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Quantification of pressure-induced hoop stress effect on fracture analysis of circumferential through-wall cracked pipes
Engineering Fracture Mechanics 69 (2002) 1249–1267 www.elsevier.com/locate/engfracmech
Quantification of pressure-induced hoop stress effect on fracture analysis of circumferential through-wall cracked pipes Yun-Jae Kim, Nam-Su Huh, Young-Jin Kim
*
SAFE Research Centre, School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu, Suwon, Kyonggi-do 440-746, Republic of Korea Received 23 April 2001; received in revised form 24 September 2001; accepted 2 October 2001
Abstract This paper provides engineering J and crack opening displacement (COD) estimation equations for through-wall cracked (TWC) pipes under internal pressure and under combined internal pressure and bending. Based on selected 3-D FE calculations for the TWC pipe under internal pressure using power law materials, elastic and plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the normalised crack length and the mean radius-to-thickness ratio. These developed GE/EPRI-type solutions are then re-formulated based on the reference stress concept. Such re-formulation not only provides simpler equations for J and COD estimation, but also can be easily extended to combined internal pressure and bending. The proposed reference stress based J and COD estimation equations are compared with elastic–plastic 3-D FE results using actual stress–strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimates. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Through-wall cracked pipe; Crack opening displacement; J -integral; Reference stress; Finite element; Internal pressure
1. Introduction Pressurised piping is an important element in power plants, and thus application of fracture mechanics analysis to such pressurised piping is important in structural integrity assessment of plant components. One example of such application is the LBB analysis of piping [1,2]. In general, application of the LBB procedure requires two steps. Firstly, the crack length corresponding to the (assumed) detectable leakage rate should be calculated for a TWC pipe. For this step, engineering methods to estimate the COD and the leak rate are needed. The second step is to perform the pipe fracture stability analysis. For this second step, an engineering method to estimate the J-integral is needed.
*
Corresponding author. Tel.: +82-31-290-5274; fax: +82-31-290-5276. E-mail address: [email protected] (Y.-J. Kim).
0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 1 3 2 - 1
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Nomenclature a half crack length, a ¼ Rm h ae effective crack length, see Eq. (38) E Young’s modulus E0 ¼ E=ð1 m2 Þ for plane strain; ¼ E for plane stress F shape factor for the elastic stress intensity factor h1 (n) plastic calibration function for Jp in the GE/EPRI approach h1 (n ¼ 1) value of h1 for elastic case (n ¼ 1) h2 (n) plastic calibration function for dp in the GE/EPRI approach h2 (n ¼ 1) value of h2 for elastic case (n ¼ 1) J J-integral Je elastically calculated value of J; Je ¼ K 2 =E0 Jp plastic component of J K linear elastic stress intensity factor n strain hardening index ð1 6 n < 1Þ for Ramberg–Osgood model, Eq. (1) n1 strain hardening index for the ERS-based COD estimation equation, Eq. (27) p, M, P internal pressure, bending moment and axial tension on a pipe pL , ML , PL limit internal pressure, moment and axial tension assuming the limiting stress of ry poR , MoR , PoR optimised reference pressure, moment and axial tension load of cracked pipe under a single loading c c poR , MoR optimised reference pressure and moment of cracked pipe under combined tension and bending Ri inner radius of pipe Rm mean radius of pipe Ro outside radius of pipe V shape factor for the elastic crack opening displacement a coefficient of Ramberg–Osgood model, see Eq. (1) d crack opening displacement at centre of crack de elastic component of crack opening displacement dp plastic component of crack opening displacement e strain, general eref reference strain at the reference stress rref eo normalising strain in the Ramberg–Osgood model, Eq. (1) eu uniform elongation k Load proportionality factor, see Eq. (33) m Poisson’s ratio h half of total crack angle r stress, general rref reference stress, see Eq. (25) for definition ry yield strength ru ultimate tensile strength Abbreviations COD crack opening displacement ERS enhanced reference stress R–O Ramberg–Osgood
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FE finite element LBB leak-before-break GE/EPRI general electric/electric power research institute TWC through-wall cracked
When fracture mechanics analysis is performed for TWC pipes subject to internal pressure, it is typical practice to transform the internal pressure into the equivalent axial load, and thus to analyse the cracked pipe subject to the axial load. In this context, existing solutions for elastic–plastic fracture analysis for cracked pipes include solutions for axial tension, but not for pressure (see for instance the GE/EPRI handbooks [3,4]). One problem associated with such an approach is that the hoop stress effect due to the pressure is neglected. The resulting analysis can be erroneous, particularly when the applied pressure is high or when the pressurised pipe is subject to elevated (creep) temperatures. More importantly, the results can be non-conservative in terms of the crack driving force. Thus, for fracture mechanics analyses of cracked pipes under internal pressure, the hoop stress effect due to internal pressure should be explicitly considered. Consideration of the hoop stress effect into the GE/EPRI-type J and COD estimation equations requires detailed 3-D FE analyses. Although the use of 3-D FE analyses is increasingly popular, performing a large number of 3-D FE analyses is currently impractical, noting that pressurised piping is typically subject to combined internal pressure and bending moment in practical situations. In this context, the enhanced reference stress method (ERSM), proposed recently by the authors [5–8], is an attractive alternative. This method is based on the reference stress approach [9], but enhancement is made in the definition of the reference stress to improve its accuracy. Reliability of this method has been checked by comparing J and COD results with published pipe test data and with extensive FE results [5,6]. One important advantage of this method is that it can be easily extended to more complex geometries and loadings. For instance, the proposed ERSM has been extended to J and COD estimations for TWC pipes under combined bending and tension [7] and to those for complex cracked pipes [8]. This paper quantifies the hoop stress effect induced by internal pressure on J and COD estimation for TWC pipes. Briefly summarising the approach taken in the present work, the GE/EPRI-type J and COD estimation equations are determined based on selected non-linear, 3-D FE analyses for internal pressure. Then these results are re-formulated in the form of the enhanced reference stress approach, which is then extended to combined pressure and moment loading. Finally the proposed J and COD estimation equations are validated against further elastic–plastic 3-D FE analyses using realistic stress–strain data.
2. GE/EPRI-type J and COD estimation formulation 2.1. FE analysis Fig. 1 depicts a circumferential TWC pipe under internal pressure p, with relevant dimensions, considered in the present work. Some important dimensions for the pipe should be noted. The mean radius and the thickness of the pipe are denoted as Rm and t, respectively. The crack length is characterised by the half crack angle, h. Two different values of Rm =t were employed in the present work, Rm =t ¼ 5 and 20, and three different values of h=p were considered, h=p ¼ 0:125, 0.25 and 0.5. Table 1 summarises the cases employed. Elastic–plastic analyses of the FE model for the circumferential TWC pipes (Fig. 1) were performed using the general-purpose FE program, A B A Q U S [10]. The tensile properties for the FE analysis are assumed to follow the R–O idealisation:
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Fig. 1. Schematic illustration for TWC pipes under internal pressure p, axial tension P and global bending M.
Table 1 Cases of the present FE analyses for internal pressure using the R–O materialsa Rm =t
h=p
n
5
0.125 0.25 0.5
1, 3, 7 1, 3, 7 1, 3, 7
20
0.125 0.25 0.5
1, 3, 7 1, 3, 7 1, 3, 7
a
Loading condition: internal pressure with crack face pressure.
n e r r ¼ þa eo ry ry
ð1Þ
where eo , ry , a and n are constants, with Eeo ¼ ry where E is Young’s modulus. The deformation plasticity option with a small geometry change continuum model was invoked. The variables a and ry are fixed to a ¼ 1 and ry ¼ 165 MPa. On the other hand, three values of the strain hardening exponent n were selected, n ¼ 1, 3 and 7, and thus leading to a total of 18 calculations. Note that the case of n ¼ 1 corresponds to the elastic case. The number of elements and nodes in a typical FE mesh are 936 elements/5561 nodes. Two elements were used through the thickness, which has been shown to provide the most reliable results for the COD calculation [11,12]. Considering the symmetric condition, only one quarter of the pipe was modelled. Fig. 2 depicts the FE mesh for h=p ¼ 0:125. To avoid problems associated with incompressibility, reduced integration 20 node elements (element type C3D20R in A B A Q U S ) were used. Internal pressure is applied as a distributed load to the inner surface of the FE model, together with an axial tension equivalent to the internal pressure applied at the end of the pipe to simulate the closed end. More importantly, to consider the effect of the crack face pressure, 50% of the internal pressure was applied to the crack face in the present work. The J-integral values were extracted from the FE results using a domain integral, as a function of the applied internal pressure. The COD values, on the other hand, were determined from the FE displacement results in the mean thickness of the centre of the crack. 2.2. Results Elastic FE calculations (with n ¼ 1) gave the elastic J, Je , from which the shape factor F for the elastic stress intensity factor K was found:
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Fig. 2. A 3-D FE mesh for the circumferentially TWC pipe.
