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Thermal fatigue damage assessment at mixing tees (elastic-plastic deformation effect on stress and strain fluctuations)
Nuclear Engineering and Design 318 (2017) 202–212
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Thermal fatigue damage assessment at mixing tees (elastic-plastic deformation effect on stress and strain fluctuations) Masayuki Kamaya ⇑, Koji Miyoshi Institute of Nuclear Safety System, Inc., 64 Sata Mihama-cho, Fukui, Japan
h i g h l i g h t s Stress and strain were analyzed for a mixing tee pipe using measured temperatures. Strain ranges obtained by elastic-plastic and elastic analyses were identical. Elastic-plastic analysis was not necessary to derive strain range.
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Article history: Received 2 August 2016 Received in revised form 13 April 2017 Accepted 13 April 2017
Keywords: Thermal stress Elastic-plastic analysis Elastic follow-up Mixing tee Fatigue damage
a b s t r a c t This study was aimed at showing a procedure for calculating ranges of stress and strain fluctuations from a given wall temperature transient to assess the fatigue life at tee junctions where fluids of different temperatures flow in. The elastic and elastic-plastic structural analyses were performed for a mixing tee pipe made of stainless steel. The pipe wall temperature transient obtained by a mock-up test was used for the analyses. Bi-linear stress-strain curves were assumed for the elastic-plastic analysis. Results obtained by the elastic and elastic-plastic analyses were compared to investigate the influence of the plastic deformation on stress and strain fluctuations. It was shown that the stress and strain fluctuations were relatively large near the boundary of the hot spot, where relatively large compressive stress was caused by hot water that came from the branch pipe. The strain range obtained by the elastic-plastic analysis was almost the same as that obtained by the elastic analysis even if plastic strain was significant. It was concluded that, since the fatigue life was correlated not with the stress range but with the strain range, the fatigue life could be predicted using the strain range obtained by the elastic analysis without considering the plastic deformation. It was also pointed out that consideration of the effects of the mean stress and mean strain was not necessary in the fatigue damage assessment. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Fatigue cracks have been found downstream from tee junctions where fluids of different temperatures flow in (Chapuliot et al., 2005; McDevitt et al., 2015). The hot and cold water mixing flow causes the fluid temperature fluctuation, which is referred to as thermal striping, and induces cyclic stress and strain at the pipe wall. In order to assess the possibility of crack initiation for a given pipe and flow conditions, the mixing flow, thermal convection from the fluid to pipe wall, thermal conduction in the pipe wall, stress and strain fluctuation and material strength for cracking have to be identified. Extensive studies have investigated the spatial and temporal temperature fluctuations at the mixing tee using mock-ups (Braillard et al., 2006; ⇑ Corresponding author. E-mail address: [email protected] (M. Kamaya). http://dx.doi.org/10.1016/j.nucengdes.2017.04.022 0029-5493/Ó 2017 Elsevier B.V. All rights reserved.
Fontes et al., 2009; Kamide et al., 2009) and computational fluid dynamics techniques (Coste et al., 2008; Gillis et al., 2013; Howard and Pasutto, 2009; Nakamura et al., 2009; Tanaka et al., 2008). The fluid temperature fluctuation near the pipe wall was measured for various pipe and flow conditions and summarized as the assessment rule issued by the Japan Society of Mechanical Engineers (2004). On the other hand, the magnitude of the stress and strain fluctuation at the pipe wall is not yet understood well, although several attempts have been made to clarify it (Stephan and Curtit, 2005; Gardin et al., 2010; Kamaya and Nakamura, 2011; Utanohara et al., 2016). Since it is not easy to measure the thermal stress and strain, the stress and strain are estimated from the structural analysis using the wall temperature identified by the fluid dynamic calculation or mock-up tests. However, there are mainly two problems to identify the stress and stress fluctuations. The first problem is estimation of heat convection from the fluid to pipe wall.
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Identifying the heat transfer coefficient is not an easy task for unstable mixing flow. Also, much effort is required to analyze the heat convection between the fluid and pipe wall (Utanohara et al., 2016). To solve this problem, in the authors’ previous study (Miyoshi et al., 2016a), the pipe wall temperatures were measured using 148 thermocouples in a mock-up test. The heat convection analysis can be skipped by measuring the pipe wall temperature directly. The used of multiple thermocouples allows the location of the maximum temperature fluctuation to be identified. The second problem is identification of elastic-plastic deformation of the pipe. It has been shown that the stress amplitude was large enough to induce cyclic plastic strain for initiating fatigue cracks in stainless steel (Kamaya and Kawakubo, 2012). In other words, fatigue cracks are not initiated without the cyclic plastic strain. Therefore, the stress and strain fluctuations should be calculated considering the plastic strain. A simple way to consider the effect of the plastic strain is to apply the elastic follow-up models (Kasahara et al., 1995; Lang et al., 2011). The constants for the elastic follow-up models are determined assuming certain loading, boundary and material conditions. The existing elastic follow-up models, however, may not be applicable to the cyclic load at mixing tees. This study was aimed at showing a procedure for identifying ranges of stress and strain fluctuations to assess the fatigue damage for a given pipe wall temperature. The material assumed in this study was stainless steel, which is a major material used for nuclear power plant piping systems. First, the stress and strain fluctuations at a mixing tee were calculated by a linearelastic finite element analysis. The pipe wall temperature transient obtained in the authors’ previous study (Miyoshi et al., 2016a) was used for the analysis in order to avoid the heat convection problem. Characteristics of the stress transient and its distribution were discussed. Second, in order to resolve the problem of identification of the pipe elastic-plastic deformation, the elastic-plastic finite element analysis was performed using the same pipe wall temperature transient. The results obtained by the elastic and elastic-plastic analyses were compared. Finally, the procedure for deriving the strain range for fatigue damage assessment from the elastic calculation was presented. Treatment of the mean stress and mean strain for predicting fatigue life was also discussed.
