Spectral Formulation for Geometrically Exact Beams: A Motion-Interpolation-Based Approach

This paper proposes a novel spectral formulation for geometrically exact beams based on motion interpolation schemes. Motion interpolation schemes based on matrix, quaternion, and geodesic metrics yields simple expressions for the sectional strains and linearized strain–motion relationships at the mesh nodes. Consequently, the expressions for the nodal forces and tangent stiffness matrices of spectral elements are simplified dramatically. Furthermore, the motion formalism is used to describe the kinematics of the problem, leading to equations of motion that present low-order nonlinearities. Spectral elements based on Gauss–Lobatto and Gauss quadrature rules are investigated. For both cases, the combination of the spectral formulation with the motion formalism leads to geometrically exact beam elements that are much simpler to implement than their counterparts based conventional finite element interpolation schemes using a classical description of kinematics. A global parameterization-free generalized-α scheme is used to integrate the equations in time. Numerical examples demonstrate the accuracy of the proposed formulation. The elements based on Gauss–Lobatto quadrature rules are shown to suffer from axial and shear locking, whereas those based on Gauss rules are locking-free. The convergence rate of (N+1)-node spectral elements based on Gauss quadrature rules is about 2N−0.5 to 2N for all three interpolation schemes. As the number of elements increases, the proposed formulation becomes more accurate than its conventional counterpart.