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Prediction of creep failure life of internally pressurised thick walled CrMoV pipes
International Journal of Pressure Vessels and Piping 76 (1999) 925–933 www.elsevier.com/locate/ijpvp
Prediction of creep failure life of internally pressurised thick walled CrMoV pipes T.H. Hyde a, W. Sun a,*, J.A. Williams b a
School of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham NG7 2RD, UK b Independent Consultant, East Leake, Leicester LE12 6LJ, UK Received 5 August 1999; received in revised form 8 November 1999; accepted 18 November 1999
Abstract Creep continuum damage finite element analyses were performed for a typical internally pressurised main steam thick-sectioned plain pipe geometry subjected to a range of axial loading conditions. The creep failure lives, for a range of loading levels, were obtained. The stress distributions showed the presence of a skeletal point or representative rupture stress. Life estimates were made using alternative stress forms and these were compared with the failure lives predicted using continuum damage analyses. For the particular ferritic material properties used in the analyses, the failure life of the pipe can be obtained accurately using the steady-state reference rupture stress or the skeletal point rupture stress. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Creep; Damage; Thick walled pipe; Failure life; Rupture stress
1. Introduction High temperature, thick walled pipes are widely used in power plants and chemical plants. One of the main concerns in the high temperature design of these pipes is the predicted service life of the pipe under creep condition. This has become of increasing importance as many plants, particularly that in the power industry, are now achieving operating periods in excess of 150,000 h. In addition, many of the problems encountered have been associated with welds [1,2]. As a result, weld replacement and repair programmes have been extensively used, thereby improving the weld quality for continued operation, but these repairs do not increase the plain pipe life [3,4]. Most design codes, for example BS5500, use the mean diameter hoop stress, s mdh, for both thick and thin wall pressurised pipes, to characterise creep rupture in a pressurised pipe, where
smdh
p 2Ro 2 T i 2T
1
over the last 40–50 years [5,6]. The mean diameter stress was selected from a range of alternative stresses for pressurised creeping pipes. Very few pipe failures have occurred in practice showing the conservatism of the approach for design purposes [7]. However, with increasing operational time and for remaining life evaluations, there is a need to reappraise the potential creep life of simple pressurised pipe structures. Continuum damage mechanics approaches can be used in conjunction with the finite element (FE) method to predict creep deformations in and failure lives of engineering components. If the material constants in the constitutive equations are known, the creep failure life, failure position and damage accumulation within a component can be assessed. Typical creep damage constitutive equations are of the form [8]
1_ cij
3 n21 1 As S tm 2 eq ij 1 2 vn
2a
sxr tm 1 1 f 1 2 vf
2b
and
in which pi is the internal pressure, Ro the outer radius and T the wall thickness. This method of pipe design is primarily based on experimental data for tubes and model pipes tested
v_ M
* Corresponding author. Tel.: 144-115-9513-809; fax: 144-115-9513800. E-mail address: [email protected] (W. Sun).
where A, n, m, M, x and f are material constants and v is the damage parameter, which varies from 0 (no damage) to 1 (failure). Such relationships can characterise the primary,
0308-0161/99/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(99)00078-2
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Nomenclature A, m, M, n constants in constitutive equations internal pressure pi r, rref, rsp, Ri, Ro radial position, radial position at reference stress point, radial position at skeletal point, inside radius and outside radius, respectively deviatoric stress Sij t, tf time and failure time T wall thickness a constant in constitutive equations 1 , 1 eq strain and equivalent strain 1Ç strain rate s , s u , s eq, s 1, s r stress, hoop, equivalent, maximum principal and rupture stresses, respectively s ax, s mdh, s rsp, s rref mean axial stress, mean diameter hoop stress, skeletal point rupture stress and steady-state reference rupture stress, respectively v , vÇ damage and damage rate x, f constants in constitutive equations secondary and tertiary components of the creep curve. The rupture stress, s r, is assumed to be a function of the maximum principal stress, s 1, and the equivalent stress, s eq, as follows:
sr as1 1 1 2 aseq
3
where a is the material constant that defines the effect of a multiaxial stress state on the rate at which damage occurs in the material. The a value ranges from 1, where the maximum principal stress is dominant, to 0, where the equivalent stress is dominant. Hayhurst and co-workers [8–12] have reported the results of analyses using continuum damage mechanics, including analyses of welded pipes. The work ranges from establishing suitable material constitutive equations to the numerical modelling of practical engineering components. Several different forms of constitutive equations have been
Fig. 1. Thick walled pipe model.
proposed [8–10]. For example, a complex two state variables model was used to model intergranular creep cavitation and coarsening of carbide precipitates [10]. However, bi-axial creep tests have shown [11] that Eqs. (2a) and (2b) are capable of accurately representing the stress dependence of the creep strain-rate components, over wide ranges of stress and temperature, for a number of metallic alloys. For instance, the one state variable Eqs. (2a) and (2b) were successfully used for the virgin CrMoV weldment at 5658C to represent the deformation and failure mechanisms of the materials [12]. In this work, creep continuum damage mechanics finite element calculations were performed for an internally pressurised thick-sectioned plain pipe, typical of those used in the main steam pipe lines of power plants. Results were obtained for closed-end conditions and a range of other axial load level to determine the expected life, when all aspects of the creep curve are incorporated. In all, three ferritic materials were used to supply the material data. The corresponding failure lives of the pipe were also predicted, based on a number of different stress levels such as the steady-state rupture reference stress, a skeletal point rupture stress and the mean diameter hoop stress. These are compared with those predicted by the finite element method using the continuum damage constitutive laws.
2. Pipe model and damage analysis The thick-sectioned pipe model, Fig. 1, has a radius ratio, Ro =Ri of 1.5555 where Ri and Ro are the inside and outside radius, respectively. It is subjected to an internal pressure, pi, and an average axial stress s ax. Under closed-end conditions, for the pipe dimensions given above, sax 0:7043 pi : The radius ratio of Ro =Ri of 1.5555 is applicable to typical main steam pipe lines of power generation plants. Identical models were used for the cases where the mean axial stress was increased to create mean axial stress to mean diameter hoop stress ratios from 0.5 to 1.0, the normal limits allowed within the design codes. A simple axisymmetric FE mesh was generated to model the creep and damage behaviour of the pipe and to obtain the failure life. A uniform deformation in the axial direction was applied, see Fig. 1. Three different sets of material properties were used in the FE damage calculations and the material constants, in Eqs. (2) and (3), are shown in Table 1. The CrMoV material (steel 1) was originally in the normalised and tempered condition, but had previously been exposed to operational service. The 2.25Cr1Mo material (steel 2) was the weld metal in a service-aged pipe weld. The material constants for steels 1 and 2 were obtained from creep testing at 6408C, [13]. The 1Cr0.5Mo material (steel 3) was from virgin stock and had been tested at 5508C, [14]. A typical operational pressure for pipes made from these materials is 16.55 MPa.
