Civil Engineering Reference
In-Depth Information
Figure 8.4.
The three frequency ranges of an oscillator response spectrum
8.2.2.
Calculation of structural responses using the modal method
Let us consider a
linear behavior
structure, with the x
soil
(t) motion imposed on a
specified number of its points (for instance, structures with stiff soil foundation for
which we can suppose that the anchoring points within the soil follow the open field
motion).
The x(r, t) relative motions with regard to the soil confirm the system (r = space
variable):
2
2
M
GG GG
x/
t
A
x/
t + Kx =
M U Ȗ (t)
[8.1]
soil
with M, A, K being the operators of inertia, viscous damping and stiffness for the
structure blocked at its anchoring points. U represents the space function “unit
translation in the earthquake direction”, and J
soil
(t) is the soil acceleration (seismic
signal).
)
n
(r), Z
n
and m
n
being the modal elastic lines, resonance pulsations and
generalized masses of the non-dampened structure with displacement limit
conditions equals zero at the anchoring points. The a
n
(t) modal contributions
confirm the following uncoupled system (when supposing that the damping operator
becomes diagonal):
2
2
2
d a /dt
2
HZ
da /dt + Ȧ a
q
/ mȖ t
[8.2]
n
n
n
n
n
n
n
n
soil
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