# Probability Study on the Thermal Stress Distribution in Thick HK40 Stainless Steel Pipe Using Finite Element Method

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Solution

#### The Stresses for Thick-Walled Cylinder Pipe under Internal Pressure (P) and Thermal Gradient ($\Delta T$)

## 3. Axisymmetric Finite Element Analysis Using ABAQUS Software

_{i}of 25 mm, and outer radius r

_{0}of 50 mm. The material properties are elastic modulus (E) of 1.38 × 10

^{5}MPa, Poisson’s ratio µ of 0.313, coefficient of thermal expansion (α) of 1.5 × 10

^{−5}(1/°C). The model was meshed with element type CAX4R, a four-noded bilinear quadrilateral element, and the mesh grid was 10 × 10 elements and was fixed in the axial direction U2 of 0, as shown in Figure 4. The loading conditions were internal pressure P = 40 MPa, inside temperature ${T}_{i}$ of 500 °C, outside temperature ${T}_{0}$ of 420 °C, thermal gradient $\Delta T$ of 80 °C. Figure 4 shows the model with meshing and applied boundary conditions in ABAQUS. Analysis on the sweep of different stress elements before and after analysis is shown in Figure 5, Figure 6, Figure 7 and Figure 8 below, along with comparisons of various stresses which are shown in Figure 9, Figure 10 and Figure 11. Table 2 and Table 3 show the comparisons of analytical and finite element analysis (FEA) using ABAQUS results by using von Mises stress.

#### 3.1. Comparison of Analytical and FEA using ABAQUS

#### 3.2. Comparison of Analytical Results and FEA Results Using Abaqus and FEM Using MATLAB

## 4. Probabilistic Study

#### 4.1. Monte Carlo Simulation

#### 4.2. Distributed Structural Parameters

#### 4.3. Lognormal Distribution

#### 4.4. Input Distribution

#### 4.4.1. Due to Variability in Material Property

^{5}Pa.

^{5}Pa. Figure 12 represents Young’s modulus due to the variation of the material properties.

#### 4.4.2. Due to Load Variability

#### Normal Distribution

#### Normal Distribution for Pressure

## 5. Probabilistic Finite Element Formulation

## 6. Derivation of Stiffness Matrix

#### Material Specifications

_{i}of 25 mm, and r

_{0}of 50 mm, respectively. The material properties of HK40 are given as follows: elastic modulus (E) of 1.38 × 10

^{5}Pa, Poisson’s ratio (µ) of 0.313, thermal expansion coefficient (α) of 1.5 × 10

^{5}(1/°C). For the numerical calculations, the random variable for Young’s modulus of elasticity is $E$.

## 7. Probabilistic Study: Output Distribution

#### 7.1. Output Distribution Due to Material Variability

#### 7.2. Output Distribution Due to Both Material and Load Variability

## 8. Probability of Failure of Von Mises Stress with Respect to Yield Strength

## 9. Stress Contours

#### Mean Stress Contours for HK40 Material

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\mathsf{\sigma}}_{\mathrm{r}}^{\mathrm{T}}$ | Radial stress |

$\Delta T$ | Thermal gradient |

${\sigma}_{z}^{T}$ | Axial stress |

${\sigma}_{\theta}^{T}$ | Circumferential stress |

${\sigma}_{\theta}^{P}$ | Hoop stress induced by pressure |

${\sigma}_{z}^{P}$ | Axial stress induced by pressure |

${\sigma}_{r}^{P}$ | Radial stress induced by pressure |

${r}_{0}$ | Outer radius (mm) |

${r}_{i}$ | Inner radius (mm) |

a | Ratio of outer to inner radius |

$\mu $ | Poisson’s ratio |

r | Radius at any position of tube wall (mm) |

E | Elastic modulus of material (MPa) |

$\alpha $ | Thermal expansion coefficient of material |

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**Figure 4.**Meshing, boundary conditions, internal pressure, and thermal gradient of axisymmetric thick pipe.

**Figure 16.**Comparison of radial stress and circumferential stress for different elements where $E$ is a random variable.

**Figure 17.**Comparison of axial stress and von Mises stress for different elements when E is a random variable.

**Figure 18.**Comparison of radial stress and axial stress for different elements when $E,P,\mathrm{and}\Delta T$ are random variables.

**Figure 19.**Comparison of circumferential stress and von Mises stress for different elements when $E,P,\text{}\mathrm{and}\text{}\Delta T$ are random variables.

