Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 149

Figure 4.13. Superposition theorem for the soil-structure interaction

If we refer to the mass, damping and stiffness matrices as [M], [C] and [K], the
equation of motion is written:

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

Mu Cu Cu Q [4.29]

f

As the source of the motion (seismic focus) is generally not included in the
model, the load vector ^ `

Q has values not equal to zero only on the outer border of

f

the model.

When there is no structure, the equation of the open field motion is similar in
shape to that described by [4.29]. With the f indexes designating the mass, damping
and stiffness matrixes related to the open field alone, this equation will be:

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

Mu Cu Ku Q [4.30]

f f f f f f f

Equation [4.30] can only be solved by making certain assumptions about the
nature and the direction of the incident field propagation.

Assuming that:

^` ^ ` ^ `

uu u [4.31]

i f