A geometrically nonlinear spring element for structural analysis of helical springs

Helical springs belong to structures with spiral shapes and large curvatures, especially, traditional views that elemental shape functions are applied to interpolate shapes of structures, will result in plenty of elements and enormous computational cost. It is proved by this paper that selecting some characteristic parameters describing their shape regulations as variables can model the helical springs with almost no errors of model. Based on this thought, a geometrically nonlinear spring element for structural analysis of helical springs is proposed. First, strains irrelevant to rigid motions of cross sections and virtual deformation power of the curved beam with geometrical nonlinearity are derived. Next, parameters that generalized strains of spring elements depend on, namely helical radius, azimuth angles, height coordinates and torsion angles at one node of each coil are chosen as variables. A special shape function is built, in contrast of traditional shape functions, they can approximate the structure of helical springs accurately using less parameters. Then, nodal forces and generalized external forces as well as equilibrium equations are given, and in order to improve the computational efficiency, the Jacobian matrices are derived. Finally, two examples are considered to evaluate the accuracy of modeling and simulation against to ANSYS. Stiffness properties of cylindrical and conical springs are analyzed by the spring element. The proposed elements can give high-precision numerical results using less parameters from the comparison and be used an effective auxiliary tool for design of springs.