asmeed1b1737-8479-20131028110317.pdf

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Liping Xue

G. E. O. Widera

Marquette University,Center for Joining and Manufacturing Assembly,

P.O. Box 1881,Milwaukee, WI 53201-1881

Zhifu SangNanjing University of Technology,

Nanjing, Jiangsu, 210009,People’s Republic of China

Flexibility Factors for Branch PipeConnections Subjected toIn-Plane and Out-of-PlaneMomentsStress intensity factor and flexibility factor are important analysis parameters for branchpipe connections subjected to in-plane and out-of-plane moments. The calculation ofstress intensity factors for a large range of unreinforced fabricated tees (0.333�d /D�1, 20�D /T�250, d /D� t /T�3) was performed by Widera and Wei [WRC Bulletin497 2004] and empirical formulas were provided based upon a parametric finite elementanalysis employing four-node shell elements. The purpose of this paper is to extend theprevious effort by Widera and Wei and calculate the in-plane and out-of-plane flexibilityfactors for the same range of geometric parameters. Similarly, empirical formulas for thedetermination of these flexibility factors are proposed. The results show that the lengthsof the branch and run pipes as well as the geometric parameters (d /D, D /T and t /T)have an effect on the calculation of flexibility factors. �DOI: 10.1115/1.2140801�

1 IntroductionBranch pipe connections are configurations commonly used in

many industries. Pipeline transportation, nuclear and power engi-neering, chemical and petrochemical engineering and aerospace,to name a few. If a branch connection is subjected to a momentloading on its branch pipe, local deformations and high stressconcentrations arise in the vicinity of the branch-to-run-pipe junc-tion. Also, the rotational deformations of a branch pipe are large incomparison with those of a branch connection modeled as a rigidjuncture �2–5�. However, the piping code ASME B31.3 Section319.3 �6� �see Appendix D of reference code �6�� normally definesthe flexibility factors as being equal to 1 on the assumption thatthe junctions are rigid. This rigid juncture interpretation can beinaccurate �7�. WRC Bulletin 297 �8� provided Figs. 59 and 60 toestimate the flexibility of branch connections resulting from radialload and in-plane and out-of-plane moments, respectively. How-ever, its application was very limited and did not describe whatboundary conditions were employed. The theory and associatedcomputer program developed by Steel �5� were employed for thedevelopment of �8�. To illustrate the potential significance of flex-ibility factors for branch pipe connections, WRC Bulletin 329 �7�provided an example to show the effect of the K’s on calculatedmoments in piping system. WRC Bulletin 463 �9� provided acomprehensive discussion of the definition of flexibility factors.Available data was discussed and the methods for developing flex-ibility factors from test data or elastic analysis were suggested.The ASME Boiler and Pressure Vessel Code, Section 3NB-3687.5 �10� provides two equations for in-plane and out-of-plane flexibility factors, but their applicability is limited to d /D�0.5, D /T�100. The code equations were developed from theresults of 25 finite element analyses, according to the authors of�2�. Fujimoto and Soh �11� performed a parametric finite elementanalysis for the calculation of in-plane and out-of-plane flexibilityfactors using 8-node isoparametric shell elements. A fixed-fixedmodel �both run pipe ends were fixed� was employed. The lengthof the branch and run pipes was fixed at 2.89 times Do. The

Contributed by the Pressure Vessel and Piping Division of ASME for publicationin the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received September 26,2005; final manuscript received November 1, 2005. Review conducted by Greg. L.
Hollinger.
Journal of Pressure Vessel Technology Copyright © 20

