Ffs Fea Pipe Elbow

International Journal of Pressure Vessels a

e

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for

t d

A s

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the straight pipe and become the weakest part of the pipingsystem [1].

be used in the assessment of an elbow.

bend radius is chosen as 100, 150, 175 and 200mm,respectively. There are 16 models (Table 1) in all.The dimensions of the elbows with LTA are 108mm

ARTICLE IN PRESShttp://www.paper.edu.cndiameters, 8mm thickness and 901, and the elbow bendradius is R 150mm. At the two ends of the elbow, two150mm straight pipes are also connected. The LTA lies in

0308-0161/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijpvp.2006.08.003

Corresponding author. Tel.: +8625 835 872 89;fax: +8625 835 872 88.

E-mail address: [email protected] (Z.-X. Duan).

Because of corrosion, erosion, mechanical damage andcrack polishing, there are local thinned areas (LTAs) in anelbow. The LTAs reduce the structural integrity, and mayaffect the safety of the pipeline. For the safety evaluation ofan elbow with LTA(s), it is signicant to calculate the limitload of an elbow with LTA(s) [2].Zhang et al. [3] took the relation of different LTA

position to the limit load into consideration and deduced

2. Finite element analysis on the plastic limit load of elbows

2.1. Finite element modeling

The dimension of the elbow without defects is a diameterof 108mm, a thickness t, and 901. At the two ends ofthe elbow, two 150mm straight pipes are connected. Thethickness t is chosen as 3, 5, 8 and 10mm, respectively. TheElbows are often considered to be the critical compo-nents in a piping system. Because of the bend radius,elbows represent different performance from a straightpipe. The elbow can not only change the direction of thepipeline, but can also absorb the force and moment causedby heat expansion because its rigidity is lower than theconnected straight pipe. The elbow is subjected to loads,such as internal pressure, moment, torsion, and deadweight. Under these loads the maximum stress in the pipingsystem occurs in the elbows and so elbows fail earlier than

Wang et al. [4] calculated many elbows with LTA bynite elements analysis and determined the inuence of theLTA dimensions on limit load.However in the above researches, an elasticperfectly-

plastic material model is used. So the results areconservative. In this research the stressstrain curve ofthe material is adopted.This paper discusses the plastic limit load of elbows

without defects, or with a local thinned area in theextrados, under internal pressure.Analysis and experiments on th

Zhi-Xiang Duan

College of Mechanical & Power Engineering, Nan

Received 8 December 2005; received in revised

Abstract

This paper discusses the plastic limit pressure of elbows withou

analysis (FEA) and experiments have been used. The results of FE

with increasing wall thickness and bend radius of the elbow. The re

the maximum error is 6.58%. By data tting of FEA, an empirica

extrados has been proposed, which is validated by experiments.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Elbow; Limit load; Finite element analysis; Local thinned area

1. Introductionnd Piping 83 (2006) 707713

plastic limit pressure of elbows

Shi-Ming Shen

University of Technology, Nanjing 210009, China

m 28 February 2006; accepted 4 August 2006

efects and with local thinned area in the extrados. Finite element

how that the limit load of elbows under internal pressure increases

s are consistent with the calculated results by the Goodall formula,

rmula for the limit load of elbows with local thinned area in the

periment

that the acceptance criteria for a straight pipe LTA cannot

www.elsevier.com/locate/ijpvp

the outer wall of the extrados of the elbow. The dimensionsof the elbows are shown in Table 2. Sixteen models arechosen for the calculations according to the Orthogonal

P D2i =D2o D2i , where Do and Di are the outer and innerdiameters, respectively.

ARTICLE IN PRESS

Nomenclature

t elbow thicknessP internal pressureP1 equivalent pressure, P D2i =D2o D2i P0 plastic limit pressure of elbow without defectsPL plastic limit pressure of elbowR elbow bend radiusr mean elbow radiusDo outer diameter of elbowDi inner diameter of elbowg axial half-angle of local thinned areay circumferential half-angle of local thinned area

C depth of local thinned areaa g/451b y/1801c C/tsf ow stress

Subscripts

T testF nite element analysisG Goodall formula4 formula (4)

Z.-X. Duan, S.-M. Shen / International Journal of Pressure Vessels and Piping 83 (2006) 707713708

http://www.paper.edu.cnDesign Table L16 (4) [5]. The dimensions of the defects aredescribed by the non-dimensional parameters a ( g/451), b( y/1801) and c ( C/t) (Fig. 1).The model of FEA is one half of the structure because of

the symmetry of geometry and loading (Fig. 1).

