# Flow-Induced Fatigue Damage of Large Francis Turbines Under Multiple Operating Loads
## 4 Flow-Induced Stress Analysis of the Francis Runner  
The calculated flow field pressure profiles of the Francis turbine from the CFD analysis can be used to conduct a flow-induced dynamics response calculation of the turbine units with FEMs. The flow-induced structural dynamic equation is described by Equation (3). \[\mathbf{M}\mathbf{\ddot{\delta}}+\mathbf{D}\mathbf{\dot{\delta}}+\mathbf{K}\mathbf{\delta}=\mathbf{F}_{f}+ \mathbf{F}_{e} \tag{3}\]  
where \(\mathbf{M}\) and \(\mathbf{\ddot{\delta}}\) are the structural mass matrix and acceleration vector of the turbine; \(\mathbf{D}\) and \(\mathbf{\dot{\delta}}\) are the damping matrix and velocity vector of the turbine; \(\mathbf{K}\) and \(\mathbf{\delta}\) are the stiffness matrix  
Figure 10: Normalized flow velocity distribution of the Francis unit at FL. Figure 9: Normalized pressure distribution of the Francis unit at FL. and displacement vector of the turbine, respectively; and \(F_{f}\) and \(F_{s}\) are the flow pressure load and other external force acting on the turbine structures. As shown in Equation (4), the flow-induced stress \(\sigma\) of the turbine unit can be further calculated with the elasticity matrix \(E\), strain-displacement matrix \(S_{\delta}\), and structural displacement \(\delta\), solved by Equation (3). \[\sigma=E\,S_{\delta}\,\delta \tag{4}\]  
According to the maximum distortion energy criterion, the von Mises stress shown in Equation (5) is adopted to calculate the three-dimensional stress of the turbine unit. \[\sigma_{mM}=\sqrt{\frac{(\sigma_{max.}-\sigma_{mid.})^{2}+(\sigma_{mid.}- \sigma_{min.})^{2}+(\sigma_{min.}-\sigma_{max.})^{2}}{2}} \tag{5}\]  
where \(\sigma_{max. }\), \(\sigma_{mid. }\), and \(\sigma_{min. }\) are the maximum principal stress, the middle principal stress, and the minimum principal stress of the Francis turbine, respectively. The hydropower plant under study in this paper is equipped with six large Francis turbine generator units, and the rated power of each unit is 180 MW. The design parameters of the large Francis hydro-generator units of the investigated power plant are presented in Table 4.  
A full-scale FE calculation model of the turbine runner is created with a high quantity of tetrahedral elements, as shown in Figure 11. To obtain accurate results, the meshes are refined at the fillets of the turbine runner blades. For the three representative operating conditions, the stress distributions of the runner are similar, as shown in Figure 12. The maximum static and dynamic stresses of the runner \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline Runner diameter & m & 5.8 \\ Runner density & kg m\({}^{-3}\) & 7700 \\ Elasticity modulus & Pa & \(2.1\times 10^{11}\) \\ Poisson’s ratio & - & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The geometrical and material parameters of the Francis runner. Figure 11: FE calculation model of the turbine runner. at LPL, HPL, and FL are located at the chamfered surface, where the blade trailing edge connects to the runner crown. As the power output increases, the Francis turbine operates under higher water flow rates and pressures. This results in greater fluid forces acting on the turbine runner, leading to increased static stress on the runner blades. The maximum static stresses of the Francis runner are 284 MPa, 352 MPa, and 381 MPa at LPL, HPL, and FL, respectively. They exceed the permissible values as half of the yield stress of the runner material of 550 MPa. Repeated dynamic loading due to pressure pulsations at different loads causes structural vibrations on the runner; the peak-to-peak values of runner dynamic stress at LPL, HPL, and FL are 15 MPa, 25 MPa, and 14.6 MPa, respectively. Higher static stresses reduce the fatigue limit of the runner material and make it easier for microcracks to initiate within the runner material, and the cracks expand faster.