The approximate solution of the elastic torsion problem of uniform bar with arbitrary cross-section

The elastic torsion of a bar is one of the most famous problems in elasticity. The torsion problem of a bar with cross-section described by an arbitrary simply connected domain with smooth boundary curve is presented. Here, we introduce Cauchy integral method to solve the torsion problem described by the Dirichlet problem for Poisson equation. The imaginary part of the analytical function on the boundary is found by reducing the problem to Fredholm integral equation of the second kind then the solution of the problem is constructed via its boundary values with the help of Cauchy integral. The integral equation is reduced to the solution of a finite linear system of equations by approximating the solution with truncated Fourier series. Scheme for improvement of the behavior of Cauchy integral for points closed to the boundary of the domain is given. An error analysis is deduced and numerical examples are presented to insure the accuracy of the method.