Three-dimensional vibration analysis of curved and twisted beams with irregular shapes of cross-sections by sub-parametric quadrature element method

This paper presents a novel three-dimensional (3D) sub-parametric quadrature element (SP-QE) method for solving the coupled dynamic behavior of curved and pre-twisted beamlike structures with irregular shapes of cross-section. The technique is an extension of the existing quadrature element method (QEM) with regular shapes by mapping the irregular solid into a regular cube. Detailed formulations are worked out. Beams with rectangular, circular, elliptical and airfoil cross-sections, various curvature and pre-twist rates, and different boundary conditions are investigated. Either Serendipity elements or Lagrange elements are considered in the mapped regular domain. Convergence studies are carried out to show the computational performance of the proposed elements. Results are compared either with the existing 3D spectral-Tchebychev (3D-ST) solutions or with the finite element data. It is shown that the proposed method can yield accurate solutions with small number of degrees of freedom. Consistent or lumped mass matrix affects little on the accuracy of solutions. Therefore, the element with lumped mass matrix can be efficiently used in dynamic analysis of solids with regular and irregular shapes.