# Out-of-Plane Experimental Study of Strengthening Slender Non-Structural Masonry Walls
## 3 Experimental Results Analysis with Discussion
### Eigenfrequencies and Damping
Another way to determine equivalent viscous damping is based on the hysteretic responses of specimens and was calculated using Equation (5). When processing the experimental results of the hysteresis, energy dissipation in both directions of loading is taken into account in the analysis of each cycle [66,71]. \[\xi_{H}=\frac{E_{D}}{2\pi\left(E_{E}^{+}+E_{E}^{-}\right)}=\frac{E_{D}}{2\pi \left(\frac{1}{2}K_{i}^{+}\,u_{max,i}^{+}+\frac{1}{2}K_{i}^{-}\,u_{min,i}^{-} \right)} \tag{5}\]  
The parameter \(E_{D}\) stands for the dissipated energy in a hysteresis loop, \(E_{E}\) stands for the elastic deformation energy (sum of the elastic deformation energy of the positive and negative direction), \(K_{i}\)' and \(K_{i}\)' represent the secant stiffness of the loop in the positive and negative direction and \(u_{max,i}^{+}\) and \(u_{min,i}^{-}\) stand for the maximum and minimum displacements of the considered hysteresis loop [66]. The development of the equivalent viscous damping coefficient \(\xi_{H}\), with respect to the mid-height amplitude displacement, is presented in Figure 16. Additionally, the values of the critical damping ratio \(\xi\) obtained with the forced vibration method are shown with the black dashed horizontal lines along the entire range of amplitudes. The linearization between the initial value of the critical damping ratio (at an amplitude of 0 mm) and the final damping value (at the maximum displacement amplitude) is presented with a red dotted line. The values of equivalent viscous damping coefficients \(\xi_{H}\) are within the range of 3.2% to 4.4% at the first crack initiation LS for all test specimens. It can be concluded that the presented strengthening systems do not have a substantial impact on the viscous damping coefficient up to the first crack initiation LS. The damping ratios \(\xi\) determined from the forced vibrations before cyclic testing have slightly lower values (2.8-3.1%) due to the intact state of the specimens. This comparison confirms that the forced vibration method could be used for the conservative in situ assessment of initial dynamic parameters. After reaching the first crack initiation LS, a similar initial peak value of viscous damping \(\xi_{H}\) is also observed in all specimens, as a consequence of the extensive energy dissipation caused by the development of plastic hinges and masonry flexural cracking. Following the complete development of plastic hinges, the value of equivalent viscous damping \(\xi_{H}\) stabilizes at a lower level. After reaching the maximum resistance LS for non-strengthened specimens, the value \(\xi_{H}\) begins to increase up to 7% (see Figure 16a). For the groups of specimens NF-A and NF-B, the beginning of increasing \(\xi_{H}\) appears slightly before the maximum resistance LS and further increases until the near collapse LS (see Figure 16b,c). The values obtained for NF-A and NF-B at near collapse LS are 6.9% and 8.6%, respectively. From the curves' shape of the NF-1C and NF-1D specimens, a more extended range of increasing damping can be observed after exceeding the displacement amplitudes at maximum resistance LS (see Figure 16d,e). The viscous damping coefficients for specimens NF-1C and NF-1D at near collapse LS achieve 13.3% and 8.7%, respectively.