Civil Engineering Reference
In-Depth Information
6.12 Appendix - elastic beams
6.12.1 Buckling of straight beams
6.12.1.1 Beams with equal end moments
The elastic buckling moment
M
cr
of the beam shown in Figure 6.3 can be deter-
minedbyfindingadeflectedandtwistedpositionwhichisoneofequilibrium.The
differential equilibrium equation of bending of the beam is
EI
z
d
2
v
d
x
2
=−
M
cr
φ
(6.81)
which states that the internal minor axis moment of resistance
EI
z
d
2
v
/
d
x
2
must
exactlybalancethedisturbingcomponent
−
M
cr
φ
oftheappliedbendingmoment
M
cr
ateverypointalongthelengthofthebeam.Thedifferentialequationoftorsion
of the beam is
d
x
−
EI
w
d
3
φ
GI
t
d
φ
d
x
3
=
M
cr
d
v
(6.82)
d
x
which states that the sum of the internal resistance to uniform torsion
GI
t
d
φ/
d
x
andtheinternalresistancetowarpingtorsion
−
EI
w
d
3
φ/
d
x
3
mustexactlybalance
thedisturbingtorque
M
cr
d
v
/
d
x
causedbytheappliedmoment
M
cr
ateverypoint
along the length of the beam.
The derivation of the left-hand side of equation 6.82 is fully discussed in Sec-
tions 10.2 and 10.3. The torsional rigidity
GI
t
in the first term determines the
beam'sresistancetouniformtorsion,forwhichtherateoftwistd
φ/
d
x
isconstant,
as shown in Figure 10.la. For thin-walled open sections, the torsion constant
I
t
is
approximately given by the summation
bt
3
/
3
I
t
≈
in which
b
is the length and
t
the thickness of each rectangular element of the
cross-section.Accurateexpressionsfor
I
t
aregivenin[3:Chapter10]fromwhich
the values for hot-rolled I-sections have been calculated [4: Chapter 10].
The warping rigidity
EI
w
in the second term of equation 6.82 determines the
additionalresistancetonon-uniformtorsion,forwhichtheflangesbendinopposite
directions,asshowninFigure10.1bandc.Whenthisflangebendingvariesalong
the length of the beam, flange shear forces are induced which exert a torque
−
EI
w
d
3
φ/
d
x
3
. For equal flanged I-beams,
I
z
d
f
4
,
in which
d
f
is the distance between flange centroids. An expression for
I
w
for a
monosymmetric I-beam is given in Figure 6.27.
I
w
=
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