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Failure criterion for steel pipe elbows under cyclic loading
EFA-02899; No of Pages 11 Engineering Failure Analysis xxx (2016) xxx–xxx
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Failure criterion for steel pipe elbows under cyclic loading Ehsan Salimi Firoozabad a, Bub-Gyu Jeon b, Hyoung-Suk Choi b, Nam-Sik Kim a,⁎ a b
Department of Civil and Environmental Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Korea KOCED Seismic Simulation Test Center, Pusan National University, Yangsan Campus Mulgeum, Yangsan, Kyungsangnam, Republic of Korea
a r t i c l e
i n f o
Article history: Received 8 January 2016 Received in revised form 18 April 2016 Accepted 5 May 2016 Available online xxxx Keywords: Steel pipe elbow Failure criteria Damage index Cyclic loading
a b s t r a c t Steel elbow components are considered to be critical parts in industrial piping system due to their probability of collapse or failure. Therefore, the structural behavior of elbows is considered with respect to failure criteria through experiments and corresponding numerical models. Thirty-eight sets of experiments were conducted on three inch pipe elbow specimens. The numerical simulation results of the specimens are in good agreement with the test results. Damage indices available in the literature are used for failure estimation of the elbows. We suggest that the damage calculated using the Park and Ang damage index, and the Banon damage index, based on only one failed specimen under any constant amplitude cyclic loading, can be defined as the failure point and used to predict the failure of the component under other loading amplitudes. Therefore, the low cycle fatigue curve of an elbow can be derived using these simulation results. We also found that the calculated damage of an elbow component under constant, non-constant, and fully or partial amplitude reversals is quite similar. © 2016 Published by Elsevier Ltd.
1. Introduction Steel pipe elbows are used in many different applications (manufacturing, hydraulics, refineries, offshore engineering, power plant construction, and other steam systems) to convey fluids such as gas, water, and oil. Elbows are considered to be critical parts in piping systems. Piping systems are often exposed to cyclic loading due to earthquakes, wind, waves, and vibrations from industrial machinery. It is well known that piping components frequently fail due to a fatigue-ratchet mechanism, rather than a plastic collapse, under reversing dynamic loads [1]. Therefore, the fatigue behavior of a pipe elbow under cyclic loading was investigated in this study in order to understand its failure criteria. Elbow components have been reported to be the most critical points of nuclear pipelines based on experimental [2] and analytical [3] investigations and also vulnerable component of piping systems in erosive environments [4]. The failure analysis of various type of steel pipe elbow due to the thermal fatigue [5], stress corrosion cracking [6], erosion [7] and buckling of an axially cracked elbow [8] has been studied in the literature. Extensive experimental studies on the structural behavior of steel elbows under monotonic loading [9–11] and cyclic loading [12–15] have been performed. Low cycle fatigue analysis and fatigue life analysis of steel elbows have been performed [16,17]. Vishnuvardhan et al. [18] and Hassan et al. [19] examined the ratcheting response and failure of elbow components. Several damage accumulation indices as a function of certain response parameters have been proposed for structural components. It has been suggested that the damage of a structure can be represented as a function of ductility and/or plastic deformation [20,21], the energy dissipation capacity [22], or a combination of both [23–25]. Bracci et al. [26] estimated the damage based on the ratio of damage consumption to damage capacity. Consenza et al. [27] defined their index as a ratio of maximum induced
⁎ Corresponding author. E-mail addresses: [email protected] (E. Salimi Firoozabad), [email protected] (B.-G. Jeon), [email protected] (H.-S. Choi), [email protected] (N.-S. Kim).
http://dx.doi.org/10.1016/j.engfailanal.2016.05.012 1350-6307/© 2016 Published by Elsevier Ltd.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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ductility to the ultimate ductility. Castiglioni [28] proposed that the reduction of the energy absorption capacity is a suitable parameter in the case of a steel component. More recently, Sucuoglu and Erberik [29] developed a hysteresis model for a deteriorating system consisting of displacement and fatigue components, and Kamaris et al. [30] defined the damage of a steel structure based on the acting axial force and bending moment on the structure. The structural failure point of a steel pipe elbow has yet to be defined. Thus, we investigated the possibility of estimating the failure point of an elbow using the damage indices available in the literature. These damage indices have either been used for reinforced concrete structures or steel structures (not a steel pipe elbow). Therefore, the applicability and correctness of these proposed indices for steel pipe elbows was carried out. Thirty-eight sets of experiments were conducted on 3 in. steel pipe elbow specimens subjected to various loading histories with internal pressure. Numerical analyses of the pipe elbows were performed and compared to the test data. The failure criterion of the elbows was represented as their damage capacity by using applicable damage indices. The procedure was also applied to 8 in. elbow components based on the experimental results reported in Varelis et al. [14]. 2. Test setup and results A tensile stress test of the material used in the specimens was performed in order to determine the elasto-plastic behavior of the material. The measured sizes of the three specimens used in test are given Table 1. The elastic modulus of the material was found to be 204,929 MPa. A photograph of the specimens and the results of the tensile stress tests for three specimens are shown in Fig. 1. A total of 38 specimens were made for the experiments (ASME [31] B36.10, carbon steel, weld pipe, SA-106, SCH. 40 (STD), diameter = 88.9 mm, thickness = 5.49 mm). The specimen cross sectional details, material descriptions, and photos are shown in Fig. 2. The first two tests were performed on an elbow subjected to monotonic loading under tension and compression. The next 20 tests were conducted under sine wave constant cyclic loading, subjected to nine different loading amplitudes (from ±20 mm to ±100 mm, as given in Table 2 Nos. 4 to 12). The next two tests were separately subjected to a constant closing in one case (compression), and opening the other case (tension), to observe elbow behavior under compression and/or