rom: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 10/29/20

proposed empirical formulas were limited to 50�D /T�300,0.5�d /D�0.95, 0.25� t /T�0.95. Williams and Clark �12� in-cluded finite element analyses results, one run end fixed, for theflexibility factors for branch moments. Three-dimensional isopara-metric 20-node solid elements and the lower order 8-node isopara-metric elements were utilized for the smaller and larger diametermodel geometries, respectively. Branch pipe and run pipe lengthswere fixed at 1.5Do and 1.5do, respectively. The models by Will-iams and Clark �12� were limited to: 1 /16�do /Do�7/16, 5 /16� t /T�3/4. Wais, Rodabaugh and Carter �13� numerically evalu-ated the flexibility factors for branch moments employing 4-nodequadrilateral shell elements. The total run segment length wasfixed at 4Do. Two boundary conditions, one run end fixed andboth run ends fixed, were employed. The average of the resultsobtained from these two cases was used to determine the flexibil-ity of the branch. The correlation equations of Wais, Rodabaughand Carter �13� were limited to: 7.50�D /T�99, 7.50�d / t�198, 0.125�d /D�1.0.

The purpose of this paper is to extend the previous effort byWidera and Wei �1� and present the development of in-plane andout-of-plane flexibility factors for the same range of geometricparameters employed in �1�. Here, the particular cases with t /T=3 were not considered previously. Based on a parametric study,the empirical formulas for the determination of flexibility factorsare proposed.

2 Scope and Method

2.1 Scope. The geometric parameters employed for this study,which cover most of the practical cases of branch pipe connec-tions, were determined by a committee of the Pressure VesselResearch Council and are tabulated in Table 1. Generally, thepresent study covers the range of geometric parameters employedin previous works �e.g., Refs. �8–12�� and extends the range to alarger t /T ratio with a higher D /T ratio.

2.2 Method. Finite element analysis is performed on 43 con-figurations �see Table 1� to investigate the relationship betweenthe various geometric parameters and flexibility factors. Aspointed out by Williams and Clark �12�, the “both ends fixed”boundary condition employed in the earlier work by Fujimoto andSoh �11� would quickly succumb to thermally induced bending
moments and in no way models any realistic piping system in the
FEBRUARY 2006, Vol. 128 / 8906 by ASME

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chemical or power industry. Therefore, finite element models withone end of the run pipe fixed and the other free are employed �seeFig. 1�. This same boundary condition was used in the calculationof stress intensity factor �SIF� due to branch bending moments inRef. �1�.

The accuracy of flexibility factors influences the accuracy ofstress intensity factors. Inaccurate flexibility factors lead to inac-curately calculated moments which result in inaccurate stress in-tensity factors. As stated by Rodabaugh �14�, it is of great interestto obtain both the stress intensity and flexibility factors from thesame finite element models. The same finite element code�COSMOS �15�� and element type �4-node shell element� as thoseused for the previous work �1� are employed for this study. Wholeshell models are utilized. The element spacing around the perim-eter of the connection is equal to 7.5° and the element lengthperpendicular to the intersecting curve is equal to t /2 for the runpipe and T /2 for the nozzle. The aspect ratios of the elements inthe connection region are less than 5. Figure 2 shows a represen-tative finite element mesh for d /D=1.0. For the sake of complete-ness, some of the results for SIFs from Ref. �1� due to branchbending moments are presented in Appendix. The guidelines formodeling cylinder-to-cylinder intersections are provided in WRCBulletin 493 �16�.

WRC Bulletin 463 �9� provided the general basic philosophyunderlying flexibility factors and a detailed description of the defi-nition of a flexibility factor for branch connection as well as themethods employed for developing flexibility factors from test dataor elastic analysis. The flexibility factors, Ki, of branch connec-tions are defined by a virtual spring rotation �2� �see Fig. 3�:

K =��n − �beam�EIn

Mdo�1�

To determine the flexibility factors Ki and Ko from the numeri-cal results, �n and �beam will need to be substituted into Eq. �1�.The basic derivation of equations for �beam due to in-plane andout-of-plane moments on the branch can be obtained from el-

Table 1 Geometry parameters for pipe branch connections

Number of models t /T D /T d /D

14a 0.333, 1, 3 20, 60, 100, 150, 250 0.33314b 0.5, 1, 3 20, 60, 100, 150, 250 0.512 0.75, 1, 3 20, 60, 100, 150 0.753 1 20, 60, 100 1.0

aModel with t /T=3, D /T=20, and d /D=0.333 is excluded.bModel with t /T=3, D /T=20, and d /D=0.5 is excluded.

