Определение прогибов физически нелинейной балки на упругом основании

Abstract

The article deals with the issues of determining the deflections of a beam lying on an elastic foundation with a constant coefficient of rigidity, while the beam is made of a material that has a non-linear relationship between stresses and deformations. The physical non-linearity of the beam material is taken into account by approximating the relationship between stresses and strains with a cubic parabola; such an approximation well describes the deformation curves of a non-linear elastic body with the same diagram of the work of the material in tension and compression. As an example, the deflections of a non-linear elastic beam of rectangular cross section, which lies on the Winkler base and carries a uniformly distributed load along the entire length, are considered for three cases of support fastenings at the edges: with two hinged supports, with two terminations, and with termination and hinged support. The method of successive approximations is used to obtain a solution to the non-linear equation for deflections, which depends on dimensionless parameters that take into account the influence of an elastic foundation, the physical non-linearity of the material, and a uniformly distributed load. The dependence of the change in the value of the maximum relative deflection on the coefficient of the bed and the distributed load, obtained by calculation, taking into account the influence of the physical non-linearity of the material for three cases of support fastenings, is given. The results of the calculations showed that the presence of additional bonds reduces the influence of the physical non-linearity of the beam material on its deflections.