Revisiting the P-δ Magnification Factor for Members Subject to End Moments

Abstract

The 2022 edition of the AISC Specification for Structural Steel Buildings allows member and system instability effects to be accounted for by using advanced analysis methods that model these effects directly or through the use of the B₁ (P-δ) and B₂ (P-Δ) multipliers. For compression members subject to end moments only, the B₁ multiplier is expressed as a function of the smaller to larger end moment ratio. This ratio is taken as positive when the member bends in reverse curvature and negative when it bends in single curvature. This code equation for B₁ often overestimates the P-δ effect for double curvature bending under a high axial force but underestimates the P-δ effect for single curvature bending under almost all values of axial force. The reason for the overestimation is because the derivation of the B₁ equation uses the equivalent moment concept and ignores the actual location where the maximum moment occurs in the member. When this condition is taken into consideration, it is shown herein that the equation is no longer applicable for members bent in reverse curvature under any value of axial force. The reason for the underestimation is because the effect of axial force on the moment ratio is ignored. By incorporating this effect in a new equation for B₁, it is shown that this underestimation is drastically reduced. The validity of this proposed B₁ multiplier is established by comparison with the corresponding theoretical values.