r

Stress Analysis of Conical Pipe-reducers

M A Rahman, Non-memberG M Zulfikar Ali, Non-memberMd R Khan,'Non-memberMd W Uddin. Non-member

NOTATION

C , D extensional rigidity, Eh; bending rigidityEh3 / [ t2 ( r -u ' ) J

I - -

( 1 - v ' ) x s / r , t / \ 1 2 P T 2 R ( t - v 2 ) i

Young's modulus. Foisson's ratio, respec-tively

reducer wall thickness, R/lz

radial and axial components of stress resul-tants, respectively

H/PR, V/PR

circumferential and meridional curvaturechanges, respectively

kg xe, kx xe

circumferential and meridional couple resul-tants, respectively

Me/PRh, Mx/PRh

rneridional and circumferential stress resul-tants, respectively

Nx/PR, Ne/PR

external pressure, PlE, xe/R

radii of the smaller and larger pipes respec-tively, connected by the reducer

radial distance of points on the undeformedmiddle surface from axis of symmetry, ro/xs

radial and axial displacements, respectively

uEh/PRz, wEhlPRz

distance measured along meridian, x/xs

total meridional length of the reducer meas-ured from its apex.

shell parameter, $1y - $, r1y* u

MA Rahman,GM ZulfikarAli, Md R Khan and Md W Uddin arewiththe Department of Mechanical Engineering, Bangladesh University ofEnginecring and Technolopr, Dhaka, Bangladesh.

This paper was received on November 7, 1996. Written discussion on thepaper will be received until January 3l, 1998.

This paper deals with the stresses in the conical pipe reducers, used for connecting pipes of unequqldiameters, under unifurm internal pressure. Computer progranx for stress analysis of general compositeshells, with necessary modifications, has been usedfor the present analysis. In this analysis, governing non-tinear equations-for axisymmetric deformation af conical reducers are solved by the method of multisegmentintegration, developed by Kalnins and Lestingi. The discrepancy between the linear anri the non-lineartheories in predicting the stresses in the reducers is discussed in the anaiysis. The effict of variation of theapex angle and the thickness ratio of the conical reducers on :he stress distribution is also discussed in thispaper.

Ke.ywords.' Stress analysls. Pipes, Conrcal reducers

y : semi-apex angle of the reducer

€x, Go : middlesurfacestrains

7 ,DE , v

€ x , € 0

6r r Oa

: e x Ehxe/ PR', eg Ehx./ PR"

: circumferential and meridional stresses re-spectlvely

(rci, oco, ; circumf"erential and meridional stresses at the

oai, oao inner and outer fibres, respectively"

0e, $ : angle between axis of symmetry and normalto undeformed and deformed middle sur-faces, respectiveli

INTR.ODUCTION

Stresses in the pipe reducers, used for connecting pipes ofunequal diameters, are completely different from those in thepipes under the same internal pressure. It is, thus, esseritial toanalyze the stresses in the reducers separately from those rnthe pipes. Further, use of reducers always involves the use offlanges for their correciion with the pipes. These flangesalways introduce additional stresses in the pipes as well as inthe transitional reducer elements. The present analysis thusinvestigates the stresses in the pipe reducers for varyingparameters"

It has been shownr'3-tr that the non-linear theory of shellsare essential for analyzing the distribution of stresses in thepressure vessel problems. This is due to the fact that, at thejunctions of two different segments of shells or at pointshaving small meridional radius of curvature, the linear theoryfails to accou'nt for the pronounced change in curvature andconsequently predicts unrealistic solutions, This fact becomesmore evident at higher loading and for thinner shells3'4.Soluiions are, thus, obtained for reducers of different thick-nesses, using both l inear and the non-linear theories in thepresent analysis, so that the shortcomings of the linear theoriesin case of reducers are verified and noted.

The multisegment method of integration developed byKalnins and Lestingir is used here to solve the non-linear shellequations for the conical reducers, This multisegment integra-tion technique is a very powerful tool for handling nonlinearshell equations where both the finite element and finite differ-

H , Vkg, kt

-Ko,k,

Mg, M,

h ,7I:1, V

fuIr, fuI*

Nr, Ne

lrr, lge

P , F , NRr, R

f,}, lo .

c [ , p , r

U , W

i , w

x r i

Xs

t45Vol78, Novemher 1997

i i - 't .

