Elastic and Plastic Design of Mitred BendsA.M. (Nol) Gresnigt

Delft University of TechnologyDelft, The Netherlands

ABSTRACT

Mitred bends were and partly still are the standard pipe bends in largediameter pipelines for water transport in The Netherlands.Especially in situations with imposed bending, the deformationcapacity is important. Failure modes in static loading are excessiveovalisation, local buckling and cracking due to insufficient ductilebehaviour at the girth weld zone.In the paper, the results are presented of a theoretical and experimentalresearch programme of mitred bends loaded by combinations ofbending moment and internal pressure. It is demonstrated that theanalytical models that were developed for smooth bends can be used asa basis for the analysis of mitred bends. Design rules for limit statedesign of mitred bends are presented as well.

KEY WORDS: Pipeline; bend; mitred bend; plastic design; deformationcapacity.

NOMENCLATURE

Du External diameterD Average diameter: D = Du-t∆Dv Change in the vertical diameter (in the plane of the bend)EI Flexural stiffness of a straight pipeEIbe Flexural stiffness of a bendix, iy Stress intensity factors for P = 0ixp, iyp Stress intensity factors for P not = 0k, kp Flexural stiffness factor for P = 0 and P = P respectivelyk*, kp* Reduced value of k, kp respectivelyMpbe Maximum moment in the bendMp Plastic moment: Mp = 4r2tσeP Difference in pressure between inside and outside of the

pipeline: P = Pi - PuR Radius of non-loaded bendr Average (or mean) radius r = D/2S Distance between individual mitrest Pipe wall thicknessWe Elastic section modules2α Angle of the not loaded bend2∆α Change of 2α

β Total bend angle in a multi - mitred bendσx Stress in longitudinal directionσy Stress in circumferential directionOther symbols are explained in the text where they first appear.

INTRODUCTION

Pipelines, offshore and onshore, are subjected to combinations ofvarious loads, such as internal or external pressure, surrounding soil,bending, normal force, shear force and sometimes torsion and localloading, e.g. due to support reactions. In many cases, the deformationcapacity is of special importance. Examples are offshore pipelinesduring installation, pipelines on uneven seabed and onshore pipelinesdue to differences in the settlement of the soil along the pipeline. Thelatter situation is very important in The Netherlands, where many riversand canals have to be crossed by pipelines. The highest water leveloccurring in such watercourses is often above the level of the adjacentland. Dykes are needed to prevent flooding. Because excavation workin dykes is normally not allowed, most pipeline profiles correspond tothe existing cross section of the dyke. After laying, the pipelines weregiven a covering of clay and top soil, see Fig. 1.In the last ten years usually directional drilling techniques are appliedwhere the pipeline crosses the dyke and watercourse underneath.

Fig. 1: Pipeline crossing over a dyke.

Application of the theory of elasticity for the analysis of pipelinecrossings has proven to be inadequate for a good insight into the actualstrength and deformation properties and thus into the actual structuralsafety. When in the early 1970s the theory of elasticity was applied in are-analysis of some pipeline crossings already in existence, several ofthem were found not to meet the requirements.

Proceedings of The Twelfth (2002) International Offshore and Polar Engineering ConferenceKitakyushu, Japan, May 26–31, 2002Copyright © 2002 by The International Society of Offshore and Polar EngineersISBN 1-880653-58-3 (Set); ISSN 1098-6189 (Set)

90

Therefore, research was carried out, resulting in a new method for thedesign and analysis of buried pipelines, which is based on limit statedesign. The theory of plasticity was introduced. By means of this newmethod, it could be established that the majority of the pipelinecrossings was safe. This resulted in considerable savings, as there wasno necessity of replacement. The main results of the research onstraight pipes and smooth bends were presented in several publications(e.g. Gresnigt, 1986, 1995). In the present paper, the results of theresearch into the structural behaviour of mitred bends are presented.

Mitred bends are often used in water transportation pipelines wherediameters are relatively large and pressures relatively low (e.g. less than10 bar). Many existing crossings of water pipelines with dykes andwatercourses contain such mitred bends. Compared to smooth bends,they are less expensive. It should be noted, however, that the pricedifference nowadays is smaller than some decades ago.

