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Plastic limit loads for pipe bends under combined bending and torsion moment
Author's Accepted Manuscript
Plastic limit loads for pipe bends under combined bending and torsion moment Jian Li, Chang-Yu Zhou, Peng Cui, Xiao-Hua He
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PII: DOI: Reference:
S0020-7403(14)00408-1 http://dx.doi.org/10.1016/j.ijmecsci.2014.12.011 MS2891
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International Journal of Mechanical Sciences
Received date: 9 October 2014 Revised date: 22 November 2014 Accepted date: 15 December 2014 Cite this article as: Jian Li, Chang-Yu Zhou, Peng Cui, Xiao-Hua He, Plastic limit loads for pipe bends under combined bending and torsion moment, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j. ijmecsci.2014.12.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Plastic limit loads for pipe bends under combined bending and torsion moment Jian Li, Chang-Yu Zhou*, Peng Cui, Xiao-Hua He School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China Abstract In present paper, 3D finite element (FE) method is used to determine plastic limit load solutions for pipe bends under combined bending and torsion moment. With a detailed analysis and comparison, a common awareness for loading effect is showing which will raise researchers concern. By the way, past solutions are not appropriate to estimate FE results. In this respect of finite element analysis, overall yielding considering the spread process of yield region from crown to the straight pipe shows these promising finite element results. A wide range of non-dimensional parameters for pipe bends are considered and plastic limit load solutions are suggested. The results show that r/t is the main factor affecting the limit loads. Plastic limit load is independent on the loading path and material constants by normalizing. Results show that the circular interaction rule is a great approximation for pipe bends under combined bending and torsion moment. A series of approaches are confirmed in order to validate our finite element method on plastic limit analysis. Based on the finite element results, approximate plastic limit load solutions are proposed. Present work will further improve the limit load solution for pipe bends under complex loading conditions. Key words: pipe bends; in-plane bending moment; torsion moment; limit loads; finite element 1. Introduction Pipe bends (or elbows) are commonly used components in a piping system, which are widely used in petroleum, chemical and nuclear power industries [1]. A detailed research on maximum loading carrying capacities of pipe bends is important in design and assessment of pipe system in power plants. Pipe bends are more flexible than straight pipes with similar dimensional parameters [2] due to the complex deformation they exhibit under bending loads. Pipe bends tend to show different mechanical properties as the interaction of geometrical nonlinearity and material nonlinearity when they are under these combined loads. Due to the self weight, valve weight, fluid weight in addition to heat expansion in the pipe system bending and torsion moment should not be overlooked. The bending stress caused by bending moment and torsion is generally greater than membrane stress only by pressure. So in the bending mode, flatting will occur, as the stress increase which even eventually produces buckling due to maximum compression stress from bending deformation and ovalization of circular cross-section of pipe bend [3] and collapse prior to the structure fracture. Meanwhile for pipe bends torsion will always exist as long as the out-of-plane bending exists. So research on pipe bends under combined bending and torsion moment is vital important to maintain the structure integrity of piping system. In present work, 3D finite element method is used to determine plastic limit load solutions for pipe bends under combined bending and torsion moment. Based on the finite element results, approximate plastic limit load solutions for limit pressures are proposed.
*Corresponding author. Tel.:+86-25-58139951; fax:+86-25-58139951. E-mail address: [email protected]
2. Research background 2.1. Review Over the past years large quantities of elastic-plastic behaviors for pipe bends have been carried out, especially for the plastic limit load solutions. Marcal [4]was the first one to present the elastic-plastic analysis for pipe bends under in-plane bending. Later Spence and Findlay [5] and Calladine [6] presented analytical limit bending solutions based on limit load theorems and plasticity mechanics theory. Goodall presented the first large deformation analysis of thin elbows under in-plane bending [7] and the lower bound solution for a thin elbow under combined loading [8]. Kitching et al.[9] determined a lower bound for the limit moment of a smooth circular pipe bend with no restrictions on the geometry using thin shell theory. Shalaby and Younan [10-12] presented a series of comprehensive studies of elastic-plastic behavior of pipe bends under in-plane bending. Hashem et al. [13-15] have performed analyses on pipe bends subjected to combined internal pressure with out-of-plane bending moment. Experiments on limit loads for pipe bends were carried out on elbows by Touboul et al. [16], Greenstreet [17], Tan et al. [18] and Chattopadhyay et al. [19, 20]. Researches above seek to study the gross plastic deformation behavior which is the fundamental failure mode for pipe structures. Orynyak and Radchenko [21] proposed an analytical method for the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. In recent years numerical method is applied in the design of engineering structures with respect to failure [22-24]. With the rapid development of the commercial software, finite element simulation to investigate the elastic-plastic behaviors for pipe bends is widely used by many researchers. This simulation method has being validated reliable and accurate compared to the past experiment data [25]. Some typical nonlinear general finite element program such as MSC.Marc [26], ANSYS [27], WARP3D [28] and ABAQUS [29] are widely adopted. On this basis, large quantities of estimated limit expressions for engineering assessment have been proposed directly in recent years, which have modified and improved some shortcomings in some design codes. Based on this procedure Hong et al.[30] and An et al.[31] proposed approximations to elastic stresses for elbows under internal pressure and in-plane bending respectively using ABAQUS. Kim and oh [32, 33] gave a detailed analysis and proposed estimated equations for limit and collapse loads of pipe bends under combined pressure and in-plane bending. They also studied the effects of attached straight pipes on finite element limit analysis for pipe bends and proposed relevant estimated solutions [34]. Hong et al.[35] quantifies the effect of internal pressure on plastic loads for elbows and proposed relevant equations. Lee et al.[36] proposed simple regression equations between the yield strength-to-elastic modulus ratio and plastic loads for elbows. Christo et al.[37-39] studied the combined effect of ovality and thinning on plastic loads of pipe bends under combined in-plane closing moment and internal pressure and found that the effect of ovality is significant while the thinning produces negligible effect, then proposed closed-form solutions to determine collapse load. Based on the research in Refs.[37-39] Buckshumiyan et al. [40,41] took in-plane opening bending into consideration and proposed estimated equations based on finite element analysis. For thick-walled elbows Kim et al. [42] conducted a detailed plastic loads analysis and gave some comparisons with thin-walled elbows, then proposed estimated equations. Large amounts of the proposed finite element based solutions from Refs.[30-42] were validated with experiment results, and results shows overall agreement regardless of some higher error. In our recently published work [43], a detailed analysis for pipe
bends under combined pressure and bending was performed by finite element analysis, which in turn can also provide some support to the relevant research work above. Studies from Refs.[1-43] focus on pressure and bending (mainly in-plane bending )load, and few references [44-46] mentioned limit load solutions for pipe bends under combined bending and torsion moment. In Ref.[44] Ayob et al. concerned the load interaction behavior of smooth piping elbows with attached long straight pipes by finite element method, then established the yield interaction behavior when an elbow is subjected to a combination loading of in-plane bending, torsion and internal pressure. Although studies in Ref.[44] is already specific for combined bending moment and torsion, their work is irrelevant with limit load solutions. In Ref.[45] Guo conducted a detailed stress analysis of pipe bends and the plastic limit load solutions of pipe bends under combined internal pressure, bending and torsion moment based on Mises yield criterion were obtained. However in Guo’s work[45] assumption of the stress and limit analysis for torsion is totally idealized as the same as straight pipes. In ASME B&PV Code [46], a general design equation of a nuclear pipe component is formed based on the stress indices which governs primary and secondary stress. 2.2. Some relevant limit load solutions 2.2.1. Pure in-plane bending Spence and Findlay [5] provided a lower bound limit moment for the pipe bend subject to in-plane bending
M I / M str
0.8λ 0.6 = 1
λ < 1.45 λ 1.45
(1)
2
Where, M str = 4r tσ s
(2)
denotes the plastic limit loads for straight pipes with the same dimension of pipe bends, σs is the material yield stress, MI is the plastic loads for a pipe bend under pure in-plane bending moment, λ is the bend characteristic of pipe bends, where,
λ = Rt / r 2 = ( R / r ) / ( r / t )
(3)
and R is bend radius of pipe bends, t is the thickness of pipe bends, r is the mean radius of cross section. Calladine [6] provided another limit load solution for pipe bends under in-plane bending.
M I / M str = 0.935λ 2/3
λ < 0.5
(4)
Goodall [8] provided a slightly high limit load solution
M I / M str = 1.04λ 2/3
λ < 0.5
(5)
Studies in Refs. [5, 6, 8] suggest that the limit bending moment depends only on the bending characteristic λ . However in some other research work result shows that a single variable λ can not reflect the limit loads tendency accurately [9, 32], and limit bending solutions are suggested dependence on both R/r and r/t. Kitching et al.[9] determined a lower bound for the limit bending moment of pipe bends using thin shell theory
M I / M str = 0.935λ 2/3 (1 − 0.36r / R ) λ < 0.5, 0 < r / R < 0.67
(6)
Recently based on 3D finite element limit analysis, Kim and Oh [32] proposed another limit load solution
M I / M str = A(λ + k )n A = 0.6453(r / t ) 0.0772 ; k = 1.5398(r / t ) −0.6755 ; n = 0.5157( r / t )
0.1 λ 0.5
(7)
0.0601
Noting that this proposed limit load solution in Eq.(7) is much high than predicted Eqs.(1), (4)-(6). This is because results by Eq.(7) considered the effect of a long attached straight pipe, and with a straight pipe attached to the pipe bend, the plastic yielding region spreads not only to the elbow but also to the attached pipe, which makes the limit bending moment higher than that for the elbow without an attachment [34]. In the limiting case of no attachment, the limit loads are found to be close to existing analytical solutions [34] by Eqs.(1), (4)-(6). 2.2.2. Pure torsion Guo [45] proposed limit load solutions based on Mises criterion
M O / M str = π / 2 3
(8)
Where, MO is the plastic loads for a pipe bend under pure torsion moment. 2.2.3. Combined in-plane bending and torsion moment Guo [45] proposed limit load solutions based on Mises criterion by stress analysis
( M LI / 3.32λ 2/3 r 2tσ s ) + ( M LO / M O ) = 1
(9)
Where, MLI is the plastic limit in-plane bending moment for a pipe bend under combined in-plane bending and torsion moment, MLO is the plastic limit torsion moment for a pipe bend under combined in-plane bending and torsion moment, and MO are from Eq. (8). ASME section III [46] code is meant to control primary plus secondary loads so as to place an upper bound on deformations [44]. Piping components such as straight pipes, bends, elbows, etc. are designed according to clauses NB/NC/ND-3600 for combined stresses due to pressure and moment, for different service levels [47]. The general design equation against plastic collapse in Class 1 piping component is given as
B1
Mr Pr + B2 i σ s t I
(10)
Where, B1 is stress index for pressure, B2 is stress index for bending,
B2 = 1.3 / λ 2/3 1
(11)
P is internal pressure, Mi is the combined force moment (combination of bending and torsion moment),
M i = M B2 + M T2
(12)
I is section modulus,
I = π r 3t
(13) MB is applied in-plane bending moment, and MT is applied torsion moment. For pipe bends without internal pressure considered, Eq. (10) can be rewrite as
1.3M i / π r 2t λ 2/3 σ s
(14)
3. Finite element analysis 3.1. Geometry The geometry of a pipe bend is shown in Fig.1. The bend angle is considered to be 90o, and the straight pipe attached to pipe bend is long enough (L>3r) to ignore the end effects on limit loads [34]. This length L is 1000 mm in this research, and the mean radius of cross section r is 100mm. The geometry parameters of pipe bends are listed in Table 1, where 9 models are chosen for in-plane bending, 16 models are chosen for torsion, and 4models are chosen for combined loading condition. 3.2. Material The material used is assumed to be elastic-perfectly plastic, and non-hardening J2 flow theory is used. The Young’s modulus E=200 GPa, Poisson’s ratio µ=0.3 and yield stress σ0=200 MPa. 3.3. Finite element model Geometrically linear finite element (FE) limit analyses were performed using ABAQUS. To reduce computing time, reduced integration elements, a 20 node quadratic brick reduced integration element (C3D20R) is used. The number of elements is 7200 and the number of nodes is 36400 for all of the models, shown in Fig.2. The static riks analysis is adopted to avoid the difficulty of convergence in FE analysis. The force moment was applied by rotation at the right top end-nodes of the attached straight pipe, constrained using the MPC (multi-point constraint) option. A rigid beam will be formed by connecting this single node to the nodes of this end. Sufficiently large rotation was applied, and the bending moment was determined from nodal forces [32]. Fig. 3 shows typical FE results of moment-rotation curves for pipe bends under pure torsion moment. Plastic limit moment can be determined apparently with maximum value in moment-rotation curve [32]. For combined loading condition, three loading sequences were considered in the investigation: proportional loading, MB-MT loading and MT-MB respectively. In proportional loading, the internal pressure and moment are applied to the model simultaneously in a proportional manner. In MB-MT loading, the in-plane bending moment is applied to in the first step then held constant, and the torsion moment is applied in the second step. In MT-MB loading, the torsion moment is applied to in the first step then held constant, and the in-plane bending moment is applied in the second step.