K2 1 ¼ Je ¼ E E
pRi 2t
2
paF 2
ð2Þ
where Ri denotes the inner radius of the pipe and a denotes the half crack length, a ¼ Rm h. Note that the plane stress condition was assumed to calculate the values of F. The resulting values of F are given in Table 2, and shown in Fig. 3(a) and (b). Fig. 3(a) and (b) also compares the present results with those from the FE analysis under axial tension, given in the GE/EPRI handbook [3]. Similarly the shape factor V, associated with the elastic COD, de , can be found from 4 pRi de ¼ aV ð3Þ E 2t Note again that the plane stress condition was assumed to calculate the values of V. The resulting values of V are given in Table 3, and shown in Fig. 3(c) and (d). Fig. 3(c) and (d) also compares the present results with those from the FE analysis under axial tension, given in the GE/EPRI handbook [3]. The results in Fig. 3 show that the present solutions for F and V are slightly higher than the GE/EPRI solutions, which results simply from the effect of the crack face pressure. Similarity between the present solutions and the GE/EPRI solutions, on the other hand, provides confidence of the present FE analysis.
Table 2 Values of the shape factor F for the stress intensity factor and the plastic influence h1 -functions for the plastic J-integral Rm =t
h=p
F
h1 ðn ¼ 1Þ
5
0.125 0.25 0.5
1.326 1.818 4.224
2.893 3.033 2.522
h1 ðn ¼ 3Þ 7.099 4.698 1.769
27.985 6.063 0.705
h1 ðn ¼ 7Þ
20
0.125 0.25 0.5
1.436 2.149 5.017
3.983 4.972 4.177
11.297 8.543 3.425
51.354 11.744 1.750
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Fig. 3. Variations of the shape factors, F and V, for the stress intensity factor and the elastic COD with h=p. The F-solutions for (a) Rm =t ¼ 5 and (b) Rm =t ¼ 20; the V-solutions for (c) Rm =t ¼ 5 and (d) Rm =t ¼ 20. The present solutions are compared with the GE/EPRI FE solutions for axial tension, given in Ref. [3]. Table 3 Values of the shape factor V for the elastic COD and the plastic influence h2 -functions for the plastic COD V
h2 ðn ¼ 1Þ
h2 ðn ¼ 3Þ
h2 ðn ¼ 7Þ
Rm =t
h=p
5
0.125 0.25 0.5
1.333 2.033 7.090
3.611 3.806 4.254
7.981 5.251 2.252
31.377 6.686 0.768
20
0.125 0.25 0.5
1.560 3.024 11.586
4.577 6.133 7.531
11.887 9.629 4.463
53.247 13.096 1.995
For the R–O materials, the plastic components of J and COD, Jp and dp , can be expressed as nþ1 h p Jp ¼ aro eo Rm ðp hÞh1 ðnÞ p pL n p dp ¼ aeo ah2 ðnÞ pL
ð4Þ
ð5Þ
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where ro is typically set to the yield stress of the material, ry and Eeo ¼ ry ; pL denotes the plastic limit pressure equivalent to the plastic limit axial tension [13]: 2t 1 1 sin h ð6Þ pL ¼ ry p h 2 sin pRm 2 In Eqs. (4) and (5), the plastic influence functions, h1 and h2 , are functions of Rm =t, the crack length h=p and the strain hardening exponent n. Values of h1 and h2 were calibrated from the present FE analysis as follows. Firstly, the plastic components of the FE J and d values were calculated by subtracting their elastic component from the total FE J and d values: 2 1 pRi JpFE ¼ J FE paF 2 ð7Þ E 2t FE dFE p ¼ d
4 E
pRi aV 2t
ð8Þ
Then the values of h1 and h2 were calibrated from Eqs. (4) and (5), respectively. Note that the calculated values of h1 and h2 depend on the load magnitude, as shown in Fig. 4. In the present work, the value was chosen as the (almost) asymptotic value at sufficiently large loads. Resulting values of h1 and h2 are tabulated in Tables 2 and 3, respectively. For the elastic case ðn ¼ 1Þ, on the other hand, the elastic component of J and COD, Je and de , can be rewritten as 2 h p Je ¼ aro eo Rm ðp hÞh1 ðn ¼ 1Þ ð9Þ p pL de ¼ aeo ah2 ðn ¼ 1Þ
p pL
ð10Þ
where h1 ðn ¼ 1Þ and h2 ðn ¼ 1Þ denote the values of h1 and h2 for elastic materials, respectively. Comparing Eq. (9) with Eq. (2) gives the values of h1 ðn ¼ 1Þ, which are tabulated in Table 2. Normalising Eq. (4) with respect to Eq. (9) gives n1 Jp h1 ðnÞ p ¼a ð11Þ h1 ðn ¼ 1Þ pL Je Variations of h1 ðnÞ=h1 ðn ¼ 1Þ with n are shown in Fig. 5(a) and (b) for Rm =t ¼ 5 and 20. Similarly, comparing Eq. (5) with Eq. (3) gives the values of h2 ðn ¼ 1Þ, which are tabulated in Table 3. Normalising Eq. (5) with respect to Eq. (10) gives n1 dp h2 ð nÞ p ¼a ð12Þ h2 ðn ¼ 1Þ pL de Variations of h2 ðnÞ=h2 ðn ¼ 1Þ with n are shown in Fig. 5(c) and (d) for Rm =t ¼ 5 and 20. The results in Fig. 5 shows that the values of h1 ðnÞ=h1 ðn ¼ 1Þ and h2 ðnÞ=h2 ðn ¼ 1Þ are quite sensitive to n, ranging from 0 to 10–15 for n ranging from 1 to 7. In principle, the GE/EPRI-type J estimation equations, given in this section, can be used to estimate J and COD for the TWC pipe under internal pressure, using appropriate interpolation/extrapolation. Detailed information on how to estimate J and COD, incorporating small scale yielding corrections based on the effective crack length concept, can be found in Refs. [3,4]. However, this type of approach has some inherent problems. First of all, a typical piping integrity assessment involves combined internal pressure
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Fig. 4. Variation of the FE results for h1 and h2 with the load magnitude for Rm =t ¼ 5 and h=p ¼ 0:125.
and bending moment. To provide relevant solutions for combined loadings, more extensive FE calculations have to be performed. Moreover, this formulation suffers from the well-known problem, namely, inaccuracy associated with the R–O idealisation. The R–O idealisation is known to be a poor approximation to tensile data for typical materials, which consequently can produce inaccuracy in the estimated J. The reference stress approach removes the above-mentioned problems. In this context, the GE/EPRI-type J estimation results, given in this section, are re-formulated in the form of the reference stress approach in Section 3.
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Fig. 5. Variations of h1 ðnÞ=h1 ðn ¼ 1Þ for the J-integral with n for (a) Rm =t ¼ 5 and (b) Rm =t ¼ 20; variations of h2 ðnÞ=h2 ðn ¼ 1Þ for the COD with n for (c) Rm =t ¼ 5 and (d) Rm =t ¼ 20.
