A general solution for torsion-induced warping in helical curved elements with non-circular solid section: Theory and application to compression springs stiffness

Real non-circular helical springs never exhibit perfectly rectangular wire cross-sections; instead, they feature fileted rectangular or obround geometries that significantly influence their torsional response and stiffness. This study presents a generalized geometrical and mechanical framework for modeling such springs, introducing two non-dimensional parameters (the shape ratio and the roundness ratio) to describe a wide range of realistic cross-sections. Based on an extensive Abaqus simulation campaign and literature data, a method was established to determine the warping factor, k warp , which was incorporated into a refined stiffness matrix for helical beam elements. A simplified analytical model was also developed to provide a practical alternative. Both models showed excellent agreement with reference numerical results, while the classical formulation consistently overestimated springs’ stiffness. An abacus is provided for evaluating the polar moment of inertia and the warping factor for common industrial wire profiles. The proposed approach enhances stiffness prediction accuracy and reliability, offering valuable insights for high-precision applications such as aerospace and robotics, and opens perspectives toward more complex non-circular geometries and anisotropic behaviors.