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Table 1 Material constants in Eqs. (2) and (3) used in FE damage calculations (stress in MPa and time in hour) A 0.5Cr0.5Mo0.25V (Steel 1) [13] 2.25Cr1Mo (Steel 2) [13] 1Cr0.5Mo (Steel 3) [14]
n 216
6:599 × 10 9:718 × 10215 1:94 × 10215
6.108 5.208 4.354
3. Creep failure life and rupture stress for pipes under closed-end condition 3.1. FE damage calculation results FE creep damage analyses were performed for the pipe using Eqs. (2) and (3), and the material properties shown in Table 1, for a range of pi values from 16.55 to 45 MPa. In addition, the influence of a on the failure behaviour of the pipe was determined, for pi 16:55 MPa; by performing a series of calculations with different a values, from 0 to 1, all
m 0 0 0
M 214
5:998 × 10 8:120 × 10213 8:325 × 10213
f
x
a
4.5 4.1 1.423
5.767 4.850 3.955
0.30 0.26 0.43
the other material constants for each material, in Table 1, being unchanged. The calculations were carried out until high damage values, e.g. v ! 1; were achieved. Examples of the variations of damage parameter, v , and equivalent strain, 1 eq, with time at the pipe outer surface for steels 1 and 2 at pi 16:55 MPa are shown in Fig. 2(a), from which it can be seen that failure times of the pipe, tf, can be easily obtained. Fig. 2(b) shows an example of damage variations through the wall thickness at different times for steel 2, from which it can be seen that the damage variations through the thickness are nearly uniform, particularly when the damage is high. The creep rupture lives obtained with different internal pressures are presented in Fig. 3, and a linear relationship between log tf and log pi is observed. This indicates that accurate predictions can be expected within the stress range for which the material constants in Eqs. (2) and (3) are applicable by linear interpolation of the failure life from Fig. 3. The effect of a on log (tf), keeping all the other material constants the same and the internal pressure maintained at 16.55 MPa, is given in Fig. 4. Unlike the behaviour previously predicted for a notched bar [15], the failure times of the pipe increase with increasing a . 3.2. Skeletal point rupture stress and steady-state reference rupture stress The representative rupture stress, Eq. (3), is shown as a function of the pipe radius position for various creep times
Fig. 2. (a) Variations of the damage parameter and equivalent creep strain at the outer surface with time, for steels 1 and 2, obtained from FE damage analyses with pi 16:55 MPa; (b) Variations of the damage through the wall thickness, at different times, for steels 2, obtained from FE damage analyses with pi 16:55 MPa:
Fig. 3. Effect of internal pressure on creep failure life using FE damage modelling for steels 1 and 2.
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Fig. 4. Effect of a on failure life using FE damage modelling for steels 1 and 2 with pi 16:55 MPa:
in Fig. 5(a) and (b) for steels 1 and 2, respectively. It is clear that the rupture stress at a particular point within the pipe wall is constant with time as would be expected from the skeletal point concept discussed by Marriot, [16], although in his case, the equivalent stress rather than the representative creep rupture stress was used. It can also be seen that, for steel 1, the rupture stress distribution does not vary significantly with time; this is mainly an effect due to the particular a value of the material, and the rupture stresses increase with time at the inside surface while decreasing at the outside surface, see Fig. 5(a). However, for steel 2, the rupture stress distribution does vary significantly with time, and the rupture stresses increase with time at the outside surface while decreasing at the inside surface, see Fig. 5(b). In all cases, the constant rupture stress position was near the centre of the pipe wall. Damage calculation results have shown that in order to obtain clearly defined skeletal point reference rupture stresses, it is marginally preferable
Fig. 5. (a) Creep rupture stress versus radial position at different times, using FE damage modelling for steel 1, with pi 16:55 MPa; (b) Creep rupture stress versus radial position at different times, using FE damage analysis for steel 2, with pi 16:55 MPa:
T.H. Hyde et al. / International Journal of Pressure Vessels and Piping 76 (1999) 925–933
Fig. 6. Steady-state creep rupture stress versus radial position for different n values, with pi 16:55 MPa: and a 0:5:
to use the stress variations through the wall thickness, at different times, in the tertiary stage, i.e. in a relatively narrow time range when t ! tf : Under the steady-state creep conditions, when the material is assumed to obey a power law creep equation of the form 1_ Asn ; a reference stress for deformation exists in the pipe, which is independent of the material constants, A and n, and occurs at a skeletal point [17]. Under closed-end condition, for an internally pressurised thick walled pipe, the steady-state stress solutions are available [18], therefore, the maximum principal stress and equivalent stress can be obtained analytically, as " # 2 2 n Ro 2=n pi s1 r su r 1 1 n r Ro =Ri 2=n 2 1 4a
seq r
p 2=n pi 3 Ro r n Ro =Ri 2=n 2 1
4b
where r is the radial position within the pipe. The steadystate creep rupture stress can be obtained from Eqs. (4) and (3). A skeletal point is shown where the rupture stress is independent of n and this stress is defined as the steady-state
Fig. 8. Effect of a on steady-state reference rupture stresses and skeletal point rupture stresses for steels 1 and 2 with pi 16:55 MPa:
reference rupture stress, sref r : An example is illustrated in Fig. 6 for the pipe with n 2; 4 and 6 and a pressure of 16.55 MPa. It is interesting to note that the steady-state reference rupture stresses are practically the same as the corresponding skeletal point rupture stresses obtained from FE damage analyses as shown in Fig. 7 for three internal pressures. More generally, the effect of a for a pi of 16.55 MPa, on both stress levels is shown in Fig. 8 and it is clear that the agreement is good for a , 0:6 and diverges to an error of around 3% when a 1:0: 3.3. Failure life prediction based on steady-state reference rupture stress The reference stress and the skeletal point stress have been widely used for creep deformation analysis of engineering components [17,19]. The current analyses have shown that there is a point, within the pipe wall, where the representative rupture stress, as defined in Eq. (3), is independent of time throughout the complete creep curve, i.e. including primary, secondary and tertiary creep. In addition, similar results have been shown for the rupture stress derived from the steady-state solutions. These stresses have been used to estimate the pipe life for comparison with the full damage analysis models. Eq. (2b) can be integrated to give the failure life, tf. When m 0; as in the case of the materials studied here, the failure life is given by: tf
Fig. 7. Variations of skeletal point rupture stresses and steady-state reference rupture stresses with internal pressure obtained for steel 1 a 0:3 and steel 2 a 0:26:
929
1 M sr x
5
If a suitable rupture stress can be determined, Eq. (5) can be used to predict the failure time under multi-axial situations. For the thick walled pipe investigated in this work, the rupture lives obtained from Eq. (5) using the steady-state reference rupture stress, or the “skeletal point” rupture stress, and the appropriate material constants, M and x , for the pipe material, are very close to those obtained from the corresponding FE damage calculations, Fig. 9. The effect of a on creep life, keeping all the other constants the same for each material, with a pressure of 16.55 MPa, is
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Fig. 9. Relationship between creep failure lives, obtained from steady-state reference rupture stresses, and internal pressure for steels 1 and 2.
