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M. D. Xue

D. F. Li

K. C. Hwang

Department of Engineering Mechanics,Tsinghua University,

Beijing, 100084,People’s Republic of China

A Thin Shell Theoretical Solutionfor Two Intersecting CylindricalShells Due to External BranchPipe Moments A theoretical solution is presented for cylindrical shells with normally intersectingnozzles subjected to three kinds of external branch pipe moments. The improved doubletrigonometric series solution is used for the particular solution of main shell subjected todistributed forces, and the modied Morley equation instead of the Donnell shallow shellequation is used for the homogeneous solution of the shell with cutout. The Goldenveizer equation instead of Timoshenko’s is used for the nozzle with a nonplanar end. The accu-rate continuity conditions at the intersection curve are adopted instead of approximateones. The presented results are in good agreement with those obtained by tests and by 3DFEM and with WRC Bulletin 297 when d / D is small. The theoretical solution can beapplied to d / D 0.8 , = d / DT 8 , and d / D t / T 2 successfully.DOI: 10.1115/1.2042471

1 IntroductionCylindrical shells attached with branch pipes shown in Fig. 1

are of common occurrence in the pressure vessel and piping in-dustry. The signicant stress concentration due to pressure andexternal moments often occurs in the vicinity of the junction. Thistopic has attracted many researchers’ attention due to its impor-tance. Since the 1960s Reidelbach 1 , Eringen et al. 2,3 , Hans-berry et al. 4 , and Lekerkerker 5 have obtained the theoreticalsolutions of two normally intersecting shells for the diameter ratio

0 = d / D 0.3 based on the Donnel’s shallow shell equation 6and on the two suppositions that the intersecting curve, , is acircle laid on the developed surface of main shell and a planecircle on the branch pipe, respectively. In order to evaluate thesignicant local stresses in a cylindrical shell due to external mo-ments on branch pipe, a thin shell theoretical solution by doubleFourier series was presented by Bijlaard 7–9 based on Timosh-enko’s equation 10 . The mathematical model adopted by Bij-laard is a cylindrical shell without branch pipe subjected to adistributed radial forces system in a square region and his solu-tions are applied by Wichman et al. to WRC Bulletin No. 10711 . Steele et al. 12 presented an approximate analytical solu-

tion of two normally intersecting cylindrical shells based on shal-low shell theory with the improved mathematical description for

. The design method obtained by Steele’s program FAST2 werepresented in WRC Bulletin No. 297 13 for d / D up to approxi-mately 0.5 and includes the effects of nozzle thickness. Moffatet al. 14,15 obtained numerical solutions on 3-D FEM and ex-perimental results. The applicable limitations of the designmethod in BS 806 based on their results are 5 D / T 70 andd / D t / T 1. Although researchers have spent great efforts toovercome the signicant difculties on mathematics and analysismethod, the design procedures for branch junctions are still inneed of improvement.

A thin shell theoretical solution 16,17 for a wide applicablerange and with higher accuracy was developed by the authors,Xue, Hwang and co-workers, supported by China National Stan-dards Committee on Pressure Vessels CNSCPV since the 1990s.

In the 1990s an analytical solution for two normally intersectingcylinders subjected to internal pressure are presented by Xue et al.18,19 and the analytical results are adopted by the Chinese Pres-

sure Vessel Design Code by Analysis JB 4732-95 20 . Later in1999 21 and in 2000 22 a theoretical solution for the tee-jointsubjected to three run pipe moments is presented. As a newprogress of the research by the authors, a theoretical solution fortwo intersecting cylindrical shells subjected to external branchpipe moments is presented in this paper.

