Calculation of Stress Intensity Factors Corresponding to Standard Residual Stress Profiles for Common Weld Joint Geometries
R H Leggatt (TWI, Cambridge, UK)
A Stacey (Health and Safety Executive, London, UK
ASME Pressure Vessel and Piping Conference Symposium on Weld Residual Stresses and Fracture 23-27 July 2000
Improved methods for including residual stresses in defect assessment procedures are becoming available. The latest generation of guidance documents, including BS 7910, API 579 and reports from the European SINTAP project, provide standard residual stress distributions for common welded joint configurations. Some also give stress intensity factor solutions for the non-linear distributions which are characteristic of residual stresses. In this paper, a standard distribution of residual stresses is referred to as a 'residual stress profile' or RSP, and K RSP
is the stress intensity factor at a defect subject to a RSP.
The new procedures for obtaining the RSP and K RSP provide a practical alternative to the traditional assumption that 'residual stresses at welds are equal to yield'. The excessive conservatism which may be associated with that assumption can be reduced or eliminated. However, these new capabilities bring with them the danger of a 'black-box' syndrome. There are many different RSPs for different joint geometries and defect orientations, and the associated stress intensity factor solutions are complex. The fracture analyst may not be aware of the nature of the assumed residual stress distribution, or of the sensitivity of the assessment to the assumed residual stresses.
Examples are given in this paper of the variation of K RSP as a function of crack size for a range of defect and weld configurations including long and short surface defects at butt and T-butt welds in plates and circumferential butt welds in cylinders. In order to demonstrate the benefits of using the standard residual stress profiles, the K RSP values are compared with the stress intensity factor for the same defect subject to uniform yield magnitude residual stresses. A new solution is given for the K RSP at a through-wall transverse crack subject to residual stresses having the trapezoidal shape given in the SINTAP and API 579 compendia for profiles of surface residual stresses.
An interesting new development in residual stress technology is the compilation of standardised residual stress profiles (RSPs) for common types of welded joints. Within the last year, three new compendia of RSPs have been published. These may be found in Appendix Q of British Standard BS 7910:1999 
, in the final report 
from Task 4 of the European collaborative project SINTAP, and in Appendix E of the draft API 579 
. Stress intensity factor solutions for the non-linear distributions which are characteristic of RSPs in welded joints are given in Appendix C of API 579 
and in a report 
from Task 2 of the SINTAP project.
This paper starts with a summary and comparison of the RSPs in the three documents listed above [1-3] . Examples are given of stress intensity factors for a range of common defect and weld combinations, mostly calculated using the profiles and stress intensity solutions given in reports [2,4] from the SINTAP project. These examples illustrate the variation of stress intensity with crack depth, and the benefits to be obtained by assuming a non-linear RSP, as opposed to the traditional assumption of uniform residual stresses equal to the yield strength. Further examples are given in another SINTAP report  .
Comparison of published residual stress profiles
summarises the RSPs available in BS 7910 
Appendix Q, the SINTAP compendium 
and API 579 
Appendix E, categorised by joint type. The first compendium of RSPs known to the present authors was compiled by Mathieson 
in 1991 for use in conjunction with the R6 assessment procedure. Very limited guidance on RSPs was given in BS PD 6493:1991 
(now superseded by BS 7910:1999 
). Bate, Green and Buttle 
reviewed the source data and RSPs in References 6
, and considered additional data with particular reference to offshore construction. They provided recommendations for revisions to the R6 compendium 
. The current British 
and European 
compendia are based on further review and refinement of References 6
, and many of the RSPs are common to both documents and are similar or identical to profiles given in the earlier documents. The profiles in API 579 
Appendix E have some similarities with the BS 7910 
and the SINTAP compendia 
: many have similar shapes; like SINTAP, API 579 gives four profiles for each joint type, namely the through-wall and surface profiles of longitudinal and transverse stresses (BS 7910 gives only through-wall profiles); and many of the same source references are quoted. However, the equations of the profiles in API 579 are entirely different from those in References 1
: it is evident that they are based on an independent review of the source data.
