Shear Stresses in Doubly Tapered Inhomogeneous Beams With Rectangular Cross Sections

Abstract This work presents an analytical investigation of how shear stresses within the cross sections of doubly tapered elastic cylinders with axially varying material properties are influenced by the continuous axial variation of both geometric and material parameters. The analytical framework is based on a set of partial differential equations derived in recent studies, which govern the stress state of the elastic solids under consideration and admit closed-form solutions only in a limited number of cases. In particular, an analytical solution is obtained for axially inhomogeneous cylinders with doubly tapered rectangular cross sections subjected to end loads. This solution enables the study of the combined effects of material inhomogeneity and taper on the cylinder’s stress state, and highlights the limitations of approaches that approximate geometric and material properties as piecewise constant along the axis. New application-oriented formulas are provided for engineering use in mechanical, civil, and aerospace structures. Numerical examples, including comparisons with benchmark finite element solutions, are presented to support and validate the analytical results.