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Bulletin of the Seismological Society of America, Vol. 71, No. 6, pp. 1743-1761, December 1981
STIFFNESS MATRICES FOR LAYERED SOILS
BY EDUARDO KAUSEL AND JOSE MANUEL Roi~,SSET
ABSTRACT
The HaskelI-Thompson transfer matrix method is used to derive layer stiffness matrices which may be interpreted and applied in the same way as stiffness matrices in conventional structural analysis. These layer stiffness matrices have several advantages over the more usual transfer matrices: (1) they are symmet- ric; (2) fewer operations are required for analysis; (3) there is an easier treatment of multiple Ioadings; (4) substructuring techniques are readily applicable; and (5} asymptotic expressions follow naturally from the expressions (very thick layers; high frequencies, etc.). While the technique presented is not more powerful than the original HaskelI-Thompson scheme, it is nevertheless an elegant complement to it. The exact expressions are given for the matrices, as well as approximations for thin layers. Also, simple examples of application are presented to illustrate the use of the method.
INTRODUCTION
The determination of the response of a soil deposit to dynamic loads, caused either by a seismic excitation or by prescribed forces at some location in the soil mass, falls mathematically into the area of wave propagation theory. The formalism to study the propagation of waves in layered media was presented by Thomson (1950) and Haskell (1953) more than 25 yr ago, and it is based on the use of transfer matrices in the frequency-wavenumber domain. The solution technique for arbitrary loadings necessitates resolving the loads in terms of their temporal and spatial Fourier transforms, assuming them to be harmonic in time and space. This corre- sponds ft)rmally to the use of the method of separation of variables to find solutions to the wave equation. Closed-form solutions are then found for simple cases by contour integration, while numerical solutions are needed for arbitrarily layered soils. The details of the procedures are well known, and need not be repeated here.
The first step in the computation for dynamic loads is then to find the harmonic displacements at the layer interfaces due to unit harmonic loads. In the transfer matrix approach, the (harmonic) displacements and internal stresses at a given interface define the state vector, which in turn is related through the transfer matrix to the state vectors at neighboring interfaces.
STATE VECTORS
Consider a layered soil system as shown in Figure 1. The interfaces are dictated by discontinuities in material properties in the vertical direction, or by the presence of external loads at a given elevation. We define then the state vectors
Z= {~x,~y,i~,~xz,~yz, iSz}T=(~ 1 (1)
for Cartesian coordinates, or
Z-~-(Up, Us, Uz, Tpz,"rsz,~z}T=(~} (2)
1743
h
EDUARDO KAUSEL AND JOS]~ MANUEL ROESSET
for cylindrical coordinates. In these expressions, u, ~, p are the displacement, shearing and normal stress components at a given elevation in the direction identified by the subindex; and T Stands for the transposed vector. The factor i = ~ has been introduced for I/z, 6z in the Cartesian coordinates case for reasons of convenience. The superscript bar, on the other hand, is a reminder that the displacement vector C and stress vector S are functions of z only, i.e., it is assumed that the variation of displacements and stresses in the horizontal plane is harmonic.
For Cartesian coordinates, the actual displacements and stresses at a point are obtained by multiplying U, S, by the factor exp i (~ot - k x - l y ) , i.e.,
P1771
r27 ° 2
1
1744
×
Fro. 1. A layered soil system.
in which w = frequency of excitation, and k and l are the wavenumbers. If we restrict our attention to a plane strain condition (i.e., plane waves), it,follows that l = 0 and the factor becomes simply exp i ( w t - k x ) .
For cylindrical coordinates, on the other hand, the variation of displacements and stresses in the azimuthal direction is obtained by multiplying up, uz, %z, az by cos /z0 and u0, ~0z by - s in /z0 (or by sin /tO and cos /tO, respectively) with /z --
STIFFNESS MATRICES FOR LAYERED SOILS 1745
0, 1, 2 , . . . . being an integer. The variation in the radial direction is obtained multiplying U, S by the matrix C (which is common to all layers)
t~
C = ~ d ~p C~, d(kp) C~
-c ,
(4)
(5)
in which C, = C, (kp) are cylindrical functions of ttth order and first, second, or third kind (Bessel, Neumann, or Hankel functions, respectively). The argument k is the wavenumber. This corresponds to the well-known decomposition of the displace- ments and stresses in a Fourier series in the azimuthal direction, and cylindrical functions in the radial direction. The variation with time is given again by the factor exp io~t.
