# An Empirical Equation for Failure Pressure Prediction of High Toughness Pipeline with Interacting Corrosion Defects Subjected to Combined Loadings Based on Artificial Neural Network

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## Abstract

**:**

^{2}value of 0.99. Extensive parametric study was subsequently conducted to determine the effects of defect spacing, defect length, defect depth, and axial compressive stress on the failure pressure of the pipe. The results of the empirical equation are comparable to the results from numerical methods for the pipes and loadings considered in this study.

## 1. Introduction

#### 1.1. Overview of Pipelines in the Oil and Gas Industry

#### 1.2. Pipeline Integrity Assessment Methods

_{UTS}, rather than the true ultimate tensile strength, σ×

_{UTS}, of the material [9], since the rupture point of a material is represented by the true ultimate tensile strength of the material [2].

- The DNV code does not incorporate axial compressive stress for the failure pressure assessment of interacting corrosion defects.
- The DNV code results in conservative failure pressure predictions.
- The DNV code results in inaccurate failure pressure predictions for high toughness pipes.

#### 1.3. Finite Element Method (FEM) for Pipeline Failure Pressure Prediction

#### 1.4. Artificial Neural Network (ANN) as a Pipeline Failure Pressure Assessment Method

## 2. Methodology

#### 2.1. Overview of Geometric Parameters, Assessment Factors, and Assumptions

#### 2.2. Development of the Finite Element Method

#### 2.3. Development of the Artificial Neural Network

^{2}) value of the ANN model was carried out.

^{2}value of 0.99, with a minimum number of hidden layers and neurons. A total of 12 ANN models were developed and each of their coefficients of determination (R

^{2}) was recorded. Initially, the ANN model was trained using one hidden layer with one node. For each subsequent ANN model, the number of neurons in the first hidden layer was increased by 1. When the number of neurons in a hidden layer reached 4, a new hidden layer with one node was added to the subsequent model. As this model is designed to receive four inputs, the maximum number hidden layers and neurons in each hidden layer was set at four, to ensure that the empirical solution that was developed was not overly complex. Table 4 summarizes the R

^{2}value obtained for each model developed.

^{2}values obtained, it was found that Models 7 and 8 resulted in the most accurate failure pressure prediction. Both the models produced a R

^{2}value of 0.99. Model 7 consisted of four neurons in the first hidden layer and three neurons in the second hidden layer, having one less neuron than Model 8. As such, Model 7 was utilized for the development of an empirical solution to ensure the simplicity of the equations without compromising the accuracy of the outcome.

^{2}) of the model. The R

^{2}value ranges from 0.0 to 1.0, with a greater R

^{2}value indicating a better goodness of fit, which is the distance between the fitted line of the ANN’s regression plot and the training data points.

#### 2.4. Material Properties

## 3. Validation of the Finite Element Method

## 4. Results and Discussion

#### 4.1. Comparison of Pipe Failure Pressure Prediction Using FEA and DNV Method

#### 4.2. Development of an Empirical Equation for the Failure Pressure Prediction of a Corroded Pipeline

- ${i}_{n,max}$ is the maximum normalized input value,
- ${i}_{n,min}$ is the minimum normalized input value,
- $i$ is the input value,
- ${i}_{min}$ is the minimum input value of the training data, and
- ${i}_{max}$ is the maximum input value of the training data.

- ${o}_{n}$ is the normalized output value,
- ${o}_{n,min}$ is the normalized minimum output value of the training data,
- ${o}_{n,max}$ is the normalized maximum output value of the training data,
- ${o}_{max}$ is the maximum output value of the training data, and
- ${o}_{min}$ is the minimum output value of the training data.