Fig.1. Schematic illustrations of a 90o pipe bend Table 1 Geometric parameters of analyzed pipe bends r=100mm L=1000mm Pure in-plane bending moment r/t=5 R/r=2, 4, 6 r/t=10 R/r=2, 4, 6 r/t=20 R/r=2, 4, 6 r/t=50 R/r=2, 4, 6 Pure torsion moment r/t=5 R/r=2,3,4,5, 6 r/t=10 R/r=2,3,4,5, 6 r/t=20 R/r=2,3,4,5, 6 r/t=50 R/r=2,3,4,5, 6 Combined in-plane plane bending and torsion moment r/t=5 R/r=2, 6 r/t=20 R/r=2, 6
λ=0.4, 0.8, 1.2 λ=0.2, 0.4, 0.6 λ=0.1, 0.2, 0.3 λ=0.04, 0.08, 0.12 λ=0.4,0.6,0.8,1,1.2 λ=0.2,0.3,0.4,0.5,0.6 λ=0.1,0.15,0.2,0.25,0.3 λ=0.04,0.06,0.08,0.1,0.12 λ=0.4, 1.2 λ=0.1, 0.3
Fig.2. Finite element mesh
1.0
MT/Mstr
r/t=5, R/r=6
0.8
r/t=5, R/r=2
0.6
r/t=20, R/r=6 r/t=20, R/r=2
0.4 0.2 0.0 0.00
0.04
0.08
0.12
0.16
0.20
θ/rad
Fig.3. Typical moment-rotation curves 4. Results and discussion 4.1. Plastic limit in-plane bending moment As is reported in section 2.1, most of the past studies emphasize on in-plane bending. In this section analysis is processed briefly mainly to validate Kim and Oh’s [32] results based on FE
（
method. Fig. 4 shows the ovalization deformation in cross section A-A seen in Fig.1
） for
in-plane bending. Because this section deformation is the most significant compared to other section spread along the axially path, this deformation in section A-A can deflect the deformation capacity and level, which will have a great importance to the plastic loads evaluation. In Fig.4 the deformed shape and undeformed circular shape are put together so as to see this ovalization shape clearly. This deformation is oriented at vertical and horizon with respect to the plane of pipe bend. Fig.5 shows the FE results and proposed results by Kim and Oh [32] in Eq.(7). In this figure, Eq.(7) can predicted FE results well for r/t≤20. However, for the thin pipe bends with r/t=50, this Eq.(7) underestimate FE results. This is only because FE based predicted Eq.(7) is intended for
0.1 λ 0.5 in Ref.[32], while FE results is for 0.04 λ 1.2 in this paper. In this aspect results by Eq.(7) can extrapolate to a high value of λ, and fail to a low value of λ. So a new predicted equation is proposed in Eq.(15), and results show better agreement.
(a) (b) Fig.4. Deformed and undeformed shapes of cross section A-A for in-plane bending: (a) for in-plane closing bending (b) for in-plane opening bending
M I / M str = 1 / A1e B1 + 1 / C1e D1 R / r
A1 = 1.1091 + 0.1657r / t ;
B1 = −0.0234 + 0.0155r / t;
C1 = 0.7886 + 0.1047r / t ; D1 = −2.3805 + 0.0129r / t
1 0.9 0.8 0.7 0.6 0.5
Hollow date: closing bending Solid data: openning bending
1 0.9 0.8 0.7 0.6 0.5 MI/Mstr
MI/Mstr
(15)
Hollow date: closing bending Solid data: openning bending
0.4
0.4 0.3 Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Eq.(7) by Kim
0.1 1.5
0.04 λ 1.2
2
2.5
3 3.5 R/r
4 4.5 5 5.5 6 6.5
0.3 Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Proposed Eq.(15)
0.1 1.5
2
2.5
3
3.5
4 4.5 5 5.5 6 6.5
R/r
(a) (b) Fig.5. Comparison of finite element solutions for pipe bends under in-plane bending moment with expected equations: (a) with Eq.(7) (b) with Eq.(15) 4.2. Plastic limit torsion moment 4.2.1. A detailed analysis for stress distribution and a common sense of torsion moment Up till now research on torsion moment capacity for pipe bends is so little in published references that no clear analysis for the basic awareness for torsion effect of pipe bends exists. Research in Ref.[45] by Guo for pipe bends with plastic limit torsion moment is idealized to straight pipes. In Guo’s[45] assumption, only torsion is existing through all the cross sections along the axial direction, and no bending region exists. In this respect only shear stress exists without other stress. So based on this assumption plastic limit torsion moment for pipe bends is the same as straight pipes. However, this assumption is not realistic as bending effect is always exists when torsion moment is applied, and this is a combined bending and torsion effects for pipe bends. Fig.6 shows the ovalization deformation in cross section A-A for torsion moment, where Fig.6 (a) is deformation shape in pipe bends, Fig.6 (b) is in straight pipes. It can be seen from Fig.6 that the shape in pipe bends is oriented at 45o angle with respect to the plane of pipe bends due to combined torsion and bending effect, while the straight pipe is just an outward expanded circular section due to pure torsion effect lonely. It can be seen obviously that this deformed shape in pipe bends is not caused only by shear stress.
(a) (b) Fig.6. Deformed and undeformed shapes of cross section A-A for torsion moment (a) in pipe bend (b) in straight pipe To have a common sense of the pipe bends behavior under torsion moment, a detailed stress analysis is conducted for a pipe bend with pipe geometry of r/t=20, R/r=2, MT=4.8KNm. Fig.7 shows the points and paths in stress analysis. Path1 to path3 are circumferential path in pipe cross section, while path4 to path6 are axial path along the direction of pipeline. Stress distribution is shown in Fig.8, 9, where Fig.8 corresponds to stress for path1 to path3 and Fig.9 for path4 to path6. According to the description in two figures, it can be concluded as follows. For path1in the left bottom end, shear stress τ is so small that can be ignored, circumferential stress σθ and axial stress σZ are obvious compared with τ, so in path1 bending is main loading effect. For path2 in right top end, uniform shear stress exists along this path, while circumferential stress and axial stress are almost 0, so in this path2 torsion is the main loading effect. For path3 in cross section A-A, all these stresses are evident showing that this cross section is combined bending and torsion moment, and local stress concentration will cause yielding resulting in pipe failure firstly. In order to find out the loading effect in axial direction along pipeline, stress in path4 to path6 is analyzed. In path5 all the stresses exist showing bending and torsion moment interaction effects. In path4 from point2 to point1, all the stresses exist are resulting from combined loading effects within 3r length of straight pipe, and beyond this region 3r shear stress is near zero which shows bending dominated effect in this region. In path6 from point3 to point4, it is just the opposite showing in Fig.9.
Path 3
Point 3
Path 2 Path 6
Point 2
Point 4
Path 5
Path 4 Path 3 Path 1 Path 1 Point 1 Fig.7. Path for stress analysis
Path 2
24
Path 1 in left bottom end Path 2 in right top end Path 3 in cross section A
Path 1 in left bottom end Path 2 in right top end Path 3 in cross section A
160 120
16
80
12
40 σθ/MPa
τ /MPa
20
8
0
-40
4 -80
0
-120
-4 0 30 60 90 120 150 180 210 240 270 300 330 360 crown extrados crown crown intrados o ϕ/ ( )
-160 0 30 60 90 120 150 180 210 240 270 300 330 360 crown crown crown extrados intrados o ϕ/ ( )
(a)
(b) 150
Path 1 in left bottom end Path 2 in right top end Path 3 in cross section A
100
σZ/MPa
50 0 -50 -100 -150 0 30 60 90 120 150 180 210 240 270 300 330 360 crown crown crown extrados intrados o ϕ/ ( )
(c) Fig.8. Stress distribution in circumferential path (path1 to path3): (a) shear stress (b) circumferential stress (c) axial stress
40
Results in Path 5
120
Axial stress σz
Combined bending and torsion
60
60
Bending dominated
20 0 -20 -40
Shear stress τ Circumferential stress σθ
Combined bending and torsion
90
Stress/MPa
Stress/MPa
Shear stress τ Circumferential stress σθ
Results in Path 4
80
Axial stress σz
30 0 -30 -60
-60
-90
-80
-120
0 Point 2
200
400
600
Path distence/mm
(a)
800
1000 Point 1
0
Point 2
15
30
45
60
Path distence/(o)
(b)
75
90
Point 3
Results in Path 6
60
Combined bending and torsion
Stress/MPa
40
Shear stress τ Circumferential stress σθ Axial stress σz Torsion dominated
20 0 -20 -40 -60 0
Point 4
200
400 600 Path distence/mm
800
1000 Point 3
(c) Fig.9. Stress distribution in axial path (path4 to path6): (a) stress distribution in path4 (b) stress distribution in path5 (c) stress distribution in path6 So it can be concluded that for pipe bends under pure torsion moment, a common awareness for loading effect should be as follows: In the geometrical bending section combined bending moment and torsion effects exist at the same time, and these combined effects also spread along attached straight pipes for as long as 3r in axial direction of straight pipes, which are showing in Fig.10.
Torsion dominated region
Combined bending and torsion region
Bending dominated region Fig.10. Force patterns for pipe bends under torsion moment with attached long straight pipes 4.2.2. Plastic limit torsion moment Fig.11 shows normalized FE results for torsion moment, where (a) shows the comparison with Eq.(8) by Guo [45], (b) shows the comparison with predicted Eq.(16) using single variable λ, (c) shows the comparison with predicted Eq.(17) by two variables R/r and r/t. Eq.(16) and Eq.(17) are predicted FE results for torsion moment in present work as follows:
1 0.9 0.8 0.7 0.6 0.5
1 0.9 0.8 0.7 MO/Mstr
MO/Mstr
0.4 0.3 Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Eq.(8) by Guo
0.1 1.8
2.4
3
3.6
4.2
0.6 0.5 0.4
Symbol: FE data Line: Predicted Eq.(16)
0.3
4.8 5.4 6 6.6
0.16
0.32
R/r
0.48 0.64 0.8
λ
(a)
(b) 1 0.9 0.8 0.7 0.6 0.5 MO/Mstr
0.4 0.3
Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Predicted Eq.(17)
0.1 1.8
2.4
3
3.6
4.2
4.8 5.4 6 6.6
R/r
(c) Fig.11. Comparisons of FE results for plastic limit torsion moment with some proposed equations: (a) with Eq.(8) (b) with Eq.(16) (c) with Eq.(17)
M O / M str = 0.8678λ 0.2862
(16)
M O / M str = 1.0682(r / t ) −0.3114 ( R / r )0.1787
(17)
It can be concluded in Fig.11 that Eq.(8) by Guo[45] overestimate our FE data in most cases. Differences between FE and Eq.(8) increase with increasing r/t and decreasing R/r. This implies that Eq.(8) by Guo [45] is not fit for predicting FE data especially for thin-walled and small bend radius pipe bends. It can also be concluded that Eq.(17) with two variables R/r and r/t can predicts FE results better than Eq. (16) with a single variable λ. This is in accordance with that for in-plane bending as is mentioned in section 2.2.1. It can be seen in Fig. 11 and Eq.(17) that r/t is the main factor affecting the limit load solutions. These limit results predicted by Eq.(17) showing in Fig.11 can be interpreted in Fig.12. In this figure, FE data on Mises stress in cross section A-A (path 3) are normalized with respect to Mises stress for straight pipes by Eq.(18), where the applied torsion is MT=4.8KNm, ro is the outer radius in cross section. Results show that the normalized peak stress is higher for thin-walled pipe bends and pipe bends with small bend radius. This high peak stress for pipe bends accelerates the yielding process compared with straight pipes and then approach plastic limits prior to straight pipes as load increasing. This means the limit load of FE result is lower compared with Eq.(8) by Guo[45] especially for thin-walled and small bend radius pipe bends.
3M T ro I
(18)
σeq(FE)/σeq(Eq.(18))
σ eq =
6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
r/t=10,R/r=2 r/t=10,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0 30 60 90 120 150 180 210 240 270 300 330 360 crown extrados crown intrados crown o ϕ/ ( )
Fig.12. Comparisons of Mises stress for pipe bends with Mises stress of straight pipes 4.3. Plastic limit loads for combined bending and torsion moment Fig.13 shows the ovalization deformation in cross section A-A for combined loading condition. In Fig.13 the deformed shape and direction are between bending and torsion moment seen from Fig.4 and Fig.6. This appearance can reflect the deformation characteristic for pipe bends under combined loads.