shown in Fig. 10. The failure lives predicted by the two techniques are practically the same with only slight differences when a exceeds 0.6 as implied by Fig. 8. 4. Creep failure life and rupture stress for pipes under internal pressure and additional axial loading The above approach was extended to the examination of combined internal pressure and end load applied to the pipe, using FE creep damage analyses, with an internal pressure of 16.55 MPa and a range of sax =smdh values, where s mdh is the mean diameter hoop stress. The material properties shown in Table 1 were used. The corresponding failure predictions, using the simplified technique, Eq. (5), based on the “skeletal point” rupture stress or steady-state reference rupture stress, were also determined for comparison. The variations of creep rupture stress with radial position Fig. 11. Plots of creep rupture stresses, obtained using FE damage modelling, with radial position at different times for steel 1 with pi 16:55 MPa: (a) sax =smdh 0:5. (b) sax =smdh 0:75. (c) sax =smdh 1:0.
Fig. 10. Variations of creep failure lives, obtained from steady-state reference rupture stresses, with a for steels 1 and 2 with pi 16:55 MPa:
for steel 1, with sax =smdh 0:5; 0.75 and 1.0, obtained from the FE damage calculations, are given in Fig. 11(a)–(c), respectively. The corresponding rupture stresses for steel 2 are shown in Fig. 12(a)–(c). It is clear that positions of constant rupture stress still exist in the presence of additional axial loading. When sax =smdh is around 0.75, there are two such points within the wall thickness of the pipe although the rupture stresses at these two points are practically the same, see Figs. 11(b) and 12(b). Since it was found that the behaviour obtained for steel 3 was similar to that obtained for steels 1 and 2, no detailed results for steel 3 have been presented in the paper; the results are, however, summarised in Table 2.
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As closed-form analytical solutions do not exist for the steady-state reference rupture stresses in pipes with arbitrary axial loads, these have been obtained from FE analyses. Examples of the variations of steady-state rupture stresses with radial position, for different n values, at a 0:3; for sax =smdh 0:5; 0.75 and 1.0, are given in Figs. 13(a)–(c), respectively. It can be seen that similar to the closed-end case, steady-state reference rupture stresses exist and can be determined. However, single crossover points are found at all values of sax =smdh from 0.5 to 1.0. Table 2 summarises the data obtained from the analyses performed with sax =smdh 0:5; 0.75 and 1.0. The variations of the skeletal point rupture stresses, ssp r ; and the steady-state reference rupture stresses, sref r ; with sax =smdh ratio, for steel 1, a 0:3; and steel 2, a 0:26; for a pressure of 16.55 MPa, are shown in Fig. 14. It can be sp seen that the sref r and sr are practically the same, in the range of 0:306 , sax =smdh ,1.0. This can be further illustrated, Fig. 15, from examination of the rupture life estimates obtained from these stresses, Eq. (5) and the material constants, M and x , Table 1. It is clear that the failure lives predicted from the “skeletal point” stresses are practically the same as those obtained from the FE damage calculations for 0:306 , sax =smdh , 1:0; as are the failure lives predicted from steady-state reference rupture stresses. 5. Discussion and conclusions
Fig. 12. Plots of creep rupture stresses, obtained using FE damage modelling, with radial position at different times for steels 2 with pi 16:55 MPa: (a) sax =smdh 0:5. (b) sax =smdh 0:75. (c) sax =smdh 1:0.
