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and
1
+
π
2
EI
w
GI
t
L
cr
N
cr
,
T
=
GI
t
i
p
,
(7.50)
where
L
cr
is the effective length of the beam-column
L
cr
=
k
cr
L
,
(7.51)
and the effective length ratio
k
cr
is the solution of
1
−
R
=
−
π
R
2
k
cr
cot
π
2
k
cr
.
(7.52)
Equation7.48isthesameasequation7.35forsimplysupportedbeam-columns,
except for the familiar use of the effective length
L
cr
instead of the actual length
L
in equations 7.49 and 7.50 defining
N
cr
,
z
and
N
cr
,
T
. Thus the solutions shown
in Figure 7.14 can also be applied to end-restrained beam-columns.
The close relationships between the flexural-torsional buckling condition of
equation7.35forunrestrainedbeam-columnswithequalendmoments(
β
m
=−
1)
and the buckling loads
N
cr
,
z
,
N
cr
,
T
of columns and moments
M
zx
of beams have
alreadybeendiscussed.Itshouldalsobenotedthatequation7.52fortheeffective
lengthsofbeam-columnswithequalendrestraintsisexactlythesameasequation
3.43 for columns when
(
1
−
R
)/
R
is substituted for
γ
1
and
γ
2
, and equation 6.52
forbeams.Thesesuggestthattheflexural-torsionalbucklingconditionforbeam-
columns with unequal end restraints could well be approximated by equations
7.48-7.51 if Figure 3.21a is used to determine the effective length ratio
k
cr
in
equation 7.51.
A further approximation may be suggested for restrained beam-columns with
unequal end moments (
β
m
=−
1), in which modifications of equations 7.45 and
7.46 (after substituting
i
p
N
cr
,
z
N
cr
,
T
for
M
zx
)
are used (with equations 7.49-7.51)
instead of equation 7.48.
7.3.2 Inelastic beam-columns
The solutions obtained in Section 7.3.1 for the flexural-torsional buckling of
straight isolated beam-columns are only valid while they remain elastic. When
the combination of the residual stresses with those induced by the in-plane load-
ing causes yielding, the effective rigidities of some sections of the member are
reduced, and buckling may occur at a load which is significantly less than the
in-plane maximum load or the elastic buckling load, as indicated in Figure 7.13.
A method of analysing the inelastic buckling of I-section beam-columns with
residual stresses has been developed [13], and used [14] to obtain the predictions
shown in Figure 7.17 for the inelastic buckling loads of isolated beam-columns
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