## 2 Materials and Methods
### Estimate of Maxwell Stiffness
For this case, the ratio \(k_{b}\)/\(k_{bb}\)* takes values between 23% and 40%, that is, a smaller influence than in the previous case but still very remarkable. It can be argued that, because the brace element is longer, the cross-section area should be increased due to buckling and robustness considerations. If that is the case, the ratio \(k_{b}\)/\(k_{bb}\)* decreases further, thus strengthening the conclusion that \(k_{b}\) is actually a fraction of \(k_{bb}\)* somewhere between, say, 10% and 50%. These limits are, of course, loosely set, based on existing data and should be taken cautiously. A relevant question is whether this conclusion can be applied to linear dampers. Unfortunately, to these authors' knowledge, no real-scale accurate data has been published for systems with this type of devices, probably because current seismic practice favors smaller exponents. In order to obtain a range of magnitude, the values in Table 1 have been used with certain assumptions and simplifications. First, non-linear dampers are replaced by linear dampers resulting in similar damper forces; this allows for a similar choice connection and brace extender design. The design forces are not listed in the reference, so \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline _Direction_ & \(T\) & _Story_ & \(L\) & \(L_{b}\) & \(A_{b}\)\({}^{2}\) & c\({}_{b}\) & \(a\) & \(k_{bd}\) & \(k_{bb}\) & \(k_{bo}\) & \(k_{bb}\) & \(k_{bb}\)* & \(k_{bb}\)* & \(k_{bb}\)/\(k_{bb}\)* \\ \hline  & **s** & & **mm** & **mm** & **mm\({}^{2}\)** & **kN-(s/mm)\({}^{0.38}\)** & **kN/mm** & **kN/mm** & **kN/mm** & **kN/mm** & **kN/mm** & **kN/mm** & **-** \\ \hline \multirow{3}{*}{X} & \multirow{3}{*}{0.536} & 4-3 & 4025 & 2429 & 9121 & 49 & 0.38 & 119 & 751 & _144_ & 60 & _453_ & _13\%_ \\  & & 2 & 3947 & 2104 & 8380 & 98 & 0.38 & 193 & 797 & 315 & 104 & _425_ & _24\%_ \\  & & 1 & 4706 & 2864 & 8380 & 98 & 0.38 & 193 & _585_ & 332 & 101 & _356_ & _28\%_ \\ \hline \multirow{3}{*}{Y} & \multirow{3}{*}{0.575} & 4-3 & 3947 & 2104 & 8380 & 98 & 0.38 & 193 & 797 & 315 & 104 & _425_ & _24\%_ \\  & & 2 & 3849 & 1542 & 15323 & 196 & 0.38 & 438 & _1987_ & 357 & 179 & _796_ & _22\%_ \\ \cline{1-1}  & & 1 & 4629 & 2322 & 15,323 & 196 & 0.38 & 438 & _1320_ & _356_ & 171 & _662_ & _26\%_ \\ \hline \hline \end{tabular}  
* Inferred data is shown in italics. \({}^{2}\) Cross-section area of brace extender. For other symbols refer to main text and Figure 4.  
\end{table} 
Table 1: Properties of non-linear viscous damping systems (after [PERSON] et al. [34]) \({}^{1}\). Figure 5: Overall stiffness \(k_{b}\) as a function of component stiffnesses \(k_{bb}\) and \(k_{bo}\). they are estimated indirectly using peak drift velocity values \(v_{0}\) derived from peak drift ratios \(\theta\) assuming harmonic vibration at the natural period \(T\):  
\[v_{0}\approx\frac{2\pi\theta h}{T}, \tag{13}\]  
where \(h\) is the corresponding interstorey height. Then, from Equation (1)  
\[c_{b,1}v_{0}=c_{b}v_{0}^{a}. \tag{14}\]  
The corresponding approximate values are listed in Table 2. A second important consideration is the value of \(k_{bd}\). This value is obtained using an assumption on the relaxation time \(\lambda\) of the linear damper. [PERSON] and [PERSON] (2018) reported values of \(\lambda=0.006\) s for dampers with \(c_{b}=15.45\) N\(\cdot\)s/mm; [PERSON] et al. (2018) reported \(\lambda=0.014\) s for dampers with \(c_{b}=201.4\) N\(\cdot\)s/mm; [PERSON] and [PERSON] (2018) reported \(\lambda=0.008\) s for dampers with \(c_{b}=17.7\) N\(\cdot\)s/mm; these values were found under forced harmonic excitation, for reduced-scaled linear dampers with small capacity, and suggest that larger values of \(c_{b}\) might be associated with larger values of \(\lambda\). However, in the absence of data to confirm this observation, a conservative value of 0.006 s (minimum of the values reported) is adopted here. The values of \(k_{bb}\), \(k_{b0}\) and \(k_{bb}\)* presented in Table 1 are unchanged, whereas \(k_{b}\) is computed using Equation (12).