Computer simulation of local and global buckling of thin-walled shells

Stability of thin-walled structures is investigated on the basis of the geometrically nonlinear theory of shells. This allows us to monitor a series of shell deformation processes under different load changes. The local and general loss of stability can be determined by observing changes in the shape of the shell curved surface before and after critical loads. The shell material can be isotropic or orthotropic, but linear-elastic. A mathematical model of deformation of a shell is the functional of total potential energy of deformation of the shell. Two methods are applied to minimize the total energy functional of deformation of the shell. One is based on the method of L-BFGS with discrete approximation of the unknown functions by NURBS-surfaces. This enables taking into account various forms of fixing of the shell contour and the complex shape of this circuit. The other employs the Ritz method and the best parameter continuation method in the continuous approximation of the unknown displacement functions and the angles of rotation of the normal. This technique allows us to find the upper and lower critical loads and the bifurcation point. A combination of these techniques makes it possible to study both the subcritical and supercritical behavior of the structure and to identify the local and global buckling of the shell and their relationship. On the curve of equilibrium states versus “load q - deflection W ” one can see all moments of buckling of the shell caused by “swatting” of some part of it. Thus, after each buckling a significant change in the shape of the curved surface is observed. For illustration purposes the changes in the shell shape at subcritical and supercritical stages are plotted against its three-dimensional surface. After the general loss of stability the shell deforms under loading without any significant change in its shape, i.e. it behaves as a plate.