# The Influence of Axial Compressive Stress and Internal Pressure on a Pipeline Network: A Review
## 2 Axial Compressive Stress
### Poisson's Effect Due to Temperature Changes  
Euler-bar buckling poses a risk, especially in conditions with high pressure and high operational temperature [30]. It occurs in both buried and unburied pipelines [30] and is caused by the frictional restraint of thermal expansion due to the variation in temperature or internal pressure. The types of buckling mechanisms include (1) inward-diamond buckling, where the pipe shows a single inward bulge (kink); (2) inward/outward-diamond buckling, where the pipe shows a single outward bulge with continued deformation resulting in a large outward depression with two small inward depressions; and (3) outward bulge buckling, where the pipe has from one up to four ripples (wrinkles) at the compressive side of the specimen. This occurs in the Arctic and Subarctic regions [22].  
A study by [PERSON] et al. [31] showed that axial compressive stress due to constrained thermal expansion results in the significant decrease in the pipe's load-carrying capacity due to severe local wrinkling. Another study by [PERSON] et al. [20] on X80 steel pipe revealed that interacting corrosion defects are more susceptable to axial compressive stress due to thermal expansion, with up to a 10% decrease in the failure pressure compared to when the pipes were subjected to internal pressure only. The study also noted that the increase in operating temperature (25-125 \({}^{\circ}\)C) caused changes in the stress field, the compressive stress field in the axial direction, which influenced the failure pressure.  
For restrained pipes, a force is required to maintain its original position from its free expanded or contracted position. [PERSON] [19] found that the restrained portion of a pipe would be subjected to axial compressive stresses when there is a moderate increase in transport fluid temperature, \(T_{2}\), compared to the nominal environment temperature, \(T_{1}\). The pipe section undergoes expansion equal to \(\alpha(T_{2}-T_{1})L\) while the hoop tensile stress results in a shrinkage equal to \(vS_{h}L/E\). This shrinkage is governed by Poisson's ratio and is also referred to as a Poisson shrinkage. By subtracting the Poisson's shrinkage from the expansion, the net expansion is given by Equation (1):  
\[\Delta=\alpha(T_{2}-T_{1})L-vS_{h}L/E \tag{1}\]  
The axial stress produced is equivalent to the stress required to compress \(\Delta\) back to the original position. Since \(S_{L}=-E\Delta/L\), Equation (1) can be expressed as Equation (2):  
\[S_{L}=-E\alpha(T_{2}-T_{1})+vS_{h} \tag{2}\]  
For the unrestrained portion of the pipe, it was stated that there were two groups of stress: self-limiting and sustained stress. Self-limiting stress from thermal stress (and other strains) is given by Equation (3), whereas the bending stress is given by Equation (4):  
\[S_{E}=\sqrt{{S_{b}}^{2}+{S_{b}}^{2}} \tag{3}\]  
\[S_{b}=\sqrt{\frac{(i_{i}M_{l})^{2}+(i_{o}M_{o})^{2}}{Z}} \tag{4}\]  
Expansion due to the pressure effect (not pressure stress, which is not included in expansion stress calculations) is significant and should not be ignored. When the pipe is exposed to transport fluid temperature, \(T_{2}\), it undergoes expansion in every direction. For the unrestrained portion of the pipe, the thermal expansion in the longitudinal direction is \(\alpha(T_{2}-T_{1})L\). When subjected to longitudinal pressure, the pipe expands \(0.5S_{h}L/E\) in the longitudinal direction but shrinks somewhat in the diametrical direction. Finally, adding radial pressure (hoop stress), the pipe expands fully in the diametrical direction but shrinks \(0.3S_{h}L/E\) in the longitudinal direction due to the Poisson effect. The net longitudinal expansion, \(\Delta\), is equal to:  
\[\Delta=\alpha(T_{2}-T_{1})L+0.5S_{h}L/E-0.3S_{h}L/E \tag{5}\]  
Since strain, \(\varepsilon=\Delta/L\) we have:  
\[\varepsilon=\alpha(T_{2}-T_{1})+0.2S_{h}/E \tag{6}\]  
The calculated strain, \(\varepsilon\), is in inches per inch unit.