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2. Analysis procedure 2.1. Pipe wall temperature transient A pipe wall temperature transient at a mixing tee has been obtained using the mock-up test loop shown in Fig. 1 (Miyoshi et al., 2014a). The test loop simulated the mixing of heated water and room temperature water at the test section, for which a detailed figure is shown in Fig. 2. The inner diameter and thickness of the main pipe were respectively 150 mm and 7.6 mm whereas they were 50 mm and 5.3 mm for the branch pipe. Both pipes were made of stainless steel. A total of 148 thermocouples were installed in the main pipe wall at the positions shown in Fig. 2 (Miyoshi et al., 2014b). The range of the measurement area was about 275 mm in the axial direction and 60° in the hoop direction. Only a half side from the axial symmetry line was measured (see A-A0 cross section in Fig. 2). The pipes were covered with heat insulator. In the test, the hot water of 60 °C was supplied from the branch pipe (vertical pipe in Fig. 2) to the main horizontal pipe, in which the cold water of 25 °C flowed. It should be noted that the objective of the mock-up test was not to initiate fatigue cracking but to measure the pipe wall temperature transient and investigate the spatial distribution of the temperature. The wall temperatures were measured at the positions 0.45 mm from the inner pipe surface. Then, the temperature distribution in the thickness direction was obtained by a heat conduction analysis using the measured temperatures. The test duration was 160 s and the results from 60 to 160 s were used in this study because it took about 60 s for the inner temperature fluctuation to penetrate the wall thickness (Miyoshi et al., 2016a). The details of the test conditions, temperature measurements and temperature analysis are described elsewhere (Miyoshi et al., 2016a). Also, detailed measured temperature data are available from Miyoshi et al. (2016b). 2.2. Structural analyses The temperature transient obtained by the mock-up test and the heat conduction analysis were used for structural analyses to calculate the stress and strain. The general purpose finite element analysis code Abaqus version 6.14 together with the finite element model shown in Fig. 3 were employed for the analyses. The model consisted of 81,856 8-node brick elements. The number of
Fig. 1. Test loop simulating mixing flow at tee junction. The loop consisted of three water tanks, two pumps and test section (mixing tee); a detailed configuration of the test section is shown in Fig. 2.
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Fig. 2. Tee junction of the mock-up test loop. The pipe wall temperatures were measured by 148 thermocouples installed at 0.45 mm from the inner surface.
Fig. 3. Finite element mesh used for the structural analyses. Fig. 4. Stress-strain curves assumed in structural analyses.
elements in the thickness direction of the main pipe was 20. The thickness of the element at the inner surface of the pipe was 0.108 mm. Only one half of the tee pipe was generated corresponding to the temperature measurement area. The symmetry displacement boundary conditions were applied for the symmetry plane to the x-direction shown in Fig. 3. The temperature difference in the mock-up test was 35 K and it was not enough to initiate fatigue cracks. Then, in the structural analyses, the enhanced thermal expansion coefficient 1.64 104 (1/K) was applied instead of the original value 1.64 105 (1/K) to simulate temperature difference of 350 K.
Young’s modulus and Poisson’s ratio were set to 198.5 GPa and 0.3, respectively. Fig. 4 shows the stress-strain curves used in the analyses. The linear stress-strain curve was used in the elastic analysis, while bi-linear stress-strain curves were applied for the elastic-plastic analysis. The yield strength Sy was set to 80– 120 MPa for the same strain hardening rate of 10 MPa/%. The yield strength was varied to investigate the effect of degree of yielding and the strain hardening rate was determined so that the bilinear curves yield about 1000 MPa at the strain of 100% (Kamaya, 2014b).
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The wall temperature has been analyzed for a duration of 100 s with the time interval of 0.02. The stress and strain were calculated for the same time interval. The stress and strain were obtained at four integration points of elements along the inner surface. The averaged values for four integration points were quoted as the stress and strain at the inner surface.
3. Analysis results 3.1. Elastic analysis 3.1.1. Stress fluctuation Fig. 5 shows the time averaged hoop and axial stresses during 100 s (hereafter, called mean stress) obtained by the elastic analy-
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sis. The compressive mean stress was observed along the symmetry line, which is the axial line at h = 0. The hot water from the branch line warmed the pipe wall locally and yielded the region of compressive mean stress, which is referred to as the hot spot. The magnitude of the compressive stress at the hot spot was larger for the axial stress than that for the hoop stress. The minimum axial mean stress at the hot spot was about 380 MPa. It should be noted again that a value 10 times larger than the thermal expansion coefficient was applied in the structural analyses to simulate the fluid temperature difference of 350 K, although the actual temperature difference was 35 K in the mock-up test. The maximum range of stress fluctuation during 100 s (hereafter, variation range) is shown in Fig. 6. The variation range became relatively large at the positon of h = 20° from the symmetry line in the hoop direction and about 75 mm in the axial direction
Fig. 5. Averaged stress during 100 s obtained by elastic analysis. ‘s’ indicates the point with the maximum hoop or axial stress range.
Fig. 6. Maximum stress range during 100 s obtained by elastic analysis. ‘s’ indicates the point with the maximum hoop or axial stress range.
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from the center of the branch pipe (z = 0). The point with the maximum variation range is indicated by a circle in the figures. The variation range exhibited a similar distribution for the axial and hoop stresses, although its magnitude was different. The variation range and the mean stress were larger for the axial direction than the hoop direction. This was due to the constraint effect brought about by the pipe geometry (Kamaya, 2014a). The deformation caused by the hot spot was constrained by the ‘‘stripe constraint” as depicted in Fig. 7(a). Since the pipe kept its cylindrical geometry, the stripe shape deformation caused by the hot spot was constrained and enhanced the axial mean stress. The piping geometry also caused the circumferential constraint (Kamaya, 2014a), which is schematically shown in Fig. 7 (b). Partial shrinkage of the hot spot was constrained by the remaining part of the pipe. The positive mean hoop stress that appeared at z < 0 was attributed to the circumferential constraint. From the stress distributions shown in Figs. 5 and 6, the stripe constraint was more pronounced than the circumferential constraint and that made the axial stress larger than the hoop stress.
