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Mechanical Loading and Residual Stress / Fracture: Pt I


Effects of Mechanical Loading on Residual Stress and Fracture: Part I: Background to the BS 7910:2013 Rules

Isabel Hadley and Simon Smith

TWI Ltd Cambridge, UK

Presented at Proceedings of the ASME 2014 Pressure Vessels & Piping Conference (PVP2014), July 20-24, 2014, Anaheim, California, USA


Failure of welded structures due to the presence of flaws is typically driven by a mixture of applied and residual stresses, yet in most cases only the former are known accurately. In as-welded structures, a typical assumption is that the magnitude of welding residual stress is bounded by the room temperature yield strength of the parent material. The UK flaw assessment procedure BS 7910:2013 also assumes that mechanical loading (either as a result of proof testing or during the initial loading of an as-welded structure) will bring about a relaxation in residual stress. Conversely, the UK structural assessment code for nuclear structures, R6, contains a warning on the ‘limited validation’ of the BS 7910 approaches for stress relaxation and suggests that they should be used ‘with caution’. The aim of this study was therefore to review the basis of the BS 7910 clauses on stress relaxation with a view to harmonising the BS 7910 and R6 rules for cases in which the original welding residual stress distribution is not known.

The residual stress relaxation clauses of BS 7910:2013 date back to the 1991 edition of PD 6493 and have not changed substantially since then. A considerable programme of work was carried out by TWI at the time to justify and validate the clause, but the full underlying details of the work have not hitherto been available in the public domain, and are described in a separate companion paper. The approach proposed in BS 7910 combines ‘global’ relaxation of residual stress (Qm) under high mechanical load with ‘local’ enhancement of crack tip driving force through the adoption of a simplified primary/secondary stress interaction factor, ρ.


There are two major fracture mechanics assessment procedures used in the UK for pressure equipment operating at ambient and low temperatures: the R6 procedure1 (principally applied to nuclear plant) and BS 79102 (previously PD 6493), which is used mainly in non-nuclear applications, including pipelines, pressure vessels and pressure piping (it is also used for non-pressurised structures such as bridges and offshore structures). The procedures are similar in many respects, although they are developed and maintained by different user groups. One of the aspects they have in common is the use of a hierarchical approach to fracture assessment, whereby the analysis becomes more accurate and less conservative as the user moves through the available ‘Options’, from Option 1 to Option 3 (older versions of BS 7910 and PD 6493 used ‘Levels’ 1-3, where the concept is similar, but not identical)3,4,5.

Both BS 7910 and R6 emphasise the importance of distinguishing between primary stresses (which contribute to failure of cracked bodies by both fracture and plastic collapse) and secondary stresses, which contribute to fracture only. Welding residual stress is typically treated as a secondary stress, and is notoriously difficult to determine reliably, whether by measurement or modelling. Consequently, the treatment of welding residual stress in both R6 and BS 7910 is, like that of fracture in general, subject to a hierarchical approach tailored to the amount of information available to the user. In R6 terminology, there are three possible ‘Levels’ of treatment of welding residual stress. A ‘Level 1’ treatment assumes, in the absence of any better information, that the residual stress is uniform across the thickness of the welded joint. ‘Level 2’ assessment makes use of idealised upper-bound residual stress profiles, based on a range of experimental data for particular types of joint. ‘Level 3’ assessment is potentially the most accurate approach, and typically involves direct measurement of residual stresses in a mock-up of the actual joint to be analysed.

An example to illustrate the use of these three levels of residual stress profile is shown in Figure 1, taken from work carried out by Eren et al6. This shows various ways of estimating/measuring residual stresses perpendicular to a 28mm thick girth weld in a large-diameter pipe spool, ie stresses parallel to the pipe axis. Based purely on the yield strength of the pipe material, a Level 1 estimate of residual stress would be 490N/mm2, acting as a uniform membrane stress across the section. A Level 2 profile, estimated from the characteristics of the weld procedure and the pipe properties, shows an area toward the inside of the pipe wall where the residual stress is lower (but still tensile – around 200N/mm2). Residual stress measurements made using neutron diffraction techniques show data well below the Level 1 and Level 2 profiles, in this case never exceeding 100N/mm2.

