## 2 Design Procedure  
The design procedure is based on the assumption that, for relatively stiff frame structures, a substantial improvement of seismic performance can be reached by incorporating a supplemental damping system with limited stiffening capacity. For more deformable structures, a supplemental stiffness contribution helps control the lateral displacements better, which prevents an overdissipation demand to the damping technology adopted.  
In the following, attention is focused on the first class of frame structures, which can be identified with low- and low-to-medium-rise multistorey building structures typical of European countries, i.e., with a height below 20 m. In order to obtain for this class of structures an approximate evaluation of the maximum fundamental translational vibration period \(T_{1}\), the simple relation provided by Eurocode 8 [46] and adopted by several national Standards, among which the Italian Technical Standards [47], can be used: \(T_{1}=C_{t}\cdot H^{\frac{3}{4}}\), where the \(C_{t}\) coefficient is set as equal to 0.075 for steel structures and 0.085 for RC ones, and \(H\) is the height of the structure measured from the foundation, expressed in meters. For \(H\) = 20 m, \(T_{1}\) = 0.71 s (steel) and \(T_{1}\) = 0.8 s (RC) is obtained. By jointly considering the two values, \(T_{1}\) = 0.8 s is assumed in the following as the approximate upper limit of the fundamental period for relatively stiff frame building structures.  
As observed above, for these structures the retrofit design objectives can be met essentially by an added damping, with a marginal role of supplemental stiffness. This prompts to select bracing systems that incorporate pressurized fluid viscous (FV) spring-dampers as a protective strategy. Indeed, the spring component of the devices, acting in series with the steel braces, remarkably constrains the total horizontal translational stiffness of these systems. This results in a moderate contribution (normally below 10%) to the lateral stiffness of the retrofitted structures.  
The \(\alpha\) reduction factors are calibrated on the mechanical characteristics of this type of devices. For dampers adding a non-negligible supplemental stiffness, the procedure follows the same steps, but different criteria for the evaluation of \(\alpha\) are required, which will be detailed in a further step of this research.  
In view of the practical application of the design method, a distinction is made between the cases where the most significant structural deficiencies are represented by poor shear or bending moment strength of columns and/or beams--where \(\alpha\) is intended as \"stress-related\" reduction factor, named \(\alpha_{s}\) in the following--and the cases having excessive horizontal displacements, where \(\alpha\) is intended as \"deformation-related\" reduction factor, \(\alpha_{d}\). Although the two types of lacks normally coexist, the former is frequently prevailing in structures with undersized member sections, albeit stiffened by structural elements (e.g., RC shear walls) or non-structural components (e.g., masonry infills interacting with the frame members), whereas the latter is likely to prevail when these elements are not present--and thus the structural system highlights high deformability to lateral loads--but the frame members have only moderate deficiencies in terms of sizes and structural details.