DOI: 10.1155/2023/6675678 ISSN:

# Corotational Finite Element Dynamic Analysis of Space Frames with Geometrically Nonlinear Behavior Based on Tait–Bryan Angles

Ahmed A. H. Elerian, Saiid A. Shebl, Hesham A. Elkaranshawy
• Mechanical Engineering
• Mechanics of Materials
• Geotechnical Engineering and Engineering Geology
• Condensed Matter Physics
• Civil and Structural Engineering

The aim of this study is to compose a corotational finite element formulation for space frames with geometrically nonlinear behavior under dynamic loads. Using a moving frame through three successive rotations similar to Euler angles is one of the oldest techniques; however, there are still some gaps that require attention, mainly due to singularity. Hence, alternative techniques had been developed, sometimes elusive and computationally expensive. In this paper, we went back to the old technique and filled the gaps. Three-coordinate systems are used, i.e., the fixed global coordinate system, the fixed local coordinate system that is attached individually to every element, and the corotational local frame for each element that moves and rotates with the element. The deformation is always small relative to the corotational frame. The successive rotations between different coordinate systems are expressed using Tait–Bryan angles. Lagrange’s equation is used to derive the equation of motion, and the stiffness and mass matrices are obtained using the Euler–Bernoulli beam model. A MATLAB code is developed based on the Newton–Raphson method and the Newmark direct integration implicit method. In traditional techniques, singularity is attained when any rotation angle in the fixed local frame approaches $\pi /2$ , and if any is greater than $\pi /2$ , the techniques could fail to specify the location of the element. In this paper, each case is treated with a proper procedure, and special handling of trigonometric formulations prevents singularity and correctly specifies the location of elements in all situations. Different examples of beams and frames are analysed. While the method is not intricate, it is timesaving, is highly effective, provides more stable and robust analysis, and gives sufficiently accurate results. Compared to the parametrization of the finite rotations technique, the method has a significant reduction in the convergence rate because it avoids the storage of joint orientation matrices.