Mixed Lagrangian formalism for dynamic analysis of self-centering systems

This paper presents two models for the analysis of self-centering concentrically braced frames (SC-CBFs) with multiple rocking sections. Both models are developed within the mixed Lagrangian formalism (MLF). For these models, MLF leads to a quadratic optimization problem at each time step where the internal forces at the end of the time step are the design variables. The first model is more general and explicitly models the truss elements. The second model is simplified and utilizes a macromodel. This model is easier to implement for initial designs, optimization, and parametric studies. For the purpose of formulating the simplified model, a new rotational contact element is developed and added to the MLF to describe the behavior of the rocking sections. Both models are compared to results obtained using the finite-element method. Although the comparison shows a good agreement, MLF is shown to allow much larger time increments for convergence. This considerably reduces the computational effort and central processing unit time. The simplified model is extended for three-dimensional analysis of SC-CBFs. This shows that the proposed models can be easily extended using relatively simple transformation matrices.