ANALYTICAL CALCULATION OF A BEAM BASED ON AN ELASTIC WINKLER FOUNDATION WITH EXPONENTIAL INHOMOGENITY
DOI:
https://doi.org/10.31650/2786-6696-2024-10-27-36Keywords:
beam, nonhomogeneous elastic foundation, exponential inhomogeneity, Winkler hypothesis, exact solution, analytical calculation.Abstract
The article is devoted to the analytical calculations for beam bending based on nonhomogeneous solid elastic Winkler base. In this paper we consider the case where the beam is subjected to a parabolic-variable transverse load and the inhomogeneity of the elastic foundation is given by an exponential function. The fundamental functions and partial solution of the corresponding differential equation of beam bending are written out in explicit closed form. These functions are dimensionless and are represented by absolutely and uniformly convergent power series. In turn, these functions are used to express the formulas for the parameters of the stress-strain state of a beam such as deflection, angle of rotation, bending moment, and shear force. The unknown integration constants in these formulas are expressed through the initial parameters, which are found after the implementation of the given boundary conditions. Thus, the calculation of a bending beam is reduced to the procedure of numerical implementation of explicit analytical formulas for the parameters of the stress-strain state.
The practical application of the obtained solutions is demonstrated by an example. A prismatic concrete beam based on an exponentially variable elastic foundation is considered. The results of the author's calculation are presented in numerical and graphical formats for the case when both ends of the beam are fixed. The numerical values obtained by the author's method are interpreted as exact values, since the applied calculation method is based on the exact solution of the corresponding differential equation. The availability of such solutions makes it possible to evaluate the accuracy of solutions obtained using various approximate methods by comparison. To such comparison, the paper presents the results of the calculation obtained by the finite element method (FEM). The absolute error of the FEM in the calculation of this structure is determined.
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