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.

Self‐centering concentrically braced frames (SC‐CBFs) have shown many advantages towards structural damage‐free seismic behavior. These systems could be designed either with a single rocking section at the base or with multiple rocking sections along their height. Initial design methodologies for both cases have been proposed with an emphasis on the former case. These methods focused on simplicity of the design process and mostly relied on parametric studies. Thus, the obtained designs may violate some displacement constraints when verified using nonlinear time history analysis (NLTHA) or lead to conservative designs that may be expensive. Furthermore, focusing on simplicity, these methods are tailored for regular structures. This paper takes a different approach that leads to minimum cost designs for irregular structures that satisfy the constraints exactly when verified using NLTHA. The proposed approach allows for multiple rocking sections along the height of the designed buildings, if these lead to cost reductions. Of course, this comes with more demanding computing resources. Nevertheless, these are kept reasonable for practical use on a personal computer. For that purpose, the NLTHA is performed using the computationally efficient Mixed Lagrangian Formalism. Furthermore, an efficient gradient‐based optimization framework is developed. The framework is applied for the design of 8, 12, and 20‐story spine SC‐CBFs and an irregular 12‐story building with a setback level. These have shown promising results while revealing also non‐traditional designs. This demonstrates the power of using optimization for the design of these lateral load resisting systems.

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