3. Proposed J and COD estimations based on the reference stress approach 3.1. Reference stress formulation Introducing another normalising (reference) pressure pref , and re-phrasing Eqs. (4) and (5) gives ( n1 ) n1 Jp h1 ð nÞ pref p ¼a h1 ðn ¼ 1Þ pL pref Je ( n1 ) n1 dp h2 ð nÞ pref p ¼a h2 ðn ¼ 1Þ pL pref de
ð13Þ
ð14Þ
Noting that h1 ðnÞ=h1 ðn ¼ 1Þ, h2 ðnÞ=h2 ðn ¼ 1Þ and pref =pL are non-dimensional variables, Eqs. (13) and (14) can be written as n1 Jp p ¼ aH1 ð15Þ pref Je
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dp p ¼ aH2 pref de
n1 ð16Þ
where non-dimensional functions, H1 and H2 , presumably depend on h=p and n. An important underlying idea of the reference stress based J and COD estimation approach is that a proper definition of pref can minimise the dependence of H1 and H2 on h=p and n in Eqs. (13) and (14) [5,9]. Suppose such a load has been found, which will be termed ‘‘optimised reference pressure’’, poR , throughout this paper. There is no concrete rule to determine poR , except engineering intuition and a trial-and-error process. The following expressions are proposed for poR : poR ¼ wðhÞpL
ð17Þ
2 h h 0:75 wðhÞ ¼ 0:45 þ 1:88 p p
ð18Þ
where the expression for pL is found from Eq. (6). Introducing these expressions for pref ¼ poR into Eqs. (15) and (16) gives the values of H1 and H2 . Variations of the resulting H1 values with n are shown in Fig. 6(a) and (b) and the corresponding results for H2 are shown in Fig. 6(c) and (d). The results in Fig. 6 firstly show that sensitivity in h1 ðnÞ=h1 ðn ¼ 1Þ and h2 ðnÞ=h2 ðn ¼ 1Þ is significantly reduced in H1 and H2 . For instance, for the range of 1 6 n 6 7, the values of h1 ðnÞ=h1 ðn ¼ 1Þ ranges from 0.2 to 13, whereas those for H1 from 0.9 to 1.3. The values of h2 ðnÞ=h2 ðn ¼ 1Þ, on the other hand, ranges from 0.1 to 12, whereas those for H2 from 0.5 to 1.2. Noting that the values of both H1 and H2 are now closer to unity, Eqs. (15) and (16) can be approximated as n1 Jp p a ð19Þ poR Je n1 dp p a poR de
ð20Þ
Noting that for the R–O materials, the plastic strain is related to the stress as n1 r r ep ¼ a E ry
ð21Þ
Eqs. (19) and (20) can be written explicitly in terms of the reference stress, rref , and the reference strain, eref , as Jp Eeref ffi ; Je rref
rref ¼
p ry poR
ð22Þ
dp Eeref ffi ; de rref
rref ¼
p ry poR
ð23Þ
In Eqs. (22) and (23), ry denotes the 0.2% proof stress, and eref is the true strain at r ¼ rref , determined from the true stress–strain data. One notable point is that the use of Eqs. (22) and (23) is general for any arbitrary stress–strain relationships, and is not restricted to R–O materials. 3.2. Proposed reference stress based J and COD estimation for TWC pipes under internal pressure Eq. (22) gives the estimate of the plastic J-integral, and the total J-integral can be estimated by adding the elastic component with a plasticity correction [14]:
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Fig. 6. Variations of H1 for the J-integral with n for (a) Rm =t ¼ 5 and (b) Rm =t ¼ 20; variations of H2 for the COD with n for (c) Rm =t ¼ 5 and (d) Rm =t ¼ 20.
J Eeref 1 ¼ þ Je 2 rref
rref ry
2
rref Eeref
ð24Þ
where rref ¼
p ry poR
ð25Þ
where poR is given in Eq. (17). Authors have recently proposed and validated the reference stress COD estimation equation [5], which is given by 8 > Eeref 1 rref 2 rref for 0 6 r < r ref y d < rref þ 2 ry Ee ð26Þ ¼ n1 1 ref > de : d rref for r 6 r y ref de ry 1
In Eq. (26), ðd=de Þ1 denotes the value of ðd=de Þ at rref =ry ¼ 1, calculated from the first equation in Eq. (26), so that Eq. (26) is continuous at rref ¼ ry . The strain hardening index n1 in Eq. (26) should be estimated from
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n1 ¼
ln½ðeu;t ru;t =EÞ=0:002 ln½ru;t =ry
ð27Þ
where ru;t and eu;t denote the true ultimate tensile stress and percentage uniform elongation at r ¼ ru , respectively. These are obtained from the corresponding engineering values using ru;t ¼ ð1 þ eu Þru eu;t ¼ ln ð1 þ eu Þ
ð28Þ
3.3. Proposed reference stress based J and COD estimation for TWC pipes under combined internal pressure and bending In the previous sub-section, reference stress based J and COD estimation equations are given for TWC pipes under internal pressure. However, a typical loading in pressurised piping is combined internal pressure and bending. The extension of the reference stress approach to combined loadings simply requires resolving two issues. One is to find the relevant elastic solutions, and the other is to determine the relevant reference stress (and thus normalising load). The elastic solutions can be easily found by the superposition principle, namely, simply adding the contributions due to internal pressure and due to bending. Furthermore, noting that the elastic solutions (such as the stress intensity factor and the elastic COD) for internal pressure are very close to those for axial tension, even existing solutions can be used (see e.g. Refs. [3,4]). Determination of the relevant reference stress or the normalising load, on the other hand, is more complicated. For TWC pipes under combined axial tension and bending, this has been done in Ref. [7], where the locus for the normalising load was given by
P PoR
2 þ
M ¼1 MoR
ð29Þ
where P and M denote the axial tension and moment, respectively. The optimised reference loads for axial tension, PoR , and moment, MoR , are given by PoR ¼ cPL ;
MoR ¼ cML 2 h h cðh=pÞ ¼ 0:82 þ 0:75 þ 0:42 p p
for h=p 6 0:5
where PL and ML are plastic limit load and moment for the TWC pipe [13]
PL ¼ 2Rm try p h 2 sin1 12 sin h ML ¼ 4R2m try cos h2 sin2 h
ð30Þ
ð31Þ
Noting the similarity between axial tension and internal pressure, the locus for the normalising load in the case of combined internal pressure and bending moment can be postulated as 2 p M þ ¼1 ð32Þ poR MoR where poR is given by Eq. (17), and MoR by Eq. (30). Determination of the relevant optimised reference load for combined internal pressure and bending is briefly described below. For this case, the load proportionality factor k can be defined as k¼
M pR2i p Rm
ð33Þ
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c Inserting Eq. (33) into Eq. (32) gives the optimised reference moment for combined loading, MoR , as 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
2 u u kpR2 Rm poR 2 2
2 7 kpR2i Rm poR 16 t i c ð34Þ þ þ 4 kpR2i Rm poR 5 MoR ¼ 4 2 MoR MoR
c or from Eq. (33), the optimised reference load for combined loading, poR , as c ¼ poR
c MoR kpR2i Rm
ð35Þ
4. Elastic–plastic FE validation 4.1. FE analysis To validate the proposed J and COD estimation equations for TWC pipes under internal pressure and under combined internal pressure and moment, additional elastic–plastic 3-D FE analyses were performed. The outer radius and the thickness of the pipe were selected as Ro ¼ 177:8 mm and t ¼ 32:3 mm, respectively, giving the value of the mean radius-to-thickness ratio, Rm =t ¼ 5. Two different crack lengths, h=p ¼ 0:125 and 0.25, were considered, which are regarded as the important range of h=p for typical LBB analyses. Regarding the material properties for the FE analysis, actual experimental uni-axial stress–strain data of Type 316 stainless steel at the temperature, T ¼ 288 °C, were taken, for which stress–strain curves are shown in Fig. 7, and the relevant data are summarised in Table 4. Elastic–plastic analyses of this FE model were performed using the general-purpose FE program, A B A Q U S [10]. The experimental true stress–plastic strain data were directly given in the FE analysis. Materials were modelled as isotropic elastic–plastic materials that obey the incremental plasticity theory,
Fig. 7. Stress–strain curve for SA312 Type 316 (288 °C) and the resulting R–O fit.
