## 3 Methodology  
To determine the plastic limit pressure, the finite element method was applied. The nominal diameter of the pipeline is DN125 (outer diameter 139.7 mm) with a wall thickness of 5.6 mm. The elbow radius is equal to 190 mm. Figure 2 displays the boundary conditions of the pipeline along with the equivalent stress of the calculated critical elbow.  
Boundary conditions and geometry are characteristic of pipelines in power plants, and besides internal pressure, the masses of valves and flanges are considered.  
In addition to finite element analysis, the structural integrity of the pipeline was assessed using the SINTAP procedure. This procedure integrates the yield load solution within the failure assessment diagram (FAD) to assess the structural integrity and ensure the safe operation of the evaluated structure. Observed results can be used for the structural integrity assessment procedure (FAD philosophy). The FAD function is given as \(K_{\mathrm{r}}=f(L_{\mathrm{r}})\), where option 1A is used as the material is expected to display a yield plateau (\(\sigma_{\mathrm{Y}}=R_{\mathrm{p},0,2}\)).  
Figure 1: Pipeline dimensions (all dimensions are in millimeters) [43].  
Figure 2: Boundary conditions: (**a**) Supports; (**b**) Loads and von [PERSON] stress distribution on critical location.  
\(K_{\rm r}\) represents the ratio between the stress intensity factor and fracture toughness of the material, while the ligament-yielding parameter \(L_{\rm r}\) is defined as the ratio between internal pressure and plastic limit pressure [37].  
Figure 3 illustrates the dimensions of defects, where \(a\) represents the crack depth, \(c\) is half the crack length, and \(W\) indicates the remaining ligament.  
The analyzed defects are described by using dimension ratios of \(c/a\) = 0.5, 1, 1.5, 2, 2.5, 3, 4, and 5, and depth/thickness ratios of \(a/t\) = 0.15, 0.3, 0.45, 0.6, and 0.75. Different \(c/a\) ratios are observed to explore behaviors that may occur with various semi-elliptical crack shapes. Additionally, different \(a/t\) ratios are examined to investigate both deeper and shallower cracks. Furthermore, such prepared input data and results can be statistically processed.  
As mentioned earlier, similarly loaded cracked pipes will exhibit local plastic collapse of the ligament as a failure mechanism. There are several approaches that can be used to determine limit load. In this study, the limit pressure as load will be obtained. To calculate the limit pressure using the finite element method, nonlinear material properties must be considered. A bilinear elastic-plastic material model with a low level of isotropic hardening was used to achieve faster convergence of the results. Material design properties for P235 are given in Table 1.  
A simplified model was created to calculate the limit pressure using the submodeling technique. This approach enables the application of a finer mesh around the crack area. Calculations were performed in Ansys Workbench 2022 [46]. The submodel meshing was performed using nearly 700.000 tetrahedral finite elements, while the crack was meshed with dominantly hexahedral elements. The mesh density increases from the edges of the model towards the crack, with the element size of 10 mm at the edges and 0.5 mm around the crack tip location. Additionally, the mesh was structured to cover at least three elements spanning the thickness of the pipe. Mesh is similar for all considered crack dimensions. Figure 4 shows the meshed submodeled region of interest containing a semi-elliptical crack.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Material Property** & **Value** \\ \hline Young’s modulus (\(E\)): & 200,000 MPa \\ Density (\(\rho\)): & 7700 kg/m\({}^{3}\) \\ Yield strength (\(R_{\rm p0,2}\)): & 235 MPa \\ Ultimate strength (\(R_{\rm m}\)): & 360 MPa \\ Poisson’s ratio in elastic range (\(\
u\)): & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of P235 steel [45].  
Figure 3: Pipe elbow with crack.