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Xue2005.PDF
M. D. Xue
D. F. Li
K. C. Hwang
Department of Engineering Mechanics,
Tsinghua University,
Beijing, 100084,
People’s Republic of China
1
A Thin Shell Theoretical Solution
for Two Intersecting Cylindrical
Shells Due to External Branch
Pipe Moments
A theoretical solution is presented for cylindrical shells with normally intersecting
nozzles subjected to three kinds of external branch pipe moments. The improved double
trigonometric series solution is used for the particular solution of main shell subjected to
distributed forces, and the modified Morley equation instead of the Donnell shallow shell
equation 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 accurate continuity conditions at the intersection curve are adopted instead of approximate
ones. The presented results are in good agreement with those obtained by tests and by 3D
FEM and with WRC Bulletin 297 when d / D is small. The theoretical solution can be
applied to d / D 艋 0.8, ␭ = d / 冑DT 艋 8, and d / D 艋 t / T 艋 2 successfully.
关DOI: 10.1115/1.2042471兴
Introduction
Cylindrical shells attached with branch pipes shown in Fig. 1
are of common occurrence in the pressure vessel and piping industry. The significant stress concentration due to pressure and
external moments often occurs in the vicinity of the junction. This
topic has attracted many researchers’ attention due to its importance. Since the 1960s Reidelbach 关1兴, Eringen et al. 关2,3兴, Hansberry et al. 关4兴, and Lekerkerker 关5兴 have obtained the theoretical
solutions of two normally intersecting shells for the diameter ratio
␳0 = d / D 艋 0.3 based on the Donnel’s shallow shell equation 关6兴
and on the two suppositions that the intersecting curve, ⌫, is a
circle laid on the developed surface of main shell and a plane
circle on the branch pipe, respectively. In order to evaluate the
significant local stresses in a cylindrical shell due to external moments on branch pipe, a thin shell theoretical solution by double
Fourier series was presented by Bijlaard 关7–9兴 based on Timoshenko’s equation 关10兴. The mathematical model adopted by Bijlaard is a cylindrical shell without branch pipe subjected to a
distributed radial forces system in a square region and his solutions are applied by Wichman et al. to WRC Bulletin No. 107
关11兴. Steele et al. 关12兴 presented an approximate analytical solution of two normally intersecting cylindrical shells based on shallow shell theory with the improved mathematical description for
⌫. The design method obtained by Steele’s program FAST2 were
presented in WRC Bulletin No. 297 关13兴 for d / D up to approximately 0.5 and includes the effects of nozzle thickness. Moffat
et al. 关14,15兴 obtained numerical solutions on 3-D FEM and experimental results. The applicable limitations of the design
method in BS 806 based on their results are 5 艋 D / T 艋 70 and
d / D 艋 t / T 艋 1. Although researchers have spent great efforts to
overcome the significant difficulties on mathematics and analysis
method, the design procedures for branch junctions are still in
need of improvement.
A thin shell theoretical solution 关16,17兴 for a wide applicable
range and with higher accuracy was developed by the authors,
Xue, Hwang and co-workers, supported by China National Standards Committee on Pressure Vessels 共CNSCPV兲 since the 1990s.
Contributed by the Pressure Vessels and Piping Division of ASME for publication
in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received: March 16,
2004; final manuscript received: June 5, 2005. Review conducted by: Dennis K.
Williams.
Journal of Pressure Vessel Technology
In the 1990s an analytical solution for two normally intersecting
cylinders subjected to internal pressure are presented by Xue et al.
关18,19兴 and the analytical results are adopted by the Chinese Pressure Vessel Design Code by Analysis JB 4732-95 关20兴. Later in
1999 关21兴 and in 2000 关22兴 a theoretical solution for the tee-joint
subjected to three run pipe moments is presented. As a new
progress of the research by the authors, a theoretical solution for
two intersecting cylindrical shells subjected to external branch
pipe moments is presented in this paper.