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Table 4 Summary of tensile properties for SA312 Type 316 stainless steel at 288 °C, used in the present FE analysis Material
E (GPa)
ry (MPa)
ru (MPa)
ERS parameters eu
n1
SA312 Type 316 (288 °C)
190
165
455
0.3
3.82
and a small geometry change continuum FE model was employed. Detailed information on the FE model can be referred to Section 2.1. For the case of internal pressure only, the magnitude of the internal pressure was increased up to a sufficiently high pressure, and the resulting J and COD values were extracted as a function of internal pressure. Note that the effect of the internal pressure on the crack face was considered by applying 50% of the internal pressure to the crack face. For the case of combined internal pressure and bending moment, it was noted that combined loading could be applied in two different ways: proportional loading and nonproportional loading. The present FE calculations employed only the proportional loading, namely, internal pressure and bending moment are increased in a proportional manner, with three different values of the load proportionality factor k (see Eq. (33) for the definition) to k ¼ 0:5, 1.0 and 2.0. 4.2. Results Fig. 8 compares the FE J and COD results for the TWC pipe under internal pressure, with the proposed J and COD estimates, Eqs. (24) and (26), for two values of h=p ¼ 0:125 and 0.25. Note that the proposed method is denoted as the ERSM in the subsequent figures. In Fig. 8, the J values are normalised with respect to ry and Rm h ð¼ aÞ, while the COD (d) values with respect to Rm h ð¼ aÞ. The load, internal pressure, is normalised with respect to the optimum reference pressure, poR (see Eq. (17)). Noting that the GE/EPRI J and COD estimations are developed in the present work (see Section 2), the resulting J and COD are also compared with the FE results. Application of the GE/EPRI method firstly requires that the material’s tensile data should be fitted using the R–O relation, see Eq. (1). In the present work, the entire true stress–strain data up to the ultimate tensile strength were fitted 1 using the R O F I T program [15], developed by Battelle. The resulting R–O parameters, a and n, are listed in Table 1, and the resulting R–O fits are compared with experimental tensile data in Fig. 7. Once the R–O parameters, a and n, are determined, then J and COD can be estimated using the effective crack length (ae ): nþ1 r2y ½ K ðae Þ 2 a p þ a ðt aÞ h1 ðnÞ ð36Þ J¼ t pL E E d¼
4 E
n pRi p ae V ðae Þ þ aeo ah2 ðnÞ pL 2t
ð37Þ
where ae was estimated from ae ¼ a þ ury ;
u¼
1 1 þ ð p=pL Þ
2
;
1 ry ¼ 2p
n1 nþ1
K ry
2 ð38Þ
1 There are other ways to fit the tensile data using the R–O relation. Typical ways include to fit the data only up 5% strain and to fit the data from 0.1% strain to 0.8eu;t , where eu;t denotes the true ultimate strain.
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Fig. 8. (a)–(d) Comparison of FE J and COD results for TWC pipes under internal pressure with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM), (ii) the GE/EPRI solutions for internal pressure, developed in the present work (present GE/EPRI), and (iii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
and the values of h1 ðnÞ and h2 ðnÞ were obtained by interpolating the present FE results (tabulated in Tables 2 and 3). The resulting values of J and COD are shown in Fig. 8, denoted as ‘‘present GE/EPRI’’. Finally it should noted that a typical practice to analyse the cracked pipe under internal pressure is to convert the internal pressure to the equivalent axial tension. In this context, the estimation formulae for the TWC pipes under axial tension, given in [3], were also used to estimate J and COD. The resulting values of J and COD are also compared in Fig. 8, denoted as ‘‘GE/EPRI-tension’’. The comparisons in Fig. 8 shows that the proposed ERSM-based J estimates are in overall good agreements with the FE results, and are on the conservative side. Note that for conservatism, the J value should be overestimated, that is, the estimated J values should be higher than the FE values. The ERSMbased COD estimations are not so accurate, compared to the FE COD results, but are still on the conservative side. Note that for conservatism, the detectable leakage crack length should be overestimated and thus the COD value should be underestimated, that is, the estimated COD values should be lower than the FE values. On the other hand, the GE/EPRI J and COD estimates are not so accurate, compared to the FE results. Such results are consistent with our earlier finding [6] and such inaccuracy is associated with the R– O fit. In fact, if different ways of fitting the R–O equation are performed, accuracy can be improved, but no guidance on the best R–O fit can be given since it depends on the material [6]. Finally the GE/EPRI J estimations based on axial tension show significant deviations from the FE J and COD results. Such deviation results from the fact that the axial tension solution neglects the hoop stress effect due to internal
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Fig. 9. (a)–(d) Comparison of FE J and COD results for TWC pipes under combined pressure and bending (k ¼ 0:5) with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM) and (ii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
pressure. The results in Fig. 8 support the well-known fact that the use of axial tension solutions for cracked pipes under internal pressure can lead to non-conservative estimates of the J-integral. The results for combined internal pressure and bending (under proportional loading conditions) are shown in Fig. 9 for the proportionality factor (see Eq. (33)), k ¼ 0:5, in Fig. 10 for k ¼ 1:0, and in Fig. 11 for k ¼ 2:0. In Figs. 9–11, the FE results for J and COD are compared with the proposed J and COD estimations. The results show overall excellent agreement, maintaining slight conservatism in both J and COD. The GE/EPRI J and COD estimates are also compared with the FE results. These estimates were based on the combined axial tension and bending solutions, given in Ref. [3]. As for the internal pressure case, deviations from the FE results are significant, resulting from neglecting the hoop stress effect due to internal pressure.
5. Concluding remarks This paper provides engineering J and COD estimation equations for TWC pipes under internal pressure and under combined internal pressure and bending. Based on selected 3-D FE calculations for the TWC
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Fig. 10. (a)–(d) Comparison of FE J and COD results for TWC pipes under combined pressure and bending (k ¼ 1:0) with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM) and (ii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
pipe under internal pressure using R–O materials, the elastic and plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the normalised crack length, b=p, and the mean radiusto-thickness ratio, Rm =t. Then these developed GE/EPRI-type solutions are re-formulated based on the reference stress concept. Such a re-formulation provides simpler equations for J and COD, which are then further extended to combined internal pressure and bending. The proposed reference stress based J and COD estimation equations are compared with elastic–plastic 3-D FE results using actual stress–strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimations. Another notable point is that caution should be exercised if a defect assessment of cracked, pressurised piping is made by considering the pipe with an axial tension equivalent to the internal pressure. Note that the majority of existing fracture mechanics analyses concentrate on finding solutions for axial tension, not for internal pressure. Such practice would be acceptable when plasticity due to internal pressure is small. On the other hand, when the plasticity due to internal pressure becomes more significant, it is well known and also is confirmed in this paper that such practice can give non-conservative estimates for crack driving forces. Another potential area for fracture mechanics analysis specific to internal pressure is the analysis of pressurised pipes operating at elevated (creep) temperatures. In such cases, time dependent fracture
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Fig. 11. (a)–(d) Comparison of FE J and COD results for TWC pipes under combined pressure and bending (k ¼ 2:0) with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM) and (ii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
mechanics parameters, for instance, the C -integral [16], should be estimated. Again the use of the axial tension solutions could provide non-conservative estimates for C . In this context, the present J and COD estimation scheme for elastic–plastic fracture can be used to estimate the C -integral and the COD rate, d_c , due to creep, using the analogy between plasticity and creep as follows [17]: 2 E K e_c
ð39Þ C ¼ E0 rref d_c e_c ¼ de ðrref =EÞ
with d_c ðt ¼ 0Þ ¼ 0
ð40Þ
where e_c is the creep strain rate at the reference stress r ¼ rref , determined from the actual creep-deformation data. The reference stress for internal pressure and for combined internal pressure and bending is defined in Eqs. (17) and (32), respectively. Validation of these estimation equations will be given in a separate paper.
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Acknowledgements The authors are grateful for the support provided by a grant from Safety and Structural Integrity Research Centre at Sungkyunkwan University.
