Civil Engineering Reference
In-Depth Information
30
120
Geo nonlinear
Geo linear
∙
1
(
t
)
x
25
80
20
40
x
1
(
t
)
15
0
10
-40
5
Geo nonlinear
Geo linear
-80
0
-120
-5
0
5
10
15
20
0
5
10
15
20
Time (s)
Time (s)
0.6
25
∙∙
1
(
Geo nonlinear
Geo linear
y
t
)
0.4
20
0.2
15
x
1
ʺ
(
t
)
0
10
-0.2
5
-0.4
Geo nonlinear
Geo linear
-0.6
0
0
5
10
15
20
0
5
10
15
20
Time (s)
Time (s)
Figure 7.17
Global response comparisons of the one-story frame with geometric nonlinearity.
Assume that themassmoment of inertia are ignored for the rotational joints, the frame can then
be statically condensed to a 2-DOF system by eliminating DOFs #3 to #6 while retaining only
DOFs #1 and #2. By following Eq. (7.147a), the condensed global stiffness matrix becomes
kN
=
m
673
:
415
−288
:
698
K
o
=
ð7
:
167Þ
−288
:
698
210
:
869
and the 2 × 12
K
0
o
and 12 × 12
K
0
o
matrices can similarly be calculated using Eqs. (7.147b)
and (7.147c).
Let the mass be 2.5 Mg on each floor and the damping be 2% for the two modes of vibration.
This gives
Mg,
C
dd
=
kN s
=
m
2
:
50
02
:
5
2
:
3479
−0
:
7403
M
dd
=
ð7
:
168Þ
−0
:
7403
1
:
1618
and by using
M
dd
and
K
o
, the two periods of vibration are computed as 1.169 s and 0.349 s.
Then using Eqs. (7.149a), (7.149b), and (7.159), the calculations of the transition matrices in
the continuous form give
2
4
3
5
2
4
3
5
2
4
3
5
0
:
0
0
:
0
1
:
0
0
:
0
0
0
−1
−1
0
:
0 0
:
0
0
:
0 0
:
0
269
:
4 −115
:
5
−115
:
5
:
3
0
:
0
0
:
0
0
:
0
1
:
0
A
=
,
H
=
,
G
=
−269
:
4 115
:
5
−0
:
939
0
:
296
115
:
5
−84
:
30
:
296
−0
:
465
ð7
:
169Þ