3.1.2. Hot spot movement Fig. 8 shows the time series stresses at the points (elements) with the maximum stress variation range, which are denoted as Ph and Pz for the hoop and axial stresses, respectively. The frequency analysis showed that there are two significant frequencies at about 4 Hz and 0.1 Hz in the stress fluctuations (Miyoshi et al., 2016a). Then in Fig. 8, the relatively low-frequency fluctuation, which was about 0.1 Hz, was superposed on the high-frequency one. The stresses had a relatively large or small peak about every 10 s. These large fluctuations are known to contribute to the fatigue damage accumulation (Kamaya and Nakamura, 2011). The maximum and minimum axial stresses during 100 s were observed at 86.28 s and 53.76 s, which are denoted as tz(max) and tz(min), respectively. The snap shots of the axial stress distributions at tz (max) and tz(min) are shown in Fig. 9. Fig. 10 shows the temperature
distributions at tz(max) and tz(min). The hot spot was clearly observed in Fig. 10 and the hot spot size was different at t = tz(max) and tz(min). Point Pz was not included in the hot spot when t = tz(max) and it was located at about the boundary of the hot spot when t = tz(min). It was demonstrated by the fluid dynamic simulation that the position of the hot spot moved in the hoop direction with the frequency of 0.1 Hz. It was deduced that the change in the hot spot size in Fig. 10 was caused by the movement of the hot spot. The region of the negative mean stress was caused by the hot spot and the region moved in the hoop direction corresponding to the movement of the hot spot. Thus, the large variation range was attributed to the hot spot movement. Although the mean stress was relatively large at the positon of h = 60° and z = 0– 100 mm in Fig. 9, the stress amplitude (stress fluctuation) was not significant at that position. Thus, the movement of the hot spot played an important role in the fatigue damage accumulation. The mean stress was negative at points Ph and Pz in the hoop and axial directions, respectively. Negative stress suppressed fatigue crack initiation and propagation. Therefore, points Ph and Pz might not be susceptible points for the crack initiation, although the magnitude of the mean stresses might be altered by plastic deformation.
3.2. Elastic-plastic analysis The elastic-plastic structural analysis was conducted assuming the bi-linear stress-strain curve of the yield strength Sy = 100 MPa. Fig. 11 shows the change in hoop and axial stresses at points Ph and Pz together with those obtained by the elastic analysis. Although the elastic analysis was carried out for 100 s, only the first 20 s was analyzed for the elastic-plastic analysis. Both the hoop and axial stresses obtained by the elastic-plastic analysis were larger than those obtained by the elastic analysis. However, the amplitude of the stress fluctuation was similar to that obtained by the elastic analysis. Fig. 12 shows the relationship between the stress
Fig. 7. Schematic drawings representing deformation constraints caused by a hot spot in a straight pipe (Kamaya, 2014a).
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As for the strain, compressive mean strain was observed in the axial direction for the stable condition, whereas the hoop mean strain converged to zero during the cyclic loading. Fig. 13 shows the relationship between the strains obtained by the elastic and elastic-plastic analyses for points Ph and Pz. The change in the strain obtained by the elastic-plastic analysis exhibited almost a linear correlation with that obtained by the elastic analysis. The maximum ranges of strain fluctuation during 20 s (hereafter, strain variation range) are shown in Fig. 14 for elements located at z 50 mm. The strain variation range obtained by the elastic analysis De(e) and that obtained by the elastic-plastic analysis De(ep) were almost identical for all locations.
4. Discussion 4.1. Plastic deformation at hot spot
Fig. 8. Time history of hoop and axial stresses at the point with the maximum stress range.
and strain in the axial and hoop directions obtained during 20 s. The mean stress shifted to zero at the beginning of the analysis.
Although the strain variation range was almost the same for the elastic and elastic-plastic analyses, the mean strain was not the same as shown in Fig. 12. Particularly, relatively large compressive strain was induced in the axial direction in the elastic-plastic analysis. Since the mean stress (elastic mean strain) converged to zero in the elastic-plastic analysis, the compressive axial mean strain was brought about by the plastic strain. In order to investigate the mean strain distribution, the analyses were performed not for the fluctuated temperature but for the static temperature distribution at t = tz(max). The thermal expansion according to the static temperature distribution was calculated based on initial uniform temperature of 0 °C without considering the temperature fluctuation. Fig. 15 shows the axial strain distributions obtained by the analyses. The elastic analysis results were the same for the static and fluctuated temperatures. The elastic-plastic analysis exhibited relatively large compressive mean strain near the hot spot. The relationship between the axial stress and strain obtained for elements at z 50 mm is shown in Fig. 16. The nonlinear correlation between the stress and strain for the elasticplastic analysis emanated from the bi-linear stress-strain curve assumed in the analysis. The magnitude of the compressive strain obtained by the elastic-plastic analysis was larger than that obtained by the elastic analysis.
Fig. 9. Axial stress distribution at the time of the maximum or minimum stress at point Pz.
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Fig. 10. Temperature distribution at the time of the maximum or minimum stress at point Pz.
Fig. 11. Time history of hoop and axial stresses during 20 s obtained by elastic and elastic-plastic analyses at point Pz (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
Fig. 12. Stress-strain relation obtained at point Ph for the hoop direction and Pz for the axial direction (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
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Fig. 13. Relationship between the strain obtained by elastic and elastic-plastic analyses at points of Ph and Pz (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
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The compressive strain was induced by the process schematically shown in Fig. 17. The hot spot was expanded by the relatively high temperature. However, the expansion was constrained by the stripe constraint and the compressive strain was induced in the axial direction. In the elastic-plastic analysis, the plastic deformation was localized in the hot spot and induced the relatively large strain, whereas the relatively large stress (mean stress) was enhanced in the elastic analysis. If the deformation is brought about by secondary loading, which is caused by displacement and constraint, the resultant strain should be the same for the elastic and elastic-plastic analyses. The enhanced compressive strain in the elastic-plastic analysis was brought about by the elastic followup behavior. Even if the stress and strain were caused by the thermal expansion, the stripe constraint enhanced the localized plastic strain. An important conclusion of the present analyses is that the strain variation range obtained by the elastic-plastic analysis was almost the same as that obtained by the elastic analysis regardless of the magnitude of the plastic strain. The elastic follow-up behavior had little influence on the strain variation range.
