Analytical solutions for predicting shear stresses in axially graded tapered beams

Abstract

This study investigates the state of stress in slender elastic cylinders with variable cross-sections and axially graded material properties. In contrast to prismatic homogeneous cylinders, the continuous variation of cross-sectional dimensions and material properties along the axis of the cylinder produces additional shear stress distributions within the cross-sections. This work sheds the light on how these stresses depend on the axial gradation of both geometry and material. The analytical approach in this paper is based on partial differential equations derived in a recent work that describe the stress state in tapered inhomogeneous elastic cylinders. A new analytical solution is presented for rectangular cross-sectioned tapered cylinders with axially graded material properties, subjected to external loads at the ends. By examining this paradigmatic case, the combined effects of taper and axial material gradation on the shear stresses across the cylinder’s cross-sections are discussed, highlighting the limitations of solutions obtained by approximating the geometric and material properties as piecewise constant along the cylinder’s axis. Numerical examples, including comparisons with benchmark solutions from a finite element method, are provided to support the analytical findings of the study. In addition to the theoretical insights, the analytical solutions developed here hold practical value in a wide range of engineering applications, such as aerospace structures, mechanical components, biomedical implants, energy harvesters, vibration control systems, and smart sensors for structural health monitoring.