| Summary: | 碩士 === 國立交通大學 === 機械工程系所 === 104 === A corotational total Lagrangian (CRTL) finite element formulation for the geometrically nonlinear dynamic analysis of asymmetric thin-walled beam with large rotations but small strain is presented.
The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear center of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. For the purpose of treating arbitrarily large rotation of node in space, the orientation of the node is described by a base coordinate system rigidly tied to each node of the discretized structure, and a nodal rotation vector is used to represent the finite rotation of the base coordinate system. The values of nodal rotation vectors are reset to zero at current configuration, thus, the values of the first and second time derivative of the nodal rotation vector are equal the values of the spatial nodal angular velocity and acceleration. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the element. The current element coordinate system is regarded as an inertial local coordinate system, not a moving coordinate system. Thus, the first and the second time derivative of the position vector defined in the element coordinates are the absolute velocity and absolute acceleration. Three rotation parameters referred to the current element coordinates are defined to determine the orientation of element cross section. The deformation of the beam element is determined by the displacements of the shear center axis and the rotations of element cross section.
The element deformation nodal forces and inertia nodal forces are systematically derived by the d'Alembert principle, the virtual work principle and consistent second order linearization in the current element coordinates. The element stiffness matrix may be obtained by differentiating the element deformation nodal forces with respect to the element nodal parameters, and the element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the element nodal parameters, and their first and second time derivatives.
An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear equations of motion. The standard Newmark method is applied to the incremental displacement and rotational vectors, and their time derivatives. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.
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