It appears that, to a great extent, the behaviour of mitred bends issimilar to the behaviour of smooth bends. Therefore, many of themodels for analysis and design developed for smooth bends can be usedas a basis for mitred bends. As in smooth bends, mitred bends are lessstiff than straight pipes of the same length and diameter and wallthickness. Further, considerable ovalisation may occur, see Fig. 2.

Fig. 2: Deformations in a mitred bend due to bending moment M.The bending direction in this figure is called "positive bending".

In case of positive bending (Fig. 2) a flattening of the cross sectionoccurs. In case of bending leading to a decrease of the bend angle(called negative bending), the ovalisation gives an increase in thevertical cross section. The increase/decrease in curvature and theflattening or increase of the vertical diameter lead to differentbehaviour in positive and negative bending, see Fig. 3.

Fig. 3: Bending moment - Change of bend angle diagram of a mitredbend in positive bending and in negative bending.

In Fig. 3, line 'a' represents the behaviour of a straight pipe of the samediameter, wall thickness and length as the mitred bend. The line 'b' isthe linear elastic solution as given in the next section. As in smoothbends, also in mitred bends the plastic moment capacity depends on thebending direction (Gresnigt, 1995).Another interesting phenomenon is that the deformation capacity inbending is considerably greater than of a straight pipe. As indicatedbefore, this is an important issue in buried pipelines in settlement areas.

In the next section, an overview is given of the existing design rules atthe start of the research programme. Much attention is paid to thesimilarities with smooth bend behaviour. Prerequisites and parametersfor the application of the smooth bend design models for the analysis ofmitred bends are identified. Then the development of analytical modelsfor mitred bends is given, followed by a summary of the design rulesresulting from the research program. Also an overview of the mainresults of the experimental testing programme is presented.

ANALYTICAL MODELS IN THE ELASTIC RANGE

Overview of existing design models

Much research has been carried out into the structural behaviour ofmitred bends. In WRC bulletin No. 208, Rodabough presented anoverview of the research carried out till 1975 (Rodabough, 1975). Healso gave background to the design rules in ANSI/ASME B31.8 (ANSI,1979).

In ANSI/ASME B31.8, a distinction is made between "widely spaced"and "closely spaced" mitred bends (Fig. 4). The distinction is that inclosely spaced bends the individual bends interact and therefore theassembly of the bend, short straight pipe and bend should be consideredas one bend. In widely spaced bends no interaction is assumed. If

)tan1( α−< rS , then the bend is assumed to be closely spaced.

Fig. 4: Widely spaced and closely spaced mitred bends.

For the bending stiffness the following rules apply (ANSI, 1979)

kEIEIbe = (1)

6/552,1

λ=k (2)

2rRt ⋅=λ (3)

For closely spaced bends:)tan1( α+< rS (4)

αcot2SR = (5)

For widely spaced bends:)1( αtgrS +≥ (6)

)cot1(2

α+= rR (7)

The length where the reduced stiffness EIbe applies is 2ℓ1 (Fig. 4):α22 1 ⋅= Rl (8)

For the maximum stress in longitudinal direction:

xe

x iWM ⋅=σ (9)

For in plane bending:

0,19,03/2 ≥= xx ii

λ(10)

91

For out of plane bending:

0,175,03/2 ≥= xx ii

λ(11)

Apart from stresses in longitudinal direction also stresses incircumferential direction occur:

ye

y iWM ⋅=σ (12)

It is noted that the real elastic stresses are about twice those given byANSI. These maximum stresses are local (peak stresses). They willcause local yielding at relatively low bending moments. Owing to theductility of the steel, redistribution of stresses will occur and thebending moment capacity is much greater than the moment at firstyielding. On the basis of this and on the basis of experimental tests (e.g.Markl, 1952) it was decided to adopt the present rules in ANSI.In case of internal pressure, the resistance to ovalisation increases andtherefore also the bending stiffness increases. In that case the factor k inEq. (2) should be replaced by kp:

3/13/761

+

=

rR

tr

EP

kk p (13)

And the stress intensity factor ixp and iyp become:

3/22/525,31

+

=

rR

tr

EP

ii xxp (14)

3/22/525,31

+

=

rR

tr

EP

ii yyp (15)

In WRC bulletin No. 208, (Rodabough, 1975), Rodaboughdemonstrates that the theoretical models of Jones, Kitching and Bondgive the best fit of test results. Their analysis mainly follows the longerexisting analysis for smooth bends (e.g. Kitching and Bond, 1970,Bond and Kitching, 1972), see Fig. 5.

