Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes

International Journal of Pressure Vessels and Piping 86 (2009) 495–507

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Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes Tae-Kwang Song a, Yun-Jae Kim a, *, Chang-Kyun Oh b, Tae-Eun Jin b, Jong-Sung Kim c a

Department of Mechanical Engineering, Korea University, 5 Ka, Anam-dong, Sungbuk-Ku, Seoul 136-713, Republic of Korea Korea Power Engineering Company, Yongin-si, Gyeonggi-do 449-713, Republic of Korea c Department of Mechanical Engineering, Sunchon National University, Sunchon, Jeonnam 540-742, Republic of Korea b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 June 2007 Received in revised form 19 February 2009 Accepted 5 March 2009

This paper firstly presents net-section limit moments for circumferential through-wall and part-through surface cracks at the interface between elbows and attached straight pipes under in-plane bending. Closed-form solutions are proposed based on fitting results from small strain FE limit analyses using elastic–perfectly plastic materials. Net-section limit moments for circumferential cracks at the interface between elbows and attached straight pipes are found to be close to those for cracks in the centre of elbows, implying that the location of the circumferential crack within an elbow has a minimal effect on the net-section limit moment. Accordingly it is also found that the assumption that the crack locates in a straight pipe could significantly overestimate the net-section limit load (and thus maximum loadcarrying capacity) of the cracked component. Based on the proposed net-section limit moment, a method to estimate elastic–plastic J based on the reference stress approach is proposed for circumferential cracks at the interface between elbows and attached straight pipes under in-plane bending. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: Circumferential crack Elbow Elastic–plastic J Finite element limit analysis In-plane bending Net-section limit load Reference stress approach

1. Introduction Developing assessment methods of crack-like defects in piping components is important in structural integrity assessment and plant life extension. Significant efforts in developing crack-like defect assessment methods have been made for the last three decades, resulting in a number of methods (see for instance Refs. [1–8]). Among them, one popular method is the net-section stress approach based on limit load analysis [3,4]. Plastic limit load results can be used directly to estimate maximum load-carrying capacities of cracked components, when the material of interest is sufficiently ductile. Furthermore, in the reference stress approach [9,10], adopted in various defect assessment codes [5–8], a limit load can be an important input to calculate the reference stress and thus to estimate elastic–plastic J. For cracked straight pipes, plastic limit load solutions have been well documented (see e.g., Refs. [11–14]). For cracked elbows, several researchers performed both experimental and numerical works to investigate the effect of cracks on plastic limit loads [15–19]. The authors also proposed closed-form

* Corresponding author. Tel.: þ82 2 3290 3372; fax: þ82 31 290 5276. E-mail address: [email protected] (Y.-J. Kim). 0308-0161/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2009.03.008

approximations of plastic limit loads for circumferential cracked elbows under in-plane bending, via small strain three-dimensional (3-D) finite element (FE) limit analyses using elastic– perfectly plastic materials [20,21]. It should be noted that, in the above works, cracks are assumed to be in the centre of the elbows. On the other hand, in piping systems, elbows are often welded to straight pipes. As welded regions are vulnerable to cracking, cracks could occur in welds between elbows and straight pipes, which need to be assessed. One might argue that the crack could be assumed to be located in the straight pipe and thus the solutions for cracked straight pipes could be used to assess cracks in welds between elbows and straight pipes. It will be shown in this work, however, that such an assumption could significantly overestimate the limit load, implying that such a procedure could lead to non-conservative results. Thus a proper crack-like defect assessment method needs to be developed for cracks in welds between elbows and straight pipes. This paper firstly presents limit moments for circumferential cracks in welds between elbows and straight pipes subject to inplane bending, via small strain FE limit analyses using elastic– perfectly plastic materials. Both circumferential part-through surface cracks and circumferential through-wall cracks (as a limiting case) are considered. Then, based on the proposed

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Nomenclature a E E0 J Je K M ML McL MiL

