# An Artificial Neural Network-Based Equation for Predicting the Remaining Strength of Mid-to-High Strength Pipelines with a Single Corrosion Defect

^{*}

## Abstract

**:**

^{2}) of 0.9975. A parametric study was subsequently performed using the equation to determine the effects of material property, defect depth, defect length, and longitudinal compressive stress on the remaining strength of pipelines with a single corrosion defect. Defect depth reduced the failure pressure by more than 65% on average, longitudinal compressive stress by more than 20% on average, and defect length by more than 21% on average.

## 1. Introduction

## 2. Materials and Methods

## 3. Methodology

#### 3.1. Finite Element Method for X65 Pipeline

_{e}) of 300 mm, a wall thickness (t) of 10 mm, and a pipe length (L) of 2000 mm. A quarter of the symmetrical pipe was modelled with a corrosion defect to reduce the needed computational resources. Figure 1 shows the dimension of the quarter pipe model.

#### 3.2. Validation of Finite Element Method

#### 3.3. Artificial Neural Network

^{−5}. The ANN development framework is shown in Figure 5.

^{2}) in Equation (1), mean squared error (MSE) in Equation (2), and mean absolute error (MAE) in Equation (3).

^{2}or the squared correlation coefficient is the evaluation of goodness-of-fit for the predicted value against actual value, where an R

^{2}value of 1.00 corresponds to a perfect fit. The MSE is the sum of squared difference between the predictions and actual values. The MAE is the mean absolute error between predictions and actual values, which measures the accuracy of the predictions.

## 4. Results

#### 4.1. Finite Element Analysis of API 5L X65 Pipeline with Single Corrosion Defect

#### 4.2. Development of an Artificial Neural Network-Based Equation

^{2}value of the assessment equations when tested against an unseen dataset was 0.9975, which indicates good correlation, as shown in Figure 8. The assessment equation had an MSE of 0.000126 and an MAE of 0.00699.

#### 4.3. Parametric Study Using the Artificial Neural Network-Based Equation

## 5. Conclusions

^{2}value of 0.9975, an MSE of 0.000126, and an MAE of 0.00699. The assessment equation was subsequently used to perform a comprehensive parametric study to investigate the effect of material property, defect depth, defect length, and longitudinal compressive stress on the remaining strength of pipelines with single corrosion defects. In conclusion:

- The higher the strength of the pipeline, the higher its failure pressure when subjected to combined loadings. Here, X80 generally had the highest failure pressure, followed by X65 and X52.
- Defect depth had the most impact on reductions in failure pressure (average reductions of −65.90% and −66.91%), while longitudinal compressive stress (average reductions of −23.99% and −20.79%) and defect length (average reductions of −21.94% and −25.00%) had similar impacts on failure pressure.
- Across all three materials, the trend for failure pressure against defect depth, defect length, and longitudinal compressive stress exhibited similar patterns.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 4.**True stress–strain curve of API 5L X65 pipeline [25].

**Figure 8.**Regression plot of predicted normalised failure pressure by the assessment equations and FEA results for an unseen dataset.

**Figure 9.**Failure pressure against normalised defect depth, with 0.0 and 0.7 normalised longitudinal compressive stresses for X52, X65, and X80.

**Figure 10.**Failure pressure against normalised defect depth with 0.2 and 1.2 normalised defect lengths for X52, X65, and X80.

**Figure 11.**Failure pressure against normalised longitudinal compressive stress with 0.2 and 0.8 normalised defect depths for X52, X65, and X80.

**Figure 12.**Failure pressure against normalised longitudinal compressive stress with 0.2 and 1.2 normalised defect lengths for X52, X65, and X80.

**Figure 13.**Failure pressure against normalised defect length with 0.2 and 0.8 normalised defect depths for X52, X65, and X80.

**Figure 14.**Failure pressure against normalised defect length with 0.0 and 0.7 normalised longitudinal compressive stresses for X52, X65, and X80.