tension. The other loading histories were chosen based on non-constant stepwise increasing cyclic loading according to the European Convention for Constructional Steelwork (ECCS) recommendations [32] (Table 2 No. 15, Fig. 3). Then, a random history starting with a small displacement (30 mm) under six cycles, increasing to 80 mm for three cycles, and then back to 30 mm were applied until failure to see the effect of geometric nonlinearity and a sudden increase of applied loading. The last case was considered for partial deformation reversals (30 mm closing following by 60 mm opening), as it is known that most cyclic loads in nature do not have fully symmetric amplitude loading. Descriptions of the loading histories are given in Table 2. All test specimens were subjected to an internal pressure of 3 MPa, and this pressure was maintained during the experiments. The experiments were repeated three times for each case of loading amplitude to reduce experimental errors and obtain more reliable results. Therefore, the number of cycles to failure (given in Table 2) is the average of results from three conducted experiments. In all the experiments, the pipe cracked and leaked at the same point, located on the outside middle (crown) of the elbow in the opening modes of cyclic loading, as shown in Fig. 4. The same results (i.e., the cracked area) are also reported in the literature [14,15,18] based on experiments with different sizes of pipe elbows. Our numerical analysis shows that the maximum strain concentration occurs in the same area, as we expected and observed during the experiments. 3. Numerical simulation The pipe elbow was modeled using finite element shell elements (shown in Fig. 5) with a beam stick model for the load point at both ends of the elbow. The beam stick length was 60 mm as the original test specimen load point, and it was coupled with the elbow structure. The material properties were obtained from the tensile test (shown in Fig. 1), the kinematic hardening rule was chosen and Poisson's ratio is 0.3. A quadrilateral standard shell finite element (S4R) was used in the analysis. The geometric nonlinearity effect for the cyclic loading was also considered in order to capture the stiffness and strength degradation. The static analysis was performed using ABAQUS 6.12 for the cyclic and monotonic loading tests, and the elbow was subjected to a 3 MPa internal pressure. Monotonic loading of the elbow was performed under both tension and compression, hereafter called Table 1 Tensile test specimens size description. Test specimen
No. 1
No. 2
No. 3
Width (mm)
18.95 18.97 18.99 19.01 19.02 5.48 5.48 5.48
19.09 19.02 19.02 19.03 19.05 5.51 5.5 5.49
18.95 18.94 18.93 18.92 18.92 5.66 5.68 5.66
Thickness (mm)
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Fig. 1. Tensile test specimens and results.
the opening mode and closing mode, respectively. The boundary conditions were assigned to the model according to the experiments, one end of the elbow is fixed and the other end is the location of applied loads in Y direction (Fig. 5). The simulation results (extracted from the same location that the loads were applied) for the elastic and plastic behavior of the structure are in good agreement with the experimental data. The comparative force-displacement curves for all the specimens
Fig. 2. The elbow specimen cross sectional and set up details.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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Table 2 Test loading description. No.
Mode
Number of performed test
Loading amplitude (mm)
Internal pressure (MPa)
1 2 3
Monotonic closing Monotonic opening Constant increasing cyclic
1 1 1
3 3 3
4 5 6 7 8 9 10 11 12 13 14 15
Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant closing Constant opening Stepwise increasing cyclic (ECCS) ½0:25δy þ ; 0:25δy − × 1 cycle
1 2 3 3 3 3 3 1 1 3 3 3
195 240 ±10 ±20 ±30 ±40 ±50 ±20 ±30 ±40 ±50 ±60 ±70 ±80 ±90 ±100 +60 −60 +1.775 to −2.175 +3.35 to −4.35
3 3 3 3 3 3 3 3 3 3 3 3
Number of cycle
10 5 5 2 2 82 45, 46 18, 19, 19 11, 12, 12 7, 9, 9 5, 6, 6 4, 5, 6 4 4 39, 51, 51 29, 35, 39 1 1
½0:50δy þ ; 0:50δy − × 1 cycle
+5.325 to −6.525
½0:75δy þ ; 0:75δy − × 1 cycle
+7.1 to −8.7
1
½1δy þ ; 1δy − × 1 cycle
+14.2 to −17.4
3
½ð2 þ 2nÞδy þ ; ð2 þ 2nÞδy − With n = 0 , 1 , 2 , … × 3 cycles
16
Random cyclic
3
17
Partial reversal cyclic
3
Leakage No No No
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
1
+28.4 to −34.8
3
+42.6 to −52.2 +56.8 to −69.6 +71 to −87 +85.2 to −104.4 ±30 ±80 ±30 +30 −60
3 3 3 0, 1, 1 6 3 11, 14, 17 13, 16, 18
3
3
Yes
Yes
under the loadings shown in Fig. 6 indicate the reliability of the numerical simulations. We note that the maximum strain concentration is in the leak area seen during the experiments (Fig. 5). 4. Damage analysis The damage of a structure can be assessed by using certain parameters. The most commonly used parameter is ductility, which relates the damage of the structure only to the maximum deformation, and is regarded as a critical design parameter by code provisions. Approaches for the determination of the plastic limit deformation such as: the tangent-intersection method, and the twice-elastic deformation (TES) method as adopted by the American Society of Mechanical Engineers (ASME) [31] for the plastic
Fig. 3. ECCS recommendation for cyclic loading history.
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Fig. 4. Crack location in specimens.
limit load material criterion. The energy dissipation capacity (in cumulative-type indices) has been included and, in some indices, linearly and/or nonlinearly combined with the maximum deformation. Stiffness and strength degradation have been incorporated in other damage indices to account for the effects of cyclic loading. These indices have been proposed mainly for reinforced concrete structures, and some have only been investigated for use with specific steel structures. Hence, in the present study, these indices were considered for use in the case of steel pipe elbows under repeated cyclic loading. Based on our results, it became clear that the failure of a steel pipe elbow cannot be expressed by deformation or hysteretic-based formulations alone (as the calculated damage would not be equal for all cases of loadings). Therefore, combined (deformation and energy dissipated) indices such as those of Park and Ang, and Banon, were examined. These two indices and their formulations for the calculation of damage of the structure (D) are given as follows: • Banon et al. [23]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !!2 !d !2 u N u X Di Ei t D¼ max −1 þ c 2 Dy F y Dy i¼1
ð1Þ
Fig. 5. The drawn pipe elbow and logarithmic strain contour.
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Fig. 6. Force-displacement graphs in the case of 20, 40, 50, 60, 70, 80 mm, ECCS, random and partial reversal respectively.
