Asymptotic Treatment of the Vlasov Effect for Composite Beams

The Vlasov effect in composite beams is studied using the variational asymptotic method. It is shown how both Timoshenko and Vlasov models can be consistently obtained based on the energy that is asymptotically correct up to the second order. Interpretation in terms of the classical strength of materials approach is provided, and interplay between finding shear center, the Timoshenko shear corrections, and Vlasov theory is discussed. In particular, it is observed that constructing a theory that has both Timoshenko and Vlasov corrections is not straightforward; instead it is suggested to construct a Timoshenko theory first, then use it to find the shear center, and finally construct a Vlasov theory with the origin at the shear center. The method is illustrated with a simple example. Introduction Traditionally, composite beam theory was developed based on the extensions of the theories for isotropic beams. This route turned out to be challenging at times due to the fact that certain assumptions which happened to work quite well for isotropic beams led to erroneous results for composites. One can separate such assumptions into two groups: 1. An assumption is correct for isotropic materials, but incorrect in general for anisotropic materials. An example of such an assumption (that seems so obvious that is actually rarely explicitly stated) “bending shell strain measures can be neglected for closed sections” (see for the evidence that this assumption can lead to significant errors for certain lay-ups). 2. An assumption is actually wrong for isotropic materials as well, but the required correction is small and thus the assumption considered to be ∗Research Engineer, School of Aerospace Engineering. Member, AIAA and AHS. †Graduate Research Assistant, School of Aerospace Engineering. Student member, AIAA. ‡Professor, School of Aerospace Engineering. Fellow, AIAA. Member, AHS and ASCE. Copyright c © 2002 by Vitali V. Volovoi, Wenbin Yu, and Dewey H. Hodges. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. valid. The error can be significantly magnified for composite beams. A good example of such an erroneous assumption is the one used for thinwalled sections: “a cross-section remains rigid in its own plane” that was used practically in all displacement-based thin-walled theories, including the classical work of Vlasov. In retrospect it is hard to explain the resilience of this error and its pervasive permeation into composite theories. Perhaps it can be contributed to the fact that beam theories are approximations to 3-D elasticity, and therefore small corrections are considered to be acceptable; see further discussion of such inconsistencies in Vlasov’s work in Ref. 4. It is heartening to see that in the last decade, this assumption has been slowly but surely eliminated, and practically all recent work on the subject takes into account the in-plane cross-sectional deformation, at least implicitly by correcting the constitutive law. Despite all the recent progress in the area of composite beam theories, however, there are certain very basic questions that are still not totally explained even for isotropic case. The present paper is devoted to one of such questions: how to construct a consistent beam theory that would appropriately take into account shear Vlasov corrections and these two corrections are related to the location of a shear center. Even for simple, thin-walled isotropic beams, such as a channel section, this question has not been fully addressed. Let us recall that in classical strength of material books the shear center is found using shell equilibrium equation, so that shear flow is found along the contour. At the same time, one can use shear flow to estimate shear correction factors applying to open sections, a procedure described in Ref. 6 for closed sections. But their resulting Timoshenko theory does not predict the torsional-shear coupling that is needed to locate the shear center. At the same time, traditionally Vlasov theory (also sometimes referred to as Wagner theory) is treated as a completely independent problem from that of finding the shear center or shear corrections.