FE creep damage analyses were performed for an internally pressurised thick walled pipe under closed-end conditions and with various additional axial loading, as allowed within the design codes. The geometry and loading conditions were typical of the main steam pipe lines in power plants. Materials data for three ferritic steels were used, to illustrate the expected effects. The creep damage analyses provided life estimates based on the time when the damage accumulations approach unity; they incorporate full creep curves and multiaxial stress effects. The representative creep rupture stress, when examined across the pipe wall, showed positions where this stress is nearly constant with time. This has
Table 2 Life estimates for the pressure plus end load models, pi 16:55 MPa; stress in MPa and time in hour Steel
1
2
3
sax =smdh
0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0
Damage
Steady-state
rsp =Ro
ssp r
Life FE
Life Eq. (5)
0.843 0.725, 0.914 0.808 0.822 0.786, 0.993 0.822 0.852 0.727, 0.912 0.806
32.10 34.65 40.33 32.2 35.0 40.4 31.63 33.85 39.91
3:43 × 104 2:20 × 104 9:15 × 103 5:97 × 104 4:08 × 104 1:98 × 104 1:39 × 106 1:06 × 106 5:59 × 105
3:42 × 104 2:20 × 104 9:17 × 103 6:0 × 104 4:0 × 104 1:99 × 104 1:40 × 106 1:07 × 106 5:58 × 105
Design
rref =Ro
sref r
Life Eq. (5)
< 0.8
32.12 34.51 40.15 32.32 34.81 40.30 31.60 33.59 39.70
3:40 × 104 2:25 × 104 9:41 × 103 5:89 × 104 4:11 × 104 2:02 × 104 1:41 × 106 1:11 × 106 5:71 × 105
< 0.8 < 0.8
s mdh
Life Eq. (5)
38.065
1:28 × 104
38.065
2:66 × 104
38.065
6:74 × 105
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Fig. 14. Variations of skeletal point rupture stresses and steady-state reference rupture stresses with sax =smdh ; obtained for steel 1 a 0:3 and steel 2 a 0:26 with pi 16:55 MPa:
constant-stress, or a skeletal point, exists within the pipe wall and, indeed, in this form can be shown to be numerically nearly independent of the Norton index, n (Fig. 6). Furthermore, there is excellent agreement between the life estimates, using steady-state solutions, and the full damage solutions for values of a , 0:6: Where a . 0:6; the estimated steady-state stress is greater than that provided from the full damage analysis and gives an over estimate of the creep life. Thus, this approach offers a potential method for the life estimation of a closed-ended, internally pressurised pipe, based on steady-state stress levels, although a knowledge of the materials constants, a , M and x and an assumed value of Norton’s index, n, is also required. This avoids the need
Fig. 13. Plots of steady-state creep rupture stresses versus radial position, for different n values, with a 0:3 and pi 16:55 MPa: (a) sax =smdh 0:5. (b) sax =smdh 0:75. (c) sax =smdh 1:0.
been previously referred to as a skeletal point, [16]. Using this stress to characterise the pipe failure provides good agreement with the life predicted from a full damage analysis. The materials data input used also covered a range of Norton’s index, n, from 4 to 6. There is a rationale in the assumption that stress levels, which are constant or nearly constant with time, will be closely related to the steady-state values under identical loading conditions. For the closed-ended condition, the maximum principal and the von Mises equivalent stresses can be obtained from analytical solutions [18], and the multiaxiality factor, a , is merely a multiplier modifying the contribution of maximum principal and equivalent stresses to the failure process. The steady-state rupture stress results clearly show the presence of a position at which a
Fig. 15. Plot of Creep failure lives against sax =smdh ; obtained from FE damage modelling, skeletal point rupture stress and steady-state reference rupture stress for steels 1 and 2 with pi 16:55 MPa:
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for detailed FE damage calculations, which are time consuming and for which specialised FE damage codes are not widely available. With additional axial loading, similar rupture stress distributions from the FE damage analysis are found to those for an internally pressurised pipe and constant stress points are observed within the pipe wall for 0:306 , sax =smdh , 1: For sax =smdh values of about 0.75, two constant stress points occur although these provide nearly identical stress levels. The steady-state FE analyses produce skeletal points/ constant stress points for the same range of sax =smdh ; but there is no second constant stress point at about sax =smdh 0:75: The failure times predicted from the skeletal point and the steady-state versions are very close to the results of the full creep damage analysis. Thus, again this provides a realistic and efficient method for the life prediction of a pipe subjected to axial loading in the range of 0:306 , sax =smdh , 1:0 and an a of 0.25–0.5. It is interesting to consider the design rules used for such pipes. Using Eq. 1, the mean diameter hoop stress for a typical internal pressure of 16.55 MPa, which is 38.065 MPa, for the pipe geometry used. The calculated and estimated stress levels for all of the materials considered are given in Table 2. For pressurised pipes, the mean diameter hoop stress is shown to be conservative by around 10 % in stress and thus, these suggested methods may be useful for remaining life studies. However, the pressure plus end load models suggest that for sax =smdh . 0:75; the mean diameter hoop stress approach may not be conservative. This is an extreme load situation for which further investigation may be necessary, particularly as it is known from weld failure statistics that certain positions within a pipework system may experience additional axial (system) loading. This paper proposes a simplified procedure that could be used to estimate the life of a plain pipe under pressure and pressure plus additional end loads. The use of steady-state analysis provides a more efficient analysis method than a full damage analysis although additional data, such as the multiaxiality factor, a , are necessary unless bounds can be prescribed for the relevant materials. Acknowledgements The authors wish to acknowledge Nuclear Electric plc, PowerGen plc, National Power plc and the Engineering and Physical Science Research Council, UK, for their financial and technical support of the work.
[2]
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between laboratory data and service experience in the cracking of CrMoV weldments. Conference on Residual Stresses and their Effect on Welded Constructions, London. Cambridge, UK: TWI, 1977. Brett SJ. In service cracking mechanism affecting 2CrMo welds in 0.5CrMoV steam pipe-work systems. Conference on Integrity of High Temperature Welds, Nottingham, UK. London: Inst Materials/I Mech E, 1998. p. 3–14 Viswanathan R, Gandy, DW. Performance of repair welds on aged Cr–Mo piping girth welds. Conference on Integrity of High Temperature Welds, Nottingham, UK. London: Inst Materials/I Mech E, 1998. p. 15–40. Brett SJ, Abson DJ, Jones RL. The repair welding of power plant without PWHT. International Conference on Integrity of High Temperature Welds, Nottingham, UK. London: Inst Materials/I Mech E, 1998. p. 53–64. Skelton WJ, Crossland B. Creep of thick walled cylinders subjected to internal pressure: Review of literature, Conf High Pressure Engineering. Proc Inst Mech Engrs 1967, 1968;182(3c):132–8. Taira S, Ohtani R. Creep rupture of internally pressurised cylinders at elevated temperatures. Bull JSME 1968;46(11):595. Aburrow AF, Cane BJ, Carmichael GDT, Dewar A, Hart RV, Heather CW, Plastow B, Williams JA, Womersley S. Creep of CrMoV piping systems. Conference on Pipework Design and Operation. London: I. Mech E, 1985. Hayhurst DR. The role of creep damage in structural mechanics. In: Wilshire B, Owen DR, editors. Engineering approach to high temperature, Swansea: Pineridge, 1983. p. 85–176. Hayhurst DR, Perrin IJ. CDM analysis of creep rupture in weldments. Proc Engng Mech 1995;1:393–6. Perrin IJ, Hayhurst DR. Creep constitutive equations for 0.5Cr0.5Mo0.25V ferritic steel in the temperature range 600– 6758C. J Strain Anal 1996;31:299–314. Hayhurst DR. Creep rupture under multiaxial states of stresses. J Mech Phys Solids 1972;20:381–90. Hall FR, Hayhurst DR. Continuum damage mechanics modelling of high temperature deformation and failure in a pipe weldment. Proc R Soc London, A 1991;443:383–403. Hyde TH, Sun W, Becker AA, Williams JA. Creep continuum damage constitutive equations for the base, weld and heat-affected zone materials of a service-aged 1/2Cr1/2Mo1/4V:2 1/2Cr1Mo multipass weld at 6408C. J Strain Anal 1997;32(4):273–85. Segle P, Tu S-T, Storesund J, Samuelson LA. Some issues in life assessment of longitudinal seam welds based on creep tests with cross-weld specimens. Int J Pres Ves Piping 1996;66:199– 222. Hyde TH, Sun W. Creep of waisted and notched cross-weld specimens. Proceedings of the Seventh Conference on Creep and Fracture of Engineering Materials and Structures, California, 1997, p. 759– 768. Marriott DL. A Review of reference stress methods for estimating creep deformation, Proceedings of the IUTAM Symposium on Creep of Structures, Gothenburg. Berlin: Springer, 1970. p. 137–52. Fairbairn J, Mackie WW. Creep of thick walled tubes subjected to internal pressure, Proceedings of the IUTAM Symposium on Creep in Structures, Gothenburg. Berlin: Springer, 1970. p. 189– 202. Boyle JT, Spence J. Stress analysis for creep. London: Butterworths, 1983. Marriott DL, Leckie FA. Some observations on the deflection of structures during creep. Proc Inst Mech Engrs 1963;178(Pt.(3L)): 115–25.
