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Procedía Engineering 130 (2015) 57 - 66

Procedía Engineering

www.elsevier.com/loeate/procedia

14th International Conference on Pressure Vessel Technology

Stress Analysis and Engineering Design of Reducer Bend under

Internal Pressure

S.-Y. Chen3 *, J. Chena, C.D. Liub

aThe Challenge Petrochemical Machinery Corporation of Maoming, Maoming, Guangdong Province 525024, P.R. China bInstitute of Pressure Vesture & Equipment of East China University of Science and Technology, Shanghai 200237, P.R. China

Abstract

Standard reducer bends has structure function of both bend and reducer pipe at the same time, but there's only few engineering application and relevant report on it. Although lacking in relevant strength design standard, it still has obvious advantage in some individual case. By comparison of analytic formulas of circumferential or longitudinal thin membrane with that of similar structure pipe specified in ASME B31.1-2012 Power Piping and ASME B31.3-2011 Process Piping separately under internal stress, we know that these two have almost the same structure form and affected factors. In order to carry out internal pressure strength design of reducer bends, by comparison and analysis of analytic formulas of circumferential and longitudinal thin membrane under internal pressure and test stress measurement under internal pressure, the result shows that analytic formulas value is apparently larger so that it is conservative to be applied in engineering. This error arises from the difference between ideal model and actual pipe. Actually, reducer bend is belonging to neither the thin wall structure nor the rotated shell of axial symmetry, but a kind of hyperboloidal shell with 2 main curvatures changed gradually complicated. Therefore, it is not appropriate to analyze such kind of reducer bends with the thin membrane theory in thin wall and axial symmetry structure. The reducer bends strength formula derived directly from circumferential stress formula of reducer bends under internal pressure is also conservative. The reducer bends fabricated with the new technology combined with half seamless and half seam has sound quality, and the internal pressure strength is sufficient to meet requirements of fittings used in engineering. ©2015PublishedbyElsevierLtd. Thisisan openaccess article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICPVT-14 Keywords: Strength design; Stress analysis; Reducer pipe; Reducer bends; Elbows

* Corresponding author. E-mail address: sys@cpmm.cn

1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

doi:10.1016/j.proeng.2015.12.175

Nomenclature

p Internal pressure, 4.0 MPa is adopted in engineering case in fig 1, 12MPa and 18MPa are selected separately for testing analysis.

6 Radial return angle, according to the testing stress analysis location, every 10 ° is determined to get the values and conversed into radian. Maximum returning angle that shall be wrap angle of the reducer bends, here is 90°according to fig.3.

$ Included angle between bends radius of the cross section and neutral axial in circular segment are

corresponding to the measure location of testing stress analysis, 0ois being in neutral line, 90ois being in extrados and -90ois being in intrados. # Angle in bottom cone of the cross section, here sin a « 1 is adopted. r\ Middle radius of large cross section of the reducing elbow, 78.5mm r Middle radius of small cross section of the reducing elbow, 39.4mm L Axial length of the reducer bends, 355mm R Bends radius at centerline of bending components, 226.2mm rh Inner radius of the large cross section, 73.75mm

t Wall thickness of the pipe components, values are obtained from measuring location with corresponding testing stress analysis.

1. Introduction

Bends and reduces are normal pipe components used widely in pressure piping such as industrial piping, utility piping and long distance transmitting piping. Although reducer bend is also the standard pipe [1, 2] and has structure function ofboth bends and reducer bends, it is not often used in engineering and there's few report about it. Figure 1 shows the complicated pipe component connection in some industrial plant of which top elbow is connected with the bottom small elbow by eccentric reducer bend and nipple.

Fig. 1. Complex piping components connection. Fig. 2. Model of bend reducer with tube on both ends.

In order to improve connection among them, the previous 3 pipe components need to be replaced by the reducer bends shown in Fig. 2.

However, there's no corresponding reducer bends strength design standard so that it can only be verified by internal pressure test of the test sample, and there's also some analysis shows that the pipe spoon used to the closing end of the pipe component in standard specified testing is too short at times and it plays a strengthening role in the pressure resistance capability, therefore, there's some uncertainty of this testing method. The minimum wall thickness of bends specified in part 102.4 of ASME B31.1-2011 [3] or reducer specified in paral04.6 were also borrowed from Minimum Wall Thickness of straight Pipe under Internal Pressure in the standard.

