The Sector Problem

On the basis of the variational method, approximate solutions of the biharmonic equation are established for the circular sector with the following properties: The stress functions f_kh_k create shear tractions on the radial boundaries; the stress functions f_kg_k create normal tractions on the radial boundaries; the stress functions F_kH_k create both shear and normal tractions on the circular boundary, and the stress functions F_kG_k create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors f_k(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors F_k(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions F_k″ + F_k constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.