Table 1. Comparison of residual stress profiles in BS 7910, SINTAP and API 579
| ||BS 7910  Appendix Q||SINTAP Compendium of RS Profiles ||API 579  Appendix E|
||Ferritic steels, austenitic steels
||Ferritic steels, austenitic steels, aluminium.
||'Currently, a distinction is not made concerning the material of construction'
||Through-wall and surface profiles.
||Through-wall and surface profiles.
|Straight full penetration butt welds
||Q.1.2 Plate butt welds
Q.1.4 Pipe axial seam welds (same as Q.1.2)
|Fig.1 Plate butt welds and pipe axial seam welds
||E4 Welds in piping and pressure vessel cylindrical shells.
Single-V (E4.3) and double-V (E.4.4) longitudinal welds
E6 Welds in storage tanks.
Single-V (E.6.3) and double-V (E.6.4) longitudinal welds
|Curved full penetration butt welds
||Q.1.3 Pipe circumferential butt welds
||Fig.3 Pipe butt welds
||E4 Welds in piping and pressure vessel cylindrical shells.
Single-V (E4.1) and double-V (E4.2) circumferential welds.
E5 Welds in spheres and pressure vessel heads
Single-V (E5.1) and double-V (E5.2) circumferential welds.
Single-V (E5.3) and double-V (E5.4) meridional welds
E6 Welds in storage tanks.
Single-V (E6.1) and double-V (E6.2) circumferential welds
|T-butt and T-fillet welds
||Q.1.5 T-butt and fillet welds. Stresses in main plate at toe of weld at plate-to-plate, tube-to-plate and tube-to-tube joints. Two alternative profiles are given.
||Fig.2 Plate T-butt welds
(whose heat input is known)
Appendix 3 Plate T-butt welds
(where heat input is not known)
Fig.4 Pipe T-butt welds
|E8 Full penetration and fillet welds at a tee joint
E8.1 Main plate
E8.2 Stay plate
|Nozzle attachment welds
||Fig.5 Set-in nozzle
Fig.6 Set-on nozzle
|E7 Full penetration welds at corner joints
(nozzles or piping branch connections)
E7.1 Corner joint
E7.2 Nozzle fillet weld
E7.3 Shell fillet weld at a reinforcing pad
||Q.1.6 Repair welds.
||Fig.7 Repair welds
||E9 Repair welds
E9.1 Seam welds
E9.2 Nozzle welds
BS 7910 and the SINTAP compendium are intended to be generally applicable for the assessment of flaws in welded structures. API 579 is concerned specifically with pressurised equipment used in the refinery and chemical industry. This difference of scope has led to some differences in the categorisation of weld types, as shown in Table 1. It should be noted that different terminologies are used for describing directions in API 579 compared with the other source documents, as summarised in Table 2. At circumferential welds, the term longitudinal refers to the cylinder axial direction in API 579, and to the circumferential direction (along the weld) in the other documents.
Table 2. Terminology for directions
| ||This paper/BS 7910/SINTAP||API 579|
|Direction of stress relative to weld
|Direction of welding in cylinders
Within the general categories of straight and curved full penetration butt welds in Table 1
, BS 7910 and SINTAP give one set of RSPs for butt welds in flat plates and pipe axial seam welds, and one set of RSPs for circumferential butt welds in pipes. API 579 does not consider butt welds in flat plates, but gives separate sets of RSPs for welds in (i) piping and pressure vessel cylindrical shells, (ii) storage tanks and (iii) spheres and pressure vessel heads. It gives separate RSPs for longitudinal, circumferential and meridional welds, and for single-V and double-V welds.
For T-butt and T-fillet welds, BS 7910 and SINTAP give separate consideration to welds in plated construction and tubular construction. API 579 does not make this distinction, but does give profiles of stresses in the main plate and the stay plate.
With regard to the actual shapes of the profiles, the through-wall RSPs in BS 7910 and SINTAP include linear, bilinear, polynominal (up to 6 th order) and cosine profiles. Those in API 579 are more consistent in shape, all being linear, bilinear or 2 nd or 3 rd order polynominals. The surface RSPs in SINTAP and API 579 are trapezoidal, i.e. uniform inside the weld and decreasing linearly to zero in parent plate adjacent to the weld.
