Approaches for Determining Limit Load and Reference Stress for Circumferential Embedded Flaws in Pipe Girth Welds
Mohamad J Cheaitani
TWI Ltd Granta Park, Great Abington, Cambridge CB21 6AL, UK
Pipeline Technology Conference, Ostend, Belgium 2009.
Abstract
This paper provides an evaluation of approaches for determining limit load (and equivalent reference stress) for use in FADbased fracture assessment of circumferential embedded flaws in pipe girth welds. Threedimensional elastic plastic finite element analyses have been conducted on pipe models containing typical circumferential embedded flaws and subject to global bending load. ‘Jbased’ limit loads (and equivalent reference stress) and global collapse limit loads have been determined from the finite element analyses and used to evaluate existing standard flat plate solutions, including those in BS 7910 and R6. A general approach for determining limit load (and equivalent reference stress) is presented. This approach is consistent with both the finite element results and reference stress J estimation scheme and, consequently, allows the development of improved assessment models.
Introduction
The use of an engineering critical assessment (ECA) to derive flaw acceptance criteria for pipeline girth welds allows the maximum tolerable size of surface and embedded circumferential planar flaws to be determined on a fitnessforpurpose basis. A typical ECA involves assessing the significance of such flaws with regard to failure mechanisms, including fracture, which the pipeline may experience during construction, commissioning and service. The most commonly used approach to assess the significance of flaws with regard to fracture and plastic collapse is the failure assessment diagram (FAD), which is based on the reference stress J estimation scheme (BSI, 2005) and (BEGL, 2001). A FADbased assessment involves the calculation of a fracture parameter (Kr,√δr or Jr) and a plastic collapse parameter, Lr, Figure 1. The fracture parameter characterises the proximity to fracture under linear elastic conditions. The plastic collapse parameter characterises the proximity to failure by yielding mechanisms and is defined as either the ratio of applied load to the limit load or, equivalently, the ratio of the reference stress to the yield stress. There is a unique relationship between a reference stress and a limit load, which enables one to be defined if the other is known. Specifically, a limit load is inversely proportional to the corresponding reference stress as follows:
The work described in this paper concerns the development of a reference stress (or limit load) model for use in FADbased fracture assessments of circumferential embedded girth weld flaws such as that shown in Figure 2. This work is considered necessary since existing reference stress solutions for embedded flaws are not consistent with reference stress solutions that are typically used to assess surface flaws because the approaches used in their derivation are different. Whereas there are several reference stress models for surface breaking flaws, including some intended specifically for circumferential flaws in pipe sections, there are fewer reference stress models for embedded flaws and these are intended for flaws in flat plates.
One of the most widely used solutions is that in BS 7910:2005 (BSI, 2005), which assumes that plastic collapse occurs locally when the net section stress on a small area surrounding the embedded flaw reaches the yield or flow strength of the material. It also assumes that tensile loads are applied through pinjointed coupling, ie that there is negligible bending restraint. The use of this solution to assess embedded flaws could lead to overly conservative results, which may in some cases be counterintuitive, eg for a given applied loading, the maximum tolerable length for an embedded flaw (whose ligament is greater than or equal to the height of one weld pass) is smaller than that for a surface breaking flaw of the same height.
The paper focuses on the evaluation of existing reference stress (and equivalent limit load) solutions for embedded flaws and the development of improved models. Existing solutions are evaluated using data generated from elasticplastic finite element analyses of pipe models containing typical circumferential embedded flaws.
Scope of Work
The scope of work includes threedimensional elasticplastic finite element analyses of pipe models containing typical circumferential embedded flaws. The pipe models are loaded by pure bending, which is the predominant loading mode during pipeline installation. The models are perfectly aligned across the section containing the flaw and the same tensile properties are assigned to the parent and weld metals. Data from the finite element analyses are used to review and evaluate a number of methods for determination of the reference stress (or limit load) for embedded flaws including conventional methods, recommended in BS 7910 (BSI, 2005) and R6 (BEGL, 2001), and novel methods. It is shown that none of the existing codified reference stress (or limit load) solutions agree well with results obtained from the finite element analyses. Therefore, a new approach using a novel method for definition of reference stress, which is fully consistent with findings from the finite element analyses, is proposed.
The remainder of this paper consists of the following sections: reference stress J estimation scheme; approaches for determining reference stresses (and limit loads) in FADbased assessments; codified reference stress (and limit load) solutions for embedded flaws; approach adopted for determining Jbased limit loads (M_{J}); scope of (and results from) finite element analyses; comments on global collapse and Jbased limit loads; comparison of flat plate solutions with Jbased solutions; plastic strain in ligament; summary and conclusions.
Reference Stress J Estimation Scheme
The evaluation of reference stress models is performed within the context of the reference stress J estimation scheme (BEGL, 2001), which is defined below.
J is estimated from the following expressions, which correspond to the materialspecific FAD (BS 7910 Level 2B/3B and R6 Option 2):
where for a given applied bending moment, M, L_{r} σ_{y} is the reference stress (denoted as σ_{ref}); ε_{ref} is the reference strain corresponding to sref and determined from the stressstrain curve of the material; σ_{y} is the yield or 0.2% proof strength of the material and E is Young’s modulus. J_{e} is the elastic value of J, at an applied moment M, determined from data obtained at an applied moment M_{o}, as follows:
