Dynamic behavior of an extended Gao beam model including shear deformation

This study develops a model of the dynamics of the extended 2D Gao beam and simulates it. Here, the static model studied by Dyniewicz, Shillor and Bajer (Meccanica, 2024), is modified by incorporating inertial terms to account for dynamic behavior. The beam model expands the 1D Gao beam, which can oscillate around a buckled position, and the Timoshenko beam, which factors in shear effects in the beam’s cross sections. The resulting model consists of two highly nonlinear wave equations, alongside specified initial and boundary conditions. A finite element method (FEM) algorithm is created and executed to analyze the system’s vibrations induced by a periodically oscillating longitudinal compressive force. The simulation results are discussed, highlighting the ways the initial conditions influence the solutions, which are graphically illustrated through phase portraits. From an engineering viewpoint, this thick Gao beam model is notable for its relative simplicity. Similarly to the Timoshenko beam model, it includes shear effects, yielding a wave-like equation of motion. Considerations of the shear are essential for accurately analyzing thicker beams, as traditional models that overlook them may fail to capture the true system behaviors. Consequently, this extended Gao model offers more realistic outcomes in dynamic scenarios.