## 4 Results and Discussion
### Calculation of the Normalized Edge Displacement Using Dynamic and Static Analysis
#### 4.4.2 Investigating Torsional Radius and Radius of Gyration on Torsional Response  
The Torsional radius, \(Tr\), and radius of gyration, \(r\), were computed for the case study consisting of 72 floor plans, described earlier in this article. These properties were computed using lateral forces and moments about the center of stiffness such that \(F_{xi}=F_{yi}=10^{6}\) kN and \(M_{t}=10^{6}\) kN.m. The torsional radius, \(Tr\), is plotted for the range of \(\Omega\) covered in this study and the results are shown in Figure 14 (_b/r_ = 1.1), Figure 15 (_b/r_ = 1.4) and Figure 16 (_b/r_ = 2.45). Each of the three figures shows the radius of gyration computed Equation (7). Figure 14: Torsional radius for buildings with (\(r\) = 21.9 m & _b/r_ = 1.10): (**a**) Building period \(T<1\) s; (**b**) Building period \(1<T<2\) s; (**c**) Building period \(T>2\) s.  
Based on Figures 14-16, the following observations are made:  
Figure 16: Torsional radius for buildings with (\(r=19.6\) m & \(b\)/\(r=2.45\)): (**a**) Building period \(T<1\) s; (**b**) Building period \(1<T<2\) s; (**c**) Building period \(T>2\) s.  
Figure 15: Torsional radius for buildings with (\(r=22.7\) m & \(b\)/\(r=1.40\)): (**a**) Building period \(T<1\) s; (**b**) Building period \(1<T<2\) s; (**c**) Building period \(T>2\) s.  
1. Torsionally flexible buildings (\(\Omega<1\)) have torsional radius that is less than the building radius of gyration. Conversely, stiff buildings (\(\Omega>1\)) have torsional radius that is greater than building radius of gyration. 2. Torsional radius of the floor plan as defined in EC 8 serves the same purpose as the uncoupled frequency ratio \(\Omega\) as indicator of susceptibility of the floor plan to accidental torsion. 3. Floor plans with unit frequency (\(\Omega=1\)) have torsional radius equal to the traditional building radius of gyration. Clearly, the torsional radius increases with an increase in \(\Omega\), as this frequency ratio is in fact an indicator of the building torsional flexibility. Floor plan and its structural system may be viewed as torsionally rigid when the torsional radius is less than the radius of gyration which occurs when \(\Omega<1.0\). On the other hand, a floor plan and its structural system is torsionally flexible when the torsional radius is larger than the radius of gyration which occurs when \(\Omega>1.0\). When \(\Omega=1\), the floor plan and its structural systems falls in the boundary between torsionally rigid and flexible systems in terms of seismic response. If the translational and torsional stiffnesses of the floor plan are such that it the translation period \(T_{x}\) (or \(T_{y}\)) is equal to the torsional period \(T_{\theta}\), the structural response is neither rigid nor flexible and the drop in torsional response demonstrated in Figures 5-13 is observed when dynamic analysis is performed. It is worthy to note that classification of buildings as torsionally rigid or flexible based on the frequency ratio and/or torsional radius, as well as the presence of the boundary frequency ratio \(\Omega=1\), are not sensitive to the fundamental period. This is confirmed by Figures 13-15 where the translation periods were examined in three regions \(T<1.0\) s, \(1.0\) s \(<T<2.0\) s, and \(T>2.0\) s.  
Accidental Torsion Design Factor-Design Correlation between Frequency Ratio and Accidental Torsion Response  
As demonstrated earlier, predicting accidental torsion response quantities using static analysis based on ASCE 7-16 and dynamic analysis procedures produce, expectedly different results. While Figures 5-13 confirm that accidental torsion response based on dynamic analysis follows specific patterns, the magnitudes of the response quantity \(\hat{u}_{b/2}\) varied significantly based on the geometric floor plan ratio \(b/r\), as well as the frequency ratio \(\Omega\). In this section, we propose an accidental torsion response spectrum, \(A_{t}\), that is function of the floor plan ratio \(b/r\) and \(\Omega\).