# Effective Assessment of Inelastic Torsional Deformation of Plan-Asymmetric Shear Wall Systems
## 4 Comparison between Predictions by the Proposed Method and Time History Analysis Results
### Model Problem and Analysis Method  
The model problem for verification of the proposed method is the same as the one illustrated in Figure 3. In the model problem, the size of the plan is given as \(a=5\) m, and the lateral stiffness of Wall Y\({}_{1}\) on the flexible side is taken as \(k=1000\) kN/m. We assumed that a mass of 500 kg is lumped at the center of mass and the yield displacement of the two lateral force resisting walls in the \(y\)-direction is defined as \(u_{y}=0.05\) m. In contrast, we also assumed that the two \(x\)-directional walls never yield. A Rayleigh damping ratio of 5% was adopted.  
For the time history analysis, each structural wall in the asymmetric structure was modeled using the equivalent truss members, as illustrated in Figure 11. In the modeling, we assumed that the horizontal and vertical trusses are rigid, and the diagonal truss members have zero compressive stiffness (tension only element) to effectively define the hysteretic behavior of the wall component. The elastic stiffness (\(k_{diag}\)) and yield force (\(P_{y,diag}\)) of the diagonal truss elements are defined as follows:  
\[k_{diag}=\frac{P_{sys}}{\Delta_{sys}}\frac{d^{2}+h^{2}}{d^{2}}=k_{wall}\frac {d^{2}+h^{2}}{d^{2}}, \tag{52}\]  
\[P_{y,diag}=P_{y,wall}\frac{\sqrt{d^{2}+h^{2}}}{d}, \tag{53}\]  
where \(k_{wall}\) is the elastic lateral stiffness of the wall considered, \(P_{y,wall}\) is its yield force. In addition, \(P_{sys}\) and \(\Delta_{sys}\) are the applied lateral force and the lateral displacement at the top of the wall, respectively. Each diagonal truss member consists of two parallel elements, as shown in Figure 12. The primary element among the two elements has an elastic-fully plastic behavior, and the values corresponding to 94% of the total wall yield force and elastic stiffness were set to its yield force and elastic lateral stiffness, respectively. The material behavior of the secondary element is modeled by a fully elastic relationship, of which the elastic stiffness is the 6% of the total wall stiffness. As a result, the structural  
Figure 10: Effect of transverse wall stiffness on the overall yielding procedure of asymmetric structures.  
wall with this diagonal truss member can exhibit the combined bilinear behavior shown in the figure, where the yield force and tangent stiffness after yielding are set to \(P_{y,diag}\) and \(0.06k_{diag}\), respectively.  
The modified Takeda model was used for the nonlinear hysteretic relationship between the time history analysis because its effectiveness has been verified by numerous experimental results. [28; 29]. Figure 13 shows the hysteretic response of the model problem when subjected to static cyclic loading. Figure 13a plots the lateral force-displacement relationships for the flexible side wall Y\({}_{1}\) and stiff side wall Y\({}_{2}\), which are distinguished by dotted and solid lines, respectively. Similarly, the relationship between the lateral displacement and rotation angle of the model problem estimated at its center of mass is plotted in Figure 13b.  
Figure 11: Modeling of an individual structural wall for the time history analysis.  
Figure 12: Diagonal truss modeling using two parallel elements.