The Design of Modern Steel Bridges - TEDI

104 The Design of Modern Steel Bridges
Figure 5.4 Plot of bending moment capacities against slenderness parameter
LT.
when
M cr ¼ elastic critical bending moment of the beam
M R ¼ the limiting moment of resistance of the beam
MU ¼ the ultimate moment of resistance of the beam cross-section based on
yielding alone, i.e. lateral-torsional buckling is prevented; M U equals
(1) the plastic moment of resistance Zp sy of the beam cross-section,
if the latter is compact, i.e. it can develop the full plastic moment of
resistance, Zp being the plastic modulus
(2) Z e s y, for a beam with ‘non-compact’ cross-section, Z e being the
elastic section modulus.
The solution for M R to this quadratic equation is
or
MR ¼ 1
2 fMU þð1 þ ZÞMcrg
MR
MU
¼ 1
2
1 þð1 þ ZÞ Mcr
MU
1
2
1
2 ½fMU þð1 þ ZÞMcrg 2
4Mcr MUŠ 1=2
1 þð1þ ZÞ Mcr
" #
2
1=2
MU
4 Mcr
MU
ð5:12Þ
Just as the Euler critical buckling stress of a strut with an effective length L e and
radius of gyration r is expressed in terms of a slenderness ratio L e/r (i.e. Euler