# Effective Assessment of Inelastic Torsional Deformation of Plan-Asymmetric Shear Wall Systems
## 2 Current Code Provisions on Torsional Design of Plan-Asymmetric Structures  
In general, most of the current seismic design code provisions are based on the force-based design approach. In this approach, the static eccentricity (\(e\)) is defined as the distance between the centers of mass and rigidity and is used to compute the torsional moment acting on a plan-asymmetric structure. Based on this definition, the torsional moment (\(T\)) can be computed by  
\[T=-Ve, \tag{1}\]  
where \(V\) is the lateral force applied to the structure. If more than one lateral force components are applied, the resulting torsional moment can be calculated by  
\[T=-\int\overline{e}\,dV, \tag{2}\]  
where \(\overline{e}\) is the eccentricity of each lateral force components. From this relation, the effective eccentricity can be derived as follows:  
\[e=-\frac{\int\overline{e}\,dV}{\int dV} \tag{3}\]  
The current seismic design codes such as IBC, Eurocode, NZS and NBCC introduce the concept of the design eccentricity (\(e_{design}\)), which can generally be written in the form of  
\[e_{design}=\gamma e+\eta b, \tag{4}\]  
where \(e\) is the eccentricity defined as the distance between the center of mass (CM) and the center of rigidity (CR), \(b\) is the floor plan length perpendicular to the excitation direction, and \(\gamma\) and \(\eta\) are the coefficients specified in the seismic design codes listed in Table 1. In Equation (4), the first term on the right-hand side accounts for the natural torsional effect, and the second term incorporates the accidental torsional effect. The design eccentricity proposed by IBC can be stated as  
\[e_{design}=1.0e\pm 0.05b. \tag{5}\]  
In this case, no dynamic amplification is considered, and the eccentricity by accidental torsion is defined as \(\pm 5\%\) of the plan length in the direction perpendicular to the applied lateral force. If the diaphragm is not rigid, and the maximum deformation at the edge of the plan considered exceeds 1.2 times the average deformation, the entire design eccentricity expressed by Equation (5) is multiplied by a torsional amplification factor (\(A_{x}\)) given as  
\[A_{x}=\frac{\delta_{\max}}{1.2\delta_{avg}}\leq 3. \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Code** & \(\gamma\) & \(\eta\) & **Note** \\ \hline IBC & 1.0 & \(\pm 0.05\) & \
\begin{tabular}{c} A torsional amplification factor \(A_{x}\) is \\ considered if the diaphragm is not rigid. \\ \end{tabular} \\ \hline NZS & 1.0 & \(\pm 0.1\) & - \\ \hline NBCC & \(1.0\pm 0.5\) & \(\pm 0.1\) & - \\ \hline Eurocode & 1.0 & 0.05 &
\begin{tabular}{c} Dynamic effect is considered by introducing \\ an additional eccentricity value. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design torsional eccentricity coefficients specified in several current seismic design codes. In Equation (6), \(\delta_{max}\) is the maximum deformation at the plan edge, and \(\delta_{avg}\) is the average value of the deformations measured at both of the plan edges. NZS suggests a design eccentricity similar to that of IBC, which is expressed by  
\[e_{design}=1.0e\pm 0.1b. \tag{7}\]  
It can be noted from Equation (7) that no dynamic effect is considered in NZS and \(\pm 10\%\) of the plan length in the direction perpendicular to the applied lateral force is used as the eccentricity by accidental torsion. The design eccentricity suggested in NBCC is similar to that of NZS, but the eccentricity amplification due to dynamic effects is also considered.