## 5 Dynamic Response of Braced Models
### Number of Cycles of NLDs from Design Specificatons  
Figure 14 shows the estimation of the number of cycles of the first story HD and UFP devices of the SBF and TBF models under the design earthquakes (100% of PGA) as a function of different ductility demands. In order to consider the complete set of ground motions the number of cycles estimation was evaluated considering the available dynamic experimental results (grey bars) and nonlinear numerical analysis (blue bars) for both HD (Figure 14a,c) and UFP (Figure 14b,d) devices, respectively. As can be observed the number of cycles at each ductility level of HD and UFP varies between the earthquakes, depending on the characteristics of each ground motion. Figure 14e,f show the average value of the mean number of dynamic cycles from experimental and from numerical results (mean dynamic num-exp). As can be observed, a good agreement was found between the mean experimental and numerical number of cycles estimated for both HD and UFP devices, with slightly and acceptable differences. In both cases the number of cycles considerably decrease at increasing level of required ductility.  
The mean values of number of cycles estimated from the design earthquakes compared with the main code requirements for TT and FPT testing of NLD devices standards [35; 36; 37; 38] is shown in Figure 15 as function of the ductility demand. The mean number of cycles confirms the trend proposed by American standards and guidelines (TT (ASCE 7-10) and QT (FEMA 461)) for quasi-static characterization tests, in accordance with characterization tests performed on UFPs (see Figure 6b).  
Figure 13: Time-history of displacement and force-displacement cycles for seismic input 187 at 100% of PGA level and photos of the rupture of (**a**) HD at the second story and (**b**) UFP at the first story.  
On the contrary the loading protocol required by European code EN 15129, also assumed by Italian code NTC 2018 and considered for the case of HDs characterization tests (see Figure 6a), seems to penalize devices excessively being not in good agreement with both experimental dynamic results. The minimum testing requirement for TT by European code [36] (TT (EN 15129)) overestimated the number of cycles at the design displacement (\(\mu\) = 5 for HD and \(\mu\) = 4 for UFP).  
The minimum required for FPCT by Italian code (FCT (NTC 2018)) at the maximum displacement ductility (\(\mu\) \(\geq\) 5 and \(\mu\) \(\geq\) 4 for SBF and TBF, respectively) is much higher than the corresponding average values observed for dynamic tests. Moreover, FPCT by Italian code is required for a minimum of four NLD dampers of the same type and size. A high number of device testing should be unnecessary due to the typically stable hysteretic behavior of yielding steel dampers. Moreover, stronger performance  
Figure 14: Number of cycles vs. ductility demand during the design earthquakes (PGA 100%) obtained by experimental dynamic tests of (**a**) HD and (**b**) UFPs and nonlinear numerical analyses of the (**c**) HD of the SBF model and (**d**) UFPs of the TBF model and corresponding mean values for (**e**) HD and (**f**) UFPs.  
requirements than the actual seismic working condition of the devices could mean that the devices become unusable after testing besides higher costs required for testing procedures.