LAUSR

Abstract The present article investigates the thermoelastic free vibration for a simply supported elliptic plate subjected to a thermal load. The realistic solution involving the Mathieu functions and also their derivatives for the heat conduction differential equation subjected to sinusoidal sectional heating on the upper face with the lower face and the curved inner surface is kept at zero temperature, and the outer curve is kept thermally insulated and is derived using the classical method. The strain energy due to bending of the middle surface of the plate undergoing large deflection was well-thought-out by neglecting the second strain invariant terms for the analysis of large amplitude (nonlinear)-free vibrations of a simply supported plate. Furthermore, nonlinear free vibration equation of elliptic structure is developed with the aid of Berger assumption and Hamilton’s principle and obtained its solution using a new integral transform involving Mathieu and modified Mathieu functions. A closed-form bending stress function obtained has been equated with those obtained by Berger’s methodology. The thermal stress components are obtained in terms of resultant bending moments and resultant forces for numerical analysis. The free-vibration mode of the corresponding nonlinear problem, the Jacobi elliptic function, is obtained from the exact resolution of the natural frequency of the simply supported elliptic plate. For the particular case by applying limiting conditions, the elliptic region can degenerate into the problem of the circular zone, and some numerical results have also been plotted in a few instances.