# Study on the Residual Strength of Nonlinear Fatigue-Damaged Pipeline Structures
## 4 Results and Discussion
### Theoretical Verification of Ultimate Internal Pressure Strength  
According to the maximum distortion-energy theory in material mechanics and the Lame formula for thick-walled tubing, the relationship between the theoretical ultimate internal pressure value \(P^{\prime}\) and the damage degree \(D_{f}\) of coiled tubing with nonlinear fatigue damage is deduced:  
\[P^{\prime}=685.719(D_{f}+0.16)^{-0.0551}\frac{K^{2}-1}{\sqrt{3\frac{R^{4}}{b_{ out}^{4}}+1}} \tag{10}\]  
where \(K\) denotes the ratio of the outer and inner diameters of the coiled tubing, \(K=R/r\); \(R\) denotes the outer radius of the coiled tubing; \(r\) denotes the inner radius of the coiled tubing; \(h_{out}\) denotes the deviation value between the outermost elements and the neutral layer; and \(D_{f}\) denotes the degree of damage to the outermost elements.  
The theoretical values of coiled tubing ultimate pressure under different damage degrees were calculated and compared with the finite element simulation values, as shown in Figure 19. The theoretical pressure-bearing value has the same variation trend as the finite element simulation value, and the theoretical value is slightly higher than the simulated value. The reasons are:  
Figure 19: Ultimate internal pressure strength of coiled tubing with nonlinear damage.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline a\({}_{1}\) & 160.774 \\ b\({}_{1}\) & \(-0.0543\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fitting coefficients of bearing strength and damage degree function equation.  
Figure 18: Relationship between ultimate internal pressure strength and damage degree of coiled tubing with nonlinear damage.  
(1) The grid size divided in the finite element software is slightly larger, causing some elements in the outermost layer to cross the boundary line between the outermost edge and the second outer edge, so the theoretical value is slightly higher than the simulated value;  
(2) The elements in the finite element software are divided from the inner wall to the outer wall in a circular manner, and the \"stepless assignment in the vertical direction\" mentioned in Section 3.1 is layered according to rectangular strips, so the theoretical value is slightly higher than the simulated value.  
Due to nonlinear damage, it is difficult to calculate the residual strength of the entire structure at various locations. Therefore, the overall residual strength is usually replaced by the residual strength of the outermost layer or the residual strength of the middle layer of the upper pipe in the neutral layer in engineering. Defining \"uniform damage 1\" and \"uniform damage 2\" as follows: if we consider the degree of outermost damage as the overall degree of damage, which is called \"uniform damage 1\", then the residual strength of the damaged pipe is replaced by the residual strength of the outermost layer; if we consider the damage degree of the middle layer of the upper pipe above the neutral layer as the overall degree of damage, which is \"uniform damage 2\", then the residual strength of the damaged pipe is replaced by the residual strength of the middle layer of the upper pipe above the neutral layer.  
Control the internal pressure to achieve the ultimate strength of the outermost elements of the structure subjected to nonlinear damage, uniform damage 1, and uniform damage 2. The maximum radial deformations of the above three types of damaged pipes are shown in Figure 20.  
The results show that the maximum radial deformation of nonlinear damage pipeline structures is smaller than that of uniform damage pipeline structures, showing a rapid decrease before \(D_{f}=20\%\) and then showing a quasi-linear slow decrease with increasing \(D_{f}\).