## 4 Strengthening Scheme and the Strengthened Model
### Model Parameters  
Four RC structures are designed to investigate the seismic performance of the RC frame structures with design deficiency before and after retrofitting. For comparison, a four-story, three-span RC frame structure (No. S-1) is designed to meet the minimum requirements of Chinese seismic code GB 50011-2010, and its column section is \(450\times 500\) mm\({}^{2}\). NS-1 is a four-story RC structure with a design deficiency in which its column section is smaller than the minimum design requirements in accordance with the design code of China and exhibits low ductility. The column sections of NS-1 are \(400\times 400\) mm\({}^{2}\) smaller than the S-1's. The low-ductility structure (NS-1) is then strengthened by enlarging the column section (RS-1) and adding steel braces (RS-2). The RC column section of RS-2 is the same as NS-1's, whereas RS-1's is \(500\times 550\) mm\({}^{2}\), the largest among the four structures. Figure 6 demonstrates the design parameters and column sections of the first three frame structures. Figure 7 illustrates the arrangement scheme of the steel braces, which are all DN 100 mm, 20 mm-thick steel pipe with Q235. Additional details are presented in Figures 6 and 7.  
A two-dimensional finite element model of structures is developed in OpenSees [37]. The fiber element model is the closest to the actual mechanical behavior in the nonlinear analysis of RC frame structures; thus, the fiber section element is used to describe the nonlinear behavior of beams and columns. The beam and column members are set with five integration points. The concrete behavior is described by the nonlinear degrading Concrete02 material model. The reinforcement behavior is modeled by the Giuffre-Menegotto-Pinto model and steel02 material [2; 37]. The steel structure design code states that the ratio of the outer diameter of the steel tube to the wall thickness should not exceed 100, and the ratio of the member length to the section height (or diameter) should be greater than 12 in the truss plane to meet the anti-buckling requirements [38]. Slenderness ratio should also be considered in brace buckling and tension [1; 3]. In this study, the ratio is 39.11. The truss element model is used to model the diagonal braces and does not buckle in the simulation. The diagonal braces are the buckling restrained. The behavior of the braces is also described using the steel02 material.  
The structures are located in a region with seismic precautionary intensity 8, seismic design category A, and site class II. The spectral acceleration at fundamental period \(S_{a}\) (\(T_{1}\)) is regarded as ground motion IM. The fundamental period T\({}_{1}\) of NS-1, RS-1, RS-2, and S-1 is 1.15, 0.90, 0.65, and 0.98 s, respectively. The spectral accelerations at the fundamental periods of the four structures under MCE according to GB 50011-2010 are 0.35, 0.44, 0.58, and 0.40 g. The aforementioned average annual collapse probability-based acceptable CMR is taken here as a collapse performance criterion. The OpenSees platform [37] is used to implement IDA for computing parameters, such as structural collapse probability. The seismic performance of these structures is analyzed to compare the considered seismic strengthening method.