The higher Levels of residual stress treatment would normally be used with the higher fracture assessment Options (see Table 1), but the two are not explicitly linked.

A new version of BS 79102 has recently been published; in this document, the treatment of welding residual stress is assumed to be either uniform across the section thickness of the component (Level 1) or non-uniform (Levels 2 and 3), and a revised compendium of upper bound residual stress profiles is given in Annex Q of the procedure7. In practice, however, the BS 7910 non-uniform stress profiles are rarely used, for a variety of reasons. One reason is that welding residual stress depends on a multitude of factors: the joint type, thickness and restraint conditions, welding process and heat input, and materials properties, to name just a few. Annex Q is simply a compilation of data from a variety of sources, and it can be difficult to know when it is appropriate to apply data derived in one joint to the analysis of another, nominally similar, joint. According to Annex Q, ‘the user should confirm that the residual stress distributions are representative of the welded joint being assessed’ (this could, of course, be taken to mean direct measurements and/or modelling, ie a Level 3 approach). Moreover, there is no information in BS 7910 on how the assumed stress distributions of Annex Q should be modified to take into account factors that could affect residual stress, such as load history, post-weld heat-treatment or crack growth.

This paper therefore concentrates on residual stresses that are treated as uniform (‘Level 1’ residual stress approach), in particular on the guidance given in BS 7910 on mechanical relaxation of residual stresses under loads, either during a proof test or during service of a welded structure. Whilst BS 7910 includes advice on these issues, R6 contains a warning on the ‘limited validation’ of the BS 7910 approaches for both cracked and uncracked structures and suggests that they should be used ‘with caution’. Unfortunately, the only alternatives are to assume the same stress profile as that for the as-welded structure (this is likely to be highly conservative) or to determine the residual stress field directly after stress relaxation using Level 3 methods (time-consuming, expensive and destructive). There is a clear need to understand the reasons for the different approaches of the two procedures, and to unify the approach in future if possible.

This paper presents the history of the relevant clauses in BS 7910, and summarises the research work underlying them. A separate companion paper8 describes validation of the stress relaxation rules against experimental data and FE analyses that were available at the time (late 1980s).

Treatment of welding residual stresses in PD 6493 and BS 7910

The current (2013) BS 7910 fracture and fatigue assessment procedures are derived mainly from earlier editions of BS 7910, the forerunner document PD 6493, the R6 and FITNET9 procedures, as summarised in Figure 2. The treatment of residual stress has evolved considerably since the first edition of PD 6493 in 1980, as shown by the brief summary below.

PD 6493: 1980

It was recognised very early on that both primary and secondary stresses (including welding residual stress) contribute to the loading of a defect. In the first edition of PD 6493, welding residual stress effects were included in fracture assessment simply by adding a uniform residual stress (typically yield magnitude, in the absence of any other information) to the primary stress and calculating the appropriate value of crack driving force, KI. Kr was calculated as a simple ratio of KI to Kmat, the fracture toughness of the material where the flaw was sited. Alternatively, the CTOD driving force, δI, was calculated from the elastic KI via the CTOD design equation and compared with the characteristic materials toughness in terms of CTOD, ie δrImat

PD 6493: 1991


During the 1980s, extensive research on the incorporation of residual stress into flaw assessment procedures was carried out, in particular by the UK nuclear power industry (in support of R6) and TWI (in support of PD 6493). As shown in Figure 2, a hierarchical approach to fracture assessment (termed Levels 1, 2 and 3) was adopted in PD 6493:1991, analogous (but not identical) to the Options 1-3 used in R6. 

Particular attention was given at this stage in the development of PD 6493 and R6 to the relative contributions of primary and secondary stresses to the crack tip driving force. Their combined effect on crack tip loading can be determined by superposition when the component is linear elastic, but the situation is more complicated when the material yields and displays a non-linear curve of stress plotted against strain.

