# Study on the Residual Strength of Nonlinear Fatigue-Damaged Pipeline Structures
## 3 Stress-Strain Analysis of Nonlinear Damaged Pipeline Structure
### Stress-Strain Analysis  
For a coiled tube without nonlinear fatigue damage, the plastic strain initially occurs on the entire inner wall annulus. As the internal pressure increases, the plastic region uniformly expands outward in a circular ring shape along the radial direction. Then, the plastic region expands to the purple-dashed ring and finally reaches the red-dashed ring. At this point, the stress on the outermost element reaches the elastic limit, as shown in Figure 7a. For the nonlinear fatigue-damaged pipe, the plastic strain initially occurs on the outermost inner wall in the vertical direction (the point of tangency between the black dotted line and the inner wall). As the internal pressure increases, the plastic zone expands outward in a straight line along the vertical direction until the stress of the outermost elements reaches the elastic limit, as shown in Figure 7b. Figure 5: Residual strength change curves. Figure 6: Calculation flow of structural strength characteristics of nonlinear fatigue-coiled tubing. Figures 4 and 5 indicate that when the degree of damage to the outermost layer reaches 80%, the nonlinear variation characteristics of the residual strength curve from the neutral layer to the outermost residual strength curve are most obvious. Therefore, the authors selected a pipe with a nonlinear damage degree of 80% to conduct stress-strain analysis on the pipe under the ultimate internal pressure condition. The solution process of ANSYS finite element software (version 2023R1) for pipes with nonuniform cross-sectional damage under internal pressure conditions is as follows:  
First, based on the residual strength model of CT110 coiled tubing, edit the customized APDL in Visual Studio Code software (version 1.85) to establish the material library with different degrees of damage. The hardening models of each layer are bilinear constitutive models with a yielding-to-tensile ratio of 0.8. Part of the customized APDL content is shown in Figure 8a. Figure 7: Plastic strain expansion characteristics of the pipe body: (**a**) expanding trend of plastic strain in coiled tubing without nonlinear fatigue damage and (**b**) expanding trend of plastic strain in coiled tubing with nonlinear fatigue damage. Figure 8: Finite element analysis process: (**a**) part of a customized APDL, (**b**) meshing result, and (**c**) boundary condition and static pressure load of a finite element model. Second, establish the physical model of the pipe in ANSYS software with the sectional dimensions set in Table 2. The axial length of the pipe is 300 mm. By dissecting the physical model into a mesh with elements of size 1.5 mm, the element type is solid185, which is an eight-node hexahedral element. The model generates a total of 145,200 elements and 174,066 nodes. The meshing results are shown in Figure 8b. Subsequently, based on the meshing result of the pipe, import the edited customized APDL with \(\text{D}_{\text{f}}=80\%\) into the ANSYS finite element software and assign material strength to the elements of each layer of the pipe. Finally, fixed constraint boundary conditions are applied to the two axial faces of the pipe, and a static pressure load is applied to the inner wall of the pipe to begin solving. The finite element model is shown in Figure 8c; the blue triangles represent the fixed constraint, and the red lines represent the pressure on the inner wall of the pipe. Set the total duration of internal pressure load loading to 1 s and divide it into 10 sub-steps for loading, with each substep lasting 0.1 s. The full Newton-Raphson iterative method is used to solve the finite element model of the pipe. When the residual force during the iterative solution process is less than the default value of the ANSYS software program, it means that the solution is convergent. Based on this convergent solution, the iterative solution is continued until all solutions are completed and converged.