Stress analysis of generally asymmetric non-prismatic beams subject to arbitrary loads

• Published in: European journal of mechanics, A, Solids Vol. 90; p. 104284
• Main Authors: Vilar, M.M.S., Hadjiloizi, D.A., Masjedi, P. Khaneh, Weaver, Paul M.
• Format: Journal Article
• Language: English
• Published: Berlin Elsevier Masson SAS 01.11.2021; Elsevier BV
• Subjects: Analytical solution; Asymmetry; Beam modelling; Boundary conditions; Closed form; Cross-sections; Exact solutions; Finite element method; Free boundaries; Internal forces; Isotropic material; Model accuracy; Non-prismatic beam; Plane stress; Rotor blades; Rotor blades (turbomachinery); Stress analysis; Stress distribution; Tapered beam; Wind turbines; Wings (aircraft); Tapered beam; Analytical solution; Beam modelling; Closed form; Non-prismatic beam
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2021.104284

Non-prismatic beams are widely employed in several engineering fields, e.g., wind turbines, rotor blades, aircraft wings, and arched bridges. While analytical solutions for variable cross-section beams are desirable, a model describing all stress components for beams with general variation of their cross-section under generalised loading remains an open and important problem to solve. To partly address this issue, we propose an analytical solution for stress recovery of untwisted, asymmetric, non-prismatic beams with smooth and continuous taper shape under general loading, considering plane stress conditions for isotropic materials undergoing small strains. The methodology follows Jourawski’s formulation, including the effect of asymmetric variable cross-section, with internal forces as known variables. We confirm the non-triviality of the stress field of non-prismatic beams, i.e., the dependency on all internal forces and beam geometry to shear and transverse stress distributions. As a particular novelty, the new formulation for transverse direct stress includes internal forces derivatives, resulting in greater accuracy than state-of-the-art models for distributed loading conditions. Also, closed-form solutions are introduced for non-prismatic and linearly tapered, generally asymmetric beams, both with rectangular cross-sections. For validation purposes, we consider three different practical beam models: a symmetric and an asymmetric, both linearly tapered, and an arched beam. The results, checked against commercial finite element analysis, show that the proposed model predicts the stress-field of non-prismatic beams under distributed loads with good levels of accuracy. Traction-free boundary condition requirements are naturally satisfied on the beam surfaces. •Recovery of the 2D stress field of non-prismatic beams under arbitrary loads.•Derivation of the transverse direct stress consistent with Cauchy-stress equilibrium.•Closed-form solutions for stresses for rectangular cross-section non-prismatic beams.•Reduction of the closed-form solution to linearly asymmetric tapered beams.