Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 152

where:

- ^ u represents the vector of the relative displacements of the system with
regard to the foundation;
- ^ I represents a unit vector giving the direction of the stress

u ;

g

- > M and > K represent the mass and stiffness matrices of the system.

To simplify matters, the damping terms are omitted;
- or in two steps by expressing the vector of relative displacements ^ u as the
sum of the “kinematic” displacements ^`

u

and of the inertial

cin

displacements ^`

u

:

iner

^` ^`^`

uu u

[4.37]

cin iner

and simultaneously solving the two equation systems [4.38] and [4.39] below:

^`> @^` ^`

Mu Ku MI

[4.38]

ª º ª º

¬ ¼ ¬ ¼ sol cin cin sol g

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

Mu Ku M ª u Iu º

[4.39]

ª º

¬ ¼ ¬

¼

inert inert st cin g

in which [M sol ] and [M st ] represent the mass matrices of the soil and structure parts:

> @

¼

[4.40]

MM ª º ª M º

¬ ¼ ¬ sol st

The equivalence of [4.36], [4.38] and [4.39] is obtained by a mere addition of the
two last values whilst taking [4.37] and [4.40] relationships into account.