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Treatment of residual stress in fracture assessment: background to the advice
given in BS 7910:2013
Article in International Journal of Pressure Vessels and Piping · November 2018
DOI: 10.1016/j.ijpvp.2018.11.005
2 authors:
J. K. Sharples
Isabel Hadley
Jacobs Engineering Group Inc
The Welding Institute
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Treatment of Residual Stress in fracture assessment:
background to the advice given in BS 7910:2013
John Sharples* and Isabel Hadley**
* Wood plc, Walton House, Birchwood Park, Warrington, Cheshire, UK, WA3 6GN
** TWI Ltd, Great Abington, Cambridge, UK, CB21 6AL (Isabel.hadley@twi.co.uk)
Published in International Journal of Pressure Vessels and Piping IJPVP), Volume 168, December
2018, Pages 323-334, https://doi.org/10.1016/j.ijpvp.2018.11.005
The UK procedure for fracture mechanics-based assessment of flaws (BS 7910) was extensively
updated in late 2013. This paper outlines how and why the clauses and Annexes relating to residual
stresses (in particular those arising from the manufacturing of welded joints) have been expanded,
and explains the origin and extent of validation of the information. A worked example of applying the
guidance to fracture assessment of a welded joint is included.
BS 7910 ‘Guide to methods for assessing the acceptability of flaws in metallic structures’ (1) is a
procedure for the assessment of flaws in metallic structures, covering failure by fracture, fatigue,
corrosion and creep. It is used to make decisions on serviceability of safety-critical plant and
structures, but also in failure investigation, life extension and design. It has been in continuous use
since 1999, when it superseded the earlier UK flaw assessment procedure BS PD 6493. The most
recent edition of the procedure was published in 2013 and amended in 2015. It evolved from a
mixture of sources, including previous editions of BS 7910/PD 6493, the UK nuclear safety
assessment procedure R6 (2) and the European SINTAP and FITNET documents (note that the two
European procedures ((3)-(4)) are no longer maintained, and have effectively been superseded by BS
One of the features of BS 7910:2013 is an expansion of Annex Q on “Residual stress distributions in
as-welded joints” with the prime aim of making it more user friendly than the previous (2005) version.
A paper previously published (5) provided a brief overview to the 2013 version of Annex Q. However,
the purpose of this paper is to document the origin and extent of validation of the current clauses and
Annexes relating to residual stresses, to identify gaps where future work may be required, and to
provide a worked example.
Clause 7.1.8, under the section on “Assessment for fracture resistance”, provides initial guidance on
residual stresses. It is first stated (Clause that such stresses may be assumed to be uniform
or non-uniform, with the former being considered within the clause and the latter being referred out to
Annex Q. Treating residual stresses as uniform (i.e. membrane) is of course generally more
conservative than considering them to be non-uniform (which is more realistic). The Clause then goes
on to state (Clause that, for a structure in the as-welded condition and for the stress direction
perpendicular to the weld, the tensile residual stress should initially be assumed to be equal to the
lesser of the room temperature yield strengths of the weld or parent metal (this is termed the
‘appropriate’ yield strength for the case of a flaw parallel to the weld). For the stress direction parallel
to the weld, the tensile residual stress should initially be assumed to be equal to the room
temperature yield strength of the material in which the flaw is located.
Weld residual stresses can be relieved either by mechanical or thermal applications, the relief process
being by plasticity effects coupled with the fact that yield strength is usually temperature dependent
(i.e. the value changes with temperature). Concerning the mechanical side, mechanical treatments
are frequently applied to engineering components, either to demonstrate sound fabrication or to
improve structural performance, the latter effect of course arising from a beneficial change in the
residual stress field. For example, in many industries, the global mechanical treatment applied to
load-bearing structures is the “proof test”. Such prior over-load tests can reduce weld residual stress
levels in components. In fact, it is stated in Clause that some global relief of residual stresses
may be assumed as a result of the primary loading applied to the as-welded structure. The membrane
residual stress may then be assumed as the lower of equations [1] and [2]:
𝑄𝑚 = 𝜎𝑌′
𝑄𝑚 = {1.4 −
𝜎𝑟𝑒𝑓 ′
} 𝜎𝑌
𝜎𝑌′ is the yield strength of the ‘appropriate material’ at the assessment temperature (except that, for
assessment temperature below ambient, it is the room temperature value);
𝜎𝑓′ is the flow strength (taken as average of yield and ultimate strengths) of the ‘appropriate material’
at the assessment temperature;
𝜎𝑟𝑒𝑓 is the reference stress.
The upper limit for 𝜎𝑟𝑒𝑓 should be 𝜎𝑓′ and the lower limit, 0.4𝜎𝑓′ .
The above equations are also contained in Annex O on “Consideration of proof testing and warm prestressing” in relation to guidance on proof or overload testing.