Fig. 1 Schematic of pipe connection

90 / Vol. 128, FEBRUARY 2006


ementary beam theory �17� based on a cantilever model.For an in-plane moment:

�n = �Y �2�

�beam =Mi

E�Ln

In+

1.0Lv

Iv� �3�

where �y is the y-direction rotation at nodes A, C �see Fig. 4� atthe top of the branch pipe �from FEA�.

For an out-of-plane moment:

�n = �X �4�

�beam =Mo

E�Ln

In+

1.3Lv

Iv� �5�

where �x is the x-direction rotation at nodes B, D �see Fig. 4� atthe top of the branch pipe �from FEA�.

In-plane and out-of-plane flexibility factors can be determined

Fig. 2 Finite element mesh „d /D=1.0…

Fig. 3 Branch flexibility concept „see Ref. †9‡…

Transactions of the ASME


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using the appropriate finite element models for the range of pa-rameters indicated in Table 1. Applicable correlation equationscan then be developed using these factors.

3 Parametric Finite Element Modeling

3.1 Effect of the Lengths of the Run and Branch Pipes. Ina parametric finite element model, it is desirable that the run andbranch pipes be long enough to eliminate their effects on theflexibility factors. The effects of lengths of the run and branchpipes on the flexibility factors for d /D=0.5 and d /D=1.0 branchconnections with D /T=100 and t /T=d /D subjected to in-planeand out-of-plane moments are investigated. The results are givenin Table 2. From Table 2, it is seen that very long finite elementmodels are necessary for converged flexibility factors. If shortermodels are utilized to calculate the flexibility factors, smaller Kvalues would be expected. The small flexibility factors in com-parison to the length of the piping system will have only minoreffects on the calculated moments. While, the larger values willreduce the magnitude of the calculated moments acting on thebranch connections.

3.2 Parametric Finite Element Modeling. On the basis ofthe results shown in Table 2, the parametric modeling employssufficiently long branch and run pipes for the finite element rep-resentations. The details of the lengths employed for shell inter-sections with various diameter ratios are given in Table 3. For thisstudy, the mean diameter of run pipe D is taken at 100 inches andd, t and T are changed as variables. The modulus of elasticity andPoisson’s ratio are taken as 30�106 psi and 0.3, respectively.

Using the method presented earlier in the Scope and Methodsection, the in-plane and out-of-plane flexibility factors for 43

Fig. 4 Rotations at the top of the branch

Table 2 Effect of lengths of run and branch pipes on flexibilityfactors „D /T=100…

d /D= t /T=0.5Lv=4D, Ln=2.5D Lv=7D, Ln=7D Lv=8D, Ln=7.5D

Ki10.5 11.8 11.8

Ko72.9 77.2 76.9

d /D= t /T=1.0Lv=5D, Ln=4.5D Lv=9D, Ln=8.5D Lv=9D, Ln=9D

Ki21.4 21.7 21.6

Ko49.1 49.4 49.4

Journal of Pressure Vessel Technology


finite element models are achieved �see Table 4�. Table 4 showsthat both in-plane and out-of-plane flexibility factors increase withincreasing t /T. The effect of t /T on the out-of-plane flexibilityfactor is striking. An out-of-plane moment produces torsion on therun pipe, making it more flexible. An in-plane moment producesbending of the run pipe which reduces flexibility. The flexibilityfactors also increase with increasing D /T because thin models aremore flexible. The effect of d /D on flexibility factors is less strik-ing than that due to t /T. Fujimoto and Soh �11� reached similarconclusions.