4.2. Material strength for stress and strain fluctuations
Fig. 14. Relationship between the maximum variation range for elastic and elasticplastic analyses obtained for elements at z 50 mm (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
The fatigue life is predicted using the obtained stress and strain fluctuations. The stress variation range for the elastic analysis was larger than that for the elastic-plastic analysis, while the strain variation range was not affected by the plastic strain. In a previous study, the fatigue life of stainless steel was investigated using prestrained stainless steel specimens (Kamaya and Kawakubo, 2015). Since the prestrained stainless steel specimens exhibited a different stress range for the same applied strain range in the fatigue test, the influence of the stress range on the fatigue life could be investigated. It was demonstrated that the fatigue life for the same strain range was almost identical regardless of the stress range. Namely, the strain range was the primary parameter to predict the fatigue life and the stress range had little influence on the fatigue life. Actually, in the component design of nuclear power plants, the fatigue damage is assessed using the strain range without considering the stress range (Chopra and Shack, 2007). Another important finding obtained by the tests using the prestrained stainless steel specimens (Kamaya and Kawakubo, 2015) was that the
Fig. 15. Axial strain distributions at t = tz(max) obtained by elastic and elastic-plastic analyses assuming static temperature distribution of t = tz(max).
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4.3. Influence of stress-strain curve
Fig. 16. Axial stress-strain relation obtained by the static elastic and elastic-plastic analyses for t = tz(max). Analysis results for elements at z 50 mm are shown.
Since the strain range is the critical parameter for predicting the fatigue life, it is important to investigate how the strain range was affected by the plastic strain. The difference in axial strain variation range obtained by the elastic analysis De(e) and that obtained by the elastic-plastic analysis De(ep) are shown in Fig. 18 for the elements located at z 50 mm. The difference in the strain variation range De(ep)–De(e) seemed to have no correlation with the magnitude of the variation range. No particular difference was found between the hoop strain and axial strain. Although the difference in the strain variation range obtained by the elastic and elastic-plastic analyses was not significant, it was less than 0.007% for Sy = 100 MPa, the difference may depend on the yield strength assumed in the analysis. Then, the maximum value of De(ep)–De(e) for all elements located at z 50 mm, which is denoted as (De(ep)–De(e))max, was investigated for various yield strengths. Fig. 19 shows the change in (De(ep)–De(e))max with the yield strength. The value of (De(ep)–De(e))max increased as the yield strength decreased. Namely, the large plastic deformation resulted in the large (De(ep)–De(e))max. The value of (De(ep)–De(e))max in the axial strain was larger than that in the hoop strain and it was 0.0082% even when Sy = 80 MPa.
Fig. 18. Difference in strain variation range obtained by elastic analysis De(e) and elastic-plastic analysis De(ep) for elements located at z 50 mm (Sy = 100 MPa for elastic–plastic analysis).
Fig. 17. A schematic drawing representing elastic follow-up behavior that occurred by deformation.
fatigue life was not affected by the mean strain (plastic strain). The fatigue life was almost the same for the same strain range regardless of the degree of prestraining. This implies that the fatigue life is not affected by the mean strain induced at the hot spot.
Fig. 19. Change in the maximum value of De(ep)–De(e) for all elements located at z 50 mm with yield strength Sy.
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ences between the elastic and elastic-plastic analyses results were compared. Then, the procedure for assessing the fatigue damage using the analysis results was discussed. The following conclusions were obtained.
Fig. 20. Correlation between the hoop and axial strains obtained by elastic analysis at point Pz.
(1) The stress and strain fluctuations were relatively large near the boundary of the hot spot. The movement of the hot spot caused the temperature fluctuation and induced the relatively large stress and strain variation ranges. (2) The stripe constraint played an important role for the relatively large axial stress and strain. Although the constraint induced significant plastic strain, it had little influence on the strain variation range. (3) The strain range obtained by the elastic-plastic analysis was almost the same as that obtained by the elastic analysis even if the plastic strain was significant. (4) The fatigue damage assessment procedure for mixing tees was addressed. The fatigue life could be predicted using the strain range obtained by the elastic analysis. It was unnecessary to consider the effect of the mean stress and mean strain (plastic strain).
4.4. Fatigue damage assessment procedure Based on the above discussion, it can be concluded that the elastic-plastic analysis does not have to be performed to obtain the strain range for given temperature fluctuations. The strain range derived by elastic analyses can be used for the fatigue damage assessment without any correction. It is not necessary to identify the degree of plastic strain because the plastic strain has little influence on the fatigue strength of stainless steel. The remaining issue in the fatigue damage assessment is treatment of the mean stress. The compressive mean stress was observed in the elastic analysis at points Pz and Ph. The negative mean stress extends the fatigue life under the same strain range, whereas the positive mean stress reduces the fatigue life (Vincent et al., 2012). It was shown that the influence of the mean stress on the fatigue life was relatively small for stainless steels (Kamaya and Kawakubo, 2015). Furthermore, the compressive mean stress was mitigated in the elastic-plastic analysis and converged to zero. In any case, the reduction in fatigue life due to the mean stress has been considered in the design fatigue curve, which is used for component designs (Chopra and Shack, 2007). Therefore, for a conservative fatigue damage assessment, the influence of the mean stress does not have to be taken into account when the design fatigue curve is used for the fatigue damage assessment. For the fatigue damage assessment, it is also required to consider the multi axial loading condition. The fatigue tests for predicting the fatigue life are usually carried out for uni-axial loading conditions. However, the stress and strain caused at the mixing tee was not uni-axial. Fig. 20 shows the relationship between the hoop and axial strains at point Pz obtained by the elastic analysis. This figure shows that the strain fluctuation was almost proportional and equi-biaxial. Fatigue tests for the equibiaxial proportional loading have been conducted for stainless steels (Tsuji and Kamaya, 2014; Kamaya, 2016). These reported test results showed that the fatigue life for bi-axial loading could be predicted by quoting the equivalent strain or principal strain. 5. Conclusions The elastic and elastic-plastic structural analyses were performed using the temperature transient obtained by the mock-up test for the mixing tee pipe made of stainless steel. Characteristics of the stress and strain changes were investigated. Also, the differ-
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Thermal fatigue damage assessment at mixing tees (elastic-plastic deformation effect on stress and strain fluctuations) Masayuki Kamaya ⇑, Koji Miyoshi Institute of Nuclear Safety System, Inc., 64 Sata Mihama-cho, Fukui, Japan
h i g h l i g h t s Stress and strain were analyzed for a mixing tee pipe using measured temperatures. Strain ranges obtained by elastic-plastic and elastic analyses were identical. Elastic-plastic analysis was not necessary to derive strain range.
