# Design Equation of Buckle Propagation Pressure for Pipe-in-Pipe Systems

^{*}

## Abstract

**:**

## 1. Introduction

_{p}) is considered another key characteristic.

_{p}are derived. Figure 1 shows the buckle propagation mode of the pipeline in the 2D scenario. Based on that, Kyriakides and Vogler [13] investigated the buckle propagation phenomenon in the PIP system. The cross-section of the pipeline contains four plastic hinges for each pipe, as shown in Figure 1a,b.

_{0}is the yield stress,

_{0i}is the yield stress of the inner pipe,

_{i}is the thickness of the inner pipe.

_{o}/t

_{o}= 1.25D

_{i}/t

_{i}[7].

## 2. FEM Validation

#### 2.1. Validation with Model Experiment

#### 2.2. Effect of Mesh Sensitivity

## 3. Effect of Inner Pipe Imperfection

#### 3.1. The Effect of Inner Pipe Ovality

_{t}and P

_{i}are listed to represent the occurrence of the contact. P

_{t}is the pressure under which the outer pipe contacts the inner pipe while P

_{i}is the pressure under which the inner surface of the inner pipe contacts itself. As shown in Table 4, in both cases, the buckle propagation pressure is the same. The difference in the values of the pressure when contact occurs between both cases is due to the geometrical features. From Figure 8, the external pressure decreases after the ultimate collapse pressure is reached. A larger distance of the gap between two pipes results in a lower value of P

_{t}. So, this pressure is smaller in the case of the same ovality direction. Additionally, the same reason leads to the difference in values of P

_{i}. Nevertheless, the difference between these two cases is insignificant. Thus, it is confirmed that the direction of the ovality of the inner pipe does not influence the buckle propagation of PIP.

#### 3.2. Effect of Eccentricity

_{0}, is defined as follows:

_{max}is the maximum offset of the inner pipe centre, as shown in Figure 10.

_{0}as 0.5 which is far beyond the acceptable range in the manufacturing stage.

## 4. Effect of the Geometry of the Inner Pipe

_{o}-t

_{o}-D

_{i}-t

_{i}’. For instance, the name of the case in the first line of Table 6 is ‘60-4-10-2’.

_{i}/t

_{i}are selected. Five cases with a constant D

_{i}of 25 mm and another five cases with the same t

_{i}as 2 mm are shown in Table 7. The ultimate collapse strength of PIP equals that of the outer pipe. Thus, because the outer pipes of cases listed in Table 7 share the same geometrical parameters, the ultimate collapse strength is not presented.

_{o}/t

_{o})/(D

_{i}/t

_{i}) is equal to 2 in both cases, the buckle propagation mode is different. It is supposed that the buckle propagation mode is influenced by the ratio of (D

_{o}/t

_{o})/(D

_{i}/t

_{i}), but the ratio is not a fixed value.

## 5. Effect of the Geometry of the Outer Pipe

_{o}/t

_{o}as those shown in Table 8, in which Case N40 is included. These cases maintain the thickness of the outer pipe at the value of 4 mm, as listed in Table 9. By comparing the results listed in Table 8 and Table 9, it is noticed that the buckling mode of Case ‘80-4-25-2’ is Mode A, whereas that of Case ‘60-3-25-2’ is Mode B, although the ratio of D

_{o}/t

_{o}is the same in both cases.

## 6. Critical Parameter of Two Different Buckle Propagation Modes

_{o}/D

_{i}for the PIP system is generally in the range of 1.3 to 2.2 [20,21,22,23,24,25]. Additional series of cases were created in the present study and calculated with outer pipe diameters of 32 mm and 36 mm. In both series, the other geometrical parameters were kept the same as in Case N40. To figure out the boundary between the two different buckle propagation modes, the thickness of the outer pipes of the cases listed in Table 9 is changed with an increment of 0.1 mm. The purpose is to illustrate the switch of the buckle propagation mode from one to another. The thickness of the outer pipe from both buckle propagation modes is summarized in Table 10.

_{o}/t

_{o})/(D

_{i}/t

_{i}) is indeed influenced by the ratio D

_{o}/D

_{i}. The relationship between (D

_{o}/t

_{o})/(D

_{i}/t

_{i}) ratio and the D

_{o}/D

_{i}ratio is shown in Figure 11, where a monotonically decreasing tendency is observed.