Ainsworth10 published a method, based on the ‘additive factor’, ρ, in 1986. He developed a complete theory of the interaction of primary and secondary stresses and proposed a simplified approach with ρ either positive or zero (even though his complete theory had predicted that ρ could be negative). This was published in Rev. 3 of R6 and also adopted in the Level 2 and Level 3 procedures of PD 6493:1991, so that:


The factor ρ is, in turn, derived from the factor γ, which is related to the relative values of secondary and primary stress intensity factor and the proximity to plastic collapse, Lr:




is the elastic value of Stress Intensity Factor (SIF) from secondary stresses alone and


is the SIF due to primary stresses alone.

TWI research on relaxation of residual stresses

At this stage in the development of assessment procedures (late 1980s), the offshore industry in particular had gained extensive experience with the application of PD 6493. The BSI committee apparently wished to adopt some of the R6 methods, preferably with some simplifications. Moreover, there was a desire to calibrate what was known about the safety factors inherent in the PD 6493:1980 procedure against the implications of using a modified R6 procedure and to ensure that the change in safety factors on moving from a PD 6493:1980 assessment to a modified R6 analysis was similar for parent materials and as-welded joints. However, the direct application of the additive factor from R6 Rev. 3 method to as-welded structures (when primary and secondary stresses were elastically summed as per PD 6493:1980) led to very high estimates of KI that appeared to be inconsistent with the behaviour observed in as-welded structures and test specimens. 

The research carried out by TWI suggested that, under conditions of high primary stress, a ‘global’ relaxation of welding residual stresses could be assumed, coupled with a ‘local’ enhancement of crack driving force that could be expressed by the plasticity interaction factor, ρ, derived from the R6 procedure. The result was that the 1991 edition of the procedure adopted both residual stress relaxation (which has the effect of decreasing the secondary stress intensity factor KIS) and the plasticity interaction factor ρ (which has the effect of increasing crack driving force through the additive term in (1).

The background to this approach is given in a paper by Garwood et al11. This paper, originally presented in 1987, compares several aspects of the R6 Rev. 3 and PD 6493:1980 procedures, and on the basis of this makes proposals for the treatment of residual stress in the 1991 edition of PD 6493.

For as-welded joints with unknown stress distributions, it was proposed that a uniform tensile residual stress (σR) equal to the ‘appropriate’ yield strength σY should be used, but that, at high levels of primary stress, a global relaxation of residual stress should be assumed (the word ‘appropriate’ in this context refers to the material to which the yield strength applies – parent or weld metal). Stress relaxation is assumed to start when the sum of the assumed residual stress and the net section stress, σn (now termed the ‘reference stress’, σref) reach the flow strength (σf) of the material. Given the assumption σf=1.2σY inherent in PD 6493 at that time, and the use of the plastic collapse parameter Sr=sref/sf (in place of LrrefY as is currently used in both BS 7910 and R6), the residual stress can be expressed as:


The temperature-sensitivity of yield/flow strength in ferritic steels is not explicitly mentioned in this source, but logically σY and σf should be assumed to be the values at the assessment temperature. It also follows that, for welded joints made at room temperature, the upper bound to σR should be the room temperature yield strength, σYRT, which could of course be either higher or lower than the value given by Equation (3).

Garwood et al also emphasised that the choice of σf=1.2σY would need to be checked, and hinted that a more ‘onerous’ definition of flow strength (eg σf=1.4σY), ie one predicting lower levels of stress relaxation, might be needed after validation checks. Using the proposed PD 6493 Level 2 fracture assessment procedure (the strip yield model, now superseded), they showed the results of seven parent material wide plate tests and 14 welded plates (HY80, HY100 and Q2 steels) analysed in accordance with the above framework, and incorporating the ρ factor in the definition of √δr. The analysis, based on CTOD test results, showed the results for all welded wide plate tests to fall outside the Level 2 Failure Assessment Line (FAL). A series of analyses were then carried out to show the consequences (in terms of the safety factor associated with the analysis) of various methods of treating residual stress, eg:

  • Using the PD 6493 Level 1 method
  • Using the σf=1.2σY assumption and allowing ‘global’ stress relaxation as shown above, but incorporating the r factor in the definition of √dr
  • Using the σf=1.4σY assumption and allowing ‘global’ stress relaxation
  • Using the σf=1.2σY assumption, allowing ‘global’ stress relaxation, but ignoring the r factor in the definition of √δr
  • Assuming σRY, ie not allowing any stress relaxation

For calculations using the PD 6493 Level 3 FAD (similar to the Option 1 FADs in BS 7910:2013 and R6), the stress relaxation equation equivalent to (3) is given by:


σY should again be interpreted here as temperature-dependent when calculating stress relaxation using Equation (4), but σYRT defines an upper limit to σR.

