| Summary: | 碩士 === 國立交通大學 === 機械工程系所 === 102 === A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of bisymmetric thin-walled Timoshenko beams 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 centroid of the end cross-sections of the beam element and the axis of centroid is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The exact kinematics of the Timoshenko beam is considered. The element nodal forces are derived using the virtual work principle with the consideration of the shear correction factor. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. A consistent second-order linearization of the element nodal forces is used here. Thus, all coupling among bending, shearing, twisting, and stretching deformations of the beam element is retained. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix.
An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is used to find the buckling load.
Numerical examples are studied and compared with the results obtained by using Euler beam element to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of the shear deformation on the loading–deflection curves and buckling load of the bisymmetric thin-walled beams.
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