## 4 Application in Crankshaft Bearing Design
### Comparison between Analytical Method and FEM
#### 4.4.2 Von Mises Stress  
The von Mises stress magnitude and distribution of the interference in sliding crankshaft bearing support system under temperature loading is solved by FE method, as shown in Figure 11.  
As for the FE calculation results, the non-uniform distribution of the stress on contact surface differs from analytical solutions, in which both the circumferential and axial stress of the contact \begin{table}
\begin{tabular}{c c c} \hline \hline
**Components** & **Element Type** & **Element Quantity** \\ \hline Cylinder & C3D4 & 1,240,988 \\ \hline Bearing sleeves (2 pieces) & C3D8 & 15,548 \\ \hline Front bearing block & C3D4 & 124,290 \\ \cline{2-3}  & C3D8 & 17,908 \\ \hline Back bearing block & C3D4 & 198,250 \\ \cline{2-3}  & C3D8 & 14,216 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Element type and quantity of the components in FE model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Contact Pair** & **FEM Solution (MPa)** & **Analytical Solution (MPa)** \\ \hline Between bearing sleeve and front bearing block & 23.94 & 25.68 \\ Between front bearing block and cylinder bearing bore & 15.91 & 12.61 \\ Between bearing sleeve and back bearing block & 24.05 & 28.19 \\ Between back bearing block and cylinder bearing bore & 15.41 & 17.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of contact pressure between analytical method and FEM. surface distributes evenly. Such characteristics of the FE solution relates to the local structural features of the components in support system. The von Mises stress distribution resulting from FE analysis indicates that for all layers of the cylinders, the von Mises stress of the inner circle is higher than that of the outer circle. This rule is also demonstrated by the analytical solution. Therefore, the von Mises stress of the inner circle of each component is paid particular attention, as shown in Table 6. For analytical solution, the obtained \(\sigma_{r}\) and \(\sigma_{\theta}\) are principal stress components, so that the von Mises stress is equal to \(\sqrt{0.5r\left(\left(\sigma_{r}-\sigma_{\theta}\right)^{2}+\sigma_{r}{}^{2}+ \sigma_{\theta}{}^{2}\right)}\). Table 6 shows that on a certain contact surface, the von Mises stress obtained by FE analysis is not evenly distributed, but varies within a range, as in Figure 11. Such non-uniformity of stress distribution results from the non axial-symmetry of the actual geometry as well as the boundary conditions, and partly due to the numerical error; while the stress on contact surface derived from analytical solution is uniformly distributed, which makes the assessment of the maximum reaction torque afforded by the contact more convenient. In addition, both the FE analysis and analytical solution indicate that the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Components the Contact** & **Contact Pair the Contact** & **FEM Solution** & **Analytical Solution** \\
**Belongs to** & **Surface Belongs to** & **(MPa)** & **(MPa)** \\ \hline \multirow{3}{*}{Cylinder(free end)} & Between front bearing & \multirow{3}{*}{30–52} & \multirow{3}{*}{28.1} \\  & block and cylinder & & \\  & bearing bore & & \\ \hline \multirow{2}{*}{Bearing block (free end)} & Between bearing sleeve & \multirow{2}{*}{30–50} & \multirow{2}{*}{30.6} \\  & and front bearing block & & \\ \hline \multirow{2}{*}{Bearing sleeve (free end)} & Between bearing sleeve & \multirow{2}{*}{310–370} & \multirow{2}{*}{357.5} \\  & and front bearing block & & \\ \hline \multirow{2}{*}{Cylinder (flywheel end)} & Between back bearing & \multirow{2}{*}{37–60} & \multirow{2}{*}{42.9} \\  & block and cylinder & & \\ \cline{1-1}  & bearing bore & & \\ \hline \multirow{2}{*}{Bearing block (flywheel end)} & Between bearing sleeve & \multirow{2}{*}{21–31} & \multirow{2}{*}{27.5} \\  & and back bearing block & & \\ \hline \multirow{2}{*}{Bearing sleeve (flywheel end)} & Between bearing sleeve & \multirow{2}{*}{300–371} & \multirow{2}{*}{392.4} \\  & and back bearing block & & \\ \hline \hline \end{tabular}
\end{table}