Page 255 - Numerical Analysis and Modelling in Geomechanics

236 SEISMIC MICROZONING USING NUMERICAL MODELLING

Figure 8.10 Scheme 2D of BESOIL (Sanò, 1996),

formulation, the most popular, the unknowns are the values of displacements and
tractions. In the second, the problem is formulated in terms of force or moment
boundary densities; it is less popular in spite of the fact that such a distribution of
forces can give a better insight on the physical phenomenon of wave propagation.
BESOIL uses the indirect method and strictly follows the works of Sanchez-
Sesma et al. (1993); Sanchez-Sesma (1978, 1987, 1990); Dravinski and
Mossessian (1987) and Kawase (1988).
The basic hypotheses are the following:

• Plane motion, i.e. the soil particle velocities and displacements, lies on a plane
(x and z in Figure 8.10). This implies that all the soil mechanical properties
are independent of y;
• The seismic source is so far from the site that also waves are plane;
• The elastic medium is divided into plane regions with homogeneous
mechanical properties, i.e. density ρ, shear modulus G, Lamé modulus λ and
damping ξ.

Consider the elastic material in a homogeneous region V with boundary S; the
displacement field at a generic internal point r can be described, in the absence
of body forces, by a boundary integral:

(8.5)

where u (r) is the i-th component of displacement at r, G (r, r′) is Green’s tensor,
ij
i
i.e. the displacement in the direction i at point r due to the unit force applied in