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Ratcheting behavior of pressurized elbow pipe with local wall thinning
International Journal of Pressure Vessels and Piping 102-103 (2013) 14e23
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Ratcheting behavior of pressurized elbow pipe with local wall thinning Hongrui Shi, Gang Chen, Yong Wang, Xu Chen* School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 July 2012 Received in revised form 9 December 2012 Accepted 19 December 2012
Ratcheting deformation is studied on elbow pipe made of Z2CND18.12N stainless steel with local wall thinning subjected to constant internal pressure and reversed in-plane bending under load control. The local wall thinning is located at extrados and crown of elbow. It is shown that local wall thinning has a very pronounced effect on the ratcheting behavior of elbows, as compared to the ratcheting behavior of sound elbow specimen. Three-dimensional elastic-plastic analyses with ANSYS in which Chaboche and CheneJiaoeKim (CJK) kinematic hardening model are carried out to evaluate structural ratcheting behaviors. A reasonable agreement has been found between the experimental and the simulated results with CJK model for the ratcheting of the elbows. The ratcheting boundary is determined by evaluating variations in the plastic strain increment with CJK model. The effects of the depth and location of local wall thinning on the ratcheting response are discussed by CJK model. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Ratcheting Elbow piping Cyclic loading Thinning wall Kinematic hardening
1. Introduction The piping fittings employed in nuclear power plants and chemical industries play a vital role in safe operation. If the pipelines subjected to internal pressure and cyclic loading, they may face progressive deformation, so-called ratcheting. For example, this can be generated by fluid pressure and temperature changes or other cyclic environmental conditions such as seismic loading [1,2]. Severe ratcheting may occur and lead to premature failure due to either accumulation of deformation [3,4] or fatigue cracks [5]. These results indicate that ratcheting may occur much earlier than the estimated life, and may result in unscheduled plant downtime. Thus, during the design of pressurized piping in nuclear power plants, ratcheting and ratcheting fatigue must be taken into consideration. Due to geometry of the elbow structure, the stress distributions in the elbow pipes are much more complex [6]. Meanwhile the material lose is enhanced at elbow components which are widely used in nuclear power plants and chemical industries. Local wall thinning due to erosion of external soil and rain water, mechanical damage and crack repair can enhance ratcheting deformation. Therefore, it is important to evaluate the ratcheting deformation of elbow pipe undergoing local wall thinning and determine the
* Corresponding author. Tel.: þ86 22 27408399; fax: þ86 22 27403389. E-mail address: [email protected] (X. Chen). 0308-0161/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2012.12.002
ratcheting boundary in order to maintain the integrity of the piping systems. The ratcheting effect has been considered in several standards, such as ASME Code Section III [7], KTA [8] and RCC-MR [9]. The aforementioned ratcheting behaviors of straight and elbow pipes have already been extensively studied in the literature [10e14]. A series of elbow experiments by applying dynamic bending were conducted in order to evaluate the ratcheting responses of the elbow components [15e18]. Moreover, ratcheting strains can also be analyzed by Elastic-Plastic Finite Element Analysis (EPFEA) [4,19,20]. The Committee of Three Dimensional Finite Element Stress Evaluation (C-TDF) in Japan proposed two alternative criteria to verify shakedown, in which an elastic perfect plasticity (EPP) model or the bilinear kinematic hardening (BKH) rule was used [21]. The ratcheting of pressurized straight and elbow pipe of low carbon steel under reversed bending was experimentally and analytically investigated by Chen et al. [19] and Gao et al. [20]. The study has shown that CheneJiaoeKim (CJK) model can reasonably simulate ratcheting strain. Degrassi et al. [22] analyzed the ratcheting responses of the piping system under seismic excitation using the bilinear, multilinear and Chaboche model in ANSYS. The authors demonstrated Chaboche model performed best in strain ratcheting simulation for elbow through adjusting ratcheting parameters based on pipe ratcheting response. The failure behavior of pipelines with wall thinning has also received considerable previous attention in the literatureliteratures. Nakamura et al. [5] experimentally examined pressurized
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Table 1 Chemical compositions of Z2CND18.12N. Chemical composition
C
Si
Mn
P
S
Ni
Cr
Mo
B
Cu
Co
N
%
0.025
0.430
1.211
0.021
0.003
12.073
17.517
2.388
0.001
0.075
0.035
0.070
Table 2 Specification of elbows. Outside diameter D0, mm
Nominal thickness t, mm
Bending radius R, mm
Bending 2 characteristic tR=rm
76
4.5
95
0.33
elbows with local wall thinning under displacement-controlled cyclic bending. It was found that the failure modes and fatigue lives of elbows seemed to be affected by a ratchet phenomenon. Miyazaki et al. [23] examined carbon steel pipes with local wall thinning under cyclic pure bending loads to evaluate their low cycle fatigue strength. In load controlled tests on these pipes, ratcheting deformation was observed, and the fatigue strength became lower than that of cracked pipes. The effect of bi-directional loading on the fatigue characteristics of pressurized 90 piping elbows with local wall thinning was investigated by Balan and Redekop [24]. The results provided extensive new information about the fatigue behavior of piping elbow subject to seismic action. Zeinoddini and Peykanu [4] studied the strain ratcheting of steel tubes with a rectangular defect under axial cycling. It was shown that the surface imperfections had a very significant effect on the ratcheting response of the defected tubes. The effects of some factors such as the stress amplitude, wall thinning and the material hardening properties on the ratcheting response of steel tubes were also investigated. The subject of ratcheting of defected elbow pipes under combined internal pressure and cyclic bending, however, has not received due attention previously. In this study, in order to
investigate the ratcheting behaviors of elbow pipes undergoing local wall thinning, ratcheting tests have been carried out using Z2CND18.12N stainless steel elbow specimens with local wall thinning. The local wall thinning areas are located at extrados part and crown part. The pressuried elbow specimens are subjected to constant internal pressure and reversed in-plane bending under load control. Then, ratcheting simulation is performed by elasticplastic finite element analyses with ANSYS in which Chaboche model and modified OhnoeWang model (CJK model) are applied. The analytical results are compared with the experimental data in order to identify the better model for structural ratcheting simulation. Ratcheting boundary is determined by evaluating variations in the plastic strain increment with CJK model. 2. Experiments 2.1. Material and specification of elbow specimens The elbow specimens were machined from Z2CND18.12N austenitic stainless steel. The chemical compositions of the material are listed in Table 1. Yield strength of the material is 388 MPa, ultimate strength is 590 MPa and Young’s modulus is 195 GPa at room temperature. The elbow samples satisfy Chinese code GB12459-90, and their dimensions are listed in Table 2. Fig. 1(a) shows the shape and geometry of elbow specimens. Fig. 1(b) and (c) shows the position and dimensions of local wall thinning which is equivalent to the defect caused by the metal loss in engineering. In this study, in order to simulate metal loss due to
Fig. 1. Shape and geometries of elbow pipe specimens: (a) Elbow specimen; (b) Elbow with local wall thinning at extrados; (c) Elbow with local wall thinning at crown.