The relaxation clause dates back to the 1991 edition of BS PD 6493 and has not changed
substantially since then. A considerable programme of experimental and analytical work on both
cracked and uncracked welded plates was carried out by TWI at the time to justify and validate the
clause (6). This included some 18 wide plate tests carried out on thin-section steel plate. A re-analysis
(7) of the original TWI work, using BS 7910:2013, has shown nothing to contradict the approach,
which represents a workable engineering solution to the problem of how to analyse residual stress
effects in as-welded structures rapidly and reasonably realistically when the as-welded stress
distribution is unknown. Of course, the accuracy of equation [2] in predicting residual stress relief
under mechanical loading depends on the value of reference stress assumed, which varies somewhat
between different procedures and is further discussed in the report (8) cited by the Case Study below.
The thermal side of residual stress relaxation is associated with post-weld-heat-treatment (PWHT)
processes. Welded structures are often subjected to PWHT in order to improve the metallurgical
properties of the weld region as well as to reduce the residual stress. Clause of BS 7910:2013
provides guidance on weld residual stresses (membrane) for PWHT structures. The degree of relief of
welding residual stress will be influenced mainly by the peak PWHT temperature and the hold time at
this temperature (9). It may be assumed that, for carbon manganese and 2¼Cr1Mo steels, the
stresses after heat treatment in an enclosed furnace within the range 5800C to 6200C are:
For the stress direction perpendicular to the weld, the residual stress should be assumed to be
20% of the lesser of the yield strengths of the weld or parent metal.
For the stress direction parallel to the weld, the residual stress should be assumed to be equal to
30% of the room temperature yield strength of the material in which the flaw is located, or 30% of
the greater of the yield strengths of the weld or parent metal for the case of a flaw which lies partly
in weld metal and partly in parent metal.
This guidance was originally developed to comply with the temperature range for PWHT of pressure
vessels built to the UK standard BS 5500. This has since been superseded by PD 5500 and EN
13445, both of which allow lower PWHT temperatures, leading to higher residual stresses than those
originally developed for BS 5500. For heat treatment at temperatures below 5800C but above 5500C:
For the stress direction perpendicular to the weld, the residual stress should be assumed to be
20% of the lesser of the yield strengths of the weld or parent metal (i.e. as above).
For the stress direction parallel to the weld, the residual stress should be assumed to be equal to
40% of the room temperature yield strength of the material in which the flaw is located, or 40% of
the greater of the yield strengths of the weld or parent metal for the case of a flaw which lies partly
in weld metal and partly in parent metal.
It is also noted that, when local PWHT is carried out, the level of residual stress may be higher than
that recommended for heat treatment carried out in an enclosed furnace. The level of residual stress
depends on factors such as the temperature attained, the width of the heated band and the width of
the insulated region; consequently, BS 7910 recommends the user either to seek expert advice or to
adopt conservative assumptions.
In terms of the background to the guidance relating to welded joints that have been subjected to
PWHT, TWI has recently reviewed all available information (Error! Reference source not found.10),
including that provided in codes such as BS EN 13445 and ASME I, II and VIII. This review has
concluded the following:
The code requirements for PWHT (in terms of temperature, hold time, heating/cooling rate,
heating method etc.) vary from code to code. Consequently, the level of relief of welding residual
stress would be expected to vary too.
It is generally suggested by the research work that the relieved welding stresses after a furnace
PWHT in accordance with the code procedures are approximately 30% of the yield strength.
The degree of stress relief after a furnace PWHT, as assumed in BS 7910:2013, is broadly
supported by the findings reported in the public domain; ie the residual stresses remaining after
PWHT are higher in the direction parallel to the weld (30% of the yield strength) than those
transverse to the weld (20% of the yield strength). An example of some of the data reviewed is
given in Figure 1, adapted from (9Error! Reference source not found.), and a further example is
given in the Case Study below.
The effects of local PWHT variables (mainly the width of the heated band (HB), the soak band
(SB) and the gradient control band (GCB)), have not been quantified with confidence due to a
number of factors involved in a local PWHT, although it appears that a HB width of the order of
5(RB)0.5 (where R and B are pipe inner radius and thickness respectively) may be a good
estimate for local heating of a circumferential weld in a cylinder. More research work (especially
experimental) on stress relief due to local PWHT was recommended at the time, and such studies
have since been initiated by TWI, centred on welding residual stresses in pipe girth welds
(11Error! Reference source not found.).
Annex Q contains recommended residual stress distributions in as-welded joints. These
recommendations are based on upper bound fits to experimental data that have been generated over
the years. Some information regarding this process is provided in (12Error! Reference source not
found.). Upper bound residual stress recommended profiles were contained in the 1999 and 2005
revisions of BS 7910. However, they were generally presented in the form of polynomial equations,
which meant that the user was not readily able to apply the profiles via the stress intensity factor
solutions contained in Annex M of the BS 7910 document, which are limited to combinations of
uniform (membrane) and through-wall bending stresses. For the 2013 version of BS 7910, Annex Q
has therefore been significantly modified as described in the following section.
