An exact method for torsion and bending problems for beams with polygonal cross-sections

We consider an elastic cylindrical beam whose cross-section is a polygonal domain 𝒢 in the w-complex plane, w = u + iυ. We assume that 𝒢 is a finite simply connected polygon with N vertices. It is well known (see [1] and [2]) that the problems of torsion and bending of this beam are reduced to the Dirichlet problem for the Laplace equation. Namely, the formulation of the torsional problem is (1)Δψ(w)  = 0,     w∈𝒢;     ψ(w)  =12|w|2,     w∈∂𝒢,https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367812027/499b2676-17a6-4805-9fd8-78c8baf0b813/content/eq861.tif"/>