. Equation solvers with sophisticated numerical algorithms are readily available in the textbook Numerical Recipes (Press et al. 1992). SURFACE DISPLACEMENTS From the earthquake engineering, structural analysis, and design to strong-motion seismology, an important aspect of the
With a0 /a = 1 in Fig. 3, x/a = x/a0, so that x/a = ⫺1 corresponds to the left rim of the surface canyon and x/a = ⫹1 to the right rim. The incident plane SH waves are assumed to be coming from the ‘‘left’’ (i.e., from x/a < 0 in all cases, except for the case of vertical incidence, when = 0). For completeness, the displacement amplitudes along the canyon surface (r = a, 兩0兩 ⱕ /2) are plotted along the horizontal xaxis. The displacement amplitudes in Fig. 3 illustrate several interesting observations of the model—some expected and some a bit unexpected. For horizontal incidence ( = 90⬚) and as frequency increases, the complexity of the displacement amplitudes increases on the left, or front, side of the canyon (x/a ⱕ 0), and the amplitudes decrease and become smoother on the right, or back, side of the canyon (x/a ⱖ 0). The zone behind the canyon and cavity is often referred to as the shadow zone. These are general trends of surface displacement resulting from the scattering and diffraction of SH waves by the surface (canyon) and subsurface (cavity) topographies. The same observations have previously been made for the case of SH waves incident on surface topographies (Cao and Lee 1989) and subsurface topographies (Lee 1977; Lee and Manoogian 1995). It is thus not surprising that the same observations are expected here. JOURNAL OF ENGINEERING MECHANICS / JUNE 1999 / 671
FIG. 3.
Surface Displacement Amplitudes, D/a = 2.5, a0/a = 1
FIG. 4.
Tunnel Displacement Amplitudes, D/a = 2.5, a0/a = 1
What is surprising, however, are the amplitudes of the resultant displacements. Cao and Lee (1989), in studying diffraction of SH waves by surface canyons with different depth-to-width ratios, observed that the maximum amplitudes are 4. Lee (1977) and Lee and Manoogian (1995), in studying diffraction 672 / JOURNAL OF ENGINEERING MECHANICS / JUNE 1999
of SH waves by subsurface cavities, circular or arbitrary in shape, at various depths, made similar observations. The only difference is that displacement amplitudes as high as 6 are observed only when the underground cavity is very close to the surface (D = 1.5a). In the present case, the underground
cavity is at a depth of 2.5 times its radius (D/a = 2.5), and the highest amplitude of displacements observed is almost 8 (7.88). Amplifications of this magnitude may be explained by the fact that, for incident waves at almost horizontal angles of incidence, some diffracted waves are more easily ‘‘trapped’’ between the bottom surface of the canyon and the top surface
FIG. 5. Surface Displacement Amplitudes at Four Angles of Incidence, ka = a /c = 12.0, D/a = 2.50, a0 /a = 1.00
FIG. 6.
of the underground cavity, resulting in a standing wave pattern there. Fig. 4 displays two 3D plots of displacement amplitudes, 兩w兩 versus frequency at points on the surface of the underground cavity, with the same ratios for a0 /a (=1) and D/a (=2.5). Points on the surface of the unlined tunnel are measured in degrees (⬚) counterclockwise from the top point of the surface. Points from 0 to 180⬚ are those on the left side facing the incident waves, whereas those from 180 to 360⬚ are the waves. As expected, consistent with previous work (Lee 1977) the displacement amplitudes on the front side of the cavity are in general more complex and higher than those on the back side, and again the complexity increases with increasing frequencies. It should also be noted that at some frequencies, at points with tunnel angle < 0⬚ (or those in the range from 270 to 360⬚) and points on the top part of the underground cavity in the shadow zone, the displacement amplitudes are as high as those in the front part. This can again be due to the standing waves between the bottom surface of the canyon and the top surface of the underground cavity—the same waves that resulted in large displacement amplitudes on the canyon surface on top (Fig. 3). It is also worthwhile noting that the displacement amplitude of 4 (or amplification of 2) corresponds to that at the corner point of a quarter-space. In fact, the displacement amplitude at the corner point of a wedge-shaped half-space with flat halfspace ( = 1), a quarter-space ( = 1/2), and 45⬚ wedge ( = 1/4) will have displacement amplitudes at the corner point of 2, 4, and 8, respectively (Lee and Sherif 1996). Figs. 3 and 4 show that at various angles of incidence ( = 0, 30, 60, and 90⬚), and between frequencies from as low as ka = 2 to as high as ka = 15, various maximum displacement amplitudes between 4 and 8 are attained, suggesting that the trapped space between the canyon bottom surface and cavity top surface can result in waves behaving as though they are reflected from spaces with wedge angles between 45⬚ ( = 1/4) and 90⬚ ( = 1/2). This ‘‘wedge’’ behavior is definitely not observed in previous studies where there is no underground cavity below the surface canyon.
Surface Displacement Amplitudes, D/a = 2.5, a0 /a = 0.25 JOURNAL OF ENGINEERING MECHANICS / JUNE 1999 / 673
FIG. 7.
Tunnel Displacement Amplitudes, D/a = 2.5, a0 /a = 0.25
Fig. 5 shows the displacement amplitudes versus frequencies at different points on the unlined tunnel for four angles of incidences that have maximum amplitudes similar to those at the half-space and canyon surfaces in Figs. 3 and 4, especially for points on the top surfaces of the tunnel. Figs. 6 and 7 are the same set of figures as Figs. 3 and 4, for displacement amplitudes corresponding to a0 /a = 0.25 and D/a = 2.50 (i.e., the case where the surface canyon radius is only a quarter of that of the underground cavity). Still the same trend of high amplifications of displacements at all surfaces are observed. CONCLUSIONS The presence of both the surface topography (canyon) and the subsurface topography (cavity) results in higher than expected amplifications of surface displacement amplitudes in the vicinity. Amplification of over 300% is observed even in the case in which the surface canyon is much smaller than the underground cavity (Fig. 6, with a0 /a = 0.25). This paper also provided the methodology for further studies involving other surface and subsurface topographies, such as that of a rigid foundation above rigid subway infrastructures (Lee et al., unpublished paper, 1998) and the case of multiple foundations above the subway (Lee and Chen 1998). ACKNOWLEDGMENTS The writers are indebted to their colleagues, Prof. M. Manoogian of Loyola Marymount University, Los Angeles, and N. Dermendjian of California State University at Northridge, Los Angeles, for critical reading of the manuscript and for many valuable suggestions. This research was supported by National Science Foundation Grant CMS-9714859.
APPENDIX.
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