Dynamic snap-through of functionally graded shallow arches under rapid surface heating

Abstract

In this paper, dynamic buckling response of a shallow arch constructed from functionally graded materials under thermal shock is investigated. It is assumed that one surface of the beam is subjected to sudden temperature elevation while the other surface is kept at reference temperature. To solve the one-dimensional Fourier heat transfer equation, Crank–Nicolson numerical method along with the central finite difference method are employed. In order to obtain the strain–deflection relations, Donnell theory of shallow arches with von Kármán simplifications are used. The governing equations are acquired based on well-known Hamilton’s principle that are converted to a set of non-linear algebraic equations using the Ritz method. Since the governing equations are non-linear equations, the Newton–Raphson method is applied and after that temporal evolution of displacements are obtained employing the Newmark time marching scheme. In the final stage, a theory to estimate the buckling load is required, which is the Bodiansky–Roth criterion in this research. To validate the formulation and solution method of this project, this work is compared with previous studies and good agreement is obtained. In addition, impacts of power law index, arch thickness, geometrical parameters, and temperature dependency on the dynamic buckling temperature are reported in this study.