MsL Mref Mo

Crack depth Young’s modulus ¼E/(1  n2) for plane strain; ¼E for plane stress J-integral Elastically calculated J, see Eq. (9) Elastic stress intensity factor In-plane moment Limit moment (general) Limit moment of an elbow with the crack in the centre of the elbow Limit moment of an elbow with the crack at the interface between the elbow and the attached straight pipe Limit moment of a cracked (straight) pipe Reference load (moment) to define the reference stress Net-section limit moment of an un-cracked elbow

net-section limit moments, a method to estimate elastic–plastic J is proposed based on the reference stress approach. Section 2 briefly describes the FE limit analysis employed in the present work. Sections 3 and 4 present plastic limit loads for circumferential through-wall cracks and circumferential part-through surface cracks, respectively. Section 4 presents a reference stress based method to estimate elastic–plastic J. The work is concluded in Section 5. 2. Finite element (FE) limit analysis 2.1. Geometry Fig. 1 depicts a 90 elbow welded to straight pipes, considered in the present work. The heterogeneous nature of weldments is not explicitly considered and thus the piping system is assumed to be homogeneous. The mean radius and thickness of the pipe are denoted by r and t, respectively, and the bend radius by R. The bend characteristic l is defined by

l ¼

Rt ðR=rÞ ¼ ; ðr=tÞ r2

MOR Mso n R r t

a 3ref 3o q l n so sref

Optimised reference moment for the reference stress Limit moment of an un-cracked straight pipe ¼ 4sor2t Strain hardening index (1  n < N) for Ramberg– Osgood model, Eq. (6) Bend radius Mean pipe radius Thickness of a pipe Coefficient of Ramberg–Osgood model, Eq. (6) Reference strain Yield strain, ¼so/E Half circumferential angle of a circumferential crack Bend characteristic, ¼Rt/r2, Eq. (1) Poisson’s ratio Limiting stress of an elastic–perfectly plastic material; 0.2% proof stress of an elastic–plastic hardening material Reference stress

complicated. Note also that, as constant-depth surface cracks are considered, results for the limiting case of a/t / 1 recover those of through-wall cracks. Based on the detailed FE limit analysis, the authors proposed a plastic limit load solution for un-cracked elbows (q/p ¼ 0, a/t ¼ 0) under in-plane bending [20]. They also proposed plastic limit load solutions for circumferential through-wall and partthrough surface cracked elbows under in-plane bending for the case when the crack is in the centre of the elbow [21]. These solutions are summarized in Appendix to compare with present FE results for cracks at the interface between elbows and attached straight pipes. 2.2. FE mesh The present work considers both circumferential through-wall and part-through surface cracks at the interface between elbows and straight pipes. For circumferential through-wall cracks, typical FE meshes employed in the present work are shown in Fig. 2a. The crack tip was designed with collapsed elements, and a ring of

(1)

to quantify the effect of the bend geometry on plastic limit loads, the above non-dimensional variables are systematically varied. Values of r/t and R/r range from r/t ¼ 5 to r/t ¼ 20 and from R/r ¼ 2 to R/r ¼ 6, respectively, whereas values of l are limited to 0.1  l  0.6. These parameter sets are believed to cover the practically relevant range of elbows in the power generating industry. The length of the attached straight pipe is chosen to be ten times the pipe mean radius, L ¼ 10r, which is sufficiently long to avoid end effects due to the applied loading [16–21]. Both circumferential part-through surface and through-wall cracks were considered. The circumferential through-wall crack is characterized by its relative crack length, q/p, where q denotes the half crack angle (Fig. 1a). The value of q/p was systematically varied from q/p ¼ 0.1 to q/p ¼ 0.75. Note that detailed FE limit analyses for the case of q/p ¼ 0 (un-cracked elbows) were performed recently by the authors [20]. For circumferential part-through surface cracks, one additional geometric variable, the relative crack depth, a/t, was further considered (Fig.1b). Note that the surface crack was assumed to have a rectangular shape (constant depth), but the value of q/p was limited to q/p ¼ 0.5. For larger values of q/p, a part of the surface crack could be subject to compression, which makes FE limit analysis

Fig. 1. Schematic illustration of a circumferential part-through surface crack at the interface between an elbow and attached straight pipe.

