Bending Theory for Fiber-Reinforced Beams

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Abstract
The Saint-Venant theory of beams cannot account for constraints against cross- sectional warping, and for this reason it gives results that are inaccurate near fixed ends and at points where the warping changes abruptly, as under concentrated side loads. In highly anisotropic materials such as fiber-reinforced composites, the cumulative effect of such inaccuracies may be significant in some circumstances. In the present paper we present an approximate theory of the bending of cross-ply fiber-reinforced composites that is simpler than the exact theory of elasticity but is capable of accommodating any given displacement boundary conditions and arbi trarily varying side loads. As an illustration, the theory is applied to the problem of a tip-loaded cantilever with a fixed end. In this problem the theoretical results agree with elementary beam theory (with a particular choice of the shear correction factor) except in a relatively long end-effect region near the clamped end. To test the theory, cantilever specimens of a cross-ply fiberglass (Scotchply 1002) were prepared and stressed in a standard Instron testing machine. Strain distributions were monitored with resistance strain gauges, and deflections were measured with dial gauges. The strain distributions in the end-effect region were found to be much closer to those predicted by the present theory than to those predicted by the Saint-Venant theory. The deflection measurements did not agree with either theory, presumably because of rotation of the specimen in the clamps.

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