On the Analytical Solution of PDEs in Bounded Domains and Applications to Inhomogeneous Tapered Elastic Solids

Slender elastic solids with tapered cross-sections are widespread in engineering applications, e.g., as components of wind turbines and bridges. They occupy a non-prismatic cylindrical bounded region of the three-dimensional Euclidean space. This makes the analytical prediction of their state of stress and strain much more difficult than in prismatic elements. A paradigmatic prismatic element for which analytical solutions are known is the circular cross-sectioned de Saint-Venant’s cylinder subject to flexure. In this paper, the flexure problem of a circular cross-sectioned cylinder with tapered inhomogeneous cross-sections is addressed. The set of partial differential equations and boundary conditions that govern its state of stress and strain, derived via a variational principle, is solved in closed form. The analytical solution obtained in terms of stresses and strains is compared with a technical solution based on the de Saint-Venant’s theory, demonstrating the inadequacy of the technical method when dealing with stress predictions in tapered inhomogeneous elements.