Based on the relevant theory, strength design formulas of the reducer bends is inferred, bearing internal pressure stress is analyzed to provide a basis for strength evaluation of the reducer bends used in the case of the Fig. 1.

2. Reducer bends internal pressure analysis

2.1. Parameter of the reducer bend

For convenient comparison, testing object is taken to be the accordance of stress analysis for the reducer bends under internal pressure, basic parameter of the testing sample see table 1. Based on the similar test sample strength, material strength of steel 20 is conversed according to the experienced formulas ab « 3.5378 HBMPa in China standard GB/T 1172-1999 Conversion of hardness and strength for ferrous metal.

Table 1. Main parameters of the test sample.

Bending radius

mm R mm

^ 168x^89x9.5 226.2 166.5 9^5

outside shell outside shell Axis Large end Small end

diameter thickness diameter thickness length

oflarge of large of small of small Hardness Tensile Hardness Tensile

end end end end strength strength

Di mm Ti mm D2 mm T2 mm mm HB MPa HB MPa

166.5 9.5 91.7 12.9 355 109 385 123 434

2.2. Stress analytic formula under internal pressure

The author in [4] proposed circumferential stress formula of the reducer bends without moment under internal pressure based on the 2 characteristics: cross section radial of the reducing bends changes as the radial bending angle changes, its conical structure, that is

p\nr - 2(r. - r,)0] 2nR + [nr. - 2(r. - r,)0]sin^ o* --X-

2tn sin a

tR + [ar, - 2(r, - r2 )6~\ sin ^

I r2 Small end (a) Side section ofreducing elbow

Extrados

Intrados

(b) Radial cross-section

Fig. 3. Circumferential stress analysis model ofreducing elbow.

The author proposed longitudinal stress formula of the reducing bends in [5] that the radial return angle 0 is taken to be reducing variable, can be obtained without moment:

here, internal radius of the cross section for reducing bends with radial turning bending angle 0 is: 2 r

r„(l---l- pff) (3)

n r„

Among which reducer angle of non-dimensional coefficient reducing angle is: r, - r2 157 - 78.8 ACA

=_iir- - °'50 (4)

Meanwhile, it is pointed out that for normal standard reducer elbow and thick wall reducer bends, the error arising from formula (2) is not more than 6.0% with the calculation result conservative, the error is within the engineering accepted scope.

Formulas for longitudinal stress in ASME B31.1-2012 paras.102.2

* = ^ <*>

and formula (2) have the similar structure form and influenced factors. Here P indicates design pressure, Do indicates external radius and tn indicates bend wall thickness.

By comparison of formula (1) with formula (2), cr, =■ 2<y6 happens at the centerline and has a same relation with

that of the cylinder under internal. But formula (1) and (2) haven't been verified by testing. Bring the relevant data into formula (1) and simplified, we can obtain the longitudinal stress:

(73.75 - 0.43440) 2x 226.2 + (73.755 - O.43440)sin0 p

o, =-x----(T^

' 2 226.2 + (78.5 --O.43440)sin0 t y '

In the spot of the intrados, there is

(kw mma 189.325 + 0.21720 p

a = (36.875 -O.21720X-x— (7)

73.85 + 0.21720 t v '

At the neutral line, there is

^ o. = (73.75 - 0.43440)-p (8)

In the spot of the extrados, there is

a = (36.875 -O.21720X 263-°75 ~°'2172g xP (9)

152.35 - 0.21720 t v '

Bring the relevant data into formula (2), circumferential stress is simplified as

at= (36.875 - 0.2180418750)P (10)

Draw a stress distribution curve of the testing value and the value by calculating formula (6) to (9), and then make a comparison.

2.3. Strength calculation under inner pressure

(1) Strength calculation formula. Based on the current existing analysis, longitudinal stress is the maximum stress of the round cross section, and longitudinal stress in the inner intrados ($ =90°) on the large end of reducer elbow is maximum which is the key of deterring strength of the reducer elbow. It is a little risky to employ rx « ru, theoretically, more than 73% is thin wall in reducer elbow standard, error from this replacing is usually 3%, therefore, it is conservative when taking sin a «1, error can be counteract with each other. Bring this simplified value into formula (1) and sort out as the following:

pr 2R - r.