All the RSPs in API 579 and about half of those in BS 7910 and SINTAP are based on upperbound curves fitted to published residual stress data obtained by measurement or numerical modelling. The remainder of the RSPs in BS 7910 and SINTAP, including the surface profiles of longitudinal stresses and the bilinear through-wall profiles at the toes of T-butt and T-fillet welds, are based on a theoretical upper bound model of the extent of the plastically deformed zone adjacent to a weld developed by Leggatt [9,10] . The dimensions of the deformed zone are given as a function of material properties, weld heat input and plate thickness. The theoretically based profiles have been compared with relevant experimental and numerical solutions in the SINTAP project  , and found to be satisfactory. The theoretical model assumes a high restraint against thermal contraction after welding, which is an appropriate assumption for longitudinal stresses in nearly all cases, and for transverse stresses at T-joints in thicker materials (i.e. thicker than about 25mm). However, it may be less realistic in thinner materials, such that the RSPs become increasingly conservative as thickness decreases.
The empirically-based RSPs are strictly valid only within the range of plate thickness and welding conditions for which data were available. They have been non-dimensionalised with respect to plate thickness or weld width. This does not reflect the influence of welding and other parameters included in the theoretical model on the width of the plastically deformed zones adjacent to the welds. Hence the empirically-based RSPs may be over-conservative for thicknesses greater than those for which source data was available, and may be non-conservative for thicknesses below the range of the source data.
Longitudinal surface cracks at straight butt weld
Longitudinal cracks are subject to transverse residual stresses. Through-wall RSPs for transverse residual stresses at butt welds are shown in Fig.1
. The residual stress is normalised with respect to the yield strength, and the depth from the surface, z, is normalised with respect to the wall thickness, t. BS 7910 
and SINTAP 
give the same profile, which is a 6 th
order polynominal and is applicable for plate butt welds and axial seam welds. API 579 gives a 2 nd
order polynominal (Eq.E.34) for single-V longitudinal welds and uniform stresses (Eq.E.43) for double-V longitudinal welds in piping and cylindrical pressure vessels, and a linear profile (Eq.E.73) for single-V welds and a 2 nd
order polynominal (Eq.E.79) for double-V welds in storage tanks.
Fig.1. Through-wall profiles of transverse residual stresses at butt weld
API 579 Eq.E.34 and E.79 are somewhat similar to the BS 7910 and SINTAP profiles, except in the region 0.4<z/t<0.9 where the API profiles are more tensile. API 579 Eq.E43 adopts the traditional conservative assumption of uniform yield magnitude stresses at double-V longitudinal welds in pipes and cylindrical pressure vessels, but Eq.E73 shows stresses decreasing linearly to zero at the root of single-V welds in storage tanks. The present authors would not be confident of always finding zero stress or less at this location: tensile transverse stresses can develop at the root of the weld as it is stretched by angular distortion during the deposition of the fill passes.
The 'normalised K RSP
', i.e. the stress intensity factor corresponding to the standard residual stresses profile divided by Equation.1.
, is plotted versus normalised crack depth for infinite longitudinal surface cracks at straight butt welds in Fig.2
. The K RSP
was calculated using a weight function solution given as Equation AI.3 in the SINTAP stress intensity factor handbook 
. The normalised K RSP
for the BS 7910 and SINTAP profiles is approximately equal to 1.0 for crack depths 0<a/t<0.4, but then starts to rise, despite the presence of compressive stresses in this region. The normalised K RSP
for uniform yield stress Equation.2.
(as in API 579 Eq.E.43) starts at 1.0, but rises faster than the K RSP
for a non-linear profile, as would be expected. This demonstrates the benefit to be obtained by using the non-linear RSP, where this is deemed to be applicable. K RSP
has not been calculated for the linear profile defined by API 579 Eq.E.73, but this may be expected to lie between the solutions shown in Fig.2
Fig.2. Normalised K RSP at longitudinal infinite surface crack at straight butt weld. Effect of residual stress profile
All the K RSP
values discussed above apply to long surface cracks at the side welded last (BS 7910), or at the outside (API 579). For cracks at the opposite face, the normalised K RSP
would tend to zero as the crack depth tends to zero for the RSP defined by API Eq.E.73 and to about 1.0 for the other profiles shown in Fig.1
. This is clearly a significant difference, and further work to confirm the validity of Eq.E.73 in this region would be desirable.
