Torsion problem: stress statement and solution by the boundary element method

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Abstract

The formulation of the problem of torsion regarding stresses and its solution by the boundary elements method are described. The main advantage of the problem formulation in stresses is direct determination of stresses in the cross-section, unlike the classical formulation, when the result of the approximate solution is the Prandtl stress function values, and the determination of stresses is brought down to numerical differentiation. The boundary integral equation of the second kind is obtained to formulate the problem with respect to stresses. The procedure for solving the problem by the boundary elements method is described, the system of solving equations is compiled. Solutions of test problems on torsion of rods with rectangular and channel cross-sections are presented. Comparison of the calculation results with known analytical solutions illustrates the reliability and permissible engineering accuracy of the obtained solutions.

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1. Introduction The torsion problem for elastic prismatic rods is one of the oldest problems in the theory of elasticity. It was mathematically formulated by Saint-Venant in the middle of the 19th century. Before the broad spread of ECM, many problems for bars with relatively simple shapes of cross-section were solved analytically. The obtained solutions were summarized in the monograph [1]. With the creation of ECM, it became possible to obtain numerical solutions to the problem of torsion for bars with an arbitrary cross-section. This led to the rapid development of numerical methods for solving torsion problems and problems of potential theory that are similar in their mathematical formulation. In spite of their considerable age, these problems are still the subject of research for many scientists and engineers. These works contain formulations and methods for solving problems for inhomogeneous bars [2; 3], nanosized bars [4], problems of dynamics [5] and others. One of the widely used numerical methods for solving torsion problems is the boundary elements method (BEM) [7-13]. This method has been actively developed since the 70s of the 20th century, but up to now its new formulations continue to appear, including those for torsion problems of bars [14-18]. The traditional mathematical formulation of the torsion problem consists of finding the stress function1 [1; 19], and the stresses themselves are subsequently found by differentiation of the stress function [20; 21]. From the point of view of determining the stresses in a cross-section in numerical solution, the formulation of the problem regarding the stress function has two significant disadvantages. Since the result of the approximate solution of the problem in such a formulation is actually the values of the stress function in the nodes of boundary elements, then the determination of stresses is reduced to numerical differentiation. This leads to an additional source of computational error. The second reason reducing the accuracy of the solution is the fact that the boundary integral equation regarding the stress function is a numerically unstable equation of the first kind2. This paper presents a formulation of the torsion problem in stresses, derives the boundary integral equation regarding stresses, and describes the procedure for solving the problem by the boundary elements method. In difference from the classical formulation, the proposed formulation leads to an integral equation of the second kind, and its numerical algorithm is a stable one2. The verification of the methodology was carried out on the example of two cross-sections by means of comparing the numerical solution of the problem with the known analytical one. 2. Methods Formulation of the torsion problem regarding stresses Consider the problem of torsion for a prismatic bar of an arbitrary cross-section under the action of two external moments which lie in the plane of its outermost cross-sections. We consider the volumetric forces to be equal to zero and the lateral surface to be free of external loads. The following coordinate system has been choosen: the z-axis coincides with the axis of torsion, i.e., the axis that remains unmoved when the bar is twisted; the x- and y-axes are mutually orthogonal and located randomly in the plane of the cross-section. The problem of torsion of a bar with cross-section S and contour Γ, is formulated in terms of the Prandtl stress function F in the following way[1] [24]:
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About the authors

Vladimir V. Lalin

Peter the Great St. Petersburg Polytechnic University; RUDN University

Author for correspondence.
Email: vllalin@yandex.ru
ORCID iD: 0000-0003-3850-424X

Dr.Sc., Professor of the Higher School of Industrial, Civil and Road Construction of the Institute of Civil Engineering

St. Petersburg, Russian Federation; Moscow, Russian Federation

Daniil A. Semenov

Peter the Great St. Petersburg Polytechnic University

Email: dan290797@gmail.com
ORCID iD: 0000-0002-9144-1412

PhD student of the Higher School of Industrial, Civil and Road Construction of the Institute of Civil Engineering

St. Petersburg, Russian Federation

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Copyright (c) 2023 Lalin V.V., Semenov D.A.

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