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Article

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

Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia
*
Author to whom correspondence should be addressed.
Metals 2020, 10(6), 749; https://doi.org/10.3390/met10060749
Received: 6 May 2020 / Revised: 31 May 2020 / Accepted: 2 June 2020 / Published: 5 June 2020
(This article belongs to the Special Issue Application of Alloys in Transport)

:

## 1. Introduction

Until now, no analytical or empirical limit load solutions were done for through-wall cracks for specimens such as PRS, especially where it concerns combined bending and shear loads. Limit load solutions based on multi-parametric analysis, as it was parametrically perform in . However, solutions for limit load can be applied in the structural integrity assessment of different components made as pipes or hollow cylinders or components, where a longitudinal crack can occur. An example of the use of limit load solutions on cylindrical and spherical parts is presented in the literature [17,18]. Meanwhile, in the case of solving the integrity of components  and welds , we need to know the limit load.
Stress strain behavior is simulated by numerical simulations using the finite element method [21,22]. With the help of numerical simulations, the maximum load of elastoplastic pipelines can be determined very precisely .
The goal of this paper is to show the approach of determining the LL solutions using finite element modelling (FEM) calculations in the range of different three-dimensional crack depths and different ring dimensions of ring size R/B = 5 to 20. Mostly we focused on a thin-walled pipe-ring specimen R/B = 20 which, under axial three-point bending, more accurately described the behavior of real thin-walled pipelines. The determination is based on the analytical stress expression for the full section of a wall, with a numerically corresponding function for a different crack depth, in order to investigate the applicability of the limit load solutions. We named the most critical analyzed section section A, which stands on the plane of load line displacement [4,5].

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

Solving the stress-strain term of the axially three-point bend ring was necessary for the determination of empirical equations of the limit load for the three-point bend ring in the axial direction. The analytical solution approach was done on the basis of the engineering strength terms for bending, shear, and torsion. As is known, due to the geometry of the ring during loading, all bending, shear, and torsional stresses are present in both wall sections A and B (Figure 1 and Figure 2). However, for analyzing the ring loading capacity, it was important to only analyze section A. Meanwhile, section B, standing on the place of the supports, is subjected to completely torsional loads, as evidenced by numerical simulations, as was already mentioned in [4,5,6,7,8]. The stress Equation (1) for determining the bending and shear stresses at point 1 of section A is schematically shown in Figure 2a where the dominant bending stresses are.
$σ A = ( 6 ⋅ F ⋅ t ⋅ e B ⋅ W 3 ) 2 + 3 ⋅ ( F 2 ⋅ B ⋅ W ) 2$
$t = 0.9 ⋅ R$
In Equation (1), F represents the loading force on each half of the ring, and acts as F/2, while on the side of supports, it is F/4. In other words, the force F from Equation (1) is actually ½ of the load on the testing machine where we axially test the ring with three point bending. t represents the torque lever of the working force F for the displacement from the support, to cross-section A, as seen in Equation (2). The span distance between supports is equal to 2t = 1.8 R (Figure 3). e represents the distance to the highest bending stresses with respect to the neutral axis of the section (B × W). Because the stresses in cross section B were not important to design the empirical terms of the limit load for section A, this part is not listed. The previous comparisons with numerical stress estimations in point 1 of section A for several different ring geometries show that the stress Equation (1) is acceptable enough for the calculation and the determination of the limit load of a full section for one sidewall of the ring. The deviation of the linear part of the stress-strain curve up to the yield point is less than or $≈$5%, even though the numerically obtained stress was Mises equivalent with included torsional stresses, which are not covered in the analytical term.

