Page 278 - Civil Engineering Formulas

212 CHAPTER NINE

For I-shaped beams, symmetrical about both the major and the minor axis or
symmetrical about the minor axis but with the compression flange larger than
the tension flange, including hybrid girders, loaded in the plane of the web:

3600 2200(M 1 /Mp)
L pd r y (9.17)
F yc
where F minimum yield stress of compression flange, ksi (MPa)
yc
M smaller of the moments, in kip (mm MPa) at the ends of the
1
unbraced length of beam
M plastic moment, in kip (mm MPa)
p
r radius of gyration, in (mm), about minor axis
y
The plastic moment M equals F Z for homogeneous sections, where Z plas-
p
y
3
3
tic modulus, in (mm ); and for hybrid girders, it may be computed from the
fully plastic distribution. M /M is positive for beams with reverse curvature.
1
p
For solid rectangular bars and symmetrical box beams:
5000 3000(M 1 /Mp) ry
L pd r y 3000 (9.18)
F y F y
The flexural design strength 0.90M is determined by the limit state of lateral-
n
torsional buckling and should be calculated for the region of the last hinge to
form and for regions not adjacent to a plastic hinge. The specification gives for-
mulas for M that depend on the geometry of the section and the bracing pro-
n
vided for the compression flange.
For compact sections bent about the major axis, for example, M depends on
n
the following unbraced lengths:
L the distance, in (mm), between points braced against lateral displace-
b
ment of the compression flange or between points braced to prevent
twist
L limiting laterally unbraced length, in (mm), for full plastic-bending
p
capacity
300r y / F yf for I shapes and channels
3750(r y /M p )/ JA for solid rectangular bars and box beams
F flange yield stress, ksi (MPa)
yf
4
4
J torsional constant, in (mm ) (see AISC “Manual of Steel Construction”
on LRFD)
2
2
A cross-sectional area, in (mm )
L limiting laterally unbraced length, in (mm), for inelastic lateral buckling
r
For I-shaped beams symmetrical about the major or the minor axis, or sym-
metrical about the minor axis with the compression flange larger than the ten-
sion flange and channels loaded in the plane of the web:
L r r y x 1 2 1 1 X 2 F L 2 (9.19)
F yw F r