2 Fundamentals of the Present Theoretical AnalysisThe applicable range of the theoretical solutions presented by

Xue et al. is expanded up to 0 = d / D 0.8 and 8 and the orderof accuracy is raised to O T / D . In comparison with the otheranalytical solutions by previous researchers, the theoretical solu-

tion is improved in the following four aspects: 1 the modiedMorley’s equation, which can be used up to = d / DT 1 withthe accuracy order O T / R , is adopted instead of Donnell’s shal-low shell equation, which is applicable to 1 with the accuracyorder O T / R ; 2 ve coordinate systems in three differentspaces, i.e., cylindrical surfaces of main shell and branch pipe astwo-dimensional spaces, respectively, and three-dimensionalspace, and the accurate geometric description of the intersectingcurve in the ve coordinate systems are used instead of previousapproximate expressions, which cause signicant error whend / D 0.3; 3 the accurate continuity conditions for forces, mo-ments, displacements, and rotations at the intersection curve of thetwo cylinders are adopted instead of approximate continuity con-ditions; 4 the great mathematical difculties caused by the ac-curate but very complicated formulations are overcome.

Because the intersection curve, , of two cylinders with largediameter ratio is a complicated space curve, the ve coordinatesystems shown in Fig. 1 are used in this paper. That is, the Car-tesian and cylindrical coordinates, x , y, z and , , z , are takenas the global systems in 3D space. Besides, the Cartesian andpolar coordinate systems, , and , , on the developed sur-face of the mean shell and the Cartesian coordinates, , on thedeveloped surface of the branch pipe are taken as Gaussian coor-dinates, which are curvilinear coordinates in both the 2D curvedsurfaces being subspaces of 3D space, respectively. A cantilevercylindrical shell attached with branch pipe subjected to threekinds of moments, M xb, M yb, and M zb, shown in Fig. 1 is a basic

Contributed by the Pressure Vessels and Piping Division of ASME for publicationin the J OURNAL OF PRESSURE VESSEL TECHNOLOGY . Manuscript received: March 16,2004; nal manuscript received: June 5, 2005. Review conducted by: Dennis K.Williams.

Journal of Pressure Vessel Technology NOVEMBER 2005, Vol. 127 / 357Copyright © 2005 by ASME

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mathematical model a for designers. Each of the basic models,category a , for three load cases can be decomposed into twocategories: the category b , i.e., the main shell on two end sup-ports under branch pipe moment, and the category c , i.e., themain shell subjected to a pair of moments on two ends. As anexample, the decomposition of the category a for load case M xbinto categories b and c is shown in Figs. 2 a –2 c . Our atten-tion is focused on the solutions of category b for three loadingcases, because the solutions of category c have been given in21,22 . Then the solutions of the basic model for the three load-

ing cases are given by superposing category b on category c .In order to obtain the solutions of category b the three types of

symmetry or antisymmetry with respect to =0 or = 0, =0and = / 2 or =0, = / 2 are considered when the solutionsare expanded in Fourier series and shown in Table 1, where thecase numbers are the same as Lekerkerker’s 5 . The case 1 is thesymmetric case with respect to both =0 and = / 2, such as theinternal pressure case.

In terms of the symmetry for case 4 or antisymmetry for case2 and 3 about =0, the boundary conditions at the two supportedends of the main shell, = ± l l = L / R 1 , are

for Case 4: T = 0, u = 0 , un = 0, M = 0 1a

for Cases 2,3: u = 0, S * = 0, Q* = 0, = 0 1b

Suppose that a tee-junction is separated at into two parts: amain shell with cutout, on which is applied a distributed boundaryforce system in equilibrium with the three kinds of moments, anda semi-innite long circle pipe with a nonplanar curved end subjected to three kinds of moments. All the general solutions for thetwo parts are decomposed into two problems: 1 a particular so-lution, which is in equilibrium with the branch pipe moment butdoes not satisfy the boundary conditions at ; 2 general solutionof the homogeneous equation of cylindrical shell. Each of thesums of the two problems with some integral constants becomesthe general solution of each part and the unknown constants couldbe determined by the continuity conditions at .