## 3. Finite Element Elastic-Plastic Analysis

$L r = σ r e f σ Y = F F Y$
The meshing of the model was done manually by changing the size of the linear quadratic finite elements and their density in the required places for the most accurate results of the cross-section A. The singular allocation of mesh elements around the crack of the model was not performed, because of focusing on just reaching the force at which the material starts to yield. The size of the elements varied from 0.1 to 0.35 mm in the vertical direction and was fixed at 0.3 mm in the horizontal direction. The linear hexagonal elements in the element library of ABAQUS 6.11-3 were used in this analysis. The element size was 0.15 at the crack front and gradually increased to 0.5 from the crack front towards the edges, using so-called continually increasing element size, as is shown in Figure 5. The shape of the elements was linear hexagonal, because this type of element is appropriate and does not consume more computation time. Other edges of the model were meshed with the number of elements at 100 and a bias ratio of 75. It was modelled with more than 2.3 million of elements, at quarter model. Figure 3 shows the boundary conditions on the lower (fixed) support, too. The properties of the interaction between the loading pin and the specimen, as well as between the support pin and the specimen were required to be introduced in the model. The discretization method was based on surface to surface contact with no adjustments for surfaces. Contact properties were described by a normal component with disallowed separation and by a tangential component with a friction coefficient of 0.1. As mentioned, a numerical analysis was carried out using a dynamic implicit procedure over a time step with period 1. The increment was set to automatic, and the simulation was performed with a maximum number of increments set to 1000. The initial and maximum increment sizes were taken as 0.01. In our study, we manually made a mesh with continuing remeshing of finite element size. We checked that stress goes smoothly in the whole volume of specimens. We have made few variations of elements, but similarly to ref. , the sensitivity analysis shows the extremely negligible influence of remeshing when the mesh is established in the proposed way.

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

The limit load determination is based on the assessment of the position of this point, see Figure 10 (stress-displacement curve of the ring R/B = 5, a/W = 0.45), and the determination of the corresponding displacement, which we estimated from Figure 9 as FY. In all cases of the rings we analyzed, FY 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

The determination of the empirical equation and final function of the limit load in dependence of crack length based on the solid (full) ring section (i.e., no cracks or notches) in Equation (1). The derivation for the limit load at point ‘‘1’’ of section A stands by simply equating σA = σY and load F = FY0:
$σ Y = ( 6 ⋅ F Y o ⋅ t ⋅ e B ⋅ W 3 ) 2 + 3 ⋅ ( F Y o 2 ⋅ B ⋅ W ) 2$
By forward derivation, we get:
$F Y 0 = 2 ⋅ σ Y ⋅ B ⋅ W 3 ⋅ 1 + 48 ⋅ ( t ⋅ e W 2 ) 2$
Equation (5) is applied for the determination of the limit load on one side of the wall of a solid section of the ring. The limit load of the entire ring with the solid section is equal to 2FY0. If instead of moments of inertia we use the resistance moment, then the formula in section A is:
$F Y ( a W ) = 2 ⋅ 2 σ Y ⋅ B ⋅ W 3 ⋅ 1 + 12 ⋅ ( t W ) 2 ⋅ ( f 0 ⋅ ( a W ) 0 + f 1 ⋅ ( a W ) 1 + f 2 ⋅ ( a W ) 2 + f 3 ⋅ ( a W ) 3 )$
The function of limit load FY (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 f0, f1, f2 and f3 represent the coefficients of the third order polynomial trend curve [4,5,6,7]. Appendix A displays the equation of the limit load FY 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.
The comparison of the LL function is for a ratio of 8, with all other ratios from 5 to 20 are displayed in Figure 12. It is obvious that the curves in accordance with the size of ring, do not sequentially follow the ratio R/B, because while we were planning the geometry of the rings, we completely randomly defined the outer radius R, depending on the ratio R/B. It shows that in some analyzed cases, the wall thickness had a much higher bending stress component compared to the torque lever than in other cases. These results show completely different behavior of the ring during loading, since we have to perform analysis with different stress-strain states not just with geometries of the ring, but also by comparing the inner and outer side of the bent ring. Figure 13 shows the distribution of the limit load FY for a ring with a crack or notch in relation with the limit load of the solid Section 2FY0 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 FY /2FY0 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

As reiterated in this article and well known for dynamically exposed structures, the limit load (LL) in addition to the stress intensity factor (SIF) are the most important quantities for structural integrity to predict the maximum load capacity and possibly the largest “allowed” crack size for material safety in regard to other exposure conditions. In the frame of the focused research here, we were mostly verifying that the new proposed ring specimens, the so-called pipe ring notch bend specimens (PRNB), can be used for measurement of fracture toughness. In one of the many required parts of this research, we calculated the ligament yielding point on the stress-strain curve using FEA with the ABAQUS non damage model for different ring sizes. This paper shows the procedure we chose to define and create an engineering practice to calculate the limit load of the axial three-point bend ring, which is simply cut from the segment or just a small piece of pipeline.
• 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

Conceptualization, A.L.; Investigation, A.L.; Methodology, N.G.; Project administration, N.G.; Software, A.L.; Supervision, N.G. All authors have read and agree to the published version of the manuscript.