Fig. 5: Mitred bend and similar smooth bend (Fig. 1 in Kitching andBond, 1970).

Kitching and Bond (1970) have compared their analysis with testresults and with ANSI rules. Table 1 gives the main data of their testspecimens.

Table 1: Test specimens as tested by Kitching and Bond (1970).

Specimen Bendangle β 2

11

rRt=λ

1Rr

1

1λr

Rtr =

a.

R1

α = 15°3 ⋅ 2α =90°β = 90°

0,177 0,343 16,5

b.

R1

α = 22,5°2 ⋅ 2α =90°β = 90°

0,1695 0,360 16,4

c.

R1

α=11,25°4 ⋅ 2α =90°β = 90°

0,0483 0,446 46,4

d. R1

α = 22,5°4 ⋅ 2α =180°β = 180°

0,1042 0,3075 31,2

It is noted that the values for λ1 and R1 in Table 1 differ from thosegiven by ANSI, Eq. (3) and (5), see Table 2.

Table 2: Values for r/R, k and ℓ1/r according to Eq. (5), (2) and (8).

SpecimenRr λ 6/5

52,1λ

=k αrR

r=1l

abcd

0,4310,6040,4460,610

0,1410,1010,04830,0525

7,8010,319,017,9

0,610,650,440,64

In Fig. 6 experimental and theoretical stress distributions according toKitching and Bond (1970) are given for specimen 'a'.

Table 3 gives an overview of the main test results together with theANSI code values and the values according to Kitching and Bond. The2 x code column refers to the above discussion on the peak stresses. Itconfirms what was said before, namely that the peak stresses are abouttwice those in the ANSI-code.

Table 3: Overview of the main test results together with the ANSIcode values and the values according to Kitching and Bond.

Flexibility factor Stress intensification factorSpecimen Present

analysis Code Experi-ment

Presentanalysis

2 *Code

Experi-ment

abcd

8,707,8034,812,7

7,8010,3019,017,9

7,958,7037,814,4

7,108,40

16,3012,60

6,608,40

13,5012,80

5,108,60

-10,50

92

From Table 3 it appears that the flexibility factor according to ANSIdoes not fit very well with the analysis of Kitching and Bond, nor withthe experiments. The stress intensity factors show better agreement iffor ANSI "2 x code" is taken.

Fig. 6: Experimental and theoretical stress distributions for testspecimen 'a' (Kitching and Bond, 1970).

Application of the rules for smooth bends

As stated before, the behaviour of mitred bends shows great similarityto the behaviour of smooth bends. Below, results are given for theflexibility factor calculation according to the models for smooth bendsas proposed by Gresnigt (1986a, 1995).

uDt1200=γ (16)

2

1)1(1*

−−−+=γ

βγkk (17)

In k* according to Eq. (17), the influence of the tangents of the bend onthe bend stiffness is taken into account (Gresnigt, 1995).

Table 4: Comparison of flexibility factors for test specimens a, b, c, d.

λ/65,1=k Smooth BendSpecimen t

Du γ λ1

k k* k k*abcd

34349463

206 o

206 o

124 o

151 o

0,1770,16950,04830,1042

9,329,74

34,1615,84

7,888,22

32,9015,84

9,8210,2535,8616,61

8,298,64

34,5316,61

From the comparison with the measured values in Table 3 it appearsthat the "smooth bend theory" with the modification for the influence ofthe tangent pipes on the stiffness factor in Eqs. (16) and (17) givesbetter results than ANSI.Also for single mitred bends as tested in the present research programmethe "smooth bend" theory (Gresnigt, 1995) gives good results. As anexample, one of the tests on mitred bends (Gresnigt, 1986c) is calculatedbelow, see also the section on experimental verification later in this paper.

