Page 257 - Numerical Analysis and Modelling in Geomechanics

238 SEISMIC MICROZONING USING NUMERICAL MODELLING
Substituting equations (8.5) and (8.6) into (8.7) and (8.8), we obtain:

(8.9)

(8.10)

which are valid in each homogenous region with the convention that m is equal
to 1 in the infinite domain and equal to zero in the inclusions. F i and v i are
respectively the stresses and displacements due to the incoming waves and ф i
physically represent the point sources distributed on the boundary, which modify
the displacement field generated by the incoming waves. The sources ф i are
unknown while G , T , f i and v i are known. Equations (8.9) and (8.10) can be
ij
ij
discretized assuming ф constant over each of N boundary segments into which
i
the entire boundary has been divided. Then imposing the continuity conditions
over the boundary between adjacent homogeneous regions and the conditions of
null stress on the interface with the air, the previous integral equations can be
transformed into a system of algebraic equations and solved in the unknown ф . i
The two major limitations to the method are the high frequency and the non-
linearity. In order to get a good approximation the dimension of each boundary
element should be a fraction (between and ) of the wavelength. In the case of a
long valley with soft soil conditions this limitation requires a great number of
elements to treat the high frequency field. A valley 1000 m long, with V shear
s
wave velocity of 240 m/s, has a wavelength of 8 m at a frequency of 30 Hz
and requires about 600 elements. Moreover it is also necessary to model the
rocky region, that is the infinite homogeneous region, at a distance in the x
direction at least 2 times the dimension of the valley on each side. The numbers
of degrees of freedom can reach very large values especially in the case of
multiple layers of soft soil. This is a great limitation, also taking into account
that the solution matrix is imaginary and sparse.
The problem of non-linearity can be overcome using a simpler program to
assess approximately the deformation of the soil under the seismic excitation, i.e.
the SHAKE program. Such deformation can be used to compute the modified
characteristic of the soil to apply to BESOIL for the final analysis.

The QUAD4M program
The QUAD-4 program is based on the Finite Element Method, FEM. It has been
implemented by Idriss et al. (1973), and updated as QUAD4M by Hudson et al.
(1993). In this method the actual continuum is represented by an assemblage of
elements interconnected at a finite number of nodal points. Details of the
formulation of the general method are available in several publications (e.g.
Desai and Abel, 1972; Zienkiewicz, 1977; Bathe, 1982).