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Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending
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International Journal of Pressure Vessels and Piping 83 (2006) 645–653 www.elsevier.com/locate/ijpvp
Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending Yun-Jae Kima,, Kuk-Hee Leea, Chi-Yong Parkb a
Department of Mechanical Engineering, Korea University, 1-5 Ka, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea b Korea Electric Power Research Institute, Yusung-gu, Daejon 305-380, Republic of Korea Received 24 April 2006; received in revised form 26 June 2006; accepted 3 July 2006
Abstract The present work presents plastic limit load solutions for thin-walled branch junctions under internal pressure and in-plane bending, based on detailed three-dimensional (3-D) finite element (FE) limit analyses using elastic–perfectly plastic materials. To assure reliability of the FE limit loads, modelling issues are addressed first, such as the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. Then the FE limit loads for branch junctions under internal pressure and in-plane bending are compared with existing limit load solutions, and new limit load solutions, improving the accuracy, are proposed based on the FE results. The proposed solutions are valid for ratios of the branch-to-run pipe radius and thickness from 0.4 to 1.0, and the mean radius-tothickness ratio of the run pipe from 10.0 to 20.0. r 2006 Elsevier Ltd. All rights reserved. Keywords: Finite element analysis; Branch junction; Internal pressure; In-plane bending; Limit load
1. Introduction Information on the limit load of piping components is important in structural integrity assessment. Such information is a direct input to estimate the maximum loadcarrying capacity of piping components [1,2]. Furthermore, based on the reference stress approach [3], it can be used to estimate creep rupture and non-linear fracture mechanics parameters (see for instance Refs. [4–6]). Due to its significance, information on limit loads for typical piping components with or without defects is widely available (see for instance Refs. [7–9]). As branch junctions are widely used in plants, numerous works on plastic limit analyses of branch junctions have been reported. Although some analytical works are available [10–14], works based on non-linear finite element (FE) analyses are increasingly popular [15–18], due to complexities associated with the geometry and loading conditions. Two Corresponding author. Tel.: +82 2 3290 3372; fax: +82 2 926 9290.
E-mail address: [email protected] (Y.-J. Kim). 0308-0161/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2006.07.002
issues need to be resolved on plastic limit analyses of branch junctions. As works based on non-linear FE analyses are popular, FE modelling issues should be resolved, but not much information has been given in the literature up to the present. As the second issue, for practical application, reliable limit load solutions need to be developed in a closed-form. Although, some solutions are available for branch junctions, the reliability of existing solutions needs to be checked. The present paper addresses the above two issues. The present work presents plastic limit load solutions for branch junctions under internal pressure and in-plane bending, based on detailed three-dimensional (3-D) FE limit analyses using elastic–perfectly plastic materials. The branch junctions considered in the present work are restricted to thin-wall cylinders. Section 2 briefly reviews existing limit load solutions for branch junctions under internal pressure and in-plane bending. The FE limit analysis is presented in Section 3, including the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. Section 4 compares the FE results with existing solutions and new limit load
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Nomenclature A B C M0 ML P0
r/R 2R/T t/T limit moment of a straight pipe in-plane limit moment of a branch junction limit pressure of a straight pipe
PL R, r t, t s0
limit pressure of a branch junction mean radius of a run (main) pipe and a branch for branch junctions thickness of a run (main) pipe and a branch for branch junctions limiting strength of an elastic–perfectly plastic material
solutions are proposed based on the FE results. The work is concluded in Section 5.
moment applied to the branch pipe), the solution by Xuan et al. [10] is given by
2. Existing limit load solutions: review
ML ¼ M0
Consider the branch junction, depicted in Fig. 1. For the main (run) pipe, the mean radius and thickness are denoted by R and T, respectively, and for the branch pipe, r and t, respectively. It is assumed that the branch junction has no weld or reinforcement around the intersection. In the literature, there are several limit load solutions for branch connections [7–14,19]. Among these, selected solutions, which are believed to be most reliable, are reviewed here. For in-plane bending (with the bending
Fig. 1. Schematics of branch junctions with relevant geometric variables.
p=2 i0:5 , pffiffiffiffi2 C f 1 A þ 0:455f 2 k B þ 0:2385Bf 22 k2 h
(1)
where
1 4 p 3 2 1 1 A ; k ¼ f1 1þ A ; f2 , 3 2 16 1 þ C3 r 2R t ; C¼ A¼ ; B¼ R T T and M0 denotes the limit moment of the branch pipe: M 0 ¼ 4s0 r2 t.
(2)
The above solution was derived from force equilibrium between the limit load and the internal force acting on the intersecting line between the main and branch pipes. In the limiting case of T-0 or R-0, Eq. (1) recovers the exact limit moment for the branch pipe, Eq. (2). Fig. 2 shows variations of the normalized limit moments, ML/M0, with t/T, for R/T ranging from R/T ¼ 10.0 to R/T ¼ 20.0. The results in Fig. 2a are for the limiting case of r/R ¼ t/T, and those in Fig. 2b are for the given r/R ¼ 0.9. The general trend is that for smaller branches (r/Rp0.3), the limit load for the branch junction is the same as that for the branch pipe, and weakening effects occur only for r/R40.3. For larger branches (r/R40.7), the limit load for the branch junction tends to saturate. Xuan et al. [10] compared
Fig. 2. Variations of existing limit loads [10] for branch junctions under in-plane bending: (a) with r/R for the limiting case of r/R ¼ t/T and (b) with t/T for the given r/R ¼ 0.9.
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Eq. (1) with experimental data and with FE data, and found overall good agreement. This will be further discussed in Section 4. Comparison of Eq. (1) with other existing solutions for branch junctions under in-plane bending can be found in Ref. [10]. For internal pressure, Budden and Goodall [19] proposed the following regression equation for the limit pressure, PL, based on FE data [18]: PL R A 1 þ Bð1 r=RÞ 1 Dð0:5 T=RÞ2 (3) ¼ s0 T 1 þ Cð1 t=TÞ with A ¼ 0:641;
B ¼ 0:908;
C ¼ 0:608;
D ¼ 1:422.
