K-Solutions for Cylinder and Sphere Through-Thickness Flaws

¹TWI Ltd, Cambridge, UK
²British Energy Generation Ltd, Gloucester, UK

Paper presented at International Conference on Pressure Vessels and Piping, PVP2007/CREEP 8 Conference July 22-26, 2007, San Antonio, USA. Paper PVP200726714.

Abstract

This paper reviews different stress intensity factor solutions for a wide range of configurations and loading conditions for a cylinder with axial and circumferential through thickness cracks and a sphere with through thickness meridional (equatorial) cracks. The most appropriate solutions to use are identified.

Introduction

One of the key inputs in defect tolerance assessments for fitness-for-service is the Mode one stress intensity factor, or SIF, K_I.^[1,2] K_I depends on the component's geometry and loading configuration. There are a number of solutions for calculating SIF given in defect assessment procedures and handbooks for cracks under different conditions.^[2-6] The concept of Leak-Before-Break (LBB), which deals with circumstances in which a crack reaches a length associated with instability, requires a profound understanding of the behaviour of through-wall cracks in cylinders and spheres. Earlier studies for evaluating SIF for cylinders and spheres are based mainly on thin-shell theory.^[5-12] More comprehensive solutions for a wider range of geometry and load configurations have also been obtained by using finite element analyses. These include work by Green and Knowles,^[13] France et al,^[14] Zang^[15] and Anderson.^[16] Defect assessment procedures such as R6,^[2] API 579^[3,17] and BS 7910^[4] use earlier analytical and finite element solutions for calculating K_I for through wall cracks in cylinders and spheres. Smith,^[18]Takahashi^[19] and Miura et al^[20] have compared the available SIF solutions for these geometries. These comparisons have identified some differences between published SIF solutions, raising the issue of the most appropriate solution to use in different circumstances.^[1,18] The study described in this paper aims to complete the earlier work^[18] by studying more cases employing different solutions.

Geometry and loading conditions

This study was confined to through wall cracks in cylinders and spheres (Figs.1 and 2). Previous work^[18] had been restricted to a range for the inner radius to thickness ratio of 5≤ R_i/t ≤10, and suggested that differences between solutions were greatest at low R_i/t. The current study considered a much wider range of R_i/t, from as low as 1 to as high as 100, if permitted by the validity limits of the solution.

In order to compare different solutions, a cylinder with a thickness of 100mm was chosen. The inside radius, R_i, was varied to get R_i/t= 3, 5, 6, 7, 8, 9, 10, (50 and/or 60 and/or 100). A sphere with a thickness of 100mm was also chosen. The inside radius, R_i was varied to get R_i/t=1, 3, 5, 10, 20, and 50. The selection of a thickness of 100 mm for cylinders and spheres does not affect comparisons of different solutions.

Two types of loading were chosen for a cylinder with a through wall axial crack: membrane and through wall bending. The magnitude of the membrane stress was chosen as 100MPa, and ±100MPa in case of through wall bending (positive on the outer surface of the cylinder). An additional loading (global bending) was considered for a cylinder with a circumferential crack. A moment of 10⁹ N.mm (or 10¹⁰ N.mm) was applied as a global bending moment in a cylinder with a circumferential crack. Spheres were subjected to membrane and through-wall bending. As for the cylinder, the membrane stress was 100MPa, and the through wall bending stress was ±100MPa (positive on the outer surface).

Comparisons of different solutions

In the following sections, comparisons between different solutions are given. However, only limited figures are shown in this paper, so as to emphasise the comparisons. More details of other comparisons (different R_i/t) can be found in Mirzaee-Sisan and Smith.^[22]

All the results given below are plotted in terms of Mode one stress intensity factor, K_I, in (MPa.m^0.5) against mid wall crack length (mm). The range of crack length compared varies for different cases. Note that the length of cylinder was assumed infinite in this study. The legends in the graphs refer to their corresponding references:

R6: based on the R6 document issued in 2001^[2]
API: API 579 issued in Jan 2000^[3]
Zang: Solutions provided within the SINTAP project^[15]
Anderson: based on original FE data provided in Anderson.^[16]
Crackwise: calculated by Crackwise software^[21] which is based on BS 7910.^[4]

