F. Khouli, F. F. Afagh and R. G. Langlois

Department of Mechanical and Aerospace Engineering,

Carleton University, Ottawa, Ontario, Canada
E-mail: fafagh@mae. carleton. ca


An application of a comprehensive and compact methodology to obtain the asymptotically-correct stiffness matrix of anisotropic, thin-walled, closed cross-section, and rotating slender beams is presented. The Variational Asymptotic Method (VAM), which utilizes small geometrical parameters inherent to thin-walled slender beams, is used to obtain the displacement and strain fields, and the cross-sectional stiffness matrix without any ad hoc assumptions. The advantage of this approach is that the asymptotically-correct and populated 4 x 4 cross-sectional stiffness matrix provides all the necessary information about the elastic behavior of the rotating beam, thereby nullifying the need for refined beam theories that incorporate higher order deformation modes, like the Vlasov’s mode. The implementation of the theory using MATLAB was validated against the Vartiational Asymptotic Beam Sectional Analysis (VABS) computer software, a two-dimensional finite element program that utilizes a more general approach to the VAM that is applicable to thick/thin-walled anisotropic cross­sections with arbitrary geometry. Sample applications of the theory to rotor blades are presented. The paper concludes with a discussion of how the presented material would be used directly in the dynamic modelling of rotating helicopter blades.


The structural dynamics modelling of rotating composite blades closely follows the advances made in capturing the elastic behavior of anisotropic, slender, and rotating beams with arbitrary cross-sectional geometry. In the past, the highly coupled structural and dynamics aspects of the model along with its strong nonlinearity proved to be an unyielding obstacle, and engineers were forced to adopt various approximations that limited the scope and applications of their models (Volovoi et al., 2001), and were later proved to be less than stellar in predicting the behavior of the rotating blades (Hodges and Patil, 2005, Volovoi et al., 2001). Some modifications were added to these models, which improved their performance, but not to the levels required for today’s advances in composite and highly flexible blades. However, these modified preliminary models are still used by the industry today despite existing limitations. One may attribute this to be the result of their simplicity in terms of their formulation, and the experience and insight into their functioning that has been accumulated over time. Performing 3D finite element analysis on the rotating composite blades is an expensive and an unfeasible option even with current computational capabilities, espe­cially when the analysis is directed at devising vibration control strategies or studying the structural dynamics interaction of the blades with other rigid/flexible multibody systems.

Recent advances in the cross-sectional modelling of anisotropic composite beams with arbitrary geometry is a major triumph in overcoming the difficulties discussed above. It was found that for slender beams, asymptotical analysis of the 3D elastic energy can split the problem into a two­dimensional analysis over the cross-section and one-dimensional analysis along the span of the


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 537-547.

© 2006 Springer. Printed in the Netherlands.

beam without any ad hoc assumptions. Utilizing other geometrical design aspects of the rotor blade, one can also arrive at a closed form solution of the stiffness properties of the cross-section. This significantly reduces the amount of effort the engineer has to spend on modelling the elastic behavior of the blade, since the asymptotically-correct development is found to be surprisingly compact for a problem that seemed to be impossibly complex from the earlier models’ point of view.

The VAM was originally developed by Berdichevsky (1982) for elastic slender rods that have an inherent small dimensionless parameter, which is the slenderness ratio defined by the ratio of the characteristic dimension of the cross-section, V, to the elastic deformation wavelength, T. Since only global elastic deformation modes that propagate along most of the beam span are of interest here, the deformation length is always of the order of the length of the beam. The theory was refined by Hodges and Cesnik (1994) and implemented in Variational Beam Sectional Analysis (VABS), a software package that utilizes the finite element method to obtain the elastic constants of any composite cross-sections with initial twist. VABS has been extensively validated against experimental data and results from other reliable 3D finite element software like ABAQUS and NASTRAN (Yu et al., 2002). Concurrently, Badir (in Berdichevsky et al., 1992) expanded the theory to thin-walled composite beams that have an additional small dimensionless parameter, which is the thinness ratio defined by the ratio of the thickness of the wall, ‘A’, to the characteristic dimension of the cross-section, V. This allowed for a simple closed form solution for the stiffness constants, which has been used to model rotor blades with active materials (Cesnik and Shin, 1998, 2001a, 2001b). However, it was later found that Badir’s work neglected the shell bending strain measure, which made it asymptotically-incorrect and produced results inconsistent with those produced by VABS for certain cross-sections. Hodges and Volovoi (2000) identified and corrected this flaw, and developed the asymptotically-correct theory for anisotropic thin-wall beams.

Rotor blades can be idealized as thin-walled closed cross-section beams while retaining high degree of fidelity. The asymptotically-correct theory has been successfully implemented using MATLAB to arrive at the elastic properties of any closed cross-section including helicopter rotor blades, which are at the centre of many research efforts of the Applied Dynamics Group at Carleton University. The versatility of the implementation allows it to obtain the elastic properties about any desired point in the plane of the cross-section like the elastic axis of the blade. Additionally, it allows for any desired material composition and distribution throughout the cross-section.