On asymptotically correct Timoshenko-like anisotropic beam theory
Introduction
The present cross-sectional beam analysis is developed in order to analyze transverse shear effects in anisotropic prismatic beams of general cross-sectional shape. These effects are important in short beam analysis, high frequency dynamics, and in composite material applications. The paper focuses in the determination of stiffness properties, including shear coefficients, for the cross-sectional analysis of general anisotropic beams when a one-dimensional (1-D) transverse shear measure is taken into consideration.
There has been a debate over the determination of the shear coefficient spread over the last century. The reader is encouraged to consult Kaneko (1975) for a good survey of the problem. Foeppl (1927), which refers to unpublished notes from 1897, obtained a value of k=5/6 for the shear coefficient for rectangular cross sections using an energy-based approach. Filon (1902) presented his shear theory based on the theory of elasticity and determined the shear coefficient both theoretically and experimentally. Timoshenko, 1921, Timoshenko, 1922 used Filon's experimental determination of the shear coefficient in his theory on the effect of transverse shear on the transverse vibration of bars. His definition of the shear coefficient was based on the average shear stress and the angle of shear at the neutral axis. Later, Timoshenko (1940) defined the shear correction factor based on geometrical assumptions as the ratio of the average strain over the section to the shear strain at the centroid. This ambiguity in the definition of the shear strain variable points out a deficiency of the geometrically-based methods. A well-known attempt to improve the results was done by Cowper (1966) whose approach gives a more involved expression for the shear correction factor based on the mean angle rotation of the cross section about the neutral axis. His results are closer to those obtained using energy methods, for example, in the case of a rectangular cross section.
Energy-based methods do not need to assume a definition for the shear strain variable. This comes naturally from the expression for the interior shear force resultant as the derivative of the strain energy with respect to the shear variable. This type of approach was used with good results by a number of authors, such as Timoshenko, 1958, Dym and Shames, 1973, Hoeborn, 1993, Renton, 1991, and Schramm et al. (1994). It must be mentioned that all the results discussed here assume a relatively slow variation of the shear force along the beam. See also Goodier, 1938, Cowper, 1966.
Thin-walled beams are a special case of importance for aeronautical structures. In this case, certain approximations can be made to simplify the problem and render relatively easily obtained approximate solutions for complex configurations; see Bauchau, 1985, Rehfield, 1985, Bauchau et al., 1987, Stemple and Lee, 1988, Chandra et al., 1990, and Rand (1998), all of which treat thin-walled beams made of composite materials. However, since none of these theories is asymptotically correct (nor are they claimed to be), a degree of uncertainty will always exist as to whether or not the results are accurate. Moreover, thin-walled beam theories are really inadequate for analysis of realistic rotor blades and wings, which typically have complex, built-up construction.
For more comprehensive treatment of anisotropic beams, an alternative to making a thin-walled beam approximation is to undertake a generalization of the Saint-Venant solutions, as done by Giavotto et al. (1983). A finite-element-based computer code called ANBA (Anisotropic Beam Analysis) or NABSA (Nonhomogeneous Anisotropic Beam Section Analysis) was developed by Borri and co-workers which renders a 6×6 cross-sectional stiffness matrix, including the two 1-D transverse shear measures (i.e., in orthogonal directions) along with extension, twist, and bending (the classical 1-D measures). Subsequent treatments by Kosmatka (1992) follow a similar approach (i.e., extended Saint-Venant solution leading to a finite-element-based cross-sectional analysis). Thus, until the present work, only a very few methods exist for determination of the cross-sectional stiffnesses of realistic anisotropic beam cross sections so that the effects of transverse shear are included in the resulting 1-D model. Unfortunately, the accuracy of such models is difficult to assess.
The variational-asymptotic method pioneered by Berdichevsky offers an accurate alternative to exact 3-D solutions when a small parameter is part of the problem. To obtain results that are consistently accurate, asymptotical correctness is the most important thing. By asymptotically correct, we mean that an expansion of the approximate solution in terms of a small parameter agrees with a similar expansion of the exact solution up to a certain order in the small parameter. When this method is applied and carried out to the extent that an asymptotically correct theory is obtained, this result is the most accurate theory possible for a given degree of complexity. This method has been applied for the transverse shear problem of prismatic isotropic beams by Berdichevsky and Kvashnina (1976) and to isotropic beams having initial twist and curvature by Berdichevsky and Starosel'skii (1983). Although it has been applied to the analysis of initially twisted and curved anisotropic beams by, for example, Cesnik (1994), Cesnik, Hodges and Sutyrin (1996a), and Cesnik and Hodges (1997), it has not been applied toward the development of a refined theory which treats transverse shear deformation of composite beams. Although Cesnik (1994) did develop a so-called `alternative theory' to treat transverse shear effects, this theory is not asymptotically correct nor was it claimed to be.
Therefore, in this paper a refined theory to treat shear deformation in composite beams is carried out via the variational-asymptotic method. The approach and the results are somewhat similar to those of a similar analysis by Sutyrin and Hodges (1996) for laminated composite plates. In that work an asymptotically correct refined theory having the form of a Reissner-like plate theory (analogous to a Timoshenko-like beam theory) does not always exist for plates made of anisotropic materials. However, by means of certain optimization procedures a Reissner-like theory can be obtained that is quite close to being asymptotically correct.
Section snippets
Refined theory formulation
The classical theory of anisotropic beams, as developed by Hodges et al. (1992) and by Cesnik and Hodges (1997), contains only four generalized strain measures: extension, torsion, and bending in two directions. The different levels of accuracy are assessed based on asymptotic series in terms of several small parameters. One such parameter is the maximum strain in the beam, ϵ. Classical theory assumes ϵ≪1. Another small parameter for beams is h/l, where h is a characteristic dimension of the
Transformation to Timoshenko-like form
The Timoshenko-like formulation that is sought here implies writing the strain energy in the formwhere S is the corresponding 6×6 stiffness matrix and is an extended column matrix of 1-D strain measures,and where the bending curvature in the relation between the curvatures in the tangential system βi and the curvatures in the cross-sectional system κi can be shown to be given by
Here, eαβ is the permutation symbol, and terms of higher
The meaning of the shear strain variable
Since there are two main approaches to the transverse shear formulation: the geometric approach and the energy approach, it makes sense to make a connection between them by extracting the geometrical definition of the shear strain variable. In the present formulation, the definition of the shear strain variable 2γ12 comes from the relation for the resultant shear force:
Let us assume, for simplicity, a homogeneous, isotropic cross section. The resultant of the shear stresses can be
Numerical results
The computer program VABS of Cesnik and Hodges (1997) has been upgraded to include the above analysis. In this section, numerical results from VABS for both isotropic and anisotropic cases are presented and compared with available published results.
Conclusions
A method capable of capturing the shear effects in beams made of arbitrary anisotropic material and of general cross-sectional shape has been developed. The method is based on asymptotic expansion of the energy in terms of a small parameter and then seeking the solution in a variational-asymptotic manner. Transverse shear effects are captured and a physical interpretation of the resulting of the shear variable was given.
It can be shown that for the situations when the stiffness matrix is
Acknowledgements
This work was supported through the Center of Excellence for Rotorcraft Technology, at Georgia Institute of Technology, sponsored by the NASA/Army National Rotorcraft Technology Center, Ames Research Center, Moffett Field, California.
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Present address: Bell Helicopter Textron, Mirabel, Que., Canada.