nafems uk 2012 final paper

A comparison of simple analytical methods for evaluating local stresses at pipe support attachments with Finite Element Analysis results.

Copyright 2012 by Chicago Bridge & Iron Company N.V (CB&I). All rights reserved. Reproduction or transmission in any form or use in any manner, in whole or in part, is prohibited without the prior written permission of CB&I.


Anindya Bhattacharya

Principal Engineer (Stress Analysis)

CB&I

Email: [email protected]

Abstract:Stress Analysis of piping systems is usually done using beam based finite element analysis and conventionally local stresses at pipe support attachments are evaluated using elementary shell theory (ring loading around circumference) . In this paper this approach will be reviewed critically with reference to results from shell based finite element analysis and stress classification on the guidelines of the ASME Pressure Vessel Code Sec VIII Division 2. Results will also be compared with more involved analytical solutions and industry standards like WRC 107. The present study will involve only circular attachments. Two FE codes, FE 107 (a special purpose FE code from Paulin Research Group, Houston, Texas) and ABAQUS (a product of Dassault Systmes) VERSION 6.9-1 have been used for the finite element analysis.

Theoretical developments leading to the methods widely used in the industry:The theoretical work for cylinder-to-cylinder intersections can be broadly classified into the way of approaching the problem. The first approach is to use the governing differential equations using thin shell theory with the applied loading based on a particular mathematical form (Double Fourier series, ring loading etc.). The second approach is based on equilibrium and compatibility considerations of two intersecting shells along a space curve. Same or different shell theories have been used for header and branch (circular attachments called trunnions, in the context of this paper). In this paper we will mainly highlight the first approach as it forms the basis for the two most widely used methods in the industry. The second approach will not be discussed in detail.



We will start with the method usually known as the Kellogg method; its origin can be traced in [3]. Since the method has its origin in the theory of thin shells, we will refer to it as Timoshenko method in this paper The Kellogg (Timoshenko) method:Governing differential equation: [1]For an Axi-symmetric loading on a circular cylinder, the governing differential equation is the well known beam on elastic foundation equation:

ZKww

Ddxd =+

4

4 (1)

where,D, is the Flexural Rigidity of the shell w = deflection in vertical DOFx = Direction of Longitudinal axis of the cylinder

Z = load in vertical direction

K = Foundation stiffness, rEt

2

E = Modulus of elasticityt = Shell thickness.r =Shell mid-surface radius

DZ

DrEtww

dxd =+ 24

4

(2)

)1(12 23

m-=

EtD

where = Poissons ratio

Introducing 22

24 )1(3

trmb -= i.e.

rt28.1

=b considering = 0.3

we therefore get



DZw

dxwd

=+ 44

44b (3)

The solution of this differential equation and boundary conditions are detailed in [1]Extending the above analysis to a case of bending of a cylindrical shell by a load uniformly distributed along a circular section [1], we get:

Maximum Bending Momentb4P

= , where P= load per unit circumference

applied through the ring of load.

Bending stress 5.1

5.0Pr17.1t

bending =s (4)

P can be defined in terms of a local radial load, Pr, and local moment Mr. This is necessary because P is a line load distributed around the circumference of the shell.

If a load Pr is divided by the attachment perimeter it becomes n

rr

Pp2

for

a nozzle of radius rn. Or a Moment divided by the Section modulus of

the attachment becomes 2n

r

rMp

Flexural stresses are added to membrane longitudinal and hoop stresses to get total stress = membrane stress in direction i + flexural stresses in direction i computed by the expression in equation (4)

To compute P, the steps to be followed are:

Computation of loads in longitudinal and circumferential directions by use of the following expressions:

Longitudinal force = (Longitudinal force x moment arm)/pr2

Where r = radius of Trunnion

Circumferential force = (Circumferential force x moment arm)/pr2

Radial force = Radial Load /2pr

Equivalent circumferential force = 2 x Circ force +1.5xRadial force



Equivalent longitudinal force = 1.5 x Radial force +Longitudinal Force

The above forces are used as P in equation (4) The reason behind the use of the factors 1.5 and 2.0 is attributed to higher flexibilities in these directions.