or using the equivalent expression
where, J_{o} is the elastic value of J determined at M_{o} (eg from an elastic finite element analysis), σ_{M} and σ_{o} are the elastic bending stresses on the pipe OD corresponding to M and M_{o}, respectively, determined using elastic section properties.
An alternative estimate of the elastic value of J may be determined as follows
which is similar to Equation [5] but uses the actual elasticplastic stress on the pipe OD (denoted σ_{M1}), rather than σ_{M}. In this case, J_{e} is not proportional to M^{2}.
The parameter L_{r}, which characterises the proximity to plastic collapse, can be expressed as follows:
where M_{L} is the limit load.
The reference stress Jestimation scheme could also be applied using alternative expressions of J, eg which correspond to simplified FADs in BS 7910 and R6.
There are a number of approaches for determining M_{L} (and the corresponding σ _{ref} ), which are described in the following sections
Approaches for Determining Limit Loads for FAD Assessments
The limit load (or plastic collapse) required for the calculation of reference stress and the parameter L_{r} is one of the most important elements of a FADbased assessment since it serves two functions:
 It ensures that the limit load of the component containing the flaw under consideration is not exceeded.
 It ensures that the relationship between elasticplastic driving force and proximity to plastic collapse is consistent with the relationship implied by the failure assessment curve.
Two potential plastic collapse modes of a flawed component can be identified:
 Local collapse: corresponds to failure, by yielding mechanisms, of the ligament adjacent to the flaw. This is deemed to occur when the stress in the remaining ligament reaches the yield strength of the material. With regard to a circumferential partthickness flaw in a girth weld (surface breaking or embedded), the significance of the circumferential extent of the remaining ligament on either side of the flaw is not well defined in any of the existing standards. Another source of uncertainty is whether bending of the section containing the flaw is restrained or not. In the absence of bending restraint, secondary bending stresses arise due to eccentric loading. This is caused by movement of the neutral axis, due to the existence of the flaw, compared with the unflawed condition.
 Global collapse: corresponds to failure, by yielding mechanisms, of the whole section containing the flaw. This is deemed to occur when the global deformation, displacement and/or rotation, of the section become unbounded. Global collapse occurs at a higher load than that corresponding to local collapse.
Most codified limit load or reference stress solutions for partthickness (surface or embedded) flaws are based on the local collapse approach. Although some standards, such as R6 (BEGL, 2001) provide also solutions based on the global collapse approach further checks, eg against finite element analyses, may be required to verify that such solutions provide safe assessments.
An alternative method for determining limit loads, which requires J data from finite element analyses, consists of defining the limit load such that it is consistent with the J data and reference stress J estimation scheme represented by Equation [2]. The main benefit of this approach is that it does not require the analyst to specify in advance whether a local or global collapse model is more suitable. The limit load is found by solving Equations [2] and [3] using J results from an elasticplastic finite element analysis of the flawed component. A simple version of this approach is recommended in Section B.6.4.3(e) of API 579 (API, 2000) and in a slightly different form in Section B.1.89 of API5791/ASMEFFS1 (API/ASME, 2007) as follows:
where P_{ref} is determined from the following relationship:
where J is the total value of J determined from an elasticplastic analysis of the flawed component; J_{e} is the elastic J determined from an elastic analysis, eg using Equations [4] or [5]; P is a characteristic applied load (or stress) such as axial force, bending moment or a combination thereof; and P_{ref} is the reference load (or stress) defined as the load at which the ratio J/J_{e} reaches the value defined by Equation [10].
If P_{ref} is used to construct a BS 7910 Level 3C FAD (with L_{r} defined according to Equation [9]), it will intersect the corresponding BS 7910 Level 2B/3B materialspecific FAD, ie give the same K_{r} value, at L_{r} = 1. Thus, the limit load P_{ref} is defined in a manner, which is consistent with the Level 2B/3B materialspecific FAD at least at L_{r} = 1.0. In this case, the limit load may depend on the strainhardening characteristics of the material.
Codified Reference Stress Solutions for Embedded Flaws
General
Whereas there are several wellestablished reference stress solutions for circumferential surface flaws in pipe girth welds, there are no such solutions for circumferential embedded flaws. Consequently, most analysts use reference stress solutions originally derived for flat plates to assess circumferential embedded flaws in pipe girth welds. The most widely used of these solutions are the reference stress equations given in BS 7910 (BSI, 2005) and R6 (BEGL, 2001). These are given in terms of the membrane stress, P_{m}, and through wall bending stress, P_{b.} However, given that in a thinwalled pipe loaded by bending, P_{b} is very small compared with P_{m}, and in a pipe loaded by tension, P_{b} is equal to zero, the reference stress equations considered below are given assuming that P_{b} is equal to zero. Note that references to equations, sections and figures in codes and standards are shown in italic.
BS 7910
The embedded flaw reference stress equation in Equation P4 of BS 7910 (BSI, 2005) is based on local collapse and assumes that tensile loads are applied through pinjointed coupling, ie that there is negligible bending restraint. The equation was derived by Willoughby and Davey (1987) assuming that the loadbearing area (or ligament) extends to the plate surfaces above and below the embedded flaw and has a length equal to the flaw length plus one plate thickness on either end of the flaw. The solution, with P_{b} set at zero, is as follows:
where p is the ligament (smallest distance between the flaw and surface), t is the wall thickness, and
where 2a is the flaw height and 2c is its length, see Figure 3 (note that the wall thickness in BS 7910 is denoted as B).
R6
R6 (BEGL, 2001) provides several reference stress solutions for embedded flaws in flat plates, which are based on local or global collapse with loads applied through pinjointed coupling (ie pin loading) or fixedgrip loading conditions. The approach used to develop these solutions is described by Lei and Budden (2004). The solutions cater for flaws located fully or partially in the tensile stress zone (determined by the location of the neutral axis). Whilst this distinction is important when assessing flat plates, it is less significant for circumferential embedded flaws in pipe girth welds, which are almost always assumed to be in the tensile stress zone. Consequently, attention is focussed below on the latter condition. The solutions are expressed using the following parameters:
where N_{L} is the limit load and other parameters are illustrated in Figure 4. The limit load solutions are given below in terms of the nondimensional parameter n_{L}.
Global collapse, pinloaded (IV.1.6.31):
Global collapse, fixedgrip tension (IV.1.6.31 with k=0):
As above (Equations [13] and [14]) but with k = 0 (limit load does not depend on crack position in the cross section).
Local collapse, pinloaded (IV.1.6.3.2), solution (a):
This solution is approached when yielding takes place across the whole loadingbearing area 2(t + c) t.
Local collapse, pinloaded (IV.1.6.3.2), solution (b):
This is approached when yielding spreads through the smallest ligament along the plate thickness. Solution (b) is always less than or equal to Solution (a).
Local collapse, fixedgrip tension:
Approach Adopted for Determining Jbased Limit Loads (M_{J})
As evident from the above, there are numerous approaches for the definition of limit load including local collapse, global collapse, or Jbased methods associated with the reference stress J estimation scheme. Each of these approaches can be applied using a number of options. For example, local collapse can be defined based on a somewhat arbitrarily postulated loadbearing area, and either fixedended or pinended supports (reflecting whether bending of the section containing the flaw is restrained or not). In practice it is difficult to assess the suitability of these solutions for use in specific applications without additional information.
The suitability of the above approaches to assess circumferential embedded flaws in pipe girth welds is evaluated in the following sections using data from elasticplastic finite element analyses of pipe models containing typical circumferential embedded flaws. A reference stress (or limit load) is deemed to be adequate if it allows the effective elasticplastic crack driving force expressed in terms of J integral (or J), to be determined using the FADbased reference stress J estimation scheme, with a reasonable degree of accuracy compared to finite element results. Therefore, the meaning of the term ‘limit load’ is extended to include any load that is used to define the reference stress within the context of a FAD calculation. Within this framework, the above approaches are assessed against a Jbased limit load, which for a pipe subjected to bending is referred to as M
_{J}. This is determined using an approach somewhat similar to that of API5791/ASMEFFS1 in that the limit load is defined to ensure consistency between the Level 3C FAD and the Level 2B/3B materialspecific FAD (Equation [3]). However, unlike the API 579 model which achieves this consistency at L
_{r} = 1.0, M
_{J} is determined such that the Level 3C FAD matches the Level 2B/3B FAD at a range of L
_{r} and applied strain values. For each model, M
_{J} is determined for each load increment (in the finite element analysis) as follows:
 L_{r} is determined by solving Equations [2] and [3].
 M_{J} is determined as M/L_{r} (according to Equation [7]), where M is the applied moment.
This approach allows M_{J} to be determined for every load increment and, consequently, allows M_{J} to be plotted as a function of any loading parameter, including applied moment, remote applied stress/strain or L_{r}.
In theory, M_{J} may depend on the position along the crack front. In the present work, attention has been restricted to solutions in the centre of the crack, which is usually the position of maximum J.
Finite Element Analyses
Threedimensional elasticplastic finite element analyses were conducted on pipe models containing a range of embedded flaw geometries. Four pipe geometries with pipe radius to thickness ratios in the range 5 to 20 were considered. In this paper, only results from Series E1 (pipe outside diameter =400mm, wall thickness = 20mm), which were conducted using the stressstrain curve shown in Figures 5a and b, are reported. The stressstrain curve was constructed from the following RambergOsgood powerlaw hardening relationship:
where:
e = true strain.
σ = true stress.
σ_{Y} = 0.2% proof strength (= 400MPa).
E = Young’s modulus (200,000MPa).
n = strain hardening exponent (= 15).
α = 0.002/e_{Y} (= 1.0).
e_{Y} = σ_{Y}/E.
Each finite element model contained an elliptical embedded flaw oriented in the circumferential direction and contained within a plane perpendicular to the pipe axis. The flaws considered had a height in the range 3 to 9mm, were located at 1.5 to 14mm from the pipe inside surface and had a length, along the ellipse major axis, in the range 25 to 250mm. The flaws were located at the 12 o’clock position to ensure that they were subjected to the largest tensile stresses when the model was loaded by bending. The models were loaded by pure bending, which is the predominant loading mode during pipeline installation. In all models, the pipes were perfectly aligned across the section containing the flaw and the same tensile properties were assigned to the finite elements representing parent and weld metals. The dimensions of the pipes and flaws considered within Series E1 are summarised in Table 1.
All analyses were performed using ABAQUS Version 6.6 using a small strain formulation, which is considered to be acceptable at least up to applied strains in the range 0.5 to 1%. The finite element meshes consist entirely of type C3D20R elements, which are 20noded, quadratic, threedimensional elements. As the flaw in the pipe is symmetric about the vertical plane and the applied load is also symmetric about this plane, it is only necessary to model half the pipe.