Scheme | Corrosion Type | Loads | Material | ANN Algorithm |
---|---|---|---|---|

Silva et al. (2007) [16] | Interacting defects (longitudinally and circumferentially aligned) | Internal pressure | X52 | Feedforward neural network |

Xu et al. (2017) [17] | Single and interacting defects (longitudinally and circumferentially aligned) | Internal pressure | X80 | Feedforward neural network |

Lu and Liang (2021) [18] | Single defects | Internal pressure and axial compressive load | X65, X70, X80 | Not specified |

Parameters | Range |
---|---|

Defect Depth, d/t | 0.2, 0.4, 0.5, 0.6, 0.8 |

Defect Length, l/D | 0.2, 0.4, 0.8, 1.2, 1.8 |

Defect Width, w/t | 2, 6, 10, 14, 18 |

$\mathrm{Longitudinal}\text{}\mathrm{Compressive}\text{}\mathrm{Stress},\phantom{\rule{0ex}{0ex}}{\sigma}_{c}/{\sigma}_{y}$ | 0.2, 0.4, 0.5, 0.6, 0.8, 1.0 |

Pipe Body—API 5L X65 (SOLID185) | Pipe End Cap—Rigid Body (SOLID186) | |
---|---|---|

Modulus of Elasticity, E | 210 GPa | 210 TPa |

Poisson’s ratio, v | 0.3 | 0.3 |

$\mathrm{Yield}\text{}\mathrm{Strength},\text{}{\sigma}_{y}$ | 464 MPa | - |

$\mathrm{Ultimate}\text{}\mathrm{tensile}\text{}\mathrm{strength},\text{}{\sigma}_{u}$ | 563 MPa | - |

$\mathrm{True}\text{}\mathrm{ultimate}\text{}\mathrm{tensile}\text{}\mathrm{strength},\text{}{\sigma}_{u}*$ | 629 MPa | - |

**Table 4.**Validation of FEM results against burst test for pipes only subjected to internal pressure and combined loadings of internal pressure and longitudinal compressive stress.

Specimen | Failure Pressure from Burst Tests (MPa) | Failure Pressure Predicted in FEM (MPa) | Absolute Percentage Difference (%) |
---|---|---|---|

LD | 19.8 | 20.1 | 1.50 |

LF | 15.0 | 15.5 | 3.67 |

Test 5 | 28.6 | 29.2 | 2.10 |

Test 6 | 28.7 | 29.6 | 3.14 |

**Table 5.**FEA of corroded X65 pipeline with single corrosion defect subjected to internal pressure and longitudinal compressive stress.

Defect Parameters | External Load | Normalised Failure Pressure | Defect Parameters | External Load | Normalised Failure Pressure | ||||
---|---|---|---|---|---|---|---|---|---|

d/t | l/D | w/t | ${\mathit{\sigma}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathit{y}}$ | d/t | l/D | w/t | ${\mathit{\sigma}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathit{y}}$ | ||

