Torsion of functionally graded hollow tubes

An analytical formulation is presented for torsional analysis of functionally graded hollow tubes of arbitrary shape. Moreover, relatively simpler formulas are presented for torsional analysis of functionally graded hollow tubes of polygonal shape. Thicknesses of all segments of the cross section are the same. Shear modulus of rigidity changes continuously across the thickness between two phases according to a power law distribution. Governing equations in terms of Prandtl's stress function are used to derive the formulas. Several examples are presented to show the accuracy and efficiency of the formulation. The obtained results are verified by accurate finite element solutions. Effects of changing thickness and volume fraction of constituent materials are also investigated. Derived formulas can be useful for analysis of thin to moderately thick-walled hollow tubes.