2
Fundamentals of the Present Theoretical Analysis
The applicable range of the theoretical solutions presented by
Xue et al. is expanded up to ␳0 = d / D 艋 0.8 and ␭ 艋 8 and the order
of accuracy is raised to O共T / D兲. In comparison with the other
analytical solutions by previous researchers, the theoretical solution is improved in the following four aspects: 共1兲 the modified
Morley’s equation, which can be used up to ␭ = d / 冑DT Ⰷ 1 with
the accuracy order O共T / R兲, is adopted instead of Donnell’s shallow shell equation, which is applicable to ␭ ⬇ 1 with the accuracy
order O共冑T / R兲; 共2兲 five coordinate systems in three different
spaces, i.e., cylindrical surfaces of main shell and branch pipe as
two-dimensional spaces, respectively, and three-dimensional
space, and the accurate geometric description of the intersecting
curve in the five coordinate systems are used instead of previous
approximate expressions, which cause significant error when
d / D ⬎ 0.3; 共3兲 the accurate continuity conditions for forces, moments, displacements, and rotations at the intersection curve of the
two cylinders are adopted instead of approximate continuity conditions; 共4兲 the great mathematical difficulties caused by the accurate but very complicated formulations are overcome.
Because the intersection curve, ⌫, of two cylinders with large
diameter ratio is a complicated space curve, the five coordinate
systems shown in Fig. 1 are used in this paper. That is, the Cartesian and cylindrical coordinates, 共x , y , z兲 and 共␳ , ␪ , z兲, are taken
as the global systems in 3D space. Besides, the Cartesian and
polar coordinate systems, 共␰ , ␸兲 and 共␣ , ␤兲, on the developed surface of the mean shell and the Cartesian coordinates, 共␪ , ␵兲 on the
developed surface of the branch pipe are taken as Gaussian coordinates, which are curvilinear coordinates in both the 2D curved
surfaces being subspaces of 3D space, respectively. A cantilever
cylindrical shell attached with branch pipe subjected to three
kinds of moments, M xb, M yb, and M zb, shown in Fig. 1 is a basic
Copyright © 2005 by ASME
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for Case 4:
for Cases 2,3:
T␰ = 0,
u␰ = 0,
u␸ = 0,
S* = 0,
un = 0,
Q* = 0,
M␰ = 0
␥␰ = 0
共1a兲
共1b兲
Suppose that a tee-junction is separated at ⌫ into two parts: a
main shell with cutout, on which is applied a distributed boundary
force system in equilibrium with the three kinds of moments, and
a semi-infinite long circle pipe with a nonplanar curved end subjected to three kinds of moments. All the general solutions for the
two parts are decomposed into two problems: 共1兲 a particular solution, which is in equilibrium with the branch pipe moment but
does not satisfy the boundary conditions at ⌫; 共2兲 general solution
of the homogeneous equation of cylindrical shell. Each of the
sums of the two problems with some integral constants becomes
the general solution of each part and the unknown constants could
be determined by the continuity conditions at ⌫.
Fig. 1 Calculated model and five coordinate systems
mathematical model 共a兲 for designers. Each of the basic models,
category 共a兲, for three load cases can be decomposed into two
categories: the category 共b兲, i.e., the main shell on two end supports under branch pipe moment, and the category 共c兲, i.e., the
main shell subjected to a pair of moments on two ends. As an
example, the decomposition of the category 共a兲 for load case M xb
into categories 共b兲 and 共c兲 is shown in Figs. 2共a兲–2共c兲. Our attention is focused on the solutions of category 共b兲 for three loading
cases, because the solutions of category 共c兲 have been given in
关21,22兴. Then the solutions of the basic model for the three loading cases are given by superposing category 共b兲 on category 共c兲.
In order to obtain the solutions of category 共b兲 the three types of
symmetry 共or antisymmetry兲 with respect to ␤ = 0 共or ␸ = 0, ␪ = 0兲
and ␤ = ␲ / 2 共or ␰ = 0, ␪ = ␲ / 2兲 are considered when the solutions
are expanded in Fourier series and shown in Table 1, where the
case numbers are the same as Lekerkerker’s 关5兴. The case 1 is the
symmetric case with respect to both ␤ = 0 and ␤ = ␲ / 2, such as the
internal pressure case.
