Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 148

The previous equations are obtained by equating the real part with the imaginary
part of equations [4.23] and [4.24] and for [4.27] by assuming a resonance situation
( Z = ~ ).

Equations [4.25] and [4.26] assume the soil-structure interaction results in:

- decreasing the characteristic pulsation Z s of the embedded basis structure (Z
Z s );
- increasing the damping of system ( ~ ![) with regard to the embedded basis
structure;

- decreasing the effective incident stress at the basis of the structure ( u u g ).

g

The conclusions are shown in Figure 4.12, which represents a circular
foundations lying on an homogenous elastic semi-space, and the relative variations
Z /Z s, [ ,

u / u g are a function of the non-dimensional parameters:

g

h Ȧ h m

s

h 1 , s = , m [4.28]

r V

3

ȡr

s

in which r is the radius of the foundations, and U and V s are respectively the volumic
density and rate of the waves S within the soil (equation [4.2]).

Figure 4.12 clearly shows that the influence of the soil-structure interaction is all
the more marked if the foundation soil is soft (increasing s) or if the structure is
massive (increasing m).

4.3.2. Expression of a soil-structure problem

Before examining the different methods employed to take soil-structure
interactions into account, the problem is worth formulating from a general point of
view. This formulation aims at dealing with phenomena due to finite elements. In
fact, the problem is so complex that resorting to digital methods cannot be avoided,
but in the rest of this chapter, we will try to point out steps that can be dealt with
analytically and those that are amenable to existing solutions.

Motion equations are obtained by referring to Figure 4.13, which schematizes a
soil-structure unit.