# The Determination of the Limit Load Solutions for the New Pipe-Ring Specimen Using Finite Element Modeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Determination of the Stress Equation for Solid Section of Wall

## 3. Finite Element Elastic-Plastic Analysis

_{Y}[4,5,6,7,8]. FITNET and R6 [14] procedures, as well as BS 7448 and BS 7910 [15] standards, provide a normalized load or a stage of structural plastification with the L

_{r}term (Equation (3)). The σ

_{ref}is a reference stress of the component with a crack, and σ

_{Y}is the yield strength of the material. Normalized load can also be written as the ratio of the current load F on the component and the limit load F

_{Y}of the component.

_{Y}) is lower in the plane stress state than it is in the plane strain state [4,5]. In the case of the plane stress state, plastification appears more distinctly at the height of the limit load compare to the case of the plane strain state, where there is no distinct plastification, and hence an appropriately sized plastic zone of elastic–plastic transition during the loading of the material. The limit load determination can be done by observing the specimen during loading. At a specific time, the LL is marked as the force when the plasticity of the crack is extended over the rest of the unbroken ligament. In the case of the ring size R/B = 8, the function of the limit load was performed for the extended crack aspect ratio a/W ranging from 0.3 to 0.8. This function is the starting point for other ring sizes and it represents one point which gives us the margin towards thin-walled pipelines (the ring R/B = 20). We have schematically shown a way of analyzing rings as three point bending with outer radius R, height of the ring W, and wall thickness B in Figure 3. The material properties of the analyzed model were obtained from the tensile tests. The whole research was divided into three main fields: experimental testing, analytical solving, and numerical analysis. Using experimental testing, we characterized the testing material used as a very ductile material with a significant Lüders Plateau at around 470 MPa yield strength [4,5]. Figure 4 shows the actual stress-strain curve with a limit yield of the material σ

_{eH}= 470 MPa, Young’s modulus of elasticity 210 GPa, a Poisson number of steel 0.3, effective fracture stress σ

_{f}= 650 MPa at 13% deformation, and a linear approximated actual stress σ

_{m}= 1842 MPa at 186% deformation. The exponent of hardening of the material is n = 0.4072.

## 4. The Ring (R/B = 20) (Thin-Walled)

_{Y}. In all cases of the rings we analyzed, F

_{Y}was much lower than from the criterion of full plastification through the unbroken ligament. The deformation at this time was only noted and visible at the contact point (s) [4,5].

## 5. Determination of the Limit Load Function

_{A}= σ

_{Y}and load F = F

_{Y}

_{0}:

_{Y}

_{0}. If instead of moments of inertia we use the resistance moment, then the formula in section A is:

_{Y}(a/W) for a pipe-ring included crack or notch is given by Equation (6) as a function of the crack aspect ratio f (a/W). Here f

_{0}, f

_{1}, f

_{2}and f

_{3}represent the coefficients of the third order polynomial trend curve [4,5,6,7]. Appendix A displays the equation of the limit load F

_{Y}as well as the normalized limit load after Equation (3), if we have been given the yield strength of a material for an axially three-point bend pipe-ring specimen, by following sizes in Appendix B. The LL function is given depending on the polynomial coefficients created based on the dimensions of the analyzed rings. Figure 11 shows the limit load function of the ring R/B = 8 and its standard height vs. thickness W/B = 2 depending on the extended crack aspect ratio a/W from 0.3 to 0.8. Meanwhile, all other rings were analyzed for a range from a/W = 0.45 to 0.55.

_{Y}for a ring with a crack or notch in relation with the limit load of the solid Section 2F

_{Y}

_{0}depending on the ratio R/B (i.e., the size of the ring to the wall thickness B). Figure 13 indicates a very discontinuous ratio F

_{Y}/2F

_{Y}

_{0}for all three crack aspect ratios in the range of R/B from 6 to 9. Until other investigations are done, such as investigating the constraint effect or analyzing the stress triaxiality, it is not possible to make any comments on the validity of the obtained results in the range of small rings. The described state indicates an uneven stress-strain state throughout the wall thickness as a result of random choice of ring dimensions and corresponding triaxiality. In such behavior, we have to deal also with a bigger effect of the plane strain state, since the thickness is larger compared to the moment arm (torque). The positive side in Figure 13 shows that for the bigger ring size, from 12 to 20, we have very nice formed continuous curves. This show the suitability for characterizing thin-walled pipelines and analyzing the sizes of rings which comply with natural gas pipelines.

