Due to their exceptional structural integrity, steel pipelines are the main component for oil and gas transmission. However, these pipelines are often affected by corrosion, despite corrosion protection, because of harsh working conditions. In addition to corrosion defects, pipelines are often subjected to multiple external loads. The combination of corrosion defects and external loads can significantly reduce the failure pressure, resulting in various failure behaviors. This reduction in failure pressure is especially critical in pipe bends as they are the weakest link in a pipeline.
The rising demand for raw materials has compelled industries to exploit resources in remote locations 
. As such, the transportation of fluid substances from such locations is facilitated by pipeline networks which have been well-established as the safest and most cost-effective mode of transportation in the industry 
. Steel pipelines have been used for decades due to their outstanding structural integrity. Underground steel pipes are regarded as the most efficient transport medium for the long-distance transmission of fluids at high temperatures and pressures 
. However, depending on the soil conditions, some pipelines are constructed aboveground. As pipelines pass through different terrains, the sophisticated network of pipes is designed to channel fluid in different directions. To achieve this, pipe bends are used at specified locations to alter the flow direction of the flowing fluid.
The influence of axial compressive stress on a pipe has proven to be the most significant among other external loads 
. This condition, coupled with pipe corrosion, which is the most common and detrimental problem with steel pipes, results in a significant loss of pipe integrity 
. Generally, steel pipelines have high yield strengths, leading to high allowable stresses. It is common to utilize pipes with yield strengths as high as 80% of the ultimate tensile strength. Despite the outstanding mechanical properties of steel pipelines, they experience various types of corrosion. The continued usage of these corroded pipes may lead to an overall reduction in the pipeline structural integrity, resulting in reduced failure pressure. In particular, pipe bends are most susceptible to failure, and are often regarded as the weakest link, as they experience unbalanced thrust forces due to the differences in the surface area of the intrados and extrados. The highest stress concentrations generally occur at the affected area’s deepest point 
The electrochemical reactions that occur in the presence of electrolytes cause corrosion in pipelines 
. Since this process is of an electrochemical nature, it also favors the detection and mitigation of the degradation of the pipe wall by current and voltage monitoring, which is associated with the corrosion rate 
. Both external and internal factors govern the corrosion rate of a pipeline 
. An increase in the electrolyte concentration, coupled with harsh operating conditions, results in an increased rate of corrosion.
For single corrosion defects, pipe failure is said to occur when disruption occurs at the defect region and the pressurized fluid begins to tear or crack through the remaining thin ligament of the pipe. For multiple interacting corrosion defects, the failure occurs when stress and strain disruptions occur on at least one of the corrosion defects. Each corrosion defect imposes its stress and strain disturbances. The regions of expansion of stress and strain disturbances are known as regions of influence of corrosion defects. Overlapping regions of influence due to adjacent interacting corrosion defects may cause a decrease in the failure pressure 
2. Axial Compressive Stress
A steel pipeline may be subjected to various external loads such as hydrodynamics stresses, axial stresses, and external collapse pressures. Past studies 
have shown that external loads cause a higher reduction in the failure pressure of a pipe. Axial compressive stress occurs when a pipeline moves 
, especially in buried pipes, as large portions of the pipes are only restrained by soil friction, whereas above-ground pipelines are generally restrained by anchors and guides. The increase in the hydrocarbon demand also encourages hydrocarbon to be explored in areas with unpredictable geotechnical conditions, such as the Subarctic and the Arctic regions. These regions often see active landslides and geotechnical movements, which may cause axial compressive stresses on these pipelines 
Pipe bends, the weakest link of the piping system, are subjected to the Bourdon effect. The bending moment results in the opening of the pipe elbow, which subjects the pipe to ovalization. As a result, the pipe experiences axial compressive stress and tensile stress. The former has a more significant impact on the pipeline’s integrity 
2.1. Poisson’s Effect Due to Temperature Changes
Euler-bar buckling poses a risk, especially in conditions with high pressure and high operational temperature 
. It occurs in both buried and unburied pipelines 
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 
A study by Roy et al. 
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 Soares et al. 
on X80 steel pipe revealed that interacting corrosion defects are more suspectable 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 °C) caused changes in the stress field, the compressive stress field in the axial direction, which influenced the failure pressure.
2.2. Geological Movements
For thin pipes, radial and shear stresses are neglected, as circumferential and axial stresses are the main stresses affecting the pipe 
. Ground movement due to earthquakes (e.g., active reverse fault), landslides, and frost heaves induce external axial force on the pipelines. Generally, a steel pipeline in direct tension due to fault rupture can safely accommodate a larger fault offset value compared to when it is strained in direct compression 
. Although good engineering practice is not to lay pipelines on volatile soils, nevertheless, adjacent soil movements affect a pipeline. Often, these soil movements are relatively small, but the effects may be large enough to affect the pipeline performance 
Under compressive geological movement, pipeline behavior would be dominated by global and local buckling 
. Axial bending of the corroded pipe region results from a differential settlement, or in the Arctic regions, the freeze-thaw action of permafrost regions coupled with axial compressive loading from the constraint due to the thermal expansion that exists when transporting high-temperature petroleum products 
. A report by the Transportation Safety Board of Canada 
revealed that some steel pipeline failures caused by axial compressive stress are due to slope movements and landslides. An example of such failure is one in 1997 when a pipeline in British Columbia, Canada ruptured, as it was subjected to excessive axial compressive stress due to soil movement in the axial direction 
. The external loads due to soil often pose several types of stresses on pipes, namely radial, shear, circumferential, and axial stresses.
