The Design of Modern Steel Bridges - TEDI

102 The Design of Modern Steel Bridges
When a beam twists and buckles laterally under a load applied on its top
flange, then the load may either move laterally with the deflected flange, or
may remain in the original plane of application. In the former case, the stability
of the beam will be further reduced by the resultant torsion with respect to the
supports. This effect is particularly pronounced in the case of deep and short
beams[3]. A simple and conservative method of dealing with this problem is to
increase l LT by 20% when the load is applied on the top flange and is free to
move laterally.
So far, the slenderness parameter l LT has been obtained for the support
condition of the compression flange held against lateral displacement but free
to rotate in plan. If there is any restraint against rotation in plan at the supports,
then l LT is reduced. Following the traditional practice for struts, a reduction
factor of 0.7 may be taken when the compression flange is fully restrained
against rotation in plan at both supports, and 0.85 when the restraint is partial
at both supports or full at one support and no restraint at the other. A more
accurate graph for the reduction factor for various degrees of rotational
restraint is given in Reference [2].
In the case of cantilevers, if the support section is held against lateral
displacement and twist, and the tip is free to twist and deflect laterally, then
its buckled shape will be the same as that of one-half of a simply supported
beam with identical support conditions; the critical value of an applied
bending moment constant along the length will be given by equation (5.1),
provided L e is taken as twice the cantilever length L. For the usual cases of
varying bending moments it may be noted that in a simply supported beam
the cross-sections with maximum lateral displacements are subjected to the
maximum bending moments, whereas in a cantilever the cross-sections with
maximum lateral displacements (i.e. near the tip) are subjected to minimum
bending moments. Hence the benefit from a varying bending moment diagram
is much more pronounced for cantilevers. Another special feature of
cantilevers in bridges is that any lateral restraint to the top flange restrains
the tension flange and thus is not as effective as a restraint to the compression
flange. The support condition of a cantilever may be either: (i) fully fixed
against rotation in both planes or (ii) continuous into an adjacent span; the
latter case provides a reduced restraint against warping and can be separated
into several cases of lateral restraint to the top and/or bottom flanges. The
critical bending moment M cr also depends significantly upon the level of
application of loading, i.e. whether at top flange or from bottom flange or
from the shear centre. M cr for a cantilever may be obtained from equations
(5.7a) and (5.8) if an appropriate effective length Le ¼ KeL is used, where L is
the actual cantilever length and K e is an effective length factor. The factor K e
depends not only upon all the features mentioned above, but also on the
geometrical parameter given by equation (5.3). A conservative set of values
for K e is given in Table 5.2, which has been derived for the case of a single
concentrated load at tip.