Torsion of a rectangular checkerboard and the analogy between rectangular and curvilinear cross-sections

The Saint-Venant torsion problem of a rectangular checkerboard is investigated using an eigenfunction expansion method. Each constituent rectangle of the checkerboard may have different dimensions with different material properties. The solutions involve an infinite series in which the coefficients can be resolved by a truncation at any desired order. Some numerical results are presented to show the effectiveness of the proposed scheme. Further, a torsional analogy is reported between the compound rectangular and curvilinear checkerboards. We show that, via the introduction of the mapping function w = Log z, the governing system for ρ, which is defined as the difference between the warping function ψ(x, y) and the function x y, for a rectangular checkerboard, except a non-homogeneous term, is analogous to that of ψ for the transformed curvilinear checkerboard. We show that the torsion solutions of a curvilinear checkerboard can be obtained from those of a rectangular one without much extra effort.