Post buckling analysis of a slender bar under transversely non-uniform temperature rise

Abstract

On the basis of Kirchhoffs assumption and geometrically nonlinear theo~y for axially extensible bars, this thesis presents an exact mathematical model for the thermal post-buckling of an elastic bar with different immovable end conditions and subjected to a transversely non-uniformly temperature rising. In order to describe accurately the post-buckling of the heated bar with axial extension, an axial stretching, !!(x) has been introduced, and the arc length, s(x), of the deformed central axis is taken as one of the basic unknown functions in the mathematical model. This is a two point boundary value problem of first order ordinary differential equations with high nonlinearity. Using the multi-segment integration method in conjunction ~ith the concept of analytical continuation, the nonlinear boundary value problems are numerically solved. The range of safe buckling temperature is determined, and some comparisons between the nonlinear and linear buckling behaviors of the bars are discussed. Some quantitative results of thermal postbuckled configurations and secondary equilibrium paths of the problem with some specific parameters, for example, buckling temperature and the slenderness ratio are obtai?ed for immovably pinned-pinned, pinned-fixed and fixed-fixed ends. The numerical results show that both the critical buckling temperature and the postbuckled temperature of the bar are sensitively influenced by the slenderness ratio, which is distinctively different from that of the post-buckling of the inextensible rod subjected to mechanical compressive loads. In addition, it is found that the magnitude of the axially constrained force reaches a maximum value at the onset of the buckling state and then decreases as the temperature increases in the post-. buckling regime. Moreover, the influences of the transverse temperature change on the thermal post-buckling deformations are examined.