## 6 Conclusions  
For a structure with insufficient earthquake resistance, this study proposed a retrofitting method based on an optimum placement of viscous dampers. The following conclusions can be drawn:  
1. The proposed retrofitting method can be applied both to satisfy the requirements of design codes and to enhance the structural collapse-resistant capacity of existing structures. An advantage of the proposed method is that the retrofitted structure has a larger earthquake collapse-resistant capacity compared with an ordinary retrofitting method entailing the same retrofitting cost. For a regular 3D frame, the computational cost is low because a 2D frame model can be adopted to determine the optimum placement pattern of the viscous dampers. Similar to other design methods for the placement of dampers, the proposed method is also a procedure based on a series of time-history analyses. However, for the 6-story RC frame structure used in this study, one time-history analysis commonly only takes about three minutes (for the 2D frame model). Therefore, the computational cost of the proposed method is acceptable.
2. The economic benefit analysis is conducted to check the convenience of the proposed retrofitting method. The owners can clearly know the economic benefits under different earthquake intensities that may happen in the future. For the 6-story RC frame structure used in this study, during its lifetime cycle, the economic benefit of retrofitting is not favorable if the structure only suffers an earthquake with a frequent earthquake intensity, while the economic benefit of retrofitting is favorable if the structure suffers an earthquake with an occasional earthquake intensity (depending on the building functions which influence the economic loss by downtime/repair time. The example building used in this study is a middle school, meaning that downtime/repair time will not induce much economic loss. However, the economic loss will be higher if it would be a commercial building.), and the economic benefit of retrofitting is significantly favorable if the structure suffers an earthquake with a rare intensity (the ratio between the cost of retrofitting and the cost of complete replacement is about 45%; the ratio between the repair time of retrofitting and the repair time of complete replacement is about 10%); in this case, the retrofitting will save householders' lives in case of earthquake. This economic benefit analysis results can be provided to households who may decide whether to perform retrofitting or not. Besides the economic benefit, environmental sustainability is also discussed by way of a carbon emission calculation. Under a rare earthquake, the original frame needs replacement, which leads to more carbon emissions than a retrofitted frame in the repair process.  
Conceptualization and funding acquisition, Writing--review and editing: S.L. Methodology and formal analysis: J.Z.  
This research project was supported by the National Key R&D Program of China (Grant No. 2016 YFC0701500), National Natural Science Foundation of China (Grant No. 51578202), and Natural Science Foundation of Heilongjiang (Grant No. YQ2019E021). The financial supports are greatly appreciated by the authors.  
The authors declare no conflict of interest.