Hankel functions are most frequently used in wave propagation problems, because they behave asymptotically like complex exponentials. For this reason, they can model [in connection with the term exp(ic0t)] waves traveling from infinity toward the origin (first Hankel functions) or from the central region toward the far-field (second Hankel functions). They show, however, a singularity for zero argument and cannot be used, in general, if the problem includes the origin.
In the transfer matrix method, the state vector at a given interface is related to that at the preceding one by the expression (Haskell, 1953)
Zj+, -- HjZj (6)
where ~ is the transfer matrix of the j t h layer. This matrix is a function of the frequency of excitation ¢o, the wavenumbers k, l, the soil properties, and the thickness of the layer. Again, in the particular case of plane waves (plane strain), the second wavenumber (l) is zero; the transfer matrix has then a structure such that motions in a vertical plane (SV-P waves) uncouple from motions in a horizontal plane (SH waves). It is interesting to note that the transfer matrix for cylindrical coordinates is identical to that of the plane strain case, and is independent of the Fourier index /~. This implies, among other things, that the solution for point loads can be derived, in principle, from the solution for the three line load cases of the plane strain case. (This is referred to as the inversion of the descent of dimensions.)
STIFFNESS MATRIX APPROACH
Referring to Figure 1, we isolate a specific layer and preserve equilibrium by application of external loads P1 = $1 at the upper interface, and P2 = -Se at the lower interface. From equation (6) we have
- P 2 = [/-/21 H 2 2 J [ P l J (7)
where Ho are submatrices of the transfer matrix/-/s- After some straightforward
1746 EDUARDO KAUSEL AND JOSl~ MANUEL ROESSET
matrix algebra, we obtain
Pl} f -HigH11 P2 = [H22H1-21Hl l - H21
or briefly
P = K U (9)
where K = stiffness matrix of the layer; P = external "load vector"; and U = displacement vector. It can be shown that K is symmetric.
In the case of a soil which consists of several layers, the global stiffness matrix is constructed by overlapping the contribution of the layer matrices at each "node" (interface) of the system. The global load vector corresponds in this case to the prescribed external stresses at the interfaces. Thus, the assemblage and solution of the equations is formally analogous to the solution of structural dynamics problems in the frequency domain. It follows then that the theorems and techniques available for these problems may also be applied to layered soils; an example is the use of substructuring techniques.
STIFFNESS MATRICES
Due to space limitations, the details of the formulation will be omitted, and only the final results will be given. Also, only the cases of plane (1 = 0) and cylindrical waves will be considered here; the formulation can, however, be extended to more general situations.
E x a c t s o l u t i o n . The elements of the 6 × 6 layer stiffness matrices given in Tables 1 to 5 were obtained solving the wave equation in Cartesian and cylindrical coordinates. For convenience, these elements are given in partitioned form. First the matrices for S V - P waves (representing rows/columns 1, 3, 4, 6) in Tables 1 to 4 and then the matrices for S H waves (representing rows/columns 2, 5) in Table 5. Also, the coupling terms are zero. The following notation is used
o~ = frequency of excitation \ ~
k = wavenumber
h = layer thickness
G = shear modulus (10) f a = C s / C p --- shear wave velocity/dilatational wave velocity /
r = x/1 - ( ~ / k C ~ ) 2
s = ~ / 1 - (o~ /kCs ) 2. /
