Torsion of bi-directional functionally graded truncated conical cylinders

Analytical solutions are presented for the torsion of bi-directional functionally graded (FG) linearly elastic truncated conical cylinders for six functional forms of the shear modulus varying in both the radial and the axial directions. Furthermore, for an arbitrary variation of the shear modulus along the two directions, we employ a weighted residual approach (WRA) to obtain accurate numerical solutions for the linear elastic boundary value problem. The influence of the variation of the shear modulus and of the cone angle on the stress distribution in the cylinder is discussed through numerical examples. The analytical solutions presented herein can serve as benchmarks for ascertaining the accuracy of approximate or numerical solutions. The WRA combined with an optimization algorithm can be used to find the shear modulus variation to maximize the torsional stiffness.