• Park and Ang [25]: Di D ¼ max Dy
! þb
N X i¼1
Ei F y Dy
! ð2Þ i
where Dy , Fy are the yield displacement and force; Di , Ei are the displacement and dissipated energy (the area under the forcedisplacement curve) in the i-th cycle; and N is the number of cycles. The constants c and d are taken as 1.1 and 0.38, respectively Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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[33] and the constant b can be adopted as 0.025 for steel structures [27]. Since the component behavior in tension and compression was different, the damage parameters were evaluated separately. Hence, the average value was taken for maximum displacement, and the tension and compression dissipated energy were summed in order to determine the energy dissipation capacity. Other damage indices have been recently proposed in the literature such as one suggested by Castiglioni [28], which is based on the reduction of the energy absorption capacity (the application condition did not apply in our study). Sucuoglu and Erberik [29] derived their index based on the energy dissipated at a constant amplitude loading, which is normalized analytically for the case in which a structure has the same response in tension and compression. However, it is not applicable in the case of different behavior in tension and compression, such as a steel pipe elbow. Kamaris et al. [30] accounted for strength and stiffness degradation in low cycle fatigue failure based on the moment and axial force capacity and response. This index is not accurate in our specific structure, which exhibits no failure after the ultimate force in tension.
4.1. Three inch specimens The elbow damage was calculated using the considered indices under all considered loading amplitudes. The results for all cases must be equal or close to each other because all the specimens failed (leaked) under the applied loading history. Closer results mean a more reliable damage index. First, the damage was obtained based on equations, and using available constant values in the literature. The constants b, c, and d, were taken as previously mentioned, and the results were compared as shown in Fig. 7. The maximum displacement presented in the damage index equations must be taken as the average of the induced displacement in closing and opening mode since the behavior of the elbow was different under each mode. The yield point was obtained from the monotonic force-displacement curve, and is defined as the reference elastic limit at the intersection between the tangent at the origin (E) of the force-displacement curve and the tangent that has a slope of E/10. The procedure to use the damage indices to express the failure of the structure was investigated under non-constant loading histories. In this case, one method is to calculate the damage directly from the damage index to predict the failure of the structure, another method is to use the low cycle fatigue curve to predict the number of cycles to failure by using Miner's rule. The component damage under non-constant loading could also be estimated by adding the percentages of damage consumed by each displacement level. Therefore, fifteen specimens were used to perform loading under five different non-constant loading histories, as
Fig. 7. The calculated Park and Ang and Banon damage for all loading case for three inch elbow.
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described in Section 2. It is found that by taking the average of maximum displacements applied to the component in each cycle under both compression and tension, the direct calculation of damages produced results that are close to the damage corresponding to the constant loading amplitude. The other methods were not used because they provided biased results. Adding the damage consumed by each displacement level produces higher rates of damage. Miner's rule also fails to predict the non-reversal or partial reversal amplitude because the number of cycles to failure in closing and opening mode under the same amplitude is different. However, it can fairly predict a fully reversed non-constant loading such as the random case with ±30 and ±80 mm. The damage results calculated for each case of loading are quite similar. Therefore, these damage indices can be used to express the “failure” state of a steel pipe elbow under constant amplitude loading. The main use of the proposed procedure is in low cycle fatigue analysis, which generally needs many experiments in order to derive a reliable low cycle fatigue curve. We suggest that the damage calculated based on one failed specimen under any constant amplitude cyclic loading can be set as the failure criterion. Hence, the number of cycles to failure under different amplitudes can be estimated based on the simulation response on the same structure that previously proved to be reliable. In this scenario, a very accurate low cycle fatigue curve can be derived by using just one experiment through analysis of the structure using the ABAQUS FEM software. The low cycle fatigue curve shown in Fig. 8, compares the experimental and numerical results, for the displacement amplitude. We already know from the experiments that the component failed in 8 cycles under 60 mm displacement. Therefore, the numerical curves given in Fig. 8 were obtained by setting the calculated damage of the elbow under 60 mm displacement as the reference failure. Hence the components subjected to other displacement amplitudes were set to reach the same level of damage as the reference failure. For example, the number of cycles to failure under a 40 mm displacement can be numerically predicted to be 19 cycles which is exactly as we obtained from the experiment. 4.2. Eight inch specimen The proposed procedure was also examined using an 8 in. component based on the experimental results performed at Delft University of Technology, and reported by Varelis et al. [14]. The specimens were subjected to seven constant loading amplitudes and one stepwise increasing loading amplitude, according to the ECCS recommendations [32]. The specimen's properties, their loading histories, and the experimental results are described in further detail by Varelis et al. [14]. We determined that the proposed procedure to use the damage capacity to estimate the failure criterion was also applicable to the 8 in. pipe elbow. The calculated damage capacity of the component under all examined loading amplitudes is shown in Fig. 9. The damage calculated for each case of loading was not as close as in the case of the 3 (based on the comparison between Figs. 7 and 9) in specimens however, the Banon index provides a slightly better result. We also observed that the average damage capacity of the 8 in. component calculated based on all constant and non-constant loading, was almost the same as the 3 in. component. It was also seen that the number of cycles to failure is quite similar from 3″ to 8″ elbows, if the maximum induced displacement is divided by the yield displacement (Fig. 10). The fitting curve for both sizes of elbow is derived with the correlation coefficient (R2) of 0.968. 5. Results and discussion The experiments reveal an interesting point about constant amplitude (±60 mm), and constant closing (+60 mm) and opening (− 60 mm). Although the number of cycles to failure in the three experiments was different, the estimated damage was similar. The energy dissipation capacity in constant closing and opening was almost twice that of the constant amplitude case;
Fig. 8. The low cycle fatigue curve.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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Fig. 9. The calculated Park and Ang and Banon damage for all loading case for eight inch elbow.