While the hyperbolic functions used for the stiffness matrices could be changed into trigonometric functions by a trivial change in the definition of r and s, care must be exercised to ensure consistency in the evaluation of the functions. The complex numbers r, s defined by equation (10) are in the first quadrant for any real k > 0, whether or not the soil has damping. For real k < 0, these numbers are in the third quadrant, (so that the product k r , k s remains in the first). Hence when k -+ 0 +, then r --~ 1, s --) 1, while when k -o 0-, r --+ -1, s --) -1. It follows that the elements of the stiffness matrices are checkerboard symmetric/antisymmetric with respect to
S T I F F N E S S MATRICES FOR L A Y E R E D SOILS 1747
T A B L E 1
SV-P WAVES--STIFFNESS MATRIX FOR NONZERO FREQUENCY, NONZERO WAVENUMBER
v > O k > O
1 -- 82 I 1_s (crss - rsC'8r)
K l 1 = ~ - ( 1 - C ~ C ~+rsS~S ~)
C r = cosh krh S ~ = sinh krh
C ~ = cosh ksh S s = sinh ksh (1) D = 2 ( 1 - c r c ~) + ~s + rs s r s ~
--(1-- CrC~ + rsS~S~) I I + S 2
~(csSr rserS s) ~-~-'-{~ ~}
K22 = same as K . , with off-diagonal signs changed
1 - s z f ! (rsSr - S~) -(or - C`) K,2 = ~ C r _ C ~ 1_ (rsS ~ _ S r)
r
K 2 , = K r
Half-space (opening downward)
K = 2 k G r ~ { r l L (1-rs) ;}-{~ ~}] (for half-space opening upward, reverse off-diagonal terms)
T A B L E 2
SV-P WAVES--STIFFNESS MATRIX FOR ZERO FREQUENCY, NONZERO WAVENUMBER
~ = 0 k > 0 ~ = k h
C = cosh ~ S = sinh
D = (1 + a~)~S 2 - ~2(1 - a2) 2
(1 + a~)S 2 (1 + a~)SC+ ~ ( 1 - a S) -
K22 = same as KI~, with off-diagonal signs changed
1 { K ( 1 - ~ 2 ) C - ( 1 + ~ 2 ) S - ~ ( I - ~ ) S } Kj2 =-~ ~(1 - ~2 )S - ( ~ ( 1 - ~ ) C + (1 + ~2)S)
K2~ = K ~
Half-space (opening downward)
K 2kG ~ 1 - a ~ 1 j
(for k < 0, reverse diagonal terms) (for half-space opening upward, reverse off-diagonal terms).
1748 EDUARDO K A U S E L AND JOSE M A N U E L ROESSET
positive/negative values of k. For negative frequencies (w < 0), the damping term in equation (36) reverses sign, i.e., fi = : f l sn(~o).
D i s c r e t e so lu t ion . If the layer thicknesses are small as compared to the wave- lengths of interest, it is possible to linearize the transcendental functions which govern the displacements in the vertical direction. This procedure was first proposed by Lysmer and Waas {1972) and later generalized by Waas (1972) and Kausel (1974), although not in the context considered here. The technique was also used by Drake
T A B L E 3
S V - P W A v E - - S T I F F N E S S MATRIX FOR NONZERO FREQUENCY AND
ZERO WAVENUMBER
K = pC~w.
t o > O
toh T ~ - c~
cot ~?
1 - cot aT 1 a
sin 1 1
a sin aT
k = 0
~h
1
sin T
cot
1 1
a sin a~
1 - cot aT a
Half-space
K = iwpC~ ( 1 _al]
T A B L E 4
S V - P WAVES--STIFFNESS MATRIX FOR ZERO FREQUENCY AND ZERO WAVENUMBER
~ = 0 k = O
K = G 07 - - - a 2 h - 1
_± a 2
Half-space
K = 0 ( the null matrix)
(1972) to study alluvial valleys, while an extension to strata of finite width was given by Schlue {1979). In principle, this technique is restricted to layered soils over rigid rock, although analysis of soils over elastic half-spaces could be accomplished with a hybrid formulation (i.e., taking the exact solution for the half-space only). The principal advantage of the method is the substitution of algebraic expressions in place of the more involved transcendental functions. Hence, the eigenvalue problems for the natural modes of wave propagation are algebraic, and may be solved by
STIFFNESS MATRICES FOR LAYERED SOILS 1749
standard techniques. The layer stiffness matrices in the discrete case may be obtained as
K = A k 2 + B k + G " ~2M, (11)
where k = wavenumber, ~ = frequency of excitation, and A, B, G, M are the matrices given in Table 6.
TABLE 5
S H WAVES
(a) Nonzero wavenumber, nonzero frequen'Cy
k > 0 , o~>0 _1 } sinh k s h - cosh k s h
Half-space K = k s G .
(b) Nonzero wavenumber, zero frequency
k > 0 , ~o=0 Same as (a), with s = 1. (s = - 1 if k < 0).