2.2. Boundary conditions

The boundary conditions of the model are shown inFig. 1. The pressure P is applied on the internal surfaces ofthe elbow. In the symmetrical plane XOY the displacementin the Z direction is restrained. In the plane of the lowerend of the model the displacement in the Y direction isrestrained. In the left node of this plane the displacement inthe X direction is restrained. An equivalent pressure P1 isapplied on the right end of the model. The value of P1 is

Table 1

FEA models of elbows without defectsNo. Dimensions Do t (mmmm) R (mm) Li

FE

01 108 3 100 1502 108 5 100 2603 108 8 100 4604 108 10 100 6005 108 3 150 1806 108 5 150 3107 108 8 150 5208 108 10 150 6709 108 3 175 1910 108 5 175 3311 108 8 175 5512 108 10 175 7113 108 3 200 2014 108 5 200 3315 108 8 200 5716 108 10 200 722.3. Meshing

Finite element analysis is performed using the softwareANSYS. The SOLID45 (3-Dimensional Structural Solid)elements are used for the analysis (Fig. 2). A three-dimensional mesh is created by taking 5 divisions in thelongitudinal direction of each straight segment andapproximately 20 divisions (more or less according to theLTA and other dimensions, but more divisions in the LTA)in the longitudinal direction of the elbow. In thecircumferential direction, about 15 divisions are takenand in the depth direction 4 or 5 divisions are taken. In theLTA, the mesh is rened taking more divisions in thecircumferential and longitudinal directions. The number ofnodes and elements are controlled by the non-dimensionalparameters a, b and c and the dimensions Do, t, R.mit loads (MPa) Error (%) eG PLG PLF=PLF

A (PLF) Goodall (PLG)

.210 15.862 4.287

.795 27.333 2.008

.000 45.973 0.060

.267 59.416 1.411

.332 19.404 5.847

.467 33.173 5.423

.616 55.168 4.851

.743 70.788 4.495

.030 20.282 6.580

.048 34.625 4.771

.922 57.467 2.764

.042 73.640 3.657

.248 20.908 3.260

.794 35.661 5.525

.068 59.109 3.577

.484 75.680 4.409

2ttmp

F

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Table 2

FEA models of elbows with LTA

No. Dimensions strain location

a b c

L01 0.05 0.10 0.10 52.48 Inner wall of intrados

L02 0.05 0.25 0.25 52.27 Inner wall of intrados

L03 0.05 0.40 0.50

L04 0.05 0.50 0.75

L05 0.125 0.10 0.25

L06 0.125 0.25 0.10

L07 0.125 0.40 0.75

L08 0.125 0.50 0.50

L09 0.25 0.10 0.50

L10 0.25 0.25 0.75

L11 0.25 0.40 0.10

L12 0.25 0.50 0.25

L13 0.375 0.10 0.75

L14 0.375 0.25 0.50

L15 0.375 0.40 0.25

L16 0.375 0.50 0.10

z x

y

A - A

A

A

P P

PP

P

P P1

P1

o

(a) Without defects (b)

z

y

Fig. 1. Model and boun

Fig. 2. Finite element mesh.

Z.-X. Duan, S.-M. Shen / International Journal of P

http://www.paper.edu.cn50.62 LTA

28.37 LTA

52.36 Inner wall of intrados


21.05 LTA

41.67 LTA

38.32 LTA

19.52 LTA

52.47 Inner wall of intrados.4.

Theheatro

YPYU

Tig

o

daLimit loads PF (MPa) Maximumressure Vessels and Piping 83 (2006) 707713 709Material model

he specimens are made from straight steel pipe of whichmaterial is the same as the elbow. The test machine isMTS-880. The stressstrain curve from tests on thiserial is used for the nite element analysis. The materialperties are

oungs modulus: E 1.7397 105MPa;oissons ratio: n 0.3;ield strength: sy 345MPa;ltimate strength: su 517MPa;

he real stressstrain curve of the material is shown in. 3. The kinematic hardening multilinear material model

52.00 LTA

18.84 LTA

37.01 LTA

52.37 LTA


With LTA

t Di

x

A - AA

A

P P

PP

PP

P1

P1

C

ry conditions.

is adopted for FEA. In the elastic stage there is a singleslope and in the plastic stage there are 6 segments (Fig. 3).

2.5. Results and analysis

The ideal limit load is dened as the load correspondingto the limit state, where the load does not increase, but thedisplacement or strain of the structure increases innitely.The hypothesis of the denition is that the structure iscomposed of elasticperfectly-plastic material and onlysmall displacements are considered. In reality, the idealmaterial does not exist because of strain-hardening andgeometry hardening or weakening. Therefore, variousmethods of determining the value of the limit load are

proposed in the engineering eld, such as the twice-elastic-slope method, 3 times-elastic-slope method, twice-elastic-defect method, tangent intersection method, zero-curvaturemethod and 0.2% residual strain method. In this paper thelimit load is determined based on the loadstrain curvesusing the twice-elastic-slope method which is described inthe ASME Boilers and Pressure Vessel Code [5]. The strainin the loadstrain curve is the maximum von-Mises strainof the elbow.The FEA results are shown in Tables 1 and 2.