:Td techniques usually fail due to nonconvergence in theipration process of these techniques in case of boundary value

6problems. This method can be used for shell meridian of anylength with discontinuity in siope or thickness where methodlike direct integration fails.

GOVERNING EQUATIONS

Reissner's2 nonlinear equations for axisymmetric deforma-tion of shells of revolution, as modified by uddin3, is used inthis analysis. The governing equations are given under:

u€ 0 = =

f11

0 = 0 0 - F

Ee = (sin fu - sin Q)/ Tu

I / ' = F c o t O + V s i n S

e = e N, - vEe- M x / D * k w

= (tt =

= D ( i e

= a cos Q (Me - Mx)/7 -

s, F T' (r/ sin o - tcos Q)The above mentioned governing equations include approxi-mations of the Kirchoff hypothesis and other plausible as-sumptions, like the middle surface strains are negiigiblecompared to unity, leading to the dropping of terms of theorder of hlR in the constitutive equations. However, theseassumptions do not introduce any significant error in thesolutions of most practical thin shell problems.

BOUNDARY CONDITIONS

For the general case of axisymmetric deformations of shellsof revolution, it was shown3 that the boundary conditions atthe edges require specification of :

H o, i, fuI* or F, uno 7 or w

d Md ;

( 1 )

(2)

(3)

{4)

(s)(6)

(7)

(16)

(r7l|

vE-)te

+ v&r)

k.

No

Ue

a , = L * € - r

r = L f o * H

s inQ - Zs in fo

cosQ - Zcosfo

d cos Q O/T - FV)

a { tacosQ - t ' te ) /7 +F Ts inQf

considering both the larger and the snnaller diameters of thereducer sufficiently large and connected to the reducersthrough flanges, it is quite justified to assume that both the

def o r rned

Fig2(a) Side view of element of sheli In deformed andundeformed state

Fig 2(b) Element of shellshowing stress resultants andcouples

(8)

(e)

d w- = C xo x

d u: - : - go x

d B ;f r = r ,d vd x

d Hd x

(10)

( 1 1 )

(r2)

(13)

(r4)

(1s)

Y

Flg 1 Mlddle surface of shell

146

7,-unde f orrned

1i4 . i ,

t lt lt l

I t

iu

IE(I) Journal - MC

ends of the reducer are fixed or clamped. Hence, conditionsat both the larger and smaller ends of the reducer are specifiedas

i l = A , 9 = 0 , w = 0 (18)

SOLUTION

The same method of multisegment integration as used byUddin3 for nonlinear analysis of pressure vessels has beenemployed with boundary conditions given in equations (18)"

The program starts with an initial arbitrary load F and an

incremental load step A F, and then solves the nonlinear

governing equation forload F with apreassigned convergence

criterion. The load F is then increased by the incremental load

step A F and solution is then obtained for this new load,through iteration, with solutions of immediate previous load

as initial values.If the solution fails to converge at any load F,

Fig 3(a) Gonlcal reducer (parameters : R/]r = thickness ratio,RrlR = reduction ratio, 1= seml-apex angle of the conicalreducer)

X=0.30

Fig 3(b) Parabolic reducer (parameters : R/lr = thlckness ra$o,Rr,/R = reduction ratiorr)

the load step is halved and the solution is again attempted.In this way, nonlinear solutions are obtained at increasedvalues of the loading parameter upto the desired level ofloading.

RESULTS AND DISCUSSION

In order to point out the dgficiencies of the results of the lineartheories, the results of both the linear and the nonlinear

Vol78, November 1997

theories are shown in the same figures. It should be mentionedhere that a diameter reduction ratio of 0.5 is considered in thisanalysis, iei = 1.0 corresponds to the larger end of the reducer

while x = 0.5 corresponds to the smaller end (Fig 3). Fromthe results as shown in Figs (4 - 8), it is evident that the lineartheory is very conservative in predicting the results. It is alsonoted that the conservativeness in estimating the stresses,stress resultants and the moment resultants increases with theincrease of thinness. Uddin3 pointed out that for shells sub-jected to internal pressure this conservativeness of the lineartheory also increased for increasing values of loading parame-ters, although, for avoiding the crowding of too many results,it is not shown in this analysis. However, it is observed thatthe prediction of nonlinear theory about the meridional stress

resultants, Nr, ir higher than that of the linear theory, whichis quite contrary to the observations in other composite shellproblems.

