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Kausel - Stiffness Matrix

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Bulletinofthe SeismologicalSocietyofAmerica,Vol.71,No.6, pp. 1743-1761,December1981

S T I F F N E S S M A T R I C E S FOR LAYERED SOILS BY EDUARDO KAUSEL AND JOSE MANUEL Roi~,SSET ABSTRACTThe 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 symmetric; (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. T he 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 corresponds 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. Th e 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. T he 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=(~1for Cartesian coordinates, or

(1)

Z-~-(Up,Us,Uz,Tpz,"rsz,~z}T=(~}1743

(2)

1744

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.,

P1771h

r27 21

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 i n /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)

(4)t~

C

=

~

d

(5)

~p C~,

d(kp) C~

-c,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 displacements 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 = - S e at the lower interface. From equation (6) we have

-P2

= [/-/21

H22J[PlJ

(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} fP2

-HigH11

= [ H 2 2 H 1 - 2 1 H l l - H21

or briefly P = KU (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 f a = C s / C p --- shear wave velocity/dilatational wave velocity /r = x/1 s = ~/1(~/kC~) 2 ( o ~ / k C s ) 2. /

\~

(10)

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

SV-P

TABLE 1

WAVES--STIFFNESS MATRIX FOR NONZERO FREQUENCY, NONZERO WAVENUMBER

v>O C r = cosh krh C ~ = cosh ksh D=2(1-crc

k>O

S ~ = sinh krh

~) + ~s + rs s r s ~

(1)

S s = sinh ksh

1 -- 82 I 1_s(crss - rsC'8r)Kl1=~ - ( 1 - C ~ C ~+rsS~S ~)

--(1-- CrC~ + rsS~S~) I

I + S2

~(csSr rserSs) ~-~-'-{~ ~}

K22 = sam

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