Vladimir Karpov, Evgeny Kobelev, Aleksandr Maslennikov


Introduction: In this paper, based on the properties of unit functions, we present accurate solutions to beam bending under various transverse loads and edge restraint conditions, using equations based on Bernoulli’s hypothesis and the hypothesis taking into account transverse shears. By comparing the analytical solutions obtained for a rectangular beam, we determined beam length-to height (L/h) ratios for cases when the difference in deflections is less than the permitted value. Thus, criteria for Bernoulli’s hypothesis application were obtained. The results of beam bending analysis can be applied when studying rod systems using the force and displacement methods. In this case, Bernoulli’s hypothesis is used. All the ratios obtained are simple and clear. However, this hypothesis is applicable for the analysis of thin-walled structures. Meanwhile, the hypothesis taking into account transverse shears can be used for structures of medium cross-section height. To ensure accurate results when studying building structures (beams, plates, shells, rod systems), the criterion of Bernoulli’s hypothesis (hypothesis of the straight normal) applicability was needed. Purpose of the study: We aimed to build a mathematical deformation model and develop a method for the analysis of bending in elastic Timoshenko beams with account for transverse shears. Methods: By applying generalized functions and direct integration of the differential equation for the bending line, we obtained analytical expressions for the deflection function under various boundary conditions. Results: Based on the proposed method, we performed beam analysis under various transverse loads and edge restraint conditions. We also evaluated the scope of Bernoulli’s hypothesis application for the main types of beams used in the analysis of rod systems by the displacement method.


Beam, bending, Kirchhoff model, transverse shear, Timoshenko model, unit functions.

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DOI: https://doi.org/10.23968/2500-0055-2022-7-3-37-43


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