## 7. Buckle Propagation Pressure of a PIP System

_{o}/t

_{o}and D

_{i}/t

_{i}were investigated. For the case with diameters of the outer and inner pipes as 60 mm and 25 mm, respectively, the buckle propagation pressure is shown in Table 11.

_{o}/D

_{i}, and will be explained later in this section.

_{o}/t

_{o}and D

_{i}/t

_{i}are in the ranges of 10 to 33 and 7 to 50, respectively. The ratio of D

_{o}/D

_{i}is in the range of 1.28 to 3.2. As pointed out in the last case in Table 11, the buckle propagation pressure did not increase to a large extent when the thickness of the inner pipe changed from 3.33 mm to 5 mm. The same phenomenon is also found in other series of calculations. So, the results of these cases are excluded from the summary.

## 8. Conclusions

- The initial imperfections of PIP, i.e., the ovality and eccentricity of the inner pipe, have an insignificant influence on the buckle propagation pressure
- The switch of two buckle propagation modes is dependent on the ratio of (D
_{o}/t_{o})/(D_{i}/t_{i}) and the critical value of the ratio is related to the ratio of D_{o}/D_{i}; instead of that in the previous literature, the critical value is considered to be constant. In practice situations of the ratio of D_{o}/D_{i}in the range of 1.2 to 2.2, a monotonically decreasing tendency of (D_{o}/t_{o})/(D_{i}/t_{i}) can be observed with respect to D_{o}/D_{i} - The buckle propagation pressure is highly influenced by the ratio of D
_{o}/t_{o}and D_{i}/t_{i}. In other words, the strength of the outer pipe compared with the inner pipe not only results in the buckle propagation mode of PIP but also leads to the stability of the buckle propagation pressure. One phenomenon observed is that during the increase of thickness of the inner pipe, after it exceeds a certain value (such as 3.33 mm in Case 60-4-25-3.33), the buckle propagation pressure will not increase any longer in line with the increase of thickness - The relationship between the buckle propagation pressure and the ratio of D/t of pipes in PIP follows the format of power functions, which includes the coupling effect of the variables. Based on the results from the present analyses, a fitted formula (Equation (7)) is proposed for predicting the buckle propagation pressure of the PIP with good accuracy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Buckle propagation mode in the 2D scenario: (

**a**) single-walled pipe, (

**b**) PIP, and (

**c**) single-walled pipe.

**Figure 7.**Mesh divisions of N-series cases from the back view: (

**a**) Case N20, (

**b**) N30, (

**c**) N30, (

**d**) N60.

**Figure 12.**Relationship between buckle propagation pressure and D/t ratios: (

**a**) outer pipe and (

**b**) inner pipe (D

_{o}= 60 mm, D

_{i}= 25 mm).

**Table 1.**Geometrical parameters and physical properties of the pipe in the model experiment; the data are from [15].

L (mm) | D (mm) | t (mm) | Δ_{0} (%) | σ_{0.5} (MPa) | E (GPa) | |
---|---|---|---|---|---|---|

Outer pipe | 1500 | 60 | 4 | 7.85 | 319.2 | 188.9 |

Inner pipe | 1500 | 25 | 2 | 0 | 319.2 | 188.9 |

Case | Element Size (mm) | Mesh Number | Total Number | |||||
---|---|---|---|---|---|---|---|---|

θ | t | z | θ | t | z | |||

N20 | outer pipe | 9.42 | 2 | 10 | 20 | 2 | 150 | 13,800 |

inner pipe | 9.82 | 2 | 10 | 8 | 1 | 150 | ||

N30 | outer pipe | 6.28 | 1.33 | 7.5 | 30 | 3 | 200 | 36,000 |

inner pipe | 6.54 | 1 | 7.5 | 12 | 2 | 200 | ||

N40 | outer pipe | 4.71 | 1 | 5 | 40 | 4 | 300 | 84,000 |

inner pipe | 4.91 | 1 | 5 | 16 | 2 | 300 | ||

N60 | outer pipe | 3.14 | 0.67 | 3.75 | 60 | 6 | 400 | 225,600 |

inner pipe | 3.27 | 0.67 | 3.75 | 24 | 3 | 400 |

Case | Ultimate Collapse Pressure (P_{co}) | Buckle Propagation Pressure (P_{p}) | ||||
---|---|---|---|---|---|---|

FEM (MPa) | Experiment (MPa) | Error (%) | FEM (MPa) | Experiment (MPa) | Error (%) | |