Prediction of creep failure life of internally pressurised thick walled CrMoV pipes T.H. Hyde a, W. Sun a,*, J.A. Williams b a
School of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham NG7 2RD, UK b Independent Consultant, East Leake, Leicester LE12 6LJ, UK Received 5 August 1999; received in revised form 8 November 1999; accepted 18 November 1999
Abstract Creep continuum damage finite element analyses were performed for a typical internally pressurised main steam thick-sectioned plain pipe geometry subjected to a range of axial loading conditions. The creep failure lives, for a range of loading levels, were obtained. The stress distributions showed the presence of a skeletal point or representative rupture stress. Life estimates were made using alternative stress forms and these were compared with the failure lives predicted using continuum damage analyses. For the particular ferritic material properties used in the analyses, the failure life of the pipe can be obtained accurately using the steady-state reference rupture stress or the skeletal point rupture stress. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Creep; Damage; Thick walled pipe; Failure life; Rupture stress
1. Introduction High temperature, thick walled pipes are widely used in power plants and chemical plants. One of the main concerns in the high temperature design of these pipes is the predicted service life of the pipe under creep condition. This has become of increasing importance as many plants, particularly that in the power industry, are now achieving operating periods in excess of 150,000 h. In addition, many of the problems encountered have been associated with welds [1,2]. As a result, weld replacement and repair programmes have been extensively used, thereby improving the weld quality for continued operation, but these repairs do not increase the plain pipe life [3,4]. Most design codes, for example BS5500, use the mean diameter hoop stress, s mdh, for both thick and thin wall pressurised pipes, to characterise creep rupture in a pressurised pipe, where
smdh
p 2Ro 2 T i 2T
1
over the last 40–50 years [5,6]. The mean diameter stress was selected from a range of alternative stresses for pressurised creeping pipes. Very few pipe failures have occurred in practice showing the conservatism of the approach for design purposes [7]. However, with increasing operational time and for remaining life evaluations, there is a need to reappraise the potential creep life of simple pressurised pipe structures. Continuum damage mechanics approaches can be used in conjunction with the finite element (FE) method to predict creep deformations in and failure lives of engineering components. If the material constants in the constitutive equations are known, the creep failure life, failure position and damage accumulation within a component can be assessed. Typical creep damage constitutive equations are of the form [8]
1_ cij
3 n21 1 As S tm 2 eq ij 1 2 vn
2a
sxr tm 1 1 f 1 2 vf
2b
and
in which pi is the internal pressure, Ro the outer radius and T the wall thickness. This method of pipe design is primarily based on experimental data for tubes and model pipes tested
v_ M
* Corresponding author. Tel.: 144-115-9513-809; fax: 144-115-9513800. E-mail address: [email protected] (W. Sun).
where A, n, m, M, x and f are material constants and v is the damage parameter, which varies from 0 (no damage) to 1 (failure). Such relationships can characterise the primary,
0308-0161/99/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(99)00078-2
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Nomenclature A, m, M, n constants in constitutive equations internal pressure pi r, rref, rsp, Ri, Ro radial position, radial position at reference stress point, radial position at skeletal point, inside radius and outside radius, respectively deviatoric stress Sij t, tf time and failure time T wall thickness a constant in constitutive equations 1 , 1 eq strain and equivalent strain 1Ç strain rate s , s u , s eq, s 1, s r stress, hoop, equivalent, maximum principal and rupture stresses, respectively s ax, s mdh, s rsp, s rref mean axial stress, mean diameter hoop stress, skeletal point rupture stress and steady-state reference rupture stress, respectively v , vÇ damage and damage rate x, f constants in constitutive equations secondary and tertiary components of the creep curve. The rupture stress, s r, is assumed to be a function of the maximum principal stress, s 1, and the equivalent stress, s eq, as follows:
sr as1 1 1 2 aseq
3
where a is the material constant that defines the effect of a multiaxial stress state on the rate at which damage occurs in the material. The a value ranges from 1, where the maximum principal stress is dominant, to 0, where the equivalent stress is dominant. Hayhurst and co-workers [8–12] have reported the results of analyses using continuum damage mechanics, including analyses of welded pipes. The work ranges from establishing suitable material constitutive equations to the numerical modelling of practical engineering components. Several different forms of constitutive equations have been
Fig. 1. Thick walled pipe model.
proposed [8–10]. For example, a complex two state variables model was used to model intergranular creep cavitation and coarsening of carbide precipitates [10]. However, bi-axial creep tests have shown [11] that Eqs. (2a) and (2b) are capable of accurately representing the stress dependence of the creep strain-rate components, over wide ranges of stress and temperature, for a number of metallic alloys. For instance, the one state variable Eqs. (2a) and (2b) were successfully used for the virgin CrMoV weldment at 5658C to represent the deformation and failure mechanisms of the materials [12]. In this work, creep continuum damage mechanics finite element calculations were performed for an internally pressurised thick-sectioned plain pipe, typical of those used in the main steam pipe lines of power plants. Results were obtained for closed-end conditions and a range of other axial load level to determine the expected life, when all aspects of the creep curve are incorporated. In all, three ferritic materials were used to supply the material data. The corresponding failure lives of the pipe were also predicted, based on a number of different stress levels such as the steady-state rupture reference stress, a skeletal point rupture stress and the mean diameter hoop stress. These are compared with those predicted by the finite element method using the continuum damage constitutive laws.
