Dynamic rupture analysis of reinforced concrete shells

Abstract

Extreme dynamic loading conditions often require the rupture analysis of reinforced and prestressed-concrete structures. The study presented in this paper extends a method of analysis of dynamic loading conditions which has proven efficient for short- and long-time loads. Another aim is to adapt the method to thin-walled structures. It is not sufficient to work only with plastic rupture and yield surfaces locally which are compared to the elastic distribution of the stress resultants; it is essential to account for the redistribution of the latter. The method proposed consists of discretizing the structure into isoparametric three-dimensional elements with 20 nodes for the concrete and one-dimensional bar elements with three nodes for the steel. The latter can also be handled with a ‘smeared’ two-dimensional membrane element. In compression a three-dimensional non-linear elastic constitutive law is introduced for the concrete, and a triaxial failure surface expressed in the stress invariants is used, determining cracking and crushing. Two- and three-dimensional cracking surfaces in which no components of stress are transmitted are accounted for. The possibility exists that, during the history of loading, cracks can close up again. For steel, a yield criterion is selected. The non-linear analysis is based on the concept of initial stress. Residual loads are calculated using information in Gauss integration points. The ultimate load is reached when the algorithm does not converge. The corresponding failure modes can be interpreted as those for which a state of equilibrium is no longer possible. The equations of motion are discretized in time, using an extension of the linear acceleration method. As in the static case, several iterations are necessary to reduce the residual load vector to a negligible quantity. A built-in circular plate is analyzed for an evenly distributed load. The results with one and several isoparametric elements in width are compared to the solution determined with the classical yield line theory. Cracks in the concrete and yielding of the steel in both directions are properly represented. In the non-linear domain moment redistribution is observed which allows for a substantial increase in the ultimate load. All states of material behaviour are observed in the final stage. A thin-walled shell consisting of a cylinder and a sphere is also examined for a non-symmetric loading involving the interaction of membrane and bending behaviour. A dynamic non-linear analysis is performed for this load, which represents the impact of an airplane on the external shield building of a nuclear power plant. The non-linear analysis does allow for a substantial saving of reinforcement steel compared to the standard design procedure. Conclusions which include pitfalls, shortcomings and suggestions for future research are specified.