In the case of the cylinder, France et al^[14] and API 579^[3] would be expected to provide the same results. But this was not the case in the recent study.^[18] Two errors in the current API 579:2000^[3] standard were identified^[18]: crack length is shown at the outer surface rather than at the mid wall, and the tables of coefficients given in API 579 are in fact in terms of R_m/t, not R_i/t. Smith^[18] showed these errors by comparing one case: a cylinder with R_i/t=5 using API 579^[3] was shown to produce the same result as a cylinder with R_i/t=4.5 using France et al's^[14] data directly. The study described in this paper uses Crackwise software^[21] to simulate further API 579:2000 solutions. Crackwise is based on BS 7910,^[4] which in turn correctly implements the France et al^[14] solution, but can be used to replicate the API 579 solutions by comparing R_i/t=9.5 (Crackwise solution) with R_i/t=10 (API 579:2000 solution) and so on. API 579^[3] solutions in all graphs, including calculation of λ (an indication of crack length), are based on the mid-wall crack length. With these assumptions, the Crackwise^[21] simulations and API 579^[3] solutions match, underpinning the earlier suggestion^[18] regarding the existence of errors in API 579.^[3]

Cylinder with axial defects

Figures 3 and 4 plot all the different SIF solutions for a cylinder with an axial through wall crack under membrane loading for R_i/t=7 and R_i/t=10. Results are shown for inner and outer surface positions. There is reasonable agreement among all solutions for R_i/t>=10, as typically shown in Fig.4. Zang^[15] and Anderson^[16] agree very well for the SIF inside the cylinder, but Zang^[15] gives slightly lower for the outside. Excellent agreement between API 579^[3] and Crackwise is evidence for the fact that the coefficients in API 579^[3] should be in terms of R_m/t, not R_i/t. Zang^[15] does not provide any solution for R_i/t<5. In the limit of a very small crack, Zang^[15] also does not provide any solution and the minimum reported coefficient corresponds to 2c/t=1, where '2c' is the crack length and 't' is the thickness. The solution in the R6 document^[2] is valid for R_m/t>=10.

Figures 5 and 6 plot the different solutions for the case of a through wall axial crack in a cylinder under through wall bending for R_i/t=6 and R_i/t=10. Different solutions agree very well for any value between R_i/t=5 to R_i/t=50, see typical comparisons shown in Fig.5 and 6. However, Zang^[15] does not provide any solution for R_i/t<5. In the limit of a very small crack, Zang^[15] also does not provide any solution and the minimum reported coefficient corresponds to 2c/t=1. The solution in the R6 document^[2] is valid for R_m/t>=10.

Cylinder with circumferential defects

Figures 7 to 10 plot different SIF solutions for a cylinder with a circumferential through wall crack under membrane stress. In case of R_i/t=3 shown in Fig.7, the current version of API 579^[3] and Anderson^[16] are shown only where API 579^[3] gives a higher value compared with Anderson.^[16] Half circumferences are shown as '180 degree' in the graphs. For 3<R_i/t<=10, Crackwise and API 579^[3] predict similar results, both of which are higher compared to Zang^[15] and Anderson,^[16] as shown in Figs.8 and 9. Zang^[15] and Anderson^[16] agree very well but Zang^[15] gives slightly lower values for 5<=R_i/t<=10. In the case of a much thinner cylinder, ie increasing the ratio of R_i/t to 50 and 100, all solutions are consistent for short cracks. Zang^[15] does not provide any solution for R_i/t<5. In the limit of a very small crack, Zang^[15] also does not provide any solution and the minimum reported coefficient corresponds to 2θ/Π = (0.006-0.06), where '2θ' is the crack length in terms of angle. Anderson^[16] produces the highest results for long cracks, as shown in Fig.10.

Figures 11 and 12 plot different SIF solutions for a cylinder with a circumferential through wall crack under through wall bending stress. In the case of R_i/t=3, shown in Fig.11, the current version of API 579^[3] and Anderson^[16] are used only. As shown in Fig.11, API 579^[3] is different from Anderson^[16] just for crack lengths close to half circumference. For 5<R_i/t<=10, all solutions agree well, but there are slight differences in K_I inside the cylinder where crack lengths are close to half circumference. Zang^[15] does not provide any solution for R_i/t<5. In the limit of a very small crack, Zang^[15] also does not provide any solution and the minimum reported coefficient corresponds to 2θ/π = (0.006-0.06). Figure 13 compares Anderson's solution^[16] for cases of R_i/t=10, 20 and 60. It has been noticed that there is an unrealistic behaviour in the SIF inside the cylinder in the case of R_i/t=20, using Anderson's solution for higher crack lengths. Figure 14 shows the variation of SIF coefficient for through wall bending using Anderson's solution,^[16] ie (G₀-2G₁) for R_i/t=10, 20 and 60. It shows that R_i/t=20 has a different trend compared to R_i/t=10 and R_i/t=60. As indicated above, the behaviour of Anderson's solution with R_i/t=20 for longer cracks (approximately at λ>2.8) is anomalous. Therefore this behaviour will affect any interpolations between R_i/t=10 and R_i/t=60, eg R_i/t=50.

