Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 150

Equation [4.31] defines the interaction displacement ^ `
i

which satisfies the

u

equation:

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

Mu Cu Ku Q [4.32]

i i i i

with:

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

QMMu ª º ª CCu º ª KKu º [4.33]

¬ ¼ ¬ ¼ ¬ ¼

i f f f f f f

The load vector ^`

i Q is determined from the open field displacements. For linear
systems, the superposition theorem is illustrated in Figure 4.13 [LYS 78]: the
interaction problem is divided into the sum of an open field soil response problem
[4.30] with a source problem [4.32] where the applied forces ^ `

i Q have components
not equal to zero only at nodes common to the soil-structure. This last problem is
analogous to a machine vibration problem. The total displacement for the integration
problem is then given by equation [4.31].

Equation [4.33] clearly shows that there is interaction as soon as a mass or
stiffness difference develops between the soil and the structure. To simplify matters,
let us suppress the damping term in the equation and restrict the problem to one of a
structure lying on the ground, being subjected to the vertical propagation of volume
waves (shear or compression). Under such conditions, in the open field, all points of
the ground surface will be driven by the same motion. If the foundations of the
structure are infinitely stiff, the last term of the equation becomes equal to zero, and
the load vector ^`

Q

is restricted to:

i

^` > @ > @ ^`

QMMu ª º

¼

[4.34]

¬

i f f

The stresses ^ `

i Q applied at the basis of the structure generate a movement of
the support that is equivalent to a force of inertia field within the structure.
Consequently, interaction only results from the inertia developed inside that
structure. This is referred to as the inertial interaction , and its effect is illustrated by
the example in section 4.3.1.

On the other hand, let us consider a structure laid in the ground, the mass of
which is equal to zero outside the ground, and equal (in value and repartition) to the