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Article history: Received 2 August 2016 Received in revised form 13 April 2017 Accepted 13 April 2017
Keywords: Thermal stress Elastic-plastic analysis Elastic follow-up Mixing tee Fatigue damage
a b s t r a c t This study was aimed at showing a procedure for calculating ranges of stress and strain fluctuations from a given wall temperature transient to assess the fatigue life at tee junctions where fluids of different temperatures flow in. The elastic and elastic-plastic structural analyses were performed for a mixing tee pipe made of stainless steel. The pipe wall temperature transient obtained by a mock-up test was used for the analyses. Bi-linear stress-strain curves were assumed for the elastic-plastic analysis. Results obtained by the elastic and elastic-plastic analyses were compared to investigate the influence of the plastic deformation on stress and strain fluctuations. It was shown that the stress and strain fluctuations were relatively large near the boundary of the hot spot, where relatively large compressive stress was caused by hot water that came from the branch pipe. The strain range obtained by the elastic-plastic analysis was almost the same as that obtained by the elastic analysis even if plastic strain was significant. It was concluded that, since the fatigue life was correlated not with the stress range but with the strain range, the fatigue life could be predicted using the strain range obtained by the elastic analysis without considering the plastic deformation. It was also pointed out that consideration of the effects of the mean stress and mean strain was not necessary in the fatigue damage assessment. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Fatigue cracks have been found downstream from tee junctions where fluids of different temperatures flow in (Chapuliot et al., 2005; McDevitt et al., 2015). The hot and cold water mixing flow causes the fluid temperature fluctuation, which is referred to as thermal striping, and induces cyclic stress and strain at the pipe wall. In order to assess the possibility of crack initiation for a given pipe and flow conditions, the mixing flow, thermal convection from the fluid to pipe wall, thermal conduction in the pipe wall, stress and strain fluctuation and material strength for cracking have to be identified. Extensive studies have investigated the spatial and temporal temperature fluctuations at the mixing tee using mock-ups (Braillard et al., 2006; ⇑ Corresponding author. E-mail address: [email protected] (M. Kamaya). http://dx.doi.org/10.1016/j.nucengdes.2017.04.022 0029-5493/Ó 2017 Elsevier B.V. All rights reserved.
Fontes et al., 2009; Kamide et al., 2009) and computational fluid dynamics techniques (Coste et al., 2008; Gillis et al., 2013; Howard and Pasutto, 2009; Nakamura et al., 2009; Tanaka et al., 2008). The fluid temperature fluctuation near the pipe wall was measured for various pipe and flow conditions and summarized as the assessment rule issued by the Japan Society of Mechanical Engineers (2004). On the other hand, the magnitude of the stress and strain fluctuation at the pipe wall is not yet understood well, although several attempts have been made to clarify it (Stephan and Curtit, 2005; Gardin et al., 2010; Kamaya and Nakamura, 2011; Utanohara et al., 2016). Since it is not easy to measure the thermal stress and strain, the stress and strain are estimated from the structural analysis using the wall temperature identified by the fluid dynamic calculation or mock-up tests. However, there are mainly two problems to identify the stress and stress fluctuations. The first problem is estimation of heat convection from the fluid to pipe wall.
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Identifying the heat transfer coefficient is not an easy task for unstable mixing flow. Also, much effort is required to analyze the heat convection between the fluid and pipe wall (Utanohara et al., 2016). To solve this problem, in the authors’ previous study (Miyoshi et al., 2016a), the pipe wall temperatures were measured using 148 thermocouples in a mock-up test. The heat convection analysis can be skipped by measuring the pipe wall temperature directly. The used of multiple thermocouples allows the location of the maximum temperature fluctuation to be identified. The second problem is identification of elastic-plastic deformation of the pipe. It has been shown that the stress amplitude was large enough to induce cyclic plastic strain for initiating fatigue cracks in stainless steel (Kamaya and Kawakubo, 2012). In other words, fatigue cracks are not initiated without the cyclic plastic strain. Therefore, the stress and strain fluctuations should be calculated considering the plastic strain. A simple way to consider the effect of the plastic strain is to apply the elastic follow-up models (Kasahara et al., 1995; Lang et al., 2011). The constants for the elastic follow-up models are determined assuming certain loading, boundary and material conditions. The existing elastic follow-up models, however, may not be applicable to the cyclic load at mixing tees. This study was aimed at showing a procedure for identifying ranges of stress and strain fluctuations to assess the fatigue damage for a given pipe wall temperature. The material assumed in this study was stainless steel, which is a major material used for nuclear power plant piping systems. First, the stress and strain fluctuations at a mixing tee were calculated by a linearelastic finite element analysis. The pipe wall temperature transient obtained in the authors’ previous study (Miyoshi et al., 2016a) was used for the analysis in order to avoid the heat convection problem. Characteristics of the stress transient and its distribution were discussed. Second, in order to resolve the problem of identification of the pipe elastic-plastic deformation, the elastic-plastic finite element analysis was performed using the same pipe wall temperature transient. The results obtained by the elastic and elastic-plastic analyses were compared. Finally, the procedure for deriving the strain range for fatigue damage assessment from the elastic calculation was presented. Treatment of the mean stress and mean strain for predicting fatigue life was also discussed.
