# Out-of-Plane Experimental Study of Strengthening Slender Non-Structural Masonry Walls
## 3 Experimental Results Analysis with Discussion
### Ductility and Effective Stiffness  
To compare the effective stiffness and ductility of non-strengthened and strengthened specimens, an idealization of hysteresis envelopes was performed with an equivalent bilinear diagram, which is defined by the energy equality principle [65; 66; 67]. The effective stiffness \(K_{sf}\), which governs the slope of the first part of the idealized equivalent bilinear diagram, is obtained as a ratio between 2/3 of the maximum experimentally achieved force \(F_{\rm max}\) and the corresponding horizontal displacement \(u_{e}\). Equation (3) was used to calculate the effective stiffness \(K_{sf}\) for each loading direction. Similarly, the ductility factor \(\mu\) was determined by Equation (4), which is defined as the ratio between the maximum displacement \(u_{\rm max}\) and the displacement at the elastic limit of the idealized diagram \(u_{\rm eff}\). All idealizations were made up to the specimens' near collapse LS. An example of characteristic  
Figure 12: Comparison of stiffness degradation curves. Figure 13: Comparison of normalized coefficients \(C_{norm}\) describing the global stiffness degradation. points and idealization are schematically presented in the case of the NF-1 specimen's hysteresis envelope in the positive direction (see Figure 14a). \[\kappa_{ef} =\frac{\frac{2}{3}F_{max}}{u_{e}} \tag{3}\] \[\mu =\frac{u_{max}}{u_{ef}} \tag{4}\]  
Figure 14: Bilinear idealization of specimens’ hysteresis envelopes: (**a**) NF-1, (**b**) NF-2, (**c**) NF-3, (**d**) NF-1A, (**e**) NF-2A, (**f**) NF-3A, (**g**) NF-1B, (**h**) NF-2B, (**i**) NF-3B, (**j**) NF-C, and (**k**) NF-D.  
Idealization curves, together with the corresponding hysteresis envelopes, are presented for all tested specimens in Figure 14. Even though the idealization is adopted from the in-plane shear tests of masonry walls, the idealized diagrams - well when compared to the experimental hysteresis envelopes from the presented out-of-plane tests. The deviation can only be noticed at specimens NF-1C and NF-1D (see Figure 14j,k). For the assessment of tested wall strengthening systems' ef' ciency and the comparison of the results, the values of displacements and idealized resistance at the characteristic points were averaged for each specimen group, including positive and negative loading directions. Based on the average values at the characteristic points, the parameters of the effective stiffness \(\mathsf{K_{eff}}\) and ductility - which represent the average idealized curve of an individual group of specimens, were also evaluated and are presented in Table 3.  
The comparison of the effective stiffness \(\mathsf{K_{eff}}\) and ductility - normalized with the average values for non-strengthened NF-good specimens' response, is presented in Figure 15. Specimens' NF-good responses show stiffer behavior against NF-poor, whereas the ductility is lower for NF-good specimens' response. Applying the strengthening system A or B increases the effective stiffness by about 50% compared to the non-strengthened (NF-good) specimens. Systems A and B increase the ductility factor - by 25% and 20%, respectively. By applying strengthening systems C and D, the greatest improvements regarding the effective stiffness and ductility can be observed. Using the strengthening system C, the effective stiffness is increased by up to 135% and the ductility by as much as 442%. With system D, these two values increase by 160% and 306%, respectively. The ductility factor is superseded with displacement-based methods in the majority of modern seismic design codes that consider performance-based design [68, 69]. The LS  
Figure 15: Comparison of: (a) average normalized effective stiffness and (b) ductility for different strengthening systems. analysis made it clear that the A and B systems have superior resistance and displacement capacity improvement at maximum resistance LS in comparison to the C and D systems.