Second-Order Axial Deflections of Imperfect 3-D Beam-Column

Beam-columns, in general, undergo axial elongation not only from the applied axial forces but also from the transverse deflections. A practical method that takes into account the effects of these transverse deflections on the total axial deformation of a beam-column δ_t is by multiplying the first-order axial stiffness AE/L by the geometrically nonlinear factor s₁ [i.e., δ_t = P/(s₁AE/L)]. A general solution for s₁ is derived for the combined effects of end moments, a uniformly distributed load, a series of concentrated loads, sidesway, and out-of-straightness. This solution requires numerical integration and is limited to 3D elastic prismatic beam-columns with doubly symmetrical cross sections or singly symmetrical 2D beam-columns under small strains. The proposed solution can be applied to the second-order and stability analyses of frames and to the evaluation of the axial load induced by transverse loads in beams built into rigid supports. These effects are particularly important in long-span structures. An example is presented to show the validity of the proposed formulation.