Fig. 7: Single mitred bend.

6,2165,1

0764,02,7526265,1

mm3,43)tan1(2

mm262)cot1(2

192

mm 2752651152mm 651

mm 152

22

1

==

=⋅==

=+=

=+=

=

===

=

λ

α

α

k

rtRλ

r

rR

α

, )/,- (r, t D

o

u

l

o

uDt 125

15265,112001200 ===γ

9,1121)1(1*2

=

−−−+=γ

αγkk (18)

According to ANSI:

9,1252,16/5 ==

λk (19)

With the "bend analysis" it is calculated:62,22=k and 94,12* =k (20)

In this case with a single mitred bend, the ANSI result agrees well withthe "bend analysis" and with test results.

Finally, some remarks about the definition of R are given. For therotation angle 2∆α between points A and B in Fig. 7, it follows:

8,25*)22(2 11 ⋅=+=∆EI

Mk

EIM

αll

(21)

If for the angle R = 2 ⋅ 262 = 524 mm was taken instead of R = 262mm, it follows:

8,1065,1

1528,02,7552465,122

==

=⋅==

λk

rtRλ

o

uDt 125

15265,112001200 ===γ

93

2,6125

191251)18,10(1*2

=

−−−+=k

8,24*)4(2 11 ⋅==∆EI

Mk

EIM

αll

(22)

It appears that with the "smooth bend" theory, the rotation 2∆α is ratherinsensitive of the value of R.This is also true for the ovalisation, because the ovalisation is almostlinear with α/α∆ .For R = 262 mm it follows:

ααα9,11* 11 ⋅=⋅=∆

EIMk

EIMα ll

(23)

For R = 524 mm it follows:

ααα4,12*2 11 ⋅=⋅=∆

EIMk

EIMα ll

(24)

For the stress intensification this is not true, because of the local natureof the stress peaks. For the stress intensification factors it isrecommended to apply:

3/28,1

λ== yx ii (25)

It is noted that the maximum stresses in longitudinal andcircumferential directions are about the same (Kitching and Bond,1970). In smooth bends, however, the maximum stresses in longitudinaldirection are about half of those in circumferential direction. Thereason for this difference is, as mentioned before, the geometricalsituation at the intersection of the pipe segments in mitred bends.

Design rules for the elastic range

The following design rules were proposed and accepted for the NEN3650 pipeline standard (NEN 3650, 1992).The scope for these rules is as follows:

o252 ≤α (26)

5,2≥rR (27)

Single mitred bends

Fig. 8: Single mitred bend.

)cot1(2

α+= rR (28)

)tan1(21 α+= r

l (29)

2rRtλ ⋅= (30)

λ65,1=k (31)

uDt1200=γ (32)

221)1(1*

−−−+=γ

αγkk (33)

3/28,1

λ== yx ii (34)

In case of internal pressure:

3/13/761

**

+

=

rR

tr

EP

kk p (35)

3/22/525,31

)(

+

===

rR

tr

EP

iiiii yxxypxp (36)

Multi-angle mitred bend with S>2ℓ1

Fig. 9: Multi-angle mitred bend with S > 2ℓ1.

If there is no interaction between the individual mitres, then the rulesfor single mitred bends of the previous section are to be followed.Whether or not such interaction occurs, follows from the following twocalculations.(a) Calculate the rotation angle between A and B, assuming individual

mitres.

)42*4( 11 ll −+= SkEIMa

ABϕ (37)

(b) Calculate the rotation angle between A and B(Fig.9), taking thewhole length as one bend:

αcot2

* SR = (38)

2*

rtRλ = (39)

λ65,1=k (40)

uDt1200=γ (41)

241)1(1*

−−−+=γ

αγkk (42)

)2*( SkEIMb

AB ⋅=ϕ (43)

If bAB

aAB ϕϕ > , then it is assumed that there is no interaction. The rules

for single mitres (a) are to be followed. In the other case the above rulesfor the bend considering it as one bend (b) must be followed. This holdsfor the flexibility factor k* and for the ovalisation.