The flexural stresses in longitudinal and circumferential directions are then computed using these equivalent forces and the membrane pressure stresses are then added to compute the total stresses.

Stresses in the Trunnion attachment is computed asAF

ZM

+ where M=

Resultant applied bending moment (square root of squares of the bending moments in longitudinal and circumferential directions) and F is the applied radial force. Z and A are the section modulus and cross sectional areas respectively of the trunnion pipe.Points to note about the background theory and about the way the above method is typically implemented in engineering design using spreadsheets:

Use of thin shell theory < 1( )10

tr

Axi-symmetric loading

Simple approach in specifying the loads are the main features of this method

Stresses in the circular attachment computed by elementary beam theory and not shell theory

Pl+Pb+Q [16] computed as a superposition of flexural stresses using beam on elastic foundation theory with the membrane stresses computed using elementary strength of materials formulas.

Computation for fatigue stresses (Pl+Pb+Q+F)[16] is usually not done using spreadsheets. This is not an inherent problem with the background theory; it has to do with the way the method is usually implemented in spreadsheets.

Multiplication factors used in circumferential and radial directions to simulate higher stresses in these directions [11] attribute to higher flexibilities in these directions.



No computation of shear stresses or Pl (local membrane stresses) and Torsional stresses.

Where things can go wrong: While thin shell theory is valid for most applications, the loading at pipe supports does not create an axi-symmetric situation. Using on thick shells where the governing differential equation requires modifications (to include shear deformation).Nozzle (in this case trunnion) stresses can be underestimated. Flexural stresses due to radial loading are underestimated.Advantages of the method: This method is simple and not much of significant variance with reference to FEA in terms of shell flexural stress when longitudinal force/circumferential force is applied at the end of the nozzle. Nozzle stresses however show values lesser than FEA/WRC 297. It can be easily used as a spreadsheet based method. There is no limitation to Dt

ratio as long as it is within the domain of thin shell theory (i.e. as

long as > 20Dt

). Upper bound of Dt

ratio can be taken same as that in

WRC 107 (=600). Although Pl stresses are not computed and since the allowable stress for Pl+Pb+Q is kept the same as Pl, there should not be a concern in overlooking Pl. Use of multiplication factors (>1.0) in circumferential loading to simulate higher stresses in circumferential direction is justified as the stiffness in circumferential direction is lower than that in the longitudinal direction as can be explained from shell theory and FE results [13].WRC 107 and 297:Another widely used method for computing local stresses in cylindrical shells (pipe support attachments on pipe) is to use simple cookbook style methods as outlined in the two documents: WRC bulletin no. 107 and 297.[8][9]

These bulletins were based on theoretical works by Professor Bijlaard (WRC 107)[5] and Professor Steele (WRC 297)[10] . Bijlaards work was based on Yuans equation [6] which again is a modification of Donnel equations [4]. Donnel equations can be derived after suitable eliminations of the variables u and v as an eighth order partial differential equation in w, which is the cylindrical shell equation under general loading (i.e. not axi-symmetric) and when external loading is applied on a pipe through a pipe support, it is certainly not axi-symmetric. Hence, in terms of applicability, the general equation is



more suited. Loading on the pipe is idealised as radial load, longitudinal moment, circumferential moment, shear force etc., hence the nozzles, as an example, are not part of the analytical model in this case. WRC 107 was developed based on Bijlaards theoretical work supplemented with experimental works. This document provides a set of curves based on non-dimensional parameters on which stresses could be computed along four different points around the loading circumference.

Development of the theoretical work for WRC 107[5]:The historical development leading to Bijlaards work can be summarized as: Differential equation for cylindrical shell subjected to general loading (Donnel) Modifications by Yuan Modifications by P.P.Bijlaard.

Donnels equation [19] is quite simple and its homogeneous equation can be expressed in complex valued displacement stress function form,it can be decomposed into two second order partial differential equations and is easy to solve in polar coordinate system for the cylindrical shell with cut out [19].