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2. Analysis procedure 2.1. Pipe wall temperature transient A pipe wall temperature transient at a mixing tee has been obtained using the mock-up test loop shown in Fig. 1 (Miyoshi et al., 2014a). The test loop simulated the mixing of heated water and room temperature water at the test section, for which a detailed figure is shown in Fig. 2. The inner diameter and thickness of the main pipe were respectively 150 mm and 7.6 mm whereas they were 50 mm and 5.3 mm for the branch pipe. Both pipes were made of stainless steel. A total of 148 thermocouples were installed in the main pipe wall at the positions shown in Fig. 2 (Miyoshi et al., 2014b). The range of the measurement area was about 275 mm in the axial direction and 60° in the hoop direction. Only a half side from the axial symmetry line was measured (see A-A0 cross section in Fig. 2). The pipes were covered with heat insulator. In the test, the hot water of 60 °C was supplied from the branch pipe (vertical pipe in Fig. 2) to the main horizontal pipe, in which the cold water of 25 °C flowed. It should be noted that the objective of the mock-up test was not to initiate fatigue cracking but to measure the pipe wall temperature transient and investigate the spatial distribution of the temperature. The wall temperatures were measured at the positions 0.45 mm from the inner pipe surface. Then, the temperature distribution in the thickness direction was obtained by a heat conduction analysis using the measured temperatures. The test duration was 160 s and the results from 60 to 160 s were used in this study because it took about 60 s for the inner temperature fluctuation to penetrate the wall thickness (Miyoshi et al., 2016a). The details of the test conditions, temperature measurements and temperature analysis are described elsewhere (Miyoshi et al., 2016a). Also, detailed measured temperature data are available from Miyoshi et al. (2016b). 2.2. Structural analyses The temperature transient obtained by the mock-up test and the heat conduction analysis were used for structural analyses to calculate the stress and strain. The general purpose finite element analysis code Abaqus version 6.14 together with the finite element model shown in Fig. 3 were employed for the analyses. The model consisted of 81,856 8-node brick elements. The number of
Fig. 1. Test loop simulating mixing flow at tee junction. The loop consisted of three water tanks, two pumps and test section (mixing tee); a detailed configuration of the test section is shown in Fig. 2.
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Fig. 2. Tee junction of the mock-up test loop. The pipe wall temperatures were measured by 148 thermocouples installed at 0.45 mm from the inner surface.
Fig. 3. Finite element mesh used for the structural analyses. Fig. 4. Stress-strain curves assumed in structural analyses.
elements in the thickness direction of the main pipe was 20. The thickness of the element at the inner surface of the pipe was 0.108 mm. Only one half of the tee pipe was generated corresponding to the temperature measurement area. The symmetry displacement boundary conditions were applied for the symmetry plane to the x-direction shown in Fig. 3. The temperature difference in the mock-up test was 35 K and it was not enough to initiate fatigue cracks. Then, in the structural analyses, the enhanced thermal expansion coefficient 1.64 104 (1/K) was applied instead of the original value 1.64 105 (1/K) to simulate temperature difference of 350 K.
Young’s modulus and Poisson’s ratio were set to 198.5 GPa and 0.3, respectively. Fig. 4 shows the stress-strain curves used in the analyses. The linear stress-strain curve was used in the elastic analysis, while bi-linear stress-strain curves were applied for the elastic-plastic analysis. The yield strength Sy was set to 80– 120 MPa for the same strain hardening rate of 10 MPa/%. The yield strength was varied to investigate the effect of degree of yielding and the strain hardening rate was determined so that the bilinear curves yield about 1000 MPa at the strain of 100% (Kamaya, 2014b).
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The wall temperature has been analyzed for a duration of 100 s with the time interval of 0.02. The stress and strain were calculated for the same time interval. The stress and strain were obtained at four integration points of elements along the inner surface. The averaged values for four integration points were quoted as the stress and strain at the inner surface.
3. Analysis results 3.1. Elastic analysis 3.1.1. Stress fluctuation Fig. 5 shows the time averaged hoop and axial stresses during 100 s (hereafter, called mean stress) obtained by the elastic analy-
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sis. The compressive mean stress was observed along the symmetry line, which is the axial line at h = 0. The hot water from the branch line warmed the pipe wall locally and yielded the region of compressive mean stress, which is referred to as the hot spot. The magnitude of the compressive stress at the hot spot was larger for the axial stress than that for the hoop stress. The minimum axial mean stress at the hot spot was about 380 MPa. It should be noted again that a value 10 times larger than the thermal expansion coefficient was applied in the structural analyses to simulate the fluid temperature difference of 350 K, although the actual temperature difference was 35 K in the mock-up test. The maximum range of stress fluctuation during 100 s (hereafter, variation range) is shown in Fig. 6. The variation range became relatively large at the positon of h = 20° from the symmetry line in the hoop direction and about 75 mm in the axial direction
Fig. 5. Averaged stress during 100 s obtained by elastic analysis. ‘s’ indicates the point with the maximum hoop or axial stress range.
Fig. 6. Maximum stress range during 100 s obtained by elastic analysis. ‘s’ indicates the point with the maximum hoop or axial stress range.
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from the center of the branch pipe (z = 0). The point with the maximum variation range is indicated by a circle in the figures. The variation range exhibited a similar distribution for the axial and hoop stresses, although its magnitude was different. The variation range and the mean stress were larger for the axial direction than the hoop direction. This was due to the constraint effect brought about by the pipe geometry (Kamaya, 2014a). The deformation caused by the hot spot was constrained by the ‘‘stripe constraint” as depicted in Fig. 7(a). Since the pipe kept its cylindrical geometry, the stripe shape deformation caused by the hot spot was constrained and enhanced the axial mean stress. The piping geometry also caused the circumferential constraint (Kamaya, 2014a), which is schematically shown in Fig. 7 (b). Partial shrinkage of the hot spot was constrained by the remaining part of the pipe. The positive mean hoop stress that appeared at z < 0 was attributed to the circumferential constraint. From the stress distributions shown in Figs. 5 and 6, the stripe constraint was more pronounced than the circumferential constraint and that made the axial stress larger than the hoop stress.