3 The General Solution for Cylindrical Shell WithCut-Out

3.1 A Particular Solution in Equilibrium With BranchPipe Moment. A thin shell theoretical solution for a main shell onend supports under a force system q z for bending cases M xb and M yb or q y for torsion cases M zb linearly distributed over asquare region dened by c / R, c / R c = R 0 / 2 in thedeveloped surface is taken as a particular solution. The vertical

force system, q z, instead of radial force system, qn, used by Bij-laard 7–9 , is statically equivalent to M xb for case 4 or M yb forcase 3 and the horizontal force system, q y, is statically equivalentto M zb for case 2 . In Bijlaard 9 a simply supported cylindricalshell is subjected to distributed linearly radial force system, qn,whose resultants include not only moment, M xb or M yb, but alsoforce, F yb. Therefore, in order to raise accuracy of the solutions inthe present paper the shell is subjected to vertical force system, q z,instead of radial force system, qn, because the latter may cause asignicant error when the diameter ratio d / D is not small. As anexample, the mathematical model of the particular solution for theload case M xb is given in Fig. 3. The particular solutions satisfythe Timoshenko equations 23 in coordinates , for the shellsubjected to three kinds of distributed loads and boundary condi-tions 1a and 1b , respectively.

In view of the deformation eld symmetric or antisymmetric

Table 1 Three types of symmetry and trigonometric functions „ n = „2n −1 … /2l ; = + L / R …

Fig. 1 Calculated model and ve coordinate systems

Fig. 2 M xb load case is decomposed into two categories „b … and „c …: „a … the basic model; „b …simply supported main shell under branch pipe moment M xb ; and „c … main shell subjected totorsion moment M xb /2.

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with respect to the plane =0 and =0 , , shown in Table 1 forthe three different cases, respectively, the Timoshenko equationswith boundary conditions 1a and 1b at = ± L / R can be solvedby expanding the displacements and external loads in double Fou-rier series as follows:

q = 0 , 2a

q = −m=0 n=1

qmn2 G N

2 m G N 3

n 2b

qn =m=0 n=1

qmn3 G N

1 m G N 3

n 2c

u =m=0 n=1

U mnG N 1 m G N

4n , 3a

u =

m=0 n=1

V mnG N 2 m G N

3n 3b

un =m=0 n=1

W mnG N 1 m G N

3n 3c

where

n =2n − 1 R

2 L; n = 1,2, ... ;

= x + L / R; G N i i =2 ,3 ,4 are shown in Table 1. In Eqs.

2a – 2c q and qn are the tangential and radial components of q zand q y, respectively, where

q =q y cos for case 2

− q z sin for cases 3,4 4a

qn =q y sin for case 2

q z cos for cases 3,4 4b

For the three cases

q z = q N , for N = 3,4 or q y = q2 , for case 2 5

q N , =q N 0 / 2, 0 / 20 0 / 2,or 0 / 2 N = 2,3,4

6

where

q 2 =3 M zb

0r 3 7a

q 3 = −3 M yb

0r 3 7b

q 4 = 2 M xb /4 sin 0 2 −

0 2 cos

0 2 rR 2 7c

By using Eqs. 4 – 7 the coefcients in Fourier series 2b and2c are obtained. Substituting Eqs. 2a – 2c and 3a – 3c into

Timoshenko equations, the coefcients of the displacements inEqs. 3a – 3c are solved.

The particular solution for resultant forces and moments in themain shell are obtained from displacements by means of geomet-ric and elastic relations 21 . The general displacements andforces at the closed curve, , can be expressed by substituting thevalues of , into Eqs. 3a – 3c and related expressions of forces and moments.

= 0 cos , 8a

= sin −1 0 sin 8b

Therefore, they are in equilibrium with M xb, M yb, or M zb, andsatisfy all the basic equations and the boundary conditions at thetwo ends of cylindrical shell, Eqs. 1a and 1b , respectively, andso could be regarded as a particular solution of the boundaryforces and displacements at the cutout of the main shell.