^ -!L (11)

2t R - r„ K '

Suppose the maxim circumferential stress is not exceeding the allowable stress of the reducer elbow at the design temperature, the above formula can be obtained:

pr, 2R - r. ,

IT Tt * ' <12>

Considering weld efficiency p , reducer bends wall thickness strength calculation formula can be sorted out as: pr 2R - r.

t >—^----("13)

2M> R - ru (li>

If formula (13) conversion is expressed in external radius, with consideration of wall thickness additional amount, minimum wall thickness calculation formula of the elbow with internal pressure has a similar structure form and affected factors as para.102.4.5 ofASME B31.1-2012.

t =-O-+ A (14)

" 2(SE /1 + Py) K '

In formula (14), P is the design internal pressure, Do is the pipe OD, SE is the maximum allowable stress in material due to internal pressure and joint efficiency at the design temperature, psi (MPa), Y is the coefficient less than 1, in the intrados area I >1, in the extrados area I<1, in the centerline 1=1, tm is the required minimum wall thickness, A is the required additional surplus wall thickness considering factors of the pipe components machine or medium corrosion.

(2) Example of calculation. Bring parameters of the above case into formula (13), among which allowable stress shall be 147MPa as per GB150.2-2011 at the temperature of 100°C, the welds shall be full 100% penetration testing, so that <¡»=1.0, compared with the actual wall thickness 9.5 on table 1, there's still 7.05mm corrosion allowance of the required wall thickness

4.0x73.75 2x226.2-73.75 „ _

t =--~ 2.49 mm

2 xl47 x 1.0 226.2 - 73.75

(3) Engineering solution. Regarding to reducer bends fabrication technology, there are 6 kinds forming technology: the 1st one that can be applicable to straight pipe with smaller reducing degree and hot bespoke-pulled

by one step; the 2nd one that can be applicable to larger reducing degree pipe bespoke-pulled step by step; the 3rd one that can be applicable to assembly welding by half pressing in lower work condition; the 4th one is the hot bespoke-pull method of eccentrically reducer bends as the pipe blanket; the 5th one is the combination of hot bespoke and assembly welding and the 6th one is 3D printing method. But no matter what products of which method, there's no corresponding strength design standard till now. Usually, longitudinal welds are not permitted to exist in elbows or reducer bends in chemical and industrial plant, strength of the reducer bends formed by electric arc additional manufacture printing can not meet requirements of the engineering, but the cost of hot bespoke-pulled to blank by model is much higher. Currently, one applicable method is that blank is hot pressed to be elbow by the traditional mature technology firstly, then open a narrow and long dovetail groove in the extrados at one end of elbow, remove part of the wall thickness, see fig.4, and then hot pressed for second time to close dovetail groove opened, and at last weld the opening. A kind of half seamless and half seaming combination structure can be manufactured by this new technology.

Fig.4. Dovetail opening in the wall of elbow.

Weld location design. Formula [13] does not consider the reducer bends under combined loading such as internal pressure, moment and torque at the same time. Under different loads, the maximum stress that determines the strength of reducer bends is in different positions, and stress constituent and its formula have differences, the product forming process and the weld position are also different, in this case, the strength calculation formula is also different. For example, a dovetail groove should be opened in the neutral line of reducer bends that mainly bearing moment in bending plane.

Weld heat treatment. In engineering, except well designed weld bevel, different technical method can be employed in closing weld of different material. PWHT can be waived for carbon steel weld; its higher strength and hardness maintained are useful for resisting abrasive corrosion for flowing fluid. Regarding to stainless steel welds, solution treatment shall be done to recovery its good corrosion resistance; Cr-Mo steel weld shall also be PWHT to relief its residual stress and to refine grains.

Equal wall thickness treatment. With regard to seamless reducer bends, hot bespoke-pulled forming is a diameter enlarging process of small diameter pipe blank, wall thickness will decrease gradually, reduction of wall thickness is increasing as the diameter become larger, using the method that gradually reduce the wall thickness of blank before forming or the method that proceed grinding treatment after forming, both of them can obtain reducer bends with equal wall thickness.