Factors such as joint type, heat input and restraint would all be expected to influence the residual
stress distribution in a welded joint, but in practice (because of the lack of data if joint types are subdivided in this way) most of the Annex Q residual stress profiles use a single upper bound profile
based on all available data for a particular type of welded joint.
The exception to this convention is pipe butt (girth) welds, where there are sufficient data to account
for the effect of heat input on the residual stress profile. There is a significant difference in behaviour
between high heat input/thin wall welds, which exhibit a tourniquet effect (i.e through wall bending,
tensile at the inner surface) and low heat input/thick wall welds, where the stress at the outer surface
is now tensile (see Figure 2). Three different profiles are provided for ferritic steel weldments, to
account for ‘high’ (>120 J/mm2), ‘low’ (≤ 50 J/mm2) and ‘medium’ (50-120J/mm2) heat inputs, and two
for austenitic stainless steels, one covering ‘low’/’medium’ and the other ‘high’ heat inputs.
An example of the underlying data for a series of circumferential pipe butt welds made using either a
low heat input (ferritic steel) or a low/medium heat input (austenitic stainless steel) is shown in Figure
3. The data show residual stresses transverse to the weld, ie along the pipe axis, and the upper
bound curve from BS 7910 is seen to describe them reasonably well.
It may be noted that the profiles in Annex Q relate to conventional weld types; Manual Metal Arc
(MMA), Tungsten Inert Gas (TIG) and Submerged Arc Welding (SAW). Work is in hand on
developing residual stress profiles for Electron Beam welds.
3.1 General requirements
In drafting Annex Q of BS 7910:2013, the committee agreed the following principles:
incorporate the most relevant up-to-date upper bound profiles,
enable the user to readily evaluate stress intensity factors for the profiles without having to go
outside the document (bearing in mind that Annex M of BS 7910 contains stress intensity
factor values only for membrane and bending stress components, whereas residual stress
profiles are commonly represented by high order polynomial equations)
make the Annex more user friendly for the assessment engineer to navigate.
3.2 Most up-to-date profiles
The through-thickness residual stress profiles provided in the 2005 version of BS 7910 (1), R6 (2),
and FITNET (4) procedures were assessed in detail in terms of understanding their validation and the
basis on which any recent changes may have been made. The profiles were considered for the
following weld types (for orientations both transverse and longitudinal to the weld):
Plate Butt and Pipe Seam Welds
Pipe Butt Welds
Plate-to-plate T-Butt Welds
Pipe-to-pipe T-Butt Welds
Repair Welds
The recommended profiles provided in BS 7910, R6 and FITNET are all based on upper bound fits to
experimental data and they originate from the same sources (e.g. (12)). Not surprisingly, the profiles
provided in these three procedures are very similar, but with some fairly minor differences seen in R6.
These differences arise from the fact that the residual stress profiles in that procedure are reviewed
on an annual basis with a view to recommending modifications, as required, on the basis of up-todate experimental data.
On the basis of consistency with the 2005 version of BS 7910 and the need to have knowledge of and
confidence in the validation data, it was decided that the residual stress profiles for the 2013 version
of BS 7910 should be based on a combination of the R6 (at the time of developing the 2013 version of
BS 7910) and earlier BS 7910 profiles (i.e. those given in BS 7910:2005).
3.3 Stress intensity factors
The recommended residual stress profiles have been partitioned into Membrane (m), Bending (b)
and Self-Balancing (sb) components by the following equations for a given stress profile (z), where
z varies from 0 to B through the wall thickness,
𝜎𝑚 =
𝜎𝑏 =
1 𝑧=𝐵
. ∫ 𝜎(𝑧). 𝑑𝑧
𝐵 𝑧=0
𝜎 (𝑧). ( − 𝑧)) 𝑑𝑧
𝐵 𝑧=0
𝜎𝑠𝑏 (𝑧) = 𝜎(𝑧) − 𝜎𝑚 − 𝜎𝑏
Figure 4 contains an example of such partitioned through-thickness transverse stresses for ferritic
pipe butt welds made with a low heat input (E1/B<50J/mm2). The stress, 𝜎𝑅𝑇 , is normalised to yield
strength, σY,, and it is shown plotted against z/B where z is the distance from the inner surface
through the wall thickness, B. The dashed curve represents the recommended residual stress profile
given by the equation:
𝑧 2
𝑧 3
𝑧 4
= 1 − (6.80 ( )) + (24.30 ( ) )-(28.68 ( ) ) + (11.18 ( ) )
The solid line is the membrane stress component evaluated from Eq. [3], the dotted line is the
bending stress component evaluated from Eq. [4] and the dashed-dotted curve is the self-balancing
component, evaluated from Eq. [5].