(c) Zero wavenumber, nonzero frequency
k = 0 , w > 0
s in~ cos~ ' ~ -- - ~ Half-space K = i~ooC~
(d) Zero wavenumber, zero frequency
k = 0 , w = 0 K = 0 (the null matrix) Half-space K = 0 .
E X A M P L E S OF APPLICATION
Externa l loads. Consider the layered soil over elastic rock shown in Figure 1. The equilibrium problem is then
K12
K12 + K~I
K2~ /~2 K~2 + K31 K~
K391 K32 + K 4
tY3 P3
t74 P4
(12)
in which the superindex identifies the layer. The stiffness matrix has a tridiagonal structure, and is symmetric.
1750 E D U A R D O K A U S E L A N D J O S E M A N U E L R O i ~ , S S E T
~9
A
- k . i . . . ~ . . ~
i i
• ~ • ° ¢ . q •
. ÷ . . I ~ . . ~ . .
I
I • + • • " ~ " " ~
I " " ÷
i
H
÷ • " -4-- •
÷ " " -k
~q
II
¢q ° . ~ . .
~l ~ H
• ÷ • • +
i
. ? • . ~ •
÷ • ÷ ,,.<
I
. ~ . . O i
• . ? •
II
3
i
+
: ¢
+
il l i II II
STIFFNESS MATRICES FOR LAYERED SOILS 1751
As a specific example, a horizontal (in-plane) line load would correspond to P1 = (1, 0, 0)T, i52 = /~ = /54 = 0. (The Fourier transform of a line load [6(x), 0, 0] T has constant spectral amplitude in the wave number domain.) The displacements at any point would thus be obtained solving, by Gaussian elimination, for U1 --) U4 as a function of k, and computing their inverse Fourier transforms, for instance, with the Fast Fourier transform (FFT) algorithm. (For real k and soils with damping, the stiffness matrix is never singular.)
For problems formulated in cylindrical coordinates, on the other hand, the procedure is entirely analogous. We would begin by expressing the external load vector P1 (P, 0) = (pp, po, pz)T at a given (j th) elevation in terms of its transform as
PJ= E T. kC .ak i~0
(13)
with T, = diag(cos #0, - s in #0, cos #0) if the loadings are symmetric with respect to the x axis, or T, = diag(sin #0, cos #0, sin #0) if the loadings are antisymmetric with respect to this axis. Also, C = C(J,). This implies
fO °~ fo 2~r ~. = a, pC T, Py dO dp (14)
with orthogonalization factor a, = 1/2~r if # = 0, and a, = 1/qr if # # 0. To compute the displacements, we would solve for ~ as a function of k from equation (12), with loading ~ .
The displacements in the spatial domain are then
U~ = y~ T, kC(J} dk. (15) #=0
Actually, since the stiffness matrix is independent of the Fourier number #, it is more advantageous to solve equation (12) only once with three unit loading vectors P / = (1, 0, 0) r, p / , = (0, 1, 0) T pj,,, = (0, 0, 1) T (solving simultaneously for various loading conditions requires little extra effort, since most of the computational work is associated with the triangularization of the stiffness matrix, not with the backsub- stitution). These transfer functions for unit loads (line loads!) would then be combined appropriately, taking into consideration the actual value of the compo- nents of Pj. The Hankel transform in (15) may be evaluated nmnerically with available fast Hankel transform routines (see, e.g., Aspel, 1979; Kausel and Bouc- kovalas, 1979), provided that the soil has material damping so that Uj has no real poles.
When using the discrete formulation instead of the continuous formulation to determine the transfer functions for either a plane-strain or cylindrical problem, it is of particular interest to take advantage of Hermite interpolation. Since
K U = P (16)
with unit load vector P, it follows that (the prime denotes the derivative d/dk)
K ' / 7 + K/.7' = 0 (17)
1752 EDUARDO KAUSEL AND JOSg MANUEL ROESSET
and in view of equation {11),
KU" -= - ( 2 k A + B) 0 (18)
Since after solving for U, the triangular form of K is available, solving for U " requires modest extra effort. Having both U and U' increases substantially the resolution of the transfer functions, and as a result, fewer points are required to resolve them in the wavenumber domain. This in turn implies significant computational savings.