- Step 1: Calculation of the normalized effective length and depth of defect.$${\left(l/D\right)}_{e}=\frac{{l}_{1}+\left({s}_{1}+{l}_{2}\right)}{t}$$$${\left(d/t\right)}_{e}=\frac{\text{}\left(\frac{{d}_{1}{l}_{1}+{d}_{2}{l}_{2}}{{l}_{1,2}}\right)}{D}$$
- Step 2: Normalization of input parameters.$${\left(s/\sqrt{D/t}\right)}_{n}=\frac{2{\left(s/\sqrt{D/t}\right)}_{i}}{3}-1$$$${\left(l/D\right)}_{e}{}_{n}=\frac{2{\left(l/D\right)}_{e}}{2.15}-1$$$${\left(d/t\right)}_{e}{}_{n}=2.5{\left(d/t\right)}_{e}-1$$$${\left(\sigma c/\sigma y\right)}_{n}=2.5{\left(\sigma c/\sigma y\right)}_{i}-1$$
- Step 3: Calculation of the normalized output value.$$\left[\begin{array}{c}{\text{h}}_{1,1}\\ {\text{h}}_{1,2}\\ {\text{h}}_{1,3}\\ {\text{h}}_{1,4}\end{array}\right]=\left[\begin{array}{c}-0.0830{\left(s/\sqrt{D/t}\right)}_{n}+0.2350{\left(l/D\right)}_{e}{}_{n}+1.4834{\left(d/t\right)}_{e}{}_{n}-1.9408{\left(\sigma c/\sigma y\right)}_{n}+2.2349\\ 0.2086{\left(s/\sqrt{D/t}\right)}_{n}-0.1390{\left(l/D\right)}_{e}{}_{n}+0.1719{\left(d/t\right)}_{e}{}_{n}+0.0065{\left(\sigma c/\sigma y\right)}_{n}-0.2576\\ -0.07195{\left(s/\sqrt{D/t}\right)}_{n}-2.645{\left(l/D\right)}_{e}{}_{n}-0.50324{\left(d/t\right)}_{e}{}_{n}-0.1314{\left(\sigma c/\sigma y\right)}_{n}-1.6399\\ -0.2208{\left(s/\sqrt{D/t}\right)}_{n}+0.1131{\left(l/D\right)}_{e}{}_{n}-0.7073{\left(d/t\right)}_{e}{}_{n}-0.0425{\left(\sigma c/\sigma y\right)}_{n}-0.6017\end{array}\right]$$$$\left[\begin{array}{c}{\mathrm{h}}_{2,1}\\ {\mathrm{h}}_{2,2}\\ {\mathrm{h}}_{2,3}\end{array}\right]=\left[\begin{array}{c}0.0208a\left({\mathrm{h}}_{1,1}\right)+1.0157a\left({\mathrm{h}}_{1,2}\right)-0.2254a\left({\mathrm{h}}_{1,3}\right)-2.2751a\left({\mathrm{h}}_{1,4}\right)-2.5736\\ 0.3627a\left({\mathrm{h}}_{1,1}\right)+3.0005a\left({\mathrm{h}}_{1,2}\right)+0.0389a\left({\mathrm{h}}_{1,3}\right)+1.9307a\left({\mathrm{h}}_{1,4}\right)+1.7659\\ 2.3000a\left({\mathrm{h}}_{1,1}\right)+0.6237a\left({\mathrm{h}}_{1,2}\right)+1.2367a\left({\mathrm{h}}_{1,3}\right)-1.7914a\left({\mathrm{h}}_{1,4}\right)-4.5105\end{array}\right].\text{}$$$${o}_{n}=-2.5645a{(\mathrm{h}}_{2,1})+0.9479a{(\mathrm{h}}_{2,2})+1.1226a{(\mathrm{h}}_{2,3})-1.2962$$

- Step 4: Denormalization of output value, ${P}_{nf,Eq}$$${P}_{nf,Eq}=0.385{o}_{n}+0.615$$
- Step 5: Calculation of failure pressure, ${P}_{f,Eq}$.$${P}_{i}=\frac{\sigma {\times}_{UTS}\text{}t}{{r}_{i}}$$$${P}_{f,Eq}={P}_{nf,Eq}\times {P}_{i}$$

