# Effective Assessment of Inelastic Torsional Deformation of Plan-Asymmetric Shear Wall Systems
## 3 Inelastic Torsional Response of Plan-Asymmetric Structures
### Torsional Response of Asymmetric Structures in the D-R Coordinate System  
In this section, the generalized form of the displacement-rotation relation of plan-asymmetric structures is derived by assuming that the structure is in the linear elastic range. From this relation, we can derive the inelastic displacement-rotation relation of plan-asymmetric structures.  
Figure 1 illustrates a representative asymmetric wall structure subjected to a lateral force in the \(y\)-direction (\(V\)) as well as its deformed shape. In the figure, Wall X\({}_{i}\) and Wall Y\({}_{i}\) are the \(i\)-th lateral load resisting walls in the \(x\)- and \(y\)-directions, respectively. The \(x\)-coordinate of Wall Y\({}_{i}\) and \(y\)-coordinate of Wall X\({}_{i}\) with respect to the center of rigidity (CR) in the undeformed configuration are denoted as \(r_{yi}\) and \(r_{xi}\), respectively. Similarly, the same quantities with respect to the center of mass (CM) are expressed as \(d_{yi}\) and \(d_{xi}\), respectively.  
The lateral force (\(V\)) and torsional moment (\(T\)) of the system can be calculated by Equations (15) and (16), respectively:  
\[V=K_{y}\Delta, \tag{15}\]  
\[T=K_{T}\varphi. \tag{16}\]  
Figure 1: Representative asymmetric wall structures under the application of a lateral force and its deformed shape.  
Here, \(K_{y}\) and \(K_{T}\) are the lateral stiffness in the \(y\)-direction and torsional stiffness of the asymmetric system, respectively. In addition, \(\Delta\) and \(\varphi\) are the lateral displacement and rotation, respectively, and they are estimated at the center of mass. The counterclockwise direction of \(\varphi\) is assumed to be positive. In the derivation of Equation (16), we assumed that the diaphragm has an infinite rigidity, and the out-of-plane deformation of the shear wall is ignored. If no external torsional moment is applied, the resulting torsional moment of the asymmetric system can be determined by multiplying the lateral force and the eccentricity in the \(x\)-direction between the centers of mass and rigidity (\(e_{x}\)), as given below:  
\[T=-Ve_{x}. \tag{17}\]  
The eccentricity in the \(x\)-direction can be computed by  
\[e_{x}=\frac{\sum k_{yi}d_{yi}}{\sum k_{yi}}, \tag{18}\]  
where \(k_{yi}\) is the lateral stiffness of the \(i\)-th wall in the \(y\)-direction (Wall Y\({}_{i}\)).  
By substituting Equations (15) and (16) into Equation (17), the rotation can be expressed in terms of the lateral displacement as  
\[\varphi=-\frac{K_{y}}{K_{T}}e_{x}\Delta. \tag{19}\]  
The lateral and torsional stiffness of the asymmetric structure can be estimated by  
\[K_{y}=\sum k_{yi}, \tag{20}\]  
\[K_{T}=\sum k_{yi}r_{yi}{}^{2}+\sum k_{xi}r_{xi}{}^{2}, \tag{21}\]  
where \(k_{xi}\) represents the lateral stiffness of the \(i\)-th wall in the \(x\)-direction. By inserting Equations (20) and (21) into Equation (19), the relationship between the lateral displacement and rotation can be obtained as follows:  
\[\varphi=-\frac{\sum k_{yi}d_{yi}}{\sum k_{yi}r_{yi}{}^{2}+\sum k_{xi}r_{xi}{} ^{2}}\Delta. \tag{22}\]  
As shown in Figure 2, this relationship can be represented by a solid line with an arrow on the lateral displacement (\(\Delta\)) and rotation (\(\varphi\)) coordinate system, which is denoted as the D-R coordinate system in this paper. This indicates that a linear relation holds between the lateral displacement and the rotation as long as no wall component yields. The yield condition for Wall Y\({}_{1}\) can be stated as  
\[\Delta-