2.3. Bourdon Effect Due to Internal Pressure
As the fluid travels across the pipe component, pipe bends are subjected to internal pressure. The difference in the surface area of the pipe bend at the intrados and extrados results in unbalanced thrust forces . These forces contribute to large stress concentrations on the wall of the pipe bend, resulting in the deformation of the cross-section of the pipe bend . In the industry, the CSA-Z662 standard is used to assess the influence of internal pressure on a pipe bend. The Tresca stress distribution is one of the criteria used for assessment. According to the Tresca criterion, plastic deformation occurs when the maximum shear stress at a point of interest reaches the maximum stress in a uniaxial tension material at yield.
3. Pipeline Residual Strength Assessment Methods
Chen et al. 
showed that corrosion defects subjected to axial compressive stress result in noticeable failure pressure differences. Therefore, simplifications made to corrosion assessment methods whereby axial compressive stress is considered to have little or no effect on pipe failure are inaccurate. For instance, Netto et al. 
showed that ASME B31G did not consider the effects of end loads (i.e., tensile loads during close-ended tests), which would result in conservative failure pressure predictions.
The exclusion of axial compressive stress influence in conventional standards and codes for the corrosion assessment methods of pipe bends also results in a highly conservative and inaccurate evaluation. Existing assessment methods only incorporate internal pressure during corrosion assessments. In addition, the research on developing corrosion assessment methods for pipe bends subjected to internal pressure and axial compressive stress is minimal. Hence, for such cases, FEM is used. In recent studies, FEM has been incorporated with ANN to develop new assessment methods that are highly accurate for corroded pipe components subjected to internal pressure and axial compressive stress 
DNV-RP-F101 (DNV) is a corrosion assessment method that was developed by Det Norske Veritas. DNV is the least conservative method for predicting the burst pressure of corroded pipelines 
for single, interacting, and complex-shaped defects subjected to internal pressure only, and single defects subjected to both internal pressure and longitudinal compressive stress. DNV states that the source of the longitudinal compressive stress may be due to longitudinal loads, bending loads, temperature loads, etc. DNV provides two alternative methods for determining the burst pressure of corroded pipes.
3.2. Finite Element Methods
Timoshenko described the elastic-plastic buckling of pipes under external pressure 
. In recent years, non-linear finite element analysis has been used as an accurate tool to predict the buckling collapse capacity of pipes under external pressure, bending, and axial force 
Vitali and Bruschi 
found that finite element methods (FEMs) such as ABAQUS and ANSYS gave reliable results compared to experimental results, especially in terms of predicting buckling modes and the bending-moment vs. curvature relations. This is possible with the proper application of FEM settings, such as ensuring large deformation effects, material non-linearities, strain anisotropy, and kinematic hardening 
3.3. Artificial Neural Networks
An Artificial Neural Network (ANN) is a powerful prediction tool in machine learning that can result in highly accurate predictions 
. They are nonlinear systems modelled efficiently using an interconnected complex assembly of nodes that utilize supervised learning algorithms to learn from a training database by recognizing and inferring from patterns without requiring an explicit set of instructions 
. Over the past decade, the use of ANN in pipe integrity assessment has become increasingly popular.
4. Effects of Axial Compressive Stress
4.1. Detrimental Effects
Axial compressive stress due to thermal expansion results in Euler-bar buckling. The peak bending moment corresponds to the maximum point of the bending moment-curvature relationship. This peak is triggered by the development of wrinkling and bulging of the pipe at the compressive side of the specimen 
. Yoosef-Ghodsi et al. 
found that the initiation of wrinkling always coincides with the maximum bending moment. The initiation of wrinkling is where the non-uniform deformation begins. Non-uniform deformation is determined when the strain in the region of wrinkle increases faster than the strain away from the wrinkle. The maximum bending moment decreases with the decrease in the material hardening ability and an increase in the outer diameter to thickness ratio.
4.2. Beneficial Effects
Chouchaoui and Pick 
conducted comprehensive studies on corroded pipes and compared the effects of the close-ended conditions and open-ended conditions. The closed-ended condition is an application of longitudinal stress on the pipe due to hoop stress, while the open-ended condition takes into account the pure circumferential forces acting on the pipe without any longitudinal stress. Their results revealed that closed-ended conditions increase the failure pressure by delaying the plastic collapse of the corrosion defect region, providing slight support for the defect ligament through the surrounding material. Based on the von Mises stress analysis, it was revealed that for close-ended conditions, the biaxial stresses created more dilatational work in comparison to deviatoric stress. The biaxial stresses delayed the point for plastic collapse to occur. This, however, caused the close-ended conditions to undergo higher deformations than the open-ended conditions. Longitudinal compression caused by bending, on the other hand, caused a decrease in the failure pressure.
When a corroded pipeline is subjected to internal pressure and axial compressive stress, the failure pressure is adversely affected, resulting in failure behaviors such as bending, buckling, wrinkling, ovalization, bursting/rupture, and axial collapse. Nevertheless, axial compressive stress on the corroded pipe may benefit the failure pressure as it provides slight support to the defect ligament through the surrounding material.