94

For the stress intensity factors, however, in all cases the rules for thesingle mitred bend are to be followed.

Multi-angle mitred bend with S < 2ℓ1

Fig. 10: Multi-angle mitred bend with S < 2ℓi.

The rules according to (b) above are to be followed.For ix and iy:

3/28,1

λ== yx ii (44)

with

2rtRλ = (45)

Internal /external pressure:Hoop stress intrados:

rRrR

trP

hoop −−⋅= 3/σ (46)

Hoop stress extrados:

rRrR

trP

hoop ++⋅= 3/σ (47)

For R the smallest value of

)cot1(2

α+= rR (48)

andαcot⋅= SR (49)

should be taken.

Soil pressure: The same Eqs. as for smooth bends can be applied(Gresnigt, 1986, 1995).

Influence of ovalisation and bending direction, out of plane bending:Here also the same Eqs. can be applied as for smooth bends.

DESIGN OF MITRED BENDS IN THE PLASTIC RANGE

As indicated in the previous section, mitred bends behave similarly tosmooth bends. Therefore it was proposed to use the same models forthe plastic range of mitred bends as for smooth bends, with somemodifications. In this section attention is paid to the prerequisites underwhich the design rules for smooth bends as published before can beapplied. For the present paper it goes too far to go into detail in thosemodels. Reference is made to Gresnigt, 1986, 1995.

According to the Eqs. for the elastic range, the bend radius equals R.

The special geometry of mitred bends requires a more severe limit onthe ovalisation than with smooth bends. In the tests that were carriedout (next section) it appeared that at large ovalisation the ovalisationconcentrates on the centre of the mitre. In this respect also the weldgeometry and the possibly lesser ductility of the weld metal and heat-affected zones are important. Some "high - lows" and other shapedeviations may cause extra distortion and local strains.

Therefore, the limit state values for the ovalisation have been reduced,compared to the smooth bend values:- Positive bending:

uDD 075,0limit =∆ (50)- Negative bending and out of plane bending:

uDD 060,0limit =∆ (51)For smooth bends these limits were set on 0,15 Du.The main reason for a somewhat smaller value for negative bending isthat at negative bending the local distortion is relatively much largerthan at positive bending.For the calculation of the strains in circumferential direction (limit statestrain) the same procedures are to be followed as for smooth bends.For the strains in longitudinal direction the same value as for thecircumferential strain should be taken (this is more than in smoothbends).The limit values for the angular rotation follow from the relationbetween the ovalisation and the angular rotation, see also the test resultsin the next section.

EXPERIMENTAL VERIFICATION

Tests and test specimens

In order to reduce costs it was decided to carry out tests on reducedscale specimens (scale factor about 1:4 to 1:8). Because suitablestandard pipe in diameters of 150 � 200 mm was not available, it wasdecided to manufacture the pipes from thin steel plate 1,65 mm (hotrolled). The pipes were rolled and welded (longitudinal seam) in a localmanufacturing shop. The standard length was 1000 mm.Two diameters were used:a. 152-1,65 mm, giving a Du/t ratio of 92.b. 122-1,65 mm, giving a Du/t ratio of 74.

Provided that the laws of similarity and scaling down are duly taken intoaccount, the conclusions drawn from tests on reduced scale specimens arevalidly applicable also to full-scale pipes.Since the goal of the tests was the study of the bending moment -curvature relationship, the bending moment - ovalisation relationship andin this case less important, the local buckling behaviour, the mainattention was focussed on the geometry such as Du/t ratio, bend angle andlength. Also important are the stress-strain diagram and the initialimperfections, such as out of roundness and "high - lows" at welds.

The D/t ratios in the tests are at the top range of what is applied inpractice. This choice was made because of the fact that thin walled pipesand bends are more sensitive to instability phenomena like local buckling.

Measured values for the yield stress in the steel plate before rolling itinto a pipe varied between 355 N/mm2 and 395 N/mm2. The averagevalue was 375 N/mm2. For the tensile strength these values were 410N/mm2 till 450 N/mm2 with an average of 435 N/mm2. Fig. 12 gives acharacteristic view of the measured stress strain relationship. Allspecimens were taken from the longitudinal direction.