Comparing Level 1 and Level 2 analyses (where the Level 2 analyses incorporate mechanical stress relaxation), Garwood et al comment that ‘The percentage decrease in safety factor inherent in moving from Level 1 to Level 2 for parent materials is thus approximately maintained for the as-welded tests using the suggested procedure […..] for stress relaxation effects in the Level 2 assessment.’ In other words, the incorporation of residual stress (including residual stress relaxation effects) led to safety margins that were consistent with those for parent materials (in which the stresses are well-defined) and those safety margins had by this stage been validated by user experience and large-scale testing.

By the time of publication of PD 6493:1991, the stress relaxation criterion had been changed to:



thus increasing the conservatism of the analysis, ie the assumed residual stress after stress relaxation is higher than suggested by (4). Consequently, the safety factor associated with the analysis of the wide plate data reported by Garwood et al would have been higher under the new rules that were ultimately incorporated into PD 6493:1991.

The approach that eventually appeared in PD 6493:1991 recommended slightly different approaches for Level 2 and for Level 3 analyses (but note that the 1991 Level 2 methods, ie those based on the strip yield model, are no longer used). It was recommended that the residual stress σR assumed in the analysis could be reduced to the lower of:


where σ’Y is the ‘appropriate material yield strength’. The implication of this clause in PD 6493:1991 is that σ’Y is the room temperature value, although this interpretation is not consistent with the assumptions that appear to underlie Equations (3) and (4) above. This is not necessarily a problem when the assessment temperature is close to ambient temperature, but could lead to inconsistencies if applied to ferritic steels at very low temperatures, for which σ’Y and s’UTS would be well above their values at room temperature.

Validation of the residual stress clauses in PD 6493:1991

Another key paper pre-dating the 1991 edition of PD 6493 is due to Leggatt12, who aimed to test the appropriateness of the proposed rules by reference to a set of full-scale test data and numerical analyses. 

Leggatt’s paper appears to have been based on a TWI report13 that was never released in full, presumably because the main findings had been published in the public domain. It is the view of the current authors that the comment in R6 about the ‘limited validation’ of the PD 6493/BS 7910 approach probably (and justifiably) stems from the fact that neither of the papers hitherto available in the public domain11,12 presents the full evidence available at the time. A companion paper8 re-visits the unpublished work, using the methods and terminology of BS 7910:2013, in order to bring the record up to date.

BS 7910:1999 to 2013

Summary of changes

In 1999, PD 6493 was revised and issued as a BS guide (BS 7910), and re-issued with minor changes in 2005. So far as the treatment of residual stresses is concerned, the main additions at this stage were:


  • The inclusion of upper-bound residual stress profiles for a number of types of welded joint (in R6, this is termed the ‘Level 2’ approach),
  • A load history annex, Annex O.

Otherwise, the approach adopted in BS 7910:1999 (Level 1 residual stress assumption, incorporating the assumptions of uniform residual stress, stress relaxation under high mechanical loads, and use of the plasticity interaction factor, ρ were retained.

The main changes introduced in the residual stress clauses of the 2013 edition are as follows7:

  • Annex Q (‘Level 2’ residual stress assumption) has been updated.
  • The Annex Q stress distributions have been resolved into membrane, bending and self-balancing components and a new K-solution for the self-balancing component added (to facilitate calculation of the stress intensity factor, KIs, when the residual stress distribution is non-linear).
  • The residual stress relaxation equations in clause 7 and Annex O have been harmonised (so that Qm cannot be lower than 0.4σY’ under conditions of high primary stress) and clarified (to remove ambiguities regarding the value of σ’Y to be used in calculating residual stress relaxation effects).

Annex R, which addresses plasticity interaction factors under conditions of combined primary and secondary loading, has also been revised to allow give users a choice between two interaction factors: the additive factor ρ (as in earlier editions) and a multiplicative factor V (as used in the R6 and FITNET procedures).

