## 2 Materials and Methods
### Numerical Study
#### 2.3.4 Description of Added Damping Systems (ADS)  
Added Damping Systems (ADSs) are included in order to obtain enhanced performance, measured as a reduction of interstorey drift ratio. Different damper exponents, damping ratios, and height-wise distributions of dampers are considered. An ADS in this study is described by a label of the type \"Q.2.1,\" with the following meaning:  
* The first letter (\"Q\" in the example) indicates the height-wise distribution (Q, R, S); the \"Q\" distribution has constant damping coefficients at all storeys; in the \"R\" and \"S\" distribution the damping coefficients decrease linearly with height up to a fraction of the first storey damping coefficient, 2/3 for the \"R\" distribution and 1/3 for the \"S\" distribution. Only complete vertical distributions are considered.
* The second number (\"2\" in the example) indicates the relative amount of damping in the first mode; for linear dampers, \"1\" indicates 11%, \"2\" indicates 23% and \"3\" indicates 34%; for non-linear dampers, the values are drift-dependent, but the damping coefficients keep a similar progression.
* The third number (\"1\" in the example) indicates the damper exponent: \"1\" indicates linear (\(a\) = 1) and \"5\" indicates non-linear (\(a\) = 0.5). All dampers in one ADS are either linear or non-linear.  
The \"Q\" distribution is a realistic option, as in some cases only one type of damper is used throughout the structure. The \"S\" distribution represents a more efficient option, as the damper coefficients are arranged loosely following the frame storey stiffness distribution. Percentages of 11%, 23%, 34% added damping ratios in the first mode were established for ADSs Q.1.1, Q.2.1 and Q.3.1 (linear dampers); values of 20% have been recommended for optimal performance [8; 44]. Non-linear damping coefficients were determined to obtain approximately similar damping ratios at \(S_{a1,C}\), resulting in higher damping ratios at design level \(S_{a1,D}\) (damping ratios for non-linear dampers are intensity-dependent). For ADSs type \"R\" and \"S\", the coefficients are arranged so that added damping ratio in the first mode \(\xi_{a1}\) is the same as in the corresponding ADS type \"Q\". Although optimized vertical distributions of damping coefficients have been proposed in the literature [14; 27; 29], this topic is out of the scope of this paper, as is the distribution of dampers among bays of the frame. Thus, the distributions in the study are chosen as simple as possible within realistic limits, and the position of dampers is limited to the central bay of the frame.  
Brace sections are defined as European hot-finished square hollow sections (SHSH) with the following criteria: (i) S355 grade steel (characteristic yield strength \(f_{y}=355\) MPa) conforming to Eurocode 3 [45] is adopted; (ii) a maximum reduced slenderness of 2 is allowed; (iii) only compact sections class 1 with a minimum wall thickness of 8 mm are considered. Buckling is assessed considering the whole centerline length. The properties of ADSs are listed in Table 3, where \(k_{b}\) has been found considering the cross-sectional properties of the brace with the complete centerline length of the damping system. Typical values for \(k_{b}\) suggested in literature are 175 to 525 kN/mm [8; 39; 44]. The values in Table 3 belong in this range and lie above \(k_{s,1}\) (63.3 kN/mm), as suggested by [PERSON] and [PERSON] [37]. In this table, \(\xi_{a,1}\) is listed at \(S_{a1,D}\) for non-linear dampers.