N20 | 34.12 | 29.54 | 15.50 | 19.50 | 14.98 | 30.17 |

N30 | 32.21 | 29.54 | 9.04 | 17.00 | 14.98 | 13.48 |

N40 | 31.39 | 29.54 | 6.27 | 16.30 | 14.98 | 8.81 |

N60 | 30.75 | 29.54 | 4.10 | 15.55 | 14.98 | 3.81 |

Direction of Ovality | P_{co} (MPa) | P_{t} (MPa) | P_{i} (MPa) | P_{p} (MPa) |
---|---|---|---|---|

a. same | 31.39 | 16.29 | 15.77 | 16.30 |

b. vertical | 31.39 | 16.95 | 15.46 | 16.30 |

_{t}: pressure under which outer pipe touches inner pipe; P

_{i}: pressure under which inner pipe contacts itself.

Case |
Δ
_{i} | e_{0} | P_{co} (MPa) | P_{t} (MPa) | P_{i} (MPa) | P_{p} (MPa) |
---|---|---|---|---|---|---|

N40 | 0 | 0 | 31.39 | 15.66 | 15.66 | 16.30 |

N40.01 | 0 | 0.1 | 31.39 | 15.95 | 15.68 | 16.33 |

N40.02 | 0 | 0.3 | 31.39 | 15.86 | 15.69 | 16.35 |

N40.03 | 0 | 0.5 | 31.39 | 15.79 | 15.69 | 16.35 |

N40.11 | 7.85 | 0.1 | 31.39 | 14.92 | 15.86 | 16.30 |

N40.12 | 7.85 | 0.3 | 31.39 | 15.28 | 15.84 | 16.35 |

N40.13 | 7.85 | 0.5 | 31.39 | 15.24 | 15.85 | 16.40 |

D_{o} (mm) | t_{o} (mm) | D_{i} (mm) | t_{i} (mm) |
---|---|---|---|

60 | 4 | 10 | 2 |

60 | 4 | 15 | 2 |

60 | 4 | 20 | 2 |

60 | 4 | 30 | 2 |

60 | 4 | 35 | 2 |

60 | 4 | 25 | 1.43 |

60 | 4 | 25 | 1.67 |

60 | 4 | 25 | 2.5 |

60 | 4 | 25 | 3.33 |

60 | 4 | 25 | 5 |

**Table 7.**Geometry and buckle propagation of PIP with the same outer pipe (D

_{o}= 60 mm, t

_{o}= 4 mm).

D_{i} (mm) | t_{i} (mm) | D_{o}/D_{i} | D_{i}/t_{i} | (D_{o}/t_{o})/(D_{i}/t_{i}) | P_{p} (MPa) | Mode |
---|---|---|---|---|---|---|

25 | 2 | 2.4 | 12.50 | 1.20 | 16.30 | A |

10 | 2 | 6 | 5.00 | 3.00 | 15.80 | B |

15 | 2 | 4 | 7.50 | 2.00 | 15.90 | A |

20 | 2 | 3 | 10.00 | 1.50 | 16.10 | A |

30 | 2 | 2 | 15.00 | 1.00 | 16.50 | A |

35 | 2 | 1.71 | 17.50 | 0.86 | 17.10 | A |

25 | 1.43 | 2.4 | 17.48 | 0.86 | 15.75 | A |

25 | 1.67 | 2.4 | 14.97 | 1.00 | 15.95 | A |

25 | 2.5 | 2.4 | 10.00 | 1.50 | 17.00 | A |

25 | 3.33 | 2.4 | 7.51 | 2.00 | 18.50 | B |

25 | 5 | 2.4 | 5.00 | 3.00 | 18.70 | B |

Case | t_{o} (mm) | D_{o}/t_{o} | P_{co} (MPa) | P_{p} (MPa) | Mode |
---|---|---|---|---|---|

60-3-25-2 | 3.0 | 20 | 19.50 | 8.75 | B |

60-3.5-25-2 | 3.5 | 17.14 | 25.10 | 11.95 | A |

60-4-25-2 (N40) | 4.0 | 15 | 31.39 | 16.30 | A |

60-4.5-25-2 | 4.5 | 13.33 | 37.80 | 21.10 | A |

60-5-25-2 | 5.0 | 12 | 44.83 | 27.00 | A |

60-5.5-25-2 | 5.5 | 10.91 | 52.09 | 33.70 | A |

60-6-25-2 | 6.0 | 10 | 59.76 | 41.20 | A |

Case | D_{o} (mm) | D_{o}/t_{o} | P_{co} (MPa) | P_{p} (MPa) | Mode |
---|---|---|---|---|---|

