On 3D exact free torsional-bending vibration and buckling of biaxially loaded isotropic and anisotropic Timoshenko beams with complex cross-section

This research reports on Three Dimensional (3D) exact free torsional-bending vibrations and buckling of Timoshenko beams possessing homogeneous and nonhomogeneous properties and complex cross-section. The proposed exact solution is derived from 3D elasticity equations. The Hamilton’s integral equation is initially employed to establish the 3D exact governing Partial Differential Equations (PDEs) of motion and buckling, taking into account the effect of biaxial load applied to Timoshenko beams possessing complex cross-section. The PDEs are then solved using integral transform and trigonometric series differentiation procedure. The current solution procedure does not require characteristic deformation function to be predetermined and has fast convergence. A root finding algorithm is proposed to screen and filter the discontinuity and breakdown points as well as imaginary roots of the determinant function associated with bending eigenfrequency and eigenbuckling matrices. Two algorithms are further developed to capture the 3D dynamic and buckling mode shapes of biaxially loaded Timoshenko beams without any restriction imposed on the complexity of cross-section and degree of material anisotropy. The accuracy and convergence of the proposed 3D exact solution are first investigated by results comparison with the experimental and theoretical results in the published literature for a particular case in which bending natural frequencies response and modes shapes of a free-free isotropic Timoshenko beam with rectangular cross-section are sought. Finally, the effect of material isotropy/anisotropy, cross-section complexity, and biaxial load on torsional-bending natural frequencies, critical buckling loads, and 3D vibration and buckling mode shapes are thoroughly explored using the proposed exact solution procedure and validated by Abaqus finite element package.