## 2 Bending and Torsional Stiffness Analysis Model of BIW  
The bending and torsional stiffness of the BIW have an important impact on vehicle performance, such as NVH (noise, vibration, and harshness), safety, handling performance and stability. They also affect the degree of lightweighting of the body, which are key parameters in the development process and represent the core competitiveness of the vehicle. The model is from ref. [23], and the pre-processing process of CAE meshing,material properties and constraint boundaries of the model is performed using Hypermesh (version 2019) software, which has about 390,000 cells. The thicknesses of 14 key sheet metal components are considered as the design variables. The FEM and component numbers are shown in Figure 1. The paired components in Figure 1 are annotated on only one side, and the initial values and ranges of the design variables are shown in Table 1.  
The loads and constraints for the bending and torsion cases are shown in Figure 2a and Figure 2b, respectively, defining the vertical (gravitational) direction of the model as the \(Z\)-direction. The forces \(F_{\text{b1}},F_{\text{b2}}\) applied to the sill beams and \(F_{\text{t1}},F_{\text{t2}}\) applied to the shock towers are \(Z\)-directed forces (N): \(F_{\text{b1}}=F_{\text{b2}}=3336\) N and \(F_{\text{t1}}=F_{\text{t2}}=1177\) N. For bending conditions, the four restraint positions (SPC1-SPC4) are located on the shock towers; for torsion conditions, the front restraint position (SPC1) is located in the middle of the front bumper frame, and the rear restraint positions (SPC2, SPC3) are located on the antiroll bar brackets.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variables** & **Symbol** & **Mean** & **Coefficient of Variation** \\ \hline A-pillar outer plate & \(t_{1}\) & 1.32 & 0.03 \\ Rear seat frame & \(t_{2}\) & 1.10 & 0.03 \\ Front floor & \(t_{3}\) & 1.62 & 0.03 \\ Rear floor & \(t_{4}\) & 1.08 & 0.03 \\ Firewall & \(t_{5}\) & 0.76 & 0.03 \\ Rear cross member & \(t_{6}\) & 1.26 & 0.03 \\ Wheel well-L 1 & \(t_{7}\) & 0.88 & 0.03 \\ Wheel well-R 2 & \(t_{8}\) & 0.88 & 0.03 \\ Rail-L & \(t_{9}\) & 1.71 & 0.03 \\ Rail-R & \(t_{10}\) & 1.71 & 0.03 \\ Rear rail-L & \(t_{11}\) & 1.65 & 0.03 \\ Rear rail-R & \(t_{12}\) & 1.65 & 0.03 \\ Unibody frame-L & \(t_{13}\) & 0.99 & 0.03 \\ Unibody frame-R & \(t_{14}\) & 1.71 & 0.03 \\ \hline \hline \end{tabular} \({}^{1}\) L means left. 2 R means right. (Unit: mm).  
\end{table}
Table 1: Design variables and probability distribution characteristics.  
Figure 2: (a) Load collectors of bending; (b) Load collectors of torsion.  
The locations of the observation points are illustrated in Figure 3. Under the bending condition, the maximum deformation (mm) in the \(Z\)-direction of the nodes below the left and right sill beams (six observation points are found uniformly on each side) are set to be \(d_{\rm L\,max}\) and \(d_{\rm R\,max}\), respectively, and, under the torsion condition, the \(Z\)-displacements (mm) at the left and right loading points of the shock towers are \(d_{\rm L}\) and \(d_{\rm R}\), respectively. Setting \(d\) as the distance (mm) between the loading points of the shock towers, the equations of bending stiffness \(K_{\rm b}\) and torsional stiffness \(K_{\rm t}\) can be obtained:  
\[\begin{cases}K_{\rm b}=\dfrac{F_{\rm b}}{(|d_{\rm L\,max}|+|d_{\rm R\,max}|)/2} \quad(\rm N/mm)\\ K_{\rm t}=\dfrac{F_{\rm t}d/1000}{\arctan(|d_{\rm L}-d_{\rm R}|/d \ )}\cdot\dfrac{\pi}{180}\quad(\rm N\cdot m/\,^{\circ})\end{cases} \tag{1}\]  
where \(F_{\rm b}=F_{\rm b1}+F_{\rm b2}\) and \(F_{\rm t}=F_{\rm t1}=F_{\rm t2}\).