Building Design and Construction

# Combined Axial Compression and Bending

A member carrying both axial and bending forces is subjected to secondary bending moments resulting from the axial force and the displacement of the neutral axis.
This effect is referred to as the P- effect. Such secondary bending moments are more critical in members where the axial force is a compressive force, because the P- secondary moment increases the deflection of the member. In ASD, the effects of these secondary moments may be neglected where the axial force is a tensile
force or where the actual compressive stress is less than 15% of the allowable compressive stress. LRFD does not include this concept.
The following design criteria apply to singly and doubly symmetrical members.

## ASD for Compression and Bending

When the computed axial stress, ƒa is less than 15% of Fa, the stress that would
be permitted if axial force alone were present, a straight-line interaction formula

where E  modulus of elasticity, 29,000 ksi Ib  actual unbraced length, in, in the plane of bending
where E  modulus of elasticity, 29,000 ksi
Ib  actual unbraced length, in, in the plane of bending
rb  corresponding radius of gyration, in
K  effective-length factor in the plane of bending
Cm  reduction factor determined from the following conditions:

1. For compression members in frames subject to joint translation (sidesway),
Cm  0.85.
2. For restrained compression members in frames braced against joint translation and not subject to transverse loading between their supports in the plane of bending, Cm  0.6  0.4M1 /M2, but not less than 0.4. M1 /M2 is the ratio of the smaller to larger moments at the ends of that portion of the member unbraced in the plane of bending under consideration. M1 /M2 is positive when the member is bent in reverse curvature, and negative when it is bent in single curvature.
3. For compression members in frames braced against joint translation in the plane of loading and subjected to transverse loading between their supports, the value of Cm may be determined by rational analysis. Instead, however, Cm may be taken as 0.85 for members whose ends are restrained, and 1.0 for ends unrestrained.
In wind and seismic design may be increased one-third. The resultant section, F e however, should not be less than that required for dead and live loads alone without the increase in allowable stress.
Additional information, including illustrations of the foregoing three conditions for determining the value of Cm, is given in the AISC ‘‘Commentary’’ on the AISC ASD ‘‘Specification for Structural Steel for Buildings.’’

## LRFD for Compression and Bending

Members subject to both axial compression and bending stresses should be proportioned
to satisfy Eq. (7.58) or (7.59), whichever is applicable.