## 6. Conclusions

- The limit load solutions of randomly chosen several different geometries, show interesting non continuous behavior depending the on the ring size, i.e., the ratio of its diameter and wall thickness, R/B, from 6 to 10.
- A plausible reason is the randomly picked diameter of the corresponding ring size ratio R/B, while in the case of more reasonable ring sizes with an increasing diameter, including dimensions taken from real pipelines, with correspondingly increasing thickness, we can expect that the LL values would slightly rise with the increasing of the ring’s size ratio R/B, for all three crack or notch depth a/W cases, as seen in Figure 13 for R/B from 12 to 20 and above, to the ratio of real-world thin-walled pipelines.
- The function we expressed to define LL in a range of different ring sizes is calculated for various crack aspect ratios from 0.45 to 0.55. We also calculated the extended range of the crack aspect ratio from 0.3 to 0.8 for one randomly chosen probe ratio R/B = 8, where the span distance between the supports is 1.8 times R, just to schematically show the calculation of the limit load if the notch or crack is not in the range of standard recommendations.
- By observing and processing all numerical results, we found spatial bending of all probes subject to the different constraint effects of limiting and spreading the yield deformation around the tip, and along the crack path. However, as we noted, stress triaxiality needs to be analyzed for a better footing to explain and completely describe the behavior of axial three-point bend ring probes.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

The general equation for calculating the limit load in dependence of the crack depth a/W from 0 to 1 and ratio of ring’s size of R/B = 8 and other ratios R/B from the a/W = 0.45 to 0.55. | ||||

${F}_{Y}\left(\frac{a}{W},\frac{R}{B}\right)=2\cdot \frac{2{\sigma}_{Y}\cdot B\cdot W}{\sqrt{3}\cdot \sqrt{1+12\cdot {\left(\frac{t}{W}\right)}^{2}}}\cdot \left({f}_{0}\cdot {\left(\frac{a}{W}\right)}^{0}+{f}_{1}\cdot {\left(\frac{a}{W}\right)}^{1}+{f}_{2}\cdot {\left(\frac{a}{W}\right)}^{2}+{f}_{3}\cdot {\left(\frac{a}{W}\right)}^{3}\right)$ | ||||

Nomenclature:- F
_{Y} - - limit load of whole ring, N
- a
- - crack length (depth), mm
- W
- - height of the ring cut from the pipe, mm
- R
- - outer radius of the ring, mm
- B
- - wall thickness of the ring, mm
- t
- σ
_{Y} - - the yield strength (proportional) of the material, MPa
- f
_{0}, f_{1}, f_{2}, f_{3} - - polynomial coefficients
| ||||

R/B | f_{0} | f_{1} | f_{2} | f_{3} |

5 | 0.3988 | 0.0835 | –0.6812 | - |

6 | –0.4321 | 3.8168 | –4.6626 | - |

7 | 1.0998 | –2.7835 | 2.1638 | - |

8 | 0.9995 | –2.5552 | 2.3803 | –0.8256 |

9 | –0.303 | 2.6802 | –3.2033 | - |

10 | 0.6145 | –0.5682 | –0.1831 | - |

12 | 1.1626 | –3.2481 | 2.7656 | - |

15 | 1.0233 | –2.585 | 1.9502 | - |

20 | 0.1511 | 1.0001 | –1.6148 | - |

## Appendix B

**Table A1.**The geometry configurations of the numerically modelled rings in the dependence by the ratio R/B between the outer radius and the wall thickness and crack depth a/W.

