# Effective Assessment of Inelastic Torsional Deformation of Plan-Asymmetric Shear Wall Systems
## 3 Inelastic Torsional Response of Plan-Asymmetric Structures
### Effect of Transverse Wall Stiffness on Yielding Procedure  
In the derivation introduced in Section 3.2, we assumed for simplicity that the two lateral force resisting walls have the same yield displacement. As a result, the wall on the flexible side always yields first, and then the wall on the stiff side. However, wall stiffness and yield displacement are determined mainly by the length of the wall. According to [24; 25], the stiffness (\(k\)) and yield displacement (\(u_{y}\)) of the lateral force resisting wall can be determined by the following two relationships:  
\[k\propto{l_{w}}^{3}, \tag{42}\]  
\[u_{y}\propto 1/{l_{w}}, \tag{43}\]  
where \(l_{w}\) denotes the wall length. In this case, the stiffness of the transverse wall may affect the overall yielding procedure of asymmetric structures, and this issue is further discussed in this section.  
An example problem illustrated in Figure 9 is considered to explore this issue. This is similar to the model problem shown in Figure 3 except that its floor plan has a square shape, not a rectangular shape. The lateral stiffness (\(k_{w1}\)), wall length (\(l_{w1}\)) and yield displacement (\(u_{y,w1}\)) of the flexible side wall \(Y_{1}\) are denoted by \(k\), \(l_{w}\) and \(u_{y}\), respectively. The same quantities for the stiff side wall \(Y_{2}\) (\(k_{w2}\), \(l_{w2}\) and \(u_{y,w2}\), respectively) are given by  
\[k_{w2}=\alpha k, \tag{44}\]  
\[l_{w2}=\sqrt[3]{\alpha}l_{w}, \tag{45}\]  
\[u_{y,w2}=u_{y}/\sqrt[3]{\alpha}, \tag{46}\]  
Figure 8: Comparison between inelastic torsional responses using the current design code and proposed approach.  
where \(\alpha\) is greater than 1 and is the stiffness ratio of the stiff and flexible side walls. Note that the parameters given by Equations (44)-(46) satisfy the conditions stated by (42) and (43). We assumed that the two transverse walls in the \(x\)-direction (Walls \(X_{1}\) and \(X_{2}\)) have the same elastic lateral stiffness as \(\beta k\) and never yield.  
By using Equations (18)-(21), the relationship between lateral displacement and rotation angle of the example problem can be expressed by  
\[\varphi=-\frac{K_{Y}}{K_{T}}e_{x}\Delta=-\frac{(\alpha-1)}{a\Big{(}\frac{4 \alpha}{\alpha+1}+2\beta\Big{)}}\Delta. \tag{47}\]  
From this equation, the displacement of the two lateral force resisting walls can be derived as  
\[u_{w1}=\Delta-d\varphi=\frac{\alpha^{2}+\alpha(2\beta+4)+(2\beta-1)}{\alpha(2 \beta+4)+2\beta}\Delta, \tag{48}\]  
\[u_{w2}=\Delta+d\varphi=\frac{-\alpha^{2}+\alpha(2\beta+4)+(2\beta+1)}{\alpha(2 \beta+4)+2\beta}\Delta. \tag{49}\]  
The flexible side wall yields first if the following condition is met:  
\[\frac{u_{w1}}{u_{y,w1}}\geq\frac{u_{w2}}{u_{y,w2}}. \tag{50}\]  
By substituting Equations (48) and (49) into Equation (50), we obtain  
\[\beta\leq\frac{\alpha^{2}\Big{(}\sqrt[3]{\alpha}+1\Big{)}-4\Big{(}\sqrt[3]{ \alpha}-1\Big{)}\alpha-\Big{(}\sqrt[3]{\alpha}+1\Big{)}}{2(\alpha+1)\Big{(} \sqrt[3]{\alpha}-1\Big{)}},\,(\alpha>1). \tag{51}\]  
The condition stated by Equation (51) is plotted in Figure 10. The horizontal and vertical axes of the figure represent the total wall stiffness in the excitation and transverse directions, respectively. The results in the figure indicate that first yielding may occur even in the stiff side wall if the total transverse wall stiffness is located in the region above the curve plotted in the figure. For this reason, some researchers have classified plan-asymmetric structures into torsionally unrestrained (TU) and torsionally restrained (TR) structures to consider this effect [11; 12; 13]. The inelastic torsional response assessment procedure proposed in this study can properly handle this effect without introducing the classification as in the previous research, but none of the current code provisions like IBC and Eurocode consider this effect.  
Figure 9: Example problem for the investigation of a transverse wall stiffness effect.