Use of generalized finite difference method for calculation of beams from nonlinear elastic material

The introduction into practice of construction of structures made of high-strength steels and other materials having a nonlinear deformation diagram caused the active development of the nonlinear theory of calculation of structures. Replacing Hooke’s law with nonlinear dependencies between stresses and strains leads to so-called physical nonlinearity. For the calculation of such structures, the experimentally obtained dependences between stresses and strains are described using analytical expressions. A number of variants of such approximations have been proposed by various researchers. In this paper, we consider the calculation of beams of symmetrical cross-section made of nonlinear elastic material, for which the dependence between stresses and strains is described by a cubic parabola. This approximation ensures the symmetry of the diagram σ with respect to the tension σ – ɛ compression, and also gives a good match with the experimental curve. The use of the generalized finite difference method for solving the problem allows to reduce the system of nonlinear differential equations to the system of algebraic equations, for the solution of which the method of successive approximations is used. Studies have shown that the proposed method of calculation makes it possible to obtain a fairly accurate solution for a small number of elements. In addition, the presented calculation algorithm is convenient for programming and numerical calculation. As an example of systems that allow calculation using the considered algorithm, beam elements of building structures can act. The calculation of a beam from a nonlinear elastic material on the action of a concentrated force is given.