For a compact section bent about the major axis, the unbraced length L b of the compression flange, where plastic hinges may form at failure, may not exceed L pd , given by Eqs. given in post . For beams bent about the minor axis and square and circular beams, L b is not restricted for plastic analysis.
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:
Lpd= ( [3600 + 22000 ( M 1 / M p ) ] / F yc ) r y
Where F yc =minimum yield stress of compression flange, ksi (MPa)
M 1 =smaller of the moments, in-kip (mm MPa) at the ends of the unbraced length of beam
M p =plastic moment, in kip (mm MPa)
R y =radius of gyration, in (mm), about minor axis
The plastic moment M p equals F y Z for homogeneous sections, where Z = plastic modulus, in l 3 (mm 3 ); and for hybrid girders, it may be computed from the fully plastic distribution. M 1 / M p is positive for beams with reverse curvature.
For solid rectangular bars and symmetrical box beams:
L pd = ( [5000 + 3000 ( M 1 / M p ) ] / F y ) r y ³ 3000 r y / F y
The flexural design strength 0.90M NSE is 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 formulas for M NSE that depend on the geometry of the section and the bracing provided for the compression flange.
For compact sections bent about the major axis, for example, M NSE depends on the following unbraced lengths:
L b =the distance, in (mm), between points braced against lateral displacement of the compression flange or between points braced to prevent twist
L p =limiting laterally unbraced length, in (mm), for full plastic-bending capacity
=300 r y / ( F yf ) ½ for I shape and channels
=3750 (r y /M p ) / (JA) ½ for solid rectangular bars and box beams
F yf =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 )
L r =limiting laterally unbraced length, in (mm), for inelastic lateral buckling
For I-shaped beams symmetrical about the major or the minor axis, or symmetrical about the minor axis with the compression flange larger than the tension flange and channels loaded in the plane of the web:
L r = r y x 1 / (F yw - F r )) ½ ( 1 + ( 1 + X 2 F 2 L ) ½ ) ½
Where F yw =specified minimum yield stress of web, ksi (MPa)
F r =compressive residual stress in flange
=10 ksi (68.9 MPa) for rolled shapes, 16.5 ksi (113.6 MPa), for welded sections
F L =smaller of F yf - F r or F yw
F yf =specified minimum yield stress of flange, ksi (MPa)
X 1 =( p / S x ) (EGJA/2) ½
X 2 =( 4 C w /l y ) (S x /GJ) 2
E=elastic modulus of the steel
G=shear modulus of elasticity
S x =section modulus about major axis, in 3 (mm 3 ) (with respect to the compression flange if that flange is larger than the tension flange)
C w =warping constant, in 6 (mm 6 ) (see AISC manual on LRFD)
l y =moment of inertia about minor axis, in 4 (mm 4 )
For the previously mentioned shapes, the limiting buckling moment M r , ksi (MPa), may be computed from
M r = F S x
For compact beams with b £ L r , bent about the major axis:
M n = C b [ M p - ( M p - M r ) ( L b - L p ) /( L r - L p )] £ M p
Where C b = 1.75 + 1.05 ( M 1 / M 2 ) + 0.3 ( M 1 / M 2 ) £ 2.3, where M 1 is the smaller and M 2 the larger end moment in the unbraced segment of the beam; M 1 /M 2 is positive for reverse curvature and equals 1.0 for unbraced cantilevers and beams with moments over much of the unbraced segment equal to or greater than the larger of the segment end moments.
For solid rectangular bars bent about the major axis:
L r =57,000 ( r y / M r ) ( JA) ½
and the limiting buckling moment is given by:
M r =F y S x
For compact beams with L b > L r , bent about the major axis:
M n =M cr £ C b M r
where M cr =critical elastic moment, kip in (MPa mm).
M cr =C b (p / Lb) (El y GJ + l y C w ( p E / L b ) 2 ) ½
For solid rectangular bars and symmetrical box sections:
M cr =57,000 C b ( JA) ½ / ( L b / r y )
For determination of the flexural strength of noncompact plate girders and other shapes not covered by the preceding requirements, see the AISC manual on LRFD.
