A MIXED VARIATIONAL PRINCIPLE AND ITS APPLICATION TO THE NONLINEAR BENDING PROBLEM OF ORTHOTROPIC TUBES .1. DEVELOPMENT OF GENERAL-THEORY AND REDUCTION TO CYLINDRICAL-SHELLS

A procedure is presented for obtaining mixed, nonlinear variational pr inciples 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-Lov e hypothesis and require the availability of curvature functions for t he given classes of shells. No other restrictions need be placed, exce pt for those related to the geometries of the shells under investigati on. Specifically, subject to the limitation of small extensional strai ns, the displacements and rotations may be large and no particular mod e of shell behavior is postulated. The variational functionals basical ly contain the strain energy of bending and the complementary energy o f the membrane force resultants. These functionals are formulated in t erms of curvature and stress functions and their Euler-Lagrange equati ons are those of normal equilibrium, Gauss compatibility and associate d boundary conditions. All may be nonlinear. Using the extended princi ple as a starting point, approximate principles and equations are deve loped in Part II for the nonlinear, nonuniform bending of orthotropic circular cylindricaltubes 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 ca n be used for problems involving tubes subjected to various beam and s hell type boundary conditions. The specific example of a clamped tube subjected to pure beam bending is calculated, using solutions of the e quations for weak nonlinearity and a Rayleigh method for strong nonlin earity. Application of some of the results to the nonlinear ''local bu ckling'' analysis of a finite-length tube subjected to bending compare favorably with published results. Besides the interest in the specifi c problem, this demonstrates the applicability of the mixed principle for obtaining direct, approximate nonlinear solutions to useful ongoin g problems, as a complement to more exact, but cumbersome, finite elem ent or series solutions.