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corrosion or mechanical damage, the local wall thinning was machined on the outside of the center of elbows which were located at two different positions, called extrados and crown, and the wall thinning at the crown was on the one side of the elbow. The sound elbow specimen without wall thinning was also used. The three types of elbow specimens were called as extrados thinning elbow, crown thinning elbow and sound elbow, respectively. The depth of local wall thinning in the thickness direction was called eroded depth (d) and was fixed at d ¼ 1 mm. The wall thinning ratio was defined as the ratio of d to the wall thickness t (d/t) and was fixed at d=tz0:22. 2.2. Experimental apparatus and testing system Experimental setup is shown in Fig. 2. Loading bars clipped the clip-head of the multiaxial test machine. Cyclic pullepush force opened and closed the elbow and made the specimen be under reversed bending of 20 kN. Loading rate was 3 kN/s. Internal pressure of 17.5 MPa was applied with another branch of the pressure system. Tests were conducted on a closed-loop servohydraulic tensionecompression testing machine. Internal pressure of 17.5 MPa is design pressure of nuclear piping. When the elbow was subjected to internal pressure of 17.5 MPa and bending load of 20 kN, the hoop stress at intrados, extrados with local wall thinning and crown with local wall thinning were 423 MPa, 261 MPa and 395 MP, simulated respectively by ANSYS.
Fig. 3. Gauge distribution.
2.3. Strain gauge location pattern In order to record the ratcheting strains, biaxial strain gauges were adhered at the local wall thinning and the specific elbow positions where the maximum bending moment and crack easily occurred [16,25,26]. The strains were monitored by a self-designed
Fig. 2. Experimental setup. Fig. 4. Strain history: (a) at intrados; (b) at crown.
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Fig. 5. Ratcheting strain of sound elbow.
strain processor and collected with loading and time by computercontrolled system synchronously. The results of previous paper [26] showed that the ratcheting strain mainly occurred at crown, intrados and 45 position at middle between crown and intrados. In addition, the extrados area was also examined emphatically in the elbow with local wall thinning at extrados. So in this study, the biaxial strain gauges were placed in hoop and axial direction at extrados, crown, 45 position and intrados area on the 3 types of elbow as shown in Fig. 3. 2.4. Experimental results and analysis 2.4.1. Ratcheting strain of sound elbow under constant internal pressure and reversed bending The strain histories at crown and intrados of sound elbow under the bending load of 20 kN and constant internal pressure of 17.5 MPa are shown in Fig. 4. The ratcheting strain at each position is obtained and compared in Fig. 5. The principal ratcheting strain illustrated in Fig. 5 is calculated from strain history by middle point method, namely
Fig. 6. Ratcheting strain of elbow with local wall thinning at extrados.
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Fig. 7. Comparison of ratcheting strain of sound elbow and elbow with local wall thinning at extrados.
εr ¼
1 ðεmax þ εmin Þ 2
(1)
where εmaxand εmin are the maximum and minimum strain of each cycle, respectively. Fig. 5 indicates the ratcheting strains are accumulated at the extrados, crown, 45 position and intrados, and mainly in hoop direction. The hoop ratcheting strain has the largest values at intrados. It can also be seen that the hoop ratcheting strain rate, i.e. the increase rate of hoop ratcheting strain with cyclic number, almost decreases with increasing cyclic number during initial several cycles of fast accumulation, and then keeps constant. 2.4.2. Ratcheting strain of elbow with local wall thinning at extrados Fig. 6 shows that the hoop ratcheting strains are obvious at the intrados, crown, 45 position, and wall thinning at extrados. The largest ratcheting strain occurs at extrados where local wall thinning is located during initial 3 cycles, and then it occurs at intrados with the increasing cycles.
Fig. 8. Ratcheting strain of elbow with local wall thinning at crown.
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during initial several cycles, and then approach the monotony upward trend with a higher constant rate, while hoop ratcheting strains in other three positions show a reasonable agreeable between sound elbow and the elbow with wall thinning at extrados.
Fig. 9. Comparison of ratcheting strain of sound elbow and elbow with local wall thinning at crown.
Comparison of ratcheting strain at same position of sound elbow and elbow with local wall thinning at extrados is presented in Fig. 7. It is shown that the local wall thinning influences both of the ratcheting strain level and the ratcheting strain rate. Compared with the hoop ratcheting strain at extrados of sound elbow, that at extrados of the elbow with wall thinning has a faster accumulation
2.4.3. Ratcheting strain of elbow with local wall thinning at crown Fig. 8 shows that the hoop ratcheting strains obviously occur at the intrados, 45 position and crown where the deepest wall thinning is located, while the axial ratcheting strains are low. The largest hoop ratcheting strain occurs at the wall thinning during initial 55 cycles of bending. As shown in Fig. 9, the local wall thinning at crown has a very pronounced effect on the hoop ratcheting strain. The hoop ratcheting strain at crown where wall thinning existed has a faster accumulation during initial several cycles, and then approach the monotony upward trend, while hoop ratcheting strain in other three positions show a reasonable agreement between sound elbow and the elbow with wall thinning at crown. On the basis of the ratcheting behavior, it appears that the hoop ratcheting strains obviously occur at the intrados, 45 position and crown while the axial ratcheting strains are considerably low. In sound elbow and the elbow with local wall thinning at extrados, the largest hoop ratcheting strain mainly occurs at intrados. In the elbow with local wall thinning at crown, the largest hoop ratcheting strain occurs at crown. Accordingly, the location of the largest hoop ratcheting is affected largely by the existence of local wall thinning at crown. The local wall thinning results in a significant ratcheting strain rate during the initial several cycles of bending and a higher ratcheting strain level.
Fig. 10. Finite element model of pressurized elbow pipe under constant pressure and symmetric cyclic bending: (a)1/4 model of sound elbow pipe; (b) 1/4 model of the elbow with local wall thinning at extrados; (c) 1/4 model of the elbow with local wall thinning at crown.