Residual stress relaxation under mechanical load

The method given in clause of the current (2013) procedure is to assume that the residual stress component, Qm, is the lower of either




where σ’Y is the yield strength of the ‘appropriate material’ at the assessment temperature (except that, for assessment temperatures below ambient, it is the room temperature value, ie σ’YRT, in Equation (8a) and σ’f is the flow strength (average of yield and ultimate strengths) of the ‘appropriate material’ at the assessment temperature. Note that the definition of flow strength has changed somewhat relative to the earlier document, PD 6493, which appears to have been based on σf=1.2σY. The inconsistency noted earlier (ie whether σ’Y refers to the appropriate yield strength at room temperature or at the temperature of assessment) appears also to have been addressed in this edition (although the use of the same term, σ’Y, to refer to two different properties is not good practice).

An example of the application of equations (8a) and (8b) is shown in Figure 3. This shows the behaviour of one of the specimens tested by Formby and discussed in the companion to this paper (8). Residual stresses were introduced into a steel sheet with room temperature yield strength of 317N/mm2. The measured residual stress in the area of the flaw (a centre through-thickness crack) was 286N/mm2. The specimen was then tested to failure at -73°C (at this temperature, the yield and tensile strengths are 390 and 590N/mm2). Failure occurred at a value Lr=1.156, σref=458.8N/mm2, so applying Equation (8b), the residual stress (Qm) after stress relief is 187N/mm2. This value may, according to BS 7910:2013, be used as an estimate of Qm in situations in which the user does not have access to measurements of residual stress.

Lines showing the trends predicted from equations (4) (original proposals for PD 6493:1991) and (7) (as adopted in PD 6493:1991) are also shown for information. The superseded equations predict greater stress relief, ie lower values of Qm after stress relief, than do the current BS 7910:2013 equations.

Residual stress relaxation after proof test

In Annex O, an equation similar to (8a) and (8b) is presented, except that σref refers to the maximum reference stress under proof test conditions, and σ’Y and σ’f refer to yield and flow strengths at the proof test temperature. This distinction is made because proof testing temperature may be different from assessment temperature, eg in the case of a pressure vessel that is proof tested at room temperature but subsequently operated at sub-zero temperature. The upper limit for sref is explicitly stated to be σ’f (ie Qm cannot be lower than 0.4σ’Y) and the lower limit 0.4σ’f (ie Qm cannot be higher than σ’Y).


  • The residual stress relaxation clauses of BS 7910:2013 date back to the 1991 edition of PD 6493 and have not changed substantially since then.
  • A considerable programme of work was carried out by TWI at the time to justify and validate the clauses, using a range of experimental and numerical work. This included analysis of work carried out in support of both the R6 and PD 6493 procedures.
  • The full underlying details of the work have not hitherto been available in the public domain, although the conclusions arising from it were published in 1988. It is believed that this is the reason that R6 describes the work as having ‘limited validation’. The approach proposed in BS 7910 combines ‘global’ relaxation of residual stress (Qm) under high mechanical load with ‘local’ enhancement of crack tip driving force through the adoption of a simplified primary/secondary stress interaction factor, ρ.


