## 2 Proposed Seismic Damper  
The proposed SS-TTD damper is constructed through the assemblage of two standardized stainless-steel tubes with a telescopic configuration (Figure 2a). All faces (i.e., the four walls) of the outer tube are regularly slit transversally to its longitudinal axis, forming strips. The strips are grooved so that one of its sides can be connected to a common plate, thus linking all in parallel. The common plate that gathers one end of each steel strip is fixed to the inner tube with plug welding at discrete points. The other end of each steel strip is connected to the corner of the outer tube. When the brace is subjected to axial forces, the relative displacements between the tubes impose a double curvature flexural deformation on the strips, which behave as a series of fixed-ended beams. These strips constitute the energy dissipating part of the SS-TTD. One end of each tube is connected to auxiliary brace members of the appropriate length that connect the SS-TTD damper to the two points of the building structure that are expected to undergo large relative horizontal displacements (Figure 2b). Typically, in frame structures, these points are two beam-column joints of consecutive spans and floor levels, as shown in Figure 2b.  
Knowing the geometry and mechanical properties of the stainless steel that form the strips, the axial yielding force \(Q_{y}\) and apparent maximum force \(Q_{B}\) of the damper are readily obtained by applying fundamental engineering principles as follows:  
\[Q_{y}\ =\ n\frac{\sigma_{y}tb^{2}}{2h^{\prime}};\ Q_{B}\ =\ n\frac{\sigma_{B}tb^{2}}{2h^{ \prime}}, \tag{1}\]  
where \(b\) and \(t\) are the strip width and thickness, respectively, and \(r\) is the radius at the extremes of the strips (see Figure 3a); \(n\) is the total number of strips in the damper; \(\sigma_{y}\) and \(\sigma_{B}\) are the steel tensile stress at yielding and maximum states, and \(h^{\prime}=h+[2r^{2}/(h+2r)]\) is the height of an equivalent strut where the actual round-shape ends are replaced by straight lines (see Figure 3b). The expression for \(h^{\prime}\) was obtained by comparing the results of numerical analyses conducted with two different finite element models (Model 1 and Model 2) that represented one steel strip. In Model 1 the strip had rounded ends, as shown in Figure 3a. In Model 2 the actual rounded ends were replaced by straight lines defined, as shown in Figure 3b. A parametric study was conducted with different combinations of \(r\), \(h\) and  
Figure 2: Proposed damper (**a**) and installation in a typical building frame (**b**).  
\(b\). For each combination, the value of the equivalent height \(h^{\prime}\) that makes the strength and stiffness of the strip approximately equal (error less than 10%) in Model 1 and in Model 2 was determined. The expression for \(h^{\prime}\) was readily obtained by numerical regression of these results.  
The yielding displacement of the struts, \(\delta_{y}\), can be estimated using the equivalent strip approximation of Figure 3b, by integration of the curvatures along the total height \(h+2r\) giving [11]:  
\[\delta_{y}\ =\ \frac{Q_{y}{h^{\prime}}^{3}}{nEtb^{3}}\left(1+3\ln\frac{h+2r}{ h^{\prime}}\right). \tag{2}\]  
Here \(E\) is the Young's modulus of the material. The total axial displacement at yielding of the overall brace-type damper is readily obtained by adding to Equation (2) the elastic axial deformation of the inner and outer tubes, and the elastic deformation of the auxiliary brace members.