## 4 Results  
After running the analysis for all crack shapes, the results showed that the typical failure pattern for all crack dimensions involves localized material yielding on both sides of the crack surface points, in a plane parallel to the crack. Localized material yielding near the crack surface is a well-known phenomenon in fracture mechanics, primarily because of the stress concentrations developed at the crack tip. This phenomenon is illustrated in Figure 5, which shows the stress distribution for two specific crack size ratios: \(c/a=1\) (\(a/t=0.15\)) and \(c/a=2.5\) (\(a/t=0.6\)). The red zone indicates stress levels exceeding the yield strength, leading to plastic collapse. There is also present a yielding of the crack tip, particularly noticeable in Figure 6, where the crack size ratio is \(c/a=2.5\) (\(a/t=0.75\)). As the crack size increases (higher \(c/a\) and \(a/t\) ratios), the stress concentration intensifies, which explains why the yielding becomes more evident in these regions, particularly for deeper cracks. Figure 4: Submodeled elbow with semi-elliptical crack. Figure 5: Plastic collapse in crack zone: (**a**) \(c/a=1\); (**b**) \(c/a=2.5\). The stress intensity factor (\(K_{\rm I}\)) was calculated using a straightforward approach based on fracture mechanics principles. The crack geometry is defined, and the working pressure applied is set as a load in the finite element model. After running the simulations for each crack configuration, the results at the crack tip, considered the critical location, were obtained. The stress intensity factor was determined based on the crack tip stress distribution, which finite element analysis (FEA) calculates by evaluating the local stress fields around the crack front. The stress intensity factor values at the crack tip were obtained directly from the FEA post-processing correlating with the boundary conditions, applied pressure, and crack geometry. Unlike the stress intensity factor, the plastic limit pressure (\(P_{\rm L}\)) was obtained by using an iterative process. In the FEA simulation, the internal pressure was set as a load, and load increases in each iterative calculation until the material around the crack tip yields through the entire thickness of the pipe. The pressure at which full yielding of the crack ligament section occurs is considered the plastic limit pressure, as it represents the point where failure due to plastic collapse is reached. To validate FEA solutions for the cracked elbow, an analytical solution for the plastic limit pressure of straight pipe \(P_{\rm LS}\) with the same diameter and thickness was calculated, based on the methodology demonstrated in [PERSON] et al. [8]. This analytical solution for the straight pipe serves as a benchmark for comparison to evaluate the accuracy of the FEA results for the more complex geometry of the cracked elbow. The comparison shows that trends in both sets of results are consistent (Table 2). As the crack depth or geometry changes, both the FEA results for the elbow and the analytical solutions for the straight pipe exhibit similar trends in pressure reduction. Elbow complex geometry creates some variation in the exact values. The primary reason for these variations is the curvature of the elbow. Elbows experience a non-uniform distribution of stress due to bending and torsional effects. When an elbow is loaded with internal pressure, the curvature of the elbow creates regions of increased stress, particularly on the intrados where the stress concentrations tend to be higher. The geometry of an elbow introduces additional complexities that affect both the plastic limit pressure and the stress intensity factor in comparison to a straight pipe. The FEA approach effectively captures the mechanical behavior under similar loading conditions. Figure 6: Equivalent von Mises stress distribution for time step when limit (yield) load is reached (\(c/a=2.5\) and \(a/t=0.75\)).