2. Pipe model and damage analysis The thick-sectioned pipe model, Fig. 1, has a radius ratio, Ro =Ri of 1.5555 where Ri and Ro are the inside and outside radius, respectively. It is subjected to an internal pressure, pi, and an average axial stress s ax. Under closed-end conditions, for the pipe dimensions given above, sax 0:7043 pi : The radius ratio of Ro =Ri of 1.5555 is applicable to typical main steam pipe lines of power generation plants. Identical models were used for the cases where the mean axial stress was increased to create mean axial stress to mean diameter hoop stress ratios from 0.5 to 1.0, the normal limits allowed within the design codes. A simple axisymmetric FE mesh was generated to model the creep and damage behaviour of the pipe and to obtain the failure life. A uniform deformation in the axial direction was applied, see Fig. 1. Three different sets of material properties were used in the FE damage calculations and the material constants, in Eqs. (2) and (3), are shown in Table 1. The CrMoV material (steel 1) was originally in the normalised and tempered condition, but had previously been exposed to operational service. The 2.25Cr1Mo material (steel 2) was the weld metal in a service-aged pipe weld. The material constants for steels 1 and 2 were obtained from creep testing at 6408C, [13]. The 1Cr0.5Mo material (steel 3) was from virgin stock and had been tested at 5508C, [14]. A typical operational pressure for pipes made from these materials is 16.55 MPa.
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Table 1 Material constants in Eqs. (2) and (3) used in FE damage calculations (stress in MPa and time in hour) A 0.5Cr0.5Mo0.25V (Steel 1) [13] 2.25Cr1Mo (Steel 2) [13] 1Cr0.5Mo (Steel 3) [14]
n 216
6:599 × 10 9:718 × 10215 1:94 × 10215
6.108 5.208 4.354
3. Creep failure life and rupture stress for pipes under closed-end condition 3.1. FE damage calculation results FE creep damage analyses were performed for the pipe using Eqs. (2) and (3), and the material properties shown in Table 1, for a range of pi values from 16.55 to 45 MPa. In addition, the influence of a on the failure behaviour of the pipe was determined, for pi 16:55 MPa; by performing a series of calculations with different a values, from 0 to 1, all
m 0 0 0
M 214
5:998 × 10 8:120 × 10213 8:325 × 10213
f
x
a
4.5 4.1 1.423
5.767 4.850 3.955
0.30 0.26 0.43
the other material constants for each material, in Table 1, being unchanged. The calculations were carried out until high damage values, e.g. v ! 1; were achieved. Examples of the variations of damage parameter, v , and equivalent strain, 1 eq, with time at the pipe outer surface for steels 1 and 2 at pi 16:55 MPa are shown in Fig. 2(a), from which it can be seen that failure times of the pipe, tf, can be easily obtained. Fig. 2(b) shows an example of damage variations through the wall thickness at different times for steel 2, from which it can be seen that the damage variations through the thickness are nearly uniform, particularly when the damage is high. The creep rupture lives obtained with different internal pressures are presented in Fig. 3, and a linear relationship between log tf and log pi is observed. This indicates that accurate predictions can be expected within the stress range for which the material constants in Eqs. (2) and (3) are applicable by linear interpolation of the failure life from Fig. 3. The effect of a on log (tf), keeping all the other material constants the same and the internal pressure maintained at 16.55 MPa, is given in Fig. 4. Unlike the behaviour previously predicted for a notched bar [15], the failure times of the pipe increase with increasing a . 3.2. Skeletal point rupture stress and steady-state reference rupture stress The representative rupture stress, Eq. (3), is shown as a function of the pipe radius position for various creep times
Fig. 2. (a) Variations of the damage parameter and equivalent creep strain at the outer surface with time, for steels 1 and 2, obtained from FE damage analyses with pi 16:55 MPa; (b) Variations of the damage through the wall thickness, at different times, for steels 2, obtained from FE damage analyses with pi 16:55 MPa:
Fig. 3. Effect of internal pressure on creep failure life using FE damage modelling for steels 1 and 2.