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through the thickness were also performed, and the results were identical. This led to a conclusion that the use of two elements was sufficient. For circumferential part-through surface cracks, the crack tip was also designed with collapsed elements, and a ring of wedgeshaped elements was used in the crack-tip region. As the crack tip region requires a sufficiently fine mesh, a total of eleven or twelve elements were used through the thickness; four elements in the cracked ligament, and seven to eight elements in the un-cracked ligament. Although a half model was considered due to symmetry, such modeling substantially increases the total number of elements in the FE models. Moreover, as 3-D solid elements (element type C3D20R within ABAQUS) were used, a number of nodes could be quite large, which makes numerical calculations rather difficult. In this respect, the shell-to-solid coupling constraint option within ABAQUS was invoked for efficient computations. As shown in Fig. 2b, only the near-tip region was modeled using solid elements (element type C3D20R), and the other region using shell elements (element type S8R). These solid and shell elements are coupled using the constraint option within ABAQUS. A typical FE mesh includes 5738 elements and 26,312 nodes. Validity of the option was checked, and the results will be shown in Section 2.4. 2.3. FE limit analysis Limit analyses were performed using ABAQUS. Materials were assumed to be elastic–perfectly plastic, and non-hardening J2 flow theory was used. As noted, the piping system considered comprised the 90 bend and the attached straight pipe of length L (Fig. 1). Bending loading was applied by rotation at the end-nodes of the pipe, constrained using the MPC (multi-point constraint) option within ABAQUS, and sufficiently large rotation was applied. The bending moment is determined directly from nodal forces of the constrained nodes. Limit analyses were performed using the small geometry change (geometrically linear) option. For elbows, the bending mode can be either opening or closing, depending on the bending direction (Fig. 1). For cracked elbows, only the crackopening bending mode was considered; opening bending for intrados cracks and closing bending for extrados cracks. Typical moment-rotation responses from the present FE limit analyses clearly showed limiting values of moments, which were chosen to be FE limit moments. 2.4. Validity of FE limit analysis using shell-to-solid coupling

Fig. 2. Typical finite element (FE) meshes: (a) circumferential through-wall crack, and (b) circumferential part-through surface crack.

wedge-shaped elements was used in the crack-tip region. To reduce the computing time, symmetry conditions were fully utilised in the FE models and thus a half model was used. To avoid problems associated with incompressibility, reduced integration elements (element type C3D20R within ABAQUS [22]) were used. A total number of elements and nodes are 1848 and 10,834, respectively. Although two elements were used through the thickness, mesh sensitivity studies using four and six elements

As noted, for FE limit analyses of circumferential part-through surface cracks, the shell-to-solid coupling constraint option within ABAQUS was utilised for efficient computation. As the shell element is used instead of the solid element, it is crucial to check the feasibility of the computational scheme used in the present work. To check its validity, this option was firstly used to model circumferential part-through surface cracks in straight pipes. A typical FE mesh is shown in Fig. 3a, having 5515 elements and 25,606 nodes. The FE analyses were performed for two different values of a/t, a/t ¼ 0.5 and 0.7, and two values of r/t, r/t ¼ 5 and 10, with the fixed q/p ¼ 0.5. The FE results were then compared with the solution proposed by Kim et al. [12], developed by fitting the results from detailed 3-D FE limit analyses. The plastic limit load solution, MsL, was given by

MLs Mos

    qa a sin q   cos    a a2 2 t t 2     ¼ 1 þ A1 þ A2 pa t t a f ðqÞ q  cos 8 t t 2q

(2)

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Fig. 3. (a) A FE mesh using the shell-to-solid coupling option for a circumferential part-through surface cracked pipe, and (b) comparison of FE results with the closed-form solution given in Ref. [12], determined from full 3-D FE models.

with

A1 ¼ 0:0741  0:1693

 

q p

A2 ¼ 0:0863  1:0127 2

 

q p

4

6

f ðqÞ ¼ 0:7854q  0:09817q þ 0:0040906q  0:000085q

8

and Mso denotes the limit moment for un-cracked straight pipes, Mso:

Mos ¼ 4so r 2 t

(3)

where so denotes the limiting stress of an elastic–perfectly plastic material. The FE limit loads are compared with Eq. (2) in Fig. 3b. For four different cases considered, the differences between the present results and Eq. (2) are within 4%. Further validation was made for constant-depth, circumferential part-through surface cracks in the centre of the elbow under in-plane bending. Two different geometries were considered, (i) r/t ¼ 10 and R/r ¼ 5, and (ii) r/t ¼ 5 and R/r ¼ 2. The value of q/p was fixed as q/p ¼ 0.5, but three different values of a/t were considered, a/t ¼ 0.3, 0.5 and 0.7. Plastic limit loads,