In terms of the symmetry 共for case 4兲 or antisymmetry 共for case
2 and 3兲 about ␰ = 0, the boundary conditions at the two supported
ends of the main shell, ␰ = ± l共l = L / R Ⰷ 1兲, are
3 The General Solution for Cylindrical Shell With
Cut-Out
3.1 A Particular Solution in Equilibrium With Branch
Pipe Moment. A thin shell theoretical solution for a main shell on
end supports under a force system qz 共for bending cases M xb and
M yb兲 or qy 共for torsion cases M zb兲 linearly distributed over a
square region defined by 兩␰兩 艋 c / R, 兩␸兩 艋 c / R 共c = R␳0 / 冑2兲 in the
developed surface is taken as a particular solution. The vertical
force system, qz, instead of radial force system, qn, used by Bijlaard 关7–9兴, is statically equivalent to M xb 共for case 4兲 or M yb 共for
case 3兲 and the horizontal force system, qy, is statically equivalent
to M zb 共for case 2兲. In Bijlaard 关9兴 a simply supported cylindrical
shell is subjected to distributed linearly radial force system, qn,
whose resultants include not only moment, M xb or M yb, but also
force, Fyb. Therefore, in order to raise accuracy of the solutions in
the present paper the shell is subjected to vertical force system, qz,
instead of radial force system, qn, because the latter may cause a
significant error when the diameter ratio d / D is not small. As an
example, the mathematical model of the particular solution for the
load case M xb is given in Fig. 3. The particular solutions satisfy
the Timoshenko equations 关23兴 in coordinates 共␰ , ␸兲 for the shell
subjected to three kinds of distributed loads and boundary conditions 共1a兲 and 共1b兲, respectively.
In view of the deformation field symmetric or antisymmetric
Fig. 2 Mxb load case is decomposed into two categories „b… and „c…: „a… the basic model; „b…
simply supported main shell under branch pipe moment Mxb; and „c… main shell subjected to
torsion moment Mxb / 2.
Table 1 Three types of symmetry and trigonometric functions „␭n = „2n − 1…␲ / 2l ; ␰⬘ = ␰ + L / R…
358 / Vol. 127, NOVEMBER 2005
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Table 2
e„j , N… in three cases „j = 1 , 2 , 3 , 4…
q␸ =
再
再
qn =
qy cos ␸
for case 2
冎
冎
共4a兲
− qz sin ␸ for cases 3,4
qy sin ␸ for case 2
qz cos ␸ for cases 3,4
共4b兲
For the three cases
qz = qN共␰, ␸兲 共for N = 3,4兲 or qy = q2共␰, ␸兲 共for case 2兲
q N共 ␰ , ␸ 兲 =
再
q共N兲 兩␰兩 艋 ␳0/冑2,兩␸兩 艋 ␳0/冑2
0
兩␰兩 ⬎ ␳0/冑2,or 兩␸兩 ⬎ ␳0/冑2
冎
共5兲
共N = 2,3,4兲
共6兲
where
Fig. 3 The analyzed models of the particular solution in the
load case Mxb. „a… The distributed force system qz equivalent to
Mxb; „b… the distributed force qn used by Bijlaard; „c… the area
on the developed surface of the main shell where is applied the
distributed forces.
q 共2兲 =
3␰ M zb
␳ 0r 3
q 共3兲 = −
with respect to the plane ␰ = 0 and ␸ = 0 , ␲, shown in Table 1 for
the three different cases, respectively, the Timoshenko equations
with boundary conditions 共1a兲 and 共1b兲 at ␰ = ± L / R can be solved
by expanding the displacements and external loads in double Fourier series as follows:
共2a兲
q␰ = 0,
⬁
⬁
兺兺q
q␸ = −
共2兲 共2兲
共3兲
mnGN 共m␸兲GN 共␭n␰⬘兲
共2b兲
m=0 n=1
⬁
qn =
⬁
兺兺q
共3兲 共1兲
共3兲
mnGN 共m␸兲GN 共␭n␰⬘兲
共2c兲
冉
q共4兲 = 冑2␸ M xb/4 sin
⬁
⬁
兺兺
UmnGN共1兲共m␸兲GN共4兲共␭n␰⬘兲,
共3a兲
m=0 n=1
⬁
u␸ =
⬁
兺兺V
共2兲
共3兲
mnGN 共m␸兲GN 共␭n␰⬘兲
共3b兲
m=0 n=1
⬁
un =
⬁
兺兺
WmnGN共1兲共m␸兲GN共3兲共␭n␰⬘兲
共3c兲
m=0 n=1
where
␭n =
共2n − 1兲␲R
;
2L
n = 1,2, . . . ;
␰⬘ = 共x + L兲 / R; G共i兲
N 共i = 2 , 3 , 4兲 are shown in Table 1. In
共2a兲–共2c兲 q␸ and qn are the tangential and radial components
and qy, respectively, where
Journal of Pressure Vessel Technology
Eqs.
of qz
3␰ M yb
␳ 0r 3
␳0
␳0
共7b兲
␳0
冑2 − 冑2 cos 冑2
冊
rR2
共7c兲
By using Eqs. 共4兲–共7兲 the coefficients in Fourier series 共2b兲 and
共2c兲 are obtained. Substituting Eqs. 共2a兲–共2c兲 and 共3a兲–共3c兲 into
Timoshenko equations, the coefficients of the displacements in
Eqs. 共3a兲–共3c兲 are solved.
