Thermomechanical nonlinear in-plane analysis of fix-ended FGM shallow arches on nonlinear elastic foundation using two-step perturbation technique H. Babaei . Y. Kiani . M. R. Eslami Received: 29 May 2018 / Accepted: 15 October 2018 Ó Springer Nature B.V. 2018 Abstract An attempt is made in this research to analyse the nonlinear response of functionally graded material shallow arches with both edges clamped. The arch is resting on a three parameter nonlinear elastic foundation during deformation and is subjected to uniform lateral pressure and uniform temperature rise. Material properties are expressed according to a power law function and are assumed to be temperature dependent. The governing equilibrium equations of the arch are established with the aid of third order shear deformation curved beam theory of Reddy and von Ka ´rma ´n type of strain–displacement relations. The obtained equations contain three coupled and nonlinear equations in terms of circumferential dis- placement, lateral displacement and cross section rotation. Considering the immovable type of edge supports, the equations are reduced to two new coupled and nonlinear equations. These equations are solved using the two step perturbation technique for the case of clamped boundary conditions. Explicit expressions are resulted which yield the deflected shape of the arch as a function of temperature elevation and uniform pressure. It is shown that the arch reveals the snap-through type of instability under certain conditions. The response of the arch is highly affected by the power law index, thermal environment, side to thickness ratio and stiffnesses of the foundation. Keywords Thermomechanical load  Shallow arches  Two-step perturbation technique  Temperature dependency  Functionally graded material  Nonlinear elastic foundation 1 Introduction Functionally graded materials (FGMs) are usually designed so that nonhomogeneous material properties vary continuously through the thickness of FGM structures from the ceramic surface to that of a metal surface. Many studies on the nonlinear free vibration, bending, buckling and post-buckling analysis for homogeneous and FGM flat beams with different boundary conditions are available in the literature. For instance, Tsiatas (2010) studied the nonlinear behavior H. Babaei Mechanical Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Iran e-mail: st_h_babaei@azad.ac.ir Y. Kiani (&) Faculty of Engineering, Shahrekord University, Shahrekord, Iran e-mail: y.kiani@sku.ac.ir M. R. Eslami Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran e-mail: eslami@aut.ac.ir 123 Int J Mech Mater Des https://doi.org/10.1007/s10999-018-9420-y