A previous study �1� investigated the effects of the length ofbranch and run pipes on SIFs. The final lengths of run and branchpipes used to calculate SIFs are tabulated in Table 7 in Appendix.From a comparison of Table 3 and Table 7 the FE models em-ployed for flexibility factors are longer than those used for SIFs.Since the shorter FE models �see Table 7� yield converged resultsfor the SIF, the calculation of SIF will not be affected if the longer

Table 3 Lengths of run and branch pipes for FE models

d /D Lv Ln

0.333 6D 4.5D0.5 7D 7D0.75 8D 8D1.0 9D 8.5D

Table 4 In-plane and out-of-plane flexibility factors

Model d /D D /T t /T Ki Ko

1 0.333 20 0.333 2.85 5.382 0.333 20 1 3.82 8.093 0.333 60 0.333 7.79 26.784 0.333 60 1 10.38 46.005 0.333 60 3 20.88 109.166 0.333 100 0.333 11.41 54.707 0.333 100 1 16.50 102.058 0.333 100 3 34.76 257.959 0.333 150 0.333 15.10 94.7510 0.333 150 1 22.83 187.0711 0.333 150 3 49.81 489.8212 0.333 250 0.333 20.56 182.1213 0.333 250 1 33.39 386.2814 0.333 250 3 75.40 1044.2615 0.5 20 0.5 3.33 7.7716 0.5 20 1 3.38 9.4117 0.5 60 0.5 8.40 37.3618 0.5 60 1 10.02 54.2919 0.5 60 3 19.25 131.3420 0.5 100 0.5 11.77 77.1521 0.5 100 1 15.03 119.1522 0.5 100 3 30.28 305.5423 0.5 150 0.5 15.12 136.9524 0.5 150 1 19.98 219.8925 0.5 150 3 41.93 583.0126 0.5 250 0.5 20.32 274.4627 0.5 250 1 27.93 459.0028 0.5 250 3 61.36 1253.0529 0.75 20 0.75 3.61 8.1530 0.75 20 1 3.84 9.1031 0.75 20 3 5.87 17.2932 0.75 60 0.75 8.00 37.5433 0.75 60 1 8.98 44.6534 0.75 60 3 16.47 107.4535 0.75 100 0.75 11.34 77.4436 0.75 100 1 12.68 94.1737 0.75 100 3 24.46 240.2738 0.75 150 0.75 14.44 137.3339 0.75 150 1 16.37 169.7740 0.75 150 3 32.59 447.5141 1.0 20 1 5.35 6.9342 1.0 60 1 14.43 26.8143 1.0 100 1 21.67 49.41

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FE models �Table 3� are used. Therefore, the lengths of branchand run pipes tabulated in Table 3 are appropriate for the calcula-tion of both the SIF and flexibility factor.

4 Correlation EquationsThe data in Table 4 was obtained from the finite element analy-

sis of the 43 models. Correlation equations for branch connec-tions, with one end of the run pipe fixed and the other free andsubjected to in-plane and out-of-plane moments, respectively, areobtained by using the data from Table 4 and the software packageSTATISTICA �18�. The resulting correlation equations for the runpipe are as follows:

Ki = 0.680� d

D�−0.242�D

T�0.802� t

T�0.622�3.437� d

D� − 7.414� d

D�2

+ 4.766� d

D�3� �6�

Table 5 Differences between FEA and correlation equations

Correlationcoefficient

VarianceR2

Max. difference between FEAand correlation equation

Eq. �6� 0.996 99.2% 28%Eq. �7� 0.999 99.9% 36%

Table 6 Comparisons of flexibility fac

Ki

d /D D /T t /TCode�10�

Fujimoto�11�

Wa�13

0.333 20 0.333 1.33 — 2.0.333 2.31 — 4.

60 0.333 4.00 — 5.1 6.92 — 9.3 12.0 — 19.

100 0.333 6.66 — 7.1 11.5 — 13.3 20.0 — 26.

150 0.333 — — —1 — — —3 — — —

250 0.333 — — —1 — — —3 — — —

0.5 20 0.5 2.00 — 2.1 2.83 — 4.