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3. Simulation of ratcheting behavior 3.1. Finite element model In order to study the ratcheting strain of elbow, elastic-plastic finite element analyses were conducted. Many efforts have been made to investigate the uniaxial/multiaxial ratcheting behavior of various materials. Several models based on the Armstrong and Frederick kinematic hardening model with modification of the dynamic recovery term have been applied to simulate the ratcheting strain, such as Chaboche model [27,28], OhnoeWang model [29,30] and modified OhnoeWang model [31]. Zhu and Hassan [32], Kobayashi et al. [33,34], Kang [35] and Postberg et al. [36,37] have incorporated these models into ABAQUS or ANSYS for analyzing ratcheting behavior of components. Chaboche model and CJK model yield good simulation results both for material and component applications. In this study, Chaboche model and CJK model were coded via UPFs (User Programmable Features) of ANSYS. The load-control experiments were simulated by FE software ANSYS. The specimen was modeled by solid95 element for elbow straight pipe and pipe end. Only a quarter of the specimen was considered due to the symmetry in geometry and loading. Internal pressure was applied to the specimen inner surface and the reversed bending load in z-direction was impressed at the central point of the pipe end. Non-linear Geometry was also considered in the analysis. Fig. 10 shows finite element model of sound elbow, the elbow with local wall thinning at extrados and the elbow with local wall thinning at crown. 3.2. Constitutive model 3.2.1. Chaboche model Chaboche et al. [27,28] proposed that several Armstrong and Frederick type rules were superimposed as
da ¼
m X
dai ¼
i¼1
m X 2 i¼1
3
Ci dεp bi ai dp ;
ðm ¼ 3Þ
(2)
The resulting kinematic hardening rule can reasonably simulate the non-linear stressestrain response. Where ai is the ith component of deviatoric back stress a, and Ci, gi are material constants, which can be determined from the stress-strain curve. For Z2CND18.12N stainless steel used in this study, three terms are adopted and the values of the parameters are: yield stresss0 ¼ 100 MPa; Young’s modulus E ¼ 1.95 105 MPa; Ci ¼ 4.0 106, 1.5 105, 2.5 103;bi ¼ 4.0 104, 0.83 103, 4.5, which are determined through simulating the composed hysteresis curve and uniaxial ratcheting response (Fig. 11). 3.2.2. CheneJiaoeKim model (CJK model) Chen, Jiao, and Kim [31] proposed a modified OhnoeWang kinematic hardening rule in which a multiaxial parameter ci was introduced. The model is shown as
a ¼
n X
ai ;
ðn ¼ 6Þ
Fig. 11. Model parameter determination and material response simulations with Chaboche model: (a) hysteresis loop representation and (b) unaxial ratcheting simulation.
CJK plasticity model are: s0 ¼ 100 MPa; E ¼ 1.95 105 MPa; mi ¼ 6; ri ¼ 40, 107, 60, 18, 60, 110; ci ¼ 0.1; gi ¼ 8000, 4000, 2000, 1500, 150, 20; which are determined by simulating the composed hysteresis curve and uniaxial ratcheting response shown in Fig. 12. Fig. 13 shows the experiment results of first 10 cyclic stressestrains curves for the pipe material and simulation by CJK model without isotropic hardening. The material shows a slight hardening during first 10 cycles.
(3)
i¼1
3.3. Numerical results
2 +ci 0 1mi * +3 * ai A 2 p 0 ai p ai 5 4 @ ai dε : dai ¼ gi r dε n : ri 3 i ai ai
(4)
0 ¼ dεp =dp ¼ 3=2s ðs aÞ, a wherepnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 i is the magnitude of ai, ai ¼ 3=2ai : ai , and ci is multiaxial parameter. The parameters of
The numerical simulation is discussed with the experimental results for sound elbow, the elbow with local wall thinning at extrados and the elbow with local wall thinning at crown subjected to internal pressure of 17.5 MPa and reversed bending load of 20 kN. Fig. 14 shows the comparison of the experimental data and the predicted results of sound elbow by Chaboche model and CJK
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Fig. 14. The hoop strain ratcheting simulation for sound elbow with Chaboche and CJK model.
model. As can be seen, CJK model has more accurately reproduced the experimental results as compared to Chaboche model which will over-predict the ratcheting strain. Fig. 15 illustrates CJK model has more reasonable simulation results of the elbow with local wall thinning at extrados as compared to Chaboche model which will over-predict the ratcheting strain. As clearly seen from Fig. 16, CJK model can reasonably simulate the hoop ratcheting strain for the elbow with local wall thinning at crown. 3.4. Ratcheting boundary determination
Fig. 12. Model parameter determination and material response simulations with CJK model: (a) hysteresis loop representation and (b) unaxial ratcheting simulation.
Fig. 13. Experimental and simulated results for uniaxial cyclic stressestrain curves.
‘Variations in equivalent plastic strain at the end of each cycle should have a decreasing trend and should become lower than the allowable limit of 104/cycle in the 10th cycle.’ is one criterion called “evaluating variation in plastic strain increments” to verify shakedown in the C-TDF method [24,38]. According to the conclusion in section 3.3, the hoop ratcheting strain of experiments
Fig. 15. The hoop strain ratcheting simulation for elbow with local wall thinning at extrados with Chaboche and CJK model.
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Fig. 16. The hoop strain ratcheting simulation for elbow with local wall thinning at crown with Chaboche and CJK model.
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Fig. 18. Ratcheting boundary at crown in the elbow with local wall thinning at crown.
3.5. Effect of the depth of local wall thinning and simulations matched well with CJK model. Therefore, CJK model was applied to determine ratcheting boundary. The following section discussed the ratcheting boundary of the pipes under different combination of pressure and bending load, DF meant the amplitude of bending load. According to the simulated results in section 3.3, the largest ratcheting strain during the first 10 cycles occurs at intrados area in sound elbow and elbow with local wall thinning at extrados. Thus the ratcheting boundary at intrados area is examined emphatically which is determined by evaluating variation in plastic strain increments and is shown in Fig. 17. According to the simulated results in section 3.3, the largest ratcheting strain during the first 10 cycles occurs at crown area in elbow with local wall thinning at crown. Thus the ratcheting boundary at crown area is examined emphatically which is obtained by evaluating variation in plastic strain increments and is shown in Fig. 18.
Fig. 17. Ratcheting boundary at intrados in sound elbow and the elbow with local wall thinning at extrados.
Because of broad and smooth local wall thinning area, the ratcheting behavior is mainly affected by the depth of local wall thinning. In this section, the effect of depth of local wall thinning on the ratcheting response is investigated by CJK model. The elbows with different depth of local wall thinning were subjected to the bending load of 20 kN and constant internal pressure of 17.5 MPa. A set of simulation with different depths of wall thinning is conducted, and the hoop ratcheting strain at the extrados where the local wall thinning existed is shown in Fig. 19. The higher level ratcheting strain occurs at deeper wall thinning, and the difference between inner and outer strain increases with increasing depth and increasing cyclic number. It shows that the increase of depth has more effect on hoop ratcheting strain at outside surface, and the ratcheting strain at inner surface is less than that at outer surface throughout. In the elbow with local wall thinning at extrados, both of extrados area and intrados area have obvious hoop ratcheting strain. In order to evaluate the effect of different depths of local wall
Fig. 19. Simulated hoop ratcheting strain with different depths of wall thinning at extrados.
