## 2 Materials and Methods
### Estimate of Maxwell Stiffness  
Viscous dampers are generally mounted using brace extenders and attached to the structure by means of strong gusset plates. Figure 4 shows a typical mounting arrangement for a single diagonal element. The stiffness and damping coefficient related to the damper axis are referred to as \(k_{b}\) and \(c_{b}\). A series of auxiliary elements (gusset plates, cleats, bolts, clevises, end plates, etc.) are needed in addition to the brace extender and damper. Elastic behavior is assumed for all elements in the set. The damper element itself presents a certain stiffness \(k_{bl}\), which in typical application is assumed as infinite. The stiffness of other elements in the set is generally uncertain at the design stage, as many of these elements are typically provided by the damper manufacturer. For other elements, the stiffness can only be grossly approximated; for instance, the flexibility of joint elements can be estimated using the component-based approach in Eurocode 3, part 1-8 [38]. For this reason, the overall stiffness is often calculated using the brace extender cross section stiffness over the total centerline length. It is assumed that the combined stiffness of the set does not deviate significantly from this value. The model is suitable for seismic analysis not involving limit states of the device, for which a more sophisticated mathematical model has been developed [39]. The actual total damping system stiffness \(k_{b}\) can be calculated as:  
\[\frac{1}{k_{b}}=\frac{1}{k_{bo}}+\frac{1}{k_{bb}}+\frac{1}{k_{bd}}. \tag{12}\]  
Figure 5 shows the resulting value of \(k_{b}\) considering only two components (\(k_{bb}\) and \(k_{bo}\)) showing that, according to Equation (12), very flexible elements in the set tend to dominate the overall stiffness, despite large stiffness values of other components. Accurate measures of real-scale damping system stiffness are not abundant in the scientific literature. In this study, experimental values reported by [PERSON] et al. [34] (listed in Table 1) are used. These data correspond to real-scale state-of-the-art tests performed in the Tokyo Institute of Technology on damping units composed of damper, brace extender and the corresponding connection elements (clevises and brackets) in both edges. The set did not include gusset plates or frontal plate connections; addition of these elements would result in further reduction of stiffness. In the table, \(k_{bb}\) has been calculated as \(A_{b}\)-\(E/L_{b}\), where \(E\) is the young modulus of steel (taken as 200 Gpa), \(k_{bo}\) has been found using Equation (10) and \(k_{bb}\)* has been estimated as \(A_{b}\)-\(E/L\), thus corresponding to the value typically adopted in practical analysis. The last column shows that the actual stiffness  
Figure 4: Stiffness of damper and auxiliary mounting elements. (\(k_{b}\)) varies between 13% and 28% of the stiffness \(k_{bb}\)* found with this usual assumption, a remarkable deviation. The very moderate values of the damper stiffness (\(k_{bd}\)) and auxiliary elements (\(k_{bo}\)) are noteworthy, as these two values dominate the total stiffness of the set \(k_{b}\). After testing, the devices were installed in a five-storey real-scale structure subjected to further tests in the E-Defense shaking table. The direction, storey of installation of device, and fundamental period \(T\) of the test structure are listed. These data have been used to find equivalent values for a longer damper unit under the assumption of similar damper forces, which leads to equal damper and mounting elements. Thus, \(k_{bd}\) and \(k_{bo}\) are unchanged, as is the difference between \(L\) and \(L_{b}\) representing the total length of damper and auxiliary elements. Length \(L\) is replaced by \(L^{\prime}\) = 8006 mm, thereby finding \(L_{b}\)\({}^{\prime}\) as \(L^{\prime}-(L-L_{b})\). \(k_{bb}\), \(k_{b}\) and \(k_{bb}\)* are recalculated accordingly.