Initial postbuckling of elastic rods subjected to thermal loads and resting on an elastic foundation

In practical applications encountered in some mechanical systems initially straight slender elements laterally supported by elastic foundations may be prevented from thermal expansion. In such conditions a compressive force is created and a serious equilibrium instability phenomenon may take place when a critical temperature gradient is imparted to the structure. If the temperature is further increased the rod may experience relatively large deflections, and a geometrically non-linear two-point boundary value problem describes this behaviour. However, if finite but moderate deflections are admitted, an approximate solution may be obtained if a classical perturbation expansion method is applied to the non-linear ordinary differential governing equations. This approach renders a set of linear equations which can be sequentially solved. This work proposes an analytical solution for the assessment of the initial post-buckling configuration of slender elastic rods supported on linear elastic foundations and subjected to a uniform temperature gradient. The rod ends are assumed hinged and immovable and the thermal strain-temperature relationship is linear. The governing equations are derived and made non-dimensional so it is seen that two parameters rule the problem: the foundation stiffness modulus and the rod slenderness ratio. Results are presented and discussed for one slenderness ratio and a range of foundation stiffness that corresponds to the fourth buckling mode. Generally an increase in the applied temperature is associated with a decrease in the compressive force, but it is shown particular conditions where an opposite behavior is observed. The initial post-buckling results for unsupported rods are compared with existing full post-buckling solution and show excellent agreement, given that in practical situations the deflections are moderate.