Figures 15 to 17 plot different SIF solutions for a cylinder with a circumferential through wall crack under global bending stress. For R_i/t=3 to R_i/t=10, a global moment of 109 N.mm has been applied, while for R_i/t=50 a global moment of 10¹⁰ N.mm has been applied. As illustrated in Figs.15 and 16, all solutions agree well for 5<=R_i/t<=10, although Anderson^[16] predicts slightly higher values compared to other solutions. For R_i/t>10, Anderson^[16] diverges from Zang^[15] and API 579 for longer cracks as shown in Fig.17. Zang^[15] does not provide any solution for R_i/t<5. In the limit of a very small crack, Zang^[15] also does not provide any solution and the minimum reported coefficient corresponds to 2θ/Π =(0.006-0.06).

Spheres with meridional defects

Different solutions for SIF calculations for a sphere with a meridional crack under 100MPa membrane stress are shown in Fig.18 and 19. The comparisons are made for the API 579,^[3] Anderson^[16] and the R6 solutions.^[2] There are good consistencies between Anderson^[16] and the R6 document.^[2] The behaviour of Anderson's solution for longer cracks (approximately at λ=6.5), where the outside SIF decreases with increasing crack length, is anomalous. Therefore for longer cracks Anderson's solutions should be used with caution.

Figures 20 and 21 show comparisons among the different solutions for a meridional crack under ±100MPa through wall bending stress. As for the previous case (meridional crack under 100MPa membrane stress), good consistency was seen between Anderson^[16] and R6 documents.^[2] However, there are slight deviations between R6^[2] and Anderson^[16] for longer cracks (see Fig.21). There is good agreement between API 579^[3] and Crackwise but both solutions are significantly different from those of Anderson^[16] and R6. Anderson^[16] and R6 solutions give much lower values compared to the current API 579 procedure.^[3]

Concluding remarks

Based on comparisons among the different solutions provided in different references, some recommendations are made below as to the most appropriate solutions for each geometry under each type of loading. It should be noted that for circumferential cracks in cylinders and meridional cracks in spheres, all recommendations are for crack lengths less than a half a circumference.

Generally, Zang and Anderson are the recommended solutions for all cylindrical cases studied in this work. Zang shows a very well behaved trend in cylinders, but it has some limitations in thick cylinders (R_i/t<5) and very small crack size. Anderson is easier to implement (compared to Zang) and has wider range of applicability but it should be used with caution for the case of long circumferential cracks, where it shows much higher values of SIF compared with Zang. A previous study by British Energy^[18] also showed that the Anderson solution^[16] in general is conservative compared to the Zang solution.^[15] From this and the previous study^[18] it appears that the coefficients in API 579^[3] should be in terms of R_m/t, not R_i/t. It is understood that this is no longer an issue in the new version of API579^[17] since this adopts Anderson's SIF solutions^[16] for cylinders and spheres.

In cylinders with an axial crack under membrane stress, there is good agreement among the different solutions. The current API procedure^[3] estimates a higher value compared with Zang^[15] and Anderson.^[16] Anderson^[16] and Zang^[15] give similar results, with some deviation for the stress intensity factor at the outer surface. It is difficult to recommend a particular solution. The Anderson^[16] solution covers R_i/t=1 to 100 whereas Zang^[15] does not provide solutions for R_i/t<5. Zang^[15] also does not provide solutions in the limit of a very small crack. In cylinders with an axial crack under through wall bending, there are negligible differences among the different solutions. It is difficult to recommend a particular solution. All solutions provide reasonable results. The Anderson^[16] solution covers R_i/t=1 to 100, whereas Zang^[15] does not provide solutions for R_i/t<5. In the limit of a very small crack, Zang^[15] also does not provide solutions.

In cylinders with a circumferential crack under membrane stress, the API procedure^[3] gives a higher SIF compared with recent Anderson^[16] and Zang^[15] solutions for a range of R_i/t=3-9. It is evident that Zang^[15] and Anderson^[16] give similar results for a range of R_i/t=5-10 while generally Zang^[15] solutions provide slightly lower values. API 579^[3], Zang^[15] and Anderson^[16] agree well for R_i/t>10 but not for long cracks. The results suggest that Anderson's^[16] and Zang's^[15] solution are good choices, but it should be noted that Zang does not cover cases where R_i/t<5 and very short cracks (see minimum reported value for 2θ/π for each R_i/t in Zang^[15]). Anderson^[16] is conservative among the different solutions for long cracks at higher R_i/t>10.

In cylinders with a circumferential crack under through wall bending, there is much less deviation among the different solutions for a range of R_i/t=5-10 and very good agreement exists among all solutions. It has been noticed that there is an unrealistic behaviour for the SIF for the inside value for the case of longer cracks (R_i/t=20) when using Anderson's solution. A pragmatic suggestion will be for any 10<R_i/t<60 and λ> (approximately) 2.8, any interpolation for Anderson's solution^[16] should be treated with care since it may generate very different SIF values compared to other solutions. For R_i/t=60, the agreement between Anderson^[16] and other solutions is good but not for longer cracks where Anderson^[16] diverges from Zang's^[15] solution.

