Civil Engineering Reference
In-Depth Information
With a time step size of
Δ
t
= 0.01 s, the transition matrices are calculated as:
,
H
=
,
G
=
0
1
0
−1
0
63
A
=
ð3
:
144aÞ
−63
−0
:
4
,
H
d
=
−0
:
00010
−0
:
00993
,
G
d
=
0
:
996856 0
:
009970
−0
:
62808 0
:
992868
0
:
00628
0
:
62551
F
d
=
e
A
Δ
t
=
ð3
:
144bÞ
and the equations for performing nonlinear dynamic analysis become
x
1
x
1
a
k
+
x
0
d
,
k
0
:
996856 0
:
009970
−0
:
62808 0
:
992868
+
−0
:
00010
−0
:
00993
0
:
00628
0
:
62551
x
1
x
1
=
ð3
:
145aÞ
k
+1
k
<
=
2
4
3
5
<
=
2
4
3
5
θ
0
1
θ
0
2
θ
0
3
θ
0
4
θ
0
5
θ
0
6
m
1
m
2
m
3
m
4
m
5
m
6
1, 040
280
40
80
−280
−80
360
270
360
270
−270
−270
280
560
80
160
−560
−160
40
80
1,040
280
−80
−280
ð3
:
145bÞ
+
=
x
1
,
k
:
;
k
:
;
k
80
160
280
560
−160
−560
−280
−560
−80
−160
560
160
−80
−160
−280
−560
160
560
<
=
θ
0
1
θ
0
2
θ
0
3
θ
0
4
θ
0
5
θ
0
6
"
#
8
7
6
7
8
7
6
7
−
6
7
−
6
7
x
0
1
,
k
=
ð3
:
145cÞ
:
;
k
Assume that the plastic hinges exhibit elastic-plastic behavior with moment capaci-
ties of
m
b
= 15.0 kN m for the beam and
m
c
= 20.0 kN m for the two columns.
This gives
Δθ
0
i
=0
m
i
=20
:
0
m
i
≤ 20
:
0
m
i
>20
:
0
,
if
then
i
=1, 2, 3, 4
ð3
:
146aÞ
Δθ
0
i
=0
m
i
=15
:
0
m
i
≤ 15
:
0
m
i
>15
:
0
,
if
then
i
=5, 6
ð3
:
146bÞ
Finally, the equations for the displacements at the DOFs #2 and #3 (i.e.
x
2,
k
and
x
3,
k
), the elastic
displacement at DOF #1 (i.e.
x
0
1
,
k
), and the absolute acceleration responses at DOF #1 (i.e.
y
1
,
k
)
can be calculated using Eqs. (3.136), (3.115), and (3.141), respectively:
