# Strength Analysis of Cylindrical Shells with Tangential Nozzles under Internal Pressure

^{*}

## Abstract

**:**

_{o}/D

_{i}), diameter-thickness ratio (D

_{i}/T) and thickness ratio (t/T). Results show that when increasing d

_{o}/D

_{i}and D

_{i}/T or decreasing t/T, the strength–weakening coefficient increases, which means that the strength–weakening effect of the tangential nozzle on the cylindrical shell increases. With sufficient simulation results, regression equations for the strength–weakening coefficient were obtained which provides a reference for the strength design of cylindrical shells with tangential nozzles under internal pressure.

## 1. Introduction

_{o}/D

_{i}, t/T and D

_{i}/T) and conducted a limit–load-based parametric study on cylinder radial nozzles under compound loads, obtaining an empirical equation describing their ultimate load relationship under such conditions through regression analysis to solve for their bearing capacity, effectively within the shells’ radial nozzles context. Xu Xinyi [7] examined four dimensionless parameters (circumferential nozzle inclination angle β, opening ratio d/D, wall thickness ratio δ

_{et}/δ

_{e}and diameter-to-thickness ratio D/δ

_{e}) influencing ultimate pressure in circumferential nozzle structures with cylindrical shells while designing an orthogonal test model for further regression analysis and fitting a formula for the limit load. Xue [8] not only investigated the relationship between various geometric parameters (d/D, D/T and t/T) and blasting pressure, but also derived the corresponding equations. Additionally, four different materials were employed to validate the proposed equations. Focusing on the open nozzle structure of cylindrical shells under external pressure, Zhang Shuling [9] systematically examined the variation pattern of critical pressure P

_{cr}under different opening ratios d/D, nozzle-to-cylinder thickness ratio δ

_{et}/δ

_{e}, cylinder diameter-to-thickness ratio D

_{o}/δ

_{e}and cylinder length-to-diameter ratio L/D

_{o}through finite element nonlinear buckling analysis. Zhang Jinwu [10] conducted finite element analysis on orthotropic titanium cylindrical shells with perforated nozzle structures. The results revealed that as δ

_{et}/δ

_{e}, D/δ

_{e}and L/D decrease, the weakening effect of the opening ratio ρ on the P

_{cr}of the cylindrical shell increases. A regression equation for critical instability pressure P

_{cr}was obtained through regression analysis. Fan Hangchao [11], by utilizing machine learning technology and redeveloping an ABAQUS plug-in toolset, developed a prediction model for nozzle connections in pressure vessels. This model enables rapid prediction of local stress at the nozzle joint by inputting parameters such as radius and wall thickness of both cylinder and nozzle (R, r, T, t; mm), as well as nozzle inclination (a; °). It provides a reliable and convenient platform for swift evaluation and optimization of pressure vessel designs.

_{o}/D

_{i}, D

_{i}/T and t/T. Regression equations for the strength–weakening coefficient were obtained, which provides a reference for the strength design of cylindrical shells with tangential nozzles under the internal pressure.

## 2. Establishment of Finite Element Models

#### 2.1. Geometric Model

_{i}is the inner diameter of the cylindrical shell, T is the wall thickness of the cylindrical shell, d

_{o}is the outer diameter of the nozzle and t is the wall thickness of the nozzle. The extension length of the cylinder and the nozzle were much larger than the requirement of the edge stress attenuation length [17].

_{i}= 1000 mm unchanged, the models were established by changing three dimensionless parameters (the diameter ratio of nozzle to cylinder 0.1 ≤ d

_{o}/D

_{i}≤ 0.3, the thickness ratio of nozzle to cylinder 0.5 ≤ t/T ≤ 1.75 and the diameter thickness ratio of cylinder 71.43 ≤ D

_{i}/T ≤ 125) for the numerical simulations. The range of structural parameters investigated in this study generally aligns with the size range of typical tangential nozzle structures in engineering. The dimensionless parameters are shown in Table 2. A total of 120 finite element analysis models were established by using the full factor analysis method.

#### 2.2. Mesh Model

_{o}/D

_{i}= 0.3, t/T = 1 and D

_{i}/T = 100, the total number of elements is 235,805, and the total number of nodes was 1,089,252.

#### 2.3. Loads and Constraints

## 3. Limit–Load Analysis

#### 3.1. Limit–Load Analysis Method

#### 3.2. Results of Limit–Load Analysis

_{i}/T, the limit–load value decreases significantly. Keeping D

_{i}/T unchanged, with the increase in t/T and the decrease in d

_{o}/D

_{i}, or in other words, the larger the wall thickness and the smaller the outer diameter of the tangential nozzle, the greater the limit load of the cylindrical shell with the tangential nozzle or the stronger the bearing capacity of the structure is, as a result of a less weakening degree of the tangential nozzle on the strength of the cylindrical shell.

#### 3.3. Strength–Weakening Coefficient and Its Influence Factors

_{0}and P are the limit load of the cylindrical shell without a nozzle and the cylindrical shell with a tangential nozzle, respectively. Obviously, the greater the strength–weakening coefficient, the greater the influence of the tangential nozzle on the load-bearing capacity of the cylindrical shell. The strength–weakening coefficient of the cylindrical shell having tangential nozzles with different diameter ratios d

_{o}/D

_{i}, thickness ratios t/T and diameter-thickness ratios D

_{i}/T under internal pressure is shown in Figure 8, Figure 9 and Figure 10.

_{o}/D

_{i}and D

_{i}/T, meaning the thicker the nozzle is, the smaller the weakening effect of the nozzle on the load-bearing capacity of the cylindrical shell is.

_{o}/D

_{i}, the strength–weakening coefficient K shows an increasing trend. But when t/T is small, i.e., the thickness of the nozzle wall is small, with the increase in d

_{o}/D

_{i}, the increasing trend of the K value gradually slows down, meaning the influence gradually becomes less significant.

_{i}/T, the strength–weakening coefficient K increases monotonically. With increasing d

_{o}/D

_{i}and decreasing t/T, or with the increase in the diameter of the nozzle and the decrease in the wall thickness of the nozzle, the K value increases significantly, i.e., the load-bearing capacity of the cylindrical shell decreases greatly.

_{o}/D

_{i}= 0.30, t/T = 0.50 and D

_{i}/T = 125, the strength–weakening coefficient reaches 61.52%, meaning the load-bearing capacity of the cylindrical shell is significantly decreased. But when d

_{o}/D

_{i}= 0.10, t/T = 1.75 and D

_{i}/T = 71.43, the strength reduction coefficient is close to 0, meaning the load-bearing capacity of the cylindrical shell is not obviously affected by the tangential nozzle.

## 4. Regression of the Strength–Weakening Coefficient

_{o}/D

_{i}, t/T and D

_{i}/T in order to facilitate the strength design of cylindrical shells with tangential nozzles under internal pressure.

#### 4.1. Regression of the Strength–Weakening Coefficient

_{o}/D

_{i}= 0.10~0.20,

_{o}/D

_{i}= 0.20~0.30,

_{o}/D

_{i}= 0.10~0.20,

_{o}/D

_{i}= 0.20~0.30,

#### 4.2. Numerical Verification of Regression Equations of the Strength–Weakening Coefficient

#### 4.3. Applicability of the Regression Equations of the Strength–Weakening Coefficient for Different Shell Diameters

_{i}= 1000 mm but, due to the use of dimensionless parameters, the results are applicable to other shell diameters.

_{o}/D

_{i}, t/T and D

_{i}/T are the same, but the shell diameters are 500 mm, 1000 mm and 2000 mm. It is seen that for the cylinder without nozzles when the diameter-thickness ratio D

_{i}/T is kept constant but the diameter D

_{i}is changed, the limit–load value is unchanged, which is 6.48 MPa. For the cylinder shells with tangential nozzles, when d

_{o}/D

_{i}, t/T and D

_{i}/T are the same, the limit load and the strength–weakening coefficient are basically the same when changing the shell diameter D

_{i}, and the relative error between the finite element solution and the formula solution for the strength–weakening coefficient is within 4%. Therefore, for the tangential nozzle structure with different shell diameters, the strength design method is also applicable when the three variables d

_{o}/D

_{i}, t/T and D

_{i}/T are within the applicable range specified in this paper.

## 5. Conclusions

- (1)
- For the strength design of cylindrical shell structure with tangential nozzles, in view of the fact that there is no accurate strength design method, this paper proposes a limit–load analysis approach based on a finite element method.
- (2)
- The weakening degree of the ultimate load-bearing capacity of cylindrical shells with tangential nozzles was defined as the strength–weakening coefficient and the influence of d
_{o}/D_{i}, t/T and D_{i}/T was studied. It was found that with increasing d_{o}/D_{i}and D_{i}/T and decreasing t/T, the strength–weakening coefficient increases or, in other words, the ultimate load-bearing capacity of the cylindrical shell decreases. - (3)
- With sufficient numerical calculation of the ultimate load-bearing capacity of the cylindrical shell with tangential nozzles under internal pressure, the regression equation of the strength–weakening coefficient was obtained which was numerically verified for accuracy. Clearly, by providing the formulas for strength evaluation, this work delves deeper into the subject compared to other similar studies in the literature and yields more practical outcomes.
- (4)
- Due to the use of dimensionless parameters, the regression equation is suitable for different shell diameters with a tangential nozzle structure. The regression equations of the strength–weakening coefficient provide a reference way for the strength design of cylindrical shell structures with tangential nozzles under internal pressure.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Three-dimensional geometric model of the cylindrical shell structure with a tangential nozzle.

**Figure 3.**Finite element mesh model of the cylindrical shell with a tangential nozzle (d

_{o}/D

_{i}= 0.3, t/T = 1, D

_{i}/T = 100). (

**a**) Overall. (

**b**) Local.

**Figure 4.**Schematic diagram of the load and constraint application (d

_{o}/D

_{i}= 0.3, t/T = 1, D

_{i}/T = 100).

**Figure 5.**Displacement–load curve of the cylindrical shell with a tangential nozzle for limit–load analysis (d

_{o}/D

_{i}= 0.3, t/T = 1, D

_{i}/T = 100).

**Figure 6.**Contours of the von Mises stress at the cylindrical shell with a tangential nozzle under the limit load (d

_{o}/D

_{i}= 0.3, t/T = 1, D

_{i}/T = 100).

**Figure 7.**Change in the limit load value of the cylindrical shell having a tangential nozzle with t/T for different d

_{o}/D

_{i}and D

_{i}/T.

**Figure 8.**Change in strength–weakening coefficient K of the cylindrical shell structure having tangential nozzles with t/T for different d

_{o}/D

_{i}and D

_{i}/T.

**Figure 9.**Change in strength–weakening coefficient K of the cylindrical shell structure having a tangential nozzle with d

_{o}/D

_{i}for different t/T and D

_{i}/T.

**Figure 10.**Change in strength–weakening coefficient K of the cylindrical shell structure having a tangential nozzle with D

_{i}/T for different d

_{o}/D

_{i}and t/T.

Material | Thickness/mm | Design Temperature/°C | Allowable Stress/MPa | Yield Strength/MPa | Tangential Modulus/MPa | Elastic Modulus/MPa | Poisson’s Specific | Density /kg·m ^{−3} |
---|---|---|---|---|---|---|---|---|

Q345R | 3~16 | 100 | 189 | 283.5 | 0 | 197,000 | 0.3 | 7850 |

20 | >16~40 | 100 | 140 | 210 | 0 | 197,000 | 0.3 | 7850 |

d_{o}/D_{i} | t/T | D_{i}/T |
---|---|---|

0.10 | 0.5 | 71.43 |

0.15 | 0.75 | 83.33 |

0.20 | 1 | 100 |

0.25 | 1.25 | 125 |

0.30 | 1.5 | |

1.75 |

D_{i}/T | Limit Load/MPa | von Mises Stress/MPa |
---|---|---|

71.43 | 9.040 | 287.47 |

83.33 | 7.765 | 286.94 |

100 | 6.480 | 286.22 |

125 | 5.190 | 285.42 |

d_{o}/D_{i} | t/T | D_{i}/T | Limit Load P/MPa | The Finite Element Solution of Strength–Weakening Coefficient K/% | The Formula Solution of Strength–Weakening Coefficient K/% | Relative Difference/% |
---|---|---|---|---|---|---|

0.13 | 0.8 | 90.91 | 5.90 | 17.18 | 17.10 | 0.50 |

0.27 | 0.8 | 111.11 | 3.21 | 45.01 | 44.44 | 1.28 |

0.18 | 1.4 | 90.91 | 6.09 | 14.49 | 14.41 | 0.56 |

0.22 | 1.4 | 111.11 | 4.52 | 22.54 | 22.30 | 1.05 |

**Table 5.**Limit load and strength–weakening coefficients of cylindrical shell structures with different shell diameters D

_{i}under the same dimensionless structure parameters.

d_{o}/D_{i} | t/T | D_{i}/T | D_{i}/mm | T/mm | Limit Load of Cylinder with Tangential Nozzle P/MPa | Limit Load of a Cylinder P_{0}/MPa | The Finite Element Solution of Strength–Weakening Coefficient K/% | The Formula Solution of Strength–Weakening Coefficient K/% | Relative Difference/% |
---|---|---|---|---|---|---|---|---|---|

0.10 | 0.75 | 100 | 500 | 5 | 5.6000 | 6.48 | 13.58 | 13.88 | 2.23 |

1000 | 10 | 5.6000 | 6.48 | 13.58 | 13.88 | 2.23 | |||

2000 | 20 | 5.6080 | 6.48 | 13.46 | 13.88 | 3.17 | |||

0.25 | 0.50 | 100 | 500 | 5 | 3.1589 | 6.48 | 51.25 | 50.13 | 2.19 |

1000 | 10 | 3.1618 | 6.48 | 51.21 | 50.13 | 2.10 | |||

2000 | 20 | 3.1694 | 6.48 | 51.09 | 50.13 | 1.88 | |||

0.15 | 1.25 | 100 | 500 | 5 | 5.5531 | 6.48 | 14.30 | 13.98 | 2.25 |

1000 | 10 | 5.5549 | 6.48 | 14.28 | 13.98 | 2.07 | |||

2000 | 20 | 5.5402 | 6.48 | 14.50 | 13.98 | 3.59 | |||

0.30 | 1.5 | 100 | 500 | 5 | 4.5802 | 6.48 | 29.32 | 28.68 | 2.17 |

1000 | 10 | 4.5962 | 6.48 | 29.07 | 28.68 | 1.34 | |||

2000 | 20 | 4.5962 | 6.48 | 29.07 | 28.68 | 1.34 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, X.; Qian, C.; Wu, Z.
Strength Analysis of Cylindrical Shells with Tangential Nozzles under Internal Pressure. *Appl. Sci.* **2024**, *14*, 2363.
https://doi.org/10.3390/app14062363

**AMA Style**

Zhao X, Qian C, Wu Z.
Strength Analysis of Cylindrical Shells with Tangential Nozzles under Internal Pressure. *Applied Sciences*. 2024; 14(6):2363.
https://doi.org/10.3390/app14062363

**Chicago/Turabian Style**

Zhao, Xiaofeng, Caifu Qian, and Zhiwei Wu.
2024. "Strength Analysis of Cylindrical Shells with Tangential Nozzles under Internal Pressure" *Applied Sciences* 14, no. 6: 2363.
https://doi.org/10.3390/app14062363