The normalised K RSP for the deepest and surface intersection points of semi-elliptical surface cracks of aspect ratio, 2c/a, equal to 3.33 and 10 are plotted in Fig.3. K RSP was evaluated using a solution for a 5 th order polynominal stress profile  , and hence it was necessary to fit a 5 th order polynominal to the 6 th order BS 7910/SINTAP profile shown in Fig.1.
Fig.3. Normalised K RSP at finite surface crack at butt weld in plate. Effect of crack aspect ratio at deepest point and surface intersection
In contrast to the K RSP
shown in Fig.2
for infinite surface cracks, the normalised K RSP
shown in Fig.3
for the deepest point of finite surface cracks falls continuously with crack depth. The stress intensity factor is always positive, despite the presence of compressive stresses in the range 0.5<z/t<0.8. The normalised K RSP
at the surface intersection points of finite surface cracks is approximately constant at about 0.8 for an aspect ratio of 2c/a = 3.33, and is initially lower but rises with crack depth for a crack of aspect ratio 2c/a = 10. The stress intensity at the surface intersection point is greater than that at the deepest point for a/t > 0.1 with 2c/a = 3.33, and for a/t > 0.5 with 2c/a = 10.
Transverse through-wall cracks at butt weld
Transverse cracks at welds are subject to longitudinal residual stresses. The characteristic trapezoidal profile of longitudinal surface residual stresses recommended in BS 7910, SINTAP and API 579 is illustrated in Fig.4
. For the purpose of assessing through-wall cracks, the usual practice is to assume that the given cross-weld profile of surface stresses is applicable throughout the thickness of the joint. The issue of interest is how the stress intensity factor changes as the crack length increases beyond the width of the tensile zone.
Fig.4. Profile across weld of longitudinal surface residual stresses
A solution for a through-wall crack in an infinite flat plate subject to a trapezoidal RSP was given in Ref 5
and is reproduced here. The solution was obtained by integrating the weight function solution given on page 5.11a of Ref 11
for a symmetrical point load, p, at a distance, y, from the centre of a crack length, 2a, in an infinite plate. For a transverse crack subjected to longitudinal stresses, Equation.3.
the point load, p, is equated with the force Equation.4.
acting on an infinitesimal length, dy, of the crack. Then Equation.5
The solution for the trapezoidal profile shown in Fig.4 is as follows:
The normalised K RSP for through-wall cracks subject to trapezoidal profiles with different widths of tensile zone from W 2/W 1 = 3 to 9 are shown in Fig.5. For a/W 1 ≤ 0.5, the crack is within the weld, and is subject to yield magnitude stresses, giving normalised K RSP=1. For a/W 1>0.5, the crack length is greater than the weld width, and the normalised K RSP falls with crack length, at a rate dependent on the total width of the tensile zone. The use of the trapezoidal RSP removes the conservatism inherent in the traditional assumption of residual stresses equal to yield strength for cracks whose width is greater than the weld width.
Fig.5. Normalised K RSP at transverse through thickness crack at butt weld in plate. Effect of residual stress profile
Longitudinal surface cracks at T-joints
Figure 6 shows the through-wall profiles of transverse residual stresses in the main plate at the toes of T-butt or T-fillet welds. The BS 7910 Eq.Q.13 and SINTAP Fig.4d RSP is a 4 th order polynominal with one peak just below the toe and a second peak near the back surface. The other profiles fall from yield stress at the toe to zero at an internal location. For the bilinear profiles given by BS 7910 Eq.Q.9 and SINTAP Fig.2d, the depth of the tensile zone, z o, is a function of the heat input of the weld pass adjacent to the weld toe and the material properties. These profiles are applicable where z o ≤t. If the validity of the bilinear RSP is accepted, then, for crack depths up to a/t = 0.5, the curved profiles of API 579 E.94, BS 7910 Eq.Q.13 and SINTAP Fig.4d would seem to be over-conservative for z o/t = 0.25, approximately correct for z o/t = 0.5, and unconservative (at greater crack depths) for z o/t = 0.75.