#### 4.3. Evaluation of the New Corroded Pipeline Failure Pressure Assessment Method

^{2}value of the new method is similar to that of the ANN, which is 0.999. This indicates that the method results in failure pressure predictions that are very close to the results obtained using FEA, which was used as the training data for the ANN model. Based on the maximum hoop stress theory, the calculation of the intact pressure of the pristine pipe model resulted in a value of 51.30 MPa, while the intact pressure obtained using FEM was 50.94 MPa. The intact pressure of the pristine pipe obtained from the newly developed failure pressure assessment method was calculated to be 51.36 MPa. The comparison of the intact pressure values obtained using the three methods are summarized in Table 8. There is good correlation between the three methods, with the percentage difference between the failure pressure obtained using the maximum hoop stress and the empirical equation is 0.12%, while the percentage difference between the failure pressure obtained using FEM and the empirical equation is 0.84%.

_{UTS}value of 718.2 MPa. Table 9 summarizes the parametric details, the failure pressure predictions using FEM and the empirical equation, and the percentage difference between the methods. A negative percentage difference indicates a conservative prediction, while a positive value indicates overestimation.

#### 4.4. Extensive Parametric Studies Using the Empirical Equation

#### 4.4.1. Effects of Defect Spacing on the Failure Pressure of a Pipe with Interacting Defects

#### 4.4.2. Effects of Defect Depth on the Failure Pressure of a Pipe with Interacting Defects

#### 4.4.3. Effects of Defect Length on the Failure Pressure of a Pipe with Interacting Defects

#### 4.4.4. Effects of Axial Compressive Stress on the Failure Pressure of a Pipe with Interacting Defects

#### 4.5. Recommendations for Future Studies

## 5. Conclusions

^{2}value of 0.99 for normalized defect spacings of 0.00 to 2.00, normalized effective defect lengths of 0.00 to 1.97, normalized effective defect depths of 0.00 to 0.80, and normalized axial compressive stress of 0.00 to 0.80, for API 5L X80 pipe grade. This equation is therefore suitable for the failure pressure prediction of high toughness pipes ranging from API 5L X80 material with normalized defect length, normalized defect depth, and normalized axial compressive stress that are within the mentioned ranges.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

ANSYS | ANSYS 16.1 Structural Product of Mechanical ANSYS Parametric Design Language (APDL) |

DNV | DNV-RP-F101 corrosion assessment method |

DOF | Degrees of freedom |

FE | Finite element |

FEA | Finite element analysis |

FEM | Finite element method |

UTS | Ultimate tensile strength |

$D$ | Pipe diameter |

$d$ | Corrosion defect depth |

${d}_{e}$ | Effective defect depth |

${H}_{1}$ | Factor for longitudinal compressive stresses |

$L$ | Pipe length |

l | Defect length |

${l}_{e}$ | Effective defect length |

${P}_{f,Eq}$ | Failure pressure of corroded pipeline using the new corrosion assessment method |

${P}_{fi,DNV}$ | Failure pressure of pipe with interacting corrosion defects using DNV |

${P}_{fs,DNV}$ | Failure pressure of pipe with single corrosion defect using DNV |

${P}_{fn,Eq}$ | Normalized failure pressure of pipe using the new corrosion assessment method |

${P}_{fn,FEA}$ | Normalized failure pressure of corroded pipeline using finite element analysis |

$r$ | Internal radius of pipe |

$StD\left(x\right)$ | standard deviation of variable x |

${s}_{c}$ | Circumferential defect spacing |

${s}_{l}$ | Longitudinal defect spacing |

$t$ | Thickness of pipe |

${\left(d/t\right)}_{e}$ | Normalized effective defect depth |

${\left(d/t\right)}_{meas}$ | Measured (relative) defect depth |

${\left(l/D\right)}_{e}$ | Normalized effective defect length |

${\epsilon}_{d}$ | Fractile value factor for the corrosion depth |

${\gamma}_{d}$ | Partial safety factor of corrosion depth |

${\gamma}_{m}$ | Model prediction partial safety factor |

$\theta $ | Ratio of circumferential length of corroded region to the nominal outside circumference of the pipe |