The geometrical imperfections were greater than what is normallyaccepted in practice. This means that in practice the behaviour can beexpected to be better than in the tests, especially for the localdeformations at the welds, because of greater weld imperfections such as"high - lows" in the test specimens than in practice.

It is noted that aspects such as deformation capacity of the pipe wall intension (limit state cracking) could not be studied with these tests. Theweld geometry, the weld material properties and the possible welddiscontinuities were not representative for real pipes.

95

Fig. 11: Set-up of the test specimens: ℓ1 = ca 500 mm, ℓ2 = ca 1000mm, ℓ = 1496 mm (Du = 152 mm), ℓ = 1504 mm (Du = 122 mm).

Fig. 11 shows the set-up of the test specimens. The length ℓ wassufficient to avoid any influence of the end plates. An estimate for suchinfluence length is ℓ':

ℓ' mm57265,1

1528152

8=⋅== ππ

tDD uu (52)

This means that the chosen length was quite sufficient.

In Table 5 an overview is given of the tests that were carried out,together with the principal dimensions (and characteristics) of the testspecimens. Apart from the two different diameters also two differentbend angles (2α = 13,5° and 19°), two different bending directions (positiveand negative bending) and two different pressures (P = 0 and P= 7 bar) werechosen, giving a total of 16 tests.

Table 5: Test specimens (Pos. = positive; Neg. = negative bending).

Test Du(mm)

2α(degrees)

Bending P(bar)

R(mm)

λ k, kp iy, iyp

51525354656655565759616258606364

152152152152152152152152122122122122122122122122

13,513,513,513,519,019,019,019,013,513,513,513,519,019,019,019,0

Pos.Pos.Neg.Neg.Pos.Pos.Neg.Neg.Pos.Pos.Neg.Neg.Pos.Pos.Neg.Neg.

0707070707070700

355355355355262262262262284284284284210210210210

0,10360,10360,10360,10360,07650,07650,07650,07650,12940,12940,12940,12940,09570,09570,09570,0957

10,058,05

10,058,05

12,9510,5512,9510,55

8,357,258,357,25

10,759,50

10,7510,75

8,155,708,155,70

10,007,40

10,007,407,055,807,055,808,607,158,608,60

Fig. 12: Measured stress-strain relationship.

Test set-up

Figs 13 to 16 show the test set-up together with the measuringequipment for the change in the rotation between two cross sections at400 mm from the centre of the mitred bend. Displacements weremeasured with dial gauges. In the centre, the ovalisation was measuredas well as the vertical displacement of the centre cross section, see Fig.14.

Fig. 13: Photograph of the test rig with test specimen 65 beforetesting, positive bending.

Fig. 14: Details of the measuring equipment. The applied measuringequipment made it possible to measure the total diameter change and itsdivision in the change of the diameter of the upper half of the crosssection and of the lower half.

96

Fig. 15: Photograph of the test rig with test specimen 65 after testing,positive bending. Note the change in the diameter in the mid-section(ovalisation) and the change of the bend angle to be recognised by thechange in the angles of the thin bars welded to the side of the pipe, seealso Figs. 13 and 14.

Fig. 16: Photograph of the test rig with test specimen 64, negativebending.

Fig. 17: Detail of test specimen 66 after testing, positive bending.Note the local deformation at the lower cross section.

Fig. 18: Detail of test specimen 64 after testing, negative bending.Local deformations are much smaller than in positive bending.

Test results

Figs. 20, 21 and 22 give measured test results together with calculatedresults with the present analytical models for test 51 (positive bending).Figs. 23, 24 and 25 give measured test results together with calculatedresults with the present analytical models for test 53 (negativebending). Calculations were carried out with the computer programme"Bend" (Gresnigt, 1995).

In the figures also the limit values for φ (Fie) and ∆D are indicated asgiven in the previous section. It can be seen that the limit values aremuch smaller than the actual deformations at the end of the tests. Thelimit values were chosen such that at those stages a local deformationas indicated in Fig. 17 does not occur yet.

