On the Geometrically Exact Low Order Modelling of a Flexible Beam: Formulation and Numerical Tests
This paper proposes a low order geometrically exact flexible beam formulation based on the utilisation of generic beam shape functions to approximate distributed kinematic properties of the deformed structure. The proposed nonlinear beam shapes approach is in contrast to the majority of geometrically nonlinear treatments in the literature in which element based --- and hence high order --- discretisations are adopted. The kinematic quantities approximated specifically pertain to shear and extensional gradients as well as local orientation parameters based on an arbitrary set of globally referenced attitude parameters. In developing the dynamic equations of motion, an Euler angle parameterisation is selected as it is found to yield fast computational performance. The resulting dynamic formulation is closed using an example shape function set satisfying the single generic kinematic constraint. The formulation is demonstrated via its application to the modelling of a series of static and dynamic test cases of both simple and non-prismatic structures; the simulated results are verified using MSC Nastran and an element-based intrinsic beam formulation. Through these examples it is shown that the nonlinear beam shapes approach is able to accurately capture the beam behaviour with a very minimal number of system states.