3.1.2. Hot spot movement Fig. 8 shows the time series stresses at the points (elements) with the maximum stress variation range, which are denoted as Ph and Pz for the hoop and axial stresses, respectively. The frequency analysis showed that there are two significant frequencies at about 4 Hz and 0.1 Hz in the stress fluctuations (Miyoshi et al., 2016a). Then in Fig. 8, the relatively low-frequency fluctuation, which was about 0.1 Hz, was superposed on the high-frequency one. The stresses had a relatively large or small peak about every 10 s. These large fluctuations are known to contribute to the fatigue damage accumulation (Kamaya and Nakamura, 2011). The maximum and minimum axial stresses during 100 s were observed at 86.28 s and 53.76 s, which are denoted as tz(max) and tz(min), respectively. The snap shots of the axial stress distributions at tz (max) and tz(min) are shown in Fig. 9. Fig. 10 shows the temperature
distributions at tz(max) and tz(min). The hot spot was clearly observed in Fig. 10 and the hot spot size was different at t = tz(max) and tz(min). Point Pz was not included in the hot spot when t = tz(max) and it was located at about the boundary of the hot spot when t = tz(min). It was demonstrated by the fluid dynamic simulation that the position of the hot spot moved in the hoop direction with the frequency of 0.1 Hz. It was deduced that the change in the hot spot size in Fig. 10 was caused by the movement of the hot spot. The region of the negative mean stress was caused by the hot spot and the region moved in the hoop direction corresponding to the movement of the hot spot. Thus, the large variation range was attributed to the hot spot movement. Although the mean stress was relatively large at the positon of h = 60° and z = 0– 100 mm in Fig. 9, the stress amplitude (stress fluctuation) was not significant at that position. Thus, the movement of the hot spot played an important role in the fatigue damage accumulation. The mean stress was negative at points Ph and Pz in the hoop and axial directions, respectively. Negative stress suppressed fatigue crack initiation and propagation. Therefore, points Ph and Pz might not be susceptible points for the crack initiation, although the magnitude of the mean stresses might be altered by plastic deformation.
3.2. Elastic-plastic analysis The elastic-plastic structural analysis was conducted assuming the bi-linear stress-strain curve of the yield strength Sy = 100 MPa. Fig. 11 shows the change in hoop and axial stresses at points Ph and Pz together with those obtained by the elastic analysis. Although the elastic analysis was carried out for 100 s, only the first 20 s was analyzed for the elastic-plastic analysis. Both the hoop and axial stresses obtained by the elastic-plastic analysis were larger than those obtained by the elastic analysis. However, the amplitude of the stress fluctuation was similar to that obtained by the elastic analysis. Fig. 12 shows the relationship between the stress
Fig. 7. Schematic drawings representing deformation constraints caused by a hot spot in a straight pipe (Kamaya, 2014a).
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As for the strain, compressive mean strain was observed in the axial direction for the stable condition, whereas the hoop mean strain converged to zero during the cyclic loading. Fig. 13 shows the relationship between the strains obtained by the elastic and elastic-plastic analyses for points Ph and Pz. The change in the strain obtained by the elastic-plastic analysis exhibited almost a linear correlation with that obtained by the elastic analysis. The maximum ranges of strain fluctuation during 20 s (hereafter, strain variation range) are shown in Fig. 14 for elements located at z 50 mm. The strain variation range obtained by the elastic analysis De(e) and that obtained by the elastic-plastic analysis De(ep) were almost identical for all locations.
4. Discussion 4.1. Plastic deformation at hot spot
Fig. 8. Time history of hoop and axial stresses at the point with the maximum stress range.
and strain in the axial and hoop directions obtained during 20 s. The mean stress shifted to zero at the beginning of the analysis.
Although the strain variation range was almost the same for the elastic and elastic-plastic analyses, the mean strain was not the same as shown in Fig. 12. Particularly, relatively large compressive strain was induced in the axial direction in the elastic-plastic analysis. Since the mean stress (elastic mean strain) converged to zero in the elastic-plastic analysis, the compressive axial mean strain was brought about by the plastic strain. In order to investigate the mean strain distribution, the analyses were performed not for the fluctuated temperature but for the static temperature distribution at t = tz(max). The thermal expansion according to the static temperature distribution was calculated based on initial uniform temperature of 0 °C without considering the temperature fluctuation. Fig. 15 shows the axial strain distributions obtained by the analyses. The elastic analysis results were the same for the static and fluctuated temperatures. The elastic-plastic analysis exhibited relatively large compressive mean strain near the hot spot. The relationship between the axial stress and strain obtained for elements at z 50 mm is shown in Fig. 16. The nonlinear correlation between the stress and strain for the elasticplastic analysis emanated from the bi-linear stress-strain curve assumed in the analysis. The magnitude of the compressive strain obtained by the elastic-plastic analysis was larger than that obtained by the elastic analysis.
Fig. 9. Axial stress distribution at the time of the maximum or minimum stress at point Pz.
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Fig. 10. Temperature distribution at the time of the maximum or minimum stress at point Pz.
Fig. 11. Time history of hoop and axial stresses during 20 s obtained by elastic and elastic-plastic analyses at point Pz (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
Fig. 12. Stress-strain relation obtained at point Ph for the hoop direction and Pz for the axial direction (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
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Fig. 13. Relationship between the strain obtained by elastic and elastic-plastic analyses at points of Ph and Pz (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
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The compressive strain was induced by the process schematically shown in Fig. 17. The hot spot was expanded by the relatively high temperature. However, the expansion was constrained by the stripe constraint and the compressive strain was induced in the axial direction. In the elastic-plastic analysis, the plastic deformation was localized in the hot spot and induced the relatively large strain, whereas the relatively large stress (mean stress) was enhanced in the elastic analysis. If the deformation is brought about by secondary loading, which is caused by displacement and constraint, the resultant strain should be the same for the elastic and elastic-plastic analyses. The enhanced compressive strain in the elastic-plastic analysis was brought about by the elastic followup behavior. Even if the stress and strain were caused by the thermal expansion, the stripe constraint enhanced the localized plastic strain. An important conclusion of the present analyses is that the strain variation range obtained by the elastic-plastic analysis was almost the same as that obtained by the elastic analysis regardless of the magnitude of the plastic strain. The elastic follow-up behavior had little influence on the strain variation range.
4.2. Material strength for stress and strain fluctuations
Fig. 14. Relationship between the maximum variation range for elastic and elasticplastic analyses obtained for elements at z 50 mm (t = 0–20 s, Sy = 100 MPa for elastic–plastic analysis).
