Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 151

mass of the soil for the part that is in the ground. The expression for loads ^ ` Q

i

becomes:

^` > @ > @ ^`

QKKu ª º [4.35]

¬ ¼

i f f

These only result from the soil and structure stiffness difference for the part
embedded in the ground. Even without any mass difference, there is interaction; it is
referred to as kinematic interaction . It results from the stiffness of the foundations
that prevents them from following motions imposed by the ground. We saw before
that it was strictly equal to zero in some cases; it can be low in others (foundations
on flexible piles) or highly significant (stiff structure strongly cross-braced and sunk
deep in the ground).

In the most general case, interaction comprises both an inertial interaction and a
kinematic interaction.

Figure 4.13 and the previous arguments illustrate the two best methods for
solving the soil-structure interaction. Figure 4.13a corresponds to the overall
methods, the solutions to which are obtained via direct resolution of equation [4.29].
They do not resort to any notion of superposition, and thus they are theoretically
suited to non-linear problems. Alternatively, sub-structure methods rely either on the
division of Figure 4.13b-4.13c, or on similar divisions to solve the problem step by
step. These methods can indeed only be applied to linear problems suitable for
superposition.

4.3.3. Superposition theorem

Dividing the soil-structure interaction into inertial and kinematic interactions (as
discussed in section 4.3.2) not only has the advantage of allowing us to illustrate the
essential phenomena, it also generates a resolution method based on the sub-
structuring principle, the validity of which relies on superposition theorem [KAU
78], [ROE 73]. This theorem establishes that the response of the model in Figure
4.14 (on the left) subjected to a ü g acceleration on its basis can be obtained the
following ways:

- either in one step by solving the equation:

> @^` > @^` > @^` g

Mu Ku MIu [4.36]