References [1] Wilkowski G, Schmidt R, Scott P, Olson R, Marschall C, Kramer G, Paul D. International Piping Integrity Research Group (IPIRG) Program—Program Final Report. NUREG/CR-6233, USNRC, 1997. [2] Hopper A, Wilkowski G, Scott P, Olson R, Rudland D, Kilinski T, Mohan R, Ghadiali N, Paul D. The Second International Piping Integrity Research Group (IPIRG-2) Program—Final Report. NUREG/CR-6452, USNRC, 1997. [3] Kumar V, German MD. Elastic–plastic fracture analysis of through-wall and surface flaws in cylinders. EPRI Report, NP-5596, 1988. [4] Zahoor A. Ductile fracture handbook. Novetech Corp; 1991. [5] Kim YJ, Budden PJ. Reference stress approximations for J and COD of circumferential through-wall cracked pipes. Int J Fract, submitted for publication. [6] Kim YJ, Huh NS, Kim YJ. Enhanced reference stress based J and COD estimation method for LBB analysis and comparison with GE/EPRI method. Fatigue Fract Engng Mater Struct 2001;24:243–54. [7] Kim YJ, Huh NS, Kim YJ. Reference stress based elastic–plastic fracture analysis for circumferential through-wall cracked pipes under combined tension and bending. Engng Fract Mech, in press. [8] Kim YJ, Huh NS, Kim YJ. Crack opening analysis of complex cracked pipes. Int J Fract 2001;111:71–86. [9] Ainsworth RA. The assessment of defects in structures of strain hardening materials. Engng Fract Mech 1984;19:633–42. [10] A B A Q U S Version 58 User’s manual. Hibbitt, RI: Karlson Sorensen; 1999. [11] Yang JS, Kim BN, Park CY, Park YS, Yoon KS. A simple method for estimating effective J-integral in LBB application to nuclear power plant piping system. In: Transactions of the 15th International Conference on Structural Mechanics on Reactor Technology, vol. 5. 1999. p. 327–34. [12] Huh NS, Kim YJ, Pyo CR, Yu YJ. Effect of finite element model on the integrity evaluation of nuclear piping. J Korea Inst Indust Safety 2000;152:51–8. [13] Miller A. Review of limit loads of structures containing defects. Int J Pressure Vessels Piping 1988;32:191–327. [14] R6: Assessment of the integrity of structures containing defects. Revision 3. British Energy Generation Ltd., 1999. [15] Computer program to calculate Ramberg–Osgood parameters for a stress–strain curve. In:Pipe fracture encyclopedia, vol. 3. Battelle; 1997. [16] Riedel H. Fracture at high temperature. New York: Springer; 1987. [17] R5: Assessment procedure for the high temperature response of structures, Issue 2, British Energy Generation Ltd., 1998.
Quantification of pressure-induced hoop stress effect on fracture analysis of circumferential through-wall cracked pipes Yun-Jae Kim, Nam-Su Huh, Young-Jin Kim
*
SAFE Research Centre, School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu, Suwon, Kyonggi-do 440-746, Republic of Korea Received 23 April 2001; received in revised form 24 September 2001; accepted 2 October 2001
Abstract This paper provides engineering J and crack opening displacement (COD) estimation equations for through-wall cracked (TWC) pipes under internal pressure and under combined internal pressure and bending. Based on selected 3-D FE calculations for the TWC pipe under internal pressure using power law materials, elastic and plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the normalised crack length and the mean radius-to-thickness ratio. These developed GE/EPRI-type solutions are then re-formulated based on the reference stress concept. Such re-formulation not only provides simpler equations for J and COD estimation, but also can be easily extended to combined internal pressure and bending. The proposed reference stress based J and COD estimation equations are compared with elastic–plastic 3-D FE results using actual stress–strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimates. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Through-wall cracked pipe; Crack opening displacement; J -integral; Reference stress; Finite element; Internal pressure
1. Introduction Pressurised piping is an important element in power plants, and thus application of fracture mechanics analysis to such pressurised piping is important in structural integrity assessment of plant components. One example of such application is the LBB analysis of piping [1,2]. In general, application of the LBB procedure requires two steps. Firstly, the crack length corresponding to the (assumed) detectable leakage rate should be calculated for a TWC pipe. For this step, engineering methods to estimate the COD and the leak rate are needed. The second step is to perform the pipe fracture stability analysis. For this second step, an engineering method to estimate the J-integral is needed.
*
Corresponding author. Tel.: +82-31-290-5274; fax: +82-31-290-5276. E-mail address: [email protected] (Y.-J. Kim).
0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 1 3 2 - 1
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Nomenclature a half crack length, a ¼ Rm h ae effective crack length, see Eq. (38) E Young’s modulus E0 ¼ E=ð1 m2 Þ for plane strain; ¼ E for plane stress F shape factor for the elastic stress intensity factor h1 (n) plastic calibration function for Jp in the GE/EPRI approach h1 (n ¼ 1) value of h1 for elastic case (n ¼ 1) h2 (n) plastic calibration function for dp in the GE/EPRI approach h2 (n ¼ 1) value of h2 for elastic case (n ¼ 1) J J-integral Je elastically calculated value of J; Je ¼ K 2 =E0 Jp plastic component of J K linear elastic stress intensity factor n strain hardening index ð1 6 n < 1Þ for Ramberg–Osgood model, Eq. (1) n1 strain hardening index for the ERS-based COD estimation equation, Eq. (27) p, M, P internal pressure, bending moment and axial tension on a pipe pL , ML , PL limit internal pressure, moment and axial tension assuming the limiting stress of ry poR , MoR , PoR optimised reference pressure, moment and axial tension load of cracked pipe under a single loading c c poR , MoR optimised reference pressure and moment of cracked pipe under combined tension and bending Ri inner radius of pipe Rm mean radius of pipe Ro outside radius of pipe V shape factor for the elastic crack opening displacement a coefficient of Ramberg–Osgood model, see Eq. (1) d crack opening displacement at centre of crack de elastic component of crack opening displacement dp plastic component of crack opening displacement e strain, general eref reference strain at the reference stress rref eo normalising strain in the Ramberg–Osgood model, Eq. (1) eu uniform elongation k Load proportionality factor, see Eq. (33) m Poisson’s ratio h half of total crack angle r stress, general rref reference stress, see Eq. (25) for definition ry yield strength ru ultimate tensile strength Abbreviations COD crack opening displacement ERS enhanced reference stress R–O Ramberg–Osgood
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FE finite element LBB leak-before-break GE/EPRI general electric/electric power research institute TWC through-wall cracked
When fracture mechanics analysis is performed for TWC pipes subject to internal pressure, it is typical practice to transform the internal pressure into the equivalent axial load, and thus to analyse the cracked pipe subject to the axial load. In this context, existing solutions for elastic–plastic fracture analysis for cracked pipes include solutions for axial tension, but not for pressure (see for instance the GE/EPRI handbooks [3,4]). One problem associated with such an approach is that the hoop stress effect due to the pressure is neglected. The resulting analysis can be erroneous, particularly when the applied pressure is high or when the pressurised pipe is subject to elevated (creep) temperatures. More importantly, the results can be non-conservative in terms of the crack driving force. Thus, for fracture mechanics analyses of cracked pipes under internal pressure, the hoop stress effect due to internal pressure should be explicitly considered. Consideration of the hoop stress effect into the GE/EPRI-type J and COD estimation equations requires detailed 3-D FE analyses. Although the use of 3-D FE analyses is increasingly popular, performing a large number of 3-D FE analyses is currently impractical, noting that pressurised piping is typically subject to combined internal pressure and bending moment in practical situations. In this context, the enhanced reference stress method (ERSM), proposed recently by the authors [5–8], is an attractive alternative. This method is based on the reference stress approach [9], but enhancement is made in the definition of the reference stress to improve its accuracy. Reliability of this method has been checked by comparing J and COD results with published pipe test data and with extensive FE results [5,6]. One important advantage of this method is that it can be easily extended to more complex geometries and loadings. For instance, the proposed ERSM has been extended to J and COD estimations for TWC pipes under combined bending and tension [7] and to those for complex cracked pipes [8]. This paper quantifies the hoop stress effect induced by internal pressure on J and COD estimation for TWC pipes. Briefly summarising the approach taken in the present work, the GE/EPRI-type J and COD estimation equations are determined based on selected non-linear, 3-D FE analyses for internal pressure. Then these results are re-formulated in the form of the enhanced reference stress approach, which is then extended to combined pressure and moment loading. Finally the proposed J and COD estimation equations are validated against further elastic–plastic 3-D FE analyses using realistic stress–strain data.
2. GE/EPRI-type J and COD estimation formulation 2.1. FE analysis Fig. 1 depicts a circumferential TWC pipe under internal pressure p, with relevant dimensions, considered in the present work. Some important dimensions for the pipe should be noted. The mean radius and the thickness of the pipe are denoted as Rm and t, respectively. The crack length is characterised by the half crack angle, h. Two different values of Rm =t were employed in the present work, Rm =t ¼ 5 and 20, and three different values of h=p were considered, h=p ¼ 0:125, 0.25 and 0.5. Table 1 summarises the cases employed. Elastic–plastic analyses of the FE model for the circumferential TWC pipes (Fig. 1) were performed using the general-purpose FE program, A B A Q U S [10]. The tensile properties for the FE analysis are assumed to follow the R–O idealisation:
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Fig. 1. Schematic illustration for TWC pipes under internal pressure p, axial tension P and global bending M.
