Civil Engineering Reference
In-Depth Information
Accurate [13]
Approximate
equations
7.53 - 7.56
1.0
L
/
i
z
300
150
100
L
/
i
z
250
200
150
m
M
0.5
L
/
i
z
N
300
200
m
m
m
= -1
= 0
= +1
0
N
0
0.5
1.0
0
1
2 0123
M
M
/
M
Iu
M
/
M
Iu
M
/
M
Iu
Figure 7.17
Inelastic flexural-torsional buckling of beam-columns.
withunequalendmoments.Itwasfoundthatthesecouldbecloselyapproximated
by modifying the elastic equations 7.45 and 7.46 to
2
1
−
M
α
bcI
M
Iu
1
−
N
N
Iz
N
N
cr
,
T
=
,
(7.53)
1
−
β
m
2
1
+
β
m
2
3
1
α
bcI
=
0.40
−
0.23
N
N
Iz
+
(7.54)
in which
M
Iu
/
M
pl
=
1.008
−
0.245
M
pl
/
M
zx
≤
1.0
(7.55)
is an approximation for the inelastic buckling of a beam in uniform bending
(equation 7.55 is similar to equation 6.23 when
β
m
=−
1), and
N
Iz
/
N
y
=
1.035
−
0.181
N
y
/
N
cr
,
z
−
0.128
N
y
/
N
cr
,
z
≤
1.0
(7.56)
provides an approximation for the inelastic buckling of a column.
7.3.3 Ultimate resistance
The ultimate resistances of real beam-columns which fail by flexural-torsional
buckling are reduced below their elastic and inelastic buckling loads by the pres-
enceofgeometricalimperfectionssuchasinitialcrookedness, justasarethoseof
columns (Section 3.4) and beams (Section 6.4).
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