## 6 Test Results
### Hysteretic Damping and Force Reduction Factor  
The hysteretic damping (\(\xi^{\mathrm{i}}\)) should be evaluated for each half-loop of the force-displacement curves according the next steps [68]. First, each half-cycle is identified, delimited by a pair of zero-force points (Figure 13). For each force-displacement half-cycle, the maximum generalized force (\(\mathrm{F}^{\mathrm{i}}_{\mathrm{max}}\)) and the maximum generalized displacement (\(\mathrm{D}^{\mathrm{i}}_{\mathrm{max}}\)) are evaluated, which allows for the calculating of the elastic strain energy (\(\mathrm{E}_{\mathrm{S0}}\)). For each half-cycle, the dissipated energy (\(\mathrm{E}^{\mathrm{i}}\)d) is computed by performing the integral of the force-displacement curve leading to the \(\mathrm{A}_{\mathrm{half-loop}}\) value (Figure 13). Finally, the hysteretic damping (\(\xi^{\mathrm{i}}\)) can be calculated by Equation (16) for each half cycle. \[\xi^{\mathrm{i}}=\frac{\mathrm{E}^{\mathrm{i}}_{\mathrm{d}}}{2\cdot\pi\cdot \mathrm{D}^{\mathrm{i}}_{\mathrm{max}}.\mathrm{F}^{\mathrm{i}}_{\mathrm{max}}} \tag{16}\]  
where \(\mathrm{F}^{\mathrm{i}}_{\mathrm{max}}\) and \(\mathrm{D}^{\mathrm{i}}_{\mathrm{max}}\) are the maximum displacements and force for each half cycle, respectively (Figure 13). The values of the hysteretic damping (\(\xi^{\mathrm{i}}\)) are given in Figure 15a,b for the ASB column (P16B) and the repaired columns (R16-UHS and R16-UH) for the negative (Figure 15a) and the positive (Figure 15b) half loops. The parameter \(\mathrm{v}^{\mathrm{i}}_{\xi}\) defined by Equation (17) estimates the variation in the damping after column repair. Figure 15: Hysteretic damping (\(\xi^{\mathrm{i}}\)) for each half loop during Tol deformation history of the ASB column (P16B) and the RR columns (R16-UHS, R16-UHP) for the negative (**a**) and the positive (**b**) half loops. Damping increase (\(\mathrm{v}^{\mathrm{i}}_{\xi}\)) for: (**c**) the column R16-UHS and (**d**) the column R16-UHP. \[\upsilon_{\xi}^{\rm i}=\frac{\xi_{\rm RR}^{\rm i}-\xi_{\rm ASB}^{\rm i}}{\xi_{\rm ASB }^{\rm i}} \tag{17}\]  
The parameter \(\upsilon^{\rm i}\xi\) was calculated for each negative and positive half loop in: (1) Figure 15c for the RR column repaired by concrete with steel fibers (UHS); and (2) Figure 15d for the RR column repaired by concrete with polymer fibers (UHP). The response of the columns subjected to low cycle fatigue excitation was evaluated considering the first five cycles (Figure 7b) during the application of the Tol part of the deformation history. The mean value of \(\upsilon^{i}\xi\) calculated during the first five cycles was 46.6% for R16-UHS and 23.8% for R16-UHP for the negative half loops, and 54.6% for R16-UHS and 34.3% for R16-UHP in the positive half loops. This means that the polymer and steel fibers may increase the hysteretic damping, but the steel fibers were more efficient. The mean value of \(\upsilon^{i}\xi\) calculated during the Tol part of the displacement history was, for the negative half loops, 69.3% for R16-UHS and 49.0% for R16-UHP; and for the positive half loops, 65.5% for R16-UHS and 61.6% for R16-UHP. The column with steel or polymer fiber showed a modest improvement in hysteretic damping after low cycle fatigue excitation in each load direction. A similarity to those discussed for the low cycles fatigue response can be drawn, considering the whole Tol displacement history. Finally, the reduction factor (\(\eta\)) of the elastic demand (elastic response spectrum), which is used in many applications of structural engineering to estimate the force reduction for effect of damping [58; 69], can be evaluated by Equation (18). \[=\sqrt{\frac{10}{5+\xi_{\rm eq}}} \tag{18}\]  
The \(\eta\) value was calculated using the equivalent damping (\(\xi_{\rm eq}\)) [68] during the first five half cycles (low cycles fatigue column response). The equivalent damping is equal to the sum of the elastic damping, usually assumed to be 5%, and the hysteretic damping (\(\xi\)).