In addition to analyses using the abovementioned stressstrain model, analyses were performed on a number of pipe models using an elasticperfectly plastic stressstrain model to determine a global collapse limit load, denoted M_{FEA}.
Results
Jbased limit loads  dependence on applied load
Analyses using the above stressstrain model enabled the Jintegral to be calculated as a function of the applied load along the crack front. The highest J values, which generally occurred at the middle of the crack front adjacent to the smaller of the two ligaments (below and above the flaw), were adopted.
The J results were used to estimate the following Jbased limit loads according to the approach described earlier:
 M_{J2B E} (consistent with Equation [2], representing the materialspecific FAD of BS 7910 (Level 2B/3B) and R6 (Option 2) with J_{e} estimated using the elastic pipe bending stress according to Equation [5]).
 M_{J2B EP} (consistent with Equation [2], representing the materialspecific FAD of BS 7910 (Level 2B/3B) and R6 (Option 2) with J_{e} estimated using the elasticplastic pipe bending stress according to Equation [6]).
The above results are shown in Figures 6 and 7 in terms of M_{J2B EP} /4tσ_{y}R_{m} ^{2} vs. L_{r} and applied strain, respectively. Here, t is the pipe thickness, R_{m} is the pipe mean radius, and s_{y} is the pipe yield strength. Plots of M_{J2B E} /4ts_{y}R_{m} ^{2} vs. L_{r} and applied strain are not included since they have broadly similar shapes to those shown in Figures 6 and 7 (but M_{J2B E} is approximately 4% higher than M_{J2B EP}).
Figures 6 and 7 indicate that M_{J2B EP} increase slightly with L_{r} (for L_{r} > 1.0) and applied strain (for strains > 0.2%). This behaviour is believed to be due to a number of factors including:
 Component loading (it can be shown that the load dependence of Jbased limit load solutions is influenced by the loading considered and is more significant for components loaded by bending than by tension).
 The fact that Jbased limit loads are not true limit loads: M_{J2B E} is lower than MFEA, which is a true global collapse limit load, by approximately 9 to 16%; M_{J2B EP} is lower than M_{FEA} by approximately 13 to 19%.
 Approximations within the reference stress J estimation scheme.
The fact that M_{J} (ie M_{J2B E} and M_{J2B EP}) increases with L_{r} and applied strain implies that a single value of M_{J} is an optimal solution only for the L_{r} or applied strain value at which it is determined. Furthermore, if M_{J} is determined at L_{r} = 1, as recommended in API 579, the use of this solution to assess load conditions where L_{r} > 1 can lead to very conservative results. Equally, if M_{J} is determined at relatively high L_{r} value, say at L_{r} = 1.2, the use of this solution to assess load conditions where L_{r} < 1.2 can lead to nonconservative results.
Such dependence of M_{J2B E} and M_{J2B EP} on L_{r} and applied strain somewhat complicates their use to assess standard solutions and/or develop new equations for determining M_{J}. However, further work has shown that this load dependence can be largely eliminated by using either of the following two approaches:
 If the estimated M_{J} for a given load level (eg corresponding to a load increment in the finite element analysis) is multiplied by σ_{M1}/σ_{M} (ratio of elasticplastic pipe stress to elastic pipe stress at the same load level), the product (M_{J2B E}x(s_{M1}/s_{M}) /4tσ_{y}R_{m} ^{2}) or (M_{J2B EP} x (σ_{M1}/σ_{M}) /4ts_{y}R_{m} ^{2}) is largely independent of L_{r} and applied strain and is nearly constant for L_{r} > 1 and applied strain >0.5%. The evidence is illustrated in Figures 8 and 9, which show (M_{J2B EP} x (σ_{M1}/σ_{M}) /4tσ_{y}R_{m} ^{2}) vs. L_{r} and applied strain, respectively. For a given applied moment, the factor (σ_{M1}/σ_{M}) depends only on the pipe cross section and its stressstrain properties and, consequently, can be determined easily.
 If the results are expressed as nondimensional reference stress (determined as σ_{ref}/σ_{M1}), it is seen that this parameter is largely independent of L_{r} and applied strain, see Figure 10. This is predictable since σ_{ref}/σ_{M1} is inversely proportional to M_{J}x(σ_{M1}/σ_{M}). Therefore the beneficial effects of multiplying M_{J} by σ_{M1}/σ_{M} (described above) are included within σ_{ref}/σ_{M1} (here M_{J} refers to both M_{J2B E} and M_{J2B EP}).
An example of the impact of the above approaches is illustrated in Figure 11 for E1BH3L50L3M0 (2a=3mm, 2c=50mm, p=3mm), which shows that the use of the correction factor (σ_{M1}/σ_{M}) or expressing the results in terms of s_{ref} /s_{M1} enables J to be estimated with greater accuracy. It can also be seen that J estimates obtained using the global collapse limit load, M_{FEA}, are significantly lower than those from finite element analysis.
The above two approaches give comparable results, but the second approach is generally preferable since it is relatively simpler to apply and is potentially easier to develop into a versatile assessment model (applicable to pipe loaded by either tension or bending).
Comments on global collapse and Jbased limit loads
To facilitate evaluation of the limits loads considered in the previous section, M_{J2B E} at an applied strain of 0.5% (denoted M_{J2B E 0.5%}) and M_{J2B EP} at an applied strain of 0.5% (denoted M_{J2B EP} 0.5%) were determined. The results are given in nondimensional form in Table 2, which also includes M_{FEA} (global collapse load determined by finite element analysis using an elastic perfectlyplastic stress strain model). The following observations to be made:
 M_{FEA} results are insensitive to crack size and ligament height and are always greater than Jbased limit loads. This implies that limit loads based on global collapse (such as M_{FEA}), could lead to nonconservative estimates of J (see also Figures 11).
 Jbased limit loads decrease as the height increases, the ligament decreases, or as the length increases, but changes in height appear to have greater influence than changes in ligament or length. For example, considering E1BH3L50L3M0 as a base case (height = 3mm, length = 50mm, ligament = 3mm), the following is observed based on results in Table 2:

 Increasing the height from 3 to 6 and 9mm leads to reductions in MJ2B E 0.5% of 3.8 and 5.5% respectively,
 Increasing the ligament from 3 to 6 and 9mm, leads to increases in MJ2B E 0.5% of 1.8 and 2.3% respectively,
 Increasing the length from 50 to 100 and 200mm leads to reductions in MJ2B E 0.5% of 1.1 and 1.8%, respectively.
 Jbased limit loads for flaws located at a given ligament from the ID are nearly equal to those for flaws located at the same ligament from the OD (for example E1BH3L50L3M0 and E1BH3L50L14M0).
 M_{J2B E} (with J_{e} based on the elastic pipe bending stress) is higher than M_{J2B EP} (with J_{e} based on the elasticplastic pipe bending stress) by approximately 4%. This is not surprising since in the former case, for a given load level, the elastic driving force (J_{e}) and f(L_{r}) are higher, L_{r} is lower and M_{J} is higher, see Equations [2] and [7].
Comparison of flat plate solutions with Jbased solutions
To facilitate comparison of the Jbased limit loads with the codified flat plate solutions reviewed earlier, the M_{J} results (M_{J2B E 0.5%} and M_{J2B EP 0.5%}) were expressed in terms of the nondimensional parameter s_{ref}/s_{M1}, where s_{M1} is the elasticplastic pipe bending stress, and compared with s_{ref}/s_{M1} determined from the flat plate solutions (as 1/n_{L}). The results are given in Table 3. The same results are reproduced as the ratio of s_{ref} from the flat plate solutions to s_{ref} from M_{J2B EP 0.5%} in Table 4. The following conclusions are drawn:
 σ_{ref}/σ_{M1} estimates based on flat plate solutions and local collapse exceed significantly σ_{ref}/σ_{M1} based on M_{J2B E 0.5%} and M_{J2B EP 0.5%}. Therefore, flat plate solutions based on local collapse lead to conservative assessments according to the following ranking (given in order of decreasing conservatism):