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Fig. 20. Simulated hoop ratcheting strain with different depths of wall thinning at crown.
thinning on the location of the largest hoop ratcheting strain, the comparison of ratcheting strain of intrados outer and extrados outer is shown in Fig. 19. The hoop ratcheting strain of intrados outer is not affected by different depths of wall thinning, and it is shown as the red cross in Fig. 19. It shows that when the depth of wall thinning is more than 1.12 mm, i.e., wall thinning ratio d/ t z 0.25, the hoop ratcheting strain at extrados outer is larger than that at intrados outer. Therefore, the largest hoop ratcheting strain occurs at extrados when wall thinning ratio is more than 0.25. Fig. 20 shows that the hoop ratcheting strain increases with increasing depth of wall thinning at crown area. It seems that the higher ratcheting strain tends to be easier to occur at outer surface than inner surface. The results in Section 3.3 show that the largest hoop ratcheting strain occurs at crown area or intrados area, therefore the hoop ratcheting strain at intrados and crown is investigated emphatically. The comparison of hoop ratcheting strain at crown outer and intrados outer is shown in Fig. 20. The hoop ratcheting strain at intrados outer is not affected by the depth of local wall thinning at crown and it is shown as the red cross in Fig. 20. When the depth of wall thinning is more than 1 mm, i.e., wall thinning ratio d=tz0:22, the hoop ratcheting strain at crown outer is larger than that at intrados outer. Therefore, the largest hoop ratcheting strain occurs at external wall of crown when local wall thinning ratio is more than 0.22. Figs. 19 and 20 show that the higher level ratcheting strain occurs at deeper wall thinning, and the ratcheting strain at inner surface is less than that at outer surface throughout, the difference between inner and outer surface becomes more with the increase of depth. The depth of local wall thinning also affects the location of the largest hoop ratcheting strain. When wall thinning ratio of wall thinning at extrados is greater than 0.25, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados. When wall thinning ratio of wall thinning at crown is more than 0.22, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados. 4. Conclusions With multi-axial test machine, ratcheting behavior is studied experimentally for pressurized Z2CND18.12N stainless steel elbows under constant internal pressure and reversed in-plane bending under load control. The local wall thinning is machined on the
outside of elbow in order to simulate metal loss. The hoop ratcheting strains obviously occur at the intrados, 45 position and crown while the axial ratcheting strains are considerably low. In sound elbow and the elbow with local wall thinning at extrados, the largest hoop ratcheting strain mainly occurs at intrados. In the elbow with local wall thinning at crown, the largest hoop ratcheting strain occurs at crown. The local wall thinning causes a significant ratcheting strain rate during the initial several cycles of bending and a higher ratcheting stain level. Ratcheting simulation is performed by EPFEA with ANSYS in which Chaboche model and CJK model are coded by user programming. Comparing the experimental data and the predicted results by Chaboche model and CJK model, it is found that CJK model has more accurately reproduced the experimental results. The ratcheting boundary of extrados area and crown area where local wall thinning existed is determined by C-TDF method with CJK model. The effect of depth of local wall thinning on the ratcheting response is investigated by CJK model. It is found that the higher ratcheting strain level and higher ratcheting strain rate occur at deeper wall thinning, and the difference between the strain of internal wall and the strain of external wall increases with increasing depth and increasing cyclic number of bending. The depth of local wall thinning also affects the location of the largest hoop ratcheting strain. When wall thinning ratio of wall thinning at extrados is greater than 0.25, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados. When wall thinning ratio of wall thinning at crown is more than 0.22, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados.
Acknowledgments The authors gratefully acknowledge financial support for this work from the National High Technology Research and Development Program of China (863 Program 2009AA04Z403) and Ph.D. Programs Foundation of Ministry of Education of China (No. 20090032110016).
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H. Shi et al. / International Journal of Pressure Vessels and Piping 102-103 (2013) 14e23 [14] Kulkarni SC, Desai YM, Kant T, Reddy GR, Prasad P, Vaze KK, et al. Uniaxial and biaxial ratcheting in piping materials-experiments and analysis. Int J Press Vessel Pip 2004;81:609e17. [15] Yahiaoui K, Moffat DG, Moreton DN. Pressurized piping elbows under simulated seismic bending: design code implications. Proc Inst Mech Eng, Part E: J Process Mech Eng 1996;210(E3):159e70. [16] Yahiaoui K, Moffat DG, Moreton DN. Damage assessment of piping elbows loaded by steady internal pressure and dynamic in-plane or out-of-plane bending. In: Proceedings of the International Conference on Pressure Vessels Technology, vol. 1. ICPVT; 1996. p. 361e75. [17] Yahiaoui K, Moffat DG, Moreton DN. Response and cyclic strain accumulation of pressurized piping elbows under dynamic in-plane bending. J Strain Anal Eng Des 1996;31(2):135e51. [18] Yahiaoui K, Moffat DG, Moreton DN. Response and cyclic strain accumulation of pressurized piping elbows under dynamic out-of-plane bending. J Strain Anal Eng Des 1996;31(2):153e66. [19] Chen X, Gao BJ, Chen G. Multiaxial ratcheting of pressurized elbows subjected to reversed in-plane bending. J Press Equip Syst 2005;3:38e44. [20] Gao BJ, Chen X, Chen G. Ratcheting and ratcheting boundary study of pressurized straight low carbon steel pipe under reversed bending. Int J Press Vessel Pip 2006;83(2):96e106. [21] Asada S, Yamashita N, Okamoto A, Nishiguchi I. Verification of alternative criteria for shakedown evaluation using flat head vessel. ASME PVP 2002;439:3e10. [22] DeGrassi G, Hofmayer C, Murphy A, Suzuki K, Namita Y. BNL nonlinear pretest seismic analysis for the NUPEC ultimate strength piping test. In: Transactions of the 17th International Conference on Structural Mechanics in Reactor Technology (SMiRT 17); 2003. p. 8. [23] Miyazaki K, Nebu A, Ishiwata M, Hasegawa K. Fracture strength and behavior of carbon steel pipes with local wall thinning subjected to cyclic bending load. Nucl Eng Des 2001;214:127e36. [24] Balan C, Redekop D. The effect of bi-directional loading on fatigue assessment of pressurized piping elbows with local thinned areas. Int J Press Vessel Pip 2005;82:235e42. [25] Stearns JC, Paul CL. Incremental plastic strain of a pipe elbow with varying bend radii. ASME PVP 1993;264:169e74.
