## 2 Methodology  
Numerical analysis of post-mainshock retrofitting of mainshock-damaged structure can be broadly classified into two commonly used methods, namely the strength reduction (SR) method [23; 48] and stepwise (SW) method [20; 22]. As the name implies, the SR method is to artificially reduce the strength of structural materials or the performance level of structure in order to promote the structure to reach a target damage state. Although SR method is simple and direct, damaged structure that is obtained by artificially setting the strength reduction is more or less different from the real damage state of the structure after earthquakes excitation. However, the SW method is to make the intact structure reach a specific damage state by performing a single earthquake excitation. Obviously, the SW method is more consistent with the actual earthquake process, and it is more reasonable to perform subsequent aftershock response analysis by adding retrofit members. Therefore, this study will adopt SW method in the post-earthquake retrofitting research of earthquake-damaged structure under aftershocks. In the SW method, the 'element birth and death' technology of numerical software is adopted in order to realize the modelling of retrofitting devices for MD-RC frames. Furthermore, because the interstory drift ratio (IDR) is an appropriate indicator of structural damage levels for RC or steel frame, the maximum IDR is selected as the evaluating indicator of structural damage state in the present study. Table 1 summarizes the structural damage states corresponding to the IDR limit values, according to China seismic code [49].  
Based on previously mentioned method and technology, a post-earthquake retrofitting evaluation framework of MD-RC frame is proposed (Figure 1) and the detailed steps are summarized, as follows: (a) build the intact structural model and perform gravity analysis (i.e., applying gravity acceleration to structural model) to simulate the real force balance state of structures in the first step; (b) determine the target damage states of MD-RC frame that are to be analyzed and perform nonlinear dynamic time history analysis of intact structure under the excitations of mainshock with different intensity to make the intact structure enter target damage states (step 2); (c) design retrofitting schemes for MD-RC frame and retrofit the MD-RC frame by designed retrofitting schemes (step 3); (d) perform a nonlinear dynamic time history analysis of the MD-RC frame with and without retrofitting schemes under aftershock excitations in step 4; and, (e) evaluate the seismic performance  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Damage States** & **Description** & **IDR 1 (\%)** \\ \hline Neglected & No damage or localized minor cracking & \textless{}0.4 \\ Minor damage & Slight cracking throughout & 0.4–1.0 \\ Moderate damage & Severe cracking, localized spalling & 1.0–2.0 \\ Severe damage & Crushing of concrete, reinforcement exposed & 2.0–4.0 \\ Collapse & Collapsed & \textgreater{}4.0 \\ \hline \hline \end{tabular} \({}^{\dagger}\) IDR: Maximum interstory drift ratio.  
\end{table}
Table 1: Structural damage states corresponding to the interstory drift ratio (IDR) limit values.  
of the MD-RC frame under given retrofitting schemes and analyze the retrofit effect of the retrofitting schemes on the MD-RC frames under aftershock excitations (step 5).