Bijlaards equations [5] which are the theoretical basis for WRC 107[8] are:



mm m

q q

mq

- + + + + -

+ + =

2 4 6 68 2

2 2 4 6 6 2 4 2

6 4

4 2 4

12(1 ) 2[ (6 )

(7 ) ]

w w wwt r x r r x

w Zr x D

(5)

mm

q m q q +

- = - + + -

2 2 2 5 54

2 2 2 2 2 2 2 4

1 ( )1 12

w w t w wr ux r x r u r x

(6)

mq q m m q

mm q q

- = + +

- -

- + +

-

3 2 2 54

2 2 2 2 4

5 5

2 3 3 3

2(2 ) [12 1

3 ](1 )

w w t r wr vr x r r x

w wr x r

(7)

where,

w, is displacement in radial direction,

u and v, are displacements along x and y direction respectivelyon the middle surface,

q, the circumferential angle

r, radius of cylinder

t, shell thickness

m, Poisson ratio

= Del Operator

There can be different approaches to the solution to this equation, but in this paper we will only highlight the approach taken by Bijlaard in his work which was to represent the displacements and loadings as double Fourier series. Fourier coefficients are a function of the type of loading (radial, circumferential moment and longitudinal moment). The following diagrams show the idealisations used by Bijlaard in simulating the applied loads for radial, longitudinal and circumferential moments.



Main features of this method:

Thin shell theory Solid attachment without hole in the pipe wall Governing differential equations are solved using Double Fourier

series representation of loads and displacements. This method is widely used but convergence may not be possible for certain boundary conditions [14]

Simply supported boundary conditions Central location of load application Load application is through solid attachments and without any

hole on pipe wall Internal pressure simulation can result in over conservatism Results usually vary significantly with reference to FE analysis

for shear and torsional moment and for <



Cautions:

Over-conservative behavior if pressure stress is simulated as a radial load

For thick shells the theory requires modification Results are generally un-conservative for shear force and

torsional moments

Valid range for dD

is less than 0.3 so for trunnion applications

where the size of the attached pipe is generally one size smaller than the header pipe, the background theory is not applicable [12]. This is mainly attributed to the sensitivity of the modelling of

the intersecting curve for > 0.3dD

[12]

Use of a linearly distributed radial force system instead of a vertical force system results in a transverse bending moment plus a force -- a fact not noted in this approach [19]

WRC Bulletin 297This was based on the theoretical work of Professor Steele and the development of equations was based on the works of Flugge -Conrad

and Sanders -Simmonds [10]. The applicable DT

was increased from

the WRC bulletin of 600 to 2500 and the issue of nozzle stresses wasincluded, i.e., this approach considered a true nozzle opening. However, our present problem does not involve an opening and hence WRC 107 is theoretically more applicable than WRC 297, although computation of the nozzle stresses has been shown in the Results section for comparison.

Difficulties in analytical solution of the problem: The analytical models for both WRC 107 and 297 are based on thin shell assumptions, so they are not valid for thick shell analysis. Bijlaard [7] made changes to his earlier work based on incorporating thick shell corrections using Flugge equations instead of Donnels. The other

difficulty is where > 0.5dD

. The main difficulty is the mathematical

complexity associated with 3D space curve of intersections which

cannot be simulated as a circle when dD

is high [11][12][15]) . Several



researchers have used different methods such as using differential formulations (e.g., use of Donnels equations for header and Flugge equation[2] for branch or Modified Morley equation for header and Goldenveizer equation for branch[11][12]) . Donnels equation which formed the starting point of Bijlaards work has serious deficiencies

when used for problems with > 0.5dD

. (Figure 6.14 of [18])



Challenges in FEA:

The issue of classification of the FE computed stresses on the line of [16] has been dealt with in numerous papers [20] [16] and will not be repeated here. In a nutshell, local membrane stresses are designated as Pl, primary + secondary stresses as Pl+Pb+Q and peak stresses as Pl+Pb+Q+F in line with [16]. Primary stresses develop to maintain equilibrium with external loads, secondary stresses to maintain compatibility of deformation (global) and peak stresses to maintain compatibility of local deformation. Pl stands for local primary stress, Pb for primary bending stress, Q for secondary stress and F for peak stress. Peak stresses are significant only from the standpoint of fatigue failure.