Table 1 Cases of the present FE analyses for internal pressure using the R–O materialsa Rm =t
h=p
n
5
0.125 0.25 0.5
1, 3, 7 1, 3, 7 1, 3, 7
20
0.125 0.25 0.5
1, 3, 7 1, 3, 7 1, 3, 7
a
Loading condition: internal pressure with crack face pressure.
n e r r ¼ þa eo ry ry
ð1Þ
where eo , ry , a and n are constants, with Eeo ¼ ry where E is Young’s modulus. The deformation plasticity option with a small geometry change continuum model was invoked. The variables a and ry are fixed to a ¼ 1 and ry ¼ 165 MPa. On the other hand, three values of the strain hardening exponent n were selected, n ¼ 1, 3 and 7, and thus leading to a total of 18 calculations. Note that the case of n ¼ 1 corresponds to the elastic case. The number of elements and nodes in a typical FE mesh are 936 elements/5561 nodes. Two elements were used through the thickness, which has been shown to provide the most reliable results for the COD calculation [11,12]. Considering the symmetric condition, only one quarter of the pipe was modelled. Fig. 2 depicts the FE mesh for h=p ¼ 0:125. To avoid problems associated with incompressibility, reduced integration 20 node elements (element type C3D20R in A B A Q U S ) were used. Internal pressure is applied as a distributed load to the inner surface of the FE model, together with an axial tension equivalent to the internal pressure applied at the end of the pipe to simulate the closed end. More importantly, to consider the effect of the crack face pressure, 50% of the internal pressure was applied to the crack face in the present work. The J-integral values were extracted from the FE results using a domain integral, as a function of the applied internal pressure. The COD values, on the other hand, were determined from the FE displacement results in the mean thickness of the centre of the crack. 2.2. Results Elastic FE calculations (with n ¼ 1) gave the elastic J, Je , from which the shape factor F for the elastic stress intensity factor K was found:
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Fig. 2. A 3-D FE mesh for the circumferentially TWC pipe.
K2 1 ¼ Je ¼ E E
pRi 2t
2
paF 2
ð2Þ
where Ri denotes the inner radius of the pipe and a denotes the half crack length, a ¼ Rm h. Note that the plane stress condition was assumed to calculate the values of F. The resulting values of F are given in Table 2, and shown in Fig. 3(a) and (b). Fig. 3(a) and (b) also compares the present results with those from the FE analysis under axial tension, given in the GE/EPRI handbook [3]. Similarly the shape factor V, associated with the elastic COD, de , can be found from 4 pRi de ¼ aV ð3Þ E 2t Note again that the plane stress condition was assumed to calculate the values of V. The resulting values of V are given in Table 3, and shown in Fig. 3(c) and (d). Fig. 3(c) and (d) also compares the present results with those from the FE analysis under axial tension, given in the GE/EPRI handbook [3]. The results in Fig. 3 show that the present solutions for F and V are slightly higher than the GE/EPRI solutions, which results simply from the effect of the crack face pressure. Similarity between the present solutions and the GE/EPRI solutions, on the other hand, provides confidence of the present FE analysis.
Table 2 Values of the shape factor F for the stress intensity factor and the plastic influence h1 -functions for the plastic J-integral Rm =t
h=p
F
h1 ðn ¼ 1Þ
5
0.125 0.25 0.5
1.326 1.818 4.224
2.893 3.033 2.522
h1 ðn ¼ 3Þ 7.099 4.698 1.769
27.985 6.063 0.705
h1 ðn ¼ 7Þ
20
0.125 0.25 0.5
1.436 2.149 5.017
3.983 4.972 4.177
11.297 8.543 3.425
51.354 11.744 1.750
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Fig. 3. Variations of the shape factors, F and V, for the stress intensity factor and the elastic COD with h=p. The F-solutions for (a) Rm =t ¼ 5 and (b) Rm =t ¼ 20; the V-solutions for (c) Rm =t ¼ 5 and (d) Rm =t ¼ 20. The present solutions are compared with the GE/EPRI FE solutions for axial tension, given in Ref. [3]. Table 3 Values of the shape factor V for the elastic COD and the plastic influence h2 -functions for the plastic COD V
h2 ðn ¼ 1Þ
h2 ðn ¼ 3Þ
h2 ðn ¼ 7Þ
Rm =t
h=p
5
0.125 0.25 0.5
1.333 2.033 7.090
3.611 3.806 4.254
7.981 5.251 2.252
31.377 6.686 0.768
20
0.125 0.25 0.5
1.560 3.024 11.586
4.577 6.133 7.531
11.887 9.629 4.463
53.247 13.096 1.995
For the R–O materials, the plastic components of J and COD, Jp and dp , can be expressed as nþ1 h p Jp ¼ aro eo Rm ðp hÞh1 ðnÞ p pL n p dp ¼ aeo ah2 ðnÞ pL
ð4Þ
ð5Þ
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where ro is typically set to the yield stress of the material, ry and Eeo ¼ ry ; pL denotes the plastic limit pressure equivalent to the plastic limit axial tension [13]: 2t 1 1 sin h ð6Þ pL ¼ ry p h 2 sin pRm 2 In Eqs. (4) and (5), the plastic influence functions, h1 and h2 , are functions of Rm =t, the crack length h=p and the strain hardening exponent n. Values of h1 and h2 were calibrated from the present FE analysis as follows. Firstly, the plastic components of the FE J and d values were calculated by subtracting their elastic component from the total FE J and d values: 2 1 pRi JpFE ¼ J FE paF 2 ð7Þ E 2t FE dFE p ¼ d
4 E
pRi aV 2t
ð8Þ
Then the values of h1 and h2 were calibrated from Eqs. (4) and (5), respectively. Note that the calculated values of h1 and h2 depend on the load magnitude, as shown in Fig. 4. In the present work, the value was chosen as the (almost) asymptotic value at sufficiently large loads. Resulting values of h1 and h2 are tabulated in Tables 2 and 3, respectively. For the elastic case ðn ¼ 1Þ, on the other hand, the elastic component of J and COD, Je and de , can be rewritten as 2 h p Je ¼ aro eo Rm ðp hÞh1 ðn ¼ 1Þ ð9Þ p pL de ¼ aeo ah2 ðn ¼ 1Þ
p pL
ð10Þ
where h1 ðn ¼ 1Þ and h2 ðn ¼ 1Þ denote the values of h1 and h2 for elastic materials, respectively. Comparing Eq. (9) with Eq. (2) gives the values of h1 ðn ¼ 1Þ, which are tabulated in Table 2. Normalising Eq. (4) with respect to Eq. (9) gives n1 Jp h1 ðnÞ p ¼a ð11Þ h1 ðn ¼ 1Þ pL Je Variations of h1 ðnÞ=h1 ðn ¼ 1Þ with n are shown in Fig. 5(a) and (b) for Rm =t ¼ 5 and 20. Similarly, comparing Eq. (5) with Eq. (3) gives the values of h2 ðn ¼ 1Þ, which are tabulated in Table 3. Normalising Eq. (5) with respect to Eq. (10) gives n1 dp h2 ð nÞ p ¼a ð12Þ h2 ðn ¼ 1Þ pL de Variations of h2 ðnÞ=h2 ðn ¼ 1Þ with n are shown in Fig. 5(c) and (d) for Rm =t ¼ 5 and 20. The results in Fig. 5 shows that the values of h1 ðnÞ=h1 ðn ¼ 1Þ and h2 ðnÞ=h2 ðn ¼ 1Þ are quite sensitive to n, ranging from 0 to 10–15 for n ranging from 1 to 7. In principle, the GE/EPRI-type J estimation equations, given in this section, can be used to estimate J and COD for the TWC pipe under internal pressure, using appropriate interpolation/extrapolation. Detailed information on how to estimate J and COD, incorporating small scale yielding corrections based on the effective crack length concept, can be found in Refs. [3,4]. However, this type of approach has some inherent problems. First of all, a typical piping integrity assessment involves combined internal pressure
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Fig. 4. Variation of the FE results for h1 and h2 with the load magnitude for Rm =t ¼ 5 and h=p ¼ 0:125.
and bending moment. To provide relevant solutions for combined loadings, more extensive FE calculations have to be performed. Moreover, this formulation suffers from the well-known problem, namely, inaccuracy associated with the R–O idealisation. The R–O idealisation is known to be a poor approximation to tensile data for typical materials, which consequently can produce inaccuracy in the estimated J. The reference stress approach removes the above-mentioned problems. In this context, the GE/EPRI-type J estimation results, given in this section, are re-formulated in the form of the reference stress approach in Section 3.
