Analysis of Elastically Coupled Non-Uniform Thin-Walled Composite Beams with Pretwist and Taper

Abstract

Helicopter rotor blades and aircraft wings can be modeled as thin-walled beams that are non-uniform and pretwisted. The non-uniformities are due to the taper of the beam arising from the fact that the thickness-to-chord ratios of the airfoils are higher at the root and lower at the tip. In addition, the wall thicknesses can be higher at the root than at the tip. Taper introduces interactions between axial strain and transverse shear strains. Pre-twist introduces a coupling between axial strain and torsion even for beams made of isotropic materials. Additional couplings can be introduced by the use of fiber reinforced composites. In order to fully model a thin-walled composite beam that has taper and pre-twist, it is necessary to model the interactions between the deformations due to axial, bending, transverse shear, and (St.Venant and Vlasov) torsion. Apart from some finite element codes, published analyses that capture these interactions are limited. In the present paper, a variationally consistent method is proposed that can analyze beams with the following features: thin- and thick-walled beams, open- or closed cross-sections, single- or multiple cells, pre-twist, taper in the cross-section profile and in wall thickness, and incorporate elastically coupled composite materials. The analysis considers the wall of the beam as a composite Reissner - Mindlin shell and includes the influence of elastic couplings and bending and transverse shear stiffnesses of the shell wall. The global beam model includes axial deformation, bending in two planes, St. Venant and constrained torsion, and transverse shear in two planes. No assumptions are made on cross-section deformability and the influences of hoop stress and hoop moment are included in the analysis. A mixed method based on a modified Hellinger-Reissner variational principle is used to model the influence of transverse shear stresses and cross-section deformation. The advantage of the present method is that it does not make any ad hoc assumptions on the cross-section warpings due to torsion and transverse shear effects. These warping functions are generated as part of the analysis, leading to a more accurate model. The analysis is applicable to thick- and thin-walled beams with open- or closed cross-sections. The cross-section stiffness matrix is obtained in closed-form. As an example, the cross-section matrix for a single-cell composite thin-walled beam is presented. It is shown that, for elastically-coupled composite beams, the transverse shear stiffness terms can influence the other responses and that ignoring these terms could lead to erroneous results. The analysis is validated by comparisons with results from elasticity theory, finite element analyses, and experiments for a variety of cross sections. These examples examine the influence of pre-twist, taper, and crosssection deformations on the axial, bending, transverse shear and torsion response of: uniform and pre-twisted solid beams with rectangular and elliptic cross-sections, uniform and tapered I-beams with pre-twist and single- and twocell box beams with pre-twist and elastic couplings.