First Principles Ritz Variational Formulation for Buckling Analysis of Polynomial Shear Deformable Beams: Analytical Solutions

This paper explores the Ritz variational method (RVM) for formulating and solving the stability of thick beams modeled using shear deformable theory. The work used a rigorous step by step approach to derive variationally consistent equations that result in vanishing transverse shear stresses at the beam’s top and bottom surfaces. Consequently, transverse shear stress modification functions are obviated. The Ritz functional is formulated as the sum of the energies of the thick beam with the transverse shear strains considered. The energy minimization principle is utilized to find the buckling equation where the buckling displacement function is expressed as a linear combination of basis functions. The study considers thick beams (a) simply supported at both ends (b) clamped at both ends and (c) clamped at one end and free at the other. Trigonometric buckling shape functions were used to derive exact buckling solutions for thick beams simply supported at both ends. One-parameter algebraic polynomial shape functions were also derived for reasonably accurate buckling solutions of simply supported thick beams. One-parameter trigonometric shape function was used to find accurate buckling load solutions for thick beam with clamped-clamped ends and with cantilevered end. Reasonably accurate buckling solution for cantilever beams were also obtained using one-parameter polynomial shape functions. For each studied boundary condition, the buckling load was found to depend on the aspect ratio lh-1. It was found for all end support cases that for the critical buckling load coefficients (Kcr) are identical to the buckling load coefficient of the Euler-Bernoulli beam theory (EBBT). For corresponding to simply supported moderately thick beams, Kcr is lower than Kcr for EBBT; illustrating the overestimation of buckling loads by EBBT and the risks inherent in using EBBT for moderately thick beam buckling analysis. For corresponding to thick beams, Kcr becomes much lower than Kcr by EBBT by as much as 13.44% for simply supported ends. This shows that for thick beam buckling, the inherent risks in using EBBT for stability analysis become much severe.