An ALE-CR formulation for nonlinear dynamics of 2D variable-length beam with unprescribed moving boundaries

Abstract

Conventional corotational (CR) beam formulations predominantly employ the Lagrangian description, which is inefficient in addressing moving boundaries. This study proposes a two-dimensional (2D) corotational formulation based on the arbitrary Lagrangian–Eulerian (ALE) description for the nonlinear dynamic analysis of beams with both prescribed and unprescribed moving boundaries. In the corotational framework, the length of the beam element is integrated into the local variables while the material coordinates of the nodes are incorporated into the global variables to describe the unprescribed material flow. Consequently, moving boundaries can be described accurately through moving nodes. An intermediate configuration is introduced to separate the material flow from the Lagrangian motion. The virtual work principle is used to derive the motion equations of the beam. The internal and inertial terms of the element, including those associated with material coordinates, are derived explicitly so that Gaussian quadrature is not required. The configurational force on the unprescribed moving boundary is effectively taken into account, which significantly influences the beam’s axial sliding behavior. The accuracy and efficiency are validated through examples involving moving loads, moving masses and moving supports. It shows the proposed ALE-CR beam formulation is capable of handling both prescribed and unprescribed moving boundaries.