Geometrically exact thin-walled beam including warping formulated on the special Euclidean group [formula omitted

Abstract

Based on a formulation on the special Euclidean group SE(3), a geometrically exact thin-walled beam with an arbitrary open cross-section is proposed to deal with the finite deformation and rotation issues. The beam strains are based on a kinematic assumption where warping deformation and Wagner effects are included such that the nonlinear behavior of a thin-walled beam is predicted accurately, particular under large torsion. To reduce the nonlinearity of rigid motion, static and dynamic equations are derived in the SE(3) framework based on the local frame approach. As the value of the iteration matrix, including the Jacobian matrix of inertial and internal forces, is invariable under arbitrary rigid motion, the number of updates required during the computation process decreases sharply, which drastically improves the computational efficiency. Furthermore, the isogeometric analysis (IGA) based on the non-uniform rational B-splines (NURBS) basis functions, which promotes the integration of computer-aided design (CAD) and computer-aided engineering (CAE), is adopted to interpolate the displacement, rotation, and warping fields separately. The interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point. To obtain the symmetric Jacobian matrix of internal forces, the linearization operation is conducted based on the previously converged configuration. A Lie group SE(3) extension of the generalized-α time integration method is utilized to solve the equations of motion for thin-walled beams. Finally, the proposed formulation is successfully tested and validated in several static and dynamic numerical examples.