Elements used for FE analysis are 8-node reduced integration shell elements as well as 20-node reduced integration brick elements. For the continuum elements, membrane and bending stresses have been segregated using linearization. The shell element has shown good results in applications involving cylinder-to-cylinder interactions specially when compared with 20-node hexahedral elements with reduced integration. The same element has been used in two FE codes (ABAQUS element type S8R). FE theory convergence theorems are in

2L or 1H norms which are difficult to implement when the exact solution is not shown and in this paper no attempt has been made to evaluate the convergence using these norms. For checking the convergence of an FE model, percentage change in stress is considered from a model with very fine mesh to gradually cruder models. Stresses are checked at Gauss points for accuracy and unaveraged. For convergence, monotonic behavior is checked with a maximum permissible variation in stress taken as 5%.The mesh size around the intersection is taken as 0.3 rt with progressive mesh grading away from it. For continuum elements four elements have been used through the thickness at and close to intersections. The objective of the FE analysis was not to catch the peak stresses which are used for fatigue evaluation, as once the Pl+Pb+Q stresses are computed, the fatigue stresses can be computed using Fatigue Strength Reduction Factors (FSRF) [16]. The paper referenced in [17] shows that modelling of welds to properly simulate joint stiffness does not have serious impact on the computed stresses and hence, welds are not part of the model. To avoid end effect, the location of the trunnion has been taken



as 5D [13] with respect to the end of the header. The worst aspect ratio around the intersection (HEX elements) was 6.0, average aspect ratio 2.0. One end of the header was fixed in all six DOFs and the other end is fixed in five DOFs, the DOF along the longitudinal axis of the header was kept free to generate longitudinal pressure stress (for models where pressure was applied). Linear and full integration elements were not used to avoid shear locking.

Results

Mesh around the intersection curve using HEX elements



Von Mises plot (30x24 HEX elements)

In the WRC 107/297 method, the length of the nozzle is not a parameter and in all tables below, the loads have been applied in the FE model at the end of the nozzle, i.e., the trunnion which in the WRC 107 /297 cases will be a force and a moment for obvious reasons. In a separate table comparison of stresses between WRC 107/297 and FEA has been shown with the loads applied at the shell-nozzle interface. The stresses are Membrane+Bending; with the Timoshenko method thecomputation is for flexural stresses only using equation (4). WRC 107 and 297 results are the maximum among the four locations specified in these documents [8 ] [9]. Computed stresses are in Mpa and are the maximum values rounded to the next integer (some exceptions being stresses at trunnions).



(30 inch header, 24 inch trunnion,. wall thickness =9.52 mm for both) Magnitude of Force =10KN, length of trunnion =100 mm, d/D=0.8, t/T=1):

Load

ing

Type

WR

C 1

07

She

llW

RC

107

Tru

nnio

n

WR

C 2

97 S

hell

WR

C 2

97 T

runn

ion

Tim

oshe

nko

She

llTi

mos

henk

o Tr

unni

on

FEA

She

ll el

emen

t(S

hell)

FEA

She

ll el

emen

ts (T

runn

ion)

FEA

Con

tinuu

m e

lem

ents

(She

ll)

FEA

Con

tinuu

m e

lem

ents

(N

ozzl

e)

Radial Force

45 N.A.

50 54 6 0.6 10 15 9 12

Longi-tudinal Force

3 N.A

6 6 3 0.4 6 5 5 3

Circum-ferential Force

16 N.A

22 20 5 0.4 3 6 3 6



Results (36 inch header, 30 inch trunnion,. wall thickness =9.52 mm for both) Magnitude of Force =10KN, length of trunnion =100 mm, d/D=0.84, t/T=1)

