# Study on the Residual Strength of Nonlinear Fatigue-Damaged Pipeline Structures
## 3 Stress-Strain Analysis of Nonlinear Damaged Pipeline Structure  
### Research Methods and Content  
During pure bending deformation of pipe, the bending strain and bending stress on either side of the neutral layer exhibit symmetry with respect to the neutral layer, and the upper part is selected as the research object. During the \"bending-straightening\" alternating deformation process, the bending strain of each layer of the upper part of the neutral layer is different, resulting in nonuniform damage to the cross-section, so the residual strength of each layer of the cross-section is reduced to different degrees. According to Equation (3), the bending strain of each part can be obtained, showing a linear increasing trend from the neutral layer to the outside, and the trend is mapped to the pipe as shown in Figure 3.  
The section size of CT110 coiled tubing is created based on the 2-7/8\" Inch coiled tubing used in the actual oil field, and the axial length is 300 mm. Dimensions and material properties are shown in Table 2.  
In order to simplify the analysis, the following basic assumptions are put forward for the coiled tubing that has been run into the wellbore and has not yet been operated:  
(1) The ellipticity of the coiled tubing section remains unchanged during the alternating deformation process. (2) During the repeated bending process, there is no wear between the coiled tubing and the drum or guide frame. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Fatigue strength coefficient & \(\sigma_{f}^{f}\) & 17,407 \\ Fatigue strength exponent & \(b\) & \(-0.51453\) \\ Fatigue ductility coefficient & \(\varepsilon_{f}^{f}\) & 0.04965 \\ Fatigue ductility exponent & \(c\) & \(-0.029159\) \\ Function fitting coefficient & \(k\) & 685.719 \\ Function fitting exponent & \(d\) & \(-0.055\) \\ Corrected value & \(\Delta C\) & 0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Residual strength model fitting coefficients. Figure 3: Variation trend of bending strain in coiled tubing section. To study the strength characteristics of continuous solid structures with nonuniform section damage, it is necessary to discretize the pipe in the upper part of the neutral layer. The residual strength model established in Section 2 with fatigue damage accumulation is used for stepless assignment in the vertical direction of the pipe. The damage accumulation is taken at an interval of 2% within the first 10% and at an interval of 5% between the last 10% and 80%. First, the bending strain \(\varepsilon_{bi}\) of each layer of the cross-section is calculated according to Equation (4), and the fatigue life \(D_{fi}\) of each layer can be obtained by bringing \(\varepsilon_{bi}\) into Equation (1). Combining the fatigue life \(D_{f,out}\) of the outermost layer with Equations (5) and (6), the maximum number of repeated bendings \(n\) accumulated with damage and the single fatigue damage weight \(\Delta D_{fi}\) of each layer can be calculated. Then, combining with \(n\) and \(\Delta D_{fi}\), the damage accumulation \(D_{fi}\) of each layer of the cross-section is obtained. Finally, the residual strength of each layer of the pipe cross-section accumulated with damage can be calculated by bringing the \(D_{fi}\) into Equation (7), as shown in Figure 4. The results show that when the damage occurs, the strength of each layer in the cross-section decreases to different degrees, showing that the residual strength of each layer decreases more with the increase in distance from the neutral layer, and the strength of the outermost layer decreases the most, with a reduction rate of 13.3%. With the accumulation of damage, the decreasing trend of the strength of each layer in the cross-section changes from rapid decreasing to slow decreasing. The variation trend of the residual strength of each layer element in the section is mapped to the pipe section, as shown in Figure 5, where \(\sigma_{0}\) is the residual strength of the outermost layer.