3.2 The Homogenous Solution for the Main Shell. The gen-eral solution of homogeneous equations for a cylindrical shellsubjected to any boundary conditions but no external load actingon the surface are obtained by solving the modied Morley equa-tion by Zhang et al. 24 which is applicable up to r / RT 1. Theradial displacement, un, and the Airy stress function, , satisfy

2 +1

2+ 2 i

2 +

1

2− 2 i

= 0 9

Here, 4 2 = 12 1− 2 1/2 R / T and = un + i 4 2 / ETR . Thesolution of Eq. 9 is

Table 2 e „ j , N … in three cases „ j =1,2 ,3 ,4 …

Fig. 3 The analyzed models of the particular solution in theload case M xb . „a … The distributed force system q z equivalent toM xb ; „b … the distributed force q n used by Bijlaard; „c … the areaon the developed surface of the main shell where is applied thedistributed forces.

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=k =e 4, N n=e 1, N

C nF kn G N 1 m 10

where m=2 k + e 2 , N and the unknown complex constants C nconsist of two parts

C n = C n1 + iC n2 11

G N 1 m are triangular functions dependent on Case number N

shown in Table 1 and

F kn = − 1 k 1 − 12 m0 J m−n − i

+ e 3, N J −m−n − i H n2 12

where

= 12 − i 2 1/2 ,

mn =0, m n

1, m = n ,

J n and H n2 are the rst kind of Bessel function and the second

kind of Hankel function, respectively. The values of e j , N j=1 ,2 ,3 ,4 are shown in Table 2.

The components of forces, moments, displacements, and rota-tions in the main shell are all expressed through the partial deriva-tives of with respect to and , see Xue et al. 16,18,21 . Theboundary general displacements and forces with unknowns C n1and C n2 a t are obtained by substituting the value of , intoEq. 10 ,

Fig. 4 Distribution of k along the line =0 deg on the outer surface of ModelORNL-1 subjected to M yb

Fig. 5 Maximum principal stress ratios around the junction of ORNL-1 sub-jected to M yb



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= 02 cos 2 + sin −1 0 sin 2 1/2 13 a

= sin −1 sin −1 0 sin / 13 b

The general solution obtained by superposing the particular so-lution on the homogeneous solution, satises all the basic equa-tions of cylindrical shell and any prescribed boundary conditionsand the resultant forces in the main shell corresponding to thegeneral solution are in equilibrium with the branch pipe moment.

The boundary displacement and force vectors, F and u , at canbe decomposed in global coordinates , , z as follows: i , i t , inbeing triad at , see Xue et al. 22

F = T i + S i t − Q in = F i + F i + F zi z 14 a

u = u i + u i + unin = u i + u i + u zi z 14 b

All the boundary forces and displacements are periodic functions

Fig. 6 Distribution of k along the line =90 deg on the outer surface of ModelORNL-1 subjected to M xb

Fig. 7 Maximum principal stress ratios around the junction of ORNL-1 sub-jected to M xb



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of with parameter 0, so that it can be expanded in Fourier seriesof and truncated at k = K m = 2K + e 2 , N and n= 2K + e 2 , N .The Fourier coefcients, which involve complicated and oscilla-tory integrands, are calculated by Filon numerical integration al-gorithm referred to in 25 .

4 The Solution for a Semi-Innite Long Circle PipeWith a Non-Planar End Subjected to Three Kinds of Moments

The membrane solution is adopted as a particular one for thebranch pipe in the three loading cases, which is given in 26 .

The homogeneous solution for the nozzle is obtained by solvingthe Goldenveizer equation 27 in terms of the displacement func-tion

8 + 4 t 4

4 4 + 8 − 2 2

6 4 2

+ 8 6

2 4 + 2

6 6

+ 4 4

2 2

+ 4 4

= 0 15

where

Fig. 8 Distribution of k along the line =60 deg on the outer surface of ModelORNL-1 subjected to M zb

Fig. 9 Maximum principal stress ratios around the junction of ORNL-1 sub-jected to M zb



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2 = 2

2 +

2

2 , t = 3 1 − 2 r 2 / t 2 1/4 16

In the three cases, can be expanded in Fourier series as follows:

=k =e 4, N l=1

8

Dklgkl G N 1 m , m = 2k + e 2, N 17

Due to the innite boundary conditions when → , the fouritems of gkl l = 1 , 2 , . . . , 8 will vanish. There remain only theother four items. The expressions of gkl are shown in 21,22 .The homogeneous solutions of displacements, resultant forces areexpressed in terms of i+ j / i j see 21 .