 R6, local collapse, solution (b), pin loading (Equation [18])
 BS 7910, local collapse, solution (a), pin loading (Equation [11])
 R6, local collapse, solution (a), pin loading (Equation [15])
 R6, local collapse, fixedgrip loading (Equation [19])
 Table 4 shows that the BS 7910 flat plate σ_{ref} solution exceeds σ_{ref} based on M_{J2B EP 0.5%} by a margin which varies depending on flaw size and ligament. The margin is higher for deeper and longer flaws. Except for the shortest flaw considered (length = 25mm) and shallow flaws located near the middle of the thickness (height = 3mm and ligament >= 6mm) the margin exceeds 5%. These results apply to the stressstrain curve considered (low work hardening). Limited results, not reported in this paper, indicate that the margin is lower for high work hardening materials. It should be noted that there are other sources of conservatism in BS 7910 assessments of circumferential embedded flaws in pipes, which include the equations used to estimate the stress intensity factor (intended for flaws in flat plates).
 The majority of σ_{ref}/σ_{M1} estimates based on flat plate solutions and global collapse (with plate width assumed equal to half the pipe mean circumference) are lower than σ_{ref}/σ_{M1} based on M_{J2B E 0.5%} and M_{J2B EP 0.5%}. Therefore, flat plate solutions based on global collapse can lead to nonconservative assessments. However, it can be shown that modifying the R6 pinloading model (Equation [13]) by adjusting the plate width, can lead to solutions that agree well with s_{ref}/s_{M1} based on M_{J2B E 0.5%} and M_{J2B EP 0.5%}.
It may be inferred from the above results that the conventional definition of local collapse solutions for flat plates, assuming that the load bearing area extends one plate thickness at either side of the flaw, leads to overestimating the reference stress for embedded flaws in pipes, which may result in overly conservative assessments. On the other hand limit loads based on global collapse of the pipe cross section (M_{FEA}) or global collapse in a flat plate model, with plate width assumed equal to half the pipe mean circumference, underestimate the reference stress compared with Jbased solutions.
Based on the above, it may be concluded that the way forward is to develop new pipespecific Jbased solutions, which represent loading conditions between global collapse and conventionallydefined local collapse. This implies that the required Jbased solutions correspond to a larger loadbearing area (defined by the extent of the ligament on either side of the flaw) than associated with conventional local collapse solutions.
In further work, simple equations to estimate M_{J2B E 0.5%} have been developed using a semianalytical approach calibrated by means of the finite element results. Additional work is currently being performed to develop equations to estimate σ_{ref} /σ_{M1}. The outcome of these activities, in terms of equations that can be used to predict M_{J} and/or σ_{ref} /σ_{M1} in routine assessments, will be published in 2010.
Plastic strain in ligament
In order to enable an assessment of strains in the ligament adjacent to embedded flaws, data on strain concentrations in the smaller of the two ligaments (below and above the flaw) were obtained for all cases. Figure 12 shows typical contour plots for the equivalent plastic strain in the ligaments adjacent to an embedded flaw. The example shown illustrates contour plots in model E1BH6L50L3M0, which indicate that strain concentration occurs in two plastic zones extending from the flaw tip towards the surface at 45 degrees with respect to the plane of the flaw.
The plastic strain at the surface on either side of the flaw corresponding to a remote axial strain on the pipe OD of 1% is given in Table 3. It can be seen that such strains, which increase as the flaw height or length increase or the ligament decreases, can be as high as 11.5% (model E1BH9L50L3). It should be noted that these strains were obtained in models which were perfectly aligned. Had axial misalignment been included, the strains in the ligament would have been even higher.
Further research is required to produce guidance on strain concentrations in the ligament in the presence of axial misalignment and strength mismatch. Such guidance can be used to assess the possibility of ligament failure due to excessive straining. This would be a separate failure criterion to Jbased fracture associated with extension of the flaw.
Summary and Conclusions
Threedimensional elasticplastic finite element analyses have been conducted on pipes containing circumferential embedded flaws. From these analyses, the elasticplastic fracture mechanics parameter J has been evaluated and used to determine limit loads consistent with the reference stress J estimation scheme. Two Jbased limit loads have been determined: M_{J2B E} and M_{J2B EP}. In addition, global collapse limit loads were obtained from elasticperfectly plastic finite element analyses (denoted as M_{FEA}).
Existing standard solutions and methods for determining limit load (and/or reference stress) estimates for circumferential embedded flaws in pipes within the context of FADbased assessments have been reviewed and evaluated against results obtained from the finite element analyses. The following conclusions are made:
a) J based limit load and reference stress
 M_{J2B E} (J_{e} based on the elastic pipe bending stress) is higher than M_{J2B EP} (J_{e} based on the elasticplastic pipe bending stress) by approximately 4%.
 Both M_{J2B E} and M_{J2B EP} are found to increase with applied load (bending moment), and hence applied strain (for strains > 0.2%), This behaviour is believed to be due to the combined effects of the loading considered, approximations within the reference stress J estimation scheme, and the fact that Jbased limit loads are not true limit loads. Also M_{J2B E} and M_{J2B EP} are both found to increase with L_{r} (for L_{r} > 1.0).
 If M_{J2B E} and M_{J2B EP} are multiplied by the ratio of the maximum elasticplastic stress to the elastic stress (s_{M1}/s_{M}) in the pipe remote from the crack at a given load level, or if the results are expressed in terms of σ_{ref}/σ_{M1}, the resulting parameters become largely independent of load level (for L_{r} > 1 and applied strain > 0.5%). This enables J to be estimated reliably for a wide range of applied loads.
b) Standard solutions and methods
 Global collapse limit loads (such as M_{FEA}) are higher than Jbased limit loads and insensitive to crack size and ligament height. Their use in FADbased assessments could lead to nonconservative estimates of J.
 Flat plate reference stress solutions based on local collapse overestimate Jbased solutions determined from M_{J2B E 0.5%} and M_{J2B EP 0.5%} (values of M_{J2B E} and M_{J2B EP} at 0.5% applied strain) and, consequently, lead to conservative estimates of J.
 In most of the cases considered, flat plate reference stress solutions based on global collapse (with a plate width equal to half the pipe circumference) underestimate Jbased solutions determined from M_{J2B E 0.5%} and M_{J2B EP 0.5%} and, consequently, can potentially lead to nonconservative assessments. However, it can be shown that modifying the R6 pinloading model (Equation [13]) by adjusting the plate width, can lead to solutions that agree well with s_{ref}/s_{M1} based on M_{J2B E 0.5%} and M_{J2B EP 0.5%}.
c) Equations for estimating Jbased limit load and/or reference stress
A new general equation for estimating M_{J2B E 0.5%} has been derived from a semianalytical approach calibrated by means of the finite element results. Additional equations are being developed for estimating σ_{ref}/σ_{M1}. When used in conjunction with BS7910 (Level 2B/3B) FAD or R6 Option 2 FAD, the new Jbased solutions give an improved estimate of J, and hence flaw assessment, compared with using standard codified limit load solutions based on local or global collapse. The new solutions will be published in 2010.
d) Plastic strain concentration
Data on plastic strain concentration in the smaller of the two ligaments adjacent to embedded flaws have been obtained. For some cases, the plastic strain at the surface nearest to the flaw exceeds 10 times the nominal remote strain on the pipe OD.
e) Future work
More work is needed to incorporate the effects of axial misalignment and strength mismatch and account for discontinuous yielding and other rates of strain hardening. More work is also required to produce guidance on strain concentration in the ligament to enable the assessment of ligament failure due to excessive straining.
Acknowledgements
The work was funded, as part the Core research Programme, by Industrial Members of TWI, whose support is gratefully acknowledged. The author also acknowledges the efforts of Dr Martin Goldthorpe, who conducted the finite element analyses, and the valuable support provided by MrJohn Wintle and Dr Simon Smith.
References
ABAQUS/Standard User’s Manuals, Version 6.6, Hibbitt, Karlsson and Sorenson Inc.
API, 2000: API 579: ‘Fitnessforservice’, First Edition, American Petroleum Institute.
API/ASME, 2007: API5791/ASMEFFS12007 ‘Fitnessforservice’, API 579 Second Edition, American Petroleum Institute (API)/American Society of Mechanical Engineers (ASME).
BSI, 2005: BS 7910:2005 including Amendment 1, 2007: ‘Guide to methods for assessing the acceptability of flaws in metallic structures’, British Standards Institution.
BEGL, 2001: R6 Revision 4 and amendments, ‘Assessment of the integrity of structures containing defects’, British Energy Generation Ltd., Gloucester, UK. (Amendments issued in subsequent years).
Lei Y and Budden P J, 2004: ‘Limit load solutions for plates with embedded cracks under combined tension and bending’, International Journal of Pressure Vessels and Piping, Vol. 81, pp589597.
Willoughby A A and Davey T G, 1987: ‘Plastic collapse at part wall flaws in plates’, in ‘Fracture mechanics: perspectives and directions. Proc. 20th national symposium, Bethlehem, PA’, USA, ASTM STP 1020, pp340409.
Table 1 Dimensions of pipe and flaw considered in finite element analysis models (series E1)