The bending moment capacity in test 51 (positive bending) was muchlower than in test 53 (negative bending), see also figure 3. The failuremode for test 51 was excessive ovalisation in the mid-section. Thefailure mode for test 53 was local buckling at one of the girth welds atabout 500 mm from the mid-section. The high bending moment incombination with the high-lows in the welds caused this local buckling.As stated before, these welds were not representative for girth welds inpractice.

400 400

M MC GE

400 400

M MC GE

Fig. 19: Positive bending for test 51 (left) and negative bending fortest 53 (right). In both tests D= 152 mm, t=1,65 mm P=0, 2α= 13,5°.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10

Test 51 Fie between C - G (rad)

M (

Nm

)

Mesured valuesCalculated with "Bend"

Fig. 20: Moment-angular rotation diagram for test 51, together withthe calculated diagram with the present analytical model. Alsoindicated with arrow, the limit value for the angular rotation ϕ-limit(Fie-limit).

97

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 5 10 15 20 25 30 35 40

Test 51 Delta D in mid section (mm)

M (

Nm

)

Mesured valuesCalculated with "Bend"

Fig. 21: Moment-ovalisation diagram for test 51, measured andcalculated results. Also indicated with arrow, the ∆D-limit (Delta D-limit).

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

0 5 10 15 20 25 30 35 40

Test 51 Delta D in mid section (mm)

Fie

(rad

)

Mesured valuesCalculated with "Bend"

Fig. 22: Relation between angular rotation and ovalisation for test 51,measured and calculated results. Also indicated with arrows the ∆D-limit (Delta D-limit) and ϕ-limit (Fie-limit).

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10

Test 53 Fie between C - G (rad)

M (

Nm

)

Mesured valuesCalculated with "Bend"

Fig. 23: Moment-angular rotation diagram for test 53, together withthe calculated diagram with the present analytical model. Alsoindicated with arrow, the limit value for the angular rotation ϕ-limit(Fie-limit). The failure mode was local buckling at a girth weld at 500mm from the mid-section.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 5 10 15 20 25 30 35 40

Test 53 Delta D in mid section (mm)

M (

Nm

)

Mesured valuesCalculated with "Bend"

Fig. 24: Moment-ovalisation diagram for test 53, measured andcalculated results. Also indicated with arrow, the ∆D-limit (Delta D-limit).

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

0 5 10 15 20 25 30 35 40

Test 53 Delta D in mid section (mm)

Fie

(rad

)

Mesured valuesCalculated with "Bend"

Fig. 25: Relation between angular rotation and ovalisation for test 53,measured and calculated results. Also indicated with arrows the ∆D-limit (Delta D-limit) and ϕ-limit (Fie-limit).

Deformation capacity.

The limits according to elastic design have been calculated andcompared with those according to plastic design on the basis of thepresent analytical models, see Table 6.

Table 6: Elastic design versus plastic design

Test Du(mm)

P(bar)

2α(degr) iy, iyp

φeℓ-lim(rad)

φpℓ-lim(rad)

φpℓ/φeℓ

51525354656655565759616258606364

152152152152152152152152122122122122122122122122

0707070707070700

13,513,513,513,519,019,019,019,013,513,513,513,519,019,019,019,0

8,155,708,155,70

10,007,40

10,007,407,055,807,055,808,607,158,608,60

0,00450,00550,00450,00550,00450,00550,00450,00550,00500,00600,00500,00600,00500,00600,00500,0050

0,0260,0260,0260,0260,0310,0310,0290,0290,0300,0300,0330,0330,0330,0330,0340,034

5,84,75,84,76,95,66,45,36,05,06,65,56,65,56,86,8

98

From Table 6 it becomes clear that the deformation capacity according toplastic design is much greater than according to elastic design, see φpℓ/φeℓin the last column.

CONCLUSIONS

1. The present analytical models enable a good description of the loaddeformation behaviour of statically loaded mitred bends in the elasticrange as well as in the plastic range. There is a good agreementbetween test results and the analytical models.

2. The behaviour of mitred bends is similar to the behaviour of smoothbends. Therefore, the analytical models that were developed forsmooth bends have been used as a basis for the present models formitred bends.