At the intersecting curve , where

= 1 − 02 sin 2 1/2 − 1 / 0 = 0, 18

the general boundary displacement and force vectors, u t , F t

rotation t , and moment M

t can be obtained easily.

u t = u t i

t + ut t i t

t + unt in

t = u t i + u

t i + u zt i z 19

F t = F t i

t + F t t i t

t + F nt in

t = F t i + F

t i + F zt i z 20

They are expanded in Fourier series of with unknowns Dkl l=1 ,2 ,3 ,4 and truncated at k = K .

Fig. 10 Distribution of stress ratios along the line =0 deg of the model d / D =0.8 subjected to M yb . „a … On the outer surface; „b … on the inner surface.



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5 The Continuity Conditions at the Intersecting CurveThe unknowns in the general solutions for both main shell and

nozzle are determined by the continuity conditions at their inter-

secting curve, , as follows:

F = − F t , F = − F

t , F z = − F zt , M = M

t 21

u = u t , u = u

t , u z = u zt , = −

t 22

The continuity conditions 21 and 22 for each harmonic Fouriercoefcient should be satised, so that the unknowns C ni in Eqs.10 and 11 and Dkl in Eq. 17 can be solved. For case 3, the

condition of uniqueness of displacements should be considered.The numbers of unknowns and equations for each case are dis-cussed in a separate paper 26 .

6 Verication of the Present Theoretical Solution

6.1 Comparison with the Test and the Numerical Resultsfor Model ORNL-1 ( d / D = t / T =0.5, D / T =100). The present the-oretical solution is veried by the test results 28 for the ORNL-1model, which is a good-quality steel model. The strain gauges onthe branch pipe are arranged in several lines running along thenozzle axially and on the main shell, in several lines, which areperpendicular to the junction curve on the developed surface of the shell. In each loading case the longitudinal and transversestresses, which are normal stresses parallel and perpendicular, re-spectively, to the gauge lines, are given in 28 . In Figs. 4–11 theabove mentioned stresses are divided by normal membrane stress

0 0 = M yb / r 2t for case 3, 0 = M xb / r 2t for case 4, and 0= M zb / 2 r 2t for Case 2, respectively and dened as dimension-less longitudinal stress, k , and transverse stress, k t .

Fig. 11 Distribution of stress ratios along the line =90 deg of the modeld / D =0.8 subjected to M xb . „a … On the outer surface; „b … on the inner surface.



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The results obtained by the present solution and by 3D FEMthe calculated FEM model by software ANSYS has 206,353

nodes and four layers of 20-nodal elements through the thicknessin close vicinity to the junction are shown in Figs. 4–9 as well.The comparison shows that the present theoretical results are invery good agreement with those by test and by FEM for bothloading cases of in-plane M yb and out-of-plane M xb bendingmoments, see Figs. 4–7. Figures 8 and 9 show that the presentresults are somewhat different from the test results, as are numeri-cal results given in 28 , but in good agreement with those by 3DFEM.

6.2 Comparison between Theoretical and Numerical Re-sults for a Model With Large Diameter Ratio d / D = t / T =0.8, D / T =100. A 3D nite element model with parameters d / D =0.8,t / T =2, and D / T =100 = d / DT = 8 is calculated by software

ANSYS to verify the applicable range for the presented theoreticalsolution. The model has 41,450 20-nodal elements and 622,722freedom degrees. The results given by the two methods for load-

ing cases of either in-plane or out-of-plane bending moment are ingood agreement as shown in Figs. 11 and 12.

6.3 Comparison of Resultant Forces and Bending Mo-ments With WRC Bulletin 297. The methods shown in WRCBulletin 297 12 based on analytical solution given by Steeleet al. 11 are currently used in pressure vessel industry within thelimits of d / D 0.5 and = d / DT 5. Figures 12 and 13 showthat the results obtained by the presented method are in agreementwith those given by WRC Bulletin 297 when d / D is small.

