## 4 Evaluation of Nonlinear Performance Curve Subjected to In-Plane Rocking Behavior
### Evaluation of Nonlinear Performance Curve after Mortar-Joint Cracking  
Figure 14 shows the relationship between the rocking strength \(P_{r2}\) (shown in Figure 12a,b), axial force, and stress distribution of solid masonry wall subjected to rocking behavior. After mortar-joint tensile failure, the stress exists only at the compression end with length \(a\), as shown in Figure 14. In this study, the rocking strength \(P_{r2}\) was calculated assuming a uniformly distributed average stress \(0.8\bar{g}_{m}{{}^{\prime}}\) (i.e., \(80\%\) of the compressive strength of the masonry prism). Since the flexural moment and axial force acting on the bottom of a wall are \(P_{r2}\,H\) and (\(N_{D}+W\)), respectively, Equation (7) must be satisfied at the rocking strength. From Equation (7), the rocking strength \(P_{r2}\) is determined by Equation (8). \[P_{r2}\cdot H=(N_{D}+W)(0.5l+0.5a) \tag{7}\]  
\[P_{r2}=(N_{D}+W)\bigg{(}\frac{l}{2H}\bigg{)}\bigg{(}1-\frac{N_{D}+W}{0.8\bar{f} _{m}^{\prime}\cdot l\cdot t}\bigg{)} \tag{8}\]  
where  
\(P_{r2}\): Rocking strength (kN);  
\(a\): Length of uniformly distributed stress block;  
\(f_{m}{{}^{\prime}}\): Compressive strength of masonry prism (=15.4 N/mm\({}^{2}\), the lowest value in cement and red bricks shown in Table 1). On the other hand, in the perforated masonry walls after mortar-joint cracking, the left and right walls of an opening individually rotate, as shown in Figure 15. Therefore, in this study, the rocking strength of perforated masonry walls after mortar-joint cracking is obtained as the sum of the strengths of each wall divided by openings, as shown in Figure 15a. Each rocking strength \(P_{r2}{{}^{\prime}}\) can be calculated from the moment equilibrium condition, as shown in Equation (9). Since the self-weight of an individual wall is very small compared to an applied axial force, it can be set as \(N_{D}{{}^{\prime}}\approx N_{D}{{}^{\prime}}+W\). Therefore, the rocking strength of an individual wall in Equation (9) can be simplified as in Equation (10). Furthermore, the effective height \(h_{e}\) shown in Figure 15b was used for the height of individual walls according to ASCE 41-17 [25]. The prime symbols in Figure 15 and Equations (9) and (10) mean each parameter of an individual wall. \[P_{r2}^{\prime}=N_{D}^{\prime}\bigg{(}\frac{l^{\prime}}{2l^{\prime}}\bigg{)} \bigg{(}1-\frac{N_{D}^{\prime}}{0.8\bar{f}_{m}^{\prime}\cdot l^{\prime}\cdot t }\bigg{)}+\big{(}N_{D}^{\prime}+W^{\prime}\big{)}\bigg{(}\frac{l^{\prime}}{2l ^{\prime}}\bigg{)}\bigg{(}1-\frac{N_{D}^{\prime}+W^{\prime}}{0.8\bar{f}_{m}^{ \prime}\cdot l^{\prime}\cdot t}\bigg{)} \tag{9}\]  
Figure 14: The relationship between rocking strength, axial force, and stress distribution of solid a masonry wall after mortar-joint cracking. \[P^{\prime}_{r2}=\left(N^{\prime}_{D}+W^{\prime}\right)\left(\frac{l^{\prime}}{h_{e} }\right)\left(1-\frac{N^{\prime}_{D}+W^{\prime}}{0.8f^{\prime}_{m}.l^{\prime} \cdot t}\right) \tag{10}\]  
where  
\(h_{e}\): Effective wall height (mm). The lateral behavior of an URM wall subjected to a rocking mechanism can be modeled with a nonlinear flexural and shear hinge, as shown in Figure 16a. In addition, each wall performance curve was replaced by a trilinear function with yielding and toe crushing points, as shown in Figure 16b [19; 25]. According to this performance curve, the ultimate deformation \(\Delta_{tc,r}\) of the rocking behavior of URM walls is determined by toe crushing. The elastic stiffness \(k\) of an URM wall subjected to the rocking behavior in Figure 16a can be calculated by Equation (11), assuming that the flexural and shear springs are connected in series. The ultimate deformation \(\Delta_{tc,r}\) at the toe crushing failure can be calculated by integrating the curvature \(\phi_{tc,r}\) along wall height [19; 25].