Linear Static Finite Element Analysis of Thin Beam under Bending Action using MATLAB

Abstract

The analysis of thin beam structures become of great interest for designing purposes, because beams are most common type of structural component. Beams are generally subjected to both, axial loads causes (axial deformation) and transverse loads causes (transverse deformation). The latter is resulted from both shear forces (shear deformation) and bending moment (flexural deformation). For linearly elastic beams, these modes of deformation can be examined independently from one another. In this research, Euler-Bernoulli beam flexural mode of deformation has been examined using finite element displacement methods (FEDM) to develop a linear static finite element computer program to predict bending behavior of thin beams under transverse loading. This bending deformation is classified as class C1 problem (continuity C1), that require continuity of both the transverse displacement and the fist derivatives of transverse displacement (slope). The finite element formulation is based on two-node thin beam bending element with two degrees of freedom per each node (transverse vertical displacement w_i^e and rotation θ_i^e). Hermite cubic polynomial displacement function used for the element unknown degrees of freedom and to satisfy this class of continuity problem, the resulting element is conforming element and the convergence will be monotonic convergence. The approach used in the deviation of the equation adopting the principle of minimum potential energy and the generalized coordinate approach. A finite element computer program was developed and implemented using MATAB R2019b adopting m-file mode of programming. The verification of the generated developed program results was compared with known published analytical exact solution results and published finite element analysis results for thin beam bending. Nine different numerical beam examples were conducted for this purpose with different loading and supporting conditions. The developed program results obtained are in good agreement with published, and there is no significance difference between the results. We conclude that using two nodes Euler-Bernoulli beam element was showed good performance after it used to analyze thin straight beams for bending. This was conformed when comparing results obtained with known published exact analytical solution