Civil Engineering Reference
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of frequencies; the dashpot theoretically explains both radiation damping and
material damping as defined in section 4.2.2. With a view to simplifying the
presentation, we assume that material damping can be neglected relative to the
radiation damping (elastic behavior of the soil), which is a valid assumption for a
homogenous medium and low to medium amplitude seismic stresses.
Figure 4.11.
Soil-structure interaction simplified model
The system in Figure 4.11 has 3 degrees of freedom:
- the horizontal displacement u
1
of mass m;
- the displacement u
0
of the foundations;
- the rotation T of the foundations,
and is subjected to a horizontal displacement of the supporting soil, with a pulsation
harmonic Z and an amplitude u
g
.
The dynamic balance equations of the system can be derived from Lagrange's
equations [CLO 75] if we take q
i
as generalized variables:
- q
1
= u, relative displacement of the mass in relation to A;
- q
2
= u
0
, displacement of the foundations;
- q
3
= T, rotation of the foundations.
The relationship between the absolute displacement and u
t
of the mass m and the
previous variables is obvious:
t
u
u + u + u + hș
[4.13]
g
0
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