Variational formulations of vibration equations of sinusoidal shear deformable beams and Eigenfrequency solutions by finite sine transform method

Abstract

This study presents analytical solutions using the finite sine transformation methodology (FSTM) for the natural dynamic solutions of  thick beams. The Euler-Bernoulli beam theory (EBBT) disregards the contributions of transverse shear strains due to the Euler-Bernoulli- Navier orthogonality hypothesis used in its formulation and is unsuitable for thick beams. It derived a variational formulation of flexural  vibration equations of sinusoidal shear deformable beams using first principles approach. The governing equation is formulated for transverse dynamic loading and in-plane compressive force as a nonhomogeneous partial differential equation (PDE). The PDE did not  need shear correction factors. The formulation yielded a cosine function shaped transverse shear strain and stress distribution which was  maximum at the neutral axis and vanished at the beam surfaces. The PDE was solved for free flexural vibration where it became  homogeneous due to the absence of forcing excitation forces. The FSTM was used for solving simply supported beams since sinusoidal  kernel complies with end conditions. The problem simplifies for harmonic excitation to an algebraic eigenvalue problem solvable using  algebraic methods. The roots are utilized to compute modal vibrations and the resonant vibration frequency at the first mode, (n = 1). The  resonant frequencies obtained are identical with past results that used theory of elasticity technique. The results for the first five vibration modes are also close to previous results obtained thick beam models for all modes and aspect ratios considered. The  effectiveness of the FSTM and its accuracy has been demonstrated for simply supported thick beam vibration problems.