calculated from full 3-D FE analysis using solid elements, were compared with the results from the FE analysis using the shellto-solid coupling constraint option. A typical FE mesh, having 5515 elements and 25,606 nodes, is shown in Fig. 4a, where the longitudinal length of the part made of 3-D solid elements was set to be exactly the same as that in other calculations. Ratios of FE limit loads from the full 3-D FE analysis to those from the FE analysis using the shell-to-solid coupling constraint option are shown in Fig. 4b. This shows that the use of the shell-to-solid coupling constraint option gives higher limit moments and the difference increases with decreasing a/t. However, differences are still less than 5%. Good agreement, shown in Figs. 3b and 4b, provides confidence in the use of present FE analyses. 3. Limit loads for cracks at the interface between elbows and straight pipes Plastic limit load solutions for cracks in the centre of elbows, proposed by the authors [20,21], are summarized in Appendix. These solutions will be compared with FE limit loads for cracks at the interface between elbows and straight pipes, based on which closed-form approximations of plastic limit loads will be proposed in this section.

Fig. 4. (a) A FE mesh using the shell-to-solid coupling option for an elbow with a circumferential part-through surface crack in the centre of the elbow, and (b) comparison of FE results with the results from full 3-D FE models.

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Fig. 5. Ratios of FE limit moments for circumferential through-wall cracks in the centre of elbows, McL, to those at the interface between elbows and straight pipes, MiL: (a) extrados cracks under closing bending, and (b) intrados cracks under opening bending.

Fig. 6. Comparisons of FE limit loads for circumferential through-wall cracks at the interface between elbows and straight pipes with Eq. (4): (a) extrados cracks under closing bending, and (b) intrados cracks under opening bending.

Fig. 7. Ratios of FE limit moments for circumferential part-through surface cracks in the centre of elbows, McL, to those at the interface between elbows and straight pipes, MiL: (a) extrados cracks under closing bending, and (b) intrados cracks under opening bending.

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Fig. 8. Comparisons of FE limit loads for circumferential part-through surface cracked elbows under in-plane bending with Eq. (5).

3.1. Circumferential through-wall cracks When cracks are at the interface between elbows and straight pipes, instead of in the centre of elbows, FE limit moments are found to be similar to those for cracks in the centre of elbows. In Fig. 5a, ratios of FE limit moments for extrados circumferential through-wall cracks in the centre of elbows to those at the interface between elbows and straight pipes are shown for four different values of r/t and R/r. The limit loads for centre cracks, McL, are higher than those for cracks at the interface between an elbow and the straight pipe, MiL, but differences are not more than 10% at

q/p ¼ 0.25. Corresponding results for intrados cracks under opening bending are shown in Fig. 5b. Again, differences are within 10%, but for intrados cracks, limit loads for centre cracks are lower than those for cracks at the interface between elbows and straight pipes. The different trend of the intrados crack from that of the extrados crack is believed to result from different plastic deformation patterns. When the crack is at the interface between elbows and straight pipes, plastic deformation is affected not only by the elbow but also by the attached straight pipe, and thus the crack location (either extrados or intrados) could affect the resulting limit moment.

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Fig. 9. Schematic illustration how to determine J in the present work: (a) part-through surface cracks and (b) through-wall cracks.

Table 1 Comparison of the influence function F for the stress intensity factor for elbows with the circumferential through-wall cracks at the interface between elbows and straight pipes with those for straight pipes. For straight pipes, values in parenthesis indicate those from Ref. [2].

q/p

r/t ¼ 5 R/r ¼ 2

Pipe bend (Extrados) Pipe bend (Intrados) Straight pipe

0.125 0.25 0.125 0.25 0.125 0.25

r/t ¼ 10 R/r ¼ 3

0.93578 1.04753 1.37776 1.44063 0.84894 1.02023 1.45544 1.46670 1.12043 (1.15928) 1.39781 (1.45543)

R/r ¼ 2

R/r ¼ 3

501

As limit loads for cracks at the interface between elbows and straight pipes are close to those for cracks in the centre of elbows, limit moment solutions for cracks in the centre of elbows, Eq. (A.2) or Eq. (A.3) in Appendix, could be in principle used to quantify limit loads for cracks at the interface between elbows and straight pipes. It is found that the use of Eq. (A.2) could give higher limit moments than FE limit loads for both extrados and intrados cracks, and thus could be non-conservative. On the other hand, the use of Eq. (A.3) gives overall satisfactory results regardless of the crack location (extrados or intrados). Thus plastic limit loads for circumferential through-wall cracks at the interface between elbows and straight pipes, MiL, can be estimated from

8   o n q q > > > þ 1:21 for 0   0:5 min 1:0; 1:78 < i ML p p   ¼ > Mo q 3 q > > for 0:5   1:0 : 2:56 1 

p

(4)

p

where Mo is the limit moment of an un-cracked elbow, given by Eq. (A.1) in Appendix. Above approximations are compared with FE results in Fig. 6.