Although Budden and Goodall [19] checked the validity of this solution by comparison with analytical solutions by Robinson [12,13] and FE data [18], the main concern seemed to be applications to thick-wall tubes. It can be seen that, in the limiting case of r-0 (t-0), the above solution does not always recover the plastic limit pressure for the tube based on the von Mises yield condition, P0: P0 R 2 ¼ pffiffiffi . s0 T 3
et al. [11] compared Eq. (5) with experimental data and with FE data, and found overall good agreement. This will be further discussed in Section 4. Note that, in the limiting case of r-0 (t-0), the above solution does recover the plastic limit pressure for the run pipe based on the von Mises yield condition, Eq. (4). Fig. 3 compares the normalized limit pressure, PL/P0, from the above two solutions, for R/T ranging from R/T ¼ 10.0 to R/T ¼ 20.0. The results in Fig. 3a are for the limiting case of r/R ¼ t/T, and show that Eq. (3) always gives a lower limit load than Eq. (5). In particular, as Eq. (3) does not give the correct limit load for the limiting case of r-0, the difference between Eq. (3) and Eq. (5) increases as r/R decreases. Fig. 3b compares the limit loads for the given r/R ¼ 0.9. The limit loads from the above two equations are similar, but those from Eq. (5) are slightly higher. Comparison of Eq. (5) with other existing solutions for branch junctions under internal pressure can be found in Ref. [11]. 3. FE limit analyses 3.1. FE model and limit analysis
(4)
More recently, Xuan et al. [11] proposed the following analytical limit pressure solution. PL: PL R 1 ¼ s0 T ð0:25 0:5h1 þ h21 þ 0:79h22 Þ0:5
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(5)
with pffiffiffiffi C 2 h1 ¼ 1 þ 0:145kA Bf 2 þ 0:3185A2 f 1 1 ; AB pffiffiffiffi C 2 h2 ¼ 0:175kA B 1 f2 AB and f1, f2, A, B and C are given in Eq. (1). Similarly to Eq. (1), Eq. (5) was derived from force equilibrium between the limit load and the internal force acting on the intersecting line between the main and branch pipes. Xuan
3-D elastic–perfectly plastic FE analyses of the branch junction, depicted in Fig. 1, were performed using ABAQUS [20]. It is assumed that the branch junction has no weld or reinforcement around the intersection. Regarding the axial length, the half-length of the run pipe is denoted as L and the length of the branch pipe as ‘. The geometric variables (R, T, r, t, L, ‘) were systematically varied, within the ranges 0.4p(r/R, t/T)p1.0 and 10.0pR/ Tp20.0. Note that such ranges correspond to thin-wall cylinders. Materials were assumed to be elastic–perfectly plastic, and non-hardening J2 flow theory was used using a small geometry change continuum FE model. Symmetry conditions were fully utilized in FE models to reduce the computing time. To avoid problems associated with incompressibility, reduced integration elements (element type C3D20R within ABAQUS) were used. Fig. 4 depicts typical FE meshes, employed in the present work; one for
Fig. 3. Variations of existing limit loads [11,19] for branch junctions under internal pressures: (a) with r/R for the limiting case of r/R ¼ t/T and (b) with t/T for the given r/R ¼ 0.9.
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Fig. 5. Typical moment-rotation curves from FE limit analyses of branch junctions under in-plane bending.
Fig. 4. Finite element meshes for: (a) r/R ¼ 0.6, t/T ¼ 0.6, and R/T ¼ 15 and (b) r/R ¼ 1.0, t/T ¼ 1.0, and R/T ¼ 15.
r/R ¼ 0.6, t/T ¼ 0.6 and R/T ¼ 15, and the other for r/R ¼ 1.0, t/T ¼ 1.0, R/T ¼ 15. For all cases, three elements are used through the thickness, and the resulting number of elements and nodes in typical FE meshes ranges from 3949 elements/20,598 nodes to 4914 elements/25,649 nodes, which are believed to be sufficiently fine for the present study. Regarding loading conditions, both internal pressure and in-plane bending moment were considered. For internal pressure, pressure was applied as a distributed load to the inner surface of the FE model, together with axial tensions equivalent to the internal pressure applied at the end of the branch and run pipes to simulate closed ends. To avoid problems associated with convergence in elastic–perfectly plastic calculations, the RIKS option within ABAQUS was invoked. Due to symmetry, only a quarter model was used. For in-plane bending cases, the nodes at the end of the branch pipe were constrained through the MPC (multi-point constraint) option within ABAQUS, and sufficiently large deformation (rotation) was directly applied. Fig. 5 shows typical moment-rotation responses from the present FE limit analyses, from which limit moments can be easily determined. 3.2. Effect of FE modelling on plastic limit loads Before presenting the FE results, some FE modelling issues are worth mentioning, as the FE results could
Fig. 6. Three boundary conditions for branch junctions under in-plane bending: (a) clamped–clamped (CL–CL), (b) clamped–simply supported (CL–SS) and (c) clamped–free (CL–F).