Load

ing

Type

WR

C 1

07

She

llW

RC

107

Tru

nnio

n

WR

C 2

97 S

hell

WR

C 2

97 T

runn

ion

Tim

oshe

nko

She

llTi

mos

henk

o Tr

unni

on

FEA

She

ll el

emen

t(S

hell)

FEA

She

llel

emen

ts (T

runn

ion)

FEA

Con

tinuu

m e

lem

ents

(She

ll)

FEA

Con

tinuu

m e

lem

ents

(Noz

zle)

Radial Force

45 N.A 51 56 6 0.44 21 15 19 15

Longi-tudinal Force

2 N.A 5 4 2 0.2 7 5 6 6


12 N.A 17 16 4 0.2 5 4 4 5



Results (36 inch header, 12 inch trunnion, and wall thickness =9.52 mm for header and 6.35 mm for trunnion) Magnitude of Force =10KN, length of trunnion =100 mm, d/D=0.34, t/T=0.67):

Load

ing

Type

WR

C 1

07

She

llW

RC

107

Tru

nnio

n

WR

C 2

97 S

hell

WR

C 2

97 T

runn

ion

Tim

oshe

nko

She

llTi

mos

henk

o Tr

unni

on

FEA

She

ll el

emen

t(S

hell)

FEA

She

ll el

emen

ts (T

runn

ion)

FEA

Con

tinuu

m e

lem

ents

(She

ll)

FEA

Con

tinuu

m e

lem

ents

(N

ozzl

e)

Radial Force

48 N.A

54 103 15 2 46 48 44 46

Longi-tudinal Force

10 N.A

16 30 11 2 16 16 13 14


31 N.A

41 75 22 2 29 31 27 29



Results (24 inch header, 20 inch trunnion, and wall thickness =9.52 mm header and 6.35 mm for trunnion) Magnitude of Force =10KN, length of trunnion =100 mm, d/D=0.84, t/T=0.67):

Load

ing

Type

WR

C 1

07S

hell

WR

C 1

07 T

runn

ion

WR

C 2

97 S

hell

WR

C 2

97 T

runn

ion

Tim

oshe

nko

She

ll

Tim

oshe

nko

Trun

nion

FEA

She

ll el

emen

t(S

hell)

FEA

She

ll el

emen

ts

(Tru

nnio

n)FE

A C

ontin

uum

el

emen

ts(S

hell)

FEA

Con

tinuu

m e

lem

ents

(N

ozzl

e)

Radial Force

44 N.A 44 90 7 1 19 20 17 19

Longi-tudinal Force

5 N.A 7 13 4 1 10 9 11 8


20 N.A 23 44 7 1 6 7 6 6



Results (24 inch header, 8 inch trunnion, and wall thickness =9.52 mm header and 8.18 mm for trunnion) Magnitude of Force =10KN, length of trunnion =100 mm, d/D=0.36, t/T=0.86):

Load

ing

Type

WR

C 1

07

She

llW

RC

107

Tru

nnio

n

WR

C 2

97 S

hell

WR

C 2

97 T

runn

ion

Tim

oshe

nko

She

llTi

mos

henk

o Tr

unni

on

FEA

She

ll el

emen

t(S

hell)

FEA

She

ll el

emen

ts

(Tru

nnio

n)

FEA

Con

tinuu

m

elem

ents

(She

ll)

FEA

Con

tinuu

m

elem

ents

(Noz

zle)

Radial Force

47 N.A

69 74 16 2 48 43 46 41

Longi-tudinal Force

21

N.A

31 34 20 4 26 21 24 19


53 N.A

77 78 40 4 46 43 44 43



36 inch header and 12 inch trunnion loads applied at shell nozzle interface (Moment=10KN-m and Force=10KN (wall thickness of shell =9.52mm, trunnion = 6.35 mm, d/D=0.34, t/T=0.67)

