Torque of Prismatic Beams of Piecewise Rectangular Cross Section

I present in this note a finite difference method and Scilab computer programs to

numerically solve the Saint-Venant theory for torsion of prismatic beams (shafts, bars) of

piecewise rectangular cross section. The purpose of this note is mostly educational, for I

believe that it is quite instructive to not only solve these problems analytically whenever

possible but also explore solutions numerically of common configurations for which one

cannot readily obtain analytical solutions. The same kind of problems occur in heat transfer,

fluid mechanics, electricity and magnetism, and indeed in any physical situation in which the

two dimensional Poisson (Laplace) equation with Dirichlet boundary conditions must be

solved. The enclosed programs can be used for problems in these fields with very little, if any,

modifications.

The cross section shapes that are treated here are U, T and I shapes. The formulation is in terms of stress functions.