Amplification problem. In the classical problem of wave amplification through layered media, solutions are found using the Haskell-Thompson algorithm for S and P waves of given amplitude and incidence angle. In this paper, on the other hand, use will be made of substructuring techniques, finding the solution for the layered system without recourse to wave content, via the known solution for an arbitrary reference system, which may be labeled the "free field" or sometimes also the "rock outcropping." The displacement and stress field associated with this reference system may be called "the free-field solution." To illustrate this concept, consider
6 . . 7
a b c
FIG. 2. A soil profile.
the soil profile shown in Figure 2a, consisting of three distinct layers over an elastic half-space which is subjected to a seismic disturbance coming from within the half- space. The displacements at the various elevations are desired. Assume also that the displacements induced by the same source on either of the systems b or c are known. The material properties of the half-space (but not of the layers) is the same in all three systems. Taking (spatial) Fourier transforms of the observed motions and internal stresses, we can compute the free-field solutions in the wavenumber domain Us, U6, S~ for system b and U7 for system c. The solution for system a may then be obtained from equation {12), with P1 = P2 =/53 = 0, and
P4 = K4~-~/6 -- S6 (19)
if using system b (free field) as reference, or
P~ = K~O7 ($7 -- 0) (20)
S T I F F N E S S MATRICES FOR L A Y E R E D SOILS 1753
if using system c (rock outcropping) as reference. On the other hand, if the layer overlying the half-space in reference system b is identical to the third layer in system a, then the solution could also be obtained from equation (12), deleting the fourth row and column, and replacing K], by
= K 2 , K 3 + K4)-lK231, K 3 K ~ I - 12~ 22 (21)
i.e., by the impedance (stiffness matrix) of the half-space as observed from elevation 3. The load vector in this case would then be P1 = P2 = 0,/~3 = K~CT~.
Note that the motion at the surface of a half-space is known in closed form for a number of wave patterns (body and surface waves); hence, the above technique can be used to study the response of layered systems to these specific input waves. It is also interesting to consider the special case when all layers in system a have the same material properties as the half-space; the motion then computed at the free surface is not exactly the same as at the outcropping, because of the delay (phase shift) and attenuation of the waves in their travel from elevation 4 to elevation 1.
Propagation modes. The natural modes of wave propagation are obtained from the eigenvalue problem that follows from setting the load vector equal to zero. Thus, in the example characterized by equation (12), we would set Pi = P2 = P~ = P4 = 0. When using the exact expressions for the layer stiffness matrices, the resulting eigenvalue problem is transcendental, has in general, infinitely many solutions, and must be solved by search techniques. On the other hand, in the case of finite strata over rigid base, it is more convenient to use the discrete solution, since the associated eigenvalue problem is algebraic in k, has a finite number of modes, and may thus be solved by standard techniques. Furthermore, it is interesting to show that the quadratic eigenvalue problem that characterizes the discrete solution
(Ak 2 + Bk + C)X = 0 (22)
(with C = G - ~2M, and the implicit understanding that the matrices for the various layers have been overlapped) may be reduced to a linear eigenvalue problem having the same dimension as the quadratic. While a quadratic eigenvalue problem may always be solved as a linear eigenvalue problem of double dimension, this is not necessary here because of the special structure of the matrices involved. To show this, we begin rearranging rows and columns by degrees of freedom rather than by interface (i.e., grouping first all horizontal, then all vertical, and finally all antiplane degrees of freedom). The resulting eigenvalue problem is then of the form
[ k2A~ + Cz kB~z kBTz k2Az + Cz Xz
k2Ay + Cy X~ (23)
with uncoupled antiplane mode Xy. The matrices Ax, Cx, etc., are tridiagonal and, except for B~z, are symmetric. The in-plane eigenvalue problem may then be transformed into
[ k B x z k2Az+Cz kXz (24)
which is a linear (although nonsymmetric) eigenvalue problem in k 2. An alternate
1754 EDUARDO KAUSEL AND JOSI~ MANUEL ROESSET
linear eigenvalue problem is also
B ~ k2Az + C~J[ Xz J =
having a characteristic matrix which is the transpose of that in equation (24). Both of these eigenvalue problems yield the same eigenvalues and have associated
"left" and "right" eigenvectors
[ Xz J and Z = kXz (26)
which are mutually orthogonal with respect to the characteristic equation. Using relatively straightforward algebra, it is possible to obtain a linear, symmetric
eigenvalue problem
BTzA;1Cx -kZAz + BT~A;1C~A;1BTz: - C~ kXz =
which is formally interesting, but computationally not very attractive. While the characteristic matrices are symmetric, they are not positive definite, so that complex eigenvalues k 2 may occur, as expected.
