## 5 Case Studies
### Engineering Experiment B: A Turbine Disc  
The turbine disc, serving as a core component in modern aircraft engines, tends to operate under complicated environments, high temperature, high pressure, heavy loads and other terms. As for its working mechanism, the heat energy will be converted into mechanical energy when the high temperature gas flows from the combustion chamber to the engine of the turbine disc, driving the engine. As a key component of an engine, the reliability level of turbine disc affects the engine performance to a great extent. The 1/37 third-stage roulette of a low-pressure compressor is selected as the investigation object, and the reliability of the life-limiting part is analyzed in reference to the fatigue failure mechanism. The reliability of a turbine disc depends primarily on the distribution of stress and strain at a dangerous point since the shape and load are completely symmetric. The mean values of each variable are chosen as parameters, and the simulation is implemented with ANSYS 18.1. The 3D model and the cloud map under equivalent stress distribution of the low-pressure compressor roulette at 1000 rad/s speed are shown in Figures 6 and 7, respectively. It can be found that stress concentration occurs at the pin hole of the low-pressure compressor roulette, and the maximum equivalent stress equals to 801 MPa. Consequently, the dangerous section of the object is the contact between the roulette and semi-hollow pin, namely the lower outside of the pin hole. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Number of Sample Points** & **Reliability Index** & **Probability of Failure** & **Relative Error/\%** \\ \hline Monte Carlo & \(10^{6}\) & 2.271446 & 0.01156 & - \\ Importance sampling & 4000 & 2.258733 & 0.01195 & 3.374 \\ Proposed approach & 1500 & 2.282497 & 0.01123 & 2.855 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the numerical example result. Figure 4: Diagram of the roof truss structure. Figure 5: Sample point addition process. The random factors affecting the failure probability of a low-pressure compressor roulette can be mainly divided into three categories, indicated in the following:  
* Material properties, such as the elastic modulus, density and Poisson's ratio of the materials utilized in roulette and pins, can reflect the material uncertainties;
* Roulette speed can reflect the uncertainty of centrifugal force;
* Uncertainties exist in the assembly force at the connection between the pin and blade. As mentioned above, eight variables are considered as random variables in this case, which represent the relevant parameters of materials and loads, respectively, and all of them obey normal distribution according to engineering experience and statistical data. In addition, eight random variables are assumed as independent normal distribution for the convenience; the mean and standard deviation are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variables** & **Physical Significance** & **Mean Value** & **Standard Deviation** & **Unit** \\ \hline \(E_{1}\) & Elastic modulus of roulette & 123,000 & 3000 & MPa \\ \(\varepsilon_{1}\) & Poisson’s ratio of roulette & 0.33 & 0.015 & / \\ \(\rho_{1}\) & Density of roulette & 4.48 & 0.2 & g/cm\({}^{3}\) \\ \(E_{2}\) & Elastic modulus of pin & 219,000 & 6000 & MPa \\ \(\varepsilon_{2}\) & Poisson’s ratio of pin & 0.3 & 0.015 & / \\ \(\rho_{2}\) & Density of pin & 7.76 & 0.3 & g/cm\({}^{3}\) \\ \(n\) & Rotational speed & 1000 & 10 & rad/s \\ \(F\) & Assembly force & 21.928 & 0.01 & kN \\ \(S_{\max}\) & Fatigue strength of material & 703.84 & / & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter and its value of the engineering experiment. Figure 6: The \(1/37\) 3D model of a low-pressure compressor roulette. Figure 7: The \(1/37\) 3D model of a low-pressure compressor roulette.