Radius (mm) | Radial Stress (σ_{r}) (MPa) | Circumferential Stress (σ_{θ}) (MPa) | Axial Stress (σ_{z}) (MPa) | Von Mises Stress (σ_{v}) (MPa) |
---|---|---|---|---|

26.25 | −35.04 | 61.70 | 8.34 | 83.93 |

28.75 | −26.99 | 53.66 | 8.34 | 70.02 |

31.25 | −20.80 | 47.46 | 8.34 | 59.33 |

33.75 | −15.93 | 42.59 | 8.34 | 50.93 |

36.25 | −12.03 | 38.70 | 8.34 | 44.21 |

38.75 | −8.86 | 35.53 | 8.34 | 38.77 |

41.25 | −6.25 | 32.92 | 8.34 | 34.29 |

43.75 | −4.08 | 30.74 | 8.34 | 30.57 |

46.25 | −2.24 | 28.91 | 8.34 | 27.44 |

48.75 | −0.69 | 27.35 | 8.34 | 24.80 |

**Table 2.**Comparison of analytical and finite element analysis (FEA) using ABAQUS results by using von Mises stress.

Radius (mm) | Analytical Results | FEA Using ABAQUS | ||||
---|---|---|---|---|---|---|

${\mathit{\sigma}}_{\mathit{r}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{\theta}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{V}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{r}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{\theta}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{V}}\left(\mathbf{MPa}\right)$ | |

26.2500 | −35.0416 | 61.7082 | 83.9361 | −35.1470 | 61.7910 | 84.0936 |

28.7500 | −26.9943 | 53.6610 | 70.0273 | −27.0669 | 53.7178 | 70.1337 |

31.2500 | −20.8000 | 47.4667 | 59.3306 | −20.8515 | 47.5068 | 59.4038 |

33.7500 | −15.9305 | 42.5972 | 50.9312 | −15.9679 | 42.6262 | 50.9821 |

36.2500 | −12.0333 | 38.7000 | 44.2184 | −12.0611 | 38.7214 | 44.2537 |

38.7500 | −8.8658 | 35.5324 | 38.7720 | −8.8868 | 35.5485 | 38.7961 |

41.2500 | −6.2565 | 32.9232 | 34.2951 | −6.2727 | 32.9354 | 34.3109 |

43.7500 | −4.0816 | 30.7483 | 30.5730 | −4.0942 | 30.7578 | 30.5825 |

46.2500 | −2.2498 | 28.9165 | 27.4476 | −2.2597 | 28.9239 | 27.4520 |

48.7500 | −0.6925 | 27.3592 | 24.8000 | −0.7004 | 27.3650 | 24.8004 |

Radius (mm) | Analytical Results | FEM Using MATLAB | ||||
---|---|---|---|---|---|---|

${\mathit{\sigma}}_{\mathit{r}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{\theta}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{V}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{r}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{\theta}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{V}}\left(\mathbf{MPa}\right)$ | |

26.250 | −35.041 | 61.708 | 83.936 | −34.950 | 61.883 | 81.636 |

28.750 | −26.994 | 53.661 | 70.027 | −26.926 | 53.768 | 68.244 |

31.250 | −20.800 | 47.466 | 59.330 | −20.748 | 47.532 | 57.917 |

33.750 | −15.930 | 42.597 | 50.931 | −15.890 | 42.635 | 49.789 |

36.250 | −12.033 | 38.700 | 44.218 | −12.002 | 38.720 | 43.280 |

38.750 | −8.865 | 35.532 | 38.772 | −8.841 | 35.540 | 37.990 |

41.250 | −6.256 | 32.923 | 34.295 | −6.237 | 32.923 | 33.636 |

43.750 | −4.081 | 30.748 | 30.573 | −4.067 | 30.742 | 30.010 |

46.250 | −2.249 | 28.916 | 27.447 | −2.238 | 28.906 | 26.963 |

48.750 | −0.692 | 27.359 | 24.800 | −0.684 | 27.346 | 24.378 |

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**MDPI and ACS Style**

Bobba, S.; Abrar, S.; Rehman, S.M. Probability Study on the Thermal Stress Distribution in Thick HK40 Stainless Steel Pipe Using Finite Element Method. *Designs* **2019**, *3*, 9.
https://doi.org/10.3390/designs3010009

**AMA Style**

Bobba S, Abrar S, Rehman SM. Probability Study on the Thermal Stress Distribution in Thick HK40 Stainless Steel Pipe Using Finite Element Method. *Designs*. 2019; 3(1):9.
https://doi.org/10.3390/designs3010009

**Chicago/Turabian Style**

Bobba, Sujith, Shaik Abrar, and Shaik Mujeebur Rehman. 2019. "Probability Study on the Thermal Stress Distribution in Thick HK40 Stainless Steel Pipe Using Finite Element Method" *Designs* 3, no. 1: 9.
https://doi.org/10.3390/designs3010009