depend on modelling. Particular issues include the effects of the length of the run (main) pipe and the boundary conditions on the FE limit loads. Consider branch connections under in-plane bending, as schematically shown in Fig. 6. Regarding boundary conditions, one could apply the clamped–clamped (CL–CL) condition (Fig. 6a), the clamped–simply supported (CL–SS) condition (Fig. 6b), or the clamped–free (CL–F) condition (Fig. 6c). Regarding the geometric variables, one could vary the relative length of the pipes, either the length of the branch pipe, ‘, or that of the run pipe, L. It is found that, as long as the length of the branch pipe, ‘, is sufficiently long to avoid the end effect
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due to the applied bending, it does not affect the limit load. Thus, the only geometric variable to be considered is the length of the run pipe, L. The effect of the ratio of the length of the run pipe to the radius, L/R, on the FE plastic limit loads is shown in Fig. 7 for branch junctions under in-plane bending, subject to three different boundary conditions. Note that the FE results in Fig. 7 are for the special case of R/r ¼ T/t. The FE plastic limit loads are normalized with respect to the analytical expression, Eq. (1). Overall trends for the clamped–clamped and the clamped–simply supported conditions are similar. The limit loads for these two boundary conditions are the same when the value of L/R is sufficiently large, as expected. For smaller branches up to r/R ¼ 0.8, the limit loads for both cases do not depend on L/R. For larger branches with r/R ¼ 0.9 and r/R ¼ 1.0, the limit loads for both cases increase with decreasing L/R. For a given L/R, the limit load for the clamped–clamped case is higher than that for the clamped–simply supported case, possibly due to the restraint effect. The end rotation in the run pipe is more restrained for the clamped–clamped case than for the clamped–simply supported case, which results in a higher limit load. Note that these results can be supported by the lower bound limit load theorem. As the limit load for a junction with clamped–simply supported conditions
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satisfies equilibrium, yield conditions and traction boundary conditions for that with clamped–clamped conditions, it must be a lower bound to the limit load for clamped–clamped conditions. The results for the clamped–free case for r/Rp0.8 are the same as those for the clamped–clamped and the clamped–simply supported cases. For r/R40.8, however, the trend is opposite; the limit loads decrease with decreasing L/R, and are lower than those for the above two cases. It should be noted, however, that the clamped–free boundary condition is less realistic than the clamped–clamped and the clamped– simply supported conditions. For internal pressure, it is found that both the boundary condition and the length of the run pipe do not affect the plastic limit pressure of the branch junction. The present results suggest that, for the FE analysis of branch junctions under pure bending, care should be exercised in the boundary condition and the length of the run pipe. The limit moment results (for in-plane bending), presented in the next section, will be those for a sufficiently long run pipe under the clamped– clamped case, which is the same as those under clamped–simply supported case. As the clamped–free case is not realistic in practical situations, it is not considered further.
Fig. 7. Effects of the normalized length of the run pipe, L/R, on plastic limit moments of branch junctions under in-plane bending for (a) the clamped–clamped (CL–CL) condition, (b) the clamped–simply supported (CL–SS) condition, and (c) the clamped–free (CL–F) condition.
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et al. [10] compared the limit loads from Eq. (1) with experimental results, and found that predictions according to Eq. (1) were slightly lower than the experimental data by up to 5%. One interesting point is that, for the comparison, they used the flow strength as 1.25 times the yield strength, but experimental loads were measured using the twiceelastic slope method from the load–displacement curves. If they had used the yield strength as the flow strength, the predictions would be lower than the experimental results by up to 30%, which is consistent with the present findings. Although Eq. (1) gives consistently lower limit moments, compared to the FE limit moments, the trend is quite clear. In this respect, a simple factor can be multiplied to Eq. (1) to give better agreement with the FE results. Such a factor is found from the FE results, and the following limit moment solution is proposed for branch junctions under in-plane bending: ML ¼ 4s0 r2 t
p=2 Q i0:5 , pffiffiffiffi2 C f 1 A þ 0:455f 2 k B þ 0:2385Bf 22 k2 h
(6)
where the factor Q is given by Q ¼ 1:11ðC 0:7Þ2 þ 1:18 ¼ 1:11
t T
0:7
2
þ 1:18.
Other factors, f1, f2, A, B and C, are given in Eq. (1). Note that the value of Q in the limiting case of t/T-0 or r/R-0 approaches 0.64, not unity, and thus the proposed expression is not correct in the limiting case of t/T-0 or r/R-0. As the proposed equation is based on the FE data, it should be applied within 0.4p(r/R, t/T) p1.0 and 10.0pR/Tp20.0. The values of Q range from Q ¼ 1.08 (at t/T ¼ 0.4 and 1.0) to Q ¼ 1.18 (at t/T ¼ 0.7) for 0.4pt/ Tp1.0 Fig. 9 compares the proposed limit moment solution, Eq. (6), with the FE results for various cases. The FE limit loads in Fig. 9 are normalized with respect to the proposed solution, Eq. (6), and thus the proximity of the data to unity indicates the accuracy. It shows that Eq. (6) gives lower limit moments than the FE data by less than 10% for all cases considered. Fig. 8. Variations of ratios of the FE limit loads to existing ones [10] for branch junctions under in-plane bending with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
4. Limit loads 4.1. In-plane bending Fig. 8 shows variations of the FE limit loads for branch junctions under in-plane bending with t/T. The FE results in Fig. 8a are for the limiting case of R/r ¼ T/t and those in Fig. 8b are for r/R ¼ 0.6, with the values of t/T ranging from t/T ¼ 0.4 to t/T ¼ 1.0. The FE limit loads are normalized with respect to the analytical expression, Eq. (1). The results in Fig. 8 show that the FE results are always higher than Eq. (1) by up to 30%, and thus Eq. (1) gives conservative moments. It should be noted that Xuan
4.2. Internal pressure Fig. 10 shows variations of the FE limit loads for branch junctions under internal pressure with t/T. The FE results in Fig. 10a are for the limiting case of R/r ¼ T/t and those in Fig. 10b are for r/R ¼ 0.6, with the values of t/T ranging from t/T ¼ 0.4 to t/T ¼ 1.0. Normalization of the FE limit loads in Fig. 10 is worth mentioning. For the data indicated as ‘‘Budden’’, the FE limit loads are normalized with respect to Eq. (3), whereas for those indicated as ‘‘Xuan et al.’’, with respect to Eq. (5). For the data indicated as ‘‘Budden’’, the values are overall higher than unity by up to 40%, suggesting that Eq. (3) gives overall lower limit loads than the FE results. On the other hand, for the data indicated as ‘‘Xuan et al.’’, the values can be lower than unity by up to 20%, suggesting that Eq. (5) can give higher (and thus non-conservative) limit loads than the