however, it was slightly higher in constant opening, during which the ratio of the maximum displacement to the yield displacement is slightly lower. The average estimated damage of the 3 in. elbow component (14.34 Park and Ang, 14.04 Banon) was almost the same as that of the 8 in. elbow (12.67 Park and Ang, 13.56 Banon), considering the difference in material properties, geometry and applied loading histories. Therefore, the estimated damage corresponding to the failure of the component under constant loading amplitude can be set as the failure point and used in low cycle fatigue analysis. In addition, this procedure works fairly well in the case of non-constant cyclic loading. Thus, it can be used in earthquake analysis. Both the 3 and 8 inch results were analyzed in order to obtain an optimized value for the constants existed in the damage indices equations. The constant b in the Park and Ang damage equation could not be optimized as the results didn't get better. On the other hand, constants c, and d in the Banon index was optimized to the values 3.3 and 0.21 respectively (the results are shown in Fig. 11). It must be noted that the optimization performed so that the average value wouldn't change after the optimization. It is possible to define the criterion for failure of an elbow component according to the average estimated damage. This approach would eliminate the need for many experiments (as previously required) to derive the low cycle fatigue curve. The proposed method will also help to overcome conservatism issues such as the twice-elastic slope (TES) recommended as the failure point by code provisions. Such recommendation cannot consider the cyclic behavior of the structure and takes a very conservative limitation point in order to be safely designed. The experiments have shown that the elbow plastic limit is far after TES based on monotonic loading and also needs for an example 82 cycles to fail subjected to 20 mm (is more than TES limit) displacement amplitude. The proposed method can also be applied to a probabilistic approach such as fragility analysis, because it estimates the median and logarithmic standard deviation of structural capacity.
Fig. 10. The low cycle fatigue fitting curve for 3 and 8 inch elbow.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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Fig. 11. Banon damage after the optimization, for all loading cases 3 and 8 in. elbow (the red columns are the averages for 3 and 8 in. respectively). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6. Conclusion The failure criteria of steel pipe elbow components were studied through experiments and corresponding numerical models. Finite element-based numerical simulations of the component under monotonic and cyclic loading were performed, and the results are found to be in good agreement with the test data. The failure criteria of the elbow components were represented as the damage capacity by using damage indices available in the literature. It is shown that the damages calculated through the Park and Ang index, and the Banon damage index, for all the considered loading histories are very similar. Hence, the failure point of a steel pipe elbow can be based on the calculated damage of the failed specimen for a constant amplitude loading. The applied procedure worked properly on both the 3 and 8 in. specimens. Thus, we conclude that the failure criterion of a pipe elbow component can be assessed based on its damage capacity. However, further investigations on different elbow sizes are recommended for a more comprehensive conclusion. We suggest that damage calculated based on only one failed specimen under any constant amplitude cyclic loading can be used as the failure point. Hence, the number of cycles to failure under any other amplitude can be estimated based on the numerical simulation response. In this scenario, an accurate low cycle fatigue curve can be derived by using just one experiment through analysis of the structure using a numerical model. Acknowledgement This work was supported by the energy efficiency and resources of Korea Institute of Energy Technology Evaluation and Planning (KIETEP) grant, funded by the Ministry of Knowledge Economy (MKE) of the Korean government (no. 2014151010170C). The authors gratefully acknowledge this support. References [1] R.W. Barnes, S.W. Tagart Jr., E.B. Branch, D.F. Landers, Proposed Revision to Section III Seismic Piping Rules, Seismic Engineering–1994: Volume 1. PVP-Volume 275–1, 1994. [2] F. Touboul, P. Sollogoub, N. Blay, Seismic behaviour of piping systems with and without defects: experimental and numerical evaluations, Nucl. Eng. Des. 192 (2) (1999) 243–260. [3] E.S. Firoozabad, B.G. Jeon, H.S. Choi, N.S. Kim, Seismic fragility analysis of seismically isolated nuclear power plants piping system, Nucl. Eng. Des. 284 (2015) 264–279. [4] H. Pouraria, J.K. Seo, J.K. Paik, Numerical study of erosion in critical components of subsea pipeline: tees vs bends, Ships Offshore Struct. (2016) 1–11. [5] A. Li, W. Wang, X. Wang, D. Zhao, Fatigue and brittle fracture of carbon steel process pipeline, Eng. Fail. Anal. 12 (4) (2005) 527–536. [6] S.L. Jiang, Y.G. Zheng, D.L. Duan, Failure analysis on weld joints between the elbow and straight pipes of a vacuum evaporator outlet, Eng. Fail. Anal. 27 (2013) 203–212. [7] N. Fujisawa, K. Wada, T. Yamagata, Numerical analysis on the wall-thinning rate of a bent pipe by liquid droplet impingement erosion, Eng. Fail. Anal. (2016). [8] Y.S. Yoo, N.S. Huh, On a leak-before-break assessment methodology for piping systems of fast breeder reactor, Eng. Fail. Anal. 33 (2013) 439–448. [9] P. Hilsenkopf, B. Boneh, P. Sollogoub, Experimental study of behavior and functional capability of ferritic steel elbows and austenitic stainless steel thin-walled elbows, Int. J. Press. Vessel. Pip. 33 (2) (1988) 111–128. [10] N. Suzuki, M. Nasu, Non-linear analysis of welded elbows subjected to in-plane bending, Comput. Struct. 32 (3) (1989) 871–881. [11] Y. Tan, V.C. Matzen, L. Yu, Correlation of test and FEA results for the nonlinear behavior of straight pipes and elbows, J. Press. Vessel. Technol. 124 (4) (2002) 465–475. [12] K. Yahiaoui, D.G. Moffat, D.N. Moreton, Response and cyclic strain accumulation of pressurized piping elbows under dynamic in-plane bending, J. Strain Anal. Eng. Des. 31 (2) (1996) 135–151. [13] G.C. Slagis, Experimental data on seismic response of piping components, J. Press. Vessel. Technol. 120 (4) (1998) 449–455. [14] G.E. Varelis, S.A. Karamanos, A.M. Gresnigt, Pipe elbows under strong cyclic loading, J. Press. Vessel. Technol. 135 (1) (2013) 011207. [15] K. Takahashi, S. Watanabe, K. Ando, Y. Urabe, A. Hidaka, M. Hisatsune, K. Miyazaki, Low cycle fatigue behaviors of elbow pipe with local wall thinning, Nucl. Eng. Des. 239 (12) (2009) 2719–2727. [16] K. Takahashi, K. Ando, K. Matsuo, Y. Urabe, Estimation of low-cycle fatigue life of elbow pipes considering the multi-axial stress effect, J. Press. Vessel. Technol. 136 (4) (2014) 041405. [17] G.E. Varelis, S.A. Karamanos, Low-cycle fatigue of pressurized steel elbows under in-plane bending, J. Press. Vessel. Technol. 137 (1) (2015) 011401.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Failure criterion for steel pipe elbows under cyclic loading Ehsan Salimi Firoozabad a, Bub-Gyu Jeon b, Hyoung-Suk Choi b, Nam-Sik Kim a,⁎ a b
Department of Civil and Environmental Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Korea KOCED Seismic Simulation Test Center, Pusan National University, Yangsan Campus Mulgeum, Yangsan, Kyungsangnam, Republic of Korea
a r t i c l e
i n f o
Article history: Received 8 January 2016 Received in revised form 18 April 2016 Accepted 5 May 2016 Available online xxxx Keywords: Steel pipe elbow Failure criteria Damage index Cyclic loading
a b s t r a c t Steel elbow components are considered to be critical parts in industrial piping system due to their probability of collapse or failure. Therefore, the structural behavior of elbows is considered with respect to failure criteria through experiments and corresponding numerical models. Thirty-eight sets of experiments were conducted on three inch pipe elbow specimens. The numerical simulation results of the specimens are in good agreement with the test results. Damage indices available in the literature are used for failure estimation of the elbows. We suggest that the damage calculated using the Park and Ang damage index, and the Banon damage index, based on only one failed specimen under any constant amplitude cyclic loading, can be defined as the failure point and used to predict the failure of the component under other loading amplitudes. Therefore, the low cycle fatigue curve of an elbow can be derived using these simulation results. We also found that the calculated damage of an elbow component under constant, non-constant, and fully or partial amplitude reversals is quite similar. © 2016 Published by Elsevier Ltd.