It was realised that stress intensity factors for the Membrane (𝐾𝐼𝑚 ) and Bending (𝐾𝐼𝑏 ) stress
components could be evaluated from the existing solutions provided in Annex M of BS 7910. The
question then remaining was how to evaluate the stress intensity factor (𝐾𝐼𝑠𝑏 ) for the self-balancing
stress component.
For surface flaws under a through-wall sinusoidal stress distribution, stress intensity factors for
extended flaws bound those values for semi-elliptical flaws of all aspect ratios. This is illustrated in
Figure 5 for a cylinder with an internal circumferential flaw, whereby normalised stress intensity factor
is plotted against flaw depth, a (normalised to B) for a sinusoidal stress distribution (tensile near the
wall surfaces, balanced by compression in the mid-wall region). These values of 𝐾𝐼 are for a unity
tensile stress level at the surfaces (and a unity compressive stress level at the mid-wall). Values are
shown for flaw depth to flaw semi-length (a/c) values of 0.2 and 1.0 as well as for an extended flaw.
The stress intensity factors were evaluated from the relevant solutions provided in the compendium
contained in R6 (2), whereby the through-wall stress distribution over the flaw depth, but in the
absence of the flaw, is represented by a polynomial equation. It may be noted that the 𝐾𝐼 values
shown in Figure 5 (for the surface flaw cases) are for the deepest point of the flaw.
Similar information to that provided in Figure 5 was evaluated for other cases as follows:
Cylinder with internal axial flaw
Cylinder with external axial flaw
Cylinder with internal circumferential flaw
Cylinder with external circumferential flaw
Figure 6 contains the highest normalised stress intensity curves for each of the five geometry cases
considered. As can be inferred from above, all these curves represent the extended flaw cases. It
can be seen that the curve for the plate provides an upper bound to the cylinder curves.
Returning now to the self-balancing component of a residual stress field, it is evident that the stress
intensity factor, 𝐾𝐼𝑠𝑏 , for this component will be significantly lower than for the membrane and bending
components (i.e. 𝐾𝐼𝑚 and 𝐾𝐼𝑏 respectively) in the majority of situations. Therefore, a good and simpleto-use upper bound solution for 𝐾𝐼𝑠𝑏 was sought, which would not result in the overall stress intensity
factor being overly-conservative, as described below.
For each residual stress profile corresponding to the various weld types as listed above, the
normalised stress intensity factor was evaluated for the self-balancing component following the same
calculation route as that referred to above in relation to Figure 5 and Figure 6. For conservatism, only
the 𝐾𝐼 solutions for extended flaws in plates were considered. Examples of the resulting normalised
stress intensity factors are shown in Figure 7 and Figure 8 for transverse residual stresses in ferritic
pipe butt welds made with a low heat input, and longitudinal residual stresses in ferritic steel pipe-topipe T-butt welds, respectively.
From the normalised stress intensity factor profiles such as those given in Figure 7 and Figure 8, the
peak value of 𝐾𝐼𝑠𝑏 can be obtained, and such values have been incorporated into the 2013 version of
Annex Q of BS 7910. If the user requires a more precise solution, then 𝐾𝐼𝑠𝑏 can be evaluated using
new stress intensity factor solutions incorporated into Annex Q. These solutions are for a surface flaw
or extended flaw in a plate under a stress field represented by a polynomial equation up to the 5 th
order. The solutions are taken from the R6 procedure, but modified so that the equations for the
stress distribution across the whole section thickness (as shown in Annex Q) can be used directly (by
contrast, the solutions in R6 are couched in terms of stress distribution across the flaw). Only the
plate solutions for a polynomial stress field are included, since they have been shown to be a
reasonable upper bound to the solutions for cylindrical geometries (see, for example, Figure 3). The
user will, of course, be able to use the newly incorporated solution to evaluate the stress intensity
factor for the full residual stress profile without having to consider the partitioned components. This
approach may be somewhat overly conservative in some cases for use with cylindrical geometries,
since (as previously indicated), the stress intensity factors for cylinders are generally lower than those
for plates.
In summary, the stress intensity factor 𝐾𝐼𝑠 due to the residual stress profile in the revised Annex Q of
BS 7910 can be evaluated as:
𝐾𝐼𝑠 = 𝐾𝐼𝑚 + 𝐾𝐼𝑏 + 𝐾𝐼𝑠𝑏
where 𝐾𝐼𝑚 and 𝐾𝐼𝑏 are obtained from Annex M, and 𝐾𝐼𝑠𝑏 can be conservatively evaluated from peak
values provided, or more accurately determined by way of the solutions from R6 that have been
incorporated into Annex Q, using the self-balancing component of the residual stress profile.
3.4 User friendliness
In Annex Q of BS 7910, the recommended residual stress profiles are now illustrated graphically
together with the σm, σb and σsb components. In addition, tabular information is provided on the σm
and σb values, the σsb equation and the peak value of 𝐾𝐼𝑠𝑏 that has been evaluated by the route
described above.
