Problems of beam bending solution on the basis of variation criterion of critical energy levels

Abstract

Introduction. The article is devoted to the development of variational formulations of structural mechanics problems using the example of the problems of bending beams. The existing variational approaches, the nonlinear theory of bending of beams, as well as the classical methods of resistance of materials, are not able to explain a number of issues related to the discrepancy between the results of theory and experiments, for example, in problems of pure and transverse bending of beams. To solve these issues, variational formulations and the criterion of critical levels of the internal potential energy of deformation, developed by the authors, are used. Materials and methods. For the internal potential energy of a deformed body, the stationarity condition at critical levels is written, which makes it possible to obtain equations of state that describe the self-stress of the structure. It is shown that a mathematical model of the state of a structure at critical levels of potential energy of deformation leads to an eigenvalue problem. The quantities characterizing the formulation of problems when formulating in generalized efforts and generalized displacements are discussed. Results. Using the examples of problems of pure bending and direct transverse bending of simple beams by a concentrated force, the formulation of the problem and the method of its solution are shown. The diagrams of deflections and bending moments are given, and the magnitudes of the amplitude values in the middle of the span are given. It is shown that for simple beams in problems of pure bending and transverse bending, the maximum values of the moments are achieved in the middle of the beam span, as in the experiment. Conclusion. The results are discussed and compared with the data obtained in the theory of flexible rods. It is noted that the dangerous section in two approaches having different physical nature is located in the middle of the beam span. The boundaries of discrepancy between the results for displacements, moments of internal forces and stresses are shown. It is noted that the results obtained according to the linear theory of strength of materials lead to a significant margin of safety. The prospects for the development of the theory of critical levels of internal potential energy of deformation, and the possibility of applying the technique to problems of structural mechanics are discussed.