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Fig. 5. Variations of h1 ðnÞ=h1 ðn ¼ 1Þ for the J-integral with n for (a) Rm =t ¼ 5 and (b) Rm =t ¼ 20; variations of h2 ðnÞ=h2 ðn ¼ 1Þ for the COD with n for (c) Rm =t ¼ 5 and (d) Rm =t ¼ 20.
3. Proposed J and COD estimations based on the reference stress approach 3.1. Reference stress formulation Introducing another normalising (reference) pressure pref , and re-phrasing Eqs. (4) and (5) gives ( n1 ) n1 Jp h1 ð nÞ pref p ¼a h1 ðn ¼ 1Þ pL pref Je ( n1 ) n1 dp h2 ð nÞ pref p ¼a h2 ðn ¼ 1Þ pL pref de
ð13Þ
ð14Þ
Noting that h1 ðnÞ=h1 ðn ¼ 1Þ, h2 ðnÞ=h2 ðn ¼ 1Þ and pref =pL are non-dimensional variables, Eqs. (13) and (14) can be written as n1 Jp p ¼ aH1 ð15Þ pref Je
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dp p ¼ aH2 pref de
n1 ð16Þ
where non-dimensional functions, H1 and H2 , presumably depend on h=p and n. An important underlying idea of the reference stress based J and COD estimation approach is that a proper definition of pref can minimise the dependence of H1 and H2 on h=p and n in Eqs. (13) and (14) [5,9]. Suppose such a load has been found, which will be termed ‘‘optimised reference pressure’’, poR , throughout this paper. There is no concrete rule to determine poR , except engineering intuition and a trial-and-error process. The following expressions are proposed for poR : poR ¼ wðhÞpL
ð17Þ
2 h h 0:75 wðhÞ ¼ 0:45 þ 1:88 p p
ð18Þ
where the expression for pL is found from Eq. (6). Introducing these expressions for pref ¼ poR into Eqs. (15) and (16) gives the values of H1 and H2 . Variations of the resulting H1 values with n are shown in Fig. 6(a) and (b) and the corresponding results for H2 are shown in Fig. 6(c) and (d). The results in Fig. 6 firstly show that sensitivity in h1 ðnÞ=h1 ðn ¼ 1Þ and h2 ðnÞ=h2 ðn ¼ 1Þ is significantly reduced in H1 and H2 . For instance, for the range of 1 6 n 6 7, the values of h1 ðnÞ=h1 ðn ¼ 1Þ ranges from 0.2 to 13, whereas those for H1 from 0.9 to 1.3. The values of h2 ðnÞ=h2 ðn ¼ 1Þ, on the other hand, ranges from 0.1 to 12, whereas those for H2 from 0.5 to 1.2. Noting that the values of both H1 and H2 are now closer to unity, Eqs. (15) and (16) can be approximated as n1 Jp p a ð19Þ poR Je n1 dp p a poR de
ð20Þ
Noting that for the R–O materials, the plastic strain is related to the stress as n1 r r ep ¼ a E ry
ð21Þ
Eqs. (19) and (20) can be written explicitly in terms of the reference stress, rref , and the reference strain, eref , as Jp Eeref ffi ; Je rref
rref ¼
p ry poR
ð22Þ
dp Eeref ffi ; de rref
rref ¼
p ry poR
ð23Þ
In Eqs. (22) and (23), ry denotes the 0.2% proof stress, and eref is the true strain at r ¼ rref , determined from the true stress–strain data. One notable point is that the use of Eqs. (22) and (23) is general for any arbitrary stress–strain relationships, and is not restricted to R–O materials. 3.2. Proposed reference stress based J and COD estimation for TWC pipes under internal pressure Eq. (22) gives the estimate of the plastic J-integral, and the total J-integral can be estimated by adding the elastic component with a plasticity correction [14]:
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Fig. 6. Variations of H1 for the J-integral with n for (a) Rm =t ¼ 5 and (b) Rm =t ¼ 20; variations of H2 for the COD with n for (c) Rm =t ¼ 5 and (d) Rm =t ¼ 20.
J Eeref 1 ¼ þ Je 2 rref
rref ry
2
rref Eeref
ð24Þ
where rref ¼
p ry poR
ð25Þ
where poR is given in Eq. (17). Authors have recently proposed and validated the reference stress COD estimation equation [5], which is given by 8 > Eeref 1 rref 2 rref for 0 6 r < r ref y d < rref þ 2 ry Ee ð26Þ ¼ n1 1 ref > de : d rref for r 6 r y ref de ry 1
In Eq. (26), ðd=de Þ1 denotes the value of ðd=de Þ at rref =ry ¼ 1, calculated from the first equation in Eq. (26), so that Eq. (26) is continuous at rref ¼ ry . The strain hardening index n1 in Eq. (26) should be estimated from
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n1 ¼
ln½ðeu;t ru;t =EÞ=0:002 ln½ru;t =ry
ð27Þ
where ru;t and eu;t denote the true ultimate tensile stress and percentage uniform elongation at r ¼ ru , respectively. These are obtained from the corresponding engineering values using ru;t ¼ ð1 þ eu Þru eu;t ¼ ln ð1 þ eu Þ
ð28Þ
3.3. Proposed reference stress based J and COD estimation for TWC pipes under combined internal pressure and bending In the previous sub-section, reference stress based J and COD estimation equations are given for TWC pipes under internal pressure. However, a typical loading in pressurised piping is combined internal pressure and bending. The extension of the reference stress approach to combined loadings simply requires resolving two issues. One is to find the relevant elastic solutions, and the other is to determine the relevant reference stress (and thus normalising load). The elastic solutions can be easily found by the superposition principle, namely, simply adding the contributions due to internal pressure and due to bending. Furthermore, noting that the elastic solutions (such as the stress intensity factor and the elastic COD) for internal pressure are very close to those for axial tension, even existing solutions can be used (see e.g. Refs. [3,4]). Determination of the relevant reference stress or the normalising load, on the other hand, is more complicated. For TWC pipes under combined axial tension and bending, this has been done in Ref. [7], where the locus for the normalising load was given by
P PoR
2 þ
M ¼1 MoR
ð29Þ
where P and M denote the axial tension and moment, respectively. The optimised reference loads for axial tension, PoR , and moment, MoR , are given by PoR ¼ cPL ;
MoR ¼ cML 2 h h cðh=pÞ ¼ 0:82 þ 0:75 þ 0:42 p p
for h=p 6 0:5
where PL and ML are plastic limit load and moment for the TWC pipe [13]
PL ¼ 2Rm try p h 2 sin1 12 sin h ML ¼ 4R2m try cos h2 sin2 h
ð30Þ
ð31Þ
Noting the similarity between axial tension and internal pressure, the locus for the normalising load in the case of combined internal pressure and bending moment can be postulated as 2 p M þ ¼1 ð32Þ poR MoR where poR is given by Eq. (17), and MoR by Eq. (30). Determination of the relevant optimised reference load for combined internal pressure and bending is briefly described below. For this case, the load proportionality factor k can be defined as k¼
M pR2i p Rm
ð33Þ
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c Inserting Eq. (33) into Eq. (32) gives the optimised reference moment for combined loading, MoR , as 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
2 u u kpR2 Rm poR 2 2
2 7 kpR2i Rm poR 16 t i c ð34Þ þ þ 4 kpR2i Rm poR 5 MoR ¼ 4 2 MoR MoR
c or from Eq. (33), the optimised reference load for combined loading, poR , as c ¼ poR
c MoR kpR2i Rm
ð35Þ
4. Elastic–plastic FE validation 4.1. FE analysis To validate the proposed J and COD estimation equations for TWC pipes under internal pressure and under combined internal pressure and moment, additional elastic–plastic 3-D FE analyses were performed. The outer radius and the thickness of the pipe were selected as Ro ¼ 177:8 mm and t ¼ 32:3 mm, respectively, giving the value of the mean radius-to-thickness ratio, Rm =t ¼ 5. Two different crack lengths, h=p ¼ 0:125 and 0.25, were considered, which are regarded as the important range of h=p for typical LBB analyses. Regarding the material properties for the FE analysis, actual experimental uni-axial stress–strain data of Type 316 stainless steel at the temperature, T ¼ 288 °C, were taken, for which stress–strain curves are shown in Fig. 7, and the relevant data are summarised in Table 4. Elastic–plastic analyses of this FE model were performed using the general-purpose FE program, A B A Q U S [10]. The experimental true stress–plastic strain data were directly given in the FE analysis. Materials were modelled as isotropic elastic–plastic materials that obey the incremental plasticity theory,
Fig. 7. Stress–strain curve for SA312 Type 316 (288 °C) and the resulting R–O fit.