σ_{c} | Axial compressive stress |

σ_{e} | Effective von Mises stress |

σ_{h} | Hoop stress |

σ_{l} | Axial/longitudinal stress |

σ_{r} | Radial stress |

σ_{y} | Yield stress |

σ_{UTS} | Ultimate tensile strength |

σ×_{UTS} | True ultimate tensile strength |

$\xi $ | Pipe usage factor |

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**Figure 2.**(

**a**) Hexahedral SOLID185 elements used to mesh the pipe body and defect region, and (

**b**) tetrahedral SOLID186 elements used to mesh the endplate.

**Figure 3.**Application of symmetrical boundary conditions, internal pressure, axial compressive stress, and DOF constraints for quarter pipe models.

**Figure 10.**True stress-strain curve for API 5L X80 steel [21].

**Figure 11.**Normalized failure pressure predictions of FEA and DNV for API 5L X80 pipe subjected to internal pressure only for interacting defects with a normalized defect spacing of 0.5.

**Figure 12.**Normalised failure pressure predictions of FEA and DNV for API 5L X80 pipe subjected to internal pressure only for interacting defects.

**Figure 13.**Probability distribution of the percentage error obtained using the newly developed failure pressure prediction method and FEM based on the parameters of the ANN training data.

**Figure 14.**Normalized failure pressure predictions based on the empirical equation against various normalized defect spacing for multiple normalized defect depths at constant normalized defect length of 1.00 and normalized axial compressive stress of 0.60.

**Figure 15.**Normalized failure pressure predictions of the new assessment method against various normalized defect depth for multiple axial compressive stresses at a constant normalized defect spacing of 0.5 and normalized defect length of 0.80.

**Figure 16.**Normalized failure pressure predictions based on the empirical equation against various normalized defect depths for multiple normalized defect lengths at constant normalized defect spacing of 0.5 and normalized axial compressive stress of 0.40.

**Figure 17.**Normalized failure pressure predictions based on the empirical equation against various normalized defect lengths for multiple axial compressive stresses at constant normalized defect spacing of 0.5 and normalized defect depth of 0.20.

**Figure 18.**Normalized failure pressure predictions based on the empirical equation against various normalized defect lengths for multiple axial compressive stresses at constant normalized defect depth of 0.80.

**Figure 19.**Normalized failure pressure predictions of the new assessment method against various normalized defect lengths for multiple normalized defect depths at constant normalized defect spacing of 0.5 and normalized axial compressive stress of 0.40.

**Figure 20.**Normalized failure pressure predictions based on the empirical equation against various normalized axial compressive stresses for multiple normalized defect depths at constant normalized defect spacing of 0.5 and normalized defect length of 0.60.

Corrosion Defect Geometry | Value(s) |
---|---|

Normalized defect width, $w/t$ | 10 |

Normalized defect spacing, $s/\sqrt{Dt}$ | 0.0, 0.5, 1.0, 2.0 |

Normalized defect depth, $d/t$ | 0.0, 0.2, 0.4, 0.6, 0.8 |

Normalized defect length, $l/D$ | 0.0, 0.2, 0.4, 0.6, 0.8 |

Normalized axial compressive stress, σ_{c}/σ_{y} | 0.0, 0.2, 0.4, 0.6, 0.8 |

Number of Element Layers | $\mathbf{Normalized}\text{}\mathbf{Failure}\text{}\mathbf{Pressure},\text{}{\mathit{P}}_{\mathit{f}}/{\mathit{P}}_{\mathit{i}}$ |
---|---|

1 | 0.92 |

2 | 0.93 |

3 | 0.95 |

4 | 0.95 |

5 | 0.95 |

$\mathit{s}/\sqrt{\mathit{D}\mathit{t}}$ | ${\left(\mathit{d}/\mathit{t}\right)}_{\mathbf{e}}$ | ${\left(\mathit{l}/\mathit{D}\right)}_{\mathbf{e}}$ | ${\mathit{\sigma}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathit{y}}$ | ||||
---|---|---|---|---|---|---|---|