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[26] Yahiaoui K, Moffat DG. Onset of ratcheting in pressurized piping elbows subjected to in-plane bending moments. Int J Press Vessel Pip 1996;68: 73e9. [27] Chaboche JL. Time independent constitutive theories for cyclic plasticity. Int J Plast 1986;2:149e88. [28] Chaboche JL. On some modifications of kinematic hardening to improve the description of ratcheting effects. Int J Plast 1991;7:661e78. [29] Ohno N, Wang JD. Kinematic hardening rules with critical state of dynamic recovery. Part I: formulations and basic features for ratcheting behavior. Int J Plast 1993;9:375e90. [30] Ohno N, Wang JD. Kinematic hardening rules with critical state of dynamic recovery. Part II: application to experiments of ratcheting behavior. Int J Plast 1993;9:391e403. [31] Chen X, Jiao R, Kim KS. On the OhnoeWang kinematic hardening rules for multiaxial ratcheting modeling of medium carbon steel. Int J Plast 2005;21: 161e84. [32] Hassan T, Zhu Y, Matzen VC. Improved ratcheting analysis of piping components. Int J Press Vessel Pip 1998;75:643e52. [33] Kobayashi M, Ohno N. Implementation of cyclic plasticity models based on a general form of kinematic hardening. Int J Numer Methods Eng 2002;53: 2217e38. [34] Kobayashi M, Mukai M, Takahashi H. Implicit integration and consistent tangent modulus of a time-dependent non-unified constitutive model. Int J Numer Methods Eng 2003;58:1523e43. [35] Kang G. A viso-plastic constitutive model for ratcheting of cyclically stable material and its finite element implementation. Mech Mater 2004; 36:299e312. [36] Postberg B, Weiß E. Simulation of ratcheting of AISI 316L(N) steel under nonproportional uniaxial loading and high number of load cycles using the Ohno and Wang nonlinear kinematic material model. Int J Press Vessel Pip 2000;77: 207e13. [37] Weiß E, Postberg B, Nicak T, Rudolphet J. Simulation of ratcheting and low cycle fatigue. Int J Press Vessel Pip 2004;81:235e42. [38] Yamamoto Y, Yamashita N, Tanaka M. Evaluation of thermal stress in plastic FEA. ASME PVP 2002;439:3e10.
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International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Ratcheting behavior of pressurized elbow pipe with local wall thinning Hongrui Shi, Gang Chen, Yong Wang, Xu Chen* School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 July 2012 Received in revised form 9 December 2012 Accepted 19 December 2012
Ratcheting deformation is studied on elbow pipe made of Z2CND18.12N stainless steel with local wall thinning subjected to constant internal pressure and reversed in-plane bending under load control. The local wall thinning is located at extrados and crown of elbow. It is shown that local wall thinning has a very pronounced effect on the ratcheting behavior of elbows, as compared to the ratcheting behavior of sound elbow specimen. Three-dimensional elastic-plastic analyses with ANSYS in which Chaboche and CheneJiaoeKim (CJK) kinematic hardening model are carried out to evaluate structural ratcheting behaviors. A reasonable agreement has been found between the experimental and the simulated results with CJK model for the ratcheting of the elbows. The ratcheting boundary is determined by evaluating variations in the plastic strain increment with CJK model. The effects of the depth and location of local wall thinning on the ratcheting response are discussed by CJK model. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Ratcheting Elbow piping Cyclic loading Thinning wall Kinematic hardening
1. Introduction The piping fittings employed in nuclear power plants and chemical industries play a vital role in safe operation. If the pipelines subjected to internal pressure and cyclic loading, they may face progressive deformation, so-called ratcheting. For example, this can be generated by fluid pressure and temperature changes or other cyclic environmental conditions such as seismic loading [1,2]. Severe ratcheting may occur and lead to premature failure due to either accumulation of deformation [3,4] or fatigue cracks [5]. These results indicate that ratcheting may occur much earlier than the estimated life, and may result in unscheduled plant downtime. Thus, during the design of pressurized piping in nuclear power plants, ratcheting and ratcheting fatigue must be taken into consideration. Due to geometry of the elbow structure, the stress distributions in the elbow pipes are much more complex [6]. Meanwhile the material lose is enhanced at elbow components which are widely used in nuclear power plants and chemical industries. Local wall thinning due to erosion of external soil and rain water, mechanical damage and crack repair can enhance ratcheting deformation. Therefore, it is important to evaluate the ratcheting deformation of elbow pipe undergoing local wall thinning and determine the
* Corresponding author. Tel.: þ86 22 27408399; fax: þ86 22 27403389. E-mail address: [email protected] (X. Chen). 0308-0161/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2012.12.002
ratcheting boundary in order to maintain the integrity of the piping systems. The ratcheting effect has been considered in several standards, such as ASME Code Section III [7], KTA [8] and RCC-MR [9]. The aforementioned ratcheting behaviors of straight and elbow pipes have already been extensively studied in the literature [10e14]. A series of elbow experiments by applying dynamic bending were conducted in order to evaluate the ratcheting responses of the elbow components [15e18]. Moreover, ratcheting strains can also be analyzed by Elastic-Plastic Finite Element Analysis (EPFEA) [4,19,20]. The Committee of Three Dimensional Finite Element Stress Evaluation (C-TDF) in Japan proposed two alternative criteria to verify shakedown, in which an elastic perfect plasticity (EPP) model or the bilinear kinematic hardening (BKH) rule was used [21]. The ratcheting of pressurized straight and elbow pipe of low carbon steel under reversed bending was experimentally and analytically investigated by Chen et al. [19] and Gao et al. [20]. The study has shown that CheneJiaoeKim (CJK) model can reasonably simulate ratcheting strain. Degrassi et al. [22] analyzed the ratcheting responses of the piping system under seismic excitation using the bilinear, multilinear and Chaboche model in ANSYS. The authors demonstrated Chaboche model performed best in strain ratcheting simulation for elbow through adjusting ratcheting parameters based on pipe ratcheting response. The failure behavior of pipelines with wall thinning has also received considerable previous attention in the literatureliteratures. Nakamura et al. [5] experimentally examined pressurized
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Table 1 Chemical compositions of Z2CND18.12N. Chemical composition
C
Si
Mn
P
S
Ni
Cr
Mo
B
Cu
Co
N
%
0.025
0.430
1.211
0.021
0.003
12.073
17.517
2.388
0.001
0.075
0.035
0.070
Table 2 Specification of elbows. Outside diameter D0, mm
Nominal thickness t, mm
Bending radius R, mm
Bending 2 characteristic tR=rm
76
4.5
95
0.33