R/r ¼ 5

0.72041 0.92676 1.11945 1.43375 1.57807 1.66596 0.43367 0.71346 1.02460 1.37376 1.45849 1.57208 1.20647 (1.20354) 1.58602 (1.58199)

3.2. Circumferential part-through surface cracks As for through-wall crack cases, FE limit moments for circumferential part-through surface cracks at the interface between elbows and straight pipes are found to be similar to those for cracks in the centre of elbows. In Fig. 7, ratios of FE limit

Fig. 10. Comparisons of FE J with the reference stress based J estimates using the reference stress defined by the FE plastic limit load determined in this work.

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Fig. 11. Comparisons of FE J with the reference stress based J estimates using the reference stress defined by the optimized reference load, Eq. (11): results for the case of n ¼ 5.

moments for circumferential part-through surface cracks in the centre of elbows to those at the interface between elbows and straight pipes are shown as a function of the relative crack depth a/t for three different values of r/t and R/r, and for two different values of q/p, q/p ¼ 0.25 and 0.5. For part-through surface cracks, differences increase with increasing a/t, and thus are smaller than those for through-wall crack cases, that is, differences are not more than 10% for all cases considered. As for through-wall crack cases, limit load solutions for circumferential part-through surface cracks at the interface between elbows and straight pipes can be also used to quantify limit loads for cracks at the interface between an elbow and the straight pipe. It was shown in Section 3.1 that for through-wall crack cases, the limit load solution for extrados cracks, Eq. (A.2), was found to be inappropriate, and that for intrados cracks, Eq. (A.3), was used to quantify limit loads for cracks at the interface between elbows and straight pipes. Similarly the limit load solution for intrados circumferential part-through surface cracks, Eq. (A.5) in Appendix, can be used to quantify limit loads for circumferential part-through surface cracks at the interface between elbows and straight pipes, regardless of the crack location (for both extrados and intrados). Thus plastic limit loads for circumferential throughwall cracks at the interface between elbows and straight pipes, MLi, can be estimated from

      MLi q q a þ 2:52 þ 0:78 ¼ min 1:0; 4:30  0:1 p p t Mo

(5)

where Mo is also given by Eq. (A.1) in Appendix. Above approximations are compared with FE results for elbows with six different values of r/t and R/r in Fig. 8. Note that Eq. (5) is valid only for 0  q/ p  0.5. 4. Reference stress based J estimates This section provides a reference stress based method of J estimates for elbows with circumferential part-through surface cracks at the interface between elbows and straight pipes. One outstanding issue in application of the reference stress approach particularly to part-through surface cracks is how to properly define the reference stress. Noting that the reference stress is typically defined by a limit load, the issue is quite complicated for part-through surface crack problems, as a limit load for the partthrough surface crack can be arbitrarily defined by a load corresponding to local yielding (ligament collapse, ‘‘local’’ limit load), or by the load corresponding to net-section yielding (‘‘global’’ limit load). In this section, a proper choice of the reference stress for circumferential part-through surface cracks at the interface between elbows and straight pipes is discussed. 4.1. Elastic–plastic finite element analyses The reference stress approach has been used to estimate nonlinear fracture mechanics parameters such as J and C* [5–10]. To develop a reference stress based method to estimate elastic–plastic J for circumferential cracks at the interface between elbows and

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Fig. 12. Comparisons of FE J with the reference stress based J estimates using the reference stress defined by the optimized reference load, Eq. (11): results for the case of n ¼ 10.

straight pipes, elastic–plastic FE analyses were performed using ABAQUS. The tensile properties for elastic–plastic FE analyses are assumed to follow   the Ramberg–Osgood (R-O) idealisation:

3 s s ¼ þa 3o so so

n

(6)

where 3o, so, a and n are constants, with E3o ¼ so where so now denotes the 0.2% proof (yield) stress and E is Young’s modulus. The deformation plasticity option with a small geometry change continuum model was invoked. Although values of E, a and so were fixed as E ¼ 200 GPa, a ¼ 1 and so ¼ 400 MPa, the results do not depend on such a choice due to proper normalization. Regarding material, two values of the strain hardening exponent n were selected, n ¼ 5 and 10. Regarding pipe geometries, values of r/t and R/r were systematically varied to produce values of l from l ¼ 0.1 to l ¼ 0.6. For crack geometries, three different values of a/t and q/p were considered, a/t ¼ 0.3, 0.5 and 0.7; and q/p ¼ 0.125, 0.25 and 0.5. For selected cases, through-wall cracks (a/t ¼ 1.0) were also considered. The J-integral values were extracted from FE results using a domain integral, as a function of the applied load. Directions for J calculations are depicted in Fig. 9 for through-wall and part-through surface cracks. For through-wall cracks, FE J values were averaged along the thickness. For constant-depth, part-through surface cracks, FE J values were extracted in the centre of the crack, as depicted in Fig. 9. 4.2. Reference stress based approximate J estimates In the reference stress approach, the J-integral is estimated from [6,9,10]

  J E3 1 sref 2 sref ¼ ref þ ; sref 2 so E3ref Je

(7)

where sref is the reference stress, defined by

sref ¼

M so ; Mref

(8)

and Je is the elastically calculated value of J:

Je ¼

K2 ; E0

(9)

where K denotes the elastic stress intensity factor; E0 ¼ E for plane stress and E0 ¼ E/(1  n2) for plane strain where n denotes Poisson’s ratio. In Eq. (7), M denotes the bending moment; Mref is a reference normalising moment; and 3ref denotes the reference strain at the reference stress s ¼ sref, determined from true stress–strain data for the material of interest. Using Eqs. (6) and (7), elastic–plastic J can be estimated if one knows Je and sref. Determination of elastic stress intensity factors is straightforward, and is not within the scope of this work. Note that elastic stress intensity factors for circumferential part-through surface cracks at the interface between elbows and attached straight pipes are not currently available, although they can be determined using appropriate numerical techniques with un-cracked body stresses. In the present work, values of Je were determined directly from the elastic FE analysis. Although investigation of K is not the main topic in this paper, it is worth comparing K values for selected cases with those for straight pipes. Table 1 compares the values of the influence

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function F for elbows with circumferential through-wall cracks at the interface between elbows and straight pipes with those for straight pipes. The influence function F is defined by

K ¼ F

M pffiffiffiffiffiffiffiffi prq;

pr2 t

(10)

values of F for straight pipes are compared with the solutions in Zahoor [2], and differences are within 0.2% for r/t ¼ 10 and 4% for r/ t ¼ 5. For interface cracks, values of F tend to increase with increasing R/r. Effects of r/t and q/p are more complex. For q/p ¼ 0.25, values of F for interface cracks are overall close to those for straight pipes. For q/p ¼ 0.125, on the other hand, values of F for interface cracks can be much lower than those for straight pipes. Determination of the reference stress is quite important, particularly for part-through surface cracks, and is the main subject of this work. A simple choice of Mref is the plastic limit load for the cracked component of interest, Mref ¼ ML [6,9,10]. Two points are worth noting at this point. The first one is that, for part-through surface cracks, limit loads can be defined by local yielding (‘‘local’’ limit load), or by net-section yielding (‘‘global’’ limit load). It has been reported that global limit loads were more relevant to define the reference stress than local ones [23–27]. The second point is that, although a simple choice of Mref is the plastic limit load for the cracked component of interest, it has been also shown that such a choice does not necessarily provide best results and thus another choice can be made for best J estimates [23,24,28–30]. The J values from FE results are compared with the estimated J using Eq. (7) in Fig. 10 for selected cases of extrados circumferential part-through surface cracks at the interface between elbows and straight pipes; q/p ¼ 0.1, 0.25 and 0.5; a/t ¼ 0.3, 0.5 and 0.7; and n ¼ 5. Note that, for selected cases, the results for through-wall cracks (q ¼ 1.0) are also included. Values of r/t and R/r were systematically varied, but only four cases are shown in Fig. 10 for the sake of space. Results are presented in the failure assessment diagram (FAD) space, i.e., (Je/J)1/2 versus M/Mref, where Mref is defined by the FE limit moment, MiL, determined in this work. Note that the proposed limit loads are global, as they are based on net-section yielding. Several features can be noted from the results in Fig. 10. The first one is that FE J results are overall below the prediction line for all cases considered, implying that estimated J values using Eq. (7) with the limit load, Eq. (4) or Eq. (5), are lower than FE results. As J values should be overestimated for conservative J estimates, the results in Fig. 10 imply that the reference stress based J estimates using Eq. (4) or Eq. (5) are overall non-conservative. On the other hand, the results in Fig. 10 also show that effects of geometric variables (a/t, q/p, r/t and l) are not significant, which is consistent with the original idea of the reference stress approach to minimize geometry effects on J estimates. The effect of a/t is almost negligible. This is quite interesting, since the effect of a/t should be significant if a local limit load is more relevant to define the reference stress. Such results are consistent with the finding in other works [23–27] that global limit loads are more relevant to define the reference stress for part-through surface cracks than local ones. Furthermore, although it is not clear from Fig. 10, careful examination leads to a conclusion that effects of r/t and R/r are not significant. On the other hand, the effect of q/p seems to be more important than effects of a/t, r/t and R/r. Differences between FE J results and estimated ones tend to decrease with decreasing q/p. The results in Fig. 10 suggest that reference stress based J estimates using Eq. (7) with the plastic limit load, Eq. (4) or Eq. (5), are insensitive to a/t, r/t and R/r, but are slightly dependent on q/p. It has been argued that the reference load Mref can be defined by another load, rather than the limit load, such that Eq. (7) can

