## 3 Evaluation of Stress and Stiffness by Means of Latin Hypercube Sampling Method
### Estimation of Stiffness under Two Conditions at Interval  
A deformation of the frame was the second aim of the analysis. It was reached basing on the frame stiffness. In order to estimate the component resistance on bending deformation, the weight of goods and carriage was applied as a kind of concentrated force in the frame, Figure 1. The translational degrees of freedom in vertical and lateral directions at front suspensions and the all translational degrees of freedom at rear suspensions were limited. Some of the 40 samples data are listed in Table 7. \(F_{\text{c}}\) and \(D\) are the applied force and the maximum deformation after loading, respectively. The contour of displacement in vertical direction for one sampling data is displayed in Figure 8. According to the simulated results of this bending condition and Equation (8), the bending stiffness of frame was of \(\left[9.65\times 10^{9},10.50\times 10^{9}\right]\) N\(\cdot\)m\({}^{2}\), Figure 9. They were exceeding the limit value and therefore the frame should be redesigned.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Number** & \(x_{1}\) **(mm)** & \(x_{2}\) **(mm)** & \(x_{3}\) **(mm)** & \(x_{4}\) **(mm)** & \(x_{5}\) **(MPa)** & \(x_{6}\) & \(F_{\text{c}}\) **(N)** & \(D\) **(mm)** \\ \hline
1 & 45.23 & 17.00 & 24.87 & 44.36 & 208.6 & 0.256 & \(\begin{array}{c}2.851\times\\ 10^{6}\end{array}\) & 2.435 \\
2 & 44.05 & 15.92 & 24.51 & 45.23 & 199.6 & 0.281 & \(\begin{array}{c}2.779\times\\ 10^{6}\end{array}\) & 2.540 \\
3 & 44.00 & 15.00 & 24.28 & 45.18 & 205.3 & 0.277 & \(\begin{array}{c}3.246\\ \times 10^{6}\end{array}\) & 2.851 \\
38 & 44.21 & 16.33 & 25.49 & 44.46 & 192.9 & 0.269 & \(\begin{array}{c}1.940\times\\ 10^{6}\end{array}\) & 2.740 \\
39 & 44.51 & 16.69 & 25.95 & 45.08 & 209.5 & 0.282 & \(\begin{array}{c}2.090\times\\ 10^{6}\end{array}\) & 2.844 \\
40 & 44.41 & 15.62 & 24.26 & 45.49 & 198.0 & 0.274 & \(\begin{array}{c}1.630\\ \times 10^{6}\end{array}\) & 3.334 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Sample data and the corresponding responses.  
Figure 6: Comparison of stress due to dynamic loading at breaking and ultimate tensile strength at the interval (unit: MPa).  
Figure 7: Comparison of stress due to dynamic torsion and ultimate tensile strength at the interval (unit: MPa).  
Similarly, the torsional stiffness assessment of frame was conducted as the approach mentioned above. The 40 sample data and their responses were obtained by finite element analysis. The contour of displacement in vertical direction for one sampling data is displayed in Figure 10. The working torsional stiffness of frame was calculated as follows: \(\left[1.35\times 10^{9},1.64\times 10^{9}\right]\) N\(\cdot\)m\({}^{2}\)/rad, and compared with the ultimate torsional stiffness in an intersection number of axis, Figure 11. They were intersected, and thus some improvement of frame is needed.  
Figure 11: Comparison of torsional stiffness and ultimate torsional stiffness at interval (unit: N\(\cdot\)m\({}^{2}\)/rad).  
Figure 8: Contour of displacement in vertical direction under bending (unit: m).  
Figure 10: Contour of displacement in vertical direction under torsion (unit: m).  
Figure 9: Comparison of bending stiffness and its ultimate value at interval (unit: N\(\cdot\)m\({}^{2}\)).