An efficient simplified approach for the nonlinear buckling analysisof frames

Abstract

In this paper, a simplified but very reliable approach for the nonlinear buckling analysis of frames subjected to either bifurcational or limit-point instability is proposed. This approach, based on slightly modified kinematic continuity conditions, is demonstrated through the buckling analysis of a simple two-bar frame. Numerical solu- tions, obtained by an accurate nonlinear buckling analysis and two simplifying variants of it, are established for a wide range of values of several parameters. Comparisons of these solutions show the simplicity, speed, and reliability of the buckling approach proposed herein. Moreover, a simple stability criterion is established for a direct determination of critical loads. HE problem of linear bifurcational analysis of plane frames has been discussed extensively by many in- vestigators during the last 50 years. An excellent review on this subject is given by Bleich. 1 However, considerable attention has recently been given to the nonlinear buckling analysis of frames.2'3 This is due primarily to the need for a precise estimation of the load-carrying capacity of frames sensitive to small initial imperfections. As is known, the presence of im- perfections may decrease the load-carrying capacity of a struc- tural system considerably, depending on the shape of the ini- tial postbuckling path. An outstanding contribution to this problem is the initial postbuckling analysis of Koiter 4; how- ever, it is only applicable to structural systems which in their primary equilibrium state exhibit bifurcational instability. Thus, for the majority of structural systems which lose their stability through a limit point, a complete nonlinear analysis is required for the estimation of their actual load- carrying capacity. Discrepancies in critical loads obtained by Koiter's initial postbuckling analysis and accurate nonlinear buckling analyses have been presented recently by Kounadis et al.,5 as well as by Simitses and Kounadis.6 The last reference presents a systematic nonlinear stability analysis applicable to a general frame composed of straight uniform bars rigidly connected to each other. However, in this analysis con- siderable difficulties arise for establishing numerical solutions for systems with more than three nonlinear equilibrium equa- tions. Thus, an approximate nonlinear buckling analysis leading to reliable results is desirable. The objective of this investigation is to present an effective and readily employed nonlinear buckling analysis of plane frames associated with the simplest possible nonlinear equa- tions. In effect, the field equations of the present analysis are those of the linear buckling theory of frames, with the unique exception that the kinematic continuity conditions for the joint translational components are slightly modified. The proposed approach has no restrictions on its range of ap- plicability and, despite the simplification of the nonlinear equations, leads to quite accurate results. The effectiveness of this approach is demonstrated through the buckling analysis of a simple two-bar frame for which numerical results are available.