For tubular components, the residual stress profiles are presented in terms of z/B, where z=0 at the
inner surface, and B is the thickness. For surface flaws emanating from the outer surface, the profiles
can be re-characterised by replacing z/B by 1-z/B in the polynomial equations, where z would then be
taken from the outer surface.
A further recommendation is that the profiles given are appropriate to ±1.5W, where W is the weld
width. It is recommended that flaws situated outside of this range need to be assessed on a case by
case basis.
An example of the layout is given in Figure 9, representing the details for longitudinal stresses in
ferritic steel pipe-to-pipe T-butt welds.
A case study is presented here, based on work carried out in support of the R6 procedure (8Error!
Reference source not found.). The work involved the re-analysis, using current BS 7910/R6
methods, of historical data generated from a welded wide plate specimen in the as-welded and PWHT
condition, and after a simulated warm prestress (WPS) treatment (13Error! Reference source not
found.). It included the determination of residual stress distributions, and the effects of those
distributions on the behaviour of welded wide plate specimens.
Experimental stress distributions are compared with the upper bound distributions given in Annex Q,
and the process of calculating secondary stress intensity factor, 𝐾𝐼𝑠 , from a stress distribution is
4.1 Materials and measurements
The material used was a 51mm thick C-Mn pressure vessel steel, with a double-vee submerged arc
butt weld at the centre (see Figure 10). The room temperature yield strength of the weld (𝜎𝑌𝑊 ) was
505MPa, compared with 358MPa for the parent plate (𝜎𝑌𝑃 ). Hence a condition of high strength
overmatch (M=𝜎𝑌𝑊 /𝜎𝑌𝑃 =1.4) was obtained, as is normal practice for welds of this type.
The residual stress distribution in the as-welded (AW) joint was determined, both parallel and
perpendicular to the joint, using the Block Removal, Splitting and Layering (BRSL) technique. This
technique allows a full through-thickness stress field to be determined, albeit destructively. Replica
joints were then subjected to various thermal and mechanical treatments, including PWHT (2 hours at
6000C) and WPS. The results discussed here relate to joints in the AW and PWHT condition only,
since these cases are covered by clause 7 and Annex Q of BS 7910.
4.2 Comparison of experimental residual stress data with advice given in BS 7910
The first stage of the study was to compare the stress distributions with the advice given in BS 7910
clause 7 and Annex Q. Figure 11 shows the stress distributions (at the weld centreline) in the
transverse direction (i.e. perpendicular to the joint), as a function of section thickness, with z=0
representing the side welded last (labelled ‘side 1’ in Figure 10). Also shown are the approximations
to the stress distribution, as advised in BS 7910. The first approximation relates to Clause
which specifies the as-welded residual stress as a membrane stress such that “for the stress direction
perpendicular to the weld, the tensile residual stress should initially be assumed to be equal to the
lesser of the room temperature yield strengths of the weld or parent metal”. The second
approximation is somewhat more involved and refers to the upper bound profiles given in Annex Q.
The approximate upper bound transverse residual stress profile for a joint of this type (equation [3]) is
taken from Figure Q.2/Equation Q4. of BS 7910) represented by the following equation:
𝑧 2
𝑧 3
= 0.9415 − 0.0319 ( ) − 8.3394 ( ) + 8.660 ( )
hence, for 𝜎𝑌 =358MPa and B=51mm
𝜎𝑅𝑇 = 337.06 − 0.224𝑧 − 1.1478𝑧 2 + 0.0234𝑧 3
𝑧 2
𝑧 3
𝜎𝑅𝑇 = 337.06 − 11.420 ( ) − 2985.5 ( ) + 3100.3 ( )
For completeness, Figure 11 includes the membrane stress after PWHT, taken from Clause
of BS 7910: “the stress perpendicular to the weld should be assumed to be 20% of the lesser of the
room temperature yield strengths of the weld or parent metal”.
It can be seen that the assumption of full yield membrane stress (clause, labelled ‘100% 𝜎𝑌𝑃 ’)
lies well above the data across the entire section thickness. This assumption, often made in the
absence of other data, has been criticised as ‘over-conservative’, but it should be noted that in
practice it is often modified to take account of primary loading, as described earlier in this paper.
The idealised upper bound profile given in Annex Q is seen to be reasonably consistent with the data,
although there is some overlap of the curves (the AW data lie slightly above the Annex Q curve)
around the mid-thickness of the specimen (it should be noted that the data shown did not form part of
the database used to generate the Annex Q curves, so can be considered as independent).
The assumption that an upper bound to the transverse stress data after PWHT can be approximated
from 20% of the yield strength of the parent metal (line labelled ‘20% 𝜎𝑌𝑃 ’) is also consistent with the
data, although in this case the data shown formed part of the original evidence for clause
(there are very limited data of this type, from real multi-pass welds, in the literature), so the argument
is somewhat circular.
