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thin shell calculation
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  M. D. XueD. F. LiK. 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 modified 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 Introduction Cylindrical shells attached with branch pipes shown in Fig. 1are of common occurrence in the pressure vessel and piping in-dustry. The significant 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   6  and 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 thesignificant 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. 107  11  . 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 and d   /   D  t   /  T   1. Although researchers have spent great efforts toovercome the significant difficulties 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 Analysis The applicable range of the theoretical solutions presented byXue 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 modifiedMorley’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   five 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 five coordinate systems are used instead of previousapproximate expressions, which cause significant error when d   /   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 difficulties 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 five 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  P RESSURE  V ESSEL  T ECHNOLOGY . Manuscript received: March 16,2004; final manuscript received: June 5, 2005. Review conducted by: Dennis K.Williams. Journal of Pressure Vessel Technology  NOVEMBER 2005, Vol. 127  / 357Copyright © 2005 by ASME Downloaded 20 Sep 2011 to 129.5.16.227. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm  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   xb into 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 in  21,22  . 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,    =0  and    =    /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  , arefor Case 4:  T     = 0,  u    = 0,  u n  = 0,  M     = 0   1 a  for Cases 2,3:  u    = 0,  S  * = 0,  Q * = 0,       = 0   1 b  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-infinite long circle pipe with a nonplanar curved end sub- jected 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 defined by       c  /   R ,       c  /   R   c =  R   0  /    2   in thedeveloped surface is taken as a particular solution. The verticalforce system,  q  z , instead of radial force system,  q n , 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,  q n ,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,  q n , because the latter may cause asignificant 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   1 a   and   1 b  , respectively.In view of the deformation field symmetric or antisymmetric Table 1 Three types of symmetry and trigonometric functions  „  n  = „ 2 n  −1 …    /2 l  ;    =   + L  /  R  … Fig. 1 Calculated model and five coordinate systemsFig. 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. 358 /   Vol. 127, NOVEMBER 2005  Transactions of the ASME Downloaded 20 Sep 2011 to 129.5.16.227. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm  with respect to the plane    =0 and    =0,   , shown in Table 1 forthe three different cases, respectively, the Timoshenko equationswith boundary conditions   1 a   and   1 b   at    =±  L  /   R  can be solvedby expanding the displacements and external loads in double Fou-rier series as follows: q    = 0,   2 a  q    = −  m =0   n =1  q mn  2  G  N   2   m    G  N   3    n      2 b  q n  =  m =0   n =1  q mn  3  G  N   1   m    G  N   3    n      2 c  u    =  m =0   n =1  U  mn G  N   1   m    G  N   4    n     ,   3 a  u    =  m =0   n =1  V  mn G  N   2   m    G  N   3    n      3 b  u n  =  m =0   n =1  W  mn G  N   1   m    G  N   3    n      3 c  where  n  =  2 n  − 1     R 2  L ;  n  = 1,2, ... ;    =   x  +  L   /   R ;  G  N   i   i =2,3,4   are shown in Table 1. In Eqs.  2 a  –  2 c   q    and  q n  are the tangential and radial components of   q  z and  q  y , respectively, where q    =  q  y  cos    for case 2−  q  z  sin    for cases 3,4    4 a  q n  =  q  y  sin    for case 2 q  z  cos    for cases 3,4    4 b  For the three cases q  z  =  q  N     ,     for  N   = 3,4   or  q  y  =  q 2    ,     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   0 r  3   7 a  q  3  = −3    M   yb   0 r  3   7 b  q  4  =    2    M   xb  /4  sin   0   2−   0   2cos   0   2  rR 2  7 c  By using Eqs.   4  –  7   the coefficients in Fourier series   2 b   and  2 c   are obtained. Substituting Eqs.   2 a  –  2 c   and   3 a  –  3 c   intoTimoshenko equations, the coefficients of the displacements inEqs.   3 a  –  3 c   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.   3 a  –  3 c   and related expressions of forces and moments.    =   0  cos   ,   8 a     = sin −1    0  sin     8 b  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.   1 a   and   1 b  , 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 modified Morley equa-tion by Zhang et al.   24   which is applicable up to  r   /     RT   1. Theradial displacement,  u n , and the Airy stress function,    , satisfy   2 +12+ 2     i        2 +12− 2     i         = 0   9  Here, 4   2 =  12  1−   2  1/2   R  /  T    and    = u n + 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 to M  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. Journal of Pressure Vessel Technology  NOVEMBER 2005, Vol. 127  / 359 Downloaded 20 Sep 2011 to 129.5.16.227. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm    =  k  = e  4,  N     n = e  1,  N    C  n F  kn     G  N   1   m     10  where  m =2 k  + e  2,  N    and the unknown complex constants  C  n consist of two parts C  n  =  C  n 1  + i C  n 2   11  G  N   1   m     are triangular functions dependent on Case number  N  shown in Table 1 and F  kn  =   − 1  k   1 −  1 2   m 0    J  m − n    −  i    +  e  3,  N    J  − m − n    −  i     H  n  2         12  where    =   1 2  −  i   2  1/2 ,   mn  =  0,  m  n 1,  m  =  n   ,  J  n  and  H  n  2  are the first kind of Bessel function and the secondkind 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  n 1 and  C  n 2  at  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  360 /   Vol. 127, NOVEMBER 2005  Transactions of the ASME Downloaded 20 Sep 2011 to 129.5.16.227. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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