Abstract
A corotational finite element formulation for two-dimensional beam elements with geometrically nonlinear behavior is presented. The formulation separates the rigid body motion from the pure deformation which is always small relative to the corotational element frame. The stiffness matrices and the mass matrices are evaluated using both Euler-Bernoulli and Timoshenko beam models to reveal the shear effect in thin and thick beams and frames. The nonlinear equilibrium equations are developed using Hamilton’s principle and are defined in the global coordinate system. A MATLAB code is developed for the numerical solution. In static analysis, the code employed an iterative method based on the full Newton-Raphson method without incremental loading, while, in dynamic analysis, the Newmark direct integration implicit method is also utilized. Several examples of flexible beams and frames with large displacements are presented. Not only is the method simple and time-saving, but it is also highly effective and highly accurate.
Highlights
Over the past few decades the geometrically nonlinear finite element analyses of flexible beams and frames with large displacements drew the attention of many authors [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]
Vi ρṘ Ti Ṙ idVi where ( )̇ is the differentiation with respect to time t, ρ is the density in the ith beam element, and Ṙ i is the velocity of a general point in the ith beam element with respect to the global coordinate system
The rigid body motion is separated from the pure deformation which is always small with respect to the corotational frame
Summary
Over the past few decades the geometrically nonlinear finite element analyses of flexible beams and frames with large displacements drew the attention of many authors [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The total Lagrangian formulation is the oldest kinematics description formulation and it is the most widely used one in finite element commercial programs such as ANSYS, ABAQUS, and MARC In this formulation, the system equations are defined in terms of a fixed global coordinate system, which is not changed through analysis. If the displacement from the current configuration to the last equilibrium configuration is large, a basic assumption is violated and the results cannot be trusted To avoid this problem, the corotational kinematical formulation was introduced. The majority of researchers neglected the shear effect in the geometrically nonlinear analysis of beam elements, so they evaluated the stiffness and mass matrices using Euler-Bernoulli beam model.
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