Computer analysis of structures has traditionally been carried out using the displacement method
combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach
is developed, which is a mixed method, where besides displacements, the stress-resultants and other variables
of state are primary unknowns. The method can potentially be used for the analysis of structures to collapse
as demonstrated by numerical examples. The evolution of the structural state in time is provided a weak
formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal
structures has a mixed Lagrangian formulation in terms of displacements and internal forces. In considering
elastic-plastic systems, it is shown to be natural to also include the time integrals of internal forces
(momentum) in the structure as configuration variables. The form of the Lagrangian 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, the Lagrangian method clearly separates 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 optimization
method.
