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:

(9.17)

where Fycminimum yield stress of compression flange, ksi (MPa)
M 1 smaller of the moments, inkip (mmMPa) at the ends of the
unbraced length of beam
Mpplastic moment, inkip (mmMPa)
ryradius of gyration, in (mm), about minor axis

The plastic moment MpequalsFyZfor homogeneous sections, where Zplas-
tic modulus, in^3 (mm^3 ); and for hybrid girders, it may be computed from the
fully plastic distribution. M 1 /Mpis positive for beams with reverse curvature.
For solid rectangular bars and symmetrical box beams:

(9.18)

Theflexural design strength0.90Mnis determined by the limit state of lateral-
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 Mnthat depend on the geometry of the section and the bracing pro-
vided for the compression flange.
For compact sections bent about the major axis, for example, Mndepends on
the following unbraced lengths:

Lb the distance, in (mm), between points braced against lateral displace-

ment of the compression flange or between points braced to prevent

twist

Lp limiting laterally unbraced length, in (mm), for full plastic-bending

capacity

 for I shapes and channels

 for solid rectangular bars and box beams

Fyf flange yield stress, ksi (MPa)

J torsional constant, in^4 (mm^4 ) (see AISC “Manual of Steel Construction”

on LRFD)

A cross-sectional area, in^2 (mm^2 )

Lr limiting laterally unbraced length, in (mm), for inelastic lateral buckling

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:

Lr (9.19)

ryx 1

FywFr^2

1  1 X 2 FL^2

3750(ry/Mp)/JA

300 ry/Fyf

Lpd

5000 3000(M 1 /Mp)

Fy

ry 3000

ry

Fy

Lpd

3600 2200(M 1 /Mp)

Fyc

ry