Nonlinear Oscillations and Buckling Prediction for Shallow Convex Shells

Abstract

Since 2010, the testing of thin-walled shallow shells has revealed that the ultimate internal stresses, under which the shells lose their stability (collapse), appeared to be lower than those predicted by the buckling theory used in American and European standards. The existing norms are based on the static theory of shallow shells proposed in 1930 with stable solutions for thin-walled structures within the nonlinear theory. These stable solutions differ significantly from the forms of equilibrium common to small initial loads. The minimum load, under which an alternative form of equilibrium exists, was used as the ultimate load. In the 1970s, this approach was proved to be unacceptable for complex loadings that were not a part of the real world in the past but are now the case with thinner products operated under complex conditions. Therefore, the initial theories on bearing capacity evaluations must be revisited. The new theory could be built on the basis of recent mathematical results establishing that the estimates made per two schemes (three-dimensional dynamic theory of elasticity and dynamic theory of shallow convex shells) were asymptotically close. Green's special function is introduced to convert the Foppl-von Karman system into a resolving integro-differential equation. The obtained nonlinear equation allows for the separation of variables and has numerous time-periodic solutions that satisfy the Duffing equation with a "soft spring". Numerical analysis enables determining the amplitude and period of oscillations depending on the properties of Green's function. This paper reviews an experimental setup in which oscillations are generated with the probing load directed normally to the surface of the shell. The experimental measurements of the resonance displacements as a function of coordinates and time enable the calculation of the safety factor for the bearing capacity of the structure using the non-destructive method under operating conditions.