The Design of Modern Steel Bridges - TEDI

108 The Design of Modern Steel Bridges
For minimum value of P, dP/dm ¼ 0. This leads to
m ¼ L 1=4
½k=EIcŠ
p
pffiffiffiffiffiffiffiffiffi
Pcr ¼ 2 EIck
L
1=4
¼ critical half-wave-length ¼ p½EIc=kŠ
m
The effective length of an equivalent strut with Euler buckling load equal to
the above value of Pcr will be given by
leading to
p 2 EIc
l 2 eff
pffiffiffiffiffiffiffiffiffi
¼ 2 EIck
leff ¼ p pffiffiffi ½EIc=kŠ
2
1=4
The above results are theoretically valid for a beam under constant bending
moment, but may be conveniently and conservatively used for beams subjected
to varying bending moments, using the maximum compressive flange force for
comparing with the P cr-value obtained from above.
If the lateral restraints are spaced at s and their flexibility, i.e. deflection due
to unit force at the restraint, is d, then k s ¼ 1/d, and
leff ¼ p pffiffiffi ½EIcsdŠ
2
1=4
The lateral restraints to the compression flange may be in the form of torsional
restraints, such that a is the rotation of the restraint due to unit torque applied
at that restraint. Then the displacement of one flange with respect to the other
due to unit forces applied at flange levels of each torsional restraint will be d 2 a,
when d is the depth of the girder between the centroids of its flanges. The
effective length of the compressive flange can then be obtained by replacing
d by d 2 a in the previous expression, i.e.
leff ¼ p pffiffi ½EIcsd
2
2 aŠ 1=4
5.3.4 Compression flange with full lateral restraints at ends and one restraint of
limited rigidity at midpoint
Assume a single sinusoidal half-wave of buckling, i.e.
y ¼ a sinðpx=LÞ
The strain energy of bending of the flange ¼ p 4 EIca 2 =4L 3