(a) (b) Fig.13. Deformed and undeformed shapes of cross section A-A for combined in-plane bending and torsion moment: (a) for combined in-plane closing bending and torsion moment (b) for combined in-plane opening bending and torsion moment 4.3.1. FE modeling parameters independency on limit load In limit load theory, the plastic limit load is independent on some model parameters, mainly independent on the loading sequence (or loading path [2]) and material constants E. In addition, Lee et al. [36] proposed σs/E independent normalized plastic limit load solution based on FE analysis. These are verified by calculating the plastic limit loads of four pipe geometry parameters ranging from 0.1 λ 0.6 for three load sequences, MB-MT loading, MT-MB loading and proportional loading respectively, which has already introduced in section 3.3. FE results are show in Figs. 13-15. In these figures bending-torsion interaction surfaces were constructed by
performing elastic-plastic analysis of each pipe bend for different load combinations, ranging from bending only to torsion only. Results in Figs. 13-15 suggest that the circular interaction rule Eq.(19) is a great approximation for pipe bends under combined bending moment and torsion. (19)
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.2 0.0 -0.2 -0.2
MLO/MO
MLO/MO
( M LI / M I ) + ( M LO / M O ) = 1
0.0
0.2
0.4
0.4
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.2 0.0 0.6
0.8
1.0
-0.2 -0.2
1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
MLIC/MIC
MLIO/MIO
1.2
1.2
1.0
1.0
0.8
0.8
0.6
MLIO/MIO
MLIC/MIC
(a) (b) Fig.14. Bending-torsion interaction curve for MB-MT loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.4 0.2
0.4
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.2
0.0 -0.2 -0.2
0.6
0.0 0.0
0.2
0.4
0.6
0.8
1.0
-0.2 -0.2
1.2
0.0
MLO/MO
0.2
0.4
0.6
0.8
1.0
1.2
MLO/MO
1.0
1.0
0.8
0.8
0.6
0.6 r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.4 0.2 0.0 0.0
MLO/MO
MLO/MO
(a) (b) Fig.15. Bending-torsion interaction curve for MT-MB loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending
0.2
0.4 0.6 MLIC/MIC
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.4 0.2
0.8
1.0
0.0 0.0
0.2
0.4 0.6 MLIO/MIO
0.8
1.0
(b) (a) Fig.16. Bending-torsion interaction curve for proportional loading: (a) combined torsion and
in-plane closing bending (b) combined torsion and in-plane opening bending Take MB-MT loading for instance, these results by Eq.(19) can be interpreted according to yielding surface and ovalization deformation showing in Figs. 17, 18. In Fig.17 the yielding region spread from pipe crown to pipe intrados, then spread to straight section till extrados, when all the bend section is overall yielding, this continues spreads along straight pipe section, finally almost all the pipe section yields. This yielding region extends with the increasing MB where the applied torsion moment is 14.4KNm. Fig.17 clearly shows that for MB≤0.4MIO, this region is small and spread slowly with MB increasing, while spread faster and faster for 0.4 MIO≤MB≤MLIO. This phenomenon implies that the circular interaction curve in Figs.14-16 is reasonable. As ovalization deformation can reflect loading capacity for pipeline, the percent ovalization C in cross section A-A for a pipe bend under combined loads is shown to explain FE data. Showing in Fig.18, C increases with the loads increasing, which shows the interaction rule in Figs.14-16 on the other hand. In Fig.18 the percent ovalization C is written as
Dmax − Dmin D D + Dmin D = max 2 C=
Where,
(20) (21)
Dmax is the maximum outside pipe diameter, and Dmin is the minimum outside pipe diameter.
(a)
(c)
(b)
(d)
(e) (f) Fig.17. Mises criterion region for a pipe bend under combined in-plane opening bending and torsion moment, where the applied torsion moment are 14.4KNm in unified: (a) MB=0 (b) MB =0.2MIO (c) MB =0.4MIO (d) MB =0.6MIO (e) MB =0.8MIO (f) MB =MLIO 0.12 0.10 Normalized M/MIO Ovality C
0.08
0 0.4 0.8
0.06 0.04 0.02 0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30
MT/MLO
Fig.18. Ovalization deformation in cross section A-A for a pipe bend under combined in-plane opening bending and torsion moment Material constants independent normalized limit load solution for combined loads is shown in Fig.19, where results show that the choice of σs and E is material independent. This also verifies the limit load theory meanwhile. 1.0
1.0
0.8
0.8 0.6 MLO/MO
MLO/MO
0.6 r/t=5, R/r=2
0.4 0.2
r/t=5, R/r=6 0.4 E=100GPa, σs=100MPa
E=100GPa, σs=100MPa E=400GPa, σs=100MPa
0.2
E=100GPa, σs=400MPa
E=100GPa, σs=400MPa 0.0 0.0
E=400GPa, σs=400MPa 0.2
0.4
0.6
MLIO/MIO
(a)
E=400GPa, σs=100MPa
0.8
1.0
0.0 0.0
E=400GPa, σs=400MPa 0.2
0.4
0.6
MLIO/MIO
(b)
0.8
1.0
1.0
1.0
0.8
0.8
0.6 MLO/MO
MLO/MO
0.6 r/t=20, R/r=2 0.4
r/t=20, R/r=6 0.4 E=100GPa, σs=100MPa
E=100GPa, σs=100MPa 0.2 0.0 0.0
E=400GPa, σs=100MPa E=100GPa, σs=400MPa
0.2
E=400GPa, σs=400MPa 0.2
0.4
0.6
0.8
1.0
0.0 0.0
MLIO/MIO
E=400GPa, σs=100MPa E=100GPa, σs=400MPa E=400GPa, σs=400MPa 0.2
0.4
0.6
0.8
1.0
MLIO/MIO
(c) (d) Fig.19. Effect of the materials constants on plastic limit loads for combined loads: (a) r/t=5, R/r=2 (b) r/t=5, R/r=6 (b) r/t=20, R/r=2 (b) r/t=20, R/r=6 4.3.2. Comparison with existing results from literature Fig.20 shows comparison of proposed result in Eq.(19) with Eq.(9) by Guo [45] and Eq.(14) in ASME [46]. This shows some differences obviously in this figure, and comparisons show that both estimate equations by Guo [45] and ASME [46] are inconsistent with FE data. In Ref.[45] by Guo, plastic limit torsion moment is considered to be the same as straight pipes as shown in section 4.2.1. This proposed solution for torsion is higher than FE data especially for thin-walled and small bend radius pipe bends. For in-plane bending Guo [45] only considered circumferential stress for stress analysis and overlooked the stiffening effect of straight pipes, which will cause a conservative estimation for thin-walled and small bend radius pipe bends. So with a circular interaction, some flat oval-shaped cures can depicts this rule by Guo [45] in Eq.(9). In ASME [46], the load moment is a simply bending moment, regardless of the bending mode and direction. In other hand, according to ASME, only circumferential stress is considered for bending in pipe crown, and when yielding appears in crown first, this state is regarded as the limiting state, which overlooks the spread procedure of yielding surface from crown to all of the pipe bends. In addition the load moment from ASME includes safety factor for design. So when considering these influencing factors, it is easy to understand why results in Fig. 20 (b) are much more conservative to FE data.
2.2 2.0
1.4 Proposed result Eq.(9) by GUO
Proposed result Eq.(14) in ASME
1.2
pipe parameters a: r/t=5, R/r=2 b: r/t=5, R/r=6 c: r/t=20, R/r=2 d: r/t=20, R/r=6
1.8 a
1.2 1.0 0.8 0.6
b
MLO/MO
MLO/MO
1.4
1.0
pipe parameters a: r/t=5, R/r=2 b: r/t=5, R/r=6 c: r/t=20, R/r=2 d: r/t=20, R/r=6
1.6
c
c a
0.8
d
0.6
b
0.4 d
0.4
0.2
0.2
0.0 0.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
MLI/MI
MLI/MI
(a) (b) Fig.20. Comparison of proposed result in Eq.(19) with some existing limit load solutions for pipe bends under combined in-plane bending and torsion moment: (a) with Eq.(9) by Guo (b) with Eq.(14) in ASME 5. Verification of finite element procedure In order to validate finite element method on plastic limit analysis, a series of verification approaches are confirmed. Detailed analysis is as follows. 5.1. Precision validation of finite element method In this section, a total of 36 FE models are calculated to validate our mesh adequacy. 4 pipe geometry parameters are chosen ranging for 0.1 λ 0.6 , and for each pipe bend the mesh node number are from 132 to 132335. Fig. 21 shows FE results showing that FE data decrease with increasing node number while results tend to converge with node increasing. It can be seen from Fig.21 clearly that this node number of 36440 in our present work can satisfy mesh adequacy. 1.0 0.9 0.8
r/t=10, R/r=6
MO/Mstr
0.7 r/t=10, R/r=2
0.6
r/t=20, R/r=6
0.5
r/t=20, R/r=2
0.4 0.3 0.2
FE results in this work ( Node number, 36440)
0.1 0.0 0
2
4
6 8 10 Node number/10000
12
14
Fig.21. Effect of mesh on plastic limit loads 5.2. Finite element yield criterion validation
Based on unified yield criterion (UYC) [48] the equivalent stress σUE is defined as
σ 1 +σ 3 1 σ UE = σ 1 − 1 + b (bσ 2 + σ 3 ), σ 2 2 σ = 1 (σ + bσ ) − σ , σ σ 1 +σ 3 UE 1 2 3 2 1+ b 2
(22)
The UYC can be simply be written as σ UE = σ s when yielding and UYC plastic limit load solution for straight pipes under internal pressure can be derived as
Pstr =
2(1 + b) t σs, 0 b 1 2+b r
(23)
With different choices of parameter b, the UYC can be simplified to the MY [49] yield criterion (b=1/3), the GM [50] yield criterion (b=0.4), the ASSY [51] yield criterion (b=0.618) etc. Fig.22 shows the comparison of FE results and analytical solutions by Eq.(23) on UYC. According to this comparison, Mises criterion based solutions lie in the middle, and shows great agreement with FE results. This implies that non-hardening J2 flow theory is used by FE procedure which gives a modest prediction of plastic limit results among these unified solutions. So this analysis validates reliability of FE yield criterion. 22 20 18
Pstr/MPa
16 14
Tresca (b=0) ASSY (b=0.168) MY (b=1/3) Mises (b=0.366) GM (b=0.4) EA (b=0.529) TSS (b=1)
12 10
FE Results
8 6 0.03
0.04
0.05
0.06
0.07
0.08
r/t
Fig.22. Comparison of FE data with analytical solutions based on different yield criterion for pressurized straight pipes 5.3. Finite element results validation with experiment data available in literature To validate our proposed expressions, experiment results would be a best validation. Some available experiment data can be obtained from Refs.[25]. Experiment data and their comparisons with FE results are listed in Table 2, where M exp indicates experiment result and M FE is FE result. In this table it can be seen clearly that FE based results can predict experiment data well, as these differences are within ±5%. As no experimental data are available for combined loading conditions, therefore reasonable explanations are tried in Section 4.3.1 in support of the solutions proposed in the present work. Noting that both pure bending, torsion and combined solutions are based on FE results, it is implicitly suggested that the accuracy of the proposed solutions for combined loading condition is credible. It can be concluded that FE based predicted equations are consistent with the available experiment data which provide confidence to present work.
