## 4 Numerical Results
### Evolution of Stress in Pipes  
The evolution of the resultant stress along the length of pipe segments due to applied loads and induced deformation is portrayed in the following figures using contour plots of the von Mises stress. The contour plots are captured at various points along the path of the load-displacement (or moment-curvature) curve, which coincide with three of the strategic stages of the load-deformation response, i.e., pre-buckling, limit load, and post-buckling. Figures 4a and 5a represent the axial load-displacement plots of YPT pipes and RHT pipes, respectively, under uniform axial compression. Figures 6a and 7a represent the moment-curvature plots of YPT pipes and RHT pipes, respectively, under uniform bending. The longitudinal distribution of the von Mises stress at the extreme of the compression side of the pipe's cross-section, which is aligned with the bending plane, is plotted in Figures 4b and 5b for YPT pipes and RHT pipes, respectively, under uniform axial compression.  
Figure 3: Comparison of FEA result with experimental result (UGA508) by Ref. [60].  
Figure 2: Meshed area for average strain measurement.  
The longitudinal stress distribution is plotted in Figures 6b and 7b for YPT pipes and RHT pipes, respectively, under uniform bending. Contour plots of the von Mises stress at specified stages of the load-deformation response are respectively presented in Figures 4c and 5c for YPT pipes and RHT pipes under uniform axial compression, while the contour plots for YPT pipes and RHT pipes under uniform bending are presented in Figures 6c and 7c, respectively. The results presented in Figures 4-7 are for DT4 pipes with \(f_{p}=0.4\); where the YPL = 1.0% for the YPT pipes and the PLUS ratio = 0.691 for the RHT pipes. The von Mises contour plots are longitudinally aligned with the longitudinal stress distribution plots such that the left end is the load end and the right end is mid-length cross-section of the pipe.  
The plots presented in Figures 4-7 indicate that the von Mises stress is constant over the entire length at the compression zone of the pipe in the pre-buckling stage (i.e., from the beginning of load application until buckling occurs). Buckling is associated with the formation of a longitudinal wrinkle, and may either precede the limit load, as in Figures 5a, 6a and 7a, or coincide with the limit load, as in Figure 4a. The limit load is typically reached when the von Mises stress in the extreme compression fibers of the pipe reach the yield stress of the pipe material, after which localization of strains and resultant stresses commences at the mid-length region of the pipe, and the remaining portions of the pipe experience a stress relief. Attainment of the limit load is essentially followed by load collapse and softening of the load-deformation response. The above plots indicate that, beyond the limit load, the von Mises stresses in the compression zone evolve such that the stresses around the mid-length cross-section advance towards the UPS, while the stresses in the remaining portions of the pipe continue to decrease.  
Figure 4: Axial load-displacement response and stress evolution in YPT DT4 pipes under uniform axial compression (YPL = 1.0%, \(f_{p}\) = 0.4).  
Figure 5: Axial load-displacement response and stress evolution in RHT DT4 pipes under uniform axial compression (PL/US = 0.691,\(f_{p}\) = 0.4).  
Figure 6: Moment-curvature response and stress evolution in YPT DT4 pipes under uniform bending (YPL = 1.0%,\(f_{p}\) = 0.4).