Pipe dimensions, mm

Flaw dimensions, mm


Outside diameter

Wall thickness

Height

Length

Ligament to ID

Ligament to OD

E1BH3L50L1.5M0

400

20

3

50

1.5

15.5

E1BH3L50L3M0

400

20

3

50

3

14

E1BH3L50L14M0

400

20

3

50

14

3

E1BH3L50L6M0

400

20

3

50

6

11

E1BH3L50L9M0

400

20

3

50

9

8

E1BH6L50L3M0

400

20

6

50

3

11

E1BH6L50L11M0

400

20

6

50

11

3

E1BH9L50L3M0

400

20

9

50

3

8

E1BH6L50L6M0

400

20

6

50

6

8

E1BH6L50L1.5M0

400

20

6

50

1.5

12.5

E1BH3L25L3M0

400

20

3

25

3

14

E1BH3L100L3M0

400

20

3

100

3

14

E1BH3L200L3M0

400

20

3

200

3

14

E1BH3L250L3M0

400

20

3

250

3

14

E1BH6L25L3M0

400

20

6

25

3

11

E1BH6L100L3M0

400

20

6

100

3

11

E1BH3L25L6M0

400

20

3

25

6

11

Table 2 Nondimensional limit load results from finite element analysis, and ligament plastic strain
Je based on:

Eq. [4]

Eq. [4]

Eq. [5]

Ligament plastic strain % (1)

Source

FEA global collapse

FEA @ 0.5% strain

FEA @ 0.5% strain

FEA @ 0.5% strain


MFEA / 4tσyRm^{2}

M_{J2A E 0.5%} /4tσyRm^{2}

M_{J2BE 0.5%} /4tσyRm^{2}

M_{J2B EP 0.5%} /4tσyRm^{2}

E1BH3L50L1.5M0

0.999

0.880

0.872

0.841

2.8

E1BH3L50L3M0

0.999

0.907

0.888

0.854

1.9

E1BH3L50L14M0

0.999

0.911

0.890

0.854

2.3

E1BH3L50L6M0

0.999

0.934

0.904

0.867

1.1

E1BH3L50L9M0

0.999

0.942

0.909

0.870

1.1

E1BH6L50L3M0

0.998

0.852

0.855

0.825

6.0

E1BH6L50L11M0

0.999

0.851

0.853

0.822

7.6

E1BH9L50L3M0

0.995

0.827

0.840

0.811

11.5

E1BH6L50L6M0



0.880

0.872

0.839

2.4

E1BH6L50L1.5M0



0.841

0.848

0.819

8.6

E1BH3L25L3M0

1.000

0.928

0.901

0.865

1.6

E1BH3L100L3M0

0.999

0.891

0.878

0.846

2.2

E1BH3L200L3M0



0.880

0.872

0.841

2.4

E1BH3L250L3M0



0.876

0.870

0.839

2.5

E1BH6L25L3M0



0.894

0.880

0.847

3.6

E1BH6L100L3M0



0.815

0.832

0.805

8.9

E1BH3L25L6M0



0.953

0.916

0.876

1.0

Note:
 Maximum plastic strain (%) on surface (ID or OD) nearest to the flaw (remote strain on OD=1%).
Table 3 σref /σM1 estimates from Jbased limit loads and flat plate solutions
Source

Eq. [4]

Eq. [5]

BS 7910

R6

R6

R6

R6

Basis of σref /σM1

M_{J2B E 0.5%} FEA @ 0.5% strain

M_{J2B EP 0.5%} FEA @ 0.5% strain

Flat plate, local, pinned

Flat plate, global, pinned

Flat plate, global, fixed

Flat plate, local, pinned

Flat plate, local, fixed

Model

(1)

(2)

(3), (4)

(3), (4)

(3), (4)

(3), (4)

(3), (4)