## Funding

This research received no external funding.

## Acknowledgments

Authors acknowledge to Slovenian Research Agency (ARRS) for financial support of Ph.D. investigation of Andrej Likeb and Reseach Program P2-0137 “Numerical and experimental analysis of mechanical systems”.

## Conflicts of Interest

The authors declare no conflict of interest.

## Appendix A

The general equation for calculating the limit load by the corresponding coefficients for the ratio of the size of the ring.
 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. $F Y ( a W , R B ) = 2 ⋅ 2 σ Y ⋅ B ⋅ W 3 ⋅ 1 + 12 ⋅ ( t W ) 2 ⋅ ( f 0 ⋅ ( a W ) 0 + f 1 ⋅ ( a W ) 1 + f 2 ⋅ ( a W ) 2 + f 3 ⋅ ( a W ) 3 )$ Nomenclature:FY - limit load of whole ring, Na   - crack length (depth), mmW  - height of the ring cut from the pipe, mmR   - outer radius of the ring, mmB   - wall thickness of the ring, mmt    - the moment arm, mm, Equation: t = 0.9R (Figure 3)σY - the yield strength (proportional) of the material, MPaf0, f1, f2, f3- polynomial coefficients R/B f0 f1 f2 f3 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.
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/Ba/WR, mmW, mmB, mma, mm
5-40168-
0.45401687.2
0.5401688
0.55401688.8
6-54189-
0.45541898.1
0.5541899
0.55541899.9
7-982814-
0.4598281412.6
0.598281414
0.5598281415.4
8.5-852010-
0.458520109
0.585201010
0.5585201011
9-72168-
0.45721687.2
0.5721688
0.55721688.8
10-60126-
0.45601265.4
0.5601266
0.55601266.6
12-5494.5-
0.455494.54.05
0.55494.54.5
0.555494.54.95
15-75105-
0.45751054.5
0.5751055
0.55751055.5
20-8084-
0.4580843.6
0.580844
0.5580844.4
Meaning of the labels: R/B—size ratio of the ring, R—outer radius of the ring, W—height of the ring, B—thickness of wall, a—crack or notch length.

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Figure 1. The scheme of the ring bending in the axial direction, with both sections A and B [4,5,6,7,8].
Figure 1. The scheme of the ring bending in the axial direction, with both sections A and B [4,5,6,7,8].
Figure 2. (a) Schematic view of the distribution of the bending stresses to the critical cross-section A and (b) schematic view of the distribution of both tangential and torsional stresses on the critical cross-section A [4,5,6,7,8].
Figure 2. (a) Schematic view of the distribution of the bending stresses to the critical cross-section A and (b) schematic view of the distribution of both tangential and torsional stresses on the critical cross-section A [4,5,6,7,8].
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 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 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 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 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 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 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 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 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.
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.
Figure 12. Comparison of matching the limit load function of R/B = 8 with other ring sizes in the range of the standard crack aspect ratio a/W = 0.45 to 0.55 [4,5,6,7].
Figure 12. Comparison of matching the limit load function of R/B = 8 with other ring sizes in the range of the standard crack aspect ratio a/W = 0.45 to 0.55 [4,5,6,7].
Figure 13. Obtained values of the limit load for all geometries from 5 to 20 and a crack or notch depth a/W in relation to the limit load of the same geometry in a solid section ring [4,5,6,7].
Figure 13. Obtained values of the limit load for all geometries from 5 to 20 and a crack or notch depth a/W in relation to the limit load of the same geometry in a solid section ring [4,5,6,7].

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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

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