The paper investigates thermal buckling and post-buckling of columns of variable cross-section along the length. Two types of variable cross-section columns, namely, sinusoidally and linearly varying cross-sections, are considered. The geometry of the cross-section is taken as symmetric about the midpoint of the length of the columns. The thermal buckling and post-buckling loads, which are the mechanical equivalent of the uniform compressive load developed when the column is subjected to high temperature, are evaluated by applying the classical Rayleigh-Ritz (RR) method. The variable cross-section of the columns is quantified through an easy-to-recognize taper ratio. The numerical results in the form of the thermal buckling loads and the thermal post-buckling loads are obtained for different taper ratios and the reference central deflection parameters. The numerical results are found to be reliable when compared with similar results of the limiting cases and from the logical physical trends in the absence of the corresponding results for the definition of the taper ratios used in this study.

Ever since Euler discovered the revolutionary phenomenon of an instability called buckling in 1757, in which the slender columns can fail at a much lesser axial compressive load when compared to its compressive strength, this instability has been the subject matter of many researches. This phenomenon of buckling instability has been extended to many structural members, like plates and shells, with many complicating effects. All these earlier studies were presented in the excellent book by Timoshenko and Gere (1961). For quite some time, of the order of roughly a couple of centuries, the structural design engineers believed that the structural members suddenly collapse just beyond reaching the corresponding buckling load, and treated it as the failure load. However, it is found that the Euler buckling considers the linear theory, and if one considers the geometric nonlinear theories, it is shown that the columns withstand an extra compressive load than the buckling load, which is called the post-buckling load, if the large lateral deflections produced at this extra load is tolerable (Dym, 1964).

It is to be noted that the compressive load is of two types, namely, the mechanically applied load, and in the case of heated column, the induced thermal load when a temperature rise occurs from its stress-free temperature (Ziegler and Rammerstorfer, 1989), and an equivalent mechanical compressive load, which is developed in the column, when the two ends of the column are not allowed to move axially. In either case, the columns (or other structural members) exhibit buckling and post-buckling phenomena. Many mathematical tools, which include the exact or approximate continuum, and powerful numerical methods have been developed to study the post-buckling of columns (including other structural members). The present study considers one-dimensional structural members like the practically important columns of variable cross-section, which will have a higher buckling load for a specified mass, when compared to the uniform columns. It is obvious that the mechanical or thermal buckling problem is simpler than the corresponding post-buckling problem, as in the evaluation of buckling loads, the use of the linear theory in terms of the strain-displacement relation is sufficient, whereas to study the post-buckling behavior, it is necessary to use the nonlinearity in the strain-displacement relation, which complicates the theoretical formulation and has to be considered.

Structural Engineering Journal, Thermal buckling, Thermal post-buckling, Tapered columns, Sinusoidally and linearly varying tapers, Rayleigh-Ritz (RR) method