Nonlinear structural analysis towards collapse simulation: A dynamical systems approach

Nonlinear analysis of structures has become increasingly important in the study of structural response to hazardous loads. Such analyses should include (i) the effects of significant material and geometric nonlinearities; (ii) various phenomenological models of structural components and (iii) the energy and momentum transfer to different parts of the structure when structural components fracture.; Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this work, considering the structure as a dynamical system, two new approaches—(i) the state space approach and (ii) the Lagrangian approach are developed. These are mixed methods, where besides displacements, the stress-resultants and other variables of state are primary unknowns.; In the state space approach, the governing equations of motion and constitutive behavior of a structure are considered as constituting a constrained dynamical system, which is represented as a system of differential algebraic equations (DAE) and solved using appropriate numerical methods. A large-deformation flexibility-based beam column element is formulated, for use with the state space approach.; In the Lagrangian approach, the evolution of the structural state in time is provided a weak formulation using Hamilton's principle. The mixed Lagrangian developed is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and is solved using an Augmented Lagrangian algorithm.; In contrast to the displacement-based method, both the state space and the Lagrangian methods clearly separate the modeling of components from the numerical solution. Phenomenological models of components essential to simulate collapse can therefore be incorporated without having to implement model-specific incremental state determination algorithms. The state determination is performed at the global level by the DAE solver and by the optimization solver in the respective methods. These methods can be coupled with suitable pre- and post-processors to develop a unified computational platform for analysis of collapsing structures.