0.2 | 0.8 | 10 | - | 0.82 | 0.5 | 0.8 | 6 | 0.5 | 0.57 |

0.4 | 0.8 | 10 | - | 0.68 | 0.5 | 0.8 | 14 | 0.5 | 0.54 |

0.5 | 0.8 | 10 | - | 0.59 | 0.5 | 0.8 | 18 | 0.5 | 0.53 |

0.6 | 0.8 | 10 | - | 0.49 | 0.5 | 0.8 | 10 | 0.2 | 0.58 |

0.8 | 0.8 | 10 | - | 0.28 | 0.5 | 0.8 | 10 | 0.4 | 0.57 |

0.5 | 0.2 | 10 | - | 0.77 | 0.5 | 0.8 | 10 | 0.6 | 0.53 |

0.5 | 0.4 | 10 | - | 0.68 | 0.5 | 0.8 | 10 | 0.7 | 0.46 |

0.5 | 1.2 | 10 | - | 0.55 | 0.5 | 0.8 | 10 | 0.8 | 0.37 |

0.5 | 1.8 | 10 | - | 0.53 | 0.5 | 0.8 | 10 | 0.9 | 0.2 |

0.5 | 0.8 | 2 | - | 0.57 | 0.5 | 0.8 | 10 | 1.0 | 0.17 |

0.5 | 0.8 | 6 | - | 0.60 | 0.2 | 0.8 | 10 | 0.7 | 0.54 |

0.5 | 0.8 | 14 | - | 0.58 | 0.4 | 0.8 | 10 | 0.7 | 0.52 |

0.5 | 0.8 | 18 | - | 0.57 | 0.6 | 0.8 | 10 | 0.7 | 0.40 |

0.2 | 0.8 | 10 | 0.5 | 0.75 | 0.8 | 0.8 | 10 | 0.7 | 0.22 |

0.4 | 0.8 | 10 | 0.5 | 0.64 | 0.5 | 0.2 | 10 | 0.7 | 0.56 |

0.5 | 0.8 | 10 | 0.5 | 0.56 | 0.5 | 0.4 | 10 | 0.7 | 0.50 |

0.6 | 0.8 | 10 | 0.5 | 0.46 | 0.5 | 1.2 | 10 | 0.7 | 0.46 |

0.8 | 0.8 | 10 | 0.5 | 0.26 | 0.5 | 1.8 | 10 | 0.7 | 0.45 |

0.5 | 0.2 | 10 | 0.5 | 0.69 | 0.5 | 0.8 | 2 | 0.7 | 0.47 |

0.5 | 0.4 | 10 | 0.5 | 0.62 | 0.5 | 0.8 | 6 | 0.7 | 0.47 |

0.5 | 1.2 | 10 | 0.5 | 0.53 | 0.5 | 0.8 | 14 | 0.7 | 0.45 |

0.5 | 1.8 | 10 | 0.5 | 0.52 | 0.5 | 0.8 | 18 | 0.7 | 0.45 |

0.5 | 0.8 | 2 | 0.5 | 0.56 |

d/t | l/D | ${\mathit{\sigma}}_{\mathit{c}}\mathbf{/}{\mathit{\sigma}}_{\mathit{y}}$ | ||||
---|---|---|---|---|---|---|

0.2 | 0.4 | 0.5 | 0.6 | 0.7 | ||

0.2 | 0.2 | 0.88 | 0.85 | 0.79 | 0.74 | 0.61 |

0.4 | 0.85 | 0.82 | 0.77 | 0.70 | 0.59 | |

0.6 | 0.84 | 0.79 | 0.75 | 0.68 | 0.56 | |

0.8 | 0.83 | 0.78 | 0.74 | 0.66 | 0.54 | |

1.2 | 0.81 | 0.77 | 0.73 | 0.65 | 0.53 | |

0.4 | 0.2 | 0.80 | 0.76 | 0.71 | 0.67 | 0.59 |

0.4 | 0.75 | 0.71 | 0.68 | 0.62 | 0.54 | |

0.6 | 0.70 | 0.68 | 0.65 | 0.61 | 0.53 | |

0.8 | 0.68 | 0.66 | 0.64 | 0.60 | 0.52 | |

1.2 | 0.65 | 0.63 | 0.62 | 0.59 | 0.52 | |

0.5 | 0.2 | 0.77 | 0.73 | 0.69 | 0.64 | 0.56 |

0.4 | 0.67 | 0.65 | 0.62 | 0.58 | 0.50 | |

0.6 | 0.61 | 0.60 | 0.58 | 0.54 | 0.47 | |

0.8 | 0.58 | 0.57 | 0.56 | 0.53 | 0.46 | |

1.2 | 0.55 | 0.54 | 0.53 | 0.52 | 0.46 | |

0.6 | 0.2 | 0.72 | 0.69 | 0.66 | 0.61 | 0.53 |

0.4 | 0.59 | 0.57 | 0.55 | 0.52 | 0.45 | |

0.6 | 0.53 | 0.51 | 0.49 | 0.47 | 0.41 | |

0.8 | 0.49 | 0.47 | 0.46 | 0.45 | 0.40 | |

1.2 | 0.45 | 0.45 | 0.44 | 0.43 | 0.39 | |

0.8 | 0.2 | 0.58 | 0.56 | 0.55 | 0.52 | 0.44 |

0.4 | 0.39 | 0.37 | 0.36 | 0.34 | 0.28 | |

0.6 | 0.32 | 0.30 | 0.29 | 0.28 | 0.24 | |

0.8 | 0.28 | 0.27 | 0.26 | 0.25 | 0.22 | |

1.2 | 0.25 | 0.24 | 0.24 | 0.23 | 0.21 |

**Table 7.**Normalised failure pressure prediction using FEA and prediction based on the assessment equations of an unseen dataset.

