Thermal Postbuckling of Slender Elastic Rods With Hinged Ends Constrained by a Linear Spring

This paper presents the critical buckling temperature and a closed-form analytical solution for the thermal postbuckling behavior of uniformly heated slender elastic rods. The boundary conditions are assumed hinged and nonmovable at one end, whereas at the other end longitudinal displacement is constrained by a linear spring. The thermal strain–temperature relationship is considered nonlinear and the material is assumed to be linearly elastic, homogeneous, and isotropic. Large displacements are included; hence, the formulation is geometrically nonlinear. A set of six first-order nonlinear ordinary differential equations with boundary conditions defined at both ends yields a complex boundary-value problem. The buckling and postbuckling solutions are respectively accomplished by assuming infinitesimal rotations and employing complete elliptical integrals. The results are presented in nondimensional graphs obtained for different values of spring stiffness, slenderness ratios, and nonlinear thermal strain coefficients. It is shown that these three parameters govern the buckling and postbuckling behavior.