Green’s Functions for Statically Indeterminate Single-Span Beams

Depending on the class in engineering practice, the problems to be solved are distinguished: static/dynamic, flat/spatial, contact/with partial or edge support, etc. The pressure of a rail on a sleeper, a column on a foundation, floor slabs on walls, a foundation on a soil foundation – all these are typical examples of practical problems that lead to the need to solve boundary value problems – mathematically and contact problems – physically. From the mathematical formulations of contact problems of structures lying on an elastic foundation, it is known that the basis for their solution is the search for the law of distribution of reactive pressures at the contact of the structure with the foundation, which depends in a complex way on the rigidity of the structure, the elastic characteristics of the foundation, external load, and the nature of the structure’s fastening. When solving many boundary-value and initial-boundary-value problems of structural mechanics and the theory of elasticity, such as solving a classical homogeneous equation by the method of eigenfunctions, under certain boundary conditions arising from the type of fastening of the beam at the ends, an important, sometimes decisive, role is played by the fundamental functions of the operator x^IV, which received their basic interpretation by Academician A. N. Krylov. However, calculations using these formulas are very difficult due to mathematical limitations and the cumbersomeness of the expressions. In the proposed work, eigenfunctions of the differential equation of bending vibrations of statically indeterminate single-span beams are used to construct the Green's function in the form of an infinite series for these eigenfunctions. Exact expressions have been constructed to determine the deflections of beams due to concentrated force. The resulting expressions are presented through elementary functions, are of a general nature and make it possible to solve various problems of statics, dynamics and stability of the beams under consideration. The authors obtained numerical results for bending moments and deflections of a clamped beam and a beam with clamped and hinged supports using the MATHEMATICA computer package.

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