Parameters | Normalised Failure Pressure | Difference | ||||
---|---|---|---|---|---|---|

UTS | d/t | l/D | ${\mathit{\sigma}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathit{y}}$ | FEA | New Equations | % |

612.0 | 0.10 | 0.7 | 0.30 | 0.82 | 0.8194 | −0.07 |

612.0 | 0.70 | 0.7 | 0.35 | 0.38 | 0.3766 | −0.88 |

612.0 | 0.80 | 0.7 | 0.50 | 0.27 | 0.2680 | −0.76 |

612.0 | 0.80 | 0.2 | 0.50 | 0.51 | 0.5078 | −0.44 |

612.0 | 0.55 | 0.5 | 0.35 | 0.55 | 0.5352 | −2.69 |

612.0 | 0.20 | 0.5 | 0.32 | 0.78 | 0.7814 | 0.18 |

612.0 | 0.70 | 0.5 | 0.25 | 0.43 | 0.4211 | −2.07 |

612.0 | 0.35 | 0.7 | 0.60 | 0.57 | 0.5836 | 2.39 |

612.0 | 0.80 | 0.3 | 0.35 | 0.43 | 0.4355 | 1.29 |

612.0 | 0.10 | 0.3 | 0.60 | 0.75 | 0.7391 | −1.46 |

629.0 | 0.70 | 0.5 | 0.25 | 0.46 | 0.4598 | −0.05 |

629.0 | 0.10 | 0.3 | 0.30 | 0.90 | 0.8597 | −4.48 |

629.0 | 0.45 | 1.1 | 0.45 | 0.59 | 0.5952 | 0.88 |

629.0 | 0.80 | 0.7 | 0.50 | 0.28 | 0.2767 | −1.18 |

629.0 | 0.70 | 0.7 | 0.35 | 0.40 | 0.4015 | 0.39 |

629.0 | 0.80 | 0.3 | 0.60 | 0.42 | 0.4208 | 0.19 |

629.0 | 0.80 | 0.7 | 0.25 | 0.30 | 0.3037 | 1.24 |

629.0 | 0.80 | 1.1 | 0.50 | 0.25 | 0.2342 | −6.33 |

629.0 | 0.30 | 0.3 | 0.30 | 0.80 | 0.8024 | 0.29 |

629.0 | 0.80 | 1.1 | 0.25 | 0.26 | 0.2595 | −0.21 |

718.2 | 0.10 | 0.7 | 0.30 | 0.93 | 0.9320 | 0.22 |

718.2 | 0.30 | 0.3 | 0.30 | 0.86 | 0.8603 | 0.04 |

718.2 | 0.55 | 0.3 | 0.60 | 0.62 | 0.6188 | −0.19 |

718.2 | 0.35 | 1.1 | 0.35 | 0.70 | 0.6970 | −0.43 |

718.2 | 0.80 | 0.3 | 0.35 | 0.45 | 0.4264 | −5.25 |

718.2 | 0.55 | 0.5 | 0.35 | 0.58 | 0.5831 | 0.53 |

718.2 | 0.80 | 0.7 | 0.60 | 0.25 | 0.2545 | 1.79 |

718.2 | 0.10 | 0.9 | 0.30 | 0.93 | 0.9246 | −0.58 |

718.2 | 0.45 | 1.1 | 0.45 | 0.59 | 0.5965 | 1.10 |

718.2 | 0.70 | 0.5 | 0.25 | 0.45 | 0.4271 | −5.09 |

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**MDPI and ACS Style**

Lo, M.; Vijaya Kumar, S.D.; Karuppanan, S.; Ovinis, M.
An Artificial Neural Network-Based Equation for Predicting the Remaining Strength of Mid-to-High Strength Pipelines with a Single Corrosion Defect. *Appl. Sci.* **2022**, *12*, 1722.
https://doi.org/10.3390/app12031722

**AMA Style**

Lo M, Vijaya Kumar SD, Karuppanan S, Ovinis M.
An Artificial Neural Network-Based Equation for Predicting the Remaining Strength of Mid-to-High Strength Pipelines with a Single Corrosion Defect. *Applied Sciences*. 2022; 12(3):1722.
https://doi.org/10.3390/app12031722

**Chicago/Turabian Style**

Lo, Michael, Suria Devi Vijaya Kumar, Saravanan Karuppanan, and Mark Ovinis.
2022. "An Artificial Neural Network-Based Equation for Predicting the Remaining Strength of Mid-to-High Strength Pipelines with a Single Corrosion Defect" *Applied Sciences* 12, no. 3: 1722.
https://doi.org/10.3390/app12031722