0.0 | 0.2 | 0.4 | 0.6 | 0.8 | |||

0.0 | 0.0 | 0.0 | 1.00 | ||||

0.0 | 0.2 | 0.4 | 0.95 | 0.94 | 0.90 | 0.84 | 0.72 |

0.8 | 0.91 | 0.91 | 0.88 | 0.80 | 0.68 | ||

1.2 | 0.89 | 0.88 | 0.86 | 0.79 | 0.66 | ||

1.6 | 0.87 | 0.87 | 0.85 | 0.78 | 0.66 | ||

0.4 | 0.4 | 0.87 | 0.86 | 0.83 | 0.76 | 0.63 | |

0.8 | 0.76 | 0.76 | 0.74 | 0.70 | 0.58 | ||

1.2 | 0.72 | 0.71 | 0.69 | 0.66 | 0.56 | ||

1.6 | 0.71 | 0.70 | 0.66 | 0.64 | 0.54 | ||

0.6 | 0.4 | 0.75 | 0.74 | 0.72 | 0.68 | 0.51 | |

0.8 | 0.59 | 0.59 | 0.57 | 0.51 | 0.45 | ||

1.2 | 0.53 | 0.51 | 0.50 | 0.47 | 0.42 | ||

1.6 | 0.48 | 0.48 | 0.47 | 0.45 | 0.41 | ||

0.8 | 0.4 | 0.58 | 0.56 | 0.52 | 0.45 | 0.41 | |

0.8 | 0.38 | 0.38 | 0.35 | 0.32 | 0.21 | ||

1.2 | 0.31 | 0.29 | 0.26 | 0.25 | 0.20 | ||

1.6 | 0.28 | 0.27 | 0.26 | 0.23 | 0.20 | ||

0.5 | 0.16 | 0.49 | 0.92 | 0.92 | 0.89 | 0.81 | 0.71 |

0.18 | 0.89 | 0.89 | 0.88 | 0.86 | 0.80 | 0.68 | |

0.19 | 1.29 | 0.87 | 0.86 | 0.85 | 0.79 | 0.66 | |

0.19 | 1.69 | 0.85 | 0.85 | 0.84 | 0.78 | 0.66 | |

0.33 | 0.49 | 0.83 | 0.80 | 0.78 | 0.73 | 0.61 | |

0.36 | 0.89 | 0.74 | 0.72 | 0.71 | 0.68 | 0.58 | |

0.37 | 1.29 | 0.70 | 0.69 | 0.68 | 0.65 | 0.56 | |

0.38 | 1.69 | 0.68 | 0.68 | 0.66 | 0.64 | 0.56 | |

0.49 | 0.49 | 0.68 | 0.66 | 0.65 | 0.62 | 0.53 | |

0.54 | 0.89 | 0.56 | 0.56 | 0.54 | 0.51 | 0.45 | |

0.56 | 1.29 | 0.51 | 0.50 | 0.49 | 0.46 | 0.42 | |

0.57 | 1.69 | 0.49 | 0.49 | 0.47 | 0.46 | 0.41 | |

0.65 | 0.49 | 0.50 | 0.49 | 0.48 | 0.47 | 0.41 | |

0.72 | 0.89 | 0.36 | 0.35 | 0.34 | 0.29 | 0.26 | |

0.74 | 1.29 | 0.30 | 0.30 | 0.28 | 0.26 | 0.24 | |

0.76 | 1.69 | 0.28 | 0.27 | 0.26 | 0.25 | 0.24 | |

1.0 | 0.14 | 0.58 | 0.93 | 0.92 | 0.89 | 0.84 | 0.71 |

0.16 | 0.98 | 0.90 | 0.90 | 0.88 | 0.83 | 0.70 | |

0.17 | 1.38 | 0.88 | 0.88 | 0.86 | 0.79 | 0.68 | |

0.18 | 1.78 | 0.88 | 0.88 | 0.85 | 0.79 | 0.66 | |

0.27 | 0.58 | 0.91 | 0.84 | 0.80 | 0.75 | 0.64 | |

0.33 | 0.98 | 0.82 | 0.73 | 0.72 | 0.70 | 0.58 | |

0.35 | 1.38 | 0.77 | 0.71 | 0.69 | 0.66 | 0.57 | |

0.36 | 1.78 | 0.75 | 0.69 | 0.66 | 0.64 | 0.56 | |

0.41 | 0.58 | 0.78 | 0.71 | 0.70 | 0.65 | 0.54 | |

0.49 | 0.98 | 0.63 | 0.58 | 0.57 | 0.54 | 0.45 | |