2.5.1. FEA results of elbows without defects

2.5.1.1. Variation of limit pressure with dimensions. Thevariation of the limit pressure PL of elbows without defectswith R/Do and t/Do is shown in Fig. 4. Here, Do is the outerdiameter.From the gures, it can be seen that when R/Do is kept

constant, the limit pressure increases with increasingthickness. The relation of limit pressure with thickness is

ARTICLE IN PRESS

Fig. 3. Real stressstrain curve of 20 g steel.

010

70

80R/Do=0.92593

R/D =1.38889

Z.-X. Duan, S.-M. Shen / International Journal of Pressure Vessels and Piping 83 (2006) 707713710

http://www.paper.edu.cn0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1t/Do20

30

40

50

60

P L

(MPa

)

o

R/Do=1.62037

R/Do=1.85185(a) PL -t/Do

Fig. 4. Variation of limit prclose to linear. When t/Do is kept constant, the limitpressure increases with increasing bend radius. Theinuence of thickness (t) on the limit pressure is greaterthan the effect of bend radius.

2.5.1.2. Comparison of FEA results with available formula.

For a thin-walled elbow, Goodall [6] proposed the formulafor the limit pressure P0

P0 sf tr

1 r=R1 r=2R . (1)

Here, r is the average radius of the elbow. sf is ow stresstaken as sfsu=2. In this paper

sf 345 517=2 431MPa

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.41015202530354045505560657075

P L

(MPa

)

R/Do

t/Do=0.02778

t/Do=0.0463

t/Do=0.07407

t/Do=0.09259

(b) PL -R/Doessure with R/Do and t/Do.

EsGttp

2

a

o

w

G5p5tlcms

ofL

P

ao

f

w

G

2d

c

P sf t 1 r=R f a; b; c, (4)

f

g

a

r

3

3

3

cashss

(((3) 108mm 8mm, g 11.251, y 901, C 6 .00mm.

3.1.2. Welding

A 901 elbow, 2 straight pipes and two at heads arewelded together to make the specimen (Fig. 6).

3.2. Experimental process

The purpose of the experiments is to determine the limitpressure of the elbows. A 60MPa pressure pump is used toprovide the pressure with water. Strain gauges aredistributed along the axis in the outer wall of the extradosand the outer wall along the circumference of thesymmetrical area. When the pressure attains a given value,the strain is measured. Finally, the loadstrain curve isdrawn, and the limit load is obtained.

3.3. Test results and analysis

ARTICLE IN PRESSf Pr

http://www.paper.edu.cnLr 1 r=2R

a; b; c ming; 0:954, (5)

18:483G2 7:108G 1:023 (6)G a0:5b0:1c3 (for the elbow without defects: b c 0, then G 0)Comparison between FEA results and results fromq. (1) is displayed in Table 1. From the table, it can beeen that the FEA results are very close to the results ofoodall, with the Goodall results being a little greater thanhe FEA results. The maximum error is 6.58%. Thushe Goodall formula may be used to calculate the limitressure of elbows with no LTA.

.5.2. Results of elbows with LTA

From Table 2, the elbows are classied into 2 groupsccording to the location of the maximum strain:Group 1, the maximum strain is located in the inner wallf the intrados;Group 2, the maximum strain is not located in the innerall of the intrados.Elbows L01, L02, L05, L06, L11 and L16 belong toroup 1. The limit pressures of these are 52.48, 52.42,2.47, 52.55, 52.36 and 52.27MPa, respectively. The limitressure of the corresponding elbow without defects is2.62MPa. Observing that the elbows belong to Group 1,he LTA is thin and small and the inuence of LTA on theimit pressure is small. The limit pressure of these elbowsan be calculated using the Goodall formula with aultiplying factor. The result of Goodall is 55.17MPa,o the factor is taken as 52.62/55.17 0.954.For elbows belonging to Group 2, the inuence of LTAn the limit pressure is obvious. The revised Goodallormula is also used. But the revised factor is related to theTA. Take the revised Goodall formula as

L P0 f a; b; c. (2)According to the inuence of a, b, c on the limit pressurend through a number of data tting exercises, f(a, b, c) isbtained as

a; b; c 18:483G2 7:108G 1:023; (3)here

a0:5b0:1c3.The tting curve is shown in Fig. 5.The limit pressure of the elbow which belongs to Groupis smaller than that of the corresponding elbow withoutefects, thus f a; b; co0:954.So the limit pressure of an elbow with or without LTA

an be described by the following equations:

Z.-X. Duan, S.-M. Shen / International Journal oBecause the range of LTA in this paper is Go0:21, theestriction of the above equations is Go0:21. s.1. Manufacture of the specimens

.1.1. Manufacture of the LTA

Because the specimen is an elbow and the LTA is in theurve of the elbow, it is difcult to ensure the dimensionsnd precision of the defect using a mechanical method. Foratisfactory test results, the LTA is manufactured throughandcraft polishing. The structure of the specimens ishown in Fig. 6. The dimensions of the 901 elbow in thepecimens are as follows:

1) 108mm 5mm, without defects;2) 108mm 5mm, g 5.6251, y 451, C 1.25mm;. ExThhowperimentsFig. 5. Fitting curve.essure Vessels and Piping 83 (2006) 707713 711e comparison between FEA results and test results isn in Table 3.