80-4-25-2 | 80 | 20 | 20.61 | 7.95 | A |

68.58-4-25-2 | 68.58 | 17.15 | 26.07 | 11.75 | A |

60-4-25-2 (N40) | 60 | 15 | 31.39 | 16.30 | A |

53.4-4-25-2 | 53.34 | 13.34 | 37.02 | 22.10 | A |

48-4-25-2 | 48 | 12 | 44.26 | 31.00 | A |

43.6-4-25-2 | 43.6 | 10.9 | 50.15 | 39.40 | A |

40-4-25-2 | 40 | 10 | 57.32 | 51.00 | A |

**Table 10.**The thickness of the outer pipe from both buckle propagation modes of cases with D

_{o}/D

_{i}less than 2.2 (D

_{i}= 25 mm, t

_{i}= 2 mm).

D_{o} (mm) | D_{o}/D_{i} | t_{o} (mm) | (D_{o}/t_{o})/(D_{i}/t_{i}) | ||
---|---|---|---|---|---|

Mode A | Mode B | Mode A | Mode B | ||

53.34 | 2.13 | 3.1 | 3 | 1.38 | 1.42 |

48 | 1.92 | 2.7 | 2.6 | 1.42 | 1.48 |

43.6 | 1.74 | 2.3 | 2.2 | 1.52 | 1.59 |

40 | 1.60 | 2.0 | 1.9 | 1.60 | 1.68 |

36 | 1.44 | 1.8 | 1.7 | 1.60 | 1.69 |

32 | 1.28 | 1.5 | 1.4 | 1.71 | 1.83 |

t_{o} (mm) | t_{i} (mm) | D_{o}/D_{i} | D_{o}/t_{o} | D_{i}/t_{i} | P_{p} (MPa) |
---|---|---|---|---|---|

4 | 2 | 2.4 | 15 | 12.5 | 16.30 |

3.2 | 2 | 2.4 | 18.75 | 12.5 | 9.95 |

3.1 | 2 | 2.4 | 19.35 | 12.5 | 9.35 |

3 | 2 | 2.4 | 20 | 12.5 | 8.75 |

2.7 | 2 | 2.4 | 22.22 | 12.5 | 7.20 |

4 | 1.43 | 2.4 | 15 | 17.48 | 15.75 |

4 | 1.67 | 2.4 | 15 | 14.97 | 15.95 |

4 | 2 | 2.4 | 15 | 12.5 | 16.30 |

4 | 2.5 | 2.4 | 15 | 10 | 17.00 |

4 | 3.33 | 2.4 | 15 | 7.51 | 18.50 |

4 | 5 | 2.4 | 15 | 5 | 18.60 |

D_{o} (mm) | D_{i} (mm) | D_{o}/t_{o} | D_{i}/t_{i} |
---|---|---|---|

80 | 25 | √ | √ |

68.58 | 25 | √ | |

60 | 25 | √ | √ |

53.34 | 25 | √ | |

48 | 25 | √ | |

43.6 | 25 | √ | √ |

40 | 25 | √ | |

36 | 25 | √ | |

32 | 25 | √ | √ |

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## Share and Cite

**MDPI and ACS Style**

Li, R.; Chen, B.-Q.; Guedes Soares, C.
Design Equation of Buckle Propagation Pressure for Pipe-in-Pipe Systems. *J. Mar. Sci. Eng.* **2023**, *11*, 622.
https://doi.org/10.3390/jmse11030622

**AMA Style**

Li R, Chen B-Q, Guedes Soares C.
Design Equation of Buckle Propagation Pressure for Pipe-in-Pipe Systems. *Journal of Marine Science and Engineering*. 2023; 11(3):622.
https://doi.org/10.3390/jmse11030622

**Chicago/Turabian Style**

Li, Ruoxuan, Bai-Qiao Chen, and C. Guedes Soares.
2023. "Design Equation of Buckle Propagation Pressure for Pipe-in-Pipe Systems" *Journal of Marine Science and Engineering* 11, no. 3: 622.
https://doi.org/10.3390/jmse11030622