R/B | a/W | R, mm | W, mm | B, mm | a, mm |
---|---|---|---|---|---|

5 | - | 40 | 16 | 8 | - |

0.45 | 40 | 16 | 8 | 7.2 | |

0.5 | 40 | 16 | 8 | 8 | |

0.55 | 40 | 16 | 8 | 8.8 | |

6 | - | 54 | 18 | 9 | - |

0.45 | 54 | 18 | 9 | 8.1 | |

0.5 | 54 | 18 | 9 | 9 | |

0.55 | 54 | 18 | 9 | 9.9 | |

7 | - | 98 | 28 | 14 | - |

0.45 | 98 | 28 | 14 | 12.6 | |

0.5 | 98 | 28 | 14 | 14 | |

0.55 | 98 | 28 | 14 | 15.4 | |

8.5 | - | 85 | 20 | 10 | - |

0.45 | 85 | 20 | 10 | 9 | |

0.5 | 85 | 20 | 10 | 10 | |

0.55 | 85 | 20 | 10 | 11 | |

9 | - | 72 | 16 | 8 | - |

0.45 | 72 | 16 | 8 | 7.2 | |

0.5 | 72 | 16 | 8 | 8 | |

0.55 | 72 | 16 | 8 | 8.8 | |

10 | - | 60 | 12 | 6 | - |

0.45 | 60 | 12 | 6 | 5.4 | |

0.5 | 60 | 12 | 6 | 6 | |

0.55 | 60 | 12 | 6 | 6.6 | |

12 | - | 54 | 9 | 4.5 | - |

0.45 | 54 | 9 | 4.5 | 4.05 | |

0.5 | 54 | 9 | 4.5 | 4.5 | |

0.55 | 54 | 9 | 4.5 | 4.95 | |

15 | - | 75 | 10 | 5 | - |

0.45 | 75 | 10 | 5 | 4.5 | |

0.5 | 75 | 10 | 5 | 5 | |

0.55 | 75 | 10 | 5 | 5.5 | |

20 | - | 80 | 8 | 4 | - |

0.45 | 80 | 8 | 4 | 3.6 | |

0.5 | 80 | 8 | 4 | 4 | |

0.55 | 80 | 8 | 4 | 4.4 |

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**Figure 3.**Schematically shown geometry of the ring R/B = 20, B = 4 mm, W = 8 mm, the span distance between the supports for FEA. The loading mode is three point bending, the scale of the sketch and the dimension do not correspond to the actual ring size).

**Figure 4.**True stress-strain curve as the material input properties for determine the numerical analysis.

**Figure 5.**The finite elements mesh on ¼ of the model, with the displayed places of densification with structural hexagonal linear elements.

**Figure 6.**At a full cross-section ratio R/B = 20, the equivalent deformation starts to expand on the upper and lower edge at 5.7 mm displacement, instead of into the middle. The middle neutral layer remains undeformed.

**Figure 7.**The equivalent deformation, where a small layer of the neutral zone still remains undeformed at 10 mm displacement. While spreading of the deformation over the upper and lower periphery of ring stopped, it now started extending into the interior.

**Figure 8.**The equivalent deformation of (

**a**) on the inner side with meeting the criterion of the limit load and (

**b**) on the outer side, where the theoretical limit load criterion is not yet met for the ratio R/B = 20 at 2.15 mm.

**Figure 9.**The load-displacement curve for a ring with the ratio R/B = 5 and a crack depth of a/W = 0.45. The figure shows the position of the points where we met specific criteria for determining the limit load on the inner and outer side and for the case of taking the Lüders slip into account.

**Figure 10.**Displayed range with points for determining the limit load from the stress-displacement curve (R/B = 5 with a/W = 0.45) the position of Lüders slip ‘‘I’’ and Lüders slip ‘‘II’’.

**Figure 11.**The limit load function for R/B = 8 and W/B = 2 in dependence on different crack aspect ratios a/W from 0.3 to 0.8.

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**MDPI and ACS Style**

Likeb, A.; Gubeljak, N.
The Determination of the Limit Load Solutions for the New Pipe-Ring Specimen Using Finite Element Modeling. *Metals* **2020**, *10*, 749.
https://doi.org/10.3390/met10060749

**AMA Style**

Likeb A, Gubeljak N.
The Determination of the Limit Load Solutions for the New Pipe-Ring Specimen Using Finite Element Modeling. *Metals*. 2020; 10(6):749.
https://doi.org/10.3390/met10060749

**Chicago/Turabian Style**

Likeb, Andrej, and Nenad Gubeljak.
2020. "The Determination of the Limit Load Solutions for the New Pipe-Ring Specimen Using Finite Element Modeling" *Metals* 10, no. 6: 749.
https://doi.org/10.3390/met10060749