Fig.6. Through-wall profiles of transverse residual stresses at toe of T-butt or T-fillet weld
Normalised K RSP
values for longitudinal surface cracks of aspect ratio 2c/a = 3.33, 10 and infinity at the toe of a T-butt weld are shown in Fig.7
. These were evaluated assuming the bilinear profile shown in Fig.6
with z o
/t = 0.5. They were evaluated using a solution from the SINTAP stress intensity factor handbook 
for a semi-elliptical surface crack in a plate subject to through-wall stress profile expressed as a 5 th
order polynominal. Hence it was necessary to fit a 5 th
order polynominal to the bilinear profile, which gave a very good fit. It can be seen that the normalised K RSP
rises with crack depth for an infinitely long surface crack, but falls for cracks of finite length.
Fig.7. Normalised K RSP at deepest point of short surface crack at toe of T-butt weld for bilinear RSP with z o/t = 0.5. Effect of aspect ratio
The normalised K RSP
for the deepest and surface intersection points of longitudinal surface cracks of aspect ratio 2c/a = 3.33 at the toes of T-butt welds subject to uniform residual stress, Img.5.
or the bilinear profile with z o
/t = 0.5 are shown in Fig.8
. The K RSP
values for the bilinear profiles were calculated for the region 0>a/t>0.5, where the profile is linear, and hence could be expressed in terms of a membrane and bending component. All K RSP
values in Fig.8
were calculated using the well-known solutions by Newman and Raju 
. The normalised K RSP
values rise with crack depth under uniform yield stresses. They fall with crack depth at the deepest point of a crack subject to the bilinear RSP, but remain approximately constant with crack depth at the surface intersection point, which is the controlling location for crack depths, a/t>0.15. Hence, the benefit of using the bilinear profile is smaller than would be expected on the basis of the stress intensity at the deepest point of the crack.
Fig.8. Normalised K RSP at short surface crack at T-butt weld. Effect of residual stress profiles
Circumferential surface cracks at circumferential butt welds
This type of crack is subject to the through-wall profile of transverse residual stresses (i.e. parallel to the cylinder axis) at circumferential butt welds. Published RSPs for these welds are shown in Fig.9
. The SINTAP compendium 
, gives different profiles for values of heat input per unit thickness, q/vt, higher or lower than 60J/mm 2
. The SINTAP low heat input profile is shown for q/vt = 60J/mm 2
: the profile is scaled down at lower values of q/vt, with a limiting scaling factor of 0.5 as q/vt tends to very low value. The SINTAP high heat profile has a cosine distribution and is not subject to scaling. The API 579 'low thickness' profile is applicable for t* ≤20mm when q/v ≤2kJ/mm, and for t* ≤50mm where q/v>2kJ/mm. The API 579 'high thickness' profile is applicable for t* ≥70mm when q/v ≤2kJ/mm, and for t* ≥100mm when q/v>2kJ/mm. t* is a modified thickness parameter dependent on R/t, and is equal to the thickness t for R/t ≤10. The RSP is scaled between the low and high thickness lines at intermediate values of t*.
Fig.9. Through-wall profiles of transverse residual stresses at single-V circumferential butt welds with q/(vt) = 60J/mm 2
Drawings in API 579 and the SINTAP compendium imply that the single sided weld is made from the outside : the profiles would be different if the weld was made from the inside.
API 579 has, in general, adopted a more conservative approach than SINTAP, in as much as the former gives entirely tensile stresses while the latter gives compressive stresses at various locations. The two approaches show some agreement with respect to the stresses at the root of the weld (i.e. at z/t=1). Both methods agree that the stress is a function of heat input, q/v, and thickness, t. API gives yield tensile stresses for low thickness conditions, and SINTAP gives yield tensile stresses for high heat input per unit thickness (i.e. for low thickness). Both methods give reduced tensile stresses at the root for high thickness or low q/vt conditions.
Although the two profiles given by SINTAP have undoubtedly been observed in practice, it is difficult to reconcile the abrupt change in profile at the transitional value of q/vt=60J/mm 2. It is also the case that some observations have been made of zero or low tensile stresses at the outside face, z/t=0. Hence, the API profiles may be considered more appropriate for surface defects located at the outside surface. On the other hand, the tensile peak shown in the SINTAP high heat input RSP at z/t=0.4 has also been observed in practice, and the API 579 profiles may be unconservative for buried defects at this location.