provide best J estimates [23,24,28–30]. Such a load will be termed to as an optimised reference load (moment), MOR. To propose an optimized reference moment, two points should be noted. The first one is that, as reference stress based J estimates are slightly sensitive only to q/p, a correction term including q/p is needed. The second one is that, as reference stress based J estimates are overall non-conservative, an additional constant is needed to reduce nonconservatism. Based on the extensive FE results, the following expression for the optimised reference moment is proposed:

MOR ¼ g

 

# "    q 2 q þ 2:0 MLi 

q M i ¼ 0:45 p L

p

p

(11)

where MLi is given by Eq. (4) or Eq. (5), depending on the crack shape. The factor g(q/p) in Eq. (11) is regarded as a correction factor for q/p and non-conservatism. The value of g decreases monotonically from g ¼ 0.9 for q/p ¼ 0.0 to g ¼ 0.79 for q/p ¼ 0.5. Note that determination of g in Eq. (11) is purely empirical based on FE results, and does not involve any theoretical background. Thus Eq. (11) should be valid only for the cases covered by the FE analysis. Estimated J values using Eq. (7) with Eq. (11) are compared with FE results in Fig. 11 for the case of n ¼ 5, and corresponding results for n ¼ 10 are shown in Fig. 12. This shows that J estimates based on the reference stress approach using the optimized reference load, Eq. (11), agree well with FE results. The proposed method is quite simple and thus is very useful in practical application.

5. Concluding remarks In the present work, net-section limit moment solutions and approximate J estimates are proposed for circumferential partthrough surface cracks in elbows under in-plane bending, via systematic, small strain FE limit analyses using elastic–perfectly plastic materials. A notable point is that cracks are assumed to be at the interface between elbows and attached straight pipes instead of being in the centre of the elbow, as elbows are typically welded to straight pipes and weldments are vulnerable to cracking. Based on the results from the FE limit analysis, closed-form approximations of net-section limit moments for circumferential part-through surface cracks (and through-wall cracks for limiting cases) in elbows under in-plane bending are firstly given. For comparisons, resulting FE limit moments are also compared with those for cracks in the centre of elbows. One interesting finding is that net-section limit moments for cracks at the interface between elbows and attached straight pipes are close to those for cracks in the centre of elbows. This implies that the location of the circumferential crack within an elbow does not affect net-section limit moment, and net-section limit moments are close to those for cracks in the centre of elbows. A more important point is that net-section limit moments for circumferential cracked elbows are overall lower than those for circumferential cracked pipes. Differences increase with decreasing relative crack depth and width. For practically interesting ranges of the circumferential crack geometry, net-section limit moments for cracks at the interface between elbows and attached straight pipes could be much lower than those for cracked straight pipes. This is an important in assessment of cracked piping components. When a crack at the interface between an elbow and attached straight pipe needs to be assessed, one might assume that the crack locates in a straight pipe. The present results, however, show that such an assessment could significantly overestimate the net-section limit load (and thus maximum load-carrying capacity) of the cracked component, and thus could be non-conservative. The present results