Finally, it is easy to derive the dispersion relation and group velocities from equation (24). Derivating this equation with respect to frequency, and premultiplying by the transposed left eigenvector Y having the same eigenvalue as Z (identified by the subindex j) , we obtain after brief algebra
dkj_ ,o ~T~Zj_ 1 do~ kj yjT/~zj group velocity
(28)
in which
fi-= [.Bxz Az (29)
A similar (but not identical) relationship was derived first by Lysmer (1970). More details on the discrete eigenvalue problem can be found in the work of Waas (1972).
DISTRIBUTED LOADS AND BODY LOADS
The technique presented so far has been restricted to cases of external loads that are "concentrated" at a given elevation; the load vectors P1 are namely the spatial transforms of loads distributed in horizontal planes. However, since the stiffness matrices are available in closed form, it is possible to derive as well the solutions for loads that are distributed across the thickness of the layers. This can be achieved by fixing the interfaces of the layer at which the external loads are prescribed, and determining the reactions necessary to enforce this fixity. Applying then forces equal and opposite to the reactions, the displacements at the interfaces can be determined.
STIFFNESS MATRICES FOR LAYERED SOILS 1755
Consider a layer of thickness h (Figure 3), which is subjected to an e lementary load di 5 = Qd~ at a point at a distance ~ from the upper interface. Identifying with the superindices 4, V = h - ~, the stiffness matr ices of layers having these thicknesses, we can write then for the two layer system
{K~I K~2
K~ K~2K~t + K~I Kh ~ i dU~ ~ = I Qd_Q . K~2J LdU2J [ dP2 J
(31)
Since dU1 = dU2 = 0 when interfaces are held fixed, we can solve for the e lementary reactions
dP~ = KI2(K~2 + KT~)-~(~d~ dR2 = K ~ (K~2 + K~)-XOd~.
(32)
On the other hand, when Q = 0 and the interfaces are not held fixed, we may condense out the intermediate auxiliary interface and recover the stiffness matr ix
h
(
) 2
FIG. 3. A layer of thickness, h.
f dP =Q d~
for the complete layer (without superindex). This implies the identities
K~2 (K~2 + K~I )--1 = -K12 (K~2)-1
K~I (K~2 + K ~ ) -~ = -K2~(K~) -~.
(33a)
(33b)
Hence, the negative fixed-end reactions are (h - ~, $ are the arguments of Q, they are not factors).
fo h - P a = K~2 (K~2)-~{~(h - ~) d~
-P~ = K~ (K~I) -1 O(~) d~
(34a)
(34b)
which should be applied in the usual manner to the relaxed interfaces to obtain the displacements.
Computa t ion of the previous integrals is not difficult, since in t h e inverse of K~2,
1756 EDUARDO KAUSEL AND JOSE MANUEL ROESSET
the hyperbolic functions appear linearly. For example, in the case of S V - P waves, the inverse is (see also Table 1)
(K~2)_1 1 2 I ~ (rss~s-S~r) c~r-c~s I - 2kG 1 - s 2 1 (35) I -(C~ r - C~ s) - - (rsS~ r - SC) $
Thus, computation of a load that varies with the n th power of ~, i.e., Q = (~/ h) nQ0, requires computation of elementary integrals of the form
fo h ~n sinh ks~ d~, f0 h ~n cosh ks~ d~
and similar ones with argument r. These integrals are easily evaluated integrating by parts.
Numerical implementation
Since the stiffness matrices involve terms with hyperbolic functions, consideration must be given in numerical applications to the cases in which the real part of the arguments is large. This situation may develop either when the wavenumber, or the frequency (or both) are large. Considering a soil with damping, the arguments kr, ks in Tables 1 and 5(a) are
2 - - + 2 ' w k r = _ ~ k 2 (_~_p)2 ( 1 - ifip)2 = ~/k2 (_~_p)2 trip (_.C~_p) (36)
k s = k 2 _ w ( 1 - i f l A 2 • k 2 - + 2ifl~ w
in which tip, fis are the fractions of linear hysteretic damping for dilatational and shear waves, respectively.
