Page 258 - Numerical Analysis and Modelling in Geomechanics

F.PERGALANI, V.PETRINI, A.PUGLIESE AND T.SANÒ 239
In earthquake response evaluations, the following set of equations are solved:

(8.11)

in which:

[M] = mass matrix for the assemblage of elements shown in Figure 8.11.
The dimension of the matrix is 2N×2N where N is the number of
nodes external to the lower boundary, i.e. the interface with the
rigid soil.
[C] = damping matrix for the assemblage of elements,
[K] = stiffness matrix for the assemblage of elements,
{u} = nodal displacements vector (dots denote differentiation with respect
to time), and
{R(t)} = earthquake load vector.

The equations of motion (8.11) are most readily solved by a direct numerical
method such as the step-by-step method (Wilson and Clough, 1962).
The solution proceeds by assigning modulus and damping values to each
element. Because these values are strain-dependent, they would not be known at
the start of the analysis and an iteration procedure is required as in case of the
SHAKE program. Thus, at the outset, values of shear modulus and damping are
estimated and the analysis is performed. Using the computed values of average
strain developed in each element, new values of modulus and damping are
determined from appropriate data relating these values to strain (Seed and Idriss,
1970; Hardin and Drnevich, 1972). Proceeding in this way, a solution is obtained
incorporating modulus and damping values for each element, which are
compatible with the average strain developed.
The main features of the QUAD4 program are the following.
The first feature regards the high frequency range. In order to get a good
response the dimension of elements should be a fraction, to , of the wavelength
of incident waves; this affects the dimensions of the matrices in equation (8.11),
that is the number of equations to solve, and the computing time. Moreover the
equivalent non-linear procedure can give imprecise results, especially in the field
of high frequencies.
The second, and more important, feature regards the lower boundary of the
FEM model, that is the interface with rigid soil. The FEM method assumes that
the boundary nodes move simultaneously, which means that the soil at the
bottom is infinitely rigid. This can be approximately right in many cases when a
well-defined contrast exists between underlain rock and soft soil. A consequence
of the previous condition is that such rigid boundaries can reflect and trap waves
going away from the soft soil. This can produce an overestimate of soil
amplification.