The particular solution for resultant forces and moments in the
main shell are obtained from displacements by means of geometric and elastic relations 关21兴. The general displacements and
forces at the closed curve, ⌫, can be expressed by substituting the
values of ␰⌫ , ␸⌫ into Eqs. 共3a兲–共3c兲 and related expressions of
forces and moments.
m=0 n=1
u␰ =
共7a兲
␰⌫ = ␳0 cos ␪ ,
共8a兲
␸⌫ = sin−1共␳0 sin ␪兲
共8b兲
Therefore, they are in equilibrium with M xb, M yb, or M zb, and
satisfy all the basic equations and the boundary conditions at the
two ends of cylindrical shell, Eqs. 共1a兲 and 共1b兲, respectively, and
so could be regarded as a particular solution of the boundary
forces and displacements at the cutout of the main shell.
3.2 The Homogenous Solution for the Main Shell. The general solution of homogeneous equations for a cylindrical shell
subjected to any boundary conditions but no external load acting
on the surface are obtained by solving the modified Morley equation by Zhang et al. 关24兴 which is applicable up to r / 冑RT Ⰷ 1. The
radial displacement, un, and the Airy stress function, ␾, satisfy
冉
ⵜ2 +
1
⳵
+ 2␮冑i
2
⳵␰
冊冉
ⵜ2 +
冊
1
⳵
− 2␮冑i
␹=0
2
⳵␰
共9兲
Here, 4␮2 = 关12共1 − ␯2兲兴1/2共R / T兲 and ␹ = un + i共4␮2 / ETR兲␾. The
solution of Eq. 共9兲 is
NOVEMBER 2005, Vol. 127 / 359
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Fig. 4 Distribution of k along the line ␪ = 0 deg on the outer surface of Model
ORNL-1 subjected to Myb
⬁
␹=
␩ = 共 21 − i␮2兲1/2 ,
⬁
兺 兺
k=e共4,N兲 n=e共1,N兲
CnFkn共␣兲GN共1兲共m␤兲
共10兲
where m = 2k + e共2 , N兲 and the unknown complex constants Cn
consist of two parts
Cn = Cn1 + iCn2
共11兲
共1兲
GN 共m␤兲 are triangular functions dependent on Case number N
shown in Table 1 and
Fkn = 共− 1兲k共1 − 2 ␦m0兲关Jm−n共冑− i␮␣兲
1
+ e共3,N兲J−m−n共冑− i␮␣兲兴Hn共2兲共␩␣兲
where
共12兲
␦mn =
共2兲
再
0, m ⫽ n
1, m = n
冎
,
Jn and Hn are the first kind of Bessel function and the second
kind of Hankel function, respectively. The values of e共j , N兲共j
= 1 , 2 , 3 , 4兲 are shown in Table 2.
The components of forces, moments, displacements, and rotations in the main shell are all expressed through the partial derivatives of ␹ with respect to ␣ and ␤, see Xue et al. 关16,18,21兴. The
boundary general displacements and forces with unknowns Cn1
and Cn2 at ⌫ are obtained by substituting the value of ␣⌫ , ␤⌫ into
Eq. 共10兲,
Fig. 5 Maximum principal stress ratios around the junction of ORNL-1 subjected to Myb
360 / Vol. 127, NOVEMBER 2005
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Fig. 6 Distribution of k along the line ␪ = 90 deg on the outer surface of Model
ORNL-1 subjected to Mxb
␣⌫ = 关␳20 cos2 ␪ + 共sin−1共␳0 sin ␪兲兲2兴1/2
共13a兲
␤⌫ = sin−1关sin−1共␳0 sin ␪兲/␣⌫兴
共13b兲
The general solution obtained by superposing the particular solution on the homogeneous solution, satisfies all the basic equations of cylindrical shell and any prescribed boundary conditions
and the resultant forces in the main shell corresponding to the
general solution are in equilibrium with the branch pipe moment.
