A mixed variational principle and its application to the nonlinear bending problem of orthotropic tubes—II. application to nonlinear bending of circular cylindrical tubes

Abstract

A procedure for deriving mixed variational principles for nonlinear shell analysis was presented in Part I and formulated in more detail for shells of weak curvatures and for circular cylindrical shells. The cylindrical shell principle is extended here to accommodate those special cases in which the retention of mixed terms in the compatibility equations is of significance. Emphasis is put on the interaction of longitudinal extensional strains with large circumferential changes of curvature. As a simple first example, it is applied to the orthotropic Brazier process, for which an Euler-Lagrange equation and perturbation solution are also derived. Variational principles have important uses for providing direct, approximate “engineering solutions” to highly nonlinear problems. Such solutions complement the more exact but cumbersome finite element or double series techniques. In the present case, reasonable simplifying assumptions are introduced in the extended principle to construct approximate principles and equations for the problem of strong, nonlinear, nonuniform bending of finite-length orthotropic tubes. As a specific example, the pure bending of a clamped, finite-length tube is studied. Approximate analysis is carried to collapse (or local buckling). Some of the local buckling results are compared with numerical results from the literature.