60 0.5 6.00 5.37 5.1 8.48 7.13 9.3 14.7 — 17.

100 0.5 10.0 7.93 8.1 14.1 10.5 12.3 24.5 — 24.

150 0.5 — 10.8 —1 — 14.4 —3 — — —

250 0.5 — 16.0 —1 — 21.3 —3 — — —

0.75 20 0.75 — — 3.1 — — 3.3 — — 7.

60 0.75 — 4.99 6.1 — 5.61 8.3 — — 16.

100 0.75 — 7.14 9.1 — 8.03 11.3 — — 22.

150 0.75 — 9.49 —1 — 10.7 —3 — — —

1 20 1 — — 3.60 1 — — 7.

100 1 — — 11.

92 / Vol. 128, FEBRUARY 2006


Ko = 0.172� d

D�0.538�D

T�1.515� t

T�0.862�5.935� d

D� − 10.454� d

D�2

+ 4.797� d

D�3� �7�

These equations hold for the following range of parameters:

0.333 � d/D � 1

20 � D/T � 250 �8�

d/D � t/T � 3

The correlation coefficients, variances and maximum percent-age differences between the FEA results and the correlation equa-tions are listed in Table 5. Table 5 shows that the correlationequations, found above, fit analysis data well as indicated by thecorrelation coefficients and variances of close to 1.

5 Discussion of ResultsTable 6 shows the comparisons of the flexibility factors from

the present study �see Table 4� to those from the ASME Code�10�, Fujimoto �11� and Wais �13�. From Table 6, one can see thatthe agreement is good for both in-plane and out-of-plane flexibil-ity factors when d /D�0.75 and D /T�100. However, the presentstudy yields larger values of K, especially Ko, for d /D=1.0,

s from different correlation equations

Ko

PresentCode�10�

Fujimoto�11�

Wais�13� Present

2.85 2.98 — 5.94 5.383.82 5.16 — 13.1 8.097.79 15.5 — 17.9 26.8

10.4 26.8 — 39.3 46.020.9 46.4 — 86.5 10911.4 33.3 — 29.8 54.716.5 57.7 — 65.7 10234.8 100 — 144 25815.1 — — — 94.822.8 — — — 18749.8 — — — 49020.6 — — — 18233.4 — — — 38675.4 — — — 1044

3.33 4.47 — 8.09 7.773.38 6.32 — 13.3 9.418.40 23.2 29.2 24.4 37.4

10.0 32.9 44.0 40.0 54.319.3 56.9 — 88.0 13111.8 50.0 54.37 40.6 77.215.0 70.7 81.84 66.8 11930.3 122 — 147 30615.1 — 89.0 — 13720.0 — 134 — 22041.9 — — — 58320.3 — 166 — 27427.9 — 249 — 45961.4 — — 1253

3.61 — — 8.13 8.153.84 — — 9.99 9.105.87 — — 22.0 17.38.00 — 23.3 24.5 37.58.98 — 27.6 30.1 44.7

16.5 — — 66.1 10711.3 — 39.0 40.8 77.4412.7 — 46.2 50.2 94.224.5 — — 110 24014.4 — 58.6 — 13716.4 — 69.5 — 16932.6 — — — 447

5.35 — — 4.18 6.9314.4 — — 12.6 26.821.7 — — 21.0 49

tor

is�

4168088411799

8329950354187

319464972918577

71800



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D /T�100. The longer pipe length found in models with greaterd /D and D /T ratios reduces the stiffness of the structures andincreases the flexibility factor. In addition, the boundary condi-tions employed for the finite element analysis influence the flex-ibility factors as well. It is obvious that compared with the one runend fixed model, the both run ends fixed model �11� increases thestiffness of the structure and reduces the flexibility factors. Thelengths of run and branch pipes and boundary conditions em-ployed for Refs. �11� and �13� can be found in the Introductionsection.