In cylinders with a circumferential crack under global bending loading, all solutions agree very closely for 3<R_i/t<=50, with Zang^[15] and Anderson^[16] giving similar results, while Anderson^[16] gives somewhat higher values. API 579^[3], Zang^[15] and Anderson^[16] match very well for higher R_i/t=50 and R_i/t=60 for short cracks. Anderson^[16] is conservative among the different solutions for long cracks at higher R_i/t>10. It should be noted that Zang does not cover cases where R_i/t<5 and very short cracks. Anderson^[16] is conservative among the different solutions for long cracks (approximately λ>4.5) at R_i/t>10.

There is high scatter between the different solutions for spheres. Generally, Anderson's solutions^[16] are recommended for all sphere cases studied in this study due to their wider range of applicability. But it should be used with caution for the case of longer cracks (approximately λ>6.5) subjected to membrane loading, where the outside SIF decreases with increasing crack length. Anderson's^[16] solutions give much lower value compared to the current API 579^[3] procedure. R6^[2] also agrees well with Anderson^[16] in its range of validity limits.

Acknowledgments

This work has been sponsored by British Energy Ltd, UK.

References

1. R5/R6 newsletter, Number 31, July 2005.
2. R6: 'Assessment of the integrity of structures containing defects', Rev 4, British Energy, 2001, (including amendment record no.5, May2006).
3. API Recommended Practice 579, Fitness-for-Service, API Publishing Services, First edition January 2000.
4. BS 7910: 'Guide on methods for assessing the acceptability of flaws in metallic structures', BSI, 2005.
5. Murakami Y: 'Stress intensity factors handbook', Pergamon Press, Oxford, 1987, pp.1356-1358.
6. Tada H, Paris P C and Irwin G: 'The stress analysis of cracks handbook', Third Edition, ASME, New York, 2000.
7. Chell G C: 'ADISC: A computer program for assessing defects in spheres and cylinders', CEGB Report TPRD/L/MT0237/M84, 1984.
8. Delale F and Erdogan F: 'Effect of transverse shear and material orthotropy in a cracked spherical cap', Int J Solids Structures, 15, 1979, pp.907-926.
9. Folias E S: 'On the effect of Initial Curvature on Cracked Sheets,' International Journal of Fracture Mechanics, Vol.5, No.4, 1969, pp.327-346.
10. Erdogan F and Kibler J J: 'Cylindrical and Spherical Shells with cracks,' International Journal of Fracture Mechanics, 5, 1969, pp.229-237.
11. Zahoor A: 'Closed form expressions for fracture mechanics analysis of cracked pipes', J Pressure Vessel Technology, 107, N2, 1985, pp203-205.
12. R-Code: 'Software for assessing the integrity of structures containing defects', Version 4.1, British Energy Generation Ltd, 2001.
13. Green D and Knowles J: 'The treatment of residual stress in fracture assessment of pressure vessels,' Journal of Pressure Vessels and Technology, Vol. 116, American Society of Mechanical Engineers, 1994, pp.345-352.
14. France C C, Green D, Sharples J K and Chivers T C: 'New stress intensity factors and crack opening area solutions for through-wall cracks in pipes and cylinders', ASME PVP Conference Fatigue and Fracture 1997, PVP-Vol. 350, pp.143-195.
15. Zang W: 'Stress intensity factor solutions for axial and circumferential through-wall cracks in cylinders', SAQ Report SINTAP/SAQ/02, 1997.
16. Anderson T L: 'Stress intensity and crack growth opening area solutions for through-wall cracks in cylinders and spheres', WRC Bulletin 478, 2003, ISSN 0043-2326.
17. API Recommended Practice 579, Fitness-for-Service, API Publishing Services, Draft, 2005.
18. Smith M C: 'SIF solutions for through-wall defects in cylinders', BE Report, 08/04/04.
19. Takahashi Y: 'Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part I: stress intensity factor and limit load solutions', Journal of Pressure Vessels and Technology, Vol. 79, 2002, pp.385-392.
20. Miura N, Takahashi Y, Shibamoto H and Inoue K: 'Comparison of stress intensity factor solutions for cylinders with axial and circumferential cracks', SMiRT18, Beijing, 2005.
21. Crackwise: 'Software of fracture and fatigue assessment procedures BS 7910) for engineering critical assessment', TWI Ltd, 2005.
22. Mirzaee-Sisan, A., Smith, S.E., 'Review of K-Solutions for through-thickness flaws in cylinders and spheres, TWI report 16492/1/06 for British Energy Ltd.