A new theory for high frequency vibrations of beams from displacements of trigonometric functions of thickness

In order to obtain an accurate displacement field of beam, displacement components can be expressed by a series of trigonometric functions and a linear function of the height coordinate for rectangular cross-sections beams with isotropic materials. Through substituting displacements into two-dimensional equations of the plane stress problem of elasticity, the one-dimensional equations have been obtained for solving the beam problems. Based on the infinite series equations, the first-order approximation equations have been deduced which compared with the Timoshenko beam theory. The results showed that two theories yield similar governing equations, displacement functions, stress components, and the strain components. The present theory not needs to introduce any correction factors, but also avoid three false assumptions of Timoshenko beam theory. The classical beam theory of Euler-Bernoulli can be deduced from our theory by ignoring the first-order shear displacement.