1. Introduction Steel pipe elbows are used in many different applications (manufacturing, hydraulics, refineries, offshore engineering, power plant construction, and other steam systems) to convey fluids such as gas, water, and oil. Elbows are considered to be critical parts in piping systems. Piping systems are often exposed to cyclic loading due to earthquakes, wind, waves, and vibrations from industrial machinery. It is well known that piping components frequently fail due to a fatigue-ratchet mechanism, rather than a plastic collapse, under reversing dynamic loads [1]. Therefore, the fatigue behavior of a pipe elbow under cyclic loading was investigated in this study in order to understand its failure criteria. Elbow components have been reported to be the most critical points of nuclear pipelines based on experimental [2] and analytical [3] investigations and also vulnerable component of piping systems in erosive environments [4]. The failure analysis of various type of steel pipe elbow due to the thermal fatigue [5], stress corrosion cracking [6], erosion [7] and buckling of an axially cracked elbow [8] has been studied in the literature. Extensive experimental studies on the structural behavior of steel elbows under monotonic loading [9–11] and cyclic loading [12–15] have been performed. Low cycle fatigue analysis and fatigue life analysis of steel elbows have been performed [16,17]. Vishnuvardhan et al. [18] and Hassan et al. [19] examined the ratcheting response and failure of elbow components. Several damage accumulation indices as a function of certain response parameters have been proposed for structural components. It has been suggested that the damage of a structure can be represented as a function of ductility and/or plastic deformation [20,21], the energy dissipation capacity [22], or a combination of both [23–25]. Bracci et al. [26] estimated the damage based on the ratio of damage consumption to damage capacity. Consenza et al. [27] defined their index as a ratio of maximum induced
⁎ Corresponding author. E-mail addresses: [email protected] (E. Salimi Firoozabad), [email protected] (B.-G. Jeon), [email protected] (H.-S. Choi), [email protected] (N.-S. Kim).
http://dx.doi.org/10.1016/j.engfailanal.2016.05.012 1350-6307/© 2016 Published by Elsevier Ltd.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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ductility to the ultimate ductility. Castiglioni [28] proposed that the reduction of the energy absorption capacity is a suitable parameter in the case of a steel component. More recently, Sucuoglu and Erberik [29] developed a hysteresis model for a deteriorating system consisting of displacement and fatigue components, and Kamaris et al. [30] defined the damage of a steel structure based on the acting axial force and bending moment on the structure. The structural failure point of a steel pipe elbow has yet to be defined. Thus, we investigated the possibility of estimating the failure point of an elbow using the damage indices available in the literature. These damage indices have either been used for reinforced concrete structures or steel structures (not a steel pipe elbow). Therefore, the applicability and correctness of these proposed indices for steel pipe elbows was carried out. Thirty-eight sets of experiments were conducted on 3 in. steel pipe elbow specimens subjected to various loading histories with internal pressure. Numerical analyses of the pipe elbows were performed and compared to the test data. The failure criterion of the elbows was represented as their damage capacity by using applicable damage indices. The procedure was also applied to 8 in. elbow components based on the experimental results reported in Varelis et al. [14]. 2. Test setup and results A tensile stress test of the material used in the specimens was performed in order to determine the elasto-plastic behavior of the material. The measured sizes of the three specimens used in test are given Table 1. The elastic modulus of the material was found to be 204,929 MPa. A photograph of the specimens and the results of the tensile stress tests for three specimens are shown in Fig. 1. A total of 38 specimens were made for the experiments (ASME [31] B36.10, carbon steel, weld pipe, SA-106, SCH. 40 (STD), diameter = 88.9 mm, thickness = 5.49 mm). The specimen cross sectional details, material descriptions, and photos are shown in Fig. 2. The first two tests were performed on an elbow subjected to monotonic loading under tension and compression. The next 20 tests were conducted under sine wave constant cyclic loading, subjected to nine different loading amplitudes (from ±20 mm to ±100 mm, as given in Table 2 Nos. 4 to 12). The next two tests were separately subjected to a constant closing in one case (compression), and opening the other case (tension), to observe elbow behavior under compression and/or tension. The other loading histories were chosen based on non-constant stepwise increasing cyclic loading according to the European Convention for Constructional Steelwork (ECCS) recommendations [32] (Table 2 No. 15, Fig. 3). Then, a random history starting with a small displacement (30 mm) under six cycles, increasing to 80 mm for three cycles, and then back to 30 mm were applied until failure to see the effect of geometric nonlinearity and a sudden increase of applied loading. The last case was considered for partial deformation reversals (30 mm closing following by 60 mm opening), as it is known that most cyclic loads in nature do not have fully symmetric amplitude loading. Descriptions of the loading histories are given in Table 2. All test specimens were subjected to an internal pressure of 3 MPa, and this pressure was maintained during the experiments. The experiments were repeated three times for each case of loading amplitude to reduce experimental errors and obtain more reliable results. Therefore, the number of cycles to failure (given in Table 2) is the average of results from three conducted experiments. In all the experiments, the pipe cracked and leaked at the same point, located on the outside middle (crown) of the elbow in the opening modes of cyclic loading, as shown in Fig. 4. The same results (i.e., the cracked area) are also reported in the literature [14,15,18] based on experiments with different sizes of pipe elbows. Our numerical analysis shows that the maximum strain concentration occurs in the same area, as we expected and observed during the experiments. 3. Numerical simulation The pipe elbow was modeled using finite element shell elements (shown in Fig. 5) with a beam stick model for the load point at both ends of the elbow. The beam stick length was 60 mm as the original test specimen load point, and it was coupled with the elbow structure. The material properties were obtained from the tensile test (shown in Fig. 1), the kinematic hardening rule was chosen and Poisson's ratio is 0.3. A quadrilateral standard shell finite element (S4R) was used in the analysis. The geometric nonlinearity effect for the cyclic loading was also considered in order to capture the stiffness and strength degradation. The static analysis was performed using ABAQUS 6.12 for the cyclic and monotonic loading tests, and the elbow was subjected to a 3 MPa internal pressure. Monotonic loading of the elbow was performed under both tension and compression, hereafter called Table 1 Tensile test specimens size description. Test specimen
No. 1
No. 2
No. 3
Width (mm)
18.95 18.97 18.99 19.01 19.02 5.48 5.48 5.48
19.09 19.02 19.02 19.03 19.05 5.51 5.5 5.49
18.95 18.94 18.93 18.92 18.92 5.66 5.68 5.66
Thickness (mm)
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Fig. 1. Tensile test specimens and results.