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Fig. 9. Variations of ratios of the FE limit loads to the proposed limit load solution, Eq. (6), for branch junctions under in-plane bending with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
Fig. 10. Variations of ratios of the FE limit loads to existing ones [11,19] for branch junctions under internal pressure with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
FE results. It can be also noted that the effect of R/T is less significant for the ‘‘Xuan et al.’’ data than the ‘‘Budden’’ data. It is not clear why Eq. (5) can give non-conservative
limit pressures. It could be due to the difference in yielding mechanisms or to simplifications for analytical derivation, assumed in their work. Xuan et al. [11] compared limit
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loads from Eq. (5) with the FE results and claimed that the predictions from Eq. (5) differed from the FE results by less than 10%. However, careful examination of Fig. 4 in their paper [11] shows that the predictions using Eq. (5) are higher (non-conservative) than the FE results. As the trend of the ‘‘Xuan et al.’’ data is also quite clear, Eq. (5) can be modified to give better agreements with the FE results. Based on the FE results, the following limit moment solution is proposed for branch junctions under internal pressure: PL R Q ¼ s0 T ð0:25 0:5h1 þ h21 þ 0:79h22 Þ0:5
(7)
with Q ¼ 1:32ðA 0:636Þ2 þ 0:906: All other factors, h1, h2, f1, f2, A, B and C, are the same as those given in Eqs. (1) and (5), except k which is now defined by k¼
1 0:78ðC þ 0:55Þ2
(8)
instead of the expression given in Eq. (1). Fig. 11 compares the proposed limit moment solution, Eq. (7), with the FE results for various cases. The FE limit loads in Fig. 11 are normalized with respect to the proposed solution, Eq. (7),
and thus the proximity of the data to unity indicates the accuracy. It shows that Eq. (7) gives lower limit pressures than the FE data by less than 10% for all cases considered. 5. Concluding remarks The present work presents plastic limit load solutions for branch junctions under internal pressure and in-plane bending, based on detailed FE limit analyses using elastic–perfectly plastic materials. To assure reliability of the FE limit loads, modelling issues are addressed first, such as the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. It is shown that for branch junctions under in-plane bending, FE limit loads can be affected by kinematic boundary conditions, particularly when the ratio of the mean radius of the branch pipe to that of the run pipe is greater than 0.8, and thus caution should be exercised. Based on systematic FE analyses for branch junctions with 0.4p(r/R, t/T) p1.0 and 10.0pR/Tp20.0 under internal pressure and in-plane bending, the FE limit loads are determined and compared with existing limit load solutions by Xuan et al. [10,11] and Budden and Goodall [19]. For in-plane bending, it is found that the analytical solution given by Xuan et al. [10] underestimates the FE
Fig. 11. Variations of ratios of the FE limit loads to the proposed limit load solution, Eq. (7), for branch junctions under internal pressure with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
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limit loads by up to 30%. For internal pressure, on the other hand, their solution in Ref. [11] could overestimate the FE limit loads by up to 20%, whereas the solution by Budden and Goodall [19] overall underestimates by up to 40%. Based on the FE results, new limit load solutions are proposed for internal pressure and for in-plane bending, by slightly modifying analytical expressions, given in Refs. [10,11]. The proposed solutions are in better agreement with the FE results, underestimating by less than 10% for all cases considered. The branch junctions considered in the present work are restricted to 0.4p(r/R, t/T) p1.0 and 10.0pR/Tp20.0, which correspond to thin-wall cylinders. Thus, the proposed equations should be applied within such ranges. Extension of the proposed solutions to other geometric variables such as to thick-wall cylinders would be interesting. On the other hand, it should be pointed out that the main objective of this work is to provide a baseline solution for limit load analysis for branch junctions with local wall thinning in pressurized water reactor nuclear power plants, where geometric variables of pipe fittings are covered by the above ranges. Results of limit loads for branch junctions with local wall thinning will be reported shortly.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
[14]
[15]
Acknowledgements This research is performed under the programme of Basic Atomic Energy Research Institute (BAERI), a part of the Nuclear R&D Programs funded by the Ministry of Science & Technology (MOST) of Korea, and under the programme of the Brain Korea 21 Project in 2006.
[16]
[17]
[18]
References [1] ASME Boiler and Pressure Vessel Code Section IX, 1992. [2] Kanninen MF, Zahoor A, Wilkoski G, Abousayed I, Marschall C, Broek D, et al. Instability predictions for circumferentially cracked
[19]
[20]
653
Type-304 stainless pipes under dynamic loading. Palo Alto, CA: EPRI NP-2347, Electric Power Research Institute; 1982. Ainsworth RA. The assessment of defects in structures of strain hardening materials. Eng Fract Mech 1984;19:633–42. R6: assessment of the integrity of structures containing defects. Revision 4, British Energy, 2001. R5: an assessment procedure for the high temperature response of structures. Issue 3, British Energy, 2003. Webster GA, Ainsworth RA. High temperature component life assessment. London: Chapman & Hall; 1994. Miller AG. Review of limit loads of structures containing defects. Int J Pressure Vessels Piping 1988;32:191–327. SINTAP final procedure, Brite Euram Project, BE95-1426, 1999. R6: assessment of the integrity of structures containing defects, Chapter IV—Compendia. Revision 4, British Energy, 2001. Xuan FZ, Li P-N, Tu ST. Limit load analysis for the piping branch junctions under in-plane moment. Int J Mech Sci 2006;48:460–7. Xuan FZ, Li P- N, Tu S- T. Limit load analysis for the piping branch junctions under internal pressure. Nucl Eng Design 2003;224:1–9. Robinson M. Lower-bound limit pressures for the cylinder–cylinder intersection: a parametric survey. J Pressure Vessel Technol 1978;100:65–73. Robinson M. Some further comments on limit pressures for the cylinder–cylinder intersection. J Pressure Vessel Technol 1980;102:119–20. Xuan FZ, Li P- N. Finite element-based limit load for piping branch junctions under combined loadings. Nucl Eng Design 2004;231:141–50. Nadarajah C, Mackenzie D, Boyle JT. Limit and shakedown analysis of nozzle/cylinder intersections under internal pressure and in-plane moment loading. Int J Pressure Vessels Piping 1996;68:261–72. Ayob AB, Moffat DG, Mistry J. Load interaction in pressurized structures using the finite element method. Int J Pressure Vessels Piping 1997;73:3–9. Chapuliot S, Moulin D, Plancq D. Mechanical behavior of a branch pipe subjected to out-of-plane bending load. J Pressure Vessel Technol 2002;124:7–13. Galt D. Parametric study of limit pressures for cylinder–cylinder intersections. Research report 56/89/07, Babcock Energy Ltd., 1989. Budden PJ, Goodall IW. Creep life of pressure vessel branch connections: the reference stress approach. Nuclear Electric report TD/SID/REP/0123, Gloucester, UK, 1990. ABAQUS Version 6.2-1. User’s manual. Rhodes Island, USA: Hibbitt, Karlsson and Sorensen, Inc.; 2001.