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suggest that a crack should be assumed to be in the centre of an elbow, rather than in a straight pipe, for appropriate assessment. Based on the proposed limit moment solutions, a reference stress based method to estimate elastic–plastic J is then proposed for circumferential part-through surface cracks in elbows under inplane bending. It is shown that geometry effects on reference stress based J estimates using the proposed limit moment are not significant, but resulting J values are overall lower (non-conservative) than FE results. A correction term is applied to re-define the reference load, and resulting J estimates agree well with FE results. The proposed method is quite simple and thus is very useful in practical application. The present FE results have restrictions in terms of pipe/crack geometries, which impose applicable limits of the proposed J estimation method. The proposed method is strictly valid for the ranges of geometries considered in the present work. Application of the proposed method outside ranges needs caution. The present FE analyses are also based on the small strain assumption. As elbows are flexible components, large geometry change effects could be important. When the large geometry change effect is considered, the bending mode (either closing or opening) also significantly affects plastic behaviour of cracked piping. The effect of the large geometry change on net-section plastic loads will be reported shortly. Finally it should be noted that elbows tend to be thicker than the attached pipes in applications. In such a case, even when a crack is present in the elbow, the straight pipe could collapse before the cracked elbow due to the thickness difference. The worst crack location will be at the weld between elbows and attached straight pipes, which is considered in the present work. Acknowledgement This research is performed under programs of Basic Atomic Energy Research Institute and Engineering Research Centre (No. R11-2007-028-00000-0), funded by Korea Science & Engineering Foundation.

505

A.1. Limit moments of un-cracked elbows, Mo Limit moment solutions for un-cracked elbows under in-plane bending are given by [20]

r 0:6755 n r 0:0772  Mo l þ 1:5398 ¼ 0:6453 t t 4so r 2 t r 0:0601 with n ¼ 0:5157 t

(A.1)

Note that the denominator in Eq. (A.1), 4sor2t, represents the limit moment for an un-cracked straight pipe. A.2. Limit moments of circumferential through-wall cracked elbows, McL When elbows with extrados circumferential through-wall cracks are subject to in-plane (closing) bending, plastic limit loads McL are given by [21]

  8 o n q q > > þ1:44 for 0   0:5 > < min 1:0;2:10

MLc ¼   Mo > q 3 > > :3:12 1

p

p

p q for 0:5   1:0 p

(A.2)

where Mo is given by Eq. (A.1). For intrados circumferential through-wall cracks under in-plane (opening) bending, plastic limit moments are given by [21]:

8   o n q q > > > þ1:21 for 0   0:5 min 1:0;1:78 < c ML p p   ¼ Mo > q 3 q > > 2:56 1 for 0:5   1:0 :

p

(A.3)

p

where Mo is also given by Eq. (A.1). These approximations are compared with FE results in Fig. A.1. A.3. Limit moments of circumferential part-through surface cracked elbows, McL

Appendix. Summary of plastic limit moment solutions

For extrados circumferential part-through surface cracks under in-plane (closing) bending, plastic limit loads McL are given by [21]

In this Appendix, plastic limit load solutions for circumferential cracked elbows under in-plane bending, proposed in Refs. [20,21], are summarized. Cracks are located in the centre of the elbow.

    

MLc q q a ¼ min 1:0; 4:4  0:1 þ 2:3 þ 1:0 p p Mo t

(A.4)

Fig. A.1. Comparisons of FE limit loads for circumferential through-wall cracks in the centre of elbows under in-plane bending with Eqs. (A.2) and (A.3): (a) extrados cracks under closing bending, and (b) intrados cracks under opening bending.

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Fig. A.2. Comparisons of FE limit loads for circumferential part-through surface cracked elbows under opening bending with Eqs. (A.4) and (A.5): (a) and (b) extrados cracks under closing bending; and (c) and (d) intrados cracks under opening bending.

For intrados circumferential through-wall cracks under in-plane (opening) bending, plastic limit moments are given by [21]:

      MLc q q a þ 2:52 þ0:78 ¼ min 1:0;4:30 0:1 p p t Mo

(A.5)

In Eqs. (A.4) and (A.5), Mo is also given by Eq. (A.1), and they are valid only for 0  q/p  0.5. In the limiting case of the through-wall crack (a/t / 1), both equations recover the solutions for throughwall cracks given above. These approximations are compared with FE results in Fig. A.2.

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