If e is some appropriate maximum value tolerable for exponential functions (say, e = 50), then
[Real (krh) [ > e does not imply [Real (ksh)[ > e
and vice versa. Nevertheless, the situation in which both [Real (krh) I, I Real (ksh) I exceed the maximum value e is also possible. For computational purposes, ill- conditioning and numerical problems are avoided in these cases by a division of the terms in the stiffness matrices (and, of course, D) by either C r, C a, or CrC ~, and going over into hyperbolic tangents; some of the terms become then negative exponentials, which tend to zero as the arguments increase. In particular, the coupling terms K12, K2~ tend to zero, implying that the interfaces behave as half- spaces. Provided then that the subroutines that generate the layer stiffness matrices account for these situations, no numerical or stability problems are encountered for very high frequencies, large wavenumbers, or thick layers. Numerical instabilities at the cutoff frequencies of the layers are also avoided by the inclusion of material damping.
STIFFNESS MATRICES FOR LAYERED SOILS 1757
Note the following
~o (a) for k >> - - kr -o k c ; ' and the system behaves essentially statically [see Tables 2 and 5(b)]
w iw (b) for ~ >> k, kr -o ~ (1 - ifi}
and the system behaves essentially as one with zero wavenumber [see Tables 3 and 5(c)]. Similar expressions can also be written for ks.
Note also that in the global stiffness matrices which are obtained by overlapping the layer stiffness matrices, there are only 2 degree s of freedom/interface for the S V - P case, and only 1 degree of freedom/interface for the S H case.
CONCLUSIONS
An alternate formulation of the Hankel-Thompson algorithm for wave amplifi- cation through layered media has been presented. The method is based on a formulation with stiffness matrices that offer some advantages over the more conventional transfer matrix method. It should be emphasized, however, that the method is not more general than Haskell-Thompson's, but merely more efficient for numerical implementation.
Use of the factor i = ~--1 in front of uz, ~z in equation (1) results in stiffness matrices which are symmetric; for a static problem, they are in addition real. By contrast, the transfer matrices are not symmetric. This implies on the one hand less storage requirements, and on the other, fewer operations (solving a system of linear equations with a bandlimited, symmetric matrix requires fewer operations than multiplication of the nonsymmetric transfer matrices). Use of the stiffness matrices is also advantageous when solving simultaneously for multiple loading and/or boundary conditions, :since most of the computational effort lies in the triangulari- zation of the stiffness matrix, not in the backsubstitution. Aiso standard techniques in structural analysis which are well known to engineers, such as dynamic conden- sation and substructu~ing, are readily applicable. Particularly attractive is the use of the discrete formulation when dealing with layered strata of finite depth. Also, no special problems (ill conditioning, stability) are encountered in the case of very thick layers, high wavenumbers, or frequencies, since uncoupling of the layers occurs naturally (K12 -~ 0) when material damping is included.
ACKNOWLEDGMENTS
This study was made possible in part by the National Science Foundation Grant ENG 79-08080 entitled "Numerical Procedures for Foundation Mechanics Problems."
REFERENCES
Apsel, R. J. (1979). Dynamic Green's functions for layered media and applications to boundary-value problems, Ph.D. Thesis, University of California at San Diego.
Drake, L. A. (1972). Love and Rayleigh waves in non-h0rizontally layered media, Bull. Seism. Soc. Am. 62, 1241-1258.
Haskell, N. A. (1953). The dispersion of surface waves on multilayered media, Bull. Seism. Soc. Am. 73, 17-34.
Kansel, E. {1974). Forced vibrations of circular foundations on layered media, Research Report R74-11, Soils Publication No. 336, Department of Civil Engineering, M.I.T., Cambridge, Massachusetts.
Kausel, E. and G. Bouckovalas (1979). Computation of Hankel transforms using the Fast Fourier
1758 EDUARDO KAUSEL AND JOSI~ MANUEL ROESSET
transform algorithm, Research Report R79-12, Order No. 636, Department of Civil Engineering, M.I.T., Cambridge, Massachusetts.
Lysmer, J. (1970). Lumped mass method for Rayleigh waves, Bull. Seism. Soc. Am. 60, 89-104. Lysmer, J. and G. Waas (1972). Shear waves in plane infinite structures. J. Eng. Mech. Div., ASCE, 18,
859-877. Schlue, J. W. {1979). Finite element matrices for seismic surface waves in three dimensional structures,
Bull. Seism. Soc. Am. 69, 1425-1438. Thomson, W. T. {1950). Transmission of elastic waves through a stratified soil medium, J. Appl. Phys.