International Journal of Pressure Vessels and Piping 83 (2006) 645–653 www.elsevier.com/locate/ijpvp
Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending Yun-Jae Kima,, Kuk-Hee Leea, Chi-Yong Parkb a
Department of Mechanical Engineering, Korea University, 1-5 Ka, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea b Korea Electric Power Research Institute, Yusung-gu, Daejon 305-380, Republic of Korea Received 24 April 2006; received in revised form 26 June 2006; accepted 3 July 2006
Abstract The present work presents plastic limit load solutions for thin-walled branch junctions under internal pressure and in-plane bending, based on detailed three-dimensional (3-D) finite element (FE) limit analyses using elastic–perfectly plastic materials. To assure reliability of the FE limit loads, modelling issues are addressed first, such as the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. Then the FE limit loads for branch junctions under internal pressure and in-plane bending are compared with existing limit load solutions, and new limit load solutions, improving the accuracy, are proposed based on the FE results. The proposed solutions are valid for ratios of the branch-to-run pipe radius and thickness from 0.4 to 1.0, and the mean radius-tothickness ratio of the run pipe from 10.0 to 20.0. r 2006 Elsevier Ltd. All rights reserved. Keywords: Finite element analysis; Branch junction; Internal pressure; In-plane bending; Limit load
1. Introduction Information on the limit load of piping components is important in structural integrity assessment. Such information is a direct input to estimate the maximum loadcarrying capacity of piping components [1,2]. Furthermore, based on the reference stress approach [3], it can be used to estimate creep rupture and non-linear fracture mechanics parameters (see for instance Refs. [4–6]). Due to its significance, information on limit loads for typical piping components with or without defects is widely available (see for instance Refs. [7–9]). As branch junctions are widely used in plants, numerous works on plastic limit analyses of branch junctions have been reported. Although some analytical works are available [10–14], works based on non-linear finite element (FE) analyses are increasingly popular [15–18], due to complexities associated with the geometry and loading conditions. Two Corresponding author. Tel.: +82 2 3290 3372; fax: +82 2 926 9290.
E-mail address: [email protected] (Y.-J. Kim). 0308-0161/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2006.07.002
issues need to be resolved on plastic limit analyses of branch junctions. As works based on non-linear FE analyses are popular, FE modelling issues should be resolved, but not much information has been given in the literature up to the present. As the second issue, for practical application, reliable limit load solutions need to be developed in a closed-form. Although, some solutions are available for branch junctions, the reliability of existing solutions needs to be checked. The present paper addresses the above two issues. The present work presents plastic limit load solutions for branch junctions under internal pressure and in-plane bending, based on detailed three-dimensional (3-D) FE limit analyses using elastic–perfectly plastic materials. The branch junctions considered in the present work are restricted to thin-wall cylinders. Section 2 briefly reviews existing limit load solutions for branch junctions under internal pressure and in-plane bending. The FE limit analysis is presented in Section 3, including the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. Section 4 compares the FE results with existing solutions and new limit load
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Nomenclature A B C M0 ML P0
r/R 2R/T t/T limit moment of a straight pipe in-plane limit moment of a branch junction limit pressure of a straight pipe
PL R, r t, t s0
limit pressure of a branch junction mean radius of a run (main) pipe and a branch for branch junctions thickness of a run (main) pipe and a branch for branch junctions limiting strength of an elastic–perfectly plastic material
solutions are proposed based on the FE results. The work is concluded in Section 5.
moment applied to the branch pipe), the solution by Xuan et al. [10] is given by
2. Existing limit load solutions: review
ML ¼ M0
Consider the branch junction, depicted in Fig. 1. For the main (run) pipe, the mean radius and thickness are denoted by R and T, respectively, and for the branch pipe, r and t, respectively. It is assumed that the branch junction has no weld or reinforcement around the intersection. In the literature, there are several limit load solutions for branch connections [7–14,19]. Among these, selected solutions, which are believed to be most reliable, are reviewed here. For in-plane bending (with the bending
Fig. 1. Schematics of branch junctions with relevant geometric variables.
p=2 i0:5 , pffiffiffiffi2 C f 1 A þ 0:455f 2 k B þ 0:2385Bf 22 k2 h
(1)
where
1 4 p 3 2 1 1 A ; k ¼ f1 1þ A ; f2 , 3 2 16 1 þ C3 r 2R t ; C¼ A¼ ; B¼ R T T and M0 denotes the limit moment of the branch pipe: M 0 ¼ 4s0 r2 t.
(2)
The above solution was derived from force equilibrium between the limit load and the internal force acting on the intersecting line between the main and branch pipes. In the limiting case of T-0 or R-0, Eq. (1) recovers the exact limit moment for the branch pipe, Eq. (2). Fig. 2 shows variations of the normalized limit moments, ML/M0, with t/T, for R/T ranging from R/T ¼ 10.0 to R/T ¼ 20.0. The results in Fig. 2a are for the limiting case of r/R ¼ t/T, and those in Fig. 2b are for the given r/R ¼ 0.9. The general trend is that for smaller branches (r/Rp0.3), the limit load for the branch junction is the same as that for the branch pipe, and weakening effects occur only for r/R40.3. For larger branches (r/R40.7), the limit load for the branch junction tends to saturate. Xuan et al. [10] compared
Fig. 2. Variations of existing limit loads [10] for branch junctions under in-plane bending: (a) with r/R for the limiting case of r/R ¼ t/T and (b) with t/T for the given r/R ¼ 0.9.
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Eq. (1) with experimental data and with FE data, and found overall good agreement. This will be further discussed in Section 4. Comparison of Eq. (1) with other existing solutions for branch junctions under in-plane bending can be found in Ref. [10]. For internal pressure, Budden and Goodall [19] proposed the following regression equation for the limit pressure, PL, based on FE data [18]: PL R A 1 þ Bð1 r=RÞ 1 Dð0:5 T=RÞ2 (3) ¼ s0 T 1 þ Cð1 t=TÞ with A ¼ 0:641;
B ¼ 0:908;
C ¼ 0:608;
D ¼ 1:422.