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Fig. 4. Effect of a on failure life using FE damage modelling for steels 1 and 2 with pi 16:55 MPa:
in Fig. 5(a) and (b) for steels 1 and 2, respectively. It is clear that the rupture stress at a particular point within the pipe wall is constant with time as would be expected from the skeletal point concept discussed by Marriot, [16], although in his case, the equivalent stress rather than the representative creep rupture stress was used. It can also be seen that, for steel 1, the rupture stress distribution does not vary significantly with time; this is mainly an effect due to the particular a value of the material, and the rupture stresses increase with time at the inside surface while decreasing at the outside surface, see Fig. 5(a). However, for steel 2, the rupture stress distribution does vary significantly with time, and the rupture stresses increase with time at the outside surface while decreasing at the inside surface, see Fig. 5(b). In all cases, the constant rupture stress position was near the centre of the pipe wall. Damage calculation results have shown that in order to obtain clearly defined skeletal point reference rupture stresses, it is marginally preferable
Fig. 5. (a) Creep rupture stress versus radial position at different times, using FE damage modelling for steel 1, with pi 16:55 MPa; (b) Creep rupture stress versus radial position at different times, using FE damage analysis for steel 2, with pi 16:55 MPa:
T.H. Hyde et al. / International Journal of Pressure Vessels and Piping 76 (1999) 925–933
Fig. 6. Steady-state creep rupture stress versus radial position for different n values, with pi 16:55 MPa: and a 0:5:
to use the stress variations through the wall thickness, at different times, in the tertiary stage, i.e. in a relatively narrow time range when t ! tf : Under the steady-state creep conditions, when the material is assumed to obey a power law creep equation of the form 1_ Asn ; a reference stress for deformation exists in the pipe, which is independent of the material constants, A and n, and occurs at a skeletal point [17]. Under closed-end condition, for an internally pressurised thick walled pipe, the steady-state stress solutions are available [18], therefore, the maximum principal stress and equivalent stress can be obtained analytically, as " # 2 2 n Ro 2=n pi s1 r su r 1 1 n r Ro =Ri 2=n 2 1 4a
seq r
p 2=n pi 3 Ro r n Ro =Ri 2=n 2 1
4b
where r is the radial position within the pipe. The steadystate creep rupture stress can be obtained from Eqs. (4) and (3). A skeletal point is shown where the rupture stress is independent of n and this stress is defined as the steady-state
Fig. 8. Effect of a on steady-state reference rupture stresses and skeletal point rupture stresses for steels 1 and 2 with pi 16:55 MPa:
reference rupture stress, sref r : An example is illustrated in Fig. 6 for the pipe with n 2; 4 and 6 and a pressure of 16.55 MPa. It is interesting to note that the steady-state reference rupture stresses are practically the same as the corresponding skeletal point rupture stresses obtained from FE damage analyses as shown in Fig. 7 for three internal pressures. More generally, the effect of a for a pi of 16.55 MPa, on both stress levels is shown in Fig. 8 and it is clear that the agreement is good for a , 0:6 and diverges to an error of around 3% when a 1:0: 3.3. Failure life prediction based on steady-state reference rupture stress The reference stress and the skeletal point stress have been widely used for creep deformation analysis of engineering components [17,19]. The current analyses have shown that there is a point, within the pipe wall, where the representative rupture stress, as defined in Eq. (3), is independent of time throughout the complete creep curve, i.e. including primary, secondary and tertiary creep. In addition, similar results have been shown for the rupture stress derived from the steady-state solutions. These stresses have been used to estimate the pipe life for comparison with the full damage analysis models. Eq. (2b) can be integrated to give the failure life, tf. When m 0; as in the case of the materials studied here, the failure life is given by: tf
Fig. 7. Variations of skeletal point rupture stresses and steady-state reference rupture stresses with internal pressure obtained for steel 1 a 0:3 and steel 2 a 0:26:
929
1 M sr x
5
If a suitable rupture stress can be determined, Eq. (5) can be used to predict the failure time under multi-axial situations. For the thick walled pipe investigated in this work, the rupture lives obtained from Eq. (5) using the steady-state reference rupture stress, or the “skeletal point” rupture stress, and the appropriate material constants, M and x , for the pipe material, are very close to those obtained from the corresponding FE damage calculations, Fig. 9. The effect of a on creep life, keeping all the other constants the same for each material, with a pressure of 16.55 MPa, is
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Fig. 9. Relationship between creep failure lives, obtained from steady-state reference rupture stresses, and internal pressure for steels 1 and 2.
shown in Fig. 10. The failure lives predicted by the two techniques are practically the same with only slight differences when a exceeds 0.6 as implied by Fig. 8. 4. Creep failure life and rupture stress for pipes under internal pressure and additional axial loading The above approach was extended to the examination of combined internal pressure and end load applied to the pipe, using FE creep damage analyses, with an internal pressure of 16.55 MPa and a range of sax =smdh values, where s mdh is the mean diameter hoop stress. The material properties shown in Table 1 were used. The corresponding failure predictions, using the simplified technique, Eq. (5), based on the “skeletal point” rupture stress or steady-state reference rupture stress, were also determined for comparison. The variations of creep rupture stress with radial position Fig. 11. Plots of creep rupture stresses, obtained using FE damage modelling, with radial position at different times for steel 1 with pi 16:55 MPa: (a) sax =smdh 0:5. (b) sax =smdh 0:75. (c) sax =smdh 1:0.
Fig. 10. Variations of creep failure lives, obtained from steady-state reference rupture stresses, with a for steels 1 and 2 with pi 16:55 MPa:
for steel 1, with sax =smdh 0:5; 0.75 and 1.0, obtained from the FE damage calculations, are given in Fig. 11(a)–(c), respectively. The corresponding rupture stresses for steel 2 are shown in Fig. 12(a)–(c). It is clear that positions of constant rupture stress still exist in the presence of additional axial loading. When sax =smdh is around 0.75, there are two such points within the wall thickness of the pipe although the rupture stresses at these two points are practically the same, see Figs. 11(b) and 12(b). Since it was found that the behaviour obtained for steel 3 was similar to that obtained for steels 1 and 2, no detailed results for steel 3 have been presented in the paper; the results are, however, summarised in Table 2.
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As closed-form analytical solutions do not exist for the steady-state reference rupture stresses in pipes with arbitrary axial loads, these have been obtained from FE analyses. Examples of the variations of steady-state rupture stresses with radial position, for different n values, at a 0:3; for sax =smdh 0:5; 0.75 and 1.0, are given in Figs. 13(a)–(c), respectively. It can be seen that similar to the closed-end case, steady-state reference rupture stresses exist and can be determined. However, single crossover points are found at all values of sax =smdh from 0.5 to 1.0. Table 2 summarises the data obtained from the analyses performed with sax =smdh 0:5; 0.75 and 1.0. The variations of the skeletal point rupture stresses, ssp r ; and the steady-state reference rupture stresses, sref r ; with sax =smdh ratio, for steel 1, a 0:3; and steel 2, a 0:26; for a pressure of 16.55 MPa, are shown in Fig. 14. It can be sp seen that the sref r and sr are practically the same, in the range of 0:306 , sax =smdh ,1.0. This can be further illustrated, Fig. 15, from examination of the rupture life estimates obtained from these stresses, Eq. (5) and the material constants, M and x , Table 1. It is clear that the failure lives predicted from the “skeletal point” stresses are practically the same as those obtained from the FE damage calculations for 0:306 , sax =smdh , 1:0; as are the failure lives predicted from steady-state reference rupture stresses. 5. Discussion and conclusions
Fig. 12. Plots of creep rupture stresses, obtained using FE damage modelling, with radial position at different times for steels 2 with pi 16:55 MPa: (a) sax =smdh 0:5. (b) sax =smdh 0:75. (c) sax =smdh 1:0.