Fig. 12 The comparison of dimensionless resultant forces and moments inthe main shell with WRCB 297 due to M yb . „a … d / t =30, t / T =1, due to M yb ; „b …d / t =100, t / T =1, due to M yb .



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7 The Maximum Stress Concentration Factors in theMain Shells for the Three Branch Moment Loading

CasesThe maximum stress concentration factors K yb, K xb, and K zb in

the main shell are dependent on the three parameters: 0= d / D , = d / DT or D / T and t / T . Here, K yb, K xb, and K zb arethe maximum stress intensities divided by the normal stress 0 or

0 , which is dened as

0 = M yb

r 2t

R

T

t

r

for in-plane bending moment and

0 = M xb

r 2t

R

T

t

r

for out-of-plane bending moment, respectively, and normal shearstress 0 is dened as 0 = M zb / 2 r 2t for torsion moment. As anexample, three sets of curves, K yb, K xb, and K zb versus and t / T when 0 =0.7 and up to 8, are given in Figs. 14 a –14 c , re-spectively. The maximum stress intensities are obtained for theloading case, M yb, at =0 deg, for M xb case, at = 90 deg, and for M zb, at 60 deg, respectively. The curves, K yb, K xb, and K zbversus and t / T when 0 =0.8 are shown in 26 .

8 ConclusionA thin shell theoretical solution of two normally intersecting

cylindrical shells subjected to three kinds of branch pipe moments

Fig. 13 The comparison of dimensionless resultant forces and moments inthe main shell with WRCB 297 due to M xb . „a … d / t =30, t / T =1, due to M xb ; „b …d / t =100, t / T =1, due to M xb .



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is presented. The results by the present method are in very goodagreement with those obtained by test and by FEM. The presentanalytical results are in good agreement with WRC Bulletin 297

when d / D is small. The present theoretical method can be appli-cable up to d / D 0.8, = d / DT 8, and d / D t / T 2 success-fully.

NomenclatureC n complex constants in the homogenous solution

for the main shelld , D diameters of the branch pipe and main shell,

respectively Dkl real constants in the homogenous solution forthe branch pipe

E , Young’s modulus and Poisson ratio,respectively

F boundary force vector at the intersecting curveG N

1 , G N 2 trigonometric functions shown in Table 1i unit vector

k , k t , k max dimensionless longitudinal, transverse, andmaximum principal stresses, respectively

K xb , K yb , K zb dimensionless maximum stress intensities forload cases M xb, M yb, and M zb, respectively

2 L length of the main shell M xb , M yb , M zb three load cases: external branch pipe moments

M component of moment in the Cartesian coordi-

nates of the main shell N load case number shown in Table 1Q* , S * boundary effective transverse and in-surface

shear forces, respectivelyr , R radii of the branch pipe and main shell,

respectivelyt , T thicknesses of the branch pipe and main shell,

respectivelyT resultant force in the Cartesian coordinates of

the main shellu boundary displacement vector at the intersect-

ing curveu , u , un component of displacement of the main shell

x , y, z global Cartesian coordinates in 3D space , polar coordinates on the developed surface of

the main shell rotation component of normal to the middlesurface of shell

, Cartesian coordinates on the developed surfaceof the branch pipe

, , z global cylindrical coordinates in 3D space 0 = d / D diameter ratio

, Cartesian coordinates on the developed surfaceof the main shell

displacement function for the branch pipe complex-valued displacement-stress function

for the main shell

Subscriptsn components in the normal direction to the

middle surface of the main shell , components in the , coordinate system for

the main shell , , z components in the 3D cylindrical coordinate

system, coincident with the normal, circular,and longitudinal directions, respectively, of thebranch pipe

, t components in the normal and tangent direc-tions to , respectively

value at the intersecting curve , components in the , coordinate system for

the main shell N =1 ,2 ,3 denote different cases

Fig. 14 The stress concentration factors versus and t / T „ 0 =0.7 …. „a … K yb for in-plane bending moment M yb ; „b … K xb forout-of-plane bending moment M xb ; „c … K zb for torsion momentM zb .



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