3. Testing stress analysis of reducer bends under internal pressure

3.1. Test plan

Specific dimension and actual measurement hardness values of the test samples are shown in table 2. Ultrasonic probe parting surface were placed paralyzed and vertically to elbow axis separately to get two readings, and calculate their average value, their difference is less than 0.5%, and the latter is a little larger. Wall thickness in the table is that measured when probe is vertical to axis of the elbow. Cross section number and letters orientation of equal division points are shown in fig5. ID in AE orientation is larger than that of CG orientation, but OD is slight smaller than that of CG orientation. The result shows the ovality of the cross section.

Table 2. Actual measuring parameter ofreducer elbow test coupon.

Cross Sec. Hardness at A Hardness at E ^ 6?,Ua', , Excircle roundness

, ,■ divisionpointsFrom A tlr „ .

wn neutral line /HB neutral line /HB Im-m of cross-section / %

• toH/mm

1 104.7 112.7 9.5 166.5 0.121 0.7

2 106.3 103.3 10 159.5 0.134 /

3 100.7 113.3 10.38 149.6 0.149 /

4 110.3 107 10.53 144.8 0.157 /

5 115.3 118 11.28 134.3 0.183 /

6 / / / (130.6) / 2.2

7 107.3 116.3 11.83 125.1 0.209 /

8 113.7 120.3 12.7 115.9 0.246 /

9 120.7 125.7 13.73 108.2 0.291 /

10 118.3 127.3 14.03 99.3 0.329 /

11 / / 12.9 91.7 0.327 0.6

Avg. value 110.8 124.2 11.69 117.7 0.221 /

o/C\ / j j A

(a) Radial cross section sequence

(è) 8 equal division points on elbow cross section

Fig. 5. Cross section and equal divisionpoints.

(2) Testing system. Strain would be measured by automatically scanning via programmable controlled YJ—33 static electric resistance strain instrument data accumulating system. Pressure increasing instrument of the bend is with the model 2D1—SY pressure test pump. Generally, each measuring point shall be stuck with a 90° rectangular electrical resistance strain disc with 2 axis and measure the longitudinal strain and circumferential strain, see Fig. 6.

Fig. 6. Test sample.

Strain value measured directly by bi-direction strain disc can be used to calculate principal stress via generalized Hook's law in generalize [6]:

E{£t +V£e)

3.2. Test results analysis

(1) Test results. The results of strain test are converted into stress by formula (15) and (16), after that stress curves will be drawn. The circumferential and longitudinal stresses at neutral line see Fig. 7 and Fig. 8. During stress calculation, according to standard, Poisson's Ratio v=0.3 and spring modulus E=1.96E5MPa will be taken.

(2) Corresponding stress curve analysis. Actual measurement value of longitudinal stress: when internal pressure is increased, longitudinal stress curve becomes abrupt; longitudinal stresses are all tensile stress, in terms of the level of stress at the maximum internal pressure, sequence of the maximum longitudinal stress from high to low are in intrados, neutral line and extrados separately. Stress curves concentration degree under different internal pressure from high to low are in intrados, extrados and neutral line respectively.

35 30 25 20 15

• 4MP a

—A— 12MPa

4MPa^ (7)

- Large end smal 1 end

20 40 60 80 100 Circumferential corner 0/

20 18 16 14 12 £io

° 6 4 2 0

—4MPa

—*— 12MPa

-1-4MPa^(2)

1 large en# >usmall

20 40 60 80 100 Longitudianl corner 9/

Fig. 7. Longitudinal stresses curves at neutral line. Fig. 8. Circumferential stresses curves at neutral line.

Actual measurement of circumferential stress: circumferential and longitudinal stress curve have the same tendency. Level of longitudinal stress has not reached 2 times that of the circumferential stress. By comparison of circumferential stress of different longitudinal 1 lines, maximum, intermedium and minimum circumferential stress exist in intrados, neutral line and extrados respectively

Actual measured stress and stress analysis: no matter whether it is longitudinal or circumferential stress, analytical values are apparently 3 to 6 times higher than that of actual measured stress which shows that analytical stress values is obvious conservative.

3.3. Errors analysiss

The analysis shows that except the unclear reason, the above errors are mainly affected by the following aspects: (1) It is assumed that stress of pipe components during stress analysis inferring process is the thin membrane stress without moment, but actually, reducer pipe test sample is with a thick wall structure. Some scholars found out that when t/r is more than 0.05, thin shell theory will not cause the unacceptable engineering error which is verified by lot comparisons and trial calculation[7]. In fact, except subjecting to internal pressure, engineering pipe components are also subjecting to the effect of moment and gravity, and the wall thickness is increased, according to the result in table 2, t/r is far larger than 0.05, and as much as 342%, radial stress of the wall thickness under internal pressure can not be ignored.