Table 2 Comparison of the estimated FE results by Eqs. (15) and (17) with existing experiment results for pipe bends under in-plane bending and torsion moment Load type Material R/r r/t σS / Error/% M FE / M FE M exp MPa /KNm /KNm
M exp
In-plane closing bending
In-plane opening bending Torsion
ASTM A-106B ASTM A-106B SS304 SS304 ASTM A-106B
1.94 2.91 1.56 2.67 2.84
7.34 7.34 12.5 8.58 11.33
239 261 272 265 345
41.69 48.36 4.38 1.65 32.43
40.82 49.22 4.52 1.71 33.74
0.98 1.01 1.03 1.04 1.04
2.08 -1.77 -3.19 -3.63 -4.03
ASTM A-106B SS 304
2.84 2.67
11.33 8.44
345 265
36.61 1.82
37.01 1.81
1.01 0.99
-1.11 0.78
6. Conclusions In present work, 3D finite element method is used to determine plastic limit load solutions for pipe bends under combined bending and torsion moment. Detailed results are investigated as follows: (1) For pipe bends under pure in-plane bending moment: FE based results proposed by Kim and Oh [32] can predict FE results well for r/t≤20. While underestimate for r/t=50, a new predicted equation is proposed and results show great agreement, which extends the range of pipe geometry λ. (2) For pipe bends under pure torsion moment: A common awareness for loading effect should be as follows: In the geometrical bending section combined bending moment and torsion effects exist at the same time, and these combined effects also spread along attached straight pipes for as long as 3r in axial direction of straight pipes. Results by Guo [45] overestimate FE data. Differences increase with increasing r/t and decreasing R/r which implies that results by Guo [45] is not appropriate to predict FE data especially for thin-walled and small bend radius pipe bends. Research by Guo [45] for pipe bends with plastic limit torsion moment is idealized to straight pipes. Results show that the normalized peak stress is higher for thin-walled pipe bends and pipe bends with small bend radius. This high peak stress for pipe bends accelerates the yielding process compared with straight pipes and then approach plastic limits prior to straight pipes as load increasing. It can also be concluded that equation with two variables R/r and r/t can predicts FE results better than that with a single variable λ. This is in accordance with that for in-plane bending. r/t is the main factor affecting the limit load solutions. (3) For pipe bends under combined in-plane bending and torsion moment: Normalized plastic limit load is independent on the loading sequence and material constants. Results show that the circular interaction rule is a great approximation for pipe bends under combined bending and torsion moment. Relevant results by Guo [45] and ASME [46] are not appropriate to predict FE results. In this respect of finite element analysis, overall yielding considering the spread process of yield
region from crown to the straight pipe shows these promising finite element results. (4) A series of verification approaches are confirmed in order to validate our finite element method on plastic limit analysis. Based on the FE data, estimated equations of limit loads for pipe bends under combined bending and torsion moment are proposed, which are justified to a good choice for limit load assessment. Acknowledgments The authors gratefully acknowledge the financial support by Graduate Student Scientific Innovative Project of Jiangsu Province (No. SJZZ_0096), China and Jiangsu Natural Science Funds (BK2008373). References [1] Chattopadhyay J, Tomar AKS. New plastic collapse moment equations of defect-free and throughwall circumferentially cracked elbows subjected to combined internal pressure and in-plane bending moment. Eng Fract Mech 2006; 73: 829–854. [2] Robertson A, Li H, Mackenzie D. Plastic collapse of pipe bends under combined internal pressure and in-plane bending. Int J Pressure Vessels Piping 2005; 82: 407-416. [3] Toshiyuki M, Yoshiaki. Proposal of failure criterion applicable to finite element analysis results for wall-thinned pipes under bending load. Nucl Eng Des 2012; 242: 34-42. [4] Marcal PV. Elastic–plastic behaviour of pipe bends with in-plane bending. J Strain Anal 1967; 2(1): 84-90. [5] Spence J, Findlay GE. Limit load for pipe bends under in-plane bending. Proceedings of the second international conference on pressure vessel technology, San Antonio; 1973. 393–399. [6] Calladine CR. Limit analysis of curved tubes. J Mech Eng Sci 1974; 16(2):85–87. [7] Goodall IW. Large deformations in plastically deformed curved tubes subjected to in-plane bending. Research Division Report RD/B/N4312. UK: Central Electricity Generating Board; 1978. [8] Goodall IW. Lower bound limit analysis of curved tubes loaded by combined internal pressure and in-plane bending moment. Research Division Report RD/B/N4360. UK: Central Electricity Generating Board; 1978. [9] Kitching R, Zarrabi K, Moore MA. Limit moment for a smooth pipe bend under in-plane bending. Int J Mech Sci 1979; 21(12): 731-738. [10] Shalaby MA, Younan M Y A. Nonlinear Analysis and Plastic Deformation of Pipe Elbows Subjected to In-Plane Bending. Int J Pressure Vessels Piping 1998; 75: 603-611. [11] Shalaby M A, Younan M Y A. Limit Loads for Pipe Elbows Subjected to In- Plane Opening Bending Moments. ASME J Pressure Vessel Technol 1999; 121: 17-23. [12] Shalaby M A, Younan MYA. Effect of Internal Pressure on Elastic-Plastic Behavior of Pipe Elbows Under In-Plane Bending Moments. ASME J Pressure Vessel Technol 1999, 121: 400-405. [13] Hashem MM, Younan MYA. Nonlinear Analysis of Pipe Bends Subjected to Out-of-Plane Moment Loading and Internal Pressure. J Press Vessel Technol 2001; 123: 253 - 258. [14] Hashem MM, Younan MYA. Limit-Load Analysis of Pipe Bends Under Out-of-Plane Moment Loading and Internal Pressure. J Press Vessel Technol 2002; 124: 32 - 37. [15] Hashem MM, Younan MYA. The Effect of Modeling Parameters on the Predicted Limit Loads for Pipe Bends Subjected to Out-of-Plane Moment Loading and Internal Pressure. J Press Vessel Technol 2000; 122: 450456. [16] Touboul F, Ben DM, Acker D. Design criteria for piping components against plastic collapse: application to pipe bend experiments. In: Liu C, Nichols RW, editors. Pressure vessel technology, proceedings of 6th international conference held in Beijing, 11-15th September 1988. p. 73-84. [17] Greenstreet WL. Experimental study of plastic response of pipe elbows. Report, ORNL/ NURE G-24, Oak Ridge National Laboratory, TN; 1978. [18] Tan Y, Wilkins K, Matzen V. Correlation of test and FEA results for elbows subjected to out-of-plane loading. Nucl Eng Des 2002; 217: 213-224. [19] Chattopadhyay J, Kushwaha HS, Roos E. Some Recent Developments on Integrity Assessment of Pipes and
Elbows. Part I: Theoretical investigations. Int J Solids Struct 2006; 43(10): 2904-2931. [20] Chattopadhyay J, Kushwaha HS, Roos E. Some Recent Developments on Integrity Assessment of Pipes and Elbows. Part II: Experimental Investigations. Int J Solids Struct 2006; 43(10): 2932-2958. [21] Orynyak IV, Radchenko SA. Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect. Int J Solids Struct 2007; 44: 1488-1510. [22] Liu YH, Zhang XF, Cen ZZ. Lower bound shakedown analysis by the symmetric Galerkin boundary element method. International Journal of Plasticity. 2005, 21(1):21-42. [23] Chen SS, Liu YH, Cen ZZ. Lower-bound limit analysis by using the EFG method and non-linear programming. International Journal for Numerical Methods in Engineering. 2008, 74:391-415. [24] Chen SS, Liu YH, Cen ZZ. Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming. Computer Methods in Applied Mechanics and Engineering. 2008, 197(45-48):3911-3921. [25] Han JJ, Lee KH, Kim NH, Kim YJ, Jerng DW, Budden P J. Comparison of existing plastic collapse load solutions with experimental data for 90o elbows. Int J Pressure Vessels Piping 2012; 89: 19-27. [26] MSC. Software Corp. MSC.Marc Volume D: User subroutines and special routines; 2005. [27] ANSYS, Version 8.1. General Purpose Finite Element Program; 2004. [28] Gullerud AS, Kopenhoefer K, Ruggieri C, Roy A, Roychoudhuri S, Dodds Jr RH. WARP3D-RELEASE 13.15, Dynamic nonlinear analysis of solids using a preconditioned conjugate gradient software architecture. User’s manual, University of Illinois, Urbana, IL, USA; 2001. [29] ABAQUS. User’s manual version 6.9; 2009. [30] Hong SP, An JH, Kim YJ, Nikbin K, Budden PJ. Approximate elastic stress estimates for elbows under internal pressure. International Journal of Mechanical Sciences 2011; 53: 526-535. [31] An JH, Hong SP, Kim YJ , Budden PJ. Elastic stresses for 90o elbows under in-plane bending. International Journal of Mechanical Sciences 2011; 53: 762-776. [32] Kim YJ, Oh CS. Limit loads for pipe bends under combined pressure and in-plane bending based on finite element limit analysis. Int J Press Vessels Piping 2006; 83: 148 - 153. [33] Kim YJ, Oh CS. Closed-form plastic collapse loads of pipe bends under combined pressure and in-plane bending. Eng Fract Mech, 2006; 73: 1437-1454. [34] Kim YJ, Oh CS. Effects of attached straight pipes on finite element limit analysis for pipe bends. Int J Press Vessels Piping 2007; 84: 177 - 184. [35] Hong SP, Kim JH, YJ Kim, Budden PJ. Effect of internal pressure on plastic loads of 90o elbows with circumferential part-through surface cracks under in-plane bending. Engng Fract Mech 2010; 77: 577-596. [36] Kuk-Hee Lee, Chang-Sik Oh, Yun-Jae Kim, Kee-Bong Yoon. Quantification of the yield strength-to-elastic modulus ratio effect on TES plastic loads from finite element limit analyses of elbows. Engng Fract Mech 2009; 76: 856-875. [37] Christo MT, Veerappan AR, Shanmugam S. Effect of ovality and variable wall thickness on collapse loads bends subjected to in-plane bending closing moment. Eng Fract Mech 2012;79: 138-148. [38] Christo MT, Veerappan AR, Shanmugam S. Comparison of plastic limit and collapse loads in pipe bends with shape imperfections under in-plane bending and an internal pressure. J Press Vess Pip 2012; 99-100:23-33. [39] Christo MT, Veerappan AR, Shanmugam S. Effect of internal pressure and shape imperfections on plastic loads of pipe bends under in-plane closing moment. Eng Fract Mech 2013; 105: 1-15. [40] Buckshumiyan A, Veerappan AR, Shanmugam S. Plastic collapse loads in shape-imperfect pipe bends under in-plane opening bending moment. Int J Press Vess Pip 2013; 111-112: 21-26. [41] Buckshumiyan A, Veerappan AR, Shanmugam S. Determination of collapse loads in pipe bends with ovality and variable wall thickness under internal pressure and in-plane opening moment. Int J Press Vess Pip 2014; xxx: 1-9. [42] Kim YJ, Je JH, Oh CS, Han JJ, Budden PJ. Plastic loads for 90° thick-walled elbows under combined pressure and bending. J Strain Anal Eng 2010; 45: 115-127. [43] Li J, Zhou CY, Xue JL, He XH. Limit loads for pipe bends under combined pressure and out-of-plane bending moment based on finite element analysis. Int J Mech Sci 2014; 88: 100-109. [44] Ayob AB, Moffat DG, Mistry J. The interaction of pressure, in-plane moment and torque loadings on piping elbows. Int J Pressure Vessels Piping 2003; 80: 861-869. [45] Guo C. Plastic limit loads for surface defect pipes and bends under combined loads of tension, bending, torsion and internal pressure. PhD thesis, East China University of Science and Technology, Shanghai, China.
1999.(in Chinese) [46] ASME. Rules for construction of nuclear facility components. New York, USA: ASME Boiler and Pressure Vessel Code Committee, Section III, NB, NC and ND, American Society of Mechanical Engineers; 2004. [47] Ghosh S, Roy P. Quantification of the uncertainty in stress index B2 for pipe bends subjected to out-of-plane bending. Int J Pressure Vessels Piping 2012; 95: 24-30. [48] Wang LZ, Zhang YQ. Plastic collapse analysis of thin-walled pipes based on unified yield criterion. Int J Mech Sci 2011; 53: 348-354. [49] Zhang SH, Song BN, Wang XN, Zhao DW. Analysis of plate rolling by MY criterion and global weighted velocity field. Appl Math Model 2014; 38: 3485-3494. [50] Zhang SH, Wang XN, Song BN, Zhao DW. Limit analysis based on GM criterion for defect-free pipe elbow under internal pressure. Int J Mech Sci 2014; 78: 91-96. [51] Zhu XK, Leis BN. Average shear stress yield criterion and its application to plastic collapse analysis of pipelines. Int J Pressure Vessels Piping 2006; 83: 663-671.
Fig.1. Schematic illustrations of a 90o pipe bend Fig.2. Finite element mesh Fig.3. Typical moment-rotation curves Fig.4. Deformed and undeformed shapes of cross section A-A for in-plane bending: (a) for in-plane closing bending (b) for in-plane opening bending Fig.5. Comparison of finite element solutions for pipe bends under in-plane bending moment with expected equations: (a) with Eq.(7) (b) with Eq.(15) Fig.6. Deformed and undeformed shapes of cross section A-A for torsion moment (a) in pipe bends (b) in straight pipes Fig.7. Path for stress analysis Fig.8. Stress distribution in circumferential path (path1 to path3): (a) shear stress (b) circumferential stress (c) axial stress Fig.9. Stress distribution in axial path (path4 to path6): (a) stress distribution in path4 (b) stress distribution in path5 (c) stress distribution in path6 Fig.10. Force patterns for pipe bends under torsion moment with attached long straight pipes Fig.11. Comparisons of FE results for plastic limit torsion moment with some proposed equations: (a) with Eq.(8) (b) with Eq.(16) (c) with Eq.(17) Fig.12. Comparisons of Mises stress for pipe bends with Mises stress of straight pipes Fig.13. Deformed and undeformed shapes of cross section A-A for combined in-plane bending and torsion moment: (a) for combined in-plane closing bending and torsion moment (b) for combined in-plane opening bending and torsion moment Fig.14. Bending-torsion interaction curve for MB-MT loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending Fig.15. Bending-torsion interaction curve for MT-MB loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending Fig.16. Bending-torsion interaction curve for proportional loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending Fig.17. Mises criterion region for a pipe bend under combined in-plane opening bending and torsion moment, where the applied torsion moment are 14.4KNm in unified: (a) MB=0 (b) MB =0.2MIO (c) MB =0.4MIO (d) MB =0.6MIO (e) MB =0.8MIO (f) MB =MLIO Fig.18. Ovalization deformation in cross section A-A for a pipe bend under combined in-plane opening bending and torsion moment Fig.19. Effect of the materials constants on plastic limit loads for combined loads: (a) r/t=5, R/r=2 (b) r/t=5, R/r=6 (b) r/t=20, R/r=2 (b) r/t=20, R/r=6
Fig.20. Comparison of proposed result in Eq.(19) with some existing limit load solutions for pipe bends under combined in-plane bending and torsion moment: (a) with Eq.(9) by Guo (b) with Eq.(14) in ASME Fig.21. Effect of mesh on plastic limit loads Fig.22. Comparison of FE data with analytical solutions based on different yield criterion for pressurized straight pipes Table 1 Geometric parameters of analyzed pipe bends r=100mm L=1000mm Pure in-plane bending moment r/t=5 R/r=2, 4, 6 r/t=10 R/r=2, 4, 6 r/t=20 R/r=2, 4, 6 r/t=50 R/r=2, 4, 6 Pure torsion moment r/t=5 R/r=2,3,4,5, 6 r/t=10 R/r=2,3,4,5, 6 r/t=20 R/r=2,3,4,5, 6 r/t=50 R/r=2,3,4,5, 6 Combined in-plane bending and torsion moment r/t=5 R/r=2, 6 r/t=20 R/r=2, 6
λ=0.4, 0.8, 1.2 λ=0.2, 0.4, 0.6 λ=0.1, 0.2, 0.3 λ=0.04, 0.08, 0.12 λ=0.4,0.6,0.8,1,1.2 λ=0.2,0.3,0.4,0.5,0.6 λ=0.1,0.15,0.2,0.25,0.3 λ=0.04,0.06,0.08,0.1,0.12 λ=0.4, 1.2 λ=0.1, 0.3
Table 2 Comparison of the estimated FE results by Eqs. (15) and (17) with existing experiment results for pipe bends under in-plane bending and torsion moment Load type Material R/r r/t σS / Error/% M FE / M exp M FE MPa /KNm /KNm
M exp
In-plane closing bending
In-plane opening bending Torsion
ASTM A-106B ASTM A-106B SS304 SS304 ASTM A-106B
1.94 2.91 1.56 2.67 2.84
7.34 7.34 12.5 8.58 11.33
239 261 272 265 345
41.69 48.36 4.38 1.65 32.43
40.82 49.22 4.52 1.71 33.74
0.98 1.01 1.03 1.04 1.04
2.08 -1.77 -3.19 -3.63 -4.03
ASTM A-106B SS 304
2.84 2.67
11.33 8.44
345 265
36.61 1.82
37.01 1.81
1.01 0.99
-1.11 0.78
Parameter r/t is the main factor influencing the plastic limit loads. Plastic limit load is independent on loading path and material constants. Estimated equations of limit loads for pipe bends under combined loads are proposed.