FE creep damage analyses were performed for an internally pressurised thick walled pipe under closed-end conditions and with various additional axial loading, as allowed within the design codes. The geometry and loading conditions were typical of the main steam pipe lines in power plants. Materials data for three ferritic steels were used, to illustrate the expected effects. The creep damage analyses provided life estimates based on the time when the damage accumulations approach unity; they incorporate full creep curves and multiaxial stress effects. The representative creep rupture stress, when examined across the pipe wall, showed positions where this stress is nearly constant with time. This has
Table 2 Life estimates for the pressure plus end load models, pi 16:55 MPa; stress in MPa and time in hour Steel
1
2
3
sax =smdh
0.5 0.75 1.0 0.5 0.75 1.0 0.5 0.75 1.0
Damage
Steady-state
rsp =Ro
ssp r
Life FE
Life Eq. (5)
0.843 0.725, 0.914 0.808 0.822 0.786, 0.993 0.822 0.852 0.727, 0.912 0.806
32.10 34.65 40.33 32.2 35.0 40.4 31.63 33.85 39.91
3:43 × 104 2:20 × 104 9:15 × 103 5:97 × 104 4:08 × 104 1:98 × 104 1:39 × 106 1:06 × 106 5:59 × 105
3:42 × 104 2:20 × 104 9:17 × 103 6:0 × 104 4:0 × 104 1:99 × 104 1:40 × 106 1:07 × 106 5:58 × 105
Design
rref =Ro
sref r
Life Eq. (5)
< 0.8
32.12 34.51 40.15 32.32 34.81 40.30 31.60 33.59 39.70
3:40 × 104 2:25 × 104 9:41 × 103 5:89 × 104 4:11 × 104 2:02 × 104 1:41 × 106 1:11 × 106 5:71 × 105
< 0.8 < 0.8
s mdh
Life Eq. (5)
38.065
1:28 × 104
38.065
2:66 × 104
38.065
6:74 × 105
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Fig. 14. Variations of skeletal point rupture stresses and steady-state reference rupture stresses with sax =smdh ; obtained for steel 1 a 0:3 and steel 2 a 0:26 with pi 16:55 MPa:
constant-stress, or a skeletal point, exists within the pipe wall and, indeed, in this form can be shown to be numerically nearly independent of the Norton index, n (Fig. 6). Furthermore, there is excellent agreement between the life estimates, using steady-state solutions, and the full damage solutions for values of a , 0:6: Where a . 0:6; the estimated steady-state stress is greater than that provided from the full damage analysis and gives an over estimate of the creep life. Thus, this approach offers a potential method for the life estimation of a closed-ended, internally pressurised pipe, based on steady-state stress levels, although a knowledge of the materials constants, a , M and x and an assumed value of Norton’s index, n, is also required. This avoids the need
Fig. 13. Plots of steady-state creep rupture stresses versus radial position, for different n values, with a 0:3 and pi 16:55 MPa: (a) sax =smdh 0:5. (b) sax =smdh 0:75. (c) sax =smdh 1:0.
been previously referred to as a skeletal point, [16]. Using this stress to characterise the pipe failure provides good agreement with the life predicted from a full damage analysis. The materials data input used also covered a range of Norton’s index, n, from 4 to 6. There is a rationale in the assumption that stress levels, which are constant or nearly constant with time, will be closely related to the steady-state values under identical loading conditions. For the closed-ended condition, the maximum principal and the von Mises equivalent stresses can be obtained from analytical solutions [18], and the multiaxiality factor, a , is merely a multiplier modifying the contribution of maximum principal and equivalent stresses to the failure process. The steady-state rupture stress results clearly show the presence of a position at which a
Fig. 15. Plot of Creep failure lives against sax =smdh ; obtained from FE damage modelling, skeletal point rupture stress and steady-state reference rupture stress for steels 1 and 2 with pi 16:55 MPa:
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for detailed FE damage calculations, which are time consuming and for which specialised FE damage codes are not widely available. With additional axial loading, similar rupture stress distributions from the FE damage analysis are found to those for an internally pressurised pipe and constant stress points are observed within the pipe wall for 0:306 , sax =smdh , 1: For sax =smdh values of about 0.75, two constant stress points occur although these provide nearly identical stress levels. The steady-state FE analyses produce skeletal points/ constant stress points for the same range of sax =smdh ; but there is no second constant stress point at about sax =smdh 0:75: The failure times predicted from the skeletal point and the steady-state versions are very close to the results of the full creep damage analysis. Thus, again this provides a realistic and efficient method for the life prediction of a pipe subjected to axial loading in the range of 0:306 , sax =smdh , 1:0 and an a of 0.25–0.5. It is interesting to consider the design rules used for such pipes. Using Eq. 1, the mean diameter hoop stress for a typical internal pressure of 16.55 MPa, which is 38.065 MPa, for the pipe geometry used. The calculated and estimated stress levels for all of the materials considered are given in Table 2. For pressurised pipes, the mean diameter hoop stress is shown to be conservative by around 10 % in stress and thus, these suggested methods may be useful for remaining life studies. However, the pressure plus end load models suggest that for sax =smdh . 0:75; the mean diameter hoop stress approach may not be conservative. This is an extreme load situation for which further investigation may be necessary, particularly as it is known from weld failure statistics that certain positions within a pipework system may experience additional axial (system) loading. This paper proposes a simplified procedure that could be used to estimate the life of a plain pipe under pressure and pressure plus additional end loads. The use of steady-state analysis provides a more efficient analysis method than a full damage analysis although additional data, such as the multiaxiality factor, a , are necessary unless bounds can be prescribed for the relevant materials. Acknowledgements The authors wish to acknowledge Nuclear Electric plc, PowerGen plc, National Power plc and the Engineering and Physical Science Research Council, UK, for their financial and technical support of the work.
[2]
[3]
[4]
[5]
[6] [7]
[8]
[9] [10]
[11] [12]
[13]
[14]
[15]
[16]
[17]
[18]
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