Implementation of fracture mechanics concepts in Dynamic progressive collapse prediction using an optimization based algorithm

Prediction of progressive collapse of buildings under extreme events is one of the challenges that the civil engineering community must face. The ability to predict progressive collapse would enable the identification of deficiencies in the common practice of structural design without having to wait for the next extreme event to occur. With such ability, new design strategies and technologies for progressive collapse prevention could be addressed. Progressive collapse prediction, where the capacity of structures for progressive collapse is assessed, faces a great number of challenges. In terms of feasibility, it should address the challenge of analyzing large scale buildings in all stages of collapse. The complex behavior of buildings during collapse often leads to issues of stability of the numerical scheme, hence to the collapse of the analysis prior to the actual analysis of collapse. Indeed, the area of dynamics of structures including phenomena expected during progressive collapse (e.g. contact, fracture) has been developed to a high level. Nevertheless, there is no theory and computational tool that can efficiently predict all stages of progressive collapse of large scale structures. The Mixed Lagrangian Formulation (MLF) could potentially provide such a theory as well as an accompanied efficient, robust and stable numerical scheme. It also considers almost all stages of collapse in a unified manner, thus, almost completes the puzzle of Progressive collapse prediction. The missing part of the puzzle, considering fracture in a unified manner, is the aim of this paper. This is developed using concepts from Fracture Mechanics for brittle material by using Griffith's theory. This, in turn, will lay the foundation for considering more complex models in the future. Additional state variables required to model fracture using fracture mechanics concepts are identified. Subsequently, appropriate stored energy and dissipation functions, which lead to Griffith's theory, are formulated. Once the stored energy and dissipation functions are formulated, Hamilton's principle is discretized in time to lead to an optimization problem at each time step. The solution of the optimization problem supplies with the states at the end of the time step. This results in a sound theory as well as an efficient, robust and stable numerical scheme for progressive collapse prediction, as supported by the examples.