The fatigue life is predicted using the obtained stress and strain fluctuations. The stress variation range for the elastic analysis was larger than that for the elastic-plastic analysis, while the strain variation range was not affected by the plastic strain. In a previous study, the fatigue life of stainless steel was investigated using prestrained stainless steel specimens (Kamaya and Kawakubo, 2015). Since the prestrained stainless steel specimens exhibited a different stress range for the same applied strain range in the fatigue test, the influence of the stress range on the fatigue life could be investigated. It was demonstrated that the fatigue life for the same strain range was almost identical regardless of the stress range. Namely, the strain range was the primary parameter to predict the fatigue life and the stress range had little influence on the fatigue life. Actually, in the component design of nuclear power plants, the fatigue damage is assessed using the strain range without considering the stress range (Chopra and Shack, 2007). Another important finding obtained by the tests using the prestrained stainless steel specimens (Kamaya and Kawakubo, 2015) was that the
Fig. 15. Axial strain distributions at t = tz(max) obtained by elastic and elastic-plastic analyses assuming static temperature distribution of t = tz(max).
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4.3. Influence of stress-strain curve
Fig. 16. Axial stress-strain relation obtained by the static elastic and elastic-plastic analyses for t = tz(max). Analysis results for elements at z 50 mm are shown.
Since the strain range is the critical parameter for predicting the fatigue life, it is important to investigate how the strain range was affected by the plastic strain. The difference in axial strain variation range obtained by the elastic analysis De(e) and that obtained by the elastic-plastic analysis De(ep) are shown in Fig. 18 for the elements located at z 50 mm. The difference in the strain variation range De(ep)–De(e) seemed to have no correlation with the magnitude of the variation range. No particular difference was found between the hoop strain and axial strain. Although the difference in the strain variation range obtained by the elastic and elastic-plastic analyses was not significant, it was less than 0.007% for Sy = 100 MPa, the difference may depend on the yield strength assumed in the analysis. Then, the maximum value of De(ep)–De(e) for all elements located at z 50 mm, which is denoted as (De(ep)–De(e))max, was investigated for various yield strengths. Fig. 19 shows the change in (De(ep)–De(e))max with the yield strength. The value of (De(ep)–De(e))max increased as the yield strength decreased. Namely, the large plastic deformation resulted in the large (De(ep)–De(e))max. The value of (De(ep)–De(e))max in the axial strain was larger than that in the hoop strain and it was 0.0082% even when Sy = 80 MPa.
Fig. 18. Difference in strain variation range obtained by elastic analysis De(e) and elastic-plastic analysis De(ep) for elements located at z 50 mm (Sy = 100 MPa for elastic–plastic analysis).
Fig. 17. A schematic drawing representing elastic follow-up behavior that occurred by deformation.
fatigue life was not affected by the mean strain (plastic strain). The fatigue life was almost the same for the same strain range regardless of the degree of prestraining. This implies that the fatigue life is not affected by the mean strain induced at the hot spot.
Fig. 19. Change in the maximum value of De(ep)–De(e) for all elements located at z 50 mm with yield strength Sy.
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ences between the elastic and elastic-plastic analyses results were compared. Then, the procedure for assessing the fatigue damage using the analysis results was discussed. The following conclusions were obtained.
Fig. 20. Correlation between the hoop and axial strains obtained by elastic analysis at point Pz.
(1) The stress and strain fluctuations were relatively large near the boundary of the hot spot. The movement of the hot spot caused the temperature fluctuation and induced the relatively large stress and strain variation ranges. (2) The stripe constraint played an important role for the relatively large axial stress and strain. Although the constraint induced significant plastic strain, it had little influence on the strain variation range. (3) The strain range obtained by the elastic-plastic analysis was almost the same as that obtained by the elastic analysis even if the plastic strain was significant. (4) The fatigue damage assessment procedure for mixing tees was addressed. The fatigue life could be predicted using the strain range obtained by the elastic analysis. It was unnecessary to consider the effect of the mean stress and mean strain (plastic strain).
4.4. Fatigue damage assessment procedure Based on the above discussion, it can be concluded that the elastic-plastic analysis does not have to be performed to obtain the strain range for given temperature fluctuations. The strain range derived by elastic analyses can be used for the fatigue damage assessment without any correction. It is not necessary to identify the degree of plastic strain because the plastic strain has little influence on the fatigue strength of stainless steel. The remaining issue in the fatigue damage assessment is treatment of the mean stress. The compressive mean stress was observed in the elastic analysis at points Pz and Ph. The negative mean stress extends the fatigue life under the same strain range, whereas the positive mean stress reduces the fatigue life (Vincent et al., 2012). It was shown that the influence of the mean stress on the fatigue life was relatively small for stainless steels (Kamaya and Kawakubo, 2015). Furthermore, the compressive mean stress was mitigated in the elastic-plastic analysis and converged to zero. In any case, the reduction in fatigue life due to the mean stress has been considered in the design fatigue curve, which is used for component designs (Chopra and Shack, 2007). Therefore, for a conservative fatigue damage assessment, the influence of the mean stress does not have to be taken into account when the design fatigue curve is used for the fatigue damage assessment. For the fatigue damage assessment, it is also required to consider the multi axial loading condition. The fatigue tests for predicting the fatigue life are usually carried out for uni-axial loading conditions. However, the stress and strain caused at the mixing tee was not uni-axial. Fig. 20 shows the relationship between the hoop and axial strains at point Pz obtained by the elastic analysis. This figure shows that the strain fluctuation was almost proportional and equi-biaxial. Fatigue tests for the equibiaxial proportional loading have been conducted for stainless steels (Tsuji and Kamaya, 2014; Kamaya, 2016). These reported test results showed that the fatigue life for bi-axial loading could be predicted by quoting the equivalent strain or principal strain. 5. Conclusions The elastic and elastic-plastic structural analyses were performed using the temperature transient obtained by the mock-up test for the mixing tee pipe made of stainless steel. Characteristics of the stress and strain changes were investigated. Also, the differ-
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