Efficient and Accurate Modeling of Non-Prismatic Beamlike Structures

Abstract

Many complex engineering structures, e.g. components of helicopters, wind turbines, aircraft wings and propellers, are beamlike and non-prismatic. Structures of this kind may be tapered, pre-twisted, and even curved in their unstressed state, and undergo large displacements and 3D cross-sectional warping. Their mechanical modeling can be addressed via non-prismatic beam elements providing the appropriate compromise between computational efficiency and accuracy. Over the years many models have been proposed for beamlike structures, but general non-prismatic cases still require investigation. Formulas valid for prismatic beams, for example, generally provide incorrect results in non-prismatic cases, as the variation in the dimensions and orientation of the transverse cross-sections produce non-trivial stress distributions absent in prismatic beams. A model suitable for the aforementioned non-prismatic elements should properly describe their shape, explicitly consider the effects of their geometric design features on their stress and strain fields, account for large displacements, and provide the known results of prismatic cases. We propose a physical-mathematical model that accounts for all such requirements. The non-prismatic beam is seen as a collection of plane figures (the transverse cross-sections) attached at a 3D curve (the beam's centre-line). The centre-line's points may undergo large displacements. The transverse cross-sections are fully deformable and may undergo warpings in and out of plane. Assuming small warping and strain fields, a variational approach provides the field equations. The model obtained enables evaluating even analytically the effects of geometric parameters (such as taper) on the stress and strain fields. Numerical examples and comparisons with the results of nonlinear 3D-FEM analyses confirm the effectiveness of the proposed modeling approach.