21, 89-93. Waas, G. (1972). Linear two-dimensional analysis of soil dynamics problems in semi-infinite layer media,
Ph.D. Thesis, University of California, Berkeley.
DEPARTMENT OF CIVIL ENGINEERING MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS 02139 (E.K.)
DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF TEXAS AUSTIN, TEXAS 78712 (J.M.R.)
Manuscript received March 12, 1981
APPENDIX
Since the t rans format ion implied by equat ions (13) and (14) is not readily evident, it will be proved here.
Consider the vectors ,~ and X re la ted by the t rans format ion
x = y T. kC. dk. ~=0
(37)
Mult iplying f rom the left by T,, and integrat ing
f02 j0 T . X dO = T.T~, dO k C X dk, =
(38)
with T~ = diag(cos/~0, - s i n ~0, cos/~0) or T. = diag(sin #0, cos/~0, sin #0), and similar expressions for T.. Bu t
fo 2~ T . T . dO = 1 6~ I~, a
(39)
with
Iu = I = ident i ty matrix, and 1 / a , -- ~r
I , = diag (1, 0, 1) or diag (0, 1, 0), 1 / a , = 2~r
if/~ ~ 0
i f# = 0.
Also, 6,u is the Kronecker delta. Subst i tu t ing (39) into (38), and changing the d u m m y index ~ to/~, we obtain
~0 2~" ~0 ~ a~, T ~ X dO = I~ k C X dk. (40)
STIFFNESS MATRICES FOR LAYERED SOILS 1759
Consider now the integral on the right-hand side of the preceding equation
x = kC.X dk, ~ 0
(41)
which relates the vectors x = (x, y, z)T and .~ = (X, Y, Z)T. In terms of the components, this equation is written as
S kp = k k-~ c ;
-c,
dk. (42)
Substituting the identities
--'~ Cp.-1 (43a) c;+vp
C / - t~ = -C,+1, (43b) kp
we obtain
{x} (:¢ fC,-I -C.+1 ][(X+Y)/2] Yz = ]o k t C , _ l C,.I - C , l (X % Y ) / 2 I dk (44)
and from here
(x + y) /2]
(X -zy) /2 ~ = s0 Hence
~0 ~ X --b y = k C u - l ( X + Y) dk (46a)
x - y = - kC.÷~(X- Y) dk (46b)
z = - kC, Z dk (46c)
which are cylindric transforms of order ~ - 1, # + 1, #. The inverse transforms, if
1760
they exist, are
EDUARDO KAUSEL AND JOS]~ MANUEL ROESSET
f0 c¢ X + Y = pC~ l(x + y) dp
X - Y = - pC~+l(x - y) dp
fj z = - p c , * z do
in which
Cit* = J , (Bessel function)
Cit* = H, (Struve's function)
Cit* = Y, (Neumann function)
(47a)
(47b)
(47c)
X = pC*x dp. (49)
In the particular case C = C (J , ) , we have C* = C, so tha t
f , = pCx dp. (50)
On the other hand, since in equation (42) the matrix is diagonal when ~ = 0, it follows tha t
fo ; I~X =Iit p C x dp = pCI , x dp (51)
and in view of equations (40), (41),
fo fo IitX = ait p C T , X dO dp. (52)
(48)
which can be writ ten briefly
~- dp, P c ; c /* c *
-- it
if C, = J , (Bessel function)
if C, -- Y~ (Neumann function)
if Cit -- Hit (Struve's function).
It can be seen tha t when Cit = J , , formulas (46) and (47) are self-reciprocating. Repeating now in inverse order the steps tha t led to equations (46), we obtain
STIFFNESS MATRICES FOR LAYERED SOILS 1761
Since T, = I , Tn, it is sufficient to write finally
fofo = a n pC T , X dO do. (53)
Equations (37)and (43) are equivalent to equations (14) and (13) respectively. Note that in equation (12), a load I~P automatically produces displacements of
the form I n [7, since the antiplane degrees of freedom in the stiffness matrix are uncoupled from the in-plane degrees of freedom. Thus, equation (15) is valid as written.