Load

ing

Type

WR

C 1

07

She

ll

WR

C 1

07 T

runn

ion

WR

C 2

97 S

hell

WR

C 2

97 T

runn

ion

FEA

She

ll el

emen

t(S

hell)

FEA

She

ll el

emen

ts (T

runn

ion)

Longitudinal moment

99 N.A 153 295 108 105

Circumferential moment

310 N.A 413 752 363 401

Radial force 48 N.A 54 103 46 48

Shear Force (long)

4 N.A 4 6 6 7

Shear force (circ)

4 N.A 4 6 10 9

Torsional Moment

13 N.A 13 19 21 23



For the following tables, applied load in longitudinal, circumferential and radial directions =10KN, pressure =18.9Barg. For the WRC 107 analysis, pressure loading has NOT been added as a radial loading at the trunnion attachment.

(30 inch header, 24 inch trunnion, wall thickness =9.52 mm for both)

WRC 107 Shell

WRC 107 Trunnion

Timoshenko

Shell

Timoshenko Trunnion

FEA (shell elements), Shell

FEA (shell elements), Trunnion

258 N.A 87 1 121 63

(36 inch header, 30 inch trunnion, wall thickness =9.52 mm for both)

WRC 107 Shell

WRC 107 Trunnion

Timoshenko

Shell

Timoshenko Trunnion



307 N.A 100 0.8 146 75

36 inch header, 12 inch trunnion, wall thickness =9.52 mm for header and 6.35 mm for trunnion

WRC 107 Shell

WRC 107 Trunnion

Timoshenko

Shell

Timoshenko Trunnion



321 N.A 125 4 170 105

Discussions and conclusions:

WRC 107 results may be higher or lower with respect to FEA results. Flexural stresses based on analytical solution using axi-symmetric ring loading are not significantly in error when compared with FEA as long as the applied loading is not radial. Stresses due to applied loading in circumferential direction are higher when compared with stresses due to applied longitudinal force. This trend, however, is always valid when the



trunnion length is around 4d, something that is typically not expected in a real life scenario where the shortest trunnion length is preferred.

Before FE analysis results are compared against analytical (includinganalytical +experimental works like WRC 107/297), the basis behind the analytical methods has to be understood, i.e., the governing differential equations, solution methods, boundary conditions and the underlying shell theory. The above methods such as Timoshenko, Bijlaard and Steele are based on assumptions relating to a specific shell theory and boundary conditions and hence the FE solution should not be considered as a numerical counterpart to the analytical solutions, i.e.,we should not expect the FE solution to monotonically approach the above solutions with mesh refinement unless the geometry and loading are the same. This is especially in view of the fact that the underlying theory used in the development of the 8-node reduced integration shell element (degenerated shell element, a development of the original Ahmad element) is not exactly in line with the shell theory used in Bijlaards ( WRC 107) or Steeles ( WRC 297) work. WRC 107/297 are based on thin shell theories and are not valid for geometries where d/D is >0.5, which is the most common case for trunnion attachments, hence the results should not be used for d/D>0.5.The results of these analyses can, however, be used for checking the correctness of an FEA analysis for d/D



hand, but in comparison with FEA, a component-by-component analysis has shown that the computation of Pl+Pb+Q stresses at intersections is not grossly incorrect as long as the applied load is not a radial one where stresses are significantly underpredicted. This method is widely used in the industry in spreadsheet form with a conservative allowable equal to the allowable for local primary membrane stresses (Pl) and local primary membrane +primary bending ( Pl+Pb) which is 1.5Sh for the stresses which by and large should be in Pl+Pb+Q category i.e., with an allowable of 3Sm [16]. This conservative allowable irons out the un-conservatism in computation of flexural stresses. Theresults of these spreadsheets should include computation of Pl+Pb+Q+F stresses which can easily be incorporated by using Fatigue Strength Reduction Factors. Shell-based analysis (8-node reduced integration shell elements have shown good results when compared with 20-node reduced integration brick elements). Theoretical work based on equilibrium and compatibility of deformation along the mathematically accurate space curve of intersection [12[14] is more accurate than WRC 107/297 results, but is not currently available in a form easily implementable in engineering design. These recent developments are based on limitations in the approaches used in WRC

107,297, with respect to dD

and the drawbacks in Donnels equation

especially with the order of accuracy [19] and dD

.

