## 2 Retrofitting Method Using Viscous Dampers  
### Structural Model Used for Verification  
In order to verify the retrofitting method, a 6-story building is used with regular configurations in plan and elevation. This frame structure was designed according to the CSDB code [3] with the design earthquake environment of Region 6 (Peak Ground Acceleration, i.e., PGA (Peak Ground Acceleration) = 0.05 g, corresponding to a 10% exceedance probability in 50 years, with a type II site condition), and now it should meet the requirements of the design earthquake environment of Region 8 (PGA = 0.2 g corresponding to a 10% exceedance probability in 50 years, with a type II site condition) due to design code revision. The geometry layout of the frame structure is shown in Figure 1. More information on the reinforced concrete (RC) frame, such as cross-section dimensions of columns and beams and usage of steel rebars, is provided in Tables 1 and 2. The frame is a flexible structure with fundamental and second periods \(T_{1}\) = 1.88 s and \(T_{2}\) = 0.62 s, respectively. The seismic response analysis is conducted using the OpenSees program [37]. The nonlinearBeamColumn elements with fiber section division are used to model beams and columns, while Concrete02 (Kent-Scott-Park) and Steel02 (Menegotto and Pinto) material models are used for the concrete and rebars [38]. The compression and tension strengths and strain at the peak stress of concrete are 20.1 Mpa, 2.01 Mpa and 0.002, respectively. The yielding strengths and elastic modulus of the longitudinal rebars are 335 Mpa and \(2.0\times 10^{5}\) Mpa. P-\(\Delta\) coordinate transformation is used to consider the geometric nonlinearity. The shear failure is not considered in the element model. This assumption will not introduce large errors, because the columns and beams are required to be designed such that flexure damage occurs before shear damage. The damping ratio is set as 0.05 in the analyses. The force provided by a viscous damper is calculated, in general, as  
\[F=Cv^{\alpha} \tag{1}\]  
where \(C\) is the damping coefficient, \(\alpha\) is the velocity index, and \(v\) is the relative velocity. In the following numerical example, \(\alpha\) is set as 1.0 corresponding to a linear viscous damper. The TwoNodeLink element with Maxwell material in OpenSees is used to model the viscous dampers [37]. The Maxwell material model comprises a series-wound stiffness spring and damping pot. The force from the damping pot is calculated by using Equation (1), and the stiffness is commonly selected within 2-3 times the value of coefficient (2.5 times is used in this study). Based on the survey of some companies' products, the commonly used damping coefficients of viscous dampers are 20, 50, 100, 150, 200, 250, 300, 350, 400, 500, 600, 750, 800, 1000, 1200, 1300, 1500, 1600, 1800, 2000, 2500, 2600, 2800, 3000, 3300, 3500, 3600, 3800, 4000, 4800, 5000, 5200, 5600, 6000, 6700, 8000, 8700, 9300, 10,000, 12,000, 15,000, 18,000, 20,000, 24,000, 26,000, 28,000, and 30,000 kN(s/m). These parameters can be referred in the retrofitting procedure. Ground motions (GMs) are selected from the PEER (Pacific Earthquake Engineering Research Center) NGA (Next Generation Attenuation) strong motion database ([[http://ngawest2.berkeley.edu](http://ngawest2.berkeley.edu)]([http://ngawest2.berkeley.edu](http://ngawest2.berkeley.edu))) to be compatible with the design spectrum in the CSDB code [39] at the main period points. Since a suite of 3 to 11 GMs is usually recommended in codes [39; 40; 41] to get an accurate assessment of structural response in seismic design procedures, a total of 7 GMs (see Table 3) are used in this study.