Two-dimensional elasticity solution for free vibration of simple-supported beams with arbitrarily and continuously varying thickness

This paper studies the analytical solution for the vibration of simply supported beams with arbitrarily and continuously varying thickness based on the two-dimensional elasticity theory. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, is derived. Frequency equation governing the free vibration of beams with variable thickness can be obtained by using the Fourier sinusoidal series expansion on the upper and lower surfaces of the beam. The present solution method ensures a rapid convergence and meets the need of high accuracy in modern precise instruments. Several examples are provided to show the application of the proposed solution method which can be used to assess the validity of various approximate solutions and numerical methods for the beams with arbitrarily and continuously varying thickness.