A unifying look at the cross-sectional analyses for composite beams

Relations between two different approaches to construct a composite beam theory are discussed. Namely, the approach that directly employs equilibrium equations is related to the asymptotic approach. First, numerical approaches that treat cross sections of general geometry, are considered. It is argued that the numerical implementations of both approaches lead to the Timoshenko beam models that are practically identical if the derivation is invoked consistently. Next, analytical approaches that treat thin-walled sections are addressed. Relative importance of the transverse shear in thin-walled beam is further discussed, and a "hybrid" Timoshenko beam theory for closed thinwalled sections is constructed. It employs first order ("classical")asymptotic approximation in combination with direct use of equilibrium equations to recover Timoshenko correction. Introduction While three-dimensional (3-D) Finite Element Analyses (FEA) are getting better, faster, and cheaper it is, perhaps too early to send all beam theories to the "dustbin of history". Yes, beam theories go back in history as far as Galileo and arguably predate any other developments in theory of structures. Despite this venerable past, there is still a great need for a compact yet precise description of elastic behavior of the bodies that can be qualified as effectively 1-D, or beam-like. The primary application of such theories would be in the multi-disciplinary fields, such as conceptual or preliminary design, as well as aeroelasticity and control. One of the obvious advantages is a beam theory is its "modularity": the 3-D elasticity problem splits into a two-dimensional 2-D cross-sectional analysis and 1-D beam problem. The results of the former can be reused again and again for various boundary conditions to solve multiple problems of the latter type. Trends be*Post Doctoral Fellow, School of Aerospace Engineering. Member,AHS. Copyright © 2001 by V.V.Volovoi. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. come more transparent, and optimization as well as parametric studies that are so important in structural design become faster and cheaper without sacrificing the fidelity of the results. The advantages of going from 3-D to 2-D are dramatic: a relatively modern PC can solve such a problem in a few seconds. Still, all these benefits non withstanding, such analyses have yet to be widely accepted in the engineering community. One of the reasons for this may be exactly the overabundance of such theories; it is difficult at times to discern the differences between them. The intent of this paper is to bring a bit more clarity and order to the multitude of existing in the literature numerical theories for composite beams. More specifically, the intent is to unite an engineering approach with a more abstract mathematical derivation, and thus, hopefully, provide simultaneous comfort for practicing engineers while satisfying the rigorous demands of theoreticians. Numerical Methods For a general geometric and material distribution over the cross section, the 2-D problem cannot be resolved analytically, and thus some discretization is needed; therefore, 2-D FEA is employed. Several composite beam theories were developed in the past 20 years." In the present paper, two approaches are briefly reviewed: a direct use of equilibrium equations to obtain the stiffness matrix for a Timoshenko model'' and variational-asymptotic method.'' Equilibrium Equations Approach Below the following notations are used: GIJ and Cij 3-D stress and strain components, respectively; Di is a tensor of generally anisotropic material properties; i — 1... 3, i = 1 indicates the direction along the beam axis (note that in the papers where this method is employed'" i — 3 corresponds to the beam axis here i = 1 is chosen for consistency throughout the paper). The main idea of the method is to to express both warping and strain measures in terms of six beam