Dynamics Of Beams Undergoing Large Rotations Accounting For Arbitrary Axial Deformation

Abstract

It is well-known that flexible beams become stiffer when subjected to high speed rotations. This is due to the
membrane-bending coupling resulting from the large displacements of the beam cross-section. This effect, often
called geometric stiffening, has been largely discussed in the last two decades. Several methodologies have been
proposed in the literature to account for the stiffening effect in the dynamics equations. However, considerable effort
is generally done to derive linear models using steady-state assumptions and membrane-bending decoupling. This
work aims first to present a brief review of the open literature on this subject. Then, a general non-linear model is
formulated using a non-linear strain-displacement relation. This model is used to deeply analyze simplified models
arising in the literature. In particular, the assumption of steady-state values for the centrifugal load is analyzed and
its consequences are discussed. Thereafter, four finite element models are proposed, one based on non-linear theory
and the others on simplified linear theories. These models are then applied to the study of a flexible beam undergoing
prescribed high speed large rotations. The analyses show that one must account for the geometric stiffening effect
to obtain realistic results. In addition, it is shown that models disregarding the axial displacement dynamics lead to
erroneous results for the axial stress in the beam, which may be of main importance in structural integrity analysis.
Hence, in the general case, geometric stiffening must be accounted for in association with the inclusion of full axialtransverse
displacements coupling dynamics in the model.