Plastic limit loads for pipe bends under combined bending and torsion moment Jian Li, Chang-Yu Zhou, Peng Cui, Xiao-Hua He
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PII: DOI: Reference:
S0020-7403(14)00408-1 http://dx.doi.org/10.1016/j.ijmecsci.2014.12.011 MS2891
To appear in:
International Journal of Mechanical Sciences
Received date: 9 October 2014 Revised date: 22 November 2014 Accepted date: 15 December 2014 Cite this article as: Jian Li, Chang-Yu Zhou, Peng Cui, Xiao-Hua He, Plastic limit loads for pipe bends under combined bending and torsion moment, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j. ijmecsci.2014.12.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Plastic limit loads for pipe bends under combined bending and torsion moment Jian Li, Chang-Yu Zhou*, Peng Cui, Xiao-Hua He School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China Abstract In present paper, 3D finite element (FE) method is used to determine plastic limit load solutions for pipe bends under combined bending and torsion moment. With a detailed analysis and comparison, a common awareness for loading effect is showing which will raise researchers concern. By the way, past solutions are not appropriate to estimate FE results. In this respect of finite element analysis, overall yielding considering the spread process of yield region from crown to the straight pipe shows these promising finite element results. A wide range of non-dimensional parameters for pipe bends are considered and plastic limit load solutions are suggested. The results show that r/t is the main factor affecting the limit loads. Plastic limit load is independent on the loading path and material constants by normalizing. Results show that the circular interaction rule is a great approximation for pipe bends under combined bending and torsion moment. A series of approaches are confirmed in order to validate our finite element method on plastic limit analysis. Based on the finite element results, approximate plastic limit load solutions are proposed. Present work will further improve the limit load solution for pipe bends under complex loading conditions. Key words: pipe bends; in-plane bending moment; torsion moment; limit loads; finite element 1. Introduction Pipe bends (or elbows) are commonly used components in a piping system, which are widely used in petroleum, chemical and nuclear power industries [1]. A detailed research on maximum loading carrying capacities of pipe bends is important in design and assessment of pipe system in power plants. Pipe bends are more flexible than straight pipes with similar dimensional parameters [2] due to the complex deformation they exhibit under bending loads. Pipe bends tend to show different mechanical properties as the interaction of geometrical nonlinearity and material nonlinearity when they are under these combined loads. Due to the self weight, valve weight, fluid weight in addition to heat expansion in the pipe system bending and torsion moment should not be overlooked. The bending stress caused by bending moment and torsion is generally greater than membrane stress only by pressure. So in the bending mode, flatting will occur, as the stress increase which even eventually produces buckling due to maximum compression stress from bending deformation and ovalization of circular cross-section of pipe bend [3] and collapse prior to the structure fracture. Meanwhile for pipe bends torsion will always exist as long as the out-of-plane bending exists. So research on pipe bends under combined bending and torsion moment is vital important to maintain the structure integrity of piping system. In present work, 3D finite element method is used to determine plastic limit load solutions for pipe bends under combined bending and torsion moment. Based on the finite element results, approximate plastic limit load solutions for limit pressures are proposed.
*Corresponding author. Tel.:+86-25-58139951; fax:+86-25-58139951. E-mail address: [email protected]
2. Research background 2.1. Review Over the past years large quantities of elastic-plastic behaviors for pipe bends have been carried out, especially for the plastic limit load solutions. Marcal [4]was the first one to present the elastic-plastic analysis for pipe bends under in-plane bending. Later Spence and Findlay [5] and Calladine [6] presented analytical limit bending solutions based on limit load theorems and plasticity mechanics theory. Goodall presented the first large deformation analysis of thin elbows under in-plane bending [7] and the lower bound solution for a thin elbow under combined loading [8]. Kitching et al.[9] determined a lower bound for the limit moment of a smooth circular pipe bend with no restrictions on the geometry using thin shell theory. Shalaby and Younan [10-12] presented a series of comprehensive studies of elastic-plastic behavior of pipe bends under in-plane bending. Hashem et al. [13-15] have performed analyses on pipe bends subjected to combined internal pressure with out-of-plane bending moment. Experiments on limit loads for pipe bends were carried out on elbows by Touboul et al. [16], Greenstreet [17], Tan et al. [18] and Chattopadhyay et al. [19, 20]. Researches above seek to study the gross plastic deformation behavior which is the fundamental failure mode for pipe structures. Orynyak and Radchenko [21] proposed an analytical method for the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. In recent years numerical method is applied in the design of engineering structures with respect to failure [22-24]. With the rapid development of the commercial software, finite element simulation to investigate the elastic-plastic behaviors for pipe bends is widely used by many researchers. This simulation method has being validated reliable and accurate compared to the past experiment data [25]. Some typical nonlinear general finite element program such as MSC.Marc [26], ANSYS [27], WARP3D [28] and ABAQUS [29] are widely adopted. On this basis, large quantities of estimated limit expressions for engineering assessment have been proposed directly in recent years, which have modified and improved some shortcomings in some design codes. Based on this procedure Hong et al.[30] and An et al.[31] proposed approximations to elastic stresses for elbows under internal pressure and in-plane bending respectively using ABAQUS. Kim and oh [32, 33] gave a detailed analysis and proposed estimated equations for limit and collapse loads of pipe bends under combined pressure and in-plane bending. They also studied the effects of attached straight pipes on finite element limit analysis for pipe bends and proposed relevant estimated solutions [34]. Hong et al.[35] quantifies the effect of internal pressure on plastic loads for elbows and proposed relevant equations. Lee et al.[36] proposed simple regression equations between the yield strength-to-elastic modulus ratio and plastic loads for elbows. Christo et al.[37-39] studied the combined effect of ovality and thinning on plastic loads of pipe bends under combined in-plane closing moment and internal pressure and found that the effect of ovality is significant while the thinning produces negligible effect, then proposed closed-form solutions to determine collapse load. Based on the research in Refs.[37-39] Buckshumiyan et al. [40,41] took in-plane opening bending into consideration and proposed estimated equations based on finite element analysis. For thick-walled elbows Kim et al. [42] conducted a detailed plastic loads analysis and gave some comparisons with thin-walled elbows, then proposed estimated equations. Large amounts of the proposed finite element based solutions from Refs.[30-42] were validated with experiment results, and results shows overall agreement regardless of some higher error. In our recently published work [43], a detailed analysis for pipe
bends under combined pressure and bending was performed by finite element analysis, which in turn can also provide some support to the relevant research work above. Studies from Refs.[1-43] focus on pressure and bending (mainly in-plane bending )load, and few references [44-46] mentioned limit load solutions for pipe bends under combined bending and torsion moment. In Ref.[44] Ayob et al. concerned the load interaction behavior of smooth piping elbows with attached long straight pipes by finite element method, then established the yield interaction behavior when an elbow is subjected to a combination loading of in-plane bending, torsion and internal pressure. Although studies in Ref.[44] is already specific for combined bending moment and torsion, their work is irrelevant with limit load solutions. In Ref.[45] Guo conducted a detailed stress analysis of pipe bends and the plastic limit load solutions of pipe bends under combined internal pressure, bending and torsion moment based on Mises yield criterion were obtained. However in Guo’s work[45] assumption of the stress and limit analysis for torsion is totally idealized as the same as straight pipes. In ASME B&PV Code [46], a general design equation of a nuclear pipe component is formed based on the stress indices which governs primary and secondary stress. 2.2. Some relevant limit load solutions 2.2.1. Pure in-plane bending Spence and Findlay [5] provided a lower bound limit moment for the pipe bend subject to in-plane bending
M I / M str
0.8λ 0.6 = 1
λ < 1.45 λ 1.45
(1)
2
Where, M str = 4r tσ s
(2)
denotes the plastic limit loads for straight pipes with the same dimension of pipe bends, σs is the material yield stress, MI is the plastic loads for a pipe bend under pure in-plane bending moment, λ is the bend characteristic of pipe bends, where,
λ = Rt / r 2 = ( R / r ) / ( r / t )
(3)
and R is bend radius of pipe bends, t is the thickness of pipe bends, r is the mean radius of cross section. Calladine [6] provided another limit load solution for pipe bends under in-plane bending.
M I / M str = 0.935λ 2/3
λ < 0.5
(4)
Goodall [8] provided a slightly high limit load solution
M I / M str = 1.04λ 2/3
λ < 0.5
(5)
Studies in Refs. [5, 6, 8] suggest that the limit bending moment depends only on the bending characteristic λ . However in some other research work result shows that a single variable λ can not reflect the limit loads tendency accurately [9, 32], and limit bending solutions are suggested dependence on both R/r and r/t. Kitching et al.[9] determined a lower bound for the limit bending moment of pipe bends using thin shell theory
M I / M str = 0.935λ 2/3 (1 − 0.36r / R ) λ < 0.5, 0 < r / R < 0.67
(6)
Recently based on 3D finite element limit analysis, Kim and Oh [32] proposed another limit load solution
M I / M str = A(λ + k )n A = 0.6453(r / t ) 0.0772 ; k = 1.5398(r / t ) −0.6755 ; n = 0.5157( r / t )
0.1 λ 0.5
(7)
0.0601
Noting that this proposed limit load solution in Eq.(7) is much high than predicted Eqs.(1), (4)-(6). This is because results by Eq.(7) considered the effect of a long attached straight pipe, and with a straight pipe attached to the pipe bend, the plastic yielding region spreads not only to the elbow but also to the attached pipe, which makes the limit bending moment higher than that for the elbow without an attachment [34]. In the limiting case of no attachment, the limit loads are found to be close to existing analytical solutions [34] by Eqs.(1), (4)-(6). 2.2.2. Pure torsion Guo [45] proposed limit load solutions based on Mises criterion
M O / M str = π / 2 3
(8)
Where, MO is the plastic loads for a pipe bend under pure torsion moment. 2.2.3. Combined in-plane bending and torsion moment Guo [45] proposed limit load solutions based on Mises criterion by stress analysis
( M LI / 3.32λ 2/3 r 2tσ s ) + ( M LO / M O ) = 1
(9)
Where, MLI is the plastic limit in-plane bending moment for a pipe bend under combined in-plane bending and torsion moment, MLO is the plastic limit torsion moment for a pipe bend under combined in-plane bending and torsion moment, and MO are from Eq. (8). ASME section III [46] code is meant to control primary plus secondary loads so as to place an upper bound on deformations [44]. Piping components such as straight pipes, bends, elbows, etc. are designed according to clauses NB/NC/ND-3600 for combined stresses due to pressure and moment, for different service levels [47]. The general design equation against plastic collapse in Class 1 piping component is given as
B1
Mr Pr + B2 i σ s t I
(10)
Where, B1 is stress index for pressure, B2 is stress index for bending,
B2 = 1.3 / λ 2/3 1
(11)
P is internal pressure, Mi is the combined force moment (combination of bending and torsion moment),
M i = M B2 + M T2
(12)
I is section modulus,
I = π r 3t
(13) MB is applied in-plane bending moment, and MT is applied torsion moment. For pipe bends without internal pressure considered, Eq. (10) can be rewrite as
1.3M i / π r 2t λ 2/3 σ s
(14)
3. Finite element analysis 3.1. Geometry The geometry of a pipe bend is shown in Fig.1. The bend angle is considered to be 90o, and the straight pipe attached to pipe bend is long enough (L>3r) to ignore the end effects on limit loads [34]. This length L is 1000 mm in this research, and the mean radius of cross section r is 100mm. The geometry parameters of pipe bends are listed in Table 1, where 9 models are chosen for in-plane bending, 16 models are chosen for torsion, and 4models are chosen for combined loading condition. 3.2. Material The material used is assumed to be elastic-perfectly plastic, and non-hardening J2 flow theory is used. The Young’s modulus E=200 GPa, Poisson’s ratio µ=0.3 and yield stress σ0=200 MPa. 3.3. Finite element model Geometrically linear finite element (FE) limit analyses were performed using ABAQUS. To reduce computing time, reduced integration elements, a 20 node quadratic brick reduced integration element (C3D20R) is used. The number of elements is 7200 and the number of nodes is 36400 for all of the models, shown in Fig.2. The static riks analysis is adopted to avoid the difficulty of convergence in FE analysis. The force moment was applied by rotation at the right top end-nodes of the attached straight pipe, constrained using the MPC (multi-point constraint) option. A rigid beam will be formed by connecting this single node to the nodes of this end. Sufficiently large rotation was applied, and the bending moment was determined from nodal forces [32]. Fig. 3 shows typical FE results of moment-rotation curves for pipe bends under pure torsion moment. Plastic limit moment can be determined apparently with maximum value in moment-rotation curve [32]. For combined loading condition, three loading sequences were considered in the investigation: proportional loading, MB-MT loading and MT-MB respectively. In proportional loading, the internal pressure and moment are applied to the model simultaneously in a proportional manner. In MB-MT loading, the in-plane bending moment is applied to in the first step then held constant, and the torsion moment is applied in the second step. In MT-MB loading, the torsion moment is applied to in the first step then held constant, and the in-plane bending moment is applied in the second step.
