[Skip to content]

TWI
Search our Site
.

A review of K-solutions for through-thickness flaws in cylinders and spheres

Ali Mirzaee Sisan1, Isabel Hadley1, Sarah E Smith1 and Mike Smith 2

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, KI.[1,2] KI 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 KI 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≤ Ri/t ≤10, and suggested that differences between solutions were greatest at low Ri/t. The current study considered a much wider range of Ri/t, from as low as 1 to as high as 100, if permitted by the validity limits of the solution.

Fig.1. Through wall axial crack in a cylinder

Fig.2. Through wall circumferential crack in a cylinder or meridional crack in a sphere

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.

 

spamsjuly07f18.gif

Fig.18. SIF for a sphere with a meridional crack (Ri/t=3, 100MPa membrane stress)

spamsjuly07f19.gif

Fig.19. SIF for a sphere with a meridional crack (Ri/t=60, 100MPa membrane stress)

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]

spamsjuly07f20.gif

Fig.20. SIF for a sphere with a meridional crack (Ri/t=3, 100MPa through wall bending stress)

spamsjuly07f21.gif

Fig.21. SIF for a sphere with a meridional crack (Ri/t=10, 100MPa through wall bending stress)

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 (Ri/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 Rm/t, not Ri/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 Ri/t=1 to 100 whereas Zang[15] does not provide solutions for Ri/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 Ri/t=1 to 100, whereas Zang[15] does not provide solutions for Ri/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 Ri/t=3-9. It is evident that Zang[15] and Anderson[16] give similar results for a range of Ri/t=5-10 while generally Zang[15] solutions provide slightly lower values. API 579[3], Zang[15] and Anderson[16] agree well for Ri/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 Ri/t<5 and very short cracks (see minimum reported value for 2π for each Ri/t in Zang[15]). Anderson[16] is conservative among the different solutions for long cracks at higher Ri/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 Ri/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 (Ri/t=20) when using Anderson's solution. A pragmatic suggestion will be for any 10<Ri/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 Ri/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<Ri/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 Ri/t=50 and Ri/t=60 for short cracks. Anderson[16] is conservative among the different solutions for long cracks at higher Ri/t>10. It should be noted that Zang does not cover cases where Ri/t<5 and very short cracks. Anderson[16] is conservative among the different solutions for long cracks (approximately λ>4.5) at Ri/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 andFracture 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 PressureVessels 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.