Abstract
The aim of the study is the further development of analytical methods for calculating the bending of beams resting on a non-homogeneous continuous Winkler elastic foundation. This paper considers the case when the beam is under the influence of a uniformly distributed constant transverse load, and the inhomogeneity of the elastic foundation is given by a power function with an arbitrary non-negative power exponent . Fundamental functions and a partial solution of the corresponding differential equation of beam bending are found in an explicit closed form. These functions are dimensionless and are represented by absolutely and uniformly convergent power series. In turn, the formulas for the parameters of the stress-strain state of the beam – deflection, angle of rotation, bending moment and transverse force – are expressed through the indicated functions. The unknown constants of integration in these formulas are expressed in terms of the initial parameters, which are after the implementation of the specified boundary conditions. Thus, the calculation of the beam for bending is reduced to the procedure of numerical implementation of explicit analytical formulas for the parameters of the stress-strain state. An example demonstrates the practical application of the obtained solutions. A prismatic concrete beam based on a cubic variable elastic foundation is considered. This case corresponds to the power value . The results of the calculation by the author's method are presented in numerical and graphical formats for the case when the left end of the beam is hinged and the right end is clamped. The numerical values obtained by the author's method are accurate, 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. For the purpose of such a comparison, the paper presents the calculation results obtained by the finite element method (FEM). The absolute error of the FEM method when calculating this design was determined.
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