Although Budden and Goodall [19] checked the validity of this solution by comparison with analytical solutions by Robinson [12,13] and FE data [18], the main concern seemed to be applications to thick-wall tubes. It can be seen that, in the limiting case of r-0 (t-0), the above solution does not always recover the plastic limit pressure for the tube based on the von Mises yield condition, P0: P0 R 2 ¼ pffiffiffi . s0 T 3
et al. [11] compared Eq. (5) with experimental data and with FE data, and found overall good agreement. This will be further discussed in Section 4. Note that, in the limiting case of r-0 (t-0), the above solution does recover the plastic limit pressure for the run pipe based on the von Mises yield condition, Eq. (4). Fig. 3 compares the normalized limit pressure, PL/P0, from the above two solutions, for R/T ranging from R/T ¼ 10.0 to R/T ¼ 20.0. The results in Fig. 3a are for the limiting case of r/R ¼ t/T, and show that Eq. (3) always gives a lower limit load than Eq. (5). In particular, as Eq. (3) does not give the correct limit load for the limiting case of r-0, the difference between Eq. (3) and Eq. (5) increases as r/R decreases. Fig. 3b compares the limit loads for the given r/R ¼ 0.9. The limit loads from the above two equations are similar, but those from Eq. (5) are slightly higher. Comparison of Eq. (5) with other existing solutions for branch junctions under internal pressure can be found in Ref. [11]. 3. FE limit analyses 3.1. FE model and limit analysis
(4)
More recently, Xuan et al. [11] proposed the following analytical limit pressure solution. PL: PL R 1 ¼ s0 T ð0:25 0:5h1 þ h21 þ 0:79h22 Þ0:5
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(5)
with pffiffiffiffi C 2 h1 ¼ 1 þ 0:145kA Bf 2 þ 0:3185A2 f 1 1 ; AB pffiffiffiffi C 2 h2 ¼ 0:175kA B 1 f2 AB and f1, f2, A, B and C are given in Eq. (1). Similarly to Eq. (1), Eq. (5) was derived from force equilibrium between the limit load and the internal force acting on the intersecting line between the main and branch pipes. Xuan
3-D elastic–perfectly plastic FE analyses of the branch junction, depicted in Fig. 1, were performed using ABAQUS [20]. It is assumed that the branch junction has no weld or reinforcement around the intersection. Regarding the axial length, the half-length of the run pipe is denoted as L and the length of the branch pipe as ‘. The geometric variables (R, T, r, t, L, ‘) were systematically varied, within the ranges 0.4p(r/R, t/T)p1.0 and 10.0pR/ Tp20.0. Note that such ranges correspond to thin-wall cylinders. Materials were assumed to be elastic–perfectly plastic, and non-hardening J2 flow theory was used using a small geometry change continuum FE model. Symmetry conditions were fully utilized in FE models to reduce the computing time. To avoid problems associated with incompressibility, reduced integration elements (element type C3D20R within ABAQUS) were used. Fig. 4 depicts typical FE meshes, employed in the present work; one for
Fig. 3. Variations of existing limit loads [11,19] for branch junctions under internal pressures: (a) with r/R for the limiting case of r/R ¼ t/T and (b) with t/T for the given r/R ¼ 0.9.
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Fig. 5. Typical moment-rotation curves from FE limit analyses of branch junctions under in-plane bending.
Fig. 4. Finite element meshes for: (a) r/R ¼ 0.6, t/T ¼ 0.6, and R/T ¼ 15 and (b) r/R ¼ 1.0, t/T ¼ 1.0, and R/T ¼ 15.
r/R ¼ 0.6, t/T ¼ 0.6 and R/T ¼ 15, and the other for r/R ¼ 1.0, t/T ¼ 1.0, R/T ¼ 15. For all cases, three elements are used through the thickness, and the resulting number of elements and nodes in typical FE meshes ranges from 3949 elements/20,598 nodes to 4914 elements/25,649 nodes, which are believed to be sufficiently fine for the present study. Regarding loading conditions, both internal pressure and in-plane bending moment were considered. For internal pressure, pressure was applied as a distributed load to the inner surface of the FE model, together with axial tensions equivalent to the internal pressure applied at the end of the branch and run pipes to simulate closed ends. To avoid problems associated with convergence in elastic–perfectly plastic calculations, the RIKS option within ABAQUS was invoked. Due to symmetry, only a quarter model was used. For in-plane bending cases, the nodes at the end of the branch pipe were constrained through the MPC (multi-point constraint) option within ABAQUS, and sufficiently large deformation (rotation) was directly applied. Fig. 5 shows typical moment-rotation responses from the present FE limit analyses, from which limit moments can be easily determined. 3.2. Effect of FE modelling on plastic limit loads Before presenting the FE results, some FE modelling issues are worth mentioning, as the FE results could
Fig. 6. Three boundary conditions for branch junctions under in-plane bending: (a) clamped–clamped (CL–CL), (b) clamped–simply supported (CL–SS) and (c) clamped–free (CL–F).
depend on modelling. Particular issues include the effects of the length of the run (main) pipe and the boundary conditions on the FE limit loads. Consider branch connections under in-plane bending, as schematically shown in Fig. 6. Regarding boundary conditions, one could apply the clamped–clamped (CL–CL) condition (Fig. 6a), the clamped–simply supported (CL–SS) condition (Fig. 6b), or the clamped–free (CL–F) condition (Fig. 6c). Regarding the geometric variables, one could vary the relative length of the pipes, either the length of the branch pipe, ‘, or that of the run pipe, L. It is found that, as long as the length of the branch pipe, ‘, is sufficiently long to avoid the end effect
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due to the applied bending, it does not affect the limit load. Thus, the only geometric variable to be considered is the length of the run pipe, L. The effect of the ratio of the length of the run pipe to the radius, L/R, on the FE plastic limit loads is shown in Fig. 7 for branch junctions under in-plane bending, subject to three different boundary conditions. Note that the FE results in Fig. 7 are for the special case of R/r ¼ T/t. The FE plastic limit loads are normalized with respect to the analytical expression, Eq. (1). Overall trends for the clamped–clamped and the clamped–simply supported conditions are similar. The limit loads for these two boundary conditions are the same when the value of L/R is sufficiently large, as expected. For smaller branches up to r/R ¼ 0.8, the limit loads for both cases do not depend on L/R. For larger branches with r/R ¼ 0.9 and r/R ¼ 1.0, the limit loads for both cases increase with decreasing L/R. For a given L/R, the limit load for the clamped–clamped case is higher than that for the clamped–simply supported case, possibly due to the restraint effect. The end rotation in the run pipe is more restrained for the clamped–clamped case than for the clamped–simply supported case, which results in a higher limit load. Note that these results can be supported by the lower bound limit load theorem. As the limit load for a junction with clamped–simply supported conditions
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satisfies equilibrium, yield conditions and traction boundary conditions for that with clamped–clamped conditions, it must be a lower bound to the limit load for clamped–clamped conditions. The results for the clamped–free case for r/Rp0.8 are the same as those for the clamped–clamped and the clamped–simply supported cases. For r/R40.8, however, the trend is opposite; the limit loads decrease with decreasing L/R, and are lower than those for the above two cases. It should be noted, however, that the clamped–free boundary condition is less realistic than the clamped–clamped and the clamped– simply supported conditions. For internal pressure, it is found that both the boundary condition and the length of the run pipe do not affect the plastic limit pressure of the branch junction. The present results suggest that, for the FE analysis of branch junctions under pure bending, care should be exercised in the boundary condition and the length of the run pipe. The limit moment results (for in-plane bending), presented in the next section, will be those for a sufficiently long run pipe under the clamped– clamped case, which is the same as those under clamped–simply supported case. As the clamped–free case is not realistic in practical situations, it is not considered further.
