# Analysis of the Impact of Forces in Hanger Rods on Power Boiler Operation
## 4 Results of the Numerical Analysis Taking into Account the Forces Measured in the Rods  
The results of the numerical analysis are presented in Figure 11. The presented model was built using elements with equivalent stiffness. The analysis time for such calculations is approximately 5 min. When analysing the time required for the calculations for the models presented in Figure 1, it can be inferred that, for the model shown in Figure 11 (which is made of 3D elements), the calculations will take several hours (assuming that the analysis time increases exponentially with the number of finite elements). Figure 11 shows the results for the global model, and it is an illustration of the boiler stress-state indicator \(\sigma_{\text{ind}}\), which is used to determine critical areas. In order to establish the indicator, analyses were carried out first to obtain stress results for the entire model of the boiler (Figure 11a). Next, the area with the highest concentration of stresses was selected, and the calculations were repeated for this part (Figure 11b). The structure under analysis was the upper part of the furnace and the separators. The stress-state indicator (\(\sigma_{\text{ind}}\)) was found as \(\sigma_{\text{ind}}=\sigma_{\text{actual}}/\sigma_{\text{max}}\), where the value of equivalent stress \(\sigma_{\text{max}}\) [MPa] was adopted as the mean value of stress in the analysed area. Equivalent stress \(\sigma_{\text{actual}}\) [MPa] is the stress value at a given point of the boiler; thus, \(\sigma_{\text{ind}}\) will take values higher or lower than 1. The aim of the analysis was to check the impact of the non-uniform tension of the hanger rods on the distribution of stresses in the pressure and non-pressure parts of the boiler. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Rod**} & \multicolumn{8}{c}{**Measurement Number [kN]**} & \multicolumn{8}{c}{**Mean Value**} \\ \cline{2-11}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **[kN]** & **MAE** \\ \hline
1 & 492 & 472 & 498 & 515 & 506 & 409 & 514 & 455 & 461 & 450 & 477.2 & 5.83\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured force values in Rod 1 for the rear wall. Figure 10: Measured forces in the rods for the cross-over ducts (COD): (**a**) COD1; (**b**) COD2; (**c**) and CODS. The results in the form of the boiler stress state present the effect of a non-uniform tension of the hanger rods on the concentration of stresses in the corners of the furnace chamber. The effect of the non-uniform tension of the rods can also be noticed on the furnace chamber walls in the form of variable stresses along the chamber width. For the tested boiler, stresses were also checked in the corner of the flange connection connecting the cross-over duct to the separators. The results of the stress state are presented in Figures 12 and 13. The greatest stress was measured in the corner of the flange that connects the cross-over Duct 2 to Separator 2 (the markings are according to Figure 4). The equivalent (von Mises) stress measured at this point exceeded the yield point of 133 MPa for a flange made of P235 GH material at an operating temperature of 400 \({}^{\circ}\)C [17]. The material properties of steel P235 GH are presented in Table 3.  
Figure 11: FEM analysis results: (**a**) equivalent stress for two selected general projections and (**b**) a stress-state indicator, with an isolated view of the furnace chamber and separators. Figure 12: Results of the numerical analysis for the upper part of the boiler: (**a**) equivalent stress for selected elements of the boiler upper part and (**b**) the equivalent stress-selected parts of Separator 2 and 3. Figure 13: Analysis results for the flange connecting the cross-over duct with the separator: (**a**) stress state; (**b**) equivalent stress; and (**c**) Detail A—flange cracking area. The stress state presented in Figure 13b reflects the real problem in the boiler (as shown in Figure 13c): it is cyclically damaged at this place.