The boundary displacement and force vectors, F and u, at ⌫ can
be decomposed in global coordinates 共␳ , ␪ , z兲 as follows: 共i␯ , it , in
being triad at ⌫, see Xue et al. 关22兴兲
F = T ␯i ␯ + S ␯i t − Q ␯i n = F ␳i ␳ + F ␪i ␪ + F zi z
共14a兲
u = u ␣i ␣ + u ␤i ␤ + u ni n = u ␳i ␳ + u ␪i ␪ + u zi z
共14b兲
All the boundary forces and displacements are periodic functions
Fig. 7 Maximum principal stress ratios around the junction of ORNL-1 subjected to Mxb
Journal of Pressure Vessel Technology
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Fig. 8 Distribution of k along the line ␪ = 60 deg on the outer surface of Model
ORNL-1 subjected to Mzb
of ␪ with parameter ␳0, so that it can be expanded in Fourier series
of ␪ and truncated at k = K共m = 2K + e共2 , N兲兲 and n = 2K + e共2 , N兲.
The Fourier coefficients, which involve complicated and oscillatory integrands, are calculated by Filon numerical integration algorithm referred to in 关25兴.
4 The Solution for a Semi-Infinite Long Circle Pipe
With a Non-Planar End Subjected to Three Kinds of
Moments
The membrane solution is adopted as a particular one for the
branch pipe in the three loading cases, which is given in 关26兴.
The homogeneous solution for the nozzle is obtained by solving
the Goldenveizer equation 关27兴 in terms of the displacement function ␺
ⵜ8␺ + 4␭t4
+
⳵ 4␺
⳵ 6␺
⳵ 6␺
⳵ 6␺
⳵ 4␺
2
+
共8
−
2
␯
兲
+
8
+
2
+
4
⳵␵4
⳵ ␵ 4⳵ ␪ 2
⳵ ␵ 2⳵ ␪ 4
⳵␪6
⳵ ␵ 2⳵ ␪ 2
⳵ 4␺
=0
⳵␪4
共15兲
where
Fig. 9 Maximum principal stress ratios around the junction of ORNL-1 subjected to Mzb
362 / Vol. 127, NOVEMBER 2005
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Fig. 10 Distribution of stress ratios along the line ␪ = 0 deg of the model d / D
= 0.8 subjected to Myb. „a… On the outer surface; „b… on the inner surface.
ⵜ2 =
⳵2
⳵2
+
,
⳵␪2 ⳵␵2
␭t = 关3共1 − ␯2兲r2/t2兴1/4
共16兲
In the three cases, ␺ can be expanded in Fourier series as follows:
⬁
␺=
8
兺 兺D
k=e共4,N兲 l=1
共1兲
klgkl共␵兲GN 共m␪兲,
共m = 2k + e共2,N兲兲 共17兲
Due to the infinite boundary conditions when ␵ → ⬁, the four
items of gkl共␵兲共l = 1 , 2 , . . . , 8兲 will vanish. There remain only the
other four items. The expressions of gkl共␵兲 are shown in 关21,22兴.
The homogeneous solutions of displacements, resultant forces are
expressed in terms of ⳵共i+j兲␺ / ⳵␪i⳵␵ j 共see 关21兴兲.
At the intersecting curve ⌫, where
Journal of Pressure Vessel Technology
␵⌫ = 关共1 − ␳20 sin2 ␪兲1/2 − 1兴/␳0 = ␵⌫共␳0, ␪兲
共18兲
the general boundary displacement and force vectors, u共t兲, F共t兲
共t兲
共t兲
rotation ␥␯ , and moment M ␯ can be obtained easily.
u共t兲 = u␯共t兲i␯共t兲 + ut共t兲it共t兲 + un共t兲in共t兲 = u␳共t兲i␳ + u␪共t兲i␪ + uz共t兲iz
共19兲
F共t兲 = F␯共t兲i␯共t兲 + Ft共t兲it共t兲 + Fn共t兲in共t兲 = F␳共t兲i␳ + F␪共t兲i␪ + Fz共t兲iz
共20兲
They are expanded in Fourier series of ␪ with unknowns Dkl共l
= 1 , 2 , 3 , 4兲 and truncated at k = K.