Fig. 7. Effects of the normalized length of the run pipe, L/R, on plastic limit moments of branch junctions under in-plane bending for (a) the clamped–clamped (CL–CL) condition, (b) the clamped–simply supported (CL–SS) condition, and (c) the clamped–free (CL–F) condition.
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et al. [10] compared the limit loads from Eq. (1) with experimental results, and found that predictions according to Eq. (1) were slightly lower than the experimental data by up to 5%. One interesting point is that, for the comparison, they used the flow strength as 1.25 times the yield strength, but experimental loads were measured using the twiceelastic slope method from the load–displacement curves. If they had used the yield strength as the flow strength, the predictions would be lower than the experimental results by up to 30%, which is consistent with the present findings. Although Eq. (1) gives consistently lower limit moments, compared to the FE limit moments, the trend is quite clear. In this respect, a simple factor can be multiplied to Eq. (1) to give better agreement with the FE results. Such a factor is found from the FE results, and the following limit moment solution is proposed for branch junctions under in-plane bending: ML ¼ 4s0 r2 t
p=2 Q i0:5 , pffiffiffiffi2 C f 1 A þ 0:455f 2 k B þ 0:2385Bf 22 k2 h
(6)
where the factor Q is given by Q ¼ 1:11ðC 0:7Þ2 þ 1:18 ¼ 1:11
t T
0:7
2
þ 1:18.
Other factors, f1, f2, A, B and C, are given in Eq. (1). Note that the value of Q in the limiting case of t/T-0 or r/R-0 approaches 0.64, not unity, and thus the proposed expression is not correct in the limiting case of t/T-0 or r/R-0. As the proposed equation is based on the FE data, it should be applied within 0.4p(r/R, t/T) p1.0 and 10.0pR/Tp20.0. The values of Q range from Q ¼ 1.08 (at t/T ¼ 0.4 and 1.0) to Q ¼ 1.18 (at t/T ¼ 0.7) for 0.4pt/ Tp1.0 Fig. 9 compares the proposed limit moment solution, Eq. (6), with the FE results for various cases. The FE limit loads in Fig. 9 are normalized with respect to the proposed solution, Eq. (6), and thus the proximity of the data to unity indicates the accuracy. It shows that Eq. (6) gives lower limit moments than the FE data by less than 10% for all cases considered. Fig. 8. Variations of ratios of the FE limit loads to existing ones [10] for branch junctions under in-plane bending with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
4. Limit loads 4.1. In-plane bending Fig. 8 shows variations of the FE limit loads for branch junctions under in-plane bending with t/T. The FE results in Fig. 8a are for the limiting case of R/r ¼ T/t and those in Fig. 8b are for r/R ¼ 0.6, with the values of t/T ranging from t/T ¼ 0.4 to t/T ¼ 1.0. The FE limit loads are normalized with respect to the analytical expression, Eq. (1). The results in Fig. 8 show that the FE results are always higher than Eq. (1) by up to 30%, and thus Eq. (1) gives conservative moments. It should be noted that Xuan
4.2. Internal pressure Fig. 10 shows variations of the FE limit loads for branch junctions under internal pressure with t/T. The FE results in Fig. 10a are for the limiting case of R/r ¼ T/t and those in Fig. 10b are for r/R ¼ 0.6, with the values of t/T ranging from t/T ¼ 0.4 to t/T ¼ 1.0. Normalization of the FE limit loads in Fig. 10 is worth mentioning. For the data indicated as ‘‘Budden’’, the FE limit loads are normalized with respect to Eq. (3), whereas for those indicated as ‘‘Xuan et al.’’, with respect to Eq. (5). For the data indicated as ‘‘Budden’’, the values are overall higher than unity by up to 40%, suggesting that Eq. (3) gives overall lower limit loads than the FE results. On the other hand, for the data indicated as ‘‘Xuan et al.’’, the values can be lower than unity by up to 20%, suggesting that Eq. (5) can give higher (and thus non-conservative) limit loads than the
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Fig. 9. Variations of ratios of the FE limit loads to the proposed limit load solution, Eq. (6), for branch junctions under in-plane bending with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
Fig. 10. Variations of ratios of the FE limit loads to existing ones [11,19] for branch junctions under internal pressure with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
FE results. It can be also noted that the effect of R/T is less significant for the ‘‘Xuan et al.’’ data than the ‘‘Budden’’ data. It is not clear why Eq. (5) can give non-conservative
limit pressures. It could be due to the difference in yielding mechanisms or to simplifications for analytical derivation, assumed in their work. Xuan et al. [11] compared limit
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loads from Eq. (5) with the FE results and claimed that the predictions from Eq. (5) differed from the FE results by less than 10%. However, careful examination of Fig. 4 in their paper [11] shows that the predictions using Eq. (5) are higher (non-conservative) than the FE results. As the trend of the ‘‘Xuan et al.’’ data is also quite clear, Eq. (5) can be modified to give better agreements with the FE results. Based on the FE results, the following limit moment solution is proposed for branch junctions under internal pressure: PL R Q ¼ s0 T ð0:25 0:5h1 þ h21 þ 0:79h22 Þ0:5
(7)
with Q ¼ 1:32ðA 0:636Þ2 þ 0:906: All other factors, h1, h2, f1, f2, A, B and C, are the same as those given in Eqs. (1) and (5), except k which is now defined by k¼
1 0:78ðC þ 0:55Þ2
(8)
instead of the expression given in Eq. (1). Fig. 11 compares the proposed limit moment solution, Eq. (7), with the FE results for various cases. The FE limit loads in Fig. 11 are normalized with respect to the proposed solution, Eq. (7),