the opening mode and closing mode, respectively. The boundary conditions were assigned to the model according to the experiments, one end of the elbow is fixed and the other end is the location of applied loads in Y direction (Fig. 5). The simulation results (extracted from the same location that the loads were applied) for the elastic and plastic behavior of the structure are in good agreement with the experimental data. The comparative force-displacement curves for all the specimens
Fig. 2. The elbow specimen cross sectional and set up details.
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Table 2 Test loading description. No.
Mode
Number of performed test
Loading amplitude (mm)
Internal pressure (MPa)
1 2 3
Monotonic closing Monotonic opening Constant increasing cyclic
1 1 1
3 3 3
4 5 6 7 8 9 10 11 12 13 14 15
Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant cyclic Constant closing Constant opening Stepwise increasing cyclic (ECCS) ½0:25δy þ ; 0:25δy − × 1 cycle
1 2 3 3 3 3 3 1 1 3 3 3
195 240 ±10 ±20 ±30 ±40 ±50 ±20 ±30 ±40 ±50 ±60 ±70 ±80 ±90 ±100 +60 −60 +1.775 to −2.175 +3.35 to −4.35
3 3 3 3 3 3 3 3 3 3 3 3
Number of cycle
10 5 5 2 2 82 45, 46 18, 19, 19 11, 12, 12 7, 9, 9 5, 6, 6 4, 5, 6 4 4 39, 51, 51 29, 35, 39 1 1
½0:50δy þ ; 0:50δy − × 1 cycle
+5.325 to −6.525
½0:75δy þ ; 0:75δy − × 1 cycle
+7.1 to −8.7
1
½1δy þ ; 1δy − × 1 cycle
+14.2 to −17.4
3
½ð2 þ 2nÞδy þ ; ð2 þ 2nÞδy − With n = 0 , 1 , 2 , … × 3 cycles
16
Random cyclic
3
17
Partial reversal cyclic
3
Leakage No No No
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
1
+28.4 to −34.8
3
+42.6 to −52.2 +56.8 to −69.6 +71 to −87 +85.2 to −104.4 ±30 ±80 ±30 +30 −60
3 3 3 0, 1, 1 6 3 11, 14, 17 13, 16, 18
3
3
Yes
Yes
under the loadings shown in Fig. 6 indicate the reliability of the numerical simulations. We note that the maximum strain concentration is in the leak area seen during the experiments (Fig. 5). 4. Damage analysis The damage of a structure can be assessed by using certain parameters. The most commonly used parameter is ductility, which relates the damage of the structure only to the maximum deformation, and is regarded as a critical design parameter by code provisions. Approaches for the determination of the plastic limit deformation such as: the tangent-intersection method, and the twice-elastic deformation (TES) method as adopted by the American Society of Mechanical Engineers (ASME) [31] for the plastic
Fig. 3. ECCS recommendation for cyclic loading history.
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Fig. 4. Crack location in specimens.
limit load material criterion. The energy dissipation capacity (in cumulative-type indices) has been included and, in some indices, linearly and/or nonlinearly combined with the maximum deformation. Stiffness and strength degradation have been incorporated in other damage indices to account for the effects of cyclic loading. These indices have been proposed mainly for reinforced concrete structures, and some have only been investigated for use with specific steel structures. Hence, in the present study, these indices were considered for use in the case of steel pipe elbows under repeated cyclic loading. Based on our results, it became clear that the failure of a steel pipe elbow cannot be expressed by deformation or hysteretic-based formulations alone (as the calculated damage would not be equal for all cases of loadings). Therefore, combined (deformation and energy dissipated) indices such as those of Park and Ang, and Banon, were examined. These two indices and their formulations for the calculation of damage of the structure (D) are given as follows: • Banon et al. [23]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !!2 !d !2 u N u X Di Ei t D¼ max −1 þ c 2 Dy F y Dy i¼1
ð1Þ
Fig. 5. The drawn pipe elbow and logarithmic strain contour.
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Fig. 6. Force-displacement graphs in the case of 20, 40, 50, 60, 70, 80 mm, ECCS, random and partial reversal respectively.
