Elastic Stability and Second-Order Analysis of Three-Dimensional Frames: Effects of Column Orientation

The elastic stability and second-order analysis of three-dimensional (3D) multicolumn systems including the effects of cross-section orientation of each column are presented in a condensed manner using the classical Timoshenko’s stability functions. This formulation is an extension of an algorithm recently presented by the writer in 2002 by which the effective length K-factor for each column, the total critical load, and second-order analysis of a 3D multicolumn system can be determined directly. Extended characteristic transcendental equations corresponding to multicolumn systems with sidesway and twist uninhibited, partially inhibited, and totally inhibited with semirigid connections are derived and discussed. The proposed method is limited to 3D multicolumn systems made up with doubly symmetrical vertical columns with the principal axes of each column cross section oriented in any direction with respect to the floor global axes and with every column sharing the same interstory sidesways (i.e., two horizontal translations and a rotation about the vertical axis). Shear and axial deformations in all members are omitted. Three comprehensive examples are presented and the calculated results compared with those obtained using SAP2000 (Version 6.1, 1997) showing: (1) the effectiveness and simplicity of the proposed approach; (2) its validity to carry out stability and second-order analyses of 3D multicolumn systems; and (3) the importance of the orientation of the cross section of the columns on the lateral response of 3D multicolumn systems. Analytical results indicate that a frame reaches its maximum overall lateral stiffness and a dominant sidesway buckling (without overall frame torsional rotation or twist about its vertical axis) when all columns are oriented with their minor axis tangent to the circumscribed circle such that the multicolumn system acts as a tube offering its maximum twist stiffness.