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Table 4 Summary of tensile properties for SA312 Type 316 stainless steel at 288 °C, used in the present FE analysis Material
E (GPa)
ry (MPa)
ru (MPa)
ERS parameters eu
n1
SA312 Type 316 (288 °C)
190
165
455
0.3
3.82
and a small geometry change continuum FE model was employed. Detailed information on the FE model can be referred to Section 2.1. For the case of internal pressure only, the magnitude of the internal pressure was increased up to a sufficiently high pressure, and the resulting J and COD values were extracted as a function of internal pressure. Note that the effect of the internal pressure on the crack face was considered by applying 50% of the internal pressure to the crack face. For the case of combined internal pressure and bending moment, it was noted that combined loading could be applied in two different ways: proportional loading and nonproportional loading. The present FE calculations employed only the proportional loading, namely, internal pressure and bending moment are increased in a proportional manner, with three different values of the load proportionality factor k (see Eq. (33) for the definition) to k ¼ 0:5, 1.0 and 2.0. 4.2. Results Fig. 8 compares the FE J and COD results for the TWC pipe under internal pressure, with the proposed J and COD estimates, Eqs. (24) and (26), for two values of h=p ¼ 0:125 and 0.25. Note that the proposed method is denoted as the ERSM in the subsequent figures. In Fig. 8, the J values are normalised with respect to ry and Rm h ð¼ aÞ, while the COD (d) values with respect to Rm h ð¼ aÞ. The load, internal pressure, is normalised with respect to the optimum reference pressure, poR (see Eq. (17)). Noting that the GE/EPRI J and COD estimations are developed in the present work (see Section 2), the resulting J and COD are also compared with the FE results. Application of the GE/EPRI method firstly requires that the material’s tensile data should be fitted using the R–O relation, see Eq. (1). In the present work, the entire true stress–strain data up to the ultimate tensile strength were fitted 1 using the R O F I T program [15], developed by Battelle. The resulting R–O parameters, a and n, are listed in Table 1, and the resulting R–O fits are compared with experimental tensile data in Fig. 7. Once the R–O parameters, a and n, are determined, then J and COD can be estimated using the effective crack length (ae ): nþ1 r2y ½ K ðae Þ 2 a p þ a ðt aÞ h1 ðnÞ ð36Þ J¼ t pL E E d¼
4 E
n pRi p ae V ðae Þ þ aeo ah2 ðnÞ pL 2t
ð37Þ
where ae was estimated from ae ¼ a þ ury ;
u¼
1 1 þ ð p=pL Þ
2
;
1 ry ¼ 2p
n1 nþ1
K ry
2 ð38Þ
1 There are other ways to fit the tensile data using the R–O relation. Typical ways include to fit the data only up 5% strain and to fit the data from 0.1% strain to 0.8eu;t , where eu;t denotes the true ultimate strain.
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Fig. 8. (a)–(d) Comparison of FE J and COD results for TWC pipes under internal pressure with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM), (ii) the GE/EPRI solutions for internal pressure, developed in the present work (present GE/EPRI), and (iii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
and the values of h1 ðnÞ and h2 ðnÞ were obtained by interpolating the present FE results (tabulated in Tables 2 and 3). The resulting values of J and COD are shown in Fig. 8, denoted as ‘‘present GE/EPRI’’. Finally it should noted that a typical practice to analyse the cracked pipe under internal pressure is to convert the internal pressure to the equivalent axial tension. In this context, the estimation formulae for the TWC pipes under axial tension, given in [3], were also used to estimate J and COD. The resulting values of J and COD are also compared in Fig. 8, denoted as ‘‘GE/EPRI-tension’’. The comparisons in Fig. 8 shows that the proposed ERSM-based J estimates are in overall good agreements with the FE results, and are on the conservative side. Note that for conservatism, the J value should be overestimated, that is, the estimated J values should be higher than the FE values. The ERSMbased COD estimations are not so accurate, compared to the FE COD results, but are still on the conservative side. Note that for conservatism, the detectable leakage crack length should be overestimated and thus the COD value should be underestimated, that is, the estimated COD values should be lower than the FE values. On the other hand, the GE/EPRI J and COD estimates are not so accurate, compared to the FE results. Such results are consistent with our earlier finding [6] and such inaccuracy is associated with the R– O fit. In fact, if different ways of fitting the R–O equation are performed, accuracy can be improved, but no guidance on the best R–O fit can be given since it depends on the material [6]. Finally the GE/EPRI J estimations based on axial tension show significant deviations from the FE J and COD results. Such deviation results from the fact that the axial tension solution neglects the hoop stress effect due to internal
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Fig. 9. (a)–(d) Comparison of FE J and COD results for TWC pipes under combined pressure and bending (k ¼ 0:5) with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM) and (ii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
pressure. The results in Fig. 8 support the well-known fact that the use of axial tension solutions for cracked pipes under internal pressure can lead to non-conservative estimates of the J-integral. The results for combined internal pressure and bending (under proportional loading conditions) are shown in Fig. 9 for the proportionality factor (see Eq. (33)), k ¼ 0:5, in Fig. 10 for k ¼ 1:0, and in Fig. 11 for k ¼ 2:0. In Figs. 9–11, the FE results for J and COD are compared with the proposed J and COD estimations. The results show overall excellent agreement, maintaining slight conservatism in both J and COD. The GE/EPRI J and COD estimates are also compared with the FE results. These estimates were based on the combined axial tension and bending solutions, given in Ref. [3]. As for the internal pressure case, deviations from the FE results are significant, resulting from neglecting the hoop stress effect due to internal pressure.
5. Concluding remarks This paper provides engineering J and COD estimation equations for TWC pipes under internal pressure and under combined internal pressure and bending. Based on selected 3-D FE calculations for the TWC
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Fig. 10. (a)–(d) Comparison of FE J and COD results for TWC pipes under combined pressure and bending (k ¼ 1:0) with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM) and (ii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
pipe under internal pressure using R–O materials, the elastic and plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the normalised crack length, b=p, and the mean radiusto-thickness ratio, Rm =t. Then these developed GE/EPRI-type solutions are re-formulated based on the reference stress concept. Such a re-formulation provides simpler equations for J and COD, which are then further extended to combined internal pressure and bending. The proposed reference stress based J and COD estimation equations are compared with elastic–plastic 3-D FE results using actual stress–strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimations. Another notable point is that caution should be exercised if a defect assessment of cracked, pressurised piping is made by considering the pipe with an axial tension equivalent to the internal pressure. Note that the majority of existing fracture mechanics analyses concentrate on finding solutions for axial tension, not for internal pressure. Such practice would be acceptable when plasticity due to internal pressure is small. On the other hand, when the plasticity due to internal pressure becomes more significant, it is well known and also is confirmed in this paper that such practice can give non-conservative estimates for crack driving forces. Another potential area for fracture mechanics analysis specific to internal pressure is the analysis of pressurised pipes operating at elevated (creep) temperatures. In such cases, time dependent fracture
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Fig. 11. (a)–(d) Comparison of FE J and COD results for TWC pipes under combined pressure and bending (k ¼ 2:0) with the engineering estimations: (i) the proposed enhanced reference stress approach (ERSM) and (ii) the GE/EPRI solutions for axial tension, given in Ref. [3] (GE/EPRI-tension).
mechanics parameters, for instance, the C -integral [16], should be estimated. Again the use of the axial tension solutions could provide non-conservative estimates for C . In this context, the present J and COD estimation scheme for elastic–plastic fracture can be used to estimate the C -integral and the COD rate, d_c , due to creep, using the analogy between plasticity and creep as follows [17]: 2 E K e_c
ð39Þ C ¼ E0 rref d_c e_c ¼ de ðrref =EÞ
with d_c ðt ¼ 0Þ ¼ 0
ð40Þ
where e_c is the creep strain rate at the reference stress r ¼ rref , determined from the actual creep-deformation data. The reference stress for internal pressure and for combined internal pressure and bending is defined in Eqs. (17) and (32), respectively. Validation of these estimation equations will be given in a separate paper.
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Acknowledgements The authors are grateful for the support provided by a grant from Safety and Structural Integrity Research Centre at Sungkyunkwan University.
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