Pure Bending, Stretching, and Twisting of Anisotropic Cylindrical Shells

Abstract

This paper considers the problem of determining stresses and deformations in elastic thin-walled, prismatical beams, subject to axial end forces and end bending and twisting moments, within the range of applicability of linear theory. The analysis, which generalizes recent work on the problem of torsion [1], is based on the differential equations of equilibrium and compatibility of thin shells in the form given by Gunther [2], together with constitutive equations given by the first-named author [3]. The technically most significant aspect of the work has to do with the analysis of the effect of anisotropy of the material, which is associated with previously not determined modes of coupling between stretching, bending, and twisting. Use of the general formulas of the theory is illustrated for a class of shells consisting of an “ordinary” material (unable to support stress moments with axes normal to the middle surface of the shell, and unable to undergo transverse shear deformation). Here explicit formulas are obtained for certain types of open as well as of closed-cross-section beams.

Saint-Venant’s problem for Cosserat shells with voids

“…We observe that the constants a 3 , a a and k 0 , respectively, represent measures of stretch, curvature and twist of the cylindrical shell considered as a beam. Thus, the relations (6.8)-(6.11) coincide with the classical results given in Reissner and Tsai (1972) (for the case of an isotropic material). The same results can be obtained from the work of Berdichevsky et al (1992) for the case of isotropic thin-walled closed-crosssection tubes (see also Ladevèze et al, 2004).…”

“…The displacement in the axial direction u 3 given by (6.6) 2 coincide with the results obtained from the work of Reissner and Tsai (1972) for isotropic materials. Also, the components u a of the displacement vector for open cylindrical shells have the same form as the solution found by Reissner and Tsai, except for the term S a [a i ,B i ](s) which appears in (6.6) 1 .…”

“…In this case, the functions x a (s) are given by (5.19) and we notice that S a [a i , B i ](s) = S a [b b , 0, C i ](s) = 0. Then, from (6.6) and (6.8)-(6.11) we deduce the solution of the extension-bending-torsion problem u a ¼ 1 2 ab a b x 2 3 À mð bc a b x c þ a 3 Þx a À k 0 ab x b x 3 ; The same solution (6.23) and (6.24) is obtained from the results of Reissner and Tsai (1972). From (6.13) and (6.14) we deduce the following solution of the flexure problem for circular closed cylindrical shells:…”

Beamlike (Saint–Venant) solutions for fully anisotropic elastic tubes of arbitrary closed cross section

“…[If the elastic coefficients are allowed to depend on y, as in Reissner and Tsai (1972), then q 0 still has the same form as (47), but the function sðyÞ becomes more complicated since it now depends on C Ã 2222 ðyÞ.] By (46), sð2pÞ ¼ sð0Þ.…”

A generalized Vlasov theory for composite beams

“…There exists an alternative approach to constructing thin-walled beam theories that avoids ad hoc kinematic assumptions and relies instead on equilibrium equations. This in general leads to more rigorous thin-walled beam theories [11,12]. Some attempts to apply this method to the development of Vlasov theory have been made [10], but the procedure is not straightforward because there are not enough equilibrium equations to solve for all the necessary quantities.…”