0.52 | 1.38 | 0.56 | 0.51 | 0.50 | 0.48 | 0.44 | |

0.54 | 1.78 | 0.53 | 0.50 | 0.49 | 0.47 | 0.44 | |

0.55 | 0.58 | 0.58 | 0.54 | 0.51 | 0.51 | 0.43 | |

0.65 | 0.98 | 0.40 | 0.36 | 0.36 | 0.33 | 0.28 | |

0.69 | 1.38 | 0.33 | 0.30 | 0.28 | 0.28 | 0.25 | |

0.72 | 1.78 | 0.30 | 0.28 | 0.26 | 0.25 | 0.24 | |

2.0 | 0.10 | 0.77 | 0.92 | 0.92 | 0.89 | 0.83 | 0.71 |

0.14 | 1.17 | 0.90 | 0.89 | 0.88 | 0.83 | 0.70 | |

0.15 | 1.57 | 0.87 | 0.88 | 0.86 | 0.79 | 0.67 | |

0.16 | 1.97 | 0.88 | 0.88 | 0.84 | 0.79 | 0.66 | |

0.21 | 0.77 | 0.91 | 0.84 | 0.80 | 0.75 | 0.63 | |

0.27 | 1.17 | 0.82 | 0.75 | 0.72 | 0.70 | 0.57 | |

0.31 | 1.57 | 0.77 | 0.71 | 0.69 | 0.66 | 0.57 | |

0.33 | 1.97 | 0.75 | 0.69 | 0.66 | 0.63 | 0.56 | |

0.31 | 0.77 | 0.78 | 0.71 | 0.70 | 0.65 | 0.54 | |

0.41 | 1.17 | 0.62 | 0.58 | 0.57 | 0.54 | 0.45 | |

0.46 | 1.57 | 0.56 | 0.51 | 0.50 | 0.48 | 0.44 | |

0.49 | 1.97 | 0.53 | 0.50 | 0.48 | 0.47 | 0.43 | |

0.42 | 0.77 | 0.58 | 0.54 | 0.51 | 0.51 | 0.43 | |

0.55 | 1.17 | 0.40 | 0.36 | 0.36 | 0.34 | 0.29 | |

0.61 | 1.57 | 0.32 | 0.30 | 0.28 | 0.28 | 0.25 | |

0.65 | 1.97 | 0.29 | 0.28 | 0.26 | 0.24 | 0.24 |

Model | No. of Hidden Layers | No. of Neurons in Hidden Layer 1 | No. of Neurons in Hidden Layer 2 | No. of Neurons in Hidden Layer 3 | R^{2} Value |
---|---|---|---|---|---|

1 | 1 | 1 | - | - | 0.93 |

2 | 1 | 2 | - | - | 0.93 |

3 | 1 | 3 | - | - | 0.94 |

4 | 1 | 4 | - | - | 0.94 |

5 | 2 | 4 | 1 | - | 0.95 |

6 | 2 | 4 | 2 | - | 0.98 |

7 | 2 | 4 | 3 | - | 0.99 |

8 | 2 | 4 | 4 | - | 0.99 |

9 | 3 | 4 | 4 | 1 | 0.97 |

10 | 3 | 4 | 4 | 2 | 0.96 |

11 | 3 | 4 | 4 | 3 | 0.93 |

12 | 3 | 4 | 4 | 4 | 0.91 |

**Table 5.**Mechanical properties of the API 5L X80 [21].

Properties | Pipe Body | Pipe End Cap |
---|---|---|

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

Poisson’s ratio, υ | 0.3 | 0.3 |

Yield strength, σ_{y} | 531.0 MPa | - |

Ultimate tensile strength, σ_{u} | 655.0 MPa | - |

True ultimate tensile strength, σ×_{u} | 718.2 MPa | - |

**Table 6.**FEM failure pressure validation against full scale burst tests by Bjorney et al. for single corrosion defect [55].