Fig.1. Schematic illustrations of a 90o pipe bend Table 1 Geometric parameters of analyzed pipe bends r=100mm L=1000mm Pure in-plane bending moment r/t=5 R/r=2, 4, 6 r/t=10 R/r=2, 4, 6 r/t=20 R/r=2, 4, 6 r/t=50 R/r=2, 4, 6 Pure torsion moment r/t=5 R/r=2,3,4,5, 6 r/t=10 R/r=2,3,4,5, 6 r/t=20 R/r=2,3,4,5, 6 r/t=50 R/r=2,3,4,5, 6 Combined in-plane plane bending and torsion moment r/t=5 R/r=2, 6 r/t=20 R/r=2, 6
λ=0.4, 0.8, 1.2 λ=0.2, 0.4, 0.6 λ=0.1, 0.2, 0.3 λ=0.04, 0.08, 0.12 λ=0.4,0.6,0.8,1,1.2 λ=0.2,0.3,0.4,0.5,0.6 λ=0.1,0.15,0.2,0.25,0.3 λ=0.04,0.06,0.08,0.1,0.12 λ=0.4, 1.2 λ=0.1, 0.3
Fig.2. Finite element mesh
1.0
MT/Mstr
r/t=5, R/r=6
0.8
r/t=5, R/r=2
0.6
r/t=20, R/r=6 r/t=20, R/r=2
0.4 0.2 0.0 0.00
0.04
0.08
0.12
0.16
0.20
θ/rad
Fig.3. Typical moment-rotation curves 4. Results and discussion 4.1. Plastic limit in-plane bending moment As is reported in section 2.1, most of the past studies emphasize on in-plane bending. In this section analysis is processed briefly mainly to validate Kim and Oh’s [32] results based on FE
（
method. Fig. 4 shows the ovalization deformation in cross section A-A seen in Fig.1
） for
in-plane bending. Because this section deformation is the most significant compared to other section spread along the axially path, this deformation in section A-A can deflect the deformation capacity and level, which will have a great importance to the plastic loads evaluation. In Fig.4 the deformed shape and undeformed circular shape are put together so as to see this ovalization shape clearly. This deformation is oriented at vertical and horizon with respect to the plane of pipe bend. Fig.5 shows the FE results and proposed results by Kim and Oh [32] in Eq.(7). In this figure, Eq.(7) can predicted FE results well for r/t≤20. However, for the thin pipe bends with r/t=50, this Eq.(7) underestimate FE results. This is only because FE based predicted Eq.(7) is intended for
0.1 λ 0.5 in Ref.[32], while FE results is for 0.04 λ 1.2 in this paper. In this aspect results by Eq.(7) can extrapolate to a high value of λ, and fail to a low value of λ. So a new predicted equation is proposed in Eq.(15), and results show better agreement.
(a) (b) Fig.4. Deformed and undeformed shapes of cross section A-A for in-plane bending: (a) for in-plane closing bending (b) for in-plane opening bending
M I / M str = 1 / A1e B1 + 1 / C1e D1 R / r
A1 = 1.1091 + 0.1657r / t ;
B1 = −0.0234 + 0.0155r / t;
C1 = 0.7886 + 0.1047r / t ; D1 = −2.3805 + 0.0129r / t
1 0.9 0.8 0.7 0.6 0.5
Hollow date: closing bending Solid data: openning bending
1 0.9 0.8 0.7 0.6 0.5 MI/Mstr
MI/Mstr
(15)
Hollow date: closing bending Solid data: openning bending
0.4
0.4 0.3 Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Eq.(7) by Kim
0.1 1.5
0.04 λ 1.2
2
2.5
3 3.5 R/r
4 4.5 5 5.5 6 6.5
0.3 Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Proposed Eq.(15)
0.1 1.5
2
2.5
3
3.5
4 4.5 5 5.5 6 6.5
R/r
(a) (b) Fig.5. Comparison of finite element solutions for pipe bends under in-plane bending moment with expected equations: (a) with Eq.(7) (b) with Eq.(15) 4.2. Plastic limit torsion moment 4.2.1. A detailed analysis for stress distribution and a common sense of torsion moment Up till now research on torsion moment capacity for pipe bends is so little in published references that no clear analysis for the basic awareness for torsion effect of pipe bends exists. Research in Ref.[45] by Guo for pipe bends with plastic limit torsion moment is idealized to straight pipes. In Guo’s[45] assumption, only torsion is existing through all the cross sections along the axial direction, and no bending region exists. In this respect only shear stress exists without other stress. So based on this assumption plastic limit torsion moment for pipe bends is the same as straight pipes. However, this assumption is not realistic as bending effect is always exists when torsion moment is applied, and this is a combined bending and torsion effects for pipe bends. Fig.6 shows the ovalization deformation in cross section A-A for torsion moment, where Fig.6 (a) is deformation shape in pipe bends, Fig.6 (b) is in straight pipes. It can be seen from Fig.6 that the shape in pipe bends is oriented at 45o angle with respect to the plane of pipe bends due to combined torsion and bending effect, while the straight pipe is just an outward expanded circular section due to pure torsion effect lonely. It can be seen obviously that this deformed shape in pipe bends is not caused only by shear stress.
(a) (b) Fig.6. Deformed and undeformed shapes of cross section A-A for torsion moment (a) in pipe bend (b) in straight pipe To have a common sense of the pipe bends behavior under torsion moment, a detailed stress analysis is conducted for a pipe bend with pipe geometry of r/t=20, R/r=2, MT=4.8KNm. Fig.7 shows the points and paths in stress analysis. Path1 to path3 are circumferential path in pipe cross section, while path4 to path6 are axial path along the direction of pipeline. Stress distribution is shown in Fig.8, 9, where Fig.8 corresponds to stress for path1 to path3 and Fig.9 for path4 to path6. According to the description in two figures, it can be concluded as follows. For path1in the left bottom end, shear stress τ is so small that can be ignored, circumferential stress σθ and axial stress σZ are obvious compared with τ, so in path1 bending is main loading effect. For path2 in right top end, uniform shear stress exists along this path, while circumferential stress and axial stress are almost 0, so in this path2 torsion is the main loading effect. For path3 in cross section A-A, all these stresses are evident showing that this cross section is combined bending and torsion moment, and local stress concentration will cause yielding resulting in pipe failure firstly. In order to find out the loading effect in axial direction along pipeline, stress in path4 to path6 is analyzed. In path5 all the stresses exist showing bending and torsion moment interaction effects. In path4 from point2 to point1, all the stresses exist are resulting from combined loading effects within 3r length of straight pipe, and beyond this region 3r shear stress is near zero which shows bending dominated effect in this region. In path6 from point3 to point4, it is just the opposite showing in Fig.9.
Path 3
Point 3
Path 2 Path 6
Point 2
Point 4
Path 5
Path 4 Path 3 Path 1 Path 1 Point 1 Fig.7. Path for stress analysis
Path 2
24
Path 1 in left bottom end Path 2 in right top end Path 3 in cross section A
Path 1 in left bottom end Path 2 in right top end Path 3 in cross section A
160 120
16
80
12
40 σθ/MPa
τ /MPa
20
8
0
-40
4 -80
0
-120
-4 0 30 60 90 120 150 180 210 240 270 300 330 360 crown extrados crown crown intrados o ϕ/ ( )
-160 0 30 60 90 120 150 180 210 240 270 300 330 360 crown crown crown extrados intrados o ϕ/ ( )
(a)
(b) 150
Path 1 in left bottom end Path 2 in right top end Path 3 in cross section A
100
σZ/MPa
50 0 -50 -100 -150 0 30 60 90 120 150 180 210 240 270 300 330 360 crown crown crown extrados intrados o ϕ/ ( )
(c) Fig.8. Stress distribution in circumferential path (path1 to path3): (a) shear stress (b) circumferential stress (c) axial stress
40
Results in Path 5
120
Axial stress σz
Combined bending and torsion
60
60
Bending dominated
20 0 -20 -40
Shear stress τ Circumferential stress σθ
Combined bending and torsion
90
Stress/MPa
Stress/MPa
Shear stress τ Circumferential stress σθ
Results in Path 4
80
Axial stress σz
30 0 -30 -60
-60
-90
-80
-120
0 Point 2
200
400
600
Path distence/mm
(a)
800
1000 Point 1
0
Point 2
15
30
45
60
Path distence/(o)
(b)
75
90
Point 3
Results in Path 6
60
Combined bending and torsion
Stress/MPa
40
Shear stress τ Circumferential stress σθ Axial stress σz Torsion dominated
20 0 -20 -40 -60 0
Point 4
200
400 600 Path distence/mm
800
1000 Point 3
(c) Fig.9. Stress distribution in axial path (path4 to path6): (a) stress distribution in path4 (b) stress distribution in path5 (c) stress distribution in path6 So it can be concluded that for pipe bends under pure torsion moment, a common awareness for loading effect should be as follows: In the geometrical bending section combined bending moment and torsion effects exist at the same time, and these combined effects also spread along attached straight pipes for as long as 3r in axial direction of straight pipes, which are showing in Fig.10.
Torsion dominated region
Combined bending and torsion region
Bending dominated region Fig.10. Force patterns for pipe bends under torsion moment with attached long straight pipes 4.2.2. Plastic limit torsion moment Fig.11 shows normalized FE results for torsion moment, where (a) shows the comparison with Eq.(8) by Guo [45], (b) shows the comparison with predicted Eq.(16) using single variable λ, (c) shows the comparison with predicted Eq.(17) by two variables R/r and r/t. Eq.(16) and Eq.(17) are predicted FE results for torsion moment in present work as follows:
1 0.9 0.8 0.7 0.6 0.5
1 0.9 0.8 0.7 MO/Mstr
MO/Mstr
0.4 0.3 Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Eq.(8) by Guo
0.1 1.8
2.4
3
3.6
4.2
0.6 0.5 0.4
Symbol: FE data Line: Predicted Eq.(16)
0.3
4.8 5.4 6 6.6
0.16
0.32
R/r
0.48 0.64 0.8
λ
(a)
(b) 1 0.9 0.8 0.7 0.6 0.5 MO/Mstr
0.4 0.3
Symbol: FE data r/t=5 r/t=10 r/t=20 r/t=50
0.2 Line: Predicted Eq.(17)
0.1 1.8
2.4
3
3.6
4.2
4.8 5.4 6 6.6
R/r
(c) Fig.11. Comparisons of FE results for plastic limit torsion moment with some proposed equations: (a) with Eq.(8) (b) with Eq.(16) (c) with Eq.(17)
M O / M str = 0.8678λ 0.2862
(16)
M O / M str = 1.0682(r / t ) −0.3114 ( R / r )0.1787
(17)
It can be concluded in Fig.11 that Eq.(8) by Guo[45] overestimate our FE data in most cases. Differences between FE and Eq.(8) increase with increasing r/t and decreasing R/r. This implies that Eq.(8) by Guo [45] is not fit for predicting FE data especially for thin-walled and small bend radius pipe bends. It can also be concluded that Eq.(17) with two variables R/r and r/t can predicts FE results better than Eq. (16) with a single variable λ. This is in accordance with that for in-plane bending as is mentioned in section 2.2.1. It can be seen in Fig. 11 and Eq.(17) that r/t is the main factor affecting the limit load solutions. These limit results predicted by Eq.(17) showing in Fig.11 can be interpreted in Fig.12. In this figure, FE data on Mises stress in cross section A-A (path 3) are normalized with respect to Mises stress for straight pipes by Eq.(18), where the applied torsion is MT=4.8KNm, ro is the outer radius in cross section. Results show that the normalized peak stress is higher for thin-walled pipe bends and pipe bends with small bend radius. This high peak stress for pipe bends accelerates the yielding process compared with straight pipes and then approach plastic limits prior to straight pipes as load increasing. This means the limit load of FE result is lower compared with Eq.(8) by Guo[45] especially for thin-walled and small bend radius pipe bends.
3M T ro I
(18)
σeq(FE)/σeq(Eq.(18))
σ eq =
6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
r/t=10,R/r=2 r/t=10,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0 30 60 90 120 150 180 210 240 270 300 330 360 crown extrados crown intrados crown o ϕ/ ( )
Fig.12. Comparisons of Mises stress for pipe bends with Mises stress of straight pipes 4.3. Plastic limit loads for combined bending and torsion moment Fig.13 shows the ovalization deformation in cross section A-A for combined loading condition. In Fig.13 the deformed shape and direction are between bending and torsion moment seen from Fig.4 and Fig.6. This appearance can reflect the deformation characteristic for pipe bends under combined loads.