NOVEMBER 2005, Vol. 127 / 363
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Fig. 11 Distribution of stress ratios along the line ␪ = 90 deg of the model
d / D = 0.8 subjected to Mxb. „a… On the outer surface; „b… on the inner surface.
5
The Continuity Conditions at the Intersecting Curve
The unknowns in the general solutions for both main shell and
nozzle are determined by the continuity conditions at their intersecting curve, ⌫, as follows:
F␳ = − F␳共t兲,
u␳ = u␳共t兲,
F␪ = − F␪共t兲,
u␪ = u␪共t兲,
Fz = − Fz共t兲,
uz = uz共t兲,
M ␯ = M ␯共t兲
␥␯ = − ␥␯共t兲
共21兲
共22兲
The continuity conditions 共21兲 and 共22兲 for each harmonic Fourier
coefficient should be satisfied, so that the unknowns Cni in Eqs.
共10兲 and 共11兲 and Dkl in Eq. 共17兲 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 discussed in a separate paper 关26兴.
364 / Vol. 127, NOVEMBER 2005
6
Verification of the Present Theoretical Solution
6.1 Comparison with the Test and the Numerical Results
for Model ORNL-1 (d / D = t / T = 0.5, D / T = 100). The present theoretical solution is verified by the test results 关28兴 for the ORNL-1
model, which is a good-quality steel model. The strain gauges on
the branch pipe are arranged in several lines running along the
nozzle axially and on the main shell, in several lines, which are
perpendicular to the junction curve on the developed surface of
the shell. In each loading case the longitudinal and transverse
stresses, which are normal stresses parallel and perpendicular, respectively, to the gauge lines, are given in 关28兴. In Figs. 4–11 the
above mentioned stresses are divided by normal membrane stress
␴0 共␴0 = M yb / ␲r2t for case 3, ␴0 = M xb / ␲r2t for case 4, and ␴0
= M zb / 2␲r2t for Case 2, respectively兲 and defined as dimensionless longitudinal stress, k␯, and transverse stress, kt.
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Fig. 12 The comparison of dimensionless resultant forces and moments in
the main shell with WRCB 297 due to Myb. „a… d / t = 30, t / T = 1, due to Myb; „b…
d / t = 100, t / T = 1, due to Myb.
The results obtained by the present solution and by 3D FEM
共the calculated FEM model by software ANSYS has 206,353
nodes and four layers of 20-nodal elements through the thickness
in close vicinity to the junction兲 are shown in Figs. 4–9 as well.
The comparison shows that the present theoretical results are in
very good agreement with those by test and by FEM for both
loading cases of in-plane 共M yb兲 and out-of-plane 共M xb兲 bending
moments, see Figs. 4–7. Figures 8 and 9 show that the present
results are somewhat different from the test results, as are numerical results given in 关28兴, but in good agreement with those by 3D
FEM.
6.2 Comparison between Theoretical and Numerical Results for a Model With Large Diameter Ratio d / D = t / T = 0.8,
D / T = 100. A 3D finite element model with parameters d / D = 0.8,
t / T = 2, and D / T = 100 共␭ = d / 冑DT = 8兲 is calculated by software
Journal of Pressure Vessel Technology
to verify the applicable range for the presented theoretical
solution. The model has 41,450 20-nodal elements and 622,722
freedom degrees. The results given by the two methods for loading cases of either in-plane or out-of-plane bending moment are in
good agreement as shown in Figs. 11 and 12.
ANSYS
6.3 Comparison of Resultant Forces and Bending Moments With WRC Bulletin 297. The methods shown in WRC
Bulletin 297 关12兴 based on analytical solution given by Steele
et al. 关11兴 are currently used in pressure vessel industry within the
limits of d / D 艋 0.5 and ␭ = d / 冑DT ⬍ 5. Figures 12 and 13 show
that the results obtained by the presented method are in agreement
with those given by WRC Bulletin 297 when d / D is small.
NOVEMBER 2005, Vol. 127 / 365
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Fig. 13 The comparison of dimensionless resultant forces and moments in
the main shell with WRCB 297 due to Mxb. „a… d / t = 30, t / T = 1, due to Mxb; „b…
d / t = 100, t / T = 1, due to Mxb.
