Civil Engineering Reference
In-Depth Information
The governing equations of the FAM in Eqs. (3.59) and (3.60) are
Þ
Δθ
00
= 175
:
5
Þ
θ
0
k
m
k
+1
+ 1755
ð
ð
Þ
x
k
+1
− 1755
ð
ð3
:
110aÞ
x
0
k
+1
=10
θ
0
k
+
Δθ
00
ð3
:
110bÞ
Finally, the absolute acceleration response can be calculated using Eq. (3.62) as
y
k
+1
= −0
:
419
x
k
+1
−17
:
55
x
k
+1
−
x
0
k
+1
ð3
:
111Þ
Assume that the earthquake ground acceleration record is the 1940 El-Centro earthquake
record as shown in Figure 3.4, and the initial displacement and velocity are both set equal
to zero (i.e.
x
0
=
x
0
= 0). It follows that the displacement, velocity, acceleration, moment,
and plastic rotation response histories can be evaluated using Eqs. (3.109) through
Eq. (3.111), and the results are presented in Figure 3.8. The responses are plotted along with
the linear responses from Example 3.1 as comparisons (note that both the present example and
Example 3.1 use the same period of 1.5 s, damping of 5%, and the same El-Centro earthquake
ground motion as the input). In addition, the initial yielding moment and the backbone curve are
plotted in the figure as appropriate.
3.3 Nonlinear Dynamic Analysis with Static Condensation
In Section 2.6, the static condensation was derived for the nonlinear static analysis of framed
structures. In Section 3.1, the static condensation was briefly introduced in the dynamic anal-
ysis of linear systems. In this section, the knowledge gained from the previous sections is com-
bined for the dynamic analysis of nonlinear systems. The goal is to condense the stiffness
matrix when the mass moment of inertia is ignored at certain DOFs. But more importantly,
it provides an avenue to invert the mass matrix embedded in the
A
matrix of Eq. (3.50) that
would otherwise be undefined with a singular mass matrix.
Consider a moment-resisting frame with
n
DOFs and
q
PHLs, the equation of motion can be
written in the matrix form as
M
0
x
(
t
)
C
0
x
(
t
)
K
K
x
′
(
t
)
M
0
g
(
t
)
dd
d
dd
d
dd
dr
d
dd
+
+
=
−
ð3
:
112Þ
0
0
x
(
t
)
0
0
x
(
t
)
K
K
x
′
(
t
)
0
0
0
r
r
rd
rr
r
where the mass matrix
M
has been defined in Eq. (3.6), the damping matrix
C
has been defined
in Eq. (3.9), and the stiffness matrix
K
has been defined in Eq. (2.142). The vector
x
0
(
t
) is the
elastic displacement response,
x
(
t
) is the velocity response,
x
(
t
) is the acceleration response,
and earthquake ground acceleration vector
g
(
t
) corresponds to the effect of ground motion
at each DOF associated with nonzero mass. The subscript
d
denotes the number of degrees
of freedom that have nonzero mass, and subscript
r
denotes the number of degrees of freedom
that have zero mass moment of inertia. This gives
n
=
d
+
r
in an
n
-DOF system.
The governing equations of the FAM presented in Eqs. (3.43) and (3.44) can similarly be
written as:
x
(
t
)
d
d
r
m
(
t
)
+
K
″
″
(
t
)
=
K
′
K
′
ð3
:
113Þ
Θ
x
(
t
)
r