Grade | Specimen | d (mm) | l (mm) | w (mm) | σ_{l} (MPa) | Burst Pressure (MPa) | FEA Failure Pressure (MPa) | Percentage Difference (%) |
---|---|---|---|---|---|---|---|---|

X52 | Test 1 | 5.15 | 243 | 154.5 | 0.0 | 23.2 | 22.95 | −1.08 |

Test 5 | 3.09 | 162 | 30.9 | 48.0 | 28.6 | 28.35 | −0.87 | |

Test 6 | 3.09 | 162 | 30.9 | 84.0 | 28.7 | 27.00 | −5.92 |

**Table 7.**FEM failure pressure validation against full scale burst tests by Benjamin et al. for interacting corrosion defects [56].

Grade | Specimen | $\mathit{d}$ | $\mathit{l}$ | $\mathit{w}\left(\mathbf{mm}\right)$ | ${\mathit{s}}_{\mathit{l}}\left(\mathbf{mm}\right)$ | ${\mathit{s}}_{\mathit{c}}\left(\mathbf{mm}\right)$ | Burst Pressure (MPa) | FEA Failure Pressure (MPa) | Percentage Difference (%) |
---|---|---|---|---|---|---|---|---|---|

X80 | IDTS 2 | 5.39 | 39.6 | 31.9 | 0.0 | 0.0 | 22.68 | 22.40 | −1.23 |

IDTS 3 | 5.32 | 39.6 | 31.9 | 20.5 | 0.0 | 20.31 | 20.12 | −0.94 | |

IDTS 4 | 5.62 | 39.6 | 32.0 | 0.0 | 9.9 | 21.14 | 20.62 | −2.46 |

Maximum Hoop Stress Theory (A) (MPa) | FEM (B) (MPa) | Newly Developed Method (C) (MPa) | Percentage Difference between (A) and (C) (%) | Percentage Difference between (B) and (C) (%) |
---|---|---|---|---|

51.30 | 50.94 | 51.36 | 0.12 | 0.84 |

$\mathit{s}/\sqrt{\mathit{D}/\mathit{t}}$ | $\mathit{d}/\mathit{t}$ | $\mathit{l}/\mathit{D}$ | $\mathit{\sigma}\mathit{c}/\mathit{\sigma}\mathit{y}$ | ${\mathit{P}}_{\mathit{n}\mathit{f},\mathit{F}\mathit{E}\mathit{A}}$ | ${\mathit{P}}_{\mathit{n}\mathit{f},\mathit{E}\mathit{q}}$ | Percentage Difference (%) |
---|---|---|---|---|---|---|