7 The Maximum Stress Concentration Factors in the
Main Shells for the Three Branch Moment Loading
Cases
The maximum stress concentration factors Kyb, Kxb, and Kzb in
the main shell are dependent on the three parameters: ␳0
= d / D , ␭ = d / 冑DT 共or D / T兲 and t / T. Here, Kyb, Kxb, and Kzb are
the maximum stress intensities divided by the normal stress ␴0 共or
␶0兲, which is defined as
␴0 =
M yb R t
␲ r 2t T r
for in-plane bending moment and
366 / Vol. 127, NOVEMBER 2005
␴0 =
M xb R t
␲ r 2t T r
for out-of-plane bending moment, respectively, and normal shear
stress ␶0 is defined as ␶0 = M zb / 2␲r2t for torsion moment. As an
example, three sets of curves, Kyb, Kxb, and Kzb versus ␭ and t / T
when ␳0 = 0.7 and ␭ up to 8, are given in Figs. 14共a兲–14共c兲, respectively. The maximum stress intensities are obtained for the
loading case, M yb, at ␪ = 0 deg, for M xb case, at ␪ = 90 deg, and for
M zb, at ␪ ⬇ 60 deg, respectively. The curves, Kyb, Kxb, and Kzb
versus ␭ and t / T when ␳0 = 0.8 are shown in 关26兴.
8
Conclusion
A thin shell theoretical solution of two normally intersecting
cylindrical shells subjected to three kinds of branch pipe moments
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when d / D is small. The present theoretical method can be applicable up to d / D 艋 0.8, ␭ = d / 冑DT 艋 8, and d / D 艋 t / T 艋 2 successfully.
Nomenclature
Cn ⫽ complex constants in the homogenous solution
for the main shell
d , D ⫽ diameters of the branch pipe and main shell,
respectively
Dkl ⫽ real constants in the homogenous solution for
the branch pipe
E , ␯ ⫽ Young’s modulus and Poisson ratio,
respectively
F ⫽ boundary force vector at the intersecting curve
共1兲
共2兲
GN , GN ⫽ trigonometric functions shown in Table 1
i ⫽ unit vector
k␯ , kt , kmax ⫽ dimensionless longitudinal, transverse, and
maximum principal stresses, respectively
Kxb , Kyb , Kzb ⫽ dimensionless maximum stress intensities for
load cases M xb, M yb, and M zb, respectively
2L ⫽ 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 coordinates of the main shell
N ⫽ load case number shown in Table 1
Q* , S* ⫽ boundary effective transverse and in-surface
shear forces, respectively
r , R ⫽ radii of the branch pipe and main shell,
respectively
t , T ⫽ thicknesses of the branch pipe and main shell,
respectively
T␰ ⫽ resultant force in the Cartesian coordinates of
the main shell
u ⫽ boundary displacement vector at the intersecting curve
u␰ , 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 middle
surface of shell
␪ , ␵ ⫽ Cartesian coordinates on the developed surface
of the branch pipe
␳ , ␪ , z ⫽ global cylindrical coordinates in 3D space
␳0 = d / D ⫽ diameter ratio
␰ , ␸ ⫽ Cartesian coordinates on the developed surface
of the main shell
␺ ⫽ displacement function for the branch pipe
␹ ⫽ complex-valued displacement-stress function
for the main shell
Subscripts
Fig. 14 The stress concentration factors versus ␭ and t / T
„␳0 = 0.7…. „a… Kyb for in-plane bending moment Myb; „b… Kxb for
out-of-plane bending moment Mxb; „c… Kzb for torsion moment
Mzb.
is presented. The results by the present method are in very good
agreement with those obtained by test and by FEM. The present
analytical results are in good agreement with WRC Bulletin 297
Journal of Pressure Vessel Technology
n ⫽ components in the normal direction to the
middle surface of the main shell
␣ , ␤ ⫽ components in the 共␣ , ␤兲 coordinate system for
the main shell
␳ , ␪ , z ⫽ components in the 3D cylindrical coordinate
system, coincident with the normal, circular,
and longitudinal directions, respectively, of the
branch pipe
␯ , t ⫽ components in the normal and tangent directions to ⌫, respectively
⌫ ⫽ value at the intersecting curve
␰ , ␸ ⫽ components in the 共␰ , ␸兲 coordinate system for
the main shell
N = 1 , 2 , 3 ⫽ denote different cases
NOVEMBER 2005, Vol. 127 / 367
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Superscripts
共t兲 ⫽ for the branch pipe
ˆ ⫽ for the particular solutions
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