and thus the proximity of the data to unity indicates the accuracy. It shows that Eq. (7) gives lower limit pressures than the FE data by less than 10% for all cases considered. 5. Concluding remarks The present work presents plastic limit load solutions for branch junctions under internal pressure and in-plane bending, based on detailed FE limit analyses using elastic–perfectly plastic materials. To assure reliability of the FE limit loads, modelling issues are addressed first, such as the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. It is shown that for branch junctions under in-plane bending, FE limit loads can be affected by kinematic boundary conditions, particularly when the ratio of the mean radius of the branch pipe to that of the run pipe is greater than 0.8, and thus caution should be exercised. Based on systematic FE analyses for branch junctions with 0.4p(r/R, t/T) p1.0 and 10.0pR/Tp20.0 under internal pressure and in-plane bending, the FE limit loads are determined and compared with existing limit load solutions by Xuan et al. [10,11] and Budden and Goodall [19]. For in-plane bending, it is found that the analytical solution given by Xuan et al. [10] underestimates the FE
Fig. 11. Variations of ratios of the FE limit loads to the proposed limit load solution, Eq. (7), for branch junctions under internal pressure with t/T: (a) for the limiting case of r/R ¼ t/T, and (b) for the given r/R ¼ 0.6.
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limit loads by up to 30%. For internal pressure, on the other hand, their solution in Ref. [11] could overestimate the FE limit loads by up to 20%, whereas the solution by Budden and Goodall [19] overall underestimates by up to 40%. Based on the FE results, new limit load solutions are proposed for internal pressure and for in-plane bending, by slightly modifying analytical expressions, given in Refs. [10,11]. The proposed solutions are in better agreement with the FE results, underestimating by less than 10% for all cases considered. The branch junctions considered in the present work are restricted to 0.4p(r/R, t/T) p1.0 and 10.0pR/Tp20.0, which correspond to thin-wall cylinders. Thus, the proposed equations should be applied within such ranges. Extension of the proposed solutions to other geometric variables such as to thick-wall cylinders would be interesting. On the other hand, it should be pointed out that the main objective of this work is to provide a baseline solution for limit load analysis for branch junctions with local wall thinning in pressurized water reactor nuclear power plants, where geometric variables of pipe fittings are covered by the above ranges. Results of limit loads for branch junctions with local wall thinning will be reported shortly.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
[14]
[15]
Acknowledgements This research is performed under the programme of Basic Atomic Energy Research Institute (BAERI), a part of the Nuclear R&D Programs funded by the Ministry of Science & Technology (MOST) of Korea, and under the programme of the Brain Korea 21 Project in 2006.
[16]
[17]
[18]
References [1] ASME Boiler and Pressure Vessel Code Section IX, 1992. [2] Kanninen MF, Zahoor A, Wilkoski G, Abousayed I, Marschall C, Broek D, et al. Instability predictions for circumferentially cracked
[19]
[20]
653
Type-304 stainless pipes under dynamic loading. Palo Alto, CA: EPRI NP-2347, Electric Power Research Institute; 1982. Ainsworth RA. The assessment of defects in structures of strain hardening materials. Eng Fract Mech 1984;19:633–42. R6: assessment of the integrity of structures containing defects. Revision 4, British Energy, 2001. R5: an assessment procedure for the high temperature response of structures. Issue 3, British Energy, 2003. Webster GA, Ainsworth RA. High temperature component life assessment. London: Chapman & Hall; 1994. Miller AG. Review of limit loads of structures containing defects. Int J Pressure Vessels Piping 1988;32:191–327. SINTAP final procedure, Brite Euram Project, BE95-1426, 1999. R6: assessment of the integrity of structures containing defects, Chapter IV—Compendia. Revision 4, British Energy, 2001. Xuan FZ, Li P-N, Tu ST. Limit load analysis for the piping branch junctions under in-plane moment. Int J Mech Sci 2006;48:460–7. Xuan FZ, Li P- N, Tu S- T. Limit load analysis for the piping branch junctions under internal pressure. Nucl Eng Design 2003;224:1–9. Robinson M. Lower-bound limit pressures for the cylinder–cylinder intersection: a parametric survey. J Pressure Vessel Technol 1978;100:65–73. Robinson M. Some further comments on limit pressures for the cylinder–cylinder intersection. J Pressure Vessel Technol 1980;102:119–20. Xuan FZ, Li P- N. Finite element-based limit load for piping branch junctions under combined loadings. Nucl Eng Design 2004;231:141–50. Nadarajah C, Mackenzie D, Boyle JT. Limit and shakedown analysis of nozzle/cylinder intersections under internal pressure and in-plane moment loading. Int J Pressure Vessels Piping 1996;68:261–72. Ayob AB, Moffat DG, Mistry J. Load interaction in pressurized structures using the finite element method. Int J Pressure Vessels Piping 1997;73:3–9. Chapuliot S, Moulin D, Plancq D. Mechanical behavior of a branch pipe subjected to out-of-plane bending load. J Pressure Vessel Technol 2002;124:7–13. Galt D. Parametric study of limit pressures for cylinder–cylinder intersections. Research report 56/89/07, Babcock Energy Ltd., 1989. Budden PJ, Goodall IW. Creep life of pressure vessel branch connections: the reference stress approach. Nuclear Electric report TD/SID/REP/0123, Gloucester, UK, 1990. ABAQUS Version 6.2-1. User’s manual. Rhodes Island, USA: Hibbitt, Karlsson and Sorensen, Inc.; 2001.