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

• Published in: Lobachevskii journal of mathematics Vol. 45; no. 8; pp. 3646 - 3656
• Main Author: Migliaccio, Giovanni
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.08.2024; Springer Nature B.V
• Subjects: Algebra; Analysis; Boundary conditions; Cross-sections; Cylinders; Elastic analysis; Euclidean geometry; Euclidean space; Exact solutions; Geometry; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Partial differential equations; Probability Theory and Stochastic Processes; Strain analysis; Three dimensional analysis; Wind turbines; tapered cylinder; material inhomogeneity; flexure problem
• DOI: 10.1134/S1995080224604405
• ISSN: 1995-0802

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.