It should be pointed out here that the stress parameters are allnormalized in terms of the loading pararneter" As a result,there is no variation of the normalized parameters with theloading for linear theory [Figs (4 - 8)].

The most interesting observation in the present analysis is theeffect of the apex angle on the predicted results. It can beobserved from the Figs (4 - 8) that the discrepancy betweenthe linear and the nonlinear theories in predicting the resultsbecomes more prominent for reducers of higher apex angle(90') than thobe of smaller apex angle (60').

f >

x

Fig 4 Effect of apex angle and thickness ratlo on meridionalbending moments in the reducer

l>

Rr /R=0 .5

Rr /R-0 .5

Nonlrnear--- Lrnear

R/h = 300

147

l z

Flg 5 Eltct of apex englc md thlckness ratlo on mcrldlona!stresc resultants In thc rrducor

.5 .6 .7 .B .9. 1.0X

Fig 6 Effect of aper angie and thickness ratio on circumferen-tial stress resultants in the reducer

r48

Flg 7 Effect of apex angle and thickness ratio on meridionalstresses in the reducer

t,This is because of the fact that the cone approaches to a plateas the apex angle is increased, losing its membrane stiffness^Consequently, conical reducers with higher apex angle de-forms substantially under load, resulting in noticeable dis-crepancies between the two theories in predicting the results.

Fig 4 shows the variation of the bending moments along thesheli meridian. The presence of bending moment, in effect,shows the disturbance in the membrane solutions of the re-ducer. In other words, the magnitude of the bending momenris ameasure of the deviation from the characteristic membranebehaviour of shells. The discrepancy between the linear_ andthe nonlinear theories in predicting the solutions is observedhere to increase with increasing thinness and increasing apexangle.

The meridional stress resultants are shown in Fig 5. As anexception, it is observed that the linear theory predicts lowervalues of the stress resultants in comparison to that of thenonlinear theory, {uite contrary to the normal expectations.

The circumferential stress resultants are shown rn Fig 6. It isclear from this figure that the linear theory overestimates theresults compared to nonlinear theory. Further, it is observedthat this stress resultant increases as apex angle of the reducerincreases and its maximum value occurs in a zone very closeto the flange at its larger end.

The variations of the meridional stresses at the inner and theouter surfaces along the reducer meridian are shown in Fig 7.The presence of bending moments at and near the two fixedends are responsible for the wavy distribution of the stressesnear the two ends. The maximum meridional stress in the

-c

E.o_

o

btl

l b -

c

Eo_

6( > h

tlr . o

i

P/E = 3x 10-5

v = 0.30

" t t r - 1

tr = '+..t

Q / h - ? n nr r r r r - u w v

n \\ \\ \

t \ \

\r\\P/E = 3x 10's

v = 0.30

Y =450

IE(I) Journal - MC

reducer occurs at the inner face of its junction with thelarger-end flange and its value is about 5.2 times the mem-brane mendional stress in a cylindrical shell of radius fr andthrckness &"

The circumferential stresses at the inner and the outer surfacesof the reducer are shown in Fig 8^ Here also" it rs observed thatthe prediction of the stresses by the linear theor,v is mucllhigher than that by the nonlinear theory.

Observation from the stress curves shown in Figs 7 and 8,

reveals that the meridional stresses at the rnner surtace (oaii

rs the rnost critical stress. It is also seen that the reclucer iscritically stressed near the two ends.

Rahman'i studred the stresses in a paraLrolic reducer tFig t 3b)lsubjected to internal pressure. The result of the analysis has

. 5 6 . 7 B . 9 1 . 0x

Flg 8 Etfect gf apex angle and thlckness ratlo on clrcumferen-tial stresses In the reducer

.25 ;

.50 .75

Fig 9 Meridional stres""i tor a parabotic reducerll

Vol78. November 1997

, \ lt\\\\l

P / E = 3 x 1 0 s

Fj/h = 300

v - 3 0

.25 .50 .75 1-00X

Flg 10 Clrcumferentlal stresses for a parabollc reducerlr

shown that, for the same diameter reduction ratio (0.50) andthickness ratio (300), the stresses in a parabolic reducerare much lower in magnitude than that in a conical reducer

[Figs (9 and 10)] at the same intensity of loading. From Fig 9,the maximum non-dimensional meridional stress at the innersurface is only 1.4 in case of a parabolic reducer, whereas itis as high as 2.6 in the identical conical reducer (Fig 7).Similarly, the circumferential stresses are also of muchsmaller magnitude in a parabolic reducer (Fig 10) than that inan identical conical reducer (Fig 8).

