APPROXIMATION OF THE ELASTIC CURVE DIFFERENTIAL EQUATION BY TRIGONOMETRIC SERIES FOR AN ISOTROPIC BEAM WITH A CONSTANT MOMENT OF INERTIA, COMPLEXLY LOADED, ACCORDING TO THE EULER-BERNOULLI THEORY

Adrian-Ioan BOTEAN

Abstract


The study of the elastic curve has been a constant concern in the field of mechanical engineering and has led to several analytical, graphical, and graph-analytical methods of analysis. In this work, an analytical method of study is presented that can be counted among the energy methods for calculating deformations, since the potential energy of deformation expression is used in the case of beams loaded to simple bending, in the field of elastic deformations, using isotropic materials. The elastic curve expressed by a fourth-order differential equation can also be approximated by an infinite trigonometric series, leading to a good convergence of the results obtained by classical methods. Starting from a series of simple loading cases, which serve to illustrate the method of solving the problem, this study examines two cases with a higher degree of complexity: in the first study, the beam is loaded on the first half unit length by a uniformly distributed load q(x), and in the second study, the beam is loaded by two concentrated forces arranged symmetrically concerning the support points.

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