# Study on the Residual Strength of Nonlinear Fatigue-Damaged Pipeline Structures
## 4 Results and Discussion
### Tensile Strength Characteristics of Nonlinear Damaged Pipe Body  
After the fatigue damage occurs, according to the calculation formula for bending strain in Section 2, the ultimate load elements of different deformation layers are established, and the entire section is integrated to obtain the ultimate tensile load \(F_{s}{}^{\prime}\) of each layer of the section:  
\[F_{s}{}^{\prime}=\begin{cases}2\int_{-R}^{R}\sqrt{R^{2}-h^{2}}\cdot\sigma_{rh} dh-\int\limits_{-r}^{r}\sqrt{r^{2}-h^{2}}\cdot\sigma_{rh}dh&0\leq h\leq r\\ 2\int_{-R}^{R}\sqrt{R^{2}-h^{2}}\cdot\sigma_{rh}dh&r<h\leq R\end{cases} \tag{9}\]  
where \(R\) denotes the outer radius of the coiled tubing, \(r\) denotes the inner radius of the coiled tubing, \(h\) denotes the deviation value between any layer and the neutral layer, and \(\sigma_{rh}\) denotes the residual strength of any layer of material.  
The variation trend of the ultimate tensile load strength of each layer element in the section is mapped to the pipe cross-section, as shown in Figure 12. The variation trend of the tensile load at each layer of the cross-section is the same as that of the residual strength of the cross-section. When the pipe is undamaged, the tensile ultimate load of each layer of the cross-section is consistent (yellow line). As damage occurs, the tensile ultimate load of each layer of the section gradually decreases, and the trend transitions from the yellow line to the blue-dashed curve. At this point, \(D_{f}\) is x, \(0<x<80\). Finally, when the degree of damage to the outermost layer reaches 80%, the tensile ultimate load curve transitions from the blue-dashed curve to the red-dashed curve.  
Figure 11: Residual strength of coiled tubing with nonlinear damage.  
Figure 12: Ultimate tensile load change curves.  
Figure 16: Equivalent plastic strain results: (**a**) \(D_{f}\) = 0, (**b**) \(D_{f}\) = 20%, (**c**) \(D_{f}\) = 40%, and (**d**) \(D_{f}\) = 80%.  
The results show that the tensile displacement increases with the increase in damage degree, and the deformation occurs at different positions of the bearing end, with the outermost layer having the largest deformation and the middle layer having the smallest deformation, exhibiting an inward concave characteristic. The results in Figure 16 indicate that when the strength of the neutral layer reaches the yield strength, the plastic strain on the outermost layer of the loading surface is the highest, and the plastic strain on the loading surface decreases from the outside to the inside. The explanation for this phenomenon is the following: combining Equation (9) and Figure 12, the ultimate tensile load of the cross-section shows a decreasing trend from the neutral layer to the outermost layer. As the tensile load gradually increases, the stress in the outermost layer first reaches the yield strength, and plastic strain occurs. As the load continues to increase, plastic strain also begins to appear in each layer. In addition, the stress strength of the middle layer elements is high, but the plastic strain is small. The reason is that for each layer element with approximately the same elongation, the neutral layer element finally exhibits plastic deformation due to its high residual strength.