• Park and Ang [25]: Di D ¼ max Dy
! þb
N X i¼1
Ei F y Dy
! ð2Þ i
where Dy , Fy are the yield displacement and force; Di , Ei are the displacement and dissipated energy (the area under the forcedisplacement curve) in the i-th cycle; and N is the number of cycles. The constants c and d are taken as 1.1 and 0.38, respectively Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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[33] and the constant b can be adopted as 0.025 for steel structures [27]. Since the component behavior in tension and compression was different, the damage parameters were evaluated separately. Hence, the average value was taken for maximum displacement, and the tension and compression dissipated energy were summed in order to determine the energy dissipation capacity. Other damage indices have been recently proposed in the literature such as one suggested by Castiglioni [28], which is based on the reduction of the energy absorption capacity (the application condition did not apply in our study). Sucuoglu and Erberik [29] derived their index based on the energy dissipated at a constant amplitude loading, which is normalized analytically for the case in which a structure has the same response in tension and compression. However, it is not applicable in the case of different behavior in tension and compression, such as a steel pipe elbow. Kamaris et al. [30] accounted for strength and stiffness degradation in low cycle fatigue failure based on the moment and axial force capacity and response. This index is not accurate in our specific structure, which exhibits no failure after the ultimate force in tension.
4.1. Three inch specimens The elbow damage was calculated using the considered indices under all considered loading amplitudes. The results for all cases must be equal or close to each other because all the specimens failed (leaked) under the applied loading history. Closer results mean a more reliable damage index. First, the damage was obtained based on equations, and using available constant values in the literature. The constants b, c, and d, were taken as previously mentioned, and the results were compared as shown in Fig. 7. The maximum displacement presented in the damage index equations must be taken as the average of the induced displacement in closing and opening mode since the behavior of the elbow was different under each mode. The yield point was obtained from the monotonic force-displacement curve, and is defined as the reference elastic limit at the intersection between the tangent at the origin (E) of the force-displacement curve and the tangent that has a slope of E/10. The procedure to use the damage indices to express the failure of the structure was investigated under non-constant loading histories. In this case, one method is to calculate the damage directly from the damage index to predict the failure of the structure, another method is to use the low cycle fatigue curve to predict the number of cycles to failure by using Miner's rule. The component damage under non-constant loading could also be estimated by adding the percentages of damage consumed by each displacement level. Therefore, fifteen specimens were used to perform loading under five different non-constant loading histories, as
Fig. 7. The calculated Park and Ang and Banon damage for all loading case for three inch elbow.
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described in Section 2. It is found that by taking the average of maximum displacements applied to the component in each cycle under both compression and tension, the direct calculation of damages produced results that are close to the damage corresponding to the constant loading amplitude. The other methods were not used because they provided biased results. Adding the damage consumed by each displacement level produces higher rates of damage. Miner's rule also fails to predict the non-reversal or partial reversal amplitude because the number of cycles to failure in closing and opening mode under the same amplitude is different. However, it can fairly predict a fully reversed non-constant loading such as the random case with ±30 and ±80 mm. The damage results calculated for each case of loading are quite similar. Therefore, these damage indices can be used to express the “failure” state of a steel pipe elbow under constant amplitude loading. The main use of the proposed procedure is in low cycle fatigue analysis, which generally needs many experiments in order to derive a reliable low cycle fatigue curve. We suggest that the damage calculated based on one failed specimen under any constant amplitude cyclic loading can be set as the failure criterion. Hence, the number of cycles to failure under different amplitudes can be estimated based on the simulation response on the same structure that previously proved to be reliable. In this scenario, a very accurate low cycle fatigue curve can be derived by using just one experiment through analysis of the structure using the ABAQUS FEM software. The low cycle fatigue curve shown in Fig. 8, compares the experimental and numerical results, for the displacement amplitude. We already know from the experiments that the component failed in 8 cycles under 60 mm displacement. Therefore, the numerical curves given in Fig. 8 were obtained by setting the calculated damage of the elbow under 60 mm displacement as the reference failure. Hence the components subjected to other displacement amplitudes were set to reach the same level of damage as the reference failure. For example, the number of cycles to failure under a 40 mm displacement can be numerically predicted to be 19 cycles which is exactly as we obtained from the experiment. 4.2. Eight inch specimen The proposed procedure was also examined using an 8 in. component based on the experimental results performed at Delft University of Technology, and reported by Varelis et al. [14]. The specimens were subjected to seven constant loading amplitudes and one stepwise increasing loading amplitude, according to the ECCS recommendations [32]. The specimen's properties, their loading histories, and the experimental results are described in further detail by Varelis et al. [14]. We determined that the proposed procedure to use the damage capacity to estimate the failure criterion was also applicable to the 8 in. pipe elbow. The calculated damage capacity of the component under all examined loading amplitudes is shown in Fig. 9. The damage calculated for each case of loading was not as close as in the case of the 3 (based on the comparison between Figs. 7 and 9) in specimens however, the Banon index provides a slightly better result. We also observed that the average damage capacity of the 8 in. component calculated based on all constant and non-constant loading, was almost the same as the 3 in. component. It was also seen that the number of cycles to failure is quite similar from 3″ to 8″ elbows, if the maximum induced displacement is divided by the yield displacement (Fig. 10). The fitting curve for both sizes of elbow is derived with the correlation coefficient (R2) of 0.968. 5. Results and discussion The experiments reveal an interesting point about constant amplitude (±60 mm), and constant closing (+60 mm) and opening (− 60 mm). Although the number of cycles to failure in the three experiments was different, the estimated damage was similar. The energy dissipation capacity in constant closing and opening was almost twice that of the constant amplitude case;
Fig. 8. The low cycle fatigue curve.
Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012
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Fig. 9. The calculated Park and Ang and Banon damage for all loading case for eight inch elbow.
however, it was slightly higher in constant opening, during which the ratio of the maximum displacement to the yield displacement is slightly lower. The average estimated damage of the 3 in. elbow component (14.34 Park and Ang, 14.04 Banon) was almost the same as that of the 8 in. elbow (12.67 Park and Ang, 13.56 Banon), considering the difference in material properties, geometry and applied loading histories. Therefore, the estimated damage corresponding to the failure of the component under constant loading amplitude can be set as the failure point and used in low cycle fatigue analysis. In addition, this procedure works fairly well in the case of non-constant cyclic loading. Thus, it can be used in earthquake analysis. Both the 3 and 8 inch results were analyzed in order to obtain an optimized value for the constants existed in the damage indices equations. The constant b in the Park and Ang damage equation could not be optimized as the results didn't get better. On the other hand, constants c, and d in the Banon index was optimized to the values 3.3 and 0.21 respectively (the results are shown in Fig. 11). It must be noted that the optimization performed so that the average value wouldn't change after the optimization. It is possible to define the criterion for failure of an elbow component according to the average estimated damage. This approach would eliminate the need for many experiments (as previously required) to derive the low cycle fatigue curve. The proposed method will also help to overcome conservatism issues such as the twice-elastic slope (TES) recommended as the failure point by code provisions. Such recommendation cannot consider the cyclic behavior of the structure and takes a very conservative limitation point in order to be safely designed. The experiments have shown that the elbow plastic limit is far after TES based on monotonic loading and also needs for an example 82 cycles to fail subjected to 20 mm (is more than TES limit) displacement amplitude. The proposed method can also be applied to a probabilistic approach such as fragility analysis, because it estimates the median and logarithmic standard deviation of structural capacity.
