Civil Engineering Reference
In-Depth Information
L
M
i
()
=
()
Y
i
t
At
(4.18)
i
ˆ
ω
ii
where
A
i
(
t
) is given by the solution of equation (3.14) in Chapter 3. Alternatively, the equation
of motion can be solved numerically in the time or frequency domain. These approaches are
known as 'direct integration method' and 'fast Fourier transform' , respectively.
(g) Compute the total elastic restoring force as follows:
N
L
M
=
∑
i
()
=
()
RK Yt
=
F
At
M
F
(4.19)
i
i
ˆ
i
i
1
(h) Compute the total seismic base shear
V
B
. It can be obtained by summing the effective earthquake
forces over the height of the structure:
N
2
L
M
=
∑
ˆ
i
()
V
=
At
(4.20)
B
i
i
i
1
(i) Compute the relative displacement with respect to the base of the structure corresponding to the
i
th mode of vibration:
L
M
i
()
=
()
x
=
F
Yt
At
F
(4.21)
i
i
i
i
ˆ
i
Equation (4.16.3) makes damping frequency -dependent. The procedure illustrated in (d) to compute
ξ
i
will usually over-damp the higher modes of vibration, thus affecting the reliability of results for
high-rise structures or systems subjected to near- fi eld earthquake ground motions. Proportional damping
can be visualized as immersion of the structure in a non- physical fl uid whose viscosity becomes infi nite
for rigid-body motion of the structure (
ω
= 0). For higher frequency modes, viscosity acts to damp rela-
tive motion of the MDOF, with increasing effect as
ω
increases. Non- physical high - frequency vibra-
tions, also known as 'noise', generated by numerical response simulation can be damped by the term
β
K
.
The term
LM
i
2
ˆ
in equation (4.20) is defi ned as the ' effective modal mass '. This quantity generally
diminishes inversely with the order of modes. For example, in regular shear frame buildings, the fun-
damental mode accounts for up to 85-90% of the total mass. Therefore, summing the response for the
fi rst two to three modes will represent the MDOF system. On the other hand, slender long- span bridges
usually respond in tens or even hundreds of modes, all of which will be required to achieve adequate
representation of the MDOF. The sum of the modal masses is the total mass of the structure; i.e.:
i
N
N
2
L
M
∑∑
i
=
M
(4.22)
ˆ
i
i
i
=
1
i
=
1
Equations (4.19) and (4.21) express the entire history of actions and deformations of MDOF struc-
tures. Lumped systems with
N
degrees of freedom possess
N
independent mode shapes. It is thus pos-
sible to express the deformed shape of the structure in terms of amplitudes of these shapes by treating
them as generalized coordinates Y(
t
) as shown in equation (4.18) .
In seismic analysis, the evaluation of maximum values of displacements and internal forces rather
than their whole time history, is often the primary purpose, especially in design. Peak responses obtained
for individual modes can be combined using statistical methods. The modal spectral (or spectral, or
