Analytical solutions for the coupled thermoelastic vibrations of Timoshenko beams by means of Green׳s functions

• The response of coupled thermoelastic vibration of a beam is analytically investigated.
• The present work is performed in a systematic manner.
• Systematic calculations are performed to fulfill our multiple purposes.
• The analytic solutions are consistent with the FEM results.
• The effects of various parameters are examined numerically.

This paper focuses on obtaining the direct expressions of steady-state two dimension temperature and displacement responses for the coupled thermoelastic vibrations of Timoshenko beams subjected to a heat flux and an external force. Coupling effects between temperature and displacement fields will be discussed conveniently through these analytical solutions. Damping effects on the transverse and rotational directions are taken into account in the vibration equations. Green׳s functions and superposition principle are used in the present study to solve the coupled thermoelastic vibration equations of a beam. The Green׳s functions for heat conduction equations and vibration equations are presented explicitly for the beams with various mechanical and thermal boundaries with the aid of eigenfunctions expansion method and Laplace transform technology. Numerical calculations are performed to validate the present solutions, and the influences of some important physical parameters on the coupling effects of the coupled multi-physics problem are discussed as well.