Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 153

Figure 4.14. Superposition theorem (from [KAU 78])

Equation [4.38] gives the response of a structure without a mass to seismic stress

u . The solution gives kinematic interaction displacements that are used as loads in
[4.39], when imaginary inertial forces are applied to the structure.

g

When resolving equation [4.39], the modeling of the soil is indifferent: it can be
represented either with finite elements or using a stiffness matrix representing
foundations and the soil defined at the soil-structure interface. This stiffness matrix
results from the condensation of all degrees of freedom of the soil at the interface
[PEC 84]; the condensation is only possible for a resolution in the frequency field.
Within such a framework, the stiffness matrix is formed from the complex moduli
(section 4.2.2), which take damping into account. The stiffness matrix consists of a
real part (representing the stiffness of the foundations) and a fictitious part that
integrates all the damping phenomena (material and radiating). The terms of the
matrix depend on the frequency.

In the case of a structure with rigid foundations, it is legitimate to replace the
stiffness matrix (NxN), N being the number of interface nodes, with a (6x6) matrix
that gives the rigid body movements of the foundations; this matrix is called an
impedance matrix, and it can be conceptually represented by springs and dashpots
depending on the frequency. The result is that the solution of the kinematic
interaction problem is completely defined by rigid body movements of the
weightless structure; the latter can then be replaced by weightless foundations
subjected to the same seismic stress.

Examining the structure of equation [4.39] reveals that the solution ^`

iner u can
be interpreted as the displacement vector that is related to a fictitious support