Civil Engineering Reference
In-Depth Information
20
40
20
I
y
(cm
4
)
A(
cm
2
)
M
py
(kNm)
Member
Section
10
3
5
8
12, 67 203 203UC 52
×
×
×
×
5259
66.3
1 5 5 .9
40
80 40
5000
20
23, 78 152 152UC 37
2210
47.1
8 5 .0
2
4
7
247 356 171UB 51
14136
64.9
2 4 6 .4
5000
6
1
358 254 102UB 25
3415
32.0
8 4 .2
6000 6000
(kN,mm units)
(b) Section properties
(a) Frame and loads
28.1
28.1
56.1
259.9
292.8
160.2
453.7
14.0
160.2
651.2
56.1
112.2
56.1
28.1
448.5
448.5
cr
p
=6.9
06
=1.403
(c) Elastic
deflections
(d) Elastic
buckling
(e) Plastic
collapse
First-order
elastic
Elastic
buckling
Second-order
elastic
First-order
plastic
Quantity Units
153.1
M
M
M
M
M
M
kNm
0
153.9
236.0
4
5
76
74
78
8
12
23
67
78
4
5
2
3
kNm
71.7
0
72.9
84.2
kNm -119.3
0
0
0
0
-130.5
-155.9
kNm -172.8
-184.7
-240.9
kNm
53.5
54.2
85.0
kNm
-62.5
-64.4
-84.2
N
N
N
N
kN
-103.3
-713.6
-100.9
-144.9
kN
-37.6
-259.9
-37.6
-56.1
kN
-136.7
-943.8
-138.3
-191.7
kN
-42.4
-292.6
-42.3
-56.1
v
v
u
u
mm
44.7
(0.003)
46.1
-
-
-
-
mm
80.1
(0.001)
83.2
mm
85.5
(0.826)
100.3
mm
121.4
(1.000)
140.4
(f) Analysis results
Figure 8.4
Analysis of an unbraced frame.
by carrying out a first-order elastic analysis of the frame under the initial loads.
Itthenusesafiniteelementmethodtodeterminetheelasticbucklingload factors
α
cr
for which the total frame stiffness vanishes [21], so that
[
K
]{
D
}−
α
cr
[
G
]{
D
}={
0
}
(8.3)
in which [
K
] is the elastic stiffness matrix, [
G
] is the stability matrix associated
withtheinitialaxialforces
{
N
i
}
,and
{
D
}
isthevectorofnodaldeformationswhich
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