# Fatigue Damage Assessment of Turbine Runner Blades Considering Sediment Wear
## 3 Numerical Calculation of Dynamic Stress on the Runner Blades of a Turbine
### Dynamic Numerical Calculation of Runner Blades  
Utilizing the fluctuating water pressure load data on the blade surface at various time points, derived from the transient flow field calculation of a Francis turbine's entire passage, a transient dynamic stress mechanical analysis of the blade is performed. This analysis employs a unidirectional fluid-structure coupling method to couple the blade surface, yielding the equivalent stress-time history data on the blade surface at different times for various degrees of wear. Figure 7 presents the equivalent stress cloud diagrams of the blade at various instances throughout the rotation cycle, under the wear condition denoted as T\({}_{0}\). It can be seen from the equivalent stress cloud diagram of the blade in the next rotation cycle under the wear degree of T\({}_{0}\) that the stress in the middle area of the blade is relatively small at different rotation times, and the stress is mostly concentrated in the area near the upper crown and the area near the lower ring at the water edge of the blade. With the rotation of the runner blade, the maximum stress area of the runner blade is always located  
Figure 7: Equivalent stress cloud diagram of runner blade under T\({}_{0}\) wear degree in a rotation cycle. in the area near the lower ring of the blade at the water exit edge, and the maximum stress in the area near the lower ring of the blade at different rotation times has obvious changes and fluctuations; this is caused by the alternating stress of the runner blade, and is also the primary reason for the crack and fatigue damage of the runner blade. Since the stress value and the amplitude of stress fluctuation in this region are the maximum values compared with other parts of the blade, the fatigue failure will be preferentially generated in this region, which is the dangerous part of the runner blade. To thoroughly understand the variation in equivalent stress within the critical areas of the runner blade, stress monitoring points were established along the worn portion of the blade's tail at locations of maximum stress for each degree of wear. These points are designated as Q1, Q2, Q3, Q4, and Q5. The positioning of the stress probes is illustrated in Figure 8. Dynamic stress data for each monitoring point over a complete cycle and across various degrees of wear were extracted and are presented in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{Time} & \multicolumn{6}{c}{Equivalent Stress at Monitoring Point Q1 (MPa)} \\ \cline{2-7}  & T\({}_{0}\) & T\({}_{1}\) & T\({}_{2}\) & T\({}_{3}\) & T\({}_{4}\) & T\({}_{5}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: Equivalent stress data of monitoring points on blades at different wear degrees in a single rotation cycle. Figure 8: Positions of monitoring points for blade wear locations. With the increase in wear time, the monitoring point will be worn out with the increase in wear amount. By extracting and analyzing the equivalent stress data of different wear times before the wear-out of each monitoring point, the specific data are shown in Table 4. It can be seen in the table that when the runner blade wear amount reaches T\({}_{2}\), the monitoring point Q\({}_{1}\) will be worn out. Therefore, only the stress-time history data under two blade wear degrees at T\({}_{0}\) and T\({}_{1}\) are available at the Q\({}_{1}\) position. When the blade wear amount reaches T\({}_{3}\), the monitoring point Q\({}_{2}\) will be worn out, so the monitoring point only has the stress-time history under the blade wear amount from T\({}_{0}\) to T\({}_{2}\). In a similar way, when the blade wear reaches T\({}_{5}\), the monitoring point Q\({}_{4}\) is worn out, and the monitoring point Q\({}_{5}\) still exists at this time. Therefore, the monitoring point Q\({}_{5}\) has all the stress-time history of the blade wear from T\({}_{0}\) to T\({}_{5}\).