(2) In fact, the reducer bends is an axial dissymmetry structure, bending moment exists in wall thickness under internal pressure. With regarding to circumferential bending moment, in one hand, it comes from area difference of pressure between extrados and intradós of the neutral line[8]; on the other hand, it comes from ovality of inner and external circle of the pipe cross section, and long and short ovality of inner and external round are not conforming and vertical to each other.

Longitudinal bending moment comes from area difference of pressure between extrados and intradós of the neutral line on one hand[9], on the other hand, it comes from area difference pressure of large end and small end section. Actually, reducer bend is belonging to neither the thin wall structure nor the rotated shell of axial symmetry, but a kind of hyperboloidal shell with 2 main curvatures changed gradually. Moment of the shell under internal pressure can not be ignored, but its mathematic calculation is extremely complicated. Therefore, it is not appropriate to analyze such kind of reducer bends by the thin membrane theory in thin wall and axial symmetry structure.

(3) According to result in table 2, cross section roundness of reducing bend pipe test-piece is up to 2.2%, there's only roundness tolerance for end of pipe component in relevant standard, but there's no roundness deviation requirement for the middle part of the elbow. So there should have been 0 - 0 °and 90° in the circumferential stress formula, just when a - 90° and sin a = 1. But, now sin a «1 in all cases without distinguishing and that is one of the errors.

4. Conclusions

(1) Stress analytical formulas value and testing stress measurement result under internal pressure are not in compliance. The Analytic formulas value is apparently larger so that it is conservative to be applied in engineering. This error mainly resulted from the total stress from thin wall membrane analysis of the formula and moment reaction of the actual measurement. This moment mainly results from the shell thickness and the dissymmetrical bi-hyperbolical shell, its mathematic calculation is extremely complicated so that its complete stress analysis is still expecting scholars who are interested in it.

(2) Based on the circumferential stress formula of the danger cross section of the reducer elbow under internal pressure, strength calculation formula of wall thickness is induced directly. Testing result shows that wall thickness calculation formula is conservative as the stress formula conservative, but it can be applied into engineering management design.

(3) Reducer bends fabricated with the new technology combined with half seamless and half seaming has a very good quality, and the strength under internal pressure is adequate so that requirements of the engineering bends can be well satisfied.

References

[1] Standard GB/T 12459-2005 "Steel Butt-welding Seamless Pipe Fittings".

[2] Standard ASME B 16.9-2012 "Factory-made Wrought Butt-welding Fittings".

[3] Standard ASME B31.1-2012 "PowerPiping".

[4] Sun-yi Chen, Cengdian Liu, Chen Jin, He Luwu. Circumferential Membrane Stress of Bend Reducer, Pressure vessel technology, 2007, 24(2): 35-39. (In Chinese)

[5] Sun-yi Chen, CengdianLiu.MeridianMembraneStressofReducingElbow.Journalofplasticityengineering, 2005, 12(2): 48-51. (In Chinese)

[6] Zhilun Xu. Elasticity (the 4 edition, and the second half volume). Beijing: Highter Education Publishing Company, 2006: 167-168, 270. (In Chinese)

[7] China Standardization Committee on Pressure Vessels. Explain on JB 4732—95 Steel Pressure Vessels—Design by Analysis. Beijing: ChinaStandard PublishingCompany, 1995:99-106.(In Chinese)

[8] Chen Sunyi, Liu Cengdian, Chen Jin, He Luwu. The Force Difference and its Circumferential Equi-moment Induce by the Inside Surface Difference Between Flank-extrados and Flank-intrados of Loop Under Internal Pressure. Proceedings of the Seventeenth National Conference on Structural Engineering,2008. (In Chinese)

[9] Chen Sunyi, Liu Cengdian, Chen Jin, He Luwu. The Force Difference and its Meridian Equi-moment Induce by the Inside Surface Difference Between Flank-extrados and Flank-intrados of Loop Under Internal Pressure, Pressure vessel technology, 2007, 24(7): 21-26. (InChinese)