# Effective Assessment of Inelastic Torsional Deformation of Plan-Asymmetric Shear Wall Systems
## 3 Inelastic Torsional Response of Plan-Asymmetric Structures
### Inelastic Torsional Response Assessment Procedure
#### 1.1.2 First yielding point  
As the lateral force increases, the wall on the flexible side yields first since (i) it has greater displacement than the one on the stiff side and (ii) the two walls have the same yield displacement.  
Figure 4: Force-displacement relationship between the lateral force resisting wall components.  
Figure 5: Yielding procedure: Linear elastic stage.  
Finally, the coordinates of the second yielding point (\(YP_{2}\)) can be written as  
\[YP_{2}=YP_{1}+\Big{[}\delta\Delta^{1},\delta\varphi^{1}\Big{]}. \tag{37}\]
5. _Second inelastic stage (final stage)_  
As in the case of the wall component on the flexible side, the lateral stiffness of the wall on the stiff side is also reduced to \(\gamma k\). After yielding of the two wall components in the \(y\)-direction, the modified lateral and torsional stiffnesses and the eccentricity of the asymmetric system can be respectively expressed by.  
\[K_{y}^{2}=\gamma k+\alpha\gamma k=\gamma K_{y}^{0}, \tag{38}\]  
\[K_{T}^{2}=a^{2}\bigg{(}\frac{4\alpha\gamma}{\alpha+1}+\frac{\beta}{2}\bigg{)}k, \tag{39}\]  
\[e_{x}^{2}=\frac{a(\alpha-1)}{(\alpha+1)}, \tag{40}\]  
where the superscript 2 indicates the second inelastic stage. Note that the eccentricity in these equations is the same as the one in the linear elastic stage. This is mainly because the lateral stiffnesses of the two walls in the \(y\)-direction are assumed to be reduced by the same rate after yielding and the transverse wall components in the \(x\)-direction do not contribute to the \(x\)-directional stiffness eccentricity. The transverse walls in the \(x\)-direction still remain in the elastic stage and now offer relatively large torsional stiffness compared to the wall components in the \(y\)-direction. Finally, the relationship between the increase in lateral displacement (\(\delta\Delta^{2}\)) and rotation (\(\delta\varphi^{2}\)) at this stage can be given by  
\[\delta\varphi^{2}=-\frac{\gamma(\alpha-1)}{a\big{(}\frac{4\alpha\gamma}{ \alpha+1}+\frac{\beta}{2}\big{)}}\delta\Delta^{2}, \tag{41}\]  
and this is plotted in Figure 7.  
Figure 6: Yielding procedure: First inelastic stage.  
Figure 7: Yielding procedure: Second inelastic stage.  
Figure 8 schematically shows the typical lateral-torsional responses of asymmetric structures obtained using three different approaches on the D-R coordinate system based on the discussion up to this point. In the figure, the dashed line represents the linear elastic torsional response given by Equation (22), and the dotted line is the amplified linear elastic torsional response, which is adopted in most of the current design codes in terms of the amplified design eccentricity, as already discussed in Section 2. The solid line represents the inelastic torsional response based on the procedure presented in this section. A comparison of these three curves shows that the amplified linear-elastic response, which is adopted in most of the current design codes, may both underestimate the torsional rotation at the initial yielding stage and overestimate it at the post yielding stage, resulting in an inaccurate estimation of the torsional response over the entire inelastic range.