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1.9

1Y

re of th

l of712

http://www.paper.edu.cn1. 90 degree elbow 2. Straight pipe 3. Flat headWithout defects

150

(a)

Fig. 6. Structuwieffrea

4.

pa

(1

(2

(3

Ta

Co

No

1

2

3

Po 108 8

R1501 2 3

108 5 10

8 8

150Z.-X. Duan, S.-M. Shen / International JournaIt is clear from Table 3 that the test results are consistentth these from the nite element analysis. The FEA isective and the formula developed from FEA results issonable.

Conclusions

From the nite element analysis and experiments in thisper, the following conclusions can be drawn:

) The FEA results are consistent with those fromexperiments, indicating that the modeling method iscorrect and it is effective to calculate the limit load ofan elbow under internal pressure using FEA.

) The results of FEA show that the limit pressure ofelbows increases with increasing wall thickness andincreasing bend radius. The results are consistent withthe results calculated by the Goodall formula, themaximum error is 6.58%, which indicates that the limitpressure of an elbow can be calculated using theGoodall formula.

) According to the FEA results, when the LTA is verythin or small, the elbow with a LTA collapses in astandard manner and the intrados of the elbow rst

(4

strththindim

Re

[1]

[2]

ble 3

mparison of results between nite element analyses and experiments

. Dimensions Test PT (MPa)

108mm 5mm, without defects 29.637108mm 5mm, a 0.125, b 0.25, c 0.25 28.58108mm 8mm, a 0.25, b 0.50, c 0.75 17.67

stscript: eF PF PT=PT; e5 P5 PT=PT.e specimens.(b)

0 degree elbow 2. Straight pipe 3. Flat head

With LTA

C 108 t

A

A - A

108 8

150

150

R150

A

1 2 3

08

8Y

Pressure Vessels and Piping 83 (2006) 707713reaches the limit state. The limit pressure of this elbowis the same as the corresponding elbow without defects.

) By data tting of FEA results, an empirical formula ofthe limit load for elbows with local thinned area in theextrados has been proposed, which is validated byexperiments.

These conclusions can be used for elbows with similaressstrain curves (the shape of the plastic stage, althoughe stress and strain values need not be the same as those toe one in the paper), because the formula in the papercludes the ow stress. Furthermore, work should takefferent material properties into consideration to attainore general conclusions.

ferences

Guo C. Plastic limit loads for surface defect pipes and bends

under combined loads of tension, bending, torsion and internal

pressure. East China University of Science and Technology;

1999.

Han L-H, LIU Z-D. Limit load analysis for local wall-

thinning pipeline under internal pressure. Press Ves Technol 1998;

15(4):14.

FEA PF (MPa) Formula (4) P4 (MPa) Error eF Error e4

31.467 31.647 0.0617 0.0678

31.62 31.65 0.1064 0.1074

19.52 18.76 0.1046 0.0617

[3] Zhang L, Wang Y, Chen J, et al. Evaluation of local thinned

pressurized elbows. Int J Press Ves Pip 2001;78:697703.

[4] Wang Y, Sun L, Fan D-S. Plastic limit load analysis of

elbow with local thinning. J Beijing Univ Chem Technol 2005;

32(1):625.

[5] ASME Boiler and Pressure Vessel Code, Section III, Division 1,

Nuclear power plant components. New York, NY: ASME; 1992.

[6] Goodall IW. Lower bound limit analysis of curved tubes loaded by

combined internal pressure and in-plane bending moment. RD/B/

N4360, CEGB; 1978.

ARTICLE IN PRESSZ.-X. Duan, S.-M. Shen / International Journal of Pressure Vessels and Piping 83 (2006) 707713 713

http://www.paper.edu.cn

Analysis and experiments on the plastic limit pressure of elbowsIntroductionFinite element analysis on the plastic limit load of elbowsFinite element modelingBoundary conditionsMeshingMaterial modelResults and analysisFEA results of elbows without defectsVariation of limit pressure with dimensionsComparison of FEA results with available formula

Results of elbows with LTA

ExperimentsManufacture of the specimensManufacture of the LTAWelding

Experimental processTest results and analysis

ConclusionsReferences

Text3:

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Ffs Fea Pipe Elbow