Torsion of uniform bars with polygon cross-section

The solution of the torsion problem of uniform bars with polygon cross-section by means of the Trefftz integral for the complex torsion function is presented. The governing equations consist of the first-order integral of the Schwarz-Christoffel mapping, and another third-order integral, known as the Trefftz integral. The known series solution of the Schwarz-Christoffel integral is used. The solution of Trefftzʹs integral is carried out in series form according to the method of von Koppenfels and Stallmann, by the separation into regular and singular corner functions. The result is in the form of explicit corner functions, Frobenius and Taylor series. The series solutions are computed with convolution number algebra. An additional Fourier analysis allows one to compute the shear in a polygon cross-section with a round hole. Torsion stiffness is computed by integrals along the boundary. An estimate of the effect of rounding of corners is given.