Normalised K RSP values corresponding to the SINTAP high heat and low heat profiles and to uniform yield stresses acting on a full circumferential defect at the root of a circumferential butt weld are shown in Fig.10. These were calculated using a weight function solution. The normalised K RSP for uniform yield residual stresses rises gradually from 1.2 at a/t=0 to 2.0 at a/t=0.6. The normalised K RSP for the SINTAP high heat RSP falls gradually with crack depth from 1.2 at a/t=0, and hence would provide some benefit in terms of reduced K RSP at crack depths a/t>0.2. The normalised K RSP for API low thickness RSP (not shown) would lie between the uniform yield stress and SINTAP high heat curves and hence would be of limited benefit.
Fig.10. Normalised K RSP at fully circumferential crack at outside surface of pipe. Effect of residual stress profile
The normalised K RSP
for the SINTAP low heat RSP falls rapidly with crack depth, and hence would show significant benefit, even for shallow cracks. The normalised K RSP
for the API 579 high thickness RSP (not shown) can be obtained by scaling down the uniform yield stress solution by 0.2, and hence would also show considerable benefit compared with the assumption of uniform yield stresses.
Summary and conclusions
- Three sets of standardised residual stress profiles for common weld joint types have recently become available.
- Many of the profiles given in BS 7910:1999  Appendix Q and the SINTAP project residual stress compendium  are identical: those in API 579  Appendix E show some similarities with the BS 7910 and SINTAP profiles, but have different equations.
- Areas where one or other of the published profiles may be unconservative or over-conservative have been discussed in this paper.
- A SINTAP project report  and API 579  provide stress intensity factor solutions applicable to non-linear residual stress profiles.
- The reduction in stress intensity factor obtained by using the published profiles instead of the traditional assumption of uniform yield magnitude residual stresses has been quantified for a range of examples.
- A new solution for the stress intensity factor at a through-wall crack subject to a trapezoidal residual stress profile is presented.
- Further investigation of the transverse residual stresses at circumferential butt welds is needed.
||half-length of through crack or depth of surface crack
||half-length of finite surface crack
||stress intensity factor
||stress intensity factor at crack subject to RSP
||residual stress profile
||weld travel speed
||width of weld
||total width of tensile zone
||depth of tensile zone
||longitudinal residual stress
||transverse residual stress
||parent metal yield strength
||weld metal yield strength
This paper was based in part on work done at TWI under sub-contract to the Health and Safety Executive, as part of SINTAP, a collaborative project supported by the European Commission. The authors acknowledge the support of their project partners and sponsors.
||'Guidance on methods for assessing the acceptability of flaws in fusion welded structures'. British Standards Institution, April 1999.
||Barthelemy J Y:
||'SINTAP Task 4. Compendium of residual stress profiles'. Final Report. Institut de Soudure, May 1999.
||'Recommended practice for fitness-for-service'. Issue 12, September 1999 (Special Release).
||Al Laham S:
||'SINTAP Task 2.6. Stress intensity factor and limit load handbook'. British Energy Generation Ltd, EPD/GEN/REP/0316/98, Issue 2, April 1998.
||Leggatt R H and Sanderson R M:
||'Stress intensity due to residual stresses TWI report SINTAP/TWI/4-6 88269/46/99, September 1999.
||Mathiesen P A R:
||'A compendium of as-welded residual stress profiles'. Nuclear Electric memorandum TD/SIB/MEM/0233, September 1991.
||'Guidance on methods for assessing the acceptability of flaws in fusion welded structures'. BSI Published Document PD 6493:1991.
||Bate S K, Green D and Buttle D J:
||'A review of residual stress distributions in welded joints for the defect assessment of offshore structures'. Offshore Technology Report OTH 95-482. London, The Stationery Office, September 1995.
||Leggatt R H:
||'Welding residual stresses'. In: The 5 th International Conference on Residual Stresses (ICRS5), Linköping, Sweden, 16-18 June, 1997. Eds Ericsson T, Oden M, Andersson A, Linköping University, Sweden. Vol.1.pp12-25.
||Leggatt R H:
||'Recommendations for revised surface residual stress profiles'. TWI report SINTAP/TWI/4-3 88269/4-3/98, June 1998.
||Tada H, Paris P and Irwin G:
||'The stress analysis of cracks handbook'. Del Research Corporation, Second Edition, 1985.
||Newman J C and Raju I S:
||'Stress-intensity factor equations for cracks in three-dimensional finite bodies subjected to tension and bending loads'. NASA Technical Memorandum 85793, April 1984.