On the importance of cross-sectional warping in solid composite beams
Introduction
Structural modeling and analysis of composite beams is essential to many engineering applications, among which helicopter blades and aircraft wings are typical representatives. It is well known that compared with isotropic beam analyses, composite materials introduce many additional design degrees of freedom which complicate their modeling and analysis mainly due to the fact that the cross-sectional warping effects play a major role. Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] are representative studies that discuss the influence of warping on the structural analysis of composite beams in general, and serve as evidence to the research effort in this field. Part of the studies is focused on the coupling effects in composite beams which are strong functions of the warping modeling. Refs. [11], [12], [13] represent studies that explored the same phenomenon denoted as shear deformation in the context of thin laminated plates.
Although numerical analyses have shown that the cross-sectional warping has a significant influence on the structural characteristics of composite beams, such analyses preclude a clear insight into the role and the relative importance of the warping components. In general, the warping components may be divided into the “out-of-plane warping” which consists of the deformation perpendicular to the cross-section, and the “in-plane warping” which consists of the deformation in the cross-sectional plane that causes changes in its shape. When isotropic materials are under discussion, the whole issue of cross-sectional warping is bypassed by invoking the axial stresses to express the cross-sectional bending moment. Unlike the case of composite materials, the axial stresses in isotropic beams do not depend on the shear strain, and therefore, bending moment–curvature relationship may be correctly predicted without warping modeling. However, in composite beams, such a technique is not possible, and as already indicated, there are many numerical indications in the literature that show that warping effects are essential for correct prediction of the structural behavior. Note that as far as the “beam behavior” is concerned, one is mainly interested in the warping influence on global beam phenomena such as the bending moment–curvature relationship or the bending curvature-twist coupling magnitude (in symmetric beams).
In order to contribute to the above effort of identifying and establishing the relative importance of the warping components, the closed form exact (linear) solution for the behavior of an orthotropic, homogeneous, solid composite beam of arbitrary geometry due to a uniform bending moment is presented in what follows. The displacements which are presented in “beam terminology”, are expressed in terms of elastic axis deformation, twist angle and three generic cross-sectional warping functions. Further on, the influence of eliminating the warping effects from the model is demonstrated and discussed.
Section snippets
A uniform beam under pure bending
Consider a uniform beam of homogeneous, orthotropic, simply connected solid cross-section of arbitrary geometry – see Fig. 1(a) and (b). The term “homogeneous” stands for the case where all material properties are constants which may be viewed as the case of a layup of identical laminae oriented at the same angle relative to the x axis (or as a single lamina layup). The orthotropic laminae are parallel to the x–y plane (as will be shown later, this assumption poses no restriction since the
Concluding remarks
The role of the cross-sectional warping components has been studied via an exact solution for solid orthotropic beam of arbitrary cross-sectional geometry that undergoes a bending moment. This solution supplies a clear and important insight regarding the role of the warping components in the determination of the structural behavior of composite beam. It has been shown that the out-of-plane warping is of primary importance and should be considered as a first-order ingredient. In general, the
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