Future work:The analysis has to be extended for non-circular attachments,attachments with reinforcement and for attachments at pipe bends.

Acknowledgement: The author would like to express his thanks to Mr. Tony Paulin of Paulin Research Group, Houston, Texas; Dr. S. Saha of Reliance Industries Ltd, India; and Professor K.C. Hwang of Tsinghua University,Department of Applied Mechanics, Peoples Republic of China, for some interesting discussions and valuable guidelines.



References:1. Theory of plates and shells, S.Timoshenko, Weinowsky Kreiger McGraw Hill Publications 19592. Stresses in Shells W. Flugge , Springer 1962

3. Design of piping systems published by M.W.Kellogg Company

4. NACA Report no. 479 Stability of Thin walled tubes under Torsion by L.H. Donnel

5. Stresses from local loadings in Cylindrical shells by P.P.Bijlaard , Trans ASME 77, (1955) 805

6. Thin Cylindrical Shells subjected to concentrated loads by W. Yuan, Quarterly of Applied Mathematics Vol 4, 1946.

7. Stresses in junction of Nozzle to Cylindrical Pressure Vessel for equal diameter of Vessel and Nozzle P.P.Bijlaard, R.J.Dohrmann,I.C.Wang, Nuclear Engineering and Design Vol 5, 1967.

8. Local stresses in spherical and cylindrical shells due to external loadings K.R.Wichman, A.G.Hooper, J.L.Mershon WRC Bulletin No. 107

9. Mehrson et al,Local stresses in Cylindrical shells due to external loadings on Nozzle WRC Bulletin No. 297

10. Stress analysis of nozzles in cylindrical vessels with external load C.R. Steele, M.L.Steele

11. A thin shell solution for two intersecting cylindrical shells due to external branch pipe moments. M.D.Xue, D.F.Li, K.C.Hwang Journal of Pressure Vessel Technology Nov.2005

12. An analytical method for Cylindrical shells with nozzles due to internal pressure and External Loads-Part 1, Theoretical foundation M.D.Xue, Q.H.Du, K.C.Hwang. Journal of Pressure Vessel Technology, June 2010.

13. Flexibility factors for branch pipe connections subjected to in-plane and out of plane moments L.Xue, G.E.O Widera, Zhifu Sang Journal of Pressure Vessel technology Feb 2006 ASME.



14. Stresses and Flexibilities for Pressure vessel attachments by F.M.G. Wong, Dissertation thesis for Master of Science in Nuclear Engineering submitted to MIT (1984)

15. The determination of Elastic stresses near Cylinder to Cylinder intersections- J.G. Lekkerkerker. Nuclear Engineering and Design Vol 20 (1972)

16. ASME Boiler and Pressure Vessel code, Sec VIII Division 2 (2007edition), ASME Publication.

17. A Finite Element-based Study on Stress Intensification Factors (SIF) for Reinforced Fabricated Tees. A. Bhattacharya NAFEMS World Congress Boston 2011

18. Beams, Plates and Shells by L.H. Donnel Mc Graw Hill Company New York 1976

19. Private communication, Professor K.C. Hwang of Tsinghua University, Department of Applied Mechanics, Peoples Republic of China

20. A two step approach of stress classification and Primary Structure method M.W.L.Y Chen and J.G. Li Trans ASME Vol. 122 February 2000

"This paper and may contain confidential or proprietary information. The information, observations, and data presented in this paper are intended for educational purposes only and do not identify, analyze, evaluate, or apply to actual, specific engineering or operating designs, applications, or processes. The comments and opinions are those of the author and not CB&I, are based on available information, and in all cases reference shall be made to the actual wording and meaning of applicable codes and standards. The use of or reliance on any information, observations, or data provided is solely at your own risk."

nafems uk 2012 final paper

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