(a) (b) Fig.13. Deformed and undeformed shapes of cross section A-A for combined in-plane bending and torsion moment: (a) for combined in-plane closing bending and torsion moment (b) for combined in-plane opening bending and torsion moment 4.3.1. FE modeling parameters independency on limit load In limit load theory, the plastic limit load is independent on some model parameters, mainly independent on the loading sequence (or loading path [2]) and material constants E. In addition, Lee et al. [36] proposed σs/E independent normalized plastic limit load solution based on FE analysis. These are verified by calculating the plastic limit loads of four pipe geometry parameters ranging from 0.1 λ 0.6 for three load sequences, MB-MT loading, MT-MB loading and proportional loading respectively, which has already introduced in section 3.3. FE results are show in Figs. 13-15. In these figures bending-torsion interaction surfaces were constructed by
performing elastic-plastic analysis of each pipe bend for different load combinations, ranging from bending only to torsion only. Results in Figs. 13-15 suggest that the circular interaction rule Eq.(19) is a great approximation for pipe bends under combined bending moment and torsion. (19)
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.2 0.0 -0.2 -0.2
MLO/MO
MLO/MO
( M LI / M I ) + ( M LO / M O ) = 1
0.0
0.2
0.4
0.4
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.2 0.0 0.6
0.8
1.0
-0.2 -0.2
1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
MLIC/MIC
MLIO/MIO
1.2
1.2
1.0
1.0
0.8
0.8
0.6
MLIO/MIO
MLIC/MIC
(a) (b) Fig.14. Bending-torsion interaction curve for MB-MT loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.4 0.2
0.4
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.2
0.0 -0.2 -0.2
0.6
0.0 0.0
0.2
0.4
0.6
0.8
1.0
-0.2 -0.2
1.2
0.0
MLO/MO
0.2
0.4
0.6
0.8
1.0
1.2
MLO/MO
1.0
1.0
0.8
0.8
0.6
0.6 r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.4 0.2 0.0 0.0
MLO/MO
MLO/MO
(a) (b) Fig.15. Bending-torsion interaction curve for MT-MB loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending
0.2
0.4 0.6 MLIC/MIC
r/t=5,R/r=2 r/t=5,R/r=6 r/t=20,R/r=2 r/t=20,R/r=6
0.4 0.2
0.8
1.0
0.0 0.0
0.2
0.4 0.6 MLIO/MIO
0.8
1.0
(b) (a) Fig.16. Bending-torsion interaction curve for proportional loading: (a) combined torsion and
in-plane closing bending (b) combined torsion and in-plane opening bending Take MB-MT loading for instance, these results by Eq.(19) can be interpreted according to yielding surface and ovalization deformation showing in Figs. 17, 18. In Fig.17 the yielding region spread from pipe crown to pipe intrados, then spread to straight section till extrados, when all the bend section is overall yielding, this continues spreads along straight pipe section, finally almost all the pipe section yields. This yielding region extends with the increasing MB where the applied torsion moment is 14.4KNm. Fig.17 clearly shows that for MB≤0.4MIO, this region is small and spread slowly with MB increasing, while spread faster and faster for 0.4 MIO≤MB≤MLIO. This phenomenon implies that the circular interaction curve in Figs.14-16 is reasonable. As ovalization deformation can reflect loading capacity for pipeline, the percent ovalization C in cross section A-A for a pipe bend under combined loads is shown to explain FE data. Showing in Fig.18, C increases with the loads increasing, which shows the interaction rule in Figs.14-16 on the other hand. In Fig.18 the percent ovalization C is written as
Dmax − Dmin D D + Dmin D = max 2 C=
Where,
(20) (21)
Dmax is the maximum outside pipe diameter, and Dmin is the minimum outside pipe diameter.
(a)
(c)
(b)
(d)
(e) (f) Fig.17. Mises criterion region for a pipe bend under combined in-plane opening bending and torsion moment, where the applied torsion moment are 14.4KNm in unified: (a) MB=0 (b) MB =0.2MIO (c) MB =0.4MIO (d) MB =0.6MIO (e) MB =0.8MIO (f) MB =MLIO 0.12 0.10 Normalized M/MIO Ovality C
0.08
0 0.4 0.8
0.06 0.04 0.02 0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30
MT/MLO
Fig.18. Ovalization deformation in cross section A-A for a pipe bend under combined in-plane opening bending and torsion moment Material constants independent normalized limit load solution for combined loads is shown in Fig.19, where results show that the choice of σs and E is material independent. This also verifies the limit load theory meanwhile. 1.0
1.0
0.8
0.8 0.6 MLO/MO
MLO/MO
0.6 r/t=5, R/r=2
0.4 0.2
r/t=5, R/r=6 0.4 E=100GPa, σs=100MPa
E=100GPa, σs=100MPa E=400GPa, σs=100MPa
0.2
E=100GPa, σs=400MPa
E=100GPa, σs=400MPa 0.0 0.0
E=400GPa, σs=400MPa 0.2
0.4
0.6
MLIO/MIO
(a)
E=400GPa, σs=100MPa
0.8
1.0
0.0 0.0
E=400GPa, σs=400MPa 0.2
0.4
0.6
MLIO/MIO
(b)
0.8
1.0
1.0
1.0
0.8
0.8
0.6 MLO/MO
MLO/MO
0.6 r/t=20, R/r=2 0.4
r/t=20, R/r=6 0.4 E=100GPa, σs=100MPa
E=100GPa, σs=100MPa 0.2 0.0 0.0
E=400GPa, σs=100MPa E=100GPa, σs=400MPa
0.2
E=400GPa, σs=400MPa 0.2
0.4
0.6
0.8
1.0
0.0 0.0
MLIO/MIO
E=400GPa, σs=100MPa E=100GPa, σs=400MPa E=400GPa, σs=400MPa 0.2
0.4
0.6
0.8
1.0
MLIO/MIO
(c) (d) Fig.19. Effect of the materials constants on plastic limit loads for combined loads: (a) r/t=5, R/r=2 (b) r/t=5, R/r=6 (b) r/t=20, R/r=2 (b) r/t=20, R/r=6 4.3.2. Comparison with existing results from literature Fig.20 shows comparison of proposed result in Eq.(19) with Eq.(9) by Guo [45] and Eq.(14) in ASME [46]. This shows some differences obviously in this figure, and comparisons show that both estimate equations by Guo [45] and ASME [46] are inconsistent with FE data. In Ref.[45] by Guo, plastic limit torsion moment is considered to be the same as straight pipes as shown in section 4.2.1. This proposed solution for torsion is higher than FE data especially for thin-walled and small bend radius pipe bends. For in-plane bending Guo [45] only considered circumferential stress for stress analysis and overlooked the stiffening effect of straight pipes, which will cause a conservative estimation for thin-walled and small bend radius pipe bends. So with a circular interaction, some flat oval-shaped cures can depicts this rule by Guo [45] in Eq.(9). In ASME [46], the load moment is a simply bending moment, regardless of the bending mode and direction. In other hand, according to ASME, only circumferential stress is considered for bending in pipe crown, and when yielding appears in crown first, this state is regarded as the limiting state, which overlooks the spread procedure of yielding surface from crown to all of the pipe bends. In addition the load moment from ASME includes safety factor for design. So when considering these influencing factors, it is easy to understand why results in Fig. 20 (b) are much more conservative to FE data.
2.2 2.0
1.4 Proposed result Eq.(9) by GUO
Proposed result Eq.(14) in ASME
1.2
pipe parameters a: r/t=5, R/r=2 b: r/t=5, R/r=6 c: r/t=20, R/r=2 d: r/t=20, R/r=6
1.8 a
1.2 1.0 0.8 0.6
b
MLO/MO
MLO/MO
1.4
1.0
pipe parameters a: r/t=5, R/r=2 b: r/t=5, R/r=6 c: r/t=20, R/r=2 d: r/t=20, R/r=6
1.6
c
c a
0.8
d
0.6
b
0.4 d
0.4
0.2
0.2
0.0 0.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
MLI/MI
MLI/MI
(a) (b) Fig.20. Comparison of proposed result in Eq.(19) with some existing limit load solutions for pipe bends under combined in-plane bending and torsion moment: (a) with Eq.(9) by Guo (b) with Eq.(14) in ASME 5. Verification of finite element procedure In order to validate finite element method on plastic limit analysis, a series of verification approaches are confirmed. Detailed analysis is as follows. 5.1. Precision validation of finite element method In this section, a total of 36 FE models are calculated to validate our mesh adequacy. 4 pipe geometry parameters are chosen ranging for 0.1 λ 0.6 , and for each pipe bend the mesh node number are from 132 to 132335. Fig. 21 shows FE results showing that FE data decrease with increasing node number while results tend to converge with node increasing. It can be seen from Fig.21 clearly that this node number of 36440 in our present work can satisfy mesh adequacy. 1.0 0.9 0.8
r/t=10, R/r=6
MO/Mstr
0.7 r/t=10, R/r=2
0.6
r/t=20, R/r=6
0.5
r/t=20, R/r=2
0.4 0.3 0.2
FE results in this work ( Node number, 36440)
0.1 0.0 0
2
4
6 8 10 Node number/10000
12
14
Fig.21. Effect of mesh on plastic limit loads 5.2. Finite element yield criterion validation
Based on unified yield criterion (UYC) [48] the equivalent stress σUE is defined as
σ 1 +σ 3 1 σ UE = σ 1 − 1 + b (bσ 2 + σ 3 ), σ 2 2 σ = 1 (σ + bσ ) − σ , σ σ 1 +σ 3 UE 1 2 3 2 1+ b 2
(22)
The UYC can be simply be written as σ UE = σ s when yielding and UYC plastic limit load solution for straight pipes under internal pressure can be derived as
Pstr =
2(1 + b) t σs, 0 b 1 2+b r
(23)
With different choices of parameter b, the UYC can be simplified to the MY [49] yield criterion (b=1/3), the GM [50] yield criterion (b=0.4), the ASSY [51] yield criterion (b=0.618) etc. Fig.22 shows the comparison of FE results and analytical solutions by Eq.(23) on UYC. According to this comparison, Mises criterion based solutions lie in the middle, and shows great agreement with FE results. This implies that non-hardening J2 flow theory is used by FE procedure which gives a modest prediction of plastic limit results among these unified solutions. So this analysis validates reliability of FE yield criterion. 22 20 18
Pstr/MPa
16 14
Tresca (b=0) ASSY (b=0.168) MY (b=1/3) Mises (b=0.366) GM (b=0.4) EA (b=0.529) TSS (b=1)
12 10
FE Results
8 6 0.03
0.04
0.05
0.06
0.07
0.08
r/t
Fig.22. Comparison of FE data with analytical solutions based on different yield criterion for pressurized straight pipes 5.3. Finite element results validation with experiment data available in literature To validate our proposed expressions, experiment results would be a best validation. Some available experiment data can be obtained from Refs.[25]. Experiment data and their comparisons with FE results are listed in Table 2, where M exp indicates experiment result and M FE is FE result. In this table it can be seen clearly that FE based results can predict experiment data well, as these differences are within ±5%. As no experimental data are available for combined loading conditions, therefore reasonable explanations are tried in Section 4.3.1 in support of the solutions proposed in the present work. Noting that both pure bending, torsion and combined solutions are based on FE results, it is implicitly suggested that the accuracy of the proposed solutions for combined loading condition is credible. It can be concluded that FE based predicted equations are consistent with the available experiment data which provide confidence to present work.
