Civil Engineering Reference
In-Depth Information
Let the mass be 200.0 Mg on each floor, giving a total mass of 1,200 Mg for the entire frame,
and the diagonal nonzero mass matrix
M
dd
can be written as
2
4
3
5
000000
000000
000000
000000
000000
000000
M
dd
= 200
:
0×
I
=
Mg
ð3
:
157Þ
Using the resulting condensed
K
stiffness matrix in Eq. (3.156) and the diagonal
M
dd
mass
matrix in Eq. (3.157), the 6 periods of vibration are calculated and summarized in
Figure 3.20. The damping is assumed to be 2% in all six modes of vibration. This gives a damp-
ing matrix of
2
3
368
−116
−4
:
6
−4
:
7
−2
:
1
−1
:
3
4
5
−116
335
−131
−9
:
5
−6
:
3
−3
:
5
−4
:
6
−131
332
−130
−7
:
5
−7
:
0
C
dd
=
kN
s
=
m
ð3
:
158Þ
−4
:
7
−9
:
5
−130
306
−111
−9
:
2
−2
:
1
−6
:
3
−7
:
5
−111
261
−100
−1
:
3
−3
:
5
−7
:
0
−9
:
2
−100
152
Assume that all 84 plastic hinges exhibit elastic-plastic behavior with moment capacity of
the
i
th plastic hinge,
m
c
,
i
, calculated as
m
c
,
i
=
f
y
×
Z
i
i
=1,…, 84
ð3
:
159Þ
where
f
y
is the yield stress of steel and
Z
i
is the plastic section modulus of the
i
th plastic hinge.
This gives
Δθ
0
i
ðÞ=0
m
i
ðÞ=
m
c
,
i
m
i
ðÞ≤
m
c
,
i
m
i
ðÞ>
m
c
,
i
if
,
then
i
=1,…, 84
ð3
:
160Þ
By subjecting the 6-story frame to the 1995 Kobe earthquake ground motion as shown
in Figure 3.2, Figures 3.21 to 3.23 show the displacement, velocity, and absolute accel-
eration responses, respectively, of selected floors. Figures 3.24 to 3.26 show the responses
and hysteresis loops of selected plastic hinges. The plastic hinge locations include the base
of the exterior column (PHL #1), base of the interior column (PHL #13), exterior hinge of
the beam on the second floor (PHL #49), and interior hinge of the beam on the third floor
(PHL #56).
Example 3.10 Ten-story Moment-Resisting Steel Frame
Consider the 10-story moment-resisting steel frame as shown in Figure 3.27. Assuming
the members are axially rigid, this gives a total of 50 DOFs (i.e.
n
= 50) and 140 PHLs
(i.e.
q
= 140). No mass moment of inertia is assumed at the rotational joints, and therefore
