Geometrically Nonlinear Analysis of Composite Beams Based Using a Space–Time Finite Element Method

In this paper, we present a transient-format time-continuous Galerkin finite element method for fully intrinsic geometrically exact beam equations that are energy-consistent. Within the grid of space and time, we derive governing equations for elements using the Galerkin method and the time finite element method, implement variable interpolation via Legendre functions, and establish an assembly process for space–time finite element equations. The key achievement is the realization of the free order variation of the program, which provides a basis for future research on adaptive algorithms. In particular, the variable order method reduces the quality requirements for the mesh. In regions with a higher degree of nonlinearity, it is easier to increase the variable order, and the result is smoother. Meanwhile, increasing the interpolation order effectively enhances computational accuracy. Introducing kinematical equations of rotation with Lagrange operators completely imposes the conservative loads on fully intrinsic equations. This means that loads in the inertial coordinate system, such as gravity, can also be iterated synchronously in the deformed coordinate system. Through a set of illustrative examples, our algorithm demonstrates effectiveness in addressing conservative loads, elastic coupling deformation, and dynamic response, demonstrating the ability to analyze elastically coupled dynamic problems pertaining to helicopter rotors.