A mixed variational principle and its application to the nonlinear bending problem of orthotropic tubes—I. Development of general theory and reduction to cylindrical shells

Abstract

A procedure is presented for obtaining mixed, nonlinear variational principles for elastic shells based on the intrinsic formulation of the shell equations. The applicability of the procedure is demonstrated by developing specific principles for shells of weak curvatures and for circular cylindrical shells in regular and extended forms. Other cases are also discussed. The principles are developed within the scope of small-strain, large-rotation theory for shells under the Kirchhoff-Love hypothesis and require the availability of curvature functions for the given classes of shells. No other restrictions need be placed, except for those related to the geometries of the shells under investigation. Specifically, subject to the limitation of small extensional strains, the displacements and rotations may be large and no particular mode of shell behavior is postulated. The variational functional basically contain the strain energy of bending and the complementary energy of the membrane force resultants. These functionals are formulated in terms of curvature and stress functions and their Euler-Lagrange equations are those of normal equilibrium, Gauss compatibility and associated boundary conditions. All may be nonlinear. Using the extended principle as a starting point, approximate principles and equations are developed in Part II for the nonlinear, nonuniform bending of orthotropic circular cylindrical tubes of finite length (extended Brazier effort). The semi-membrane approximation, with membrane-type shear deformation retained, is used in the analysis, plus some added restrictions of the Rayleigh-Ritz type on the curvature and stress fields. The results can be used for problems involving tubes subjected to various beam and shell type boundary conditions. The specific example of a clamped tube subjected to pure beam bending is calculated, using solutions of the equations for weak nonlinearity and a Rayleigh method for strong non-linearity. Application of some of the results to the nonlinear “local buckling” analysis of a finite-length tube subjected to bending compare favorably with published results. Besides the interest in the specific problem, this demonstrates the applicability of the mixed principle for obtaining direct, approximate nonlinear solutions to useful ongoing problems, as a complement to more exact, but cumbersome, finite element or series solutions.