4.3 Calculation of 𝑲𝒔𝑰 from residual stress profiles
Uniaxial wide plate tests were subsequently carried out on as-welded plates containing an artificial
surface-breaking flaw parallel to the weld, and located at the weld centreline (see Table 1 and Figure
10 for details of plate and flaw geometry). Consequently, the results can be used to illustrate how the
residual stress distributions given in Annex Q (or, indeed, the actual stress distributions determined
experimentally) can be used to determine the applied stress intensity factor from secondary stresses,
𝐾𝐼𝑠 , as described earlier.
The loading direction of the wide plate test was perpendicular to the weld (and to the flaw) – as a first
approximation, only the transverse residual stresses, as shown in Figure 11, were considered in this
study, although some effect from interaction between longitudinal and transverse stresses would be
expected in practice.
The complete transverse stress distribution, along with its decomposition into membrane, bending
and self-balancing components, is shown in Figure 12 (adapted from Figure Q.2 of BS 7910,
assuming B=51mm and Y=358MPa).
As previously explained, Annex Q now provides three methods for calculating the stress intensity
factor due to secondary stresses:
a. Apply the whole of equation (Q.4) of BS 7910 (equation [8]) using the K-solution given in
equations (Q.17) and (Q.18) of BS 7910 (reproduced here as equations [11] and [12]):
𝑎 𝑖
𝑎 2𝑐
𝐾𝐼 = √𝜋𝑎 ∑ 𝜎𝑖 ( ) 𝑓𝑖 ( , )
𝐵 𝑎
𝑎 2𝑐
where 𝑓𝑖 ( , ) is tabulated in BS 7910
for 0≤ z ≤B
𝑧 𝑖
𝜎 = 𝜎(𝑧) = ∑ 𝜎𝑖 ( )
b. Use the partitioned stress distribution (see Figure 12 and Table 2, reproduced from Figure Q.2
and Table Q.2 respectively of BS 7910) and the appropriate solutions from Annex M to calculate
𝐾𝐼𝑚 and 𝐾𝐼𝑏 . Use equations [11] and [12] along with the self-balancing component of the
distribution (sb) to calculate 𝐾𝐼𝑠𝑏 , then 𝐾𝐼𝑠 = 𝐾𝐼𝑚 + 𝐾𝐼𝑏 + 𝐾𝐼𝑠𝑏 . This should give approximately the
same result as approach a.
c. Calculate 𝐾𝐼𝑚 and 𝐾𝐼𝑏 as in approach b. above, and add the upper bound value of 𝐾𝐼𝑠𝑏 , as given in
Table Q.2 of BS 7910 (Table 2 here). This will, in general, give a higher value of 𝐾𝐼𝑠 than
approaches a. and b., but does not require the use of equations [11] and [12].
Note that the stress distribution is defined from a reference z=0 which, for the case of a flat plate,
corresponds to the side of the plate welded last. From the macro-section of the weld (Figure 10), it is
clear that the side containing the flaw (labelled ‘side 1’) was actually welded last; consequently the
reference points for the stress distribution and the weight function used to calculate 𝐾𝐼𝑠 are the same,
ie the stress distribution does not need to be reversed.
𝒂 𝟐𝒄
From Table Q.11 of BS 7910, the coefficients 𝑓𝑖 ( , ) for a/B=0.5 and 2c/a~10 can be estimated by
𝑩 𝒂
interpolation as shown in Table 3. Then, combining this information with equations [10]-[12]:
𝐾𝐼𝑆 = √𝜋𝑎 [
(337.06 ∗ 0.50 ∗ 1.439) − (11.420 ∗ 0.51 ∗ 0.77) − (2985.5 ∗ 0.52 ∗ 0.548) +
(3100.3 ∗ 0.53 ∗ 0.434)
= 2147Nmm-3/2 for a=25.5mm and =90o
At =0o, 𝐾𝐼𝑆 =1997 Nmm-3/2, using the second set of coefficients from Table 3.
A similar process can be used to derive 𝐾𝐼𝑠𝑏 from the equation for 𝜎𝑠𝑏 given in Table 2, whilst 𝐾𝐼𝑚 and
𝐾𝐼𝑏 can be calculated from Annex M of BS 7910 in the usual way.
Results of the three approaches recommended in Annex Q are shown in Table 4, along with:
the simple approximation given by clause, ie 𝜎𝑅𝑇 = 𝜎𝑌 = 358𝑀𝑃𝑎
a fifth solution (not currently covered by BS 7910) in which the actual data are approximated by
an 8th order polynomial and the K-solution solved via a weight function method due to Glinka and
Shen (14Error! Reference source not found.).
