Numerical analysis of the branching of solutions to nonlinear equations for cylindrical shells

Abstract

A nonlinear boundary-value problem for shells is formulated. The problem statement permits us to analyze the behavior of the postbuckling solutions in a finite-dimensional space with the dimension independent of the discretization technique. A branching pattern is established for the solutions of the problem in the case of a uniform external pressure. Four characteristic types of solution are recognized in the case of an arbitrary external pressure. The associated buckling loads are determined. It is found out that five significant parameters are sufficient for a qualitative analysis of a cylindrical shell Keywords: nonlinear boundary-value problem, shell theory, internal and external pressure, buckling loads Introduction. The behavior of thin-walled shells under predominantly compressive stress was addressed in a great many studies, which is due to the wide use of such thin-walled structures in aircraft and mechanical engineering. The majority of these studies involve an analysis of the bifurcation points on the prebuckling solution branch of the corresponding nonlinear boundary-value problem. Secondary branching in local modes discovered in problems of uniform hoop [9] and axial [11] compression and the experimentally observed variety of deformation modes of shells [1,7] call for a more careful study into the general properties of the solutions of nonlinear systems. This would make it possible to predict and justify real buckling loads.A stability analysis of continuous conservative systems usually reduces to an analysis of infinite-dimensional discrete systems based on numerical procedures (the finite-element method, finite-difference methods, and direct variational methods) [4]. The development of computational methods of finding postbuckling solutions of nonlinear boundary-value problems (the parameter continuation method and a technique of moving to a postbuckling branch in the neighborhood of a bifurcation point) made it possible to analyze the postbuckling behavior of cylindrical shells. Noteworthy are the studies [16,21], where the dynamics of buckling is modeled. In [16], a group-theoretical analysis of a nonlinear boundary-value problem for shells was performed, and the structure of postbuckling branches was determined. Of special importance are the studies [10, 11], which set forth a "jump" method for finding the postbuckling branches of a nonlinear boundary-value problem. This method made it possible to obtain secondary postbuckling branches in the axial-compression problem for a cylindrical shell. The jump occurs in a Euclidean space obtained from a finite-element approximation of the governing partial differential equations.Also worthy of mention are the paper [13], where the Mushtari-Donnell-Vlasov equations were solved analytically, and the papers [5, 6], which used the solution to study the prebuckling and postbuckling stress-strain states of noncircular cylindrical shells. The papers [12,15,18,19] outline methods of solving geometrically nonlinear problems and specific results on the nonlinear ...

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