0.25 | 0.65 | 0.09 | 0.30 | 0.93 | 0.91 | −2.28 |

0.25 | 0.65 | 0.09 | 0.60 | 0.84 | 0.84 | 0.02 |

0.25 | 1.45 | 0.10 | 0.30 | 0.91 | 0.89 | −1.72 |

0.25 | 1.85 | 0.10 | 0.30 | 0.90 | 0.89 | −1.05 |

0.25 | 1.85 | 0.10 | 0.60 | 0.83 | 0.81 | −2.17 |

0.25 | 0.65 | 0.28 | 0.30 | 0.82 | 0.78 | −4.51 |

0.25 | 0.65 | 0.28 | 0.60 | 0.72 | 0.73 | 0.35 |

0.25 | 1.45 | 0.34 | 0.60 | 0.68 | 0.66 | −2.32 |

0.25 | 1.45 | 0.44 | 0.60 | 0.59 | 0.57 | −3.21 |

0.25 | 1.45 | 0.48 | 0.30 | 0.58 | 0.55 | −4.39 |

0.25 | 0.65 | 0.51 | 0.60 | 0.54 | 0.53 | −1.32 |

0.25 | 1.45 | 0.53 | 0.60 | 0.51 | 0.48 | −5.86 |

0.25 | 0.45 | 0.72 | 0.50 | 0.40 | 0.37 | −7.54 |

0.25 | 0.65 | 0.74 | 0.60 | 0.31 | 0.28 | −8.99 |

0.25 | 1.45 | 0.77 | 0.50 | 0.26 | 0.24 | −8.02 |

0.25 | 1.45 | 0.77 | 0.60 | 0.26 | 0.23 | −8.63 |

0.30 | 1.05 | 0.47 | 0.30 | 0.62 | 0.57 | −7.34 |

0.30 | 0.65 | 0.27 | 0.30 | 0.83 | 0.79 | −4.63 |

0.40 | 1.47 | 0.43 | 0.60 | 0.58 | 0.58 | −0.39 |

0.40 | 1.87 | 0.53 | 0.50 | 0.52 | 0.49 | −5.63 |

0.70 | 0.73 | 0.37 | 0.30 | 0.73 | 0.70 | −4.28 |

0.70 | 1.53 | 0.50 | 0.60 | 0.53 | 0.50 | −5.25 |

0.70 | 1.93 | 0.75 | 0.50 | 0.26 | 0.24 | −5.90 |

0.90 | 0.76 | 0.43 | 0.50 | 0.63 | 0.61 | −3.62 |

0.90 | 1.56 | 0.20 | 0.30 | 0.85 | 0.82 | −3.58 |

1.00 | 0.58 | 0.40 | 0.20 | 0.71 | 0.71 | −0.09 |

0.50 | 0.89 | 0.60 | 0.40 | 0.45 | 0.43 | −4.09 |

0.80 | 1.75 | 0.30 | 0.70 | 0.66 | 0.64 | −2.35 |

0.20 | 0.45 | 0.20 | 0.30 | 0.89 | 0.86 | −3.46 |

0.30 | 0.65 | 0.40 | 0.60 | 0.68 | 0.63 | −6.95 |

0.40 | 1.45 | 0.60 | 0.30 | 0.45 | 0.43 | −5.47 |

0.80 | 1.05 | 0.30 | 0.60 | 0.72 | 0.69 | −3.52 |

0.90 | 0.65 | 0.20 | 0.30 | 0.89 | 0.85 | −4.72 |

1.00 | 1.47 | 0.40 | 0.60 | 0.60 | 0.59 | −0.90 |

1.20 | 1.87 | 0.60 | 0.40 | 0.41 | 0.39 | −4.94 |

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## Share and Cite

**MDPI and ACS Style**

Vijaya Kumar, S.D.; Karuppanan, S.; Ovinis, M.
An Empirical Equation for Failure Pressure Prediction of High Toughness Pipeline with Interacting Corrosion Defects Subjected to Combined Loadings Based on Artificial Neural Network. *Mathematics* **2021**, *9*, 2582.
https://doi.org/10.3390/math9202582

**AMA Style**

Vijaya Kumar SD, Karuppanan S, Ovinis M.
An Empirical Equation for Failure Pressure Prediction of High Toughness Pipeline with Interacting Corrosion Defects Subjected to Combined Loadings Based on Artificial Neural Network. *Mathematics*. 2021; 9(20):2582.
https://doi.org/10.3390/math9202582

**Chicago/Turabian Style**

Vijaya Kumar, Suria Devi, Saravanan Karuppanan, and Mark Ovinis.
2021. "An Empirical Equation for Failure Pressure Prediction of High Toughness Pipeline with Interacting Corrosion Defects Subjected to Combined Loadings Based on Artificial Neural Network" *Mathematics* 9, no. 20: 2582.
https://doi.org/10.3390/math9202582