End Effects in Spatial Beam Elements: Comparing the Variational Asymptotic Method and the Eigenwarping Method

Spatial beam elements can be used for efficient simulation and optimisation of flexure mechanisms undergoing large deflections. However, for flexures with low slenderness ratio, end effects can become relevant. These effects can be captured by introducing additional warping fields in the formulation. In this work, we compare the warping fields generated by the Variational Asymptotic Method (VAM) to the warping fields generated by a recently developed method, referred to as the Eigenwarping method (EWM). VAM constructs an expansion of the cross sectional warping, where it is shown that each term has an asymptotically smaller contribution to the internal elastic energy in terms of the slenderness ratio h/L. By truncating the expansion at a given order, an accurate representation of the beam behaviour is obtained for only a small number of required warping coordinates. The EWM considers the decomposition of the cross sectional constitutive behaviour in non-decaying fields associated with an
elastic line and decaying cross sectional warping fields. The resulting set of decaying warping fields can be truncated based on the associated decay length. In this paper, an alternate interpretation of VAM is presented that allows better comparison to the EWM. The warping fields obtained from both methods are compared to each other and to classical results. Additionally, a beam element with warping fields is introduced. Using the beam element, simulation results obtained with the warping fields obtained from VAM and EWM can be compared. The case of a notched leaf flexure is introduced, which contains a discontinuity in the cross section. Both methods can describe the behaviour in the continuum and at fixed boundaries with only a small number of additional warping fields. However, a large number of warping fields is required to accurately describe the behaviour at the notch boundary.