E1BH3L50L1_5M0

1.077

1.117

1.176

1.022

1.013

1.168

1.091

E1BH3L50L3M0

1.057

1.099

1.158

1.020

1.013

1.150

1.091

E1BH3L50L14M0

1.052

1.096

1.158

1.020

1.013

1.150

1.091

E1BH3L50L6M0

1.038

1.082

1.124

1.016

1.013

1.117

1.091

E1BH3L50L9M0

1.031

1.077

1.103

1.013

1.013

1.096

1.091

E1BH6L50L3M0

1.098

1.138

1.351

1.037

1.026

1.308

1.200

E1BH6L50L11M0

1.097

1.138

1.351

1.037

1.026

1.308

1.200

E1BH9L50L3M0

1.117

1.157

1.586

1.049

1.039

1.459

1.333

E1BH6L50L6M0

1.075

1.117

1.260

1.028

1.026

1.225

1.200

E1BH6L50L1_5M0

1.107

1.146

1.404

1.041

1.026

1.357

1.200

E1BH3L25L3M0

1.042

1.086

1.106

1.010

1.006

1.099

1.061

E1BH3L100L3M0

1.069

1.110

1.209

1.041

1.026

1.202

1.120

E1BH3L200L3M0

1.077

1.117

1.249

1.085

1.053

1.244

1.143

E1BH3L250L3M0

1.080

1.120

1.260

1.109

1.067

1.255

1.149

E1BH6L25L3M0

1.066

1.108

1.227

1.018

1.013

1.194

1.130

E1BH6L100L3M0

1.128

1.167

1.480

1.076

1.053

1.437

1.273

E1BH3L25L6M0

1.024

1.071

1.084

1.008

1.006

1.078

1.061

Notes:
 K based on elastic stress, Lr uses elasticplastic stress and is Jbased at 0.5% strain.
 K based on elasticplastic stress, Lr uses elasticplastic stress and is Jbased at 0.5% strain.
 Collapse of load bearing area around crack front.
 All plate solutions: Width = Π x mean pipe radius.
Table 4 Ratio of sref from code flat plate solution to sref from MJ2B EP 0.5%
Basis of sref from code flat plate solution

BS 7910: local, pinned (2), (3)

R6: global, pinned (2), (3)

R6: global, fixed (2), (3)

R6: local, pinned (2), (3)


sref (BS 7910) / sref (MJ2B EP 0.5%)

sref (R6) / sref (MJ2B EP 0.5%)

sref (R6) / sref (MJ2B EP 0.5%)

sref (R6) / sref (MJ2B EP 0.5%)

effects of ligament size (2a=3, 2c=50, p=1.5 to 9)

E1BH3L50L1_5M0

1.05

0.91

0.91

1.05

E1BH3L50L3M0

1.05

0.93

0.92

1.05

E1BH3L50L6M0

1.04

0.94

0.94

1.03

E1BH3L50L9M0

1.02

0.94

0.94

1.02

effects of ligament size (2a=6, 2c=50, p=1.5 to 6)

E1BH6L50L1_5M0

1.23

0.91

0.90

1.18

E1BH6L50L3M0

1.19

0.91

0.90

1.15

E1BH6L50L6M0

1.13

0.92

0.92

1.10

effects of ligament size (2a=3, 2c=25, p=3 to 6)

E1BH3L25L3M0

1.02

0.93

0.93

1.01

E1BH3L25L6M0

1.01

0.94

0.94

1.01

effects of height (2a=3 to 9, 2c=50, p=3)

E1BH3L50L3M0

1.05

0.93

0.92

1.05

E1BH6L50L3M0

1.19

0.91

0.90

1.15

E1BH9L50L3M0

1.37

0.91

0.90

1.26

effects of height (2a=3 to 6, 2c=25, p=3)

E1BH3L25L3M0

1.02

0.93

0.93

1.01

E1BH6L25L3M0

1.11

0.92

0.91

1.08

effects of length (2a=3, 2c=25 to 250, p=3)

E1BH3L25L3M0

1.02

0.93

0.93

1.01

E1BH3L50L3M0

1.05

0.93

0.92

1.05

E1BH3L100L3M0

1.09

0.94

0.92

1.08

E1BH3L200L3M0

1.12

0.97

0.94

1.11

E1BH3L250L3M0

1.13

0.99

0.95

1.12

effects of length (2a=6, 2c=25 to 100, p=3)

E1BH6L25L3M0

1.11

0.92

0.91

1.08

E1BH6L50L3M0

1.19

0.91

0.90

1.15

E1BH6L100L3M0

1.27

0.92

0.90

1.23

Notes:
 MJ2B EP 0.5% based on elasticplastic stress and is determined at 0.5% strain.
 Collapse of load bearing area around crack front.
 All plate solutions: Width = p ´ mean pipe radius.
Figure 1 A typical failure assessment diagram (FAD).
Figure 1 A typical failure assessment diagram (FAD).
Figure 2 Idealised curved elliptical embedded flaw in a pipe (located at 12 o'clock position)
Figure 2 Idealised curved elliptical embedded flaw in a pipe (located at 12 o’clock position)
Figure 3 Idealised elliptical embedded flaw in a flat plate, used in BS 7910 (BSI, 2005).
Figure 3 Idealised elliptical embedded flaw in a flat plate, used in BS 7910 (BSI, 2005).
Figure 4 Idealised rectangular embedded flaw in a flat plate subjected to tension and/or bending loading, used in R6 (BEGL, 2001).
Figure 4 Idealised rectangular embedded flaw in a flat plate subjected to tension and/or bending loading, used in R6 (BEGL, 2001).
Figure 5 Stress strain model used in all but two of the finite element analyses curve up to: a) 10% strain; b) 0.8% strain.
Figure 5 Stress strain model used in all but two of the finite element analyses curve up to:
a) 10% strain;
b) 0.8% strain.
Figure 6 M_{J2B EP} /4tσyRm^{2} (Je based on elasticplastic pipe bending stress) vs. Lr.
Figure 7 M_{J2B EP} /4tσyRm2 (Je based on elasticplastic pipe bending stress) vs. remote strain (%).
Figure 8 M_{J2B EP} x (σM1/σM) / 4tσyRm^{2} (Je based on elasticplastic pipe bending stress) vs. Lr.
Figure 9 M_{J2B EP} x (σM1/σM) / 4tσyRm^{2} (Je based on elasticplastic pipe bending stress) vs. remote strain (%).
Figure 10 Nondimensional reference stress (σref/σM1) with Je based on elasticplastic pipe bending stress) vs. remote strain %.
Figure 11 J resultsfor model E1BH3L50L3M0 (from FEA and based on estimates of limit moment).
Figure 12 Contour plots for the equivalent plastic strain in the ligaments adjacent to an embedded flaw (model E1BH6L50L3M0).