Civil Engineering Reference
In-Depth Information
First, assume the column has other properties of
C
= 0.0 Mg s/m,
I
= 52.64 m
4
, a yield
displacement of
d
y
= 15 cm, and an elastic-plastic moment versus plastic rotation relationship.
This gives the following structural properties:
3
= 157
:
92 kN
=
m
K
=3
EI
=
L
3
= 3 × 1000 × 52
:
64
=
1ðÞ
ð3
:
99aÞ
K
0
=3
EI
=
L
2
= 3 × 1000 × 52
:
64
=
1ðÞ
2
= 1579
:
2kN
ð3
:
99bÞ
K
00
=3
EI
=
L
= 3 × 1000 × 52
:
64
=
10 = 15, 792 kNm
ð3
:
99cÞ
p
M
=
K
p
1
=
157
:
92
T
=2
π
=2
π
=0
:
5s
ð3
:
99dÞ
m
p
=
d
y
×3
EI
=
L
2
=0
:
15 × 1579
:
2 = 236
:
9kNm
ð3
:
99eÞ
Δθ
00
=0
m
k
= 236
:
9
m
k
≤ 236
:
9
if
m
k
> 236
:
9
,
then
ð3
:
99fÞ
With a time step size of the earthquake ground motion of
Δ
t
= 0.01 s, the matrices in
Eq. (3.56) are evaluated as:
,
,
0
:
992115 0
:
009974
−1
:
57498 0
:
992115
−0
:
000100
−0
:
009921
0
:
01575
1
:
5667
F
d
=
H
d
=
G
d
=
ð3
:
100Þ
The recursive equation as given in Eq. (3.57) becomes
=
x
k
x
k
+
a
k
+
x
0
k
x
k
+1
x
k
+1
0
:
992115 0
:
009974
−1
:
57498 0
:
992115
−0
:
000100
−0
:
009921
0
:
01575
1
:
5667
ð3
:
101Þ
and the governing equations of the FAM in Eqs. (3.59) and (3.60) are
Þ
Δθ
00
= 1579
:
2
Þ
θ
0
k
m
k
+1
+ 15792
ð
ð
Þ
x
k
+1
− 15792
ð
ð3
:
102aÞ
x
0
k
+1
=10
θ
0
k
+
Δθ
00
ð3
:
102bÞ
Finally, the absolute acceleration response can be calculated using Eq. (3.62) as
y
k
+1
= −157
:
92
x
k
+1
−
x
0
k
+1
ð3
:
103Þ
Assume that the earthquake ground acceleration record is the 1995 Kobe earthquake
record as shown in Figure 3.2, 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. (3.101) through
Eq. (3.103), and the results are presented in Figure 3.6. 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 0.5 s, damping of 0%, and the same Kobe earthquake
ground motion as the input). In addition, the yielding envelope and the backbone curve are
plotted in the figure as appropriate.
