# Study on the Residual Strength of Nonlinear Fatigue-Damaged Pipeline Structures
## 2 Residual Strength Model of Materials with Fatigue Damage Accumulation
### Residual Strength Model  
First, it is necessary to determine the bending strain of each layer in the coiled tubing cross-section during the repeated bending process, then calculate the fatigue life of each layer with damage according to the strain amplitude, and finally obtain the residual strength model of the material with fatigue damage accumulation. Under the action of bending moment, the coiled tubing begins to deform from the initial straight pipe state (blue straight line) toward the guide frame until it completely fits into the guide frame (black curve). The deformation process is shown in Figure 2.  
Assuming that the bending curvature of the coiled tubing on the drum and the guide frame is the same, the fatigue calculation of the coiled tubing can be superimposed by  
Figure 1: Peak stress response curve of CT110 steel. Figure 2: Schematic diagram of coiled tubing deformation. Miner's law according to the constant strain amplitude. The bending strain of the coiled tubing cross-section in the vertical direction to each layer at the drum and guide frame is:  
\[\epsilon_{bh}=\frac{h}{\frac{d_{c}}{2}+\frac{d^{\prime}}{2}}=\frac{h}{r_{b}} \tag{3}\]  
where \(\epsilon_{bh}\) denotes the bending strain of the vertical direction of the coiled tubing section towards each layer, \(h\) denotes the deviation value between any layer and the neutral layer, \(d_{c}\) denotes the diameter of the coiled tubing, \(d^{\prime}\) denotes the diameter of the drum and guide frame, and \(r_{b}\) denotes the bending radius of the coiled tubing. In particular, the maximum bending strain \(\epsilon_{bmax}\) of coiled tubing is:  
\[\epsilon_{bmax}=\frac{\frac{d_{c}}{2}}{\frac{d_{c}}{2}+\frac{d^{\prime}}{2}}= \frac{\frac{d_{c}}{2}}{r_{b}} \tag{4}\]  
Combined Equations (1) and (4) to obtain the coiled tubing service fatigue life. Although nonlinear fatigue damage accumulation models consider the loading sequence of different stress levels and overcome the shortcomings of linear damage accumulation models, due to their complex forms and unclear meanings in some physical models, it is difficult to determine whether the structure has experienced fatigue failure. In addition, it has been assumed in the article that the bending radius of the pipe when bent on the drum and guide frame is the same, so there is no need to consider the loading sequence of the load. Based on the above considerations, we choose [PERSON]'s linear damage accumulation model as the damage accumulation model for the pipe. For the influence condition of constant strain amplitude, according to [PERSON]'s linear damage theory, the weight of each cycle on the damage to coiled tubing material is:  
\[\Delta D_{f}=\frac{1}{N_{f}} \tag{5}\]  
where \(\Delta D_{f}\) denotes the damage amount of a single fatigue cycle and \(N_{f}\) denotes the fatigue life. The number of fatigue cycles \(n\) can be determined from the damage amount of a single fatigue cycle:  
\[n=\frac{D_{f}}{\Delta D_{f}} \tag{6}\]  
where \(D_{f}\) denotes the degree of damage to the coiled tubing. Fitting the peak stress response curve in Figure 1 using the Belehradek function:  
\[\sigma_{r}=k(D_{f}+\Delta C)^{d} \tag{7}\]  
where \(\sigma_{r}\) denotes the residual strength of the material, \(D_{f}\) denotes the damage degree of material, \(k\) denotes the function fitting coefficient, \(d\) denotes the function fitting exponent, and \(\Delta C\) denotes the corrected value. Combined with Equations (1), (3), and (4)-(7), the residual strength model of any layer of material with loading times can be established:  
\[\left\{ \begin{array}{l}\epsilon_{bh}=\frac{h}{r_{b}}\\ \epsilon_{a}=\frac{\sigma_{r}^{\prime}}{E}\Big{(}2N_{fh}\Big{)}^{b}+\epsilon_ {f}{}^{\prime}\Big{(}2N_{fh}\Big{)}^{c}\\ \Delta D_{fh}=\frac{1}{N_{fh}}\\ D_{f}=n\Delta D_{f}\\ \sigma_{rh}=k(D_{fh}+\Delta C)^{d}\end{array} \right.