elbows with local wall thinning under displacement-controlled cyclic bending. It was found that the failure modes and fatigue lives of elbows seemed to be affected by a ratchet phenomenon. Miyazaki et al. [23] examined carbon steel pipes with local wall thinning under cyclic pure bending loads to evaluate their low cycle fatigue strength. In load controlled tests on these pipes, ratcheting deformation was observed, and the fatigue strength became lower than that of cracked pipes. The effect of bi-directional loading on the fatigue characteristics of pressurized 90 piping elbows with local wall thinning was investigated by Balan and Redekop [24]. The results provided extensive new information about the fatigue behavior of piping elbow subject to seismic action. Zeinoddini and Peykanu [4] studied the strain ratcheting of steel tubes with a rectangular defect under axial cycling. It was shown that the surface imperfections had a very significant effect on the ratcheting response of the defected tubes. The effects of some factors such as the stress amplitude, wall thinning and the material hardening properties on the ratcheting response of steel tubes were also investigated. The subject of ratcheting of defected elbow pipes under combined internal pressure and cyclic bending, however, has not received due attention previously. In this study, in order to
investigate the ratcheting behaviors of elbow pipes undergoing local wall thinning, ratcheting tests have been carried out using Z2CND18.12N stainless steel elbow specimens with local wall thinning. The local wall thinning areas are located at extrados part and crown part. The pressuried elbow specimens are subjected to constant internal pressure and reversed in-plane bending under load control. Then, ratcheting simulation is performed by elasticplastic finite element analyses with ANSYS in which Chaboche model and modified OhnoeWang model (CJK model) are applied. The analytical results are compared with the experimental data in order to identify the better model for structural ratcheting simulation. Ratcheting boundary is determined by evaluating variations in the plastic strain increment with CJK model. 2. Experiments 2.1. Material and specification of elbow specimens The elbow specimens were machined from Z2CND18.12N austenitic stainless steel. The chemical compositions of the material are listed in Table 1. Yield strength of the material is 388 MPa, ultimate strength is 590 MPa and Young’s modulus is 195 GPa at room temperature. The elbow samples satisfy Chinese code GB12459-90, and their dimensions are listed in Table 2. Fig. 1(a) shows the shape and geometry of elbow specimens. Fig. 1(b) and (c) shows the position and dimensions of local wall thinning which is equivalent to the defect caused by the metal loss in engineering. In this study, in order to simulate metal loss due to
Fig. 1. Shape and geometries of elbow pipe specimens: (a) Elbow specimen; (b) Elbow with local wall thinning at extrados; (c) Elbow with local wall thinning at crown.
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corrosion or mechanical damage, the local wall thinning was machined on the outside of the center of elbows which were located at two different positions, called extrados and crown, and the wall thinning at the crown was on the one side of the elbow. The sound elbow specimen without wall thinning was also used. The three types of elbow specimens were called as extrados thinning elbow, crown thinning elbow and sound elbow, respectively. The depth of local wall thinning in the thickness direction was called eroded depth (d) and was fixed at d ¼ 1 mm. The wall thinning ratio was defined as the ratio of d to the wall thickness t (d/t) and was fixed at d=tz0:22. 2.2. Experimental apparatus and testing system Experimental setup is shown in Fig. 2. Loading bars clipped the clip-head of the multiaxial test machine. Cyclic pullepush force opened and closed the elbow and made the specimen be under reversed bending of 20 kN. Loading rate was 3 kN/s. Internal pressure of 17.5 MPa was applied with another branch of the pressure system. Tests were conducted on a closed-loop servohydraulic tensionecompression testing machine. Internal pressure of 17.5 MPa is design pressure of nuclear piping. When the elbow was subjected to internal pressure of 17.5 MPa and bending load of 20 kN, the hoop stress at intrados, extrados with local wall thinning and crown with local wall thinning were 423 MPa, 261 MPa and 395 MP, simulated respectively by ANSYS.
Fig. 3. Gauge distribution.
2.3. Strain gauge location pattern In order to record the ratcheting strains, biaxial strain gauges were adhered at the local wall thinning and the specific elbow positions where the maximum bending moment and crack easily occurred [16,25,26]. The strains were monitored by a self-designed
Fig. 2. Experimental setup. Fig. 4. Strain history: (a) at intrados; (b) at crown.
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Fig. 5. Ratcheting strain of sound elbow.
strain processor and collected with loading and time by computercontrolled system synchronously. The results of previous paper [26] showed that the ratcheting strain mainly occurred at crown, intrados and 45 position at middle between crown and intrados. In addition, the extrados area was also examined emphatically in the elbow with local wall thinning at extrados. So in this study, the biaxial strain gauges were placed in hoop and axial direction at extrados, crown, 45 position and intrados area on the 3 types of elbow as shown in Fig. 3. 2.4. Experimental results and analysis 2.4.1. Ratcheting strain of sound elbow under constant internal pressure and reversed bending The strain histories at crown and intrados of sound elbow under the bending load of 20 kN and constant internal pressure of 17.5 MPa are shown in Fig. 4. The ratcheting strain at each position is obtained and compared in Fig. 5. The principal ratcheting strain illustrated in Fig. 5 is calculated from strain history by middle point method, namely
Fig. 6. Ratcheting strain of elbow with local wall thinning at extrados.
17
Fig. 7. Comparison of ratcheting strain of sound elbow and elbow with local wall thinning at extrados.
εr ¼
1 ðεmax þ εmin Þ 2
(1)
where εmaxand εmin are the maximum and minimum strain of each cycle, respectively. Fig. 5 indicates the ratcheting strains are accumulated at the extrados, crown, 45 position and intrados, and mainly in hoop direction. The hoop ratcheting strain has the largest values at intrados. It can also be seen that the hoop ratcheting strain rate, i.e. the increase rate of hoop ratcheting strain with cyclic number, almost decreases with increasing cyclic number during initial several cycles of fast accumulation, and then keeps constant. 2.4.2. Ratcheting strain of elbow with local wall thinning at extrados Fig. 6 shows that the hoop ratcheting strains are obvious at the intrados, crown, 45 position, and wall thinning at extrados. The largest ratcheting strain occurs at extrados where local wall thinning is located during initial 3 cycles, and then it occurs at intrados with the increasing cycles.
Fig. 8. Ratcheting strain of elbow with local wall thinning at crown.
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during initial several cycles, and then approach the monotony upward trend with a higher constant rate, while hoop ratcheting strains in other three positions show a reasonable agreeable between sound elbow and the elbow with wall thinning at extrados.