Table 2 Comparison of the estimated FE results by Eqs. (15) and (17) with existing experiment results for pipe bends under in-plane bending and torsion moment Load type Material R/r r/t σS / Error/% M FE / M FE M exp MPa /KNm /KNm
M exp
In-plane closing bending
In-plane opening bending Torsion
ASTM A-106B ASTM A-106B SS304 SS304 ASTM A-106B
1.94 2.91 1.56 2.67 2.84
7.34 7.34 12.5 8.58 11.33
239 261 272 265 345
41.69 48.36 4.38 1.65 32.43
40.82 49.22 4.52 1.71 33.74
0.98 1.01 1.03 1.04 1.04
2.08 -1.77 -3.19 -3.63 -4.03
ASTM A-106B SS 304
2.84 2.67
11.33 8.44
345 265
36.61 1.82
37.01 1.81
1.01 0.99
-1.11 0.78
6. Conclusions In present work, 3D finite element method is used to determine plastic limit load solutions for pipe bends under combined bending and torsion moment. Detailed results are investigated as follows: (1) For pipe bends under pure in-plane bending moment: FE based results proposed by Kim and Oh [32] can predict FE results well for r/t≤20. While underestimate for r/t=50, a new predicted equation is proposed and results show great agreement, which extends the range of pipe geometry λ. (2) For pipe bends under pure torsion moment: A common awareness for loading effect should be as follows: In the geometrical bending section combined bending moment and torsion effects exist at the same time, and these combined effects also spread along attached straight pipes for as long as 3r in axial direction of straight pipes. Results by Guo [45] overestimate FE data. Differences increase with increasing r/t and decreasing R/r which implies that results by Guo [45] is not appropriate to predict FE data especially for thin-walled and small bend radius pipe bends. Research by Guo [45] for pipe bends with plastic limit torsion moment is idealized to straight pipes. Results show that the normalized peak stress is higher for thin-walled pipe bends and pipe bends with small bend radius. This high peak stress for pipe bends accelerates the yielding process compared with straight pipes and then approach plastic limits prior to straight pipes as load increasing. It can also be concluded that equation with two variables R/r and r/t can predicts FE results better than that with a single variable λ. This is in accordance with that for in-plane bending. r/t is the main factor affecting the limit load solutions. (3) For pipe bends under combined in-plane bending and torsion moment: Normalized plastic limit load is independent on the loading sequence and material constants. Results show that the circular interaction rule is a great approximation for pipe bends under combined bending and torsion moment. Relevant results by Guo [45] and ASME [46] are not appropriate to predict FE results. In this respect of finite element analysis, overall yielding considering the spread process of yield
region from crown to the straight pipe shows these promising finite element results. (4) A series of verification approaches are confirmed in order to validate our finite element method on plastic limit analysis. Based on the FE data, estimated equations of limit loads for pipe bends under combined bending and torsion moment are proposed, which are justified to a good choice for limit load assessment. Acknowledgments The authors gratefully acknowledge the financial support by Graduate Student Scientific Innovative Project of Jiangsu Province (No. SJZZ_0096), China and Jiangsu Natural Science Funds (BK2008373). References [1] Chattopadhyay J, Tomar AKS. New plastic collapse moment equations of defect-free and throughwall circumferentially cracked elbows subjected to combined internal pressure and in-plane bending moment. Eng Fract Mech 2006; 73: 829–854. [2] Robertson A, Li H, Mackenzie D. Plastic collapse of pipe bends under combined internal pressure and in-plane bending. Int J Pressure Vessels Piping 2005; 82: 407-416. [3] Toshiyuki M, Yoshiaki. Proposal of failure criterion applicable to finite element analysis results for wall-thinned pipes under bending load. Nucl Eng Des 2012; 242: 34-42. [4] Marcal PV. Elastic–plastic behaviour of pipe bends with in-plane bending. J Strain Anal 1967; 2(1): 84-90. [5] Spence J, Findlay GE. Limit load for pipe bends under in-plane bending. Proceedings of the second international conference on pressure vessel technology, San Antonio; 1973. 393–399. [6] Calladine CR. Limit analysis of curved tubes. J Mech Eng Sci 1974; 16(2):85–87. [7] Goodall IW. Large deformations in plastically deformed curved tubes subjected to in-plane bending. Research Division Report RD/B/N4312. UK: Central Electricity Generating Board; 1978. [8] Goodall IW. Lower bound limit analysis of curved tubes loaded by combined internal pressure and in-plane bending moment. Research Division Report RD/B/N4360. UK: Central Electricity Generating Board; 1978. [9] Kitching R, Zarrabi K, Moore MA. Limit moment for a smooth pipe bend under in-plane bending. Int J Mech Sci 1979; 21(12): 731-738. [10] Shalaby MA, Younan M Y A. Nonlinear Analysis and Plastic Deformation of Pipe Elbows Subjected to In-Plane Bending. Int J Pressure Vessels Piping 1998; 75: 603-611. [11] Shalaby M A, Younan M Y A. Limit Loads for Pipe Elbows Subjected to In- Plane Opening Bending Moments. ASME J Pressure Vessel Technol 1999; 121: 17-23. [12] Shalaby M A, Younan MYA. Effect of Internal Pressure on Elastic-Plastic Behavior of Pipe Elbows Under In-Plane Bending Moments. ASME J Pressure Vessel Technol 1999, 121: 400-405. [13] Hashem MM, Younan MYA. Nonlinear Analysis of Pipe Bends Subjected to Out-of-Plane Moment Loading and Internal Pressure. J Press Vessel Technol 2001; 123: 253 - 258. [14] Hashem MM, Younan MYA. Limit-Load Analysis of Pipe Bends Under Out-of-Plane Moment Loading and Internal Pressure. J Press Vessel Technol 2002; 124: 32 - 37. [15] Hashem MM, Younan MYA. The Effect of Modeling Parameters on the Predicted Limit Loads for Pipe Bends Subjected to Out-of-Plane Moment Loading and Internal Pressure. J Press Vessel Technol 2000; 122: 450456. [16] Touboul F, Ben DM, Acker D. Design criteria for piping components against plastic collapse: application to pipe bend experiments. In: Liu C, Nichols RW, editors. Pressure vessel technology, proceedings of 6th international conference held in Beijing, 11-15th September 1988. p. 73-84. [17] Greenstreet WL. Experimental study of plastic response of pipe elbows. Report, ORNL/ NURE G-24, Oak Ridge National Laboratory, TN; 1978. [18] Tan Y, Wilkins K, Matzen V. Correlation of test and FEA results for elbows subjected to out-of-plane loading. Nucl Eng Des 2002; 217: 213-224. [19] Chattopadhyay J, Kushwaha HS, Roos E. Some Recent Developments on Integrity Assessment of Pipes and
Elbows. Part I: Theoretical investigations. Int J Solids Struct 2006; 43(10): 2904-2931. [20] Chattopadhyay J, Kushwaha HS, Roos E. Some Recent Developments on Integrity Assessment of Pipes and Elbows. Part II: Experimental Investigations. Int J Solids Struct 2006; 43(10): 2932-2958. [21] Orynyak IV, Radchenko SA. Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect. Int J Solids Struct 2007; 44: 1488-1510. [22] Liu YH, Zhang XF, Cen ZZ. Lower bound shakedown analysis by the symmetric Galerkin boundary element method. International Journal of Plasticity. 2005, 21(1):21-42. [23] Chen SS, Liu YH, Cen ZZ. Lower-bound limit analysis by using the EFG method and non-linear programming. International Journal for Numerical Methods in Engineering. 2008, 74:391-415. [24] Chen SS, Liu YH, Cen ZZ. Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming. Computer Methods in Applied Mechanics and Engineering. 2008, 197(45-48):3911-3921. [25] Han JJ, Lee KH, Kim NH, Kim YJ, Jerng DW, Budden P J. Comparison of existing plastic collapse load solutions with experimental data for 90o elbows. Int J Pressure Vessels Piping 2012; 89: 19-27. [26] MSC. Software Corp. MSC.Marc Volume D: User subroutines and special routines; 2005. [27] ANSYS, Version 8.1. General Purpose Finite Element Program; 2004. [28] Gullerud AS, Kopenhoefer K, Ruggieri C, Roy A, Roychoudhuri S, Dodds Jr RH. WARP3D-RELEASE 13.15, Dynamic nonlinear analysis of solids using a preconditioned conjugate gradient software architecture. User’s manual, University of Illinois, Urbana, IL, USA; 2001. [29] ABAQUS. User’s manual version 6.9; 2009. [30] Hong SP, An JH, Kim YJ, Nikbin K, Budden PJ. Approximate elastic stress estimates for elbows under internal pressure. International Journal of Mechanical Sciences 2011; 53: 526-535. [31] An JH, Hong SP, Kim YJ , Budden PJ. Elastic stresses for 90o elbows under in-plane bending. International Journal of Mechanical Sciences 2011; 53: 762-776. [32] Kim YJ, Oh CS. Limit loads for pipe bends under combined pressure and in-plane bending based on finite element limit analysis. Int J Press Vessels Piping 2006; 83: 148 - 153. [33] Kim YJ, Oh CS. Closed-form plastic collapse loads of pipe bends under combined pressure and in-plane bending. Eng Fract Mech, 2006; 73: 1437-1454. [34] Kim YJ, Oh CS. Effects of attached straight pipes on finite element limit analysis for pipe bends. Int J Press Vessels Piping 2007; 84: 177 - 184. [35] Hong SP, Kim JH, YJ Kim, Budden PJ. Effect of internal pressure on plastic loads of 90o elbows with circumferential part-through surface cracks under in-plane bending. Engng Fract Mech 2010; 77: 577-596. [36] Kuk-Hee Lee, Chang-Sik Oh, Yun-Jae Kim, Kee-Bong Yoon. Quantification of the yield strength-to-elastic modulus ratio effect on TES plastic loads from finite element limit analyses of elbows. Engng Fract Mech 2009; 76: 856-875. [37] Christo MT, Veerappan AR, Shanmugam S. Effect of ovality and variable wall thickness on collapse loads bends subjected to in-plane bending closing moment. Eng Fract Mech 2012;79: 138-148. [38] Christo MT, Veerappan AR, Shanmugam S. Comparison of plastic limit and collapse loads in pipe bends with shape imperfections under in-plane bending and an internal pressure. J Press Vess Pip 2012; 99-100:23-33. [39] Christo MT, Veerappan AR, Shanmugam S. Effect of internal pressure and shape imperfections on plastic loads of pipe bends under in-plane closing moment. Eng Fract Mech 2013; 105: 1-15. [40] Buckshumiyan A, Veerappan AR, Shanmugam S. Plastic collapse loads in shape-imperfect pipe bends under in-plane opening bending moment. Int J Press Vess Pip 2013; 111-112: 21-26. [41] Buckshumiyan A, Veerappan AR, Shanmugam S. Determination of collapse loads in pipe bends with ovality and variable wall thickness under internal pressure and in-plane opening moment. Int J Press Vess Pip 2014; xxx: 1-9. [42] Kim YJ, Je JH, Oh CS, Han JJ, Budden PJ. Plastic loads for 90° thick-walled elbows under combined pressure and bending. J Strain Anal Eng 2010; 45: 115-127. [43] Li J, Zhou CY, Xue JL, He XH. Limit loads for pipe bends under combined pressure and out-of-plane bending moment based on finite element analysis. Int J Mech Sci 2014; 88: 100-109. [44] Ayob AB, Moffat DG, Mistry J. The interaction of pressure, in-plane moment and torque loadings on piping elbows. Int J Pressure Vessels Piping 2003; 80: 861-869. [45] Guo C. Plastic limit loads for surface defect pipes and bends under combined loads of tension, bending, torsion and internal pressure. PhD thesis, East China University of Science and Technology, Shanghai, China.
1999.(in Chinese) [46] ASME. Rules for construction of nuclear facility components. New York, USA: ASME Boiler and Pressure Vessel Code Committee, Section III, NB, NC and ND, American Society of Mechanical Engineers; 2004. [47] Ghosh S, Roy P. Quantification of the uncertainty in stress index B2 for pipe bends subjected to out-of-plane bending. Int J Pressure Vessels Piping 2012; 95: 24-30. [48] Wang LZ, Zhang YQ. Plastic collapse analysis of thin-walled pipes based on unified yield criterion. Int J Mech Sci 2011; 53: 348-354. [49] Zhang SH, Song BN, Wang XN, Zhao DW. Analysis of plate rolling by MY criterion and global weighted velocity field. Appl Math Model 2014; 38: 3485-3494. [50] Zhang SH, Wang XN, Song BN, Zhao DW. Limit analysis based on GM criterion for defect-free pipe elbow under internal pressure. Int J Mech Sci 2014; 78: 91-96. [51] Zhu XK, Leis BN. Average shear stress yield criterion and its application to plastic collapse analysis of pipelines. Int J Pressure Vessels Piping 2006; 83: 663-671.
Fig.1. Schematic illustrations of a 90o pipe bend Fig.2. Finite element mesh Fig.3. Typical moment-rotation curves Fig.4. Deformed and undeformed shapes of cross section A-A for in-plane bending: (a) for in-plane closing bending (b) for in-plane opening bending Fig.5. Comparison of finite element solutions for pipe bends under in-plane bending moment with expected equations: (a) with Eq.(7) (b) with Eq.(15) Fig.6. Deformed and undeformed shapes of cross section A-A for torsion moment (a) in pipe bends (b) in straight pipes Fig.7. Path for stress analysis Fig.8. Stress distribution in circumferential path (path1 to path3): (a) shear stress (b) circumferential stress (c) axial stress Fig.9. Stress distribution in axial path (path4 to path6): (a) stress distribution in path4 (b) stress distribution in path5 (c) stress distribution in path6 Fig.10. Force patterns for pipe bends under torsion moment with attached long straight pipes Fig.11. Comparisons of FE results for plastic limit torsion moment with some proposed equations: (a) with Eq.(8) (b) with Eq.(16) (c) with Eq.(17) Fig.12. Comparisons of Mises stress for pipe bends with Mises stress of straight pipes Fig.13. Deformed and undeformed shapes of cross section A-A for combined in-plane bending and torsion moment: (a) for combined in-plane closing bending and torsion moment (b) for combined in-plane opening bending and torsion moment Fig.14. Bending-torsion interaction curve for MB-MT loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending Fig.15. Bending-torsion interaction curve for MT-MB loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending Fig.16. Bending-torsion interaction curve for proportional loading: (a) combined torsion and in-plane closing bending (b) combined torsion and in-plane opening bending Fig.17. Mises criterion region for a pipe bend under combined in-plane opening bending and torsion moment, where the applied torsion moment are 14.4KNm in unified: (a) MB=0 (b) MB =0.2MIO (c) MB =0.4MIO (d) MB =0.6MIO (e) MB =0.8MIO (f) MB =MLIO Fig.18. Ovalization deformation in cross section A-A for a pipe bend under combined in-plane opening bending and torsion moment Fig.19. Effect of the materials constants on plastic limit loads for combined loads: (a) r/t=5, R/r=2 (b) r/t=5, R/r=6 (b) r/t=20, R/r=2 (b) r/t=20, R/r=6
Fig.20. Comparison of proposed result in Eq.(19) with some existing limit load solutions for pipe bends under combined in-plane bending and torsion moment: (a) with Eq.(9) by Guo (b) with Eq.(14) in ASME Fig.21. Effect of mesh on plastic limit loads Fig.22. Comparison of FE data with analytical solutions based on different yield criterion for pressurized straight pipes Table 1 Geometric parameters of analyzed pipe bends r=100mm L=1000mm Pure in-plane bending moment r/t=5 R/r=2, 4, 6 r/t=10 R/r=2, 4, 6 r/t=20 R/r=2, 4, 6 r/t=50 R/r=2, 4, 6 Pure torsion moment r/t=5 R/r=2,3,4,5, 6 r/t=10 R/r=2,3,4,5, 6 r/t=20 R/r=2,3,4,5, 6 r/t=50 R/r=2,3,4,5, 6 Combined in-plane bending and torsion moment r/t=5 R/r=2, 6 r/t=20 R/r=2, 6
λ=0.4, 0.8, 1.2 λ=0.2, 0.4, 0.6 λ=0.1, 0.2, 0.3 λ=0.04, 0.08, 0.12 λ=0.4,0.6,0.8,1,1.2 λ=0.2,0.3,0.4,0.5,0.6 λ=0.1,0.15,0.2,0.25,0.3 λ=0.04,0.06,0.08,0.1,0.12 λ=0.4, 1.2 λ=0.1, 0.3
Table 2 Comparison of the estimated FE results by Eqs. (15) and (17) with existing experiment results for pipe bends under in-plane bending and torsion moment Load type Material R/r r/t σS / Error/% M FE / M exp M FE MPa /KNm /KNm
M exp
In-plane closing bending
In-plane opening bending Torsion
ASTM A-106B ASTM A-106B SS304 SS304 ASTM A-106B
1.94 2.91 1.56 2.67 2.84
7.34 7.34 12.5 8.58 11.33
239 261 272 265 345
41.69 48.36 4.38 1.65 32.43
40.82 49.22 4.52 1.71 33.74
0.98 1.01 1.03 1.04 1.04
2.08 -1.77 -3.19 -3.63 -4.03
ASTM A-106B SS 304
2.84 2.67
11.33 8.44
345 265
36.61 1.82
37.01 1.81
1.01 0.99
-1.11 0.78
Parameter r/t is the main factor influencing the plastic limit loads. Plastic limit load is independent on loading path and material constants. Estimated equations of limit loads for pipe bends under combined loads are proposed.