  1. R6 - Assessment of the integrity of structures containing defects, Revision 4, EDF Energy Nuclear Generation Ltd, as amended.
  2. BS 7910:2013 - ‘Guide to methods for assessing the acceptability of flaws in metallic structures’, BSI Standards Limited, 2013.
  3. I Hadley and HG Pisarski: ‘Overview of BS 7910:2013’, FESI ESIA12 - 12th International Conference on Engineering Structural Integrity Assessment, 28-29 May 2013, Manchester, UK.
  4. Isabel Hadley: PVP2011-57307, ‘Progress towards the revision of BS 7910’, Proceedings of the PVP2011 Pressure Vessels and Piping Division Conference, 17-21 July 2011, Baltimore, Maryland, USA.
  5. Isabel Hadley, Bob Ainsworth, Peter Budden, John Sharples: PVP2010-25582, ‘The Future of the BS 7910 Flaw Assessment Procedures’, Proceedings of the ASME 2010 Pressure Vessels and Piping Division Conference, (PVP2010), July 18-22, 2010, Bellevue, WA, USA.
  6. SE Eren, AM Paradowska and I Hadley, 'Evaluation of Residual Stresses in Narrow Gap Welded Gas Transmission Pipe Spools', presented at 7th MECA - SENS Conference, by Neutron & Synchrotron diffraction techniques, Sydney, Australia, 10-12 Sept 2013.
  7. Sharples et al (2011): Sharples J, Gill P, Wei L and Bate S, ‘Revised Guidance on Residual Stresses in BS 7910’, Proceedings of the ASME 2011 Pressure Vessels & Piping Division Conference (PVP2011), July 17 - 21, 2011, Baltimore, Maryland, USA. Paper No. PVP2011-57071.
  8. Isabel Hadley and Simon D Smith: PVP2014-28092, ‘Effects of Mechanical Loading on Residual Stress and Fracture: Validation of the BS 7910:2013 Rules’.
  9. FITNET Fitness-for-Service (FFS) - Procedure (Volume 1) ISBN 978-3-940923-00-4 (Koçak, M, Webster, S, Janosch, JJ, Ainsworth, RA and Koers, R) and FITNET Fitness-for-Service (FFS) - Annex (Volume 2) ISBN 978-3-940923-01-1, (Koçak, M, Hadley, I, Szavai, S, Tkach, Y and Taylor, N) both printed by GKSS Research Center, Geesthacht, 2008: see
  10. Ainsworth (1986): Ainsworth R A, ‘The treatment of thermal and residual stresses in fracture assessments’, Engineering Fracture Mechanics 24 1 65-76.
  11. Garwood et al (1989): Garwood S J, Willoughby A A, Leggatt R H and Jutla T, ‘Crack Tip Opening Displacement (CTOD) methods for fracture mechanics assessments: proposals for revisions to PD 6493’, The assessment of cracked components by fracture mechanics, EGF4, ed L H Larsson, 1989, Mechanical Engineering Publications, London, pp267-301 (Proceedings of the 6th advanced seminar on fracture mechanics (ASFM6) organized by the Commission of the European Communities, Joint Research Centre, Ispra, Italy, 28 Sept-2 Oct 1987).
  12. Leggatt R H (1988): ‘Investigation of proposed procedures for the inclusion of residual stresses in the revised fracture sections of PD6493’, Proceedings of the international conference on Residual Stresses (ICRS2), Nancy, November 23-25, 1988.
  13. Leggatt R H (1987): ‘Investigation of proposed procedures for the inclusion of residual stresses in the revised fracture sections of PD 6493’, TWI report 7011/01.87/584.2.

Table 1 BS 7910 approaches to residual stress assumptions for as-welded joints

Fracture assessment Level (PD 6493 and BS 7910:1999 and 2005)

Fracture assessment Option (BS 7910:2013)

Assuming uniform membrane residual stress (‘Level 1’)

Using an idealised upper bound residual stress profile (‘Level 2’)

After application of mechanical load




Qm is equal to the room temperature yield strength of the material in which the flaw is located

Not addressed

Qm may be reduced by a pressure test ‘or other form of mechanical pre-loading’. Annex O is cited

Plasticity interaction factor, r, not required (because of conservative simplifications elsewhere)

2 and 3


Qm is equal to the ‘appropriate’ yield strength, typically the room temperature yield strength of either the parent or the weld metal

Refers to Annex Q, but does not permit modification of distribution to take account of stress relaxation as a result of high temperature, overload or primary/secondary stress interaction

Clause; allows ‘global’ relaxation of residual stress as per Equations (8a) and (8b).

Cites Annex O if the structure has been subjected to a pressure test

Plasticity interaction factor (r) used (1999 and 2005 editions).

r or V may be used to represent plasticity interaction.

Figure 1 Example of residual stress distributions at Levels 1, 2 and 3.
Figure 1 Example of residual stress distributions at Levels 1, 2 and 3.
Figure 2 Summary of the development of BS 7910.
Figure 2 Summary of the development of BS 7910.
Figure 3 Example of the application of BS 7910:2013 rules to relief of residual stress.
Figure 3 Example of the application of BS 7910:2013 rules to relief of residual stress.

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