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Analytical Solution to the Plane Bending Task of the Multilayer Beam with a Circular Axis under Normal Uniform Loading

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Strength of Materials Aims and scope

Circular bars are widespread elements in building and mechanical engineering. They allow one to enhance the strength of the multilayer composite materials along with their weight reduction. However, their analytical modeling is developed rather poorly, which creates additional obstacles in their efficient design. The paper presents the results of the exact analytical solution to the plane bending task of elasticity theory for the multilayer beam with a circular axis under normal uniform loading on its longitudinal surfaces considering the cylindrical orthotropy of its layers’ materials. The problem has been solved using the methods of linear elasticity theory based on the continual approach where the circular bar is considered as a solid with the elastic characteristics varying in the radial direction. The solution for the shear stresses is assumed using the known distribution in the cross section. Therefore, the solution to the problem is reduced to the determination of two unknown functions of stress distribution throughout the section height. The system of linear differential equations has been obtained for the determination of the sought functions and unknown functions of integration in the process of the solution to the equations of plane theory considering the variable elastic characteristics. General analytical solutions for uniform and discontinuous-nonuniform layers from cylindrically orthotropic and isotropic materials have been developed. The influence of transverse shear deformation and compression on the characteristics of the stress-strain state of a cantilever three-layer beam with a semicircular axis is theoretically investigated. The obtained relations allow us to predict the stress-strain state of circular multilayer beams, as well as to build applied methods for solving problems of strength, rigidity, and optimal design of such structural elements.

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Authors and Affiliations

  1. Poltava State Agrarian Academy, Poltava, Ukraine

    S. B. Koval’chuk

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  1. S. B. Koval’chuk
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Correspondence to S. B. Koval’chuk.

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Translated from Problemy Prochnosti, No. 5, pp. 84 – 102, September – October, 2020.

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Koval’chuk, S.B. Analytical Solution to the Plane Bending Task of the Multilayer Beam with a Circular Axis under Normal Uniform Loading. Strength Mater 52, 762–778 (2020). https://doi.org/10.1007/s11223-020-00230-6

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