Of the three methods available in BS 7910 Annex Q, approach a. gives the lowest value of KIs and
approach c. the highest, as would be expected. These results are bounded by the simple
approximation 𝜎𝑅𝑇 = 𝜎𝑌 and the optimum solution based on an 8th-order polynomial.
As with this latest version of BS 7910, the drafting committee will be responsible for agreeing on how
the guidance on residual stresses can be further improved for the next edition of the code planned in
a few years’ time. Aspects likely to be included are:
It would seem pertinent to consider the use of best estimate profiles as an alternative option to the
conservative upper bound ones. This would likely incorporate guidance on the determination of
best estimate profiles by both experimental and analytical methods. More recently published data
should, of course, be included, along with the historical data used to develop the upper bound
profiles given in the current Annex Q.
It would seem logical to introduce the most up to date stress intensity factors which would include
those (for various geometries) based on representing the stress distribution by a polynomial
equation as has been incorporated into Annex Q (for a plate geometry only). The solutions
should be intended for wider adoption (e.g. when considering thermal stresses) and should hence
be contained in Annex M, the annex containing stress intensity factor solutions.
As previously noted, the current Annex Q residual stress profiles are upper bound curve fits to
available experimental data relevant to each type of welded joint. A more scientific approach is
currently being considered, particularly in relation to development of the R6 procedures, whereby
the reliability (and quality) of the various data sets used in this process is estimated with a view to
“weightings” being placed on the data. This development could of course be beneficial in future in
re-formulating the Annex Q profiles, and should include recently-published data.
Further work (eg (11Error! Reference source not found.)) has recently been undertaken on the
effect of local PWHT on residual stresses, the output of which could usefully be incorporated in
the next edition of BS 7910.
Recommended residual stress profiles for alternative welding process, such as electron beam
(EB, previously referred to), should be included.
The treatment of residual stress in the 2013 edition of BS 7910, outlined primarily in clause 7.1.8 and
Annex Q of the document, represents an important step forward relative to earlier editions. The layout
of the document has been improved and the profiles themselves updated. The stress distributions are
expressed in terms of polynomial equations representing an upper bound to experimental data. A new
method for deriving secondary stress intensity factor using a weight function method is given. An
alternative ‘simplified’ approach, which does not require the direct use of a weight function, is to
‘partition’ the stress distributions into membrane, through-wall bending and self-balancing
components. The stress intensity factors associated with the first two components can then be
derived from Annex M of BS 7910, whilst the third can be read directly from tabulated upper-bound
values given in Annex Q.
Finally, the treatment of residual stress is illustrated through a worked example involving welded
plates in the as-welded and PWHT condition. Measured residual stresses are compared with the
recommendations of BS 7910, and various routes for the calculation of secondary stress intensity
factor are illustrated.
The contribution of all members of the residual stress panel is gratefully acknowledged.
1. BS 7910:2013 (incorporating Amendment 1 and Corrigenda 1-2): ‘Guide to methods for assessing
the acceptability of flaws in metallic structures’, British Standards Institution, 2013
2. R6 - Assessment of the Integrity of Structures Containing Defects: Revision 4, 2001, as amended
3. http://www.eurofitnet.org/ and http://www.eurofitnet.org/sintap_index.html
4. FITNET Fitness-for-Service (FFS) – Procedure (Volume 1) ISBN 978-3-940923-00-4, Koçak, M,
Webster, S, Janosch, JJ, Ainsworth, RA, 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, Taylor, N,
printed by GKSS Research Center, Geesthacht, 2008 (out of print)
5. John Sharples, Peter Gill, Liwu Wei and Steve Bate, ‘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
6. Isabel Hadley and Simon Smith, ‘Effects of Mechanical Loading on Residual Stress and Fracture:
Part I: Background to the BS 7910:2013 Rules’, Proceedings of the ASME 2014 Pressure Vessels
and Piping Conference PVP2014, July 20 – 24, 2014, Anaheim, California, USA. Paper No.
7. Isabel Hadley and Simon Smith, ‘Effects of Mechanical Loading on Residual Stress and Fracture:
Part II: Validation of the BS 7910:2013 Rules’, Proceedings of the ASME 2014 Pressure Vessels
and Piping Conference PVP2014, July 20 – 24, 2014, Anaheim, California, USA. Paper No.
8. Isabel Hadley, ‘Effects of warm prestress on Residual Stress and Fracture in Uniaxially Loaded
Welded Joints – TWI support to the R6 Procedure’, TWI Industrial Member report 1093/2017
9. R Leggatt: ‘Relaxation of Residual Stresses during Postweld Heat Treatment of Submerged-Arc
Welds in A C-Mn-Nb-Al Steel, TWI member report 288/1985
10. Liwu Wei and Magdalena Nowak: ‘A review of post-weld heat-treatment effects of welding
residual stress’, TWI report 23887/1/14, April 2014/TWI Industrial Member Report 1071/2016
11. Yao Ren, Anna Paradowska, Bin Wang and Elvin Eren: ‘Residual Stress State of X65 Pipeline
Girth Welds Before and After Local and Furnace Post Weld Heat Treatment’, Proceedings of the
ASME 2016 Pressure Vessels and Piping Conference, PVP2016, July 17-20, 2016, Vancouver,
British Columbia, Canada (PVP2016-63378)
12. 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”, OTH 482, HMSO (1997).