Fig. 10. The low cycle fatigue fitting curve for 3 and 8 inch elbow.
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Fig. 11. Banon damage after the optimization, for all loading cases 3 and 8 in. elbow (the red columns are the averages for 3 and 8 in. respectively). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6. Conclusion The failure criteria of steel pipe elbow components were studied through experiments and corresponding numerical models. Finite element-based numerical simulations of the component under monotonic and cyclic loading were performed, and the results are found to be in good agreement with the test data. The failure criteria of the elbow components were represented as the damage capacity by using damage indices available in the literature. It is shown that the damages calculated through the Park and Ang index, and the Banon damage index, for all the considered loading histories are very similar. Hence, the failure point of a steel pipe elbow can be based on the calculated damage of the failed specimen for a constant amplitude loading. The applied procedure worked properly on both the 3 and 8 in. specimens. Thus, we conclude that the failure criterion of a pipe elbow component can be assessed based on its damage capacity. However, further investigations on different elbow sizes are recommended for a more comprehensive conclusion. We suggest that damage calculated based on only one failed specimen under any constant amplitude cyclic loading can be used as the failure point. Hence, the number of cycles to failure under any other amplitude can be estimated based on the numerical simulation response. In this scenario, an accurate low cycle fatigue curve can be derived by using just one experiment through analysis of the structure using a numerical model. Acknowledgement This work was supported by the energy efficiency and resources of Korea Institute of Energy Technology Evaluation and Planning (KIETEP) grant, funded by the Ministry of Knowledge Economy (MKE) of the Korean government (no. 2014151010170C). The authors gratefully acknowledge this support. References [1] R.W. Barnes, S.W. Tagart Jr., E.B. Branch, D.F. Landers, Proposed Revision to Section III Seismic Piping Rules, Seismic Engineering–1994: Volume 1. PVP-Volume 275–1, 1994. [2] F. Touboul, P. Sollogoub, N. Blay, Seismic behaviour of piping systems with and without defects: experimental and numerical evaluations, Nucl. Eng. Des. 192 (2) (1999) 243–260. [3] E.S. Firoozabad, B.G. Jeon, H.S. Choi, N.S. Kim, Seismic fragility analysis of seismically isolated nuclear power plants piping system, Nucl. Eng. Des. 284 (2015) 264–279. [4] H. Pouraria, J.K. Seo, J.K. Paik, Numerical study of erosion in critical components of subsea pipeline: tees vs bends, Ships Offshore Struct. (2016) 1–11. [5] A. Li, W. Wang, X. Wang, D. Zhao, Fatigue and brittle fracture of carbon steel process pipeline, Eng. Fail. Anal. 12 (4) (2005) 527–536. [6] S.L. Jiang, Y.G. Zheng, D.L. Duan, Failure analysis on weld joints between the elbow and straight pipes of a vacuum evaporator outlet, Eng. Fail. Anal. 27 (2013) 203–212. [7] N. Fujisawa, K. Wada, T. Yamagata, Numerical analysis on the wall-thinning rate of a bent pipe by liquid droplet impingement erosion, Eng. Fail. Anal. (2016). [8] Y.S. Yoo, N.S. Huh, On a leak-before-break assessment methodology for piping systems of fast breeder reactor, Eng. Fail. Anal. 33 (2013) 439–448. [9] P. Hilsenkopf, B. Boneh, P. Sollogoub, Experimental study of behavior and functional capability of ferritic steel elbows and austenitic stainless steel thin-walled elbows, Int. J. Press. Vessel. Pip. 33 (2) (1988) 111–128. [10] N. Suzuki, M. Nasu, Non-linear analysis of welded elbows subjected to in-plane bending, Comput. Struct. 32 (3) (1989) 871–881. [11] Y. Tan, V.C. Matzen, L. Yu, Correlation of test and FEA results for the nonlinear behavior of straight pipes and elbows, J. Press. Vessel. Technol. 124 (4) (2002) 465–475. [12] K. Yahiaoui, D.G. Moffat, D.N. Moreton, Response and cyclic strain accumulation of pressurized piping elbows under dynamic in-plane bending, J. Strain Anal. Eng. Des. 31 (2) (1996) 135–151. [13] G.C. Slagis, Experimental data on seismic response of piping components, J. Press. Vessel. Technol. 120 (4) (1998) 449–455. [14] G.E. Varelis, S.A. Karamanos, A.M. Gresnigt, Pipe elbows under strong cyclic loading, J. Press. Vessel. Technol. 135 (1) (2013) 011207. [15] K. Takahashi, S. Watanabe, K. Ando, Y. Urabe, A. Hidaka, M. Hisatsune, K. Miyazaki, Low cycle fatigue behaviors of elbow pipe with local wall thinning, Nucl. Eng. Des. 239 (12) (2009) 2719–2727. [16] K. Takahashi, K. Ando, K. Matsuo, Y. Urabe, Estimation of low-cycle fatigue life of elbow pipes considering the multi-axial stress effect, J. Press. Vessel. Technol. 136 (4) (2014) 041405. [17] G.E. Varelis, S.A. Karamanos, Low-cycle fatigue of pressurized steel elbows under in-plane bending, J. Press. Vessel. Technol. 137 (1) (2015) 011401.
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Please cite this article as: E. Salimi Firoozabad, et al., Failure criterion for steel pipe elbows under cyclic loading, Engineering Failure Analysis (2016), http://dx.doi.org/10.1016/j.engfailanal.2016.05.012