Comparing the conical reducers with their counterpart para-bolic reducers, it is observed that the parabolic reducers arefar superior to their conical counterparts in respect of stressesin them forthe s:rme internal pressure. Of course, stress shouldnot be the only factor in concluding the superiority of theparabolic reducers over their coniial counterparts. For exam-ple, factors like ease of fabrication and loss of energy by fluidflowing through them should also be taken into considerationin evaluating the ultimate superiority of one kind of reducerover the other.

CONCLUSIONS

It is concluded that the nonlinear theory is essential for ana-lyzing the stress problems of reducers, specially of thinnerreducers. In most of the cases, the linear theory fails to accountfor the effect ofchange in curvature and consequently over-estimates the stresses developed in the reducers under uniforminternal pressure. The pipe reducers are critically stressed atand near the flange connected ends because ofthe presence ofthe bending moments. An important observation of this workis that the more the apex angle the more is the discrepancybetween the linear and the nonlinear theories in predicting theresults. The result of this analysis has also revealed the factthat, between a conical and an identical paraboli.c reducer, thelatter one can withstand much hither internal pressure.

REFERENCES

l. A Kalnins and J E lrstingi. 'On nonlinear analysis of Elastic Shells ofRevolution.' Joumtl of Applied Meclanics Transactions of ASME, vol 34,1967, pp 59-64.

2. E Reissner. 'On the Theory of Thin Elastic Shells.' H Reissner AnniversaryVolumc, J W Edwards, Ann Arboq Michigan, 1949,p 1231.

3. Md W Uddin. 'L-arge Deflection Analysis of Composite Shells of Revolu-tron.' PhD Thcsis, Carleton University, Canada, 1969.

4. Md W Uddin. 'l,arge Deformation Analysis of Ellipsoidal Head PressureVessels-' Composites and Structures, vol 23, no 4, 1986, pp 487-495.

c.

oco_

il

f b '

..\- , L

l c

_co:o-

o

oI I

l b '

l:\r CL

l-)

r o

-C

CE.

€t)t l

r t )

-10

ocl

oc0

P/E = 3x

v = 0 3 0

rn -5,u R/h = 300

v - l t r O[ - - r w

oci

o

P / E = 3 x 1 0 5 oR/h = 300

al

v = 0 3 0

1 0 0

r49

e J - r ' r d F r r - r r

f-F

fld w Uddin. 'A computer Program for Nonrinear Analysis of prcssure

f^e1yls.' Interrutrittnul Jourrutl of pressures ve.rsehr and piping, vol zz,r1986, pp27l-30F.

6' P Dutta. tstability and stress Analysis of roroidal pipe Reducers.' M scEnlg ine e ri n g T he s is, B un g tade s h u niv e rs ity t tf En g i ne e ri ng uwl Te c hru fu t g y,Dhaka, 1996.

7. I Famili and R R Archer. 'Finite Asymmetric Deformation of shallowSpherical Shells.' AIAA Journal, vol 3, 1965, pp 506-520.

8. G A Thurston. 'Newton's Method Appried to problems in Nonlinear

Mechanics.' lournal of Applied Meclunic.s, June l 965, pp 3g3-3gS.

9.. E L Reiss, H J Greenberg and H B Kelley. 'Nonlinear Deflertrons ofshallow Spherical shells under Extemal pressure.' kturnal cf Aerospu:eScience,.Series 24, 1951, pp 533-5a3.

10. R E Ball. 'A Geometrically Nonlinear Analysis of Arbitrary lradedshef f s of Revol ution.' Natiorutl A e rcrutut i c s,f spuc e A ge ncy, cR gug, I g sg.

I l. M A Rahman. 'stability and stress Analysis of parabolic pipe-re<tucer.'M sc Engrneeiing Tlusi.s, Bangkdesh university tf Engineeirg and rech-nobgy, Dhaka 1000, Bangladesh, 1994.

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