13. SJ Garwood, K Bell and DJ Smith, ‘The influence of proof testing on fracture performance at low
temperatures’, PVP-Vol.244, Pressure Vessel Fracture, Fatigue and Life Management, 45-50,
ASME 1992
14. Glinka G and Shen G, ‘Universal features of weight functions for cracks in mode I’Engineering
Fracture Mechanics, Vol. 40 pp1135-1146, 1991.
Table 1 Plate and flaw dimensions
Plate thickness, B, mm
Plate width, W, mm
Flaw height, a, mm
Flaw length, 2c, mm
Flaw Aspect ratio, 2c/a
Relative flaw height, a/B
Table 2 Components of normalised transverse stress and the maximum values of 𝑲𝒔𝒃
𝑰 (𝑲𝑰 𝒎𝒂𝒙 ) for
plate butt welds, taken from Table Q.2 of BS 7910
𝐾𝐼𝑠𝑏 𝑚𝑎𝑥
𝜎𝑌 √𝐵
0.3415 + 0.5461 ( ) − 8.3394 ( ) + 8.66 ( )
( = 0.289 − 0.578 )
Table 3 Geometry functions for a/B=0.5 and 2c/a=10, taken from Tables Q.11 of BS 7910
90o (Table Q.11)
(Table Q.12)
Table 4 Secondary stress intensity factors associated with welding residual stress
𝜎𝑅𝑇 = 𝜎𝑌 = 358𝑀𝑃𝑎 (clause
a. Annex Q – complete
distribution (Figure Q.2)
b. Annex Q – Partitioning (Figure
c. Annex Q – simplified (Table
d. Full weight function method,
based directly on experimental
data (not covered by BS 7910)
m=111.3MPa for calculation of KIm
b=103.5MPa for calculation of KIb
Figure 1 Example of stress relief due to PWHT in a thick-section overmatched butt weld (9),(10), and
interpretation in terms of BS 7910:2013 clauses. The solid and dashed ellipses bound the data for
transverse and longitudinal stresses respectively
Figure 2 Example of transverse residual stress distributions in pipe butt (girth) welds as a function of
welding heat input
Figure 3 Example of data and upper bound profile from Annex Q of BS 7910. The profile shown is for
a circumferential pipe butt weld (girth weld) made in ferritic steel with a low heat input
(E1/B<50J/mm2); the same profile is also applicable to austenitic stainless steels with low/medium
heat input (E1/B<120J/mm2). Individual datasets are indicated by different symbols. Stresses are
transverse to the weld, ie parallel to the pipe axis
Figure 4 Plot of decomposed components of transverse stresses in ferritic pipe butt welds made with
a low heat input
Figure 5 Normalised stress intensity factors for cylinder with internal circumferential flaw under
sinusoidal through-wall stress distribution
Figure 6 Highest normalised stress intensity factor curves for all cases under sinusoidal through-wall
stress distribution – for extended surface flaw in all cases
Figure 7 Normalised stress intensity factor for self-balancing stress component of transverse residual
stresses in ferritic pipe butt welds made with a low heat input 𝐸1 /𝐵 ≤ 50 J/mm2
Figure 8 Normalised stress intensity factor for self-balancing stress component of longitudinal
residual stresses in ferritic steel pipe-to-pipe T-butt welds
For longitudinal stresses (ferritic steels) the following equation applies.
𝑧 2
𝑧 3
𝑧 4
= 1.025 + (3.478 ( )) − (27.861 ( ) )+(45.788 ( ) ) − (21.799 ( ) )
 RL
Figure Q.7.Typical longitudinal residual stress distribution for pipe-to-pipe T-Butt Welds
Below is a breakdown of the stress components:
( = 0.307 − 0.614 )
𝑧 2
𝑧 3
𝑧 4
0.1541 + 4.0918 ( ) − 27.861 ( ) + 45.788 ( ) − 21.799 ( )
𝐾𝐼𝑠𝑏 𝑚𝑎𝑥
𝜎𝑌 √𝐵
Figure 9 Example of layout for longitudinal stresses in pipe-to-pipe T-butt welds (ferritic steel)
Figure 10 Macrosection of weld considered (crack position also shown)
Figure 11 Residual stress distributions transverse to the weld before and after PWHT, and a
comparison with advice given in BS 7910
Figure 12 Residual stress distribution perpendicular to the weld and its decomposition into
membrane, bending and self-balancing components (from Figure Q.2 of BS 7910).
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