3. The strength and deformation capacity of mitred bends is much largerthan can be calculated with elastic theory. The present plastic analysismodels enable the calculation of the deformation behaviour until alimit state occurs.

4. The behaviour of mitred bends depends very much on the bendingdirection. In case of "negative bending", the bending moment can beso large that yielding of the adjacent tangents will occur. Localbuckling in the vicinity of the mitred bend may be the governingfailure mode.

5. An obvious prerequisite for the utilisation of the present models is, ofcourse, that the pipe material is ductile, not only in the plate material,but also in the welds. In view of the strain concentrations at the centreof the mitres, it is important to ensure overmatching weld metal. Thiswill lead to less strain in the less ductile weld zones. It is noted thatthis overmatching requirement is a general requirement for welds, butin this case of extra importance.

6. The rules as presented in this paper were used in NEN 3650 by theDutch Standards Committee for pipelines.

7. For practical application, the "Bend" computer model can be used(Diana Manual, 2001).

REFERENCES

ANSI/ASME B 31.8 (1979). "Liquid Petroleum Transportation PipingSystems," The American Society of Mechanical Engineers, NewYork.

Bond, MP and Kitching, R (1971a). "Multi-mitred and single-mitredbends subjected to internal pressure," Int. Journal of Mech. Sci.,Vol. 13, pp. 471 � 488.

Bond, MP and Kitching, R (1971b). "Stress and flexibility factors formulti-mitred bends subjected to out-of-plane bending," Journal ofStrain Analysis, Vol. 6, no. 4, pp. 213 � 225.

Bond, MP and Kitching, R (1972). "Stress and flexibility factors formulti-mitred pipe bends subjected to internal pressure combinedwith external loadings," Journal of Strain Analysis, Vol. 7, no. 2,pp. 97 � 108.

Gresnigt, AM (1986a). "Plastic design of buried steel pipelines insettlement areas," HERON, Vol 31, No. 4, Delft.

Gresnigt, AM (1986b). "Design rules for mitred bends in buried steelpipelines," (in Dutch). IBBC-TNO report BI-86-114, Delft.

Gresnigt, AM (1986c). "Test results of tests on mitred bends andcomparison with the design rules," (in Dutch). IBBC-TNO reportBI-86-124, Delft.

Gresnigt, AM and Foeken, RJ van (1995). "Strength and deformationcapacity of bends in pipelines," International Journal of Offshore andPolar Engineering. Vol. 5, Number 4, December 1995.

Jones, N and Kitching, R (1966). "A theoretical study of in-planebending of a single unreinforced mitred bend," Journal of StrainAnalysis, Vol. 1, no. 3, pp. 264 � 276.

Jones, N (1966). "On the design of pipe-bends," Nuclear Engineeringand Design 4 (1966), pp. 399 - 405. North-Holland PublishingComp., Amsterdam.

Kitching, R (1965). "Mitre bends subjected to in-plane bendingmoments," Int. Journal of Mech. Sci., Vol. 7, pp. 551 � 575, 1965.

Kitching, R and Bond, MP (1970). "Flexibility and stress factors formitred bends under in-plane loading," Int. Journal of Mech. Sci.,Vol. 12, pp. 267 � 285.

Markl, ARC (1952). "Fatigue tests of piping components,"Transactions of the ASME. April 1952, pp. 287 - 303.

NEN 3650, (1992). "Requirements for steel pipeline transportationsystems," Nederlands Normalisatie Instituut (Dutch StandardsInstitute), Delft.

Rodabaugh, EC, and George HH (1957). "Effect of internal pressure onflexibility and stress intensification factors of curved pipe or weldingelbows," Transactions of the ASME, Vol 79.

Rodabaugh, EC (1975). "Review of data on mitre joints in piping toestablish maximum angularity for fabrication of girth butt welds,"Welding Research Council Bulletin, no. 208, August 1975.

Sobieszczanski, Jaroslaw. (1970). "Strength of a pipe mitred bend,"Journal of Engineering for Industry. November 1970, pp. 767- 73.

Diana Manual. (2001). TNO - Building and Construction Research, Delft.

99