Initial Postcritical Analysis of Asymmetric Bifurcation in Frames

Abstract

Asymmetry of the shape, supports, or loading of frames causes the initial post critical behavior to exhibit asymmetric bifurcation. Its source is the second‐order axial shortening of columns due to lateral deflection, which induces asymmetric shear forces and bending moments transmitted into the column from the adjacent beams. As an alternative to Koiter, Roorda, and Chilver's solutions based on the nonlinear differential equations for deflections and perturbation method, the paper presents a simpler and more general method of analysis that uses the stiffness matrices and stability functions from linear stability theory of frames. The usual linear matrix equilibrium equations are enhanced by additional terms that are quadratic in joint rotations, with coefficients that depend on the derivatives of the stability functions. The solution is accurate up to the second order in deflections. As an example, Koiter and Roorda's L‐frame is analyzed and the results are shown to agree with the previous solutions as well as Roorda's experiments. It is concluded that the decrease of the maximum load due to asymmetric bifurcation is not insignificant and should be taken into account in design.