Fig. 9. Comparison of ratcheting strain of sound elbow and elbow with local wall thinning at crown.
Comparison of ratcheting strain at same position of sound elbow and elbow with local wall thinning at extrados is presented in Fig. 7. It is shown that the local wall thinning influences both of the ratcheting strain level and the ratcheting strain rate. Compared with the hoop ratcheting strain at extrados of sound elbow, that at extrados of the elbow with wall thinning has a faster accumulation
2.4.3. Ratcheting strain of elbow with local wall thinning at crown Fig. 8 shows that the hoop ratcheting strains obviously occur at the intrados, 45 position and crown where the deepest wall thinning is located, while the axial ratcheting strains are low. The largest hoop ratcheting strain occurs at the wall thinning during initial 55 cycles of bending. As shown in Fig. 9, the local wall thinning at crown has a very pronounced effect on the hoop ratcheting strain. The hoop ratcheting strain at crown where wall thinning existed has a faster accumulation during initial several cycles, and then approach the monotony upward trend, while hoop ratcheting strain in other three positions show a reasonable agreement between sound elbow and the elbow with wall thinning at crown. On the basis of the ratcheting behavior, it appears that the hoop ratcheting strains obviously occur at the intrados, 45 position and crown while the axial ratcheting strains are considerably low. In sound elbow and the elbow with local wall thinning at extrados, the largest hoop ratcheting strain mainly occurs at intrados. In the elbow with local wall thinning at crown, the largest hoop ratcheting strain occurs at crown. Accordingly, the location of the largest hoop ratcheting is affected largely by the existence of local wall thinning at crown. The local wall thinning results in a significant ratcheting strain rate during the initial several cycles of bending and a higher ratcheting strain level.
Fig. 10. Finite element model of pressurized elbow pipe under constant pressure and symmetric cyclic bending: (a)1/4 model of sound elbow pipe; (b) 1/4 model of the elbow with local wall thinning at extrados; (c) 1/4 model of the elbow with local wall thinning at crown.
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3. Simulation of ratcheting behavior 3.1. Finite element model In order to study the ratcheting strain of elbow, elastic-plastic finite element analyses were conducted. Many efforts have been made to investigate the uniaxial/multiaxial ratcheting behavior of various materials. Several models based on the Armstrong and Frederick kinematic hardening model with modification of the dynamic recovery term have been applied to simulate the ratcheting strain, such as Chaboche model [27,28], OhnoeWang model [29,30] and modified OhnoeWang model [31]. Zhu and Hassan [32], Kobayashi et al. [33,34], Kang [35] and Postberg et al. [36,37] have incorporated these models into ABAQUS or ANSYS for analyzing ratcheting behavior of components. Chaboche model and CJK model yield good simulation results both for material and component applications. In this study, Chaboche model and CJK model were coded via UPFs (User Programmable Features) of ANSYS. The load-control experiments were simulated by FE software ANSYS. The specimen was modeled by solid95 element for elbow straight pipe and pipe end. Only a quarter of the specimen was considered due to the symmetry in geometry and loading. Internal pressure was applied to the specimen inner surface and the reversed bending load in z-direction was impressed at the central point of the pipe end. Non-linear Geometry was also considered in the analysis. Fig. 10 shows finite element model of sound elbow, the elbow with local wall thinning at extrados and the elbow with local wall thinning at crown. 3.2. Constitutive model 3.2.1. Chaboche model Chaboche et al. [27,28] proposed that several Armstrong and Frederick type rules were superimposed as
da ¼
m X
dai ¼
i¼1
m X 2 i¼1
3
Ci dεp bi ai dp ;
ðm ¼ 3Þ
(2)
The resulting kinematic hardening rule can reasonably simulate the non-linear stressestrain response. Where ai is the ith component of deviatoric back stress a, and Ci, gi are material constants, which can be determined from the stress-strain curve. For Z2CND18.12N stainless steel used in this study, three terms are adopted and the values of the parameters are: yield stresss0 ¼ 100 MPa; Young’s modulus E ¼ 1.95 105 MPa; Ci ¼ 4.0 106, 1.5 105, 2.5 103;bi ¼ 4.0 104, 0.83 103, 4.5, which are determined through simulating the composed hysteresis curve and uniaxial ratcheting response (Fig. 11). 3.2.2. CheneJiaoeKim model (CJK model) Chen, Jiao, and Kim [31] proposed a modified OhnoeWang kinematic hardening rule in which a multiaxial parameter ci was introduced. The model is shown as
a ¼
n X
ai ;
ðn ¼ 6Þ
Fig. 11. Model parameter determination and material response simulations with Chaboche model: (a) hysteresis loop representation and (b) unaxial ratcheting simulation.
CJK plasticity model are: s0 ¼ 100 MPa; E ¼ 1.95 105 MPa; mi ¼ 6; ri ¼ 40, 107, 60, 18, 60, 110; ci ¼ 0.1; gi ¼ 8000, 4000, 2000, 1500, 150, 20; which are determined by simulating the composed hysteresis curve and uniaxial ratcheting response shown in Fig. 12. Fig. 13 shows the experiment results of first 10 cyclic stressestrains curves for the pipe material and simulation by CJK model without isotropic hardening. The material shows a slight hardening during first 10 cycles.
(3)
i¼1
3.3. Numerical results
2 +ci 0 1mi * +3 * ai A 2 p 0 ai p ai 5 4 @ ai dε : dai ¼ gi r dε n : ri 3 i ai ai
(4)
0 ¼ dεp =dp ¼ 3=2s ðs aÞ, a wherepnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 i is the magnitude of ai, ai ¼ 3=2ai : ai , and ci is multiaxial parameter. The parameters of
The numerical simulation is discussed with the experimental results for sound elbow, the elbow with local wall thinning at extrados and the elbow with local wall thinning at crown subjected to internal pressure of 17.5 MPa and reversed bending load of 20 kN. Fig. 14 shows the comparison of the experimental data and the predicted results of sound elbow by Chaboche model and CJK
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Fig. 14. The hoop strain ratcheting simulation for sound elbow with Chaboche and CJK model.
model. As can be seen, CJK model has more accurately reproduced the experimental results as compared to Chaboche model which will over-predict the ratcheting strain. Fig. 15 illustrates CJK model has more reasonable simulation results of the elbow with local wall thinning at extrados as compared to Chaboche model which will over-predict the ratcheting strain. As clearly seen from Fig. 16, CJK model can reasonably simulate the hoop ratcheting strain for the elbow with local wall thinning at crown. 3.4. Ratcheting boundary determination
Fig. 12. Model parameter determination and material response simulations with CJK model: (a) hysteresis loop representation and (b) unaxial ratcheting simulation.
