The static response of sandwich exact conical helices via MFEM

Abstract

This paper investigates the internal forces of sandwich exact conical helices using the mixed finite elements based on Timoshenko beam theory and a numerical method of cross-sectional warping inclusion. The exact conical axis geometry is derived over Archimedean and logarithmic planar curves. The exact curved mixed finite element has two nodes and twenty-four degrees of freedom in total. The constitutive equations of sandwich sections are derived by modifying the constitutive equations of an orthotropic continuum based on the classical rod theory. The cross-sectional warping included torsional rigidity is taken into account using a numerical method in the mixed finite element formulation. A convergence analysis for the support reactions of sandwich exact conical helices having fixed-fixed boundary conditions is performed. The results are compared with the displacement type finite element results of computer aided design programs. Finally, the internal forces of sandwich exact conical helices over Archimedean and logarithmic planar curves are obtained for various conicity ratios, pitch angles and the number of active turns. In order to investigate the influence of cross-sectional warping on precision, the warping included results are compared to the conventional torsional rigidity results. In the scope of the investigated internal forces, the maximum influence of cross-sectional warping on the results is obtained for the normal force. The influence of cross-sectionalwarping on the normal force increases as the pitch angle or the conicity ratio decreases. Besides, the maximum influence of the conical helix geometry on the warping included internal forces is obtained for the torque.