Civil Engineering Reference
In-Depth Information
Top flange unrestrained
Bottom flange prevented from twisting
Figure 6.19
End distortion of a beam with an unrestrained top flange.
buckling solutions in [16, 34, 37] or by using a computer program [18] to carry
out an elastic buckling analysis.
Another situation for which end twist rotation is not prevented is illustrated in
Figure 6.19, where the bottom flange of a beam is simply supported at its end
and prevented from twisting but the top flange is unrestrained. In this case, beam
bucklingmaybeaccentuatedbydistortionofthecross-sectionwhichresultsinthe
webbendingshowninFigure6.19.Studiesofthedistortionalbuckling[38-40]of
beams such as that shown in Figure 6.19 have suggested that the reduction in the
buckling capacity can be allowed for approximately by using an effective length
factor
k
cr
=
1
+
d
f
/
6
L
t
f
/
t
w
3
1
+
b
f
/
d
f
/
2,
(6.56)
in which
t
f
and
t
w
are the flange and web thicknesses and
b
f
is the flange width.
Theprecedingdiscussionhasdealtwiththeeffectsofeachtypeofendrestraint,
butmostbeamsinrigid-jointedstructureshavealltypesofelasticrestraintacting
simultaneously. Many such cases of combined restraints have been analysed, and
tabular, graphical, or approximate solutions are given in [12, 16, 33, 35, 41].
6.7 Cantilevers and overhanging beams
6.7.1 Cantilevers
Thesupportconditionsofcantileversdifferfromthoseofbeamsinthatacantilever
is usually assumed to be completely fixed at one end (so that lateral deflection,
lateralrotation,andwarpingareprevented)andcompletelyfree(todeflect,rotate,
andwarp)attheother.Theelasticbucklingsolutionforsuchacantileverinuniform
bendingcausedbyanendmoment
M
whichrotates
φ
L
withtheendofthecantilever
[16] can be obtained from the solution given by equations 6.2 and 6.3 for simply
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