Fig. 13. Experimental and simulated results for uniaxial cyclic stressestrain curves.
‘Variations in equivalent plastic strain at the end of each cycle should have a decreasing trend and should become lower than the allowable limit of 104/cycle in the 10th cycle.’ is one criterion called “evaluating variation in plastic strain increments” to verify shakedown in the C-TDF method [24,38]. According to the conclusion in section 3.3, the hoop ratcheting strain of experiments
Fig. 15. The hoop strain ratcheting simulation for elbow with local wall thinning at extrados with Chaboche and CJK model.
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Fig. 16. The hoop strain ratcheting simulation for elbow with local wall thinning at crown with Chaboche and CJK model.
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Fig. 18. Ratcheting boundary at crown in the elbow with local wall thinning at crown.
3.5. Effect of the depth of local wall thinning and simulations matched well with CJK model. Therefore, CJK model was applied to determine ratcheting boundary. The following section discussed the ratcheting boundary of the pipes under different combination of pressure and bending load, DF meant the amplitude of bending load. According to the simulated results in section 3.3, the largest ratcheting strain during the first 10 cycles occurs at intrados area in sound elbow and elbow with local wall thinning at extrados. Thus the ratcheting boundary at intrados area is examined emphatically which is determined by evaluating variation in plastic strain increments and is shown in Fig. 17. According to the simulated results in section 3.3, the largest ratcheting strain during the first 10 cycles occurs at crown area in elbow with local wall thinning at crown. Thus the ratcheting boundary at crown area is examined emphatically which is obtained by evaluating variation in plastic strain increments and is shown in Fig. 18.
Fig. 17. Ratcheting boundary at intrados in sound elbow and the elbow with local wall thinning at extrados.
Because of broad and smooth local wall thinning area, the ratcheting behavior is mainly affected by the depth of local wall thinning. In this section, the effect of depth of local wall thinning on the ratcheting response is investigated by CJK model. The elbows with different depth of local wall thinning were subjected to the bending load of 20 kN and constant internal pressure of 17.5 MPa. A set of simulation with different depths of wall thinning is conducted, and the hoop ratcheting strain at the extrados where the local wall thinning existed is shown in Fig. 19. The higher level ratcheting strain occurs at deeper wall thinning, and the difference between inner and outer strain increases with increasing depth and increasing cyclic number. It shows that the increase of depth has more effect on hoop ratcheting strain at outside surface, and the ratcheting strain at inner surface is less than that at outer surface throughout. In the elbow with local wall thinning at extrados, both of extrados area and intrados area have obvious hoop ratcheting strain. In order to evaluate the effect of different depths of local wall
Fig. 19. Simulated hoop ratcheting strain with different depths of wall thinning at extrados.
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Fig. 20. Simulated hoop ratcheting strain with different depths of wall thinning at crown.
thinning on the location of the largest hoop ratcheting strain, the comparison of ratcheting strain of intrados outer and extrados outer is shown in Fig. 19. The hoop ratcheting strain of intrados outer is not affected by different depths of wall thinning, and it is shown as the red cross in Fig. 19. It shows that when the depth of wall thinning is more than 1.12 mm, i.e., wall thinning ratio d/ t z 0.25, the hoop ratcheting strain at extrados outer is larger than that at intrados outer. Therefore, the largest hoop ratcheting strain occurs at extrados when wall thinning ratio is more than 0.25. Fig. 20 shows that the hoop ratcheting strain increases with increasing depth of wall thinning at crown area. It seems that the higher ratcheting strain tends to be easier to occur at outer surface than inner surface. The results in Section 3.3 show that the largest hoop ratcheting strain occurs at crown area or intrados area, therefore the hoop ratcheting strain at intrados and crown is investigated emphatically. The comparison of hoop ratcheting strain at crown outer and intrados outer is shown in Fig. 20. The hoop ratcheting strain at intrados outer is not affected by the depth of local wall thinning at crown and it is shown as the red cross in Fig. 20. When the depth of wall thinning is more than 1 mm, i.e., wall thinning ratio d=tz0:22, the hoop ratcheting strain at crown outer is larger than that at intrados outer. Therefore, the largest hoop ratcheting strain occurs at external wall of crown when local wall thinning ratio is more than 0.22. Figs. 19 and 20 show that the higher level ratcheting strain occurs at deeper wall thinning, and the ratcheting strain at inner surface is less than that at outer surface throughout, the difference between inner and outer surface becomes more with the increase of depth. The depth of local wall thinning also affects the location of the largest hoop ratcheting strain. When wall thinning ratio of wall thinning at extrados is greater than 0.25, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados. When wall thinning ratio of wall thinning at crown is more than 0.22, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados. 4. Conclusions With multi-axial test machine, ratcheting behavior is studied experimentally for pressurized Z2CND18.12N stainless steel elbows under constant internal pressure and reversed in-plane bending under load control. The local wall thinning is machined on the
outside of elbow in order to simulate metal loss. The hoop ratcheting strains obviously occur at the intrados, 45 position and crown while the axial ratcheting strains are considerably low. In sound elbow and the elbow with local wall thinning at extrados, the largest hoop ratcheting strain mainly occurs at intrados. In the elbow with local wall thinning at crown, the largest hoop ratcheting strain occurs at crown. The local wall thinning causes a significant ratcheting strain rate during the initial several cycles of bending and a higher ratcheting stain level. Ratcheting simulation is performed by EPFEA with ANSYS in which Chaboche model and CJK model are coded by user programming. Comparing the experimental data and the predicted results by Chaboche model and CJK model, it is found that CJK model has more accurately reproduced the experimental results. The ratcheting boundary of extrados area and crown area where local wall thinning existed is determined by C-TDF method with CJK model. The effect of depth of local wall thinning on the ratcheting response is investigated by CJK model. It is found that the higher ratcheting strain level and higher ratcheting strain rate occur at deeper wall thinning, and the difference between the strain of internal wall and the strain of external wall increases with increasing depth and increasing cyclic number of bending. The depth of local wall thinning also affects the location of the largest hoop ratcheting strain. When wall thinning ratio of wall thinning at extrados is greater than 0.25, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados. When wall thinning ratio of wall thinning at crown is more than 0.22, the largest hoop ratcheting strain occurs at local wall thinning, otherwise, it occurs at intrados.
Acknowledgments The authors gratefully acknowledge financial support for this work from the National High Technology Research and Development Program of China (863 Program 2009AA04Z403) and Ph.D. Programs Foundation of Ministry of Education of China (No. 20090032110016).
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