6 ConclusionsAn extensive parametric finite element analysis of large diam-

eter ratio branch connections subjected to in-plane and out-of-plane moments was carried out. The following general conclu-sions are drawn:

1� The length of the branch and run pipes does affect theflexibility factors, especially the out-of-plane flexibilityfactor. Therefore, the proper length for the FE modelshould be ascertained prior to performing the analysis.

2� Geometric parameters have an effect on flexibility fac-tors. Both in-plane and out-of-plane flexibility factors in-crease with increasing t /T and D /T. In comparison withD /T and t /T, the effect of d /D on flexibility factors isless striking.

3� The FE predicted in-plane and out-of-plane flexibilityfactors presented in Table 4 can serve as the basis of thedesign of branch pipe connections under branch bendingmoments within the range of parameters stated in �8�.

Nomenclatured � mean diameter of branch pipe, in.

do � outside diameter of branch pipe, in.D � mean diameter of run pipe, in.

Do � outside diameter of run pipe, in.t � nominal wall thickness of branch pipe, in.

T � nominal wall thickness of run pipe, in.Ln � length of branch pipe, in.Lv � distance between the intersection of vessel and

nozzle axes and the fixed end of the vessel, in.K � flexibility factorKi � in-plane flexibility factorKo � out-of-plane flexibility factorM � moment, in.-lbMi � in-plane moment, in.-lbMo � out-of-plane moment, in.-lb

E � Young’s modulus of elasticity, psiI � moment of inertia, in.4

In � moment of inertia of branch pipe, =��do4

−di4� /64, in.4

Iv � moment of inertia of run pipe, =��Do4−Di

4� /64,in.4

� � Poisson’s ratio� � rotation, rad

�n � rotation at the top of the branch pipe as calcu-lated by finite element analysis, rad

�beam � nominal rotation at the top of the branch pipeaccording to beam theory, rad

�x ,�y � rotations at the top of the branch pipe �fromFEA�, rad

� translation at the top of the branch pipe, in.x ,y � translations at the top of the branch pipe �from

FEA�, in.

Journal of Pressure Vessel Technology


AppendixStress Intensity Factors for Pipe Branch Connections Sub-

jected to In-Plane and Out-of-Plane Moments. The finite ele-ment commercial code COSMOS is used to carry out a parametricstudy of pipe branch connections subjected to in-plane and out-of-plane moments on the branch pipe. The geometry parametersfor the calculation of stress intensity factors are the same as thosetabulated in Table 1. Here, 4-node shell elements are used basedon cantilever models with one run end fixed and the other free.The choice of these geometry parameters is based on the sugges-tions from PVRC committee members.

The benchmark studies were carried out before the parametricstudy �see Ref. �1� for detailed information�. Based on the excel-lent agreement between the finite element results and the experi-mental data as well as the investigation of lengths of the branchand run pipes on the finite element results, the parametric model-ing was as follows:

A� A 4-node shell element model is used.B� A long run pipe and a long branch pipe are used. The

details of the lengths employed for branch connectionswith various diameter ratios are tabulated in Table 7.

With the FEA data from forty-three models, the following cor-relation equations are developed using STATISTICA �18�. Theproposed equations fit the FEA data points well as indicated bythe correlation coefficients shown in Table 8.

For in-plane moment on the branch:

Sn

0= �− 1.119 + 11.23� d

D� − 19.67� d

D�2

+ 11.32� d

D�3�

��D

T�0.4763

�A1�

Snm

0= �− 0.074 + 1.505� d

D� − 2.731� d

D�2

+ 1.775� d

D�3�

��D

T�0.6526� t

T�0.109

�A2�

Table 7 Length of run and branch pipes for FE models

d /D Lv Ln

0.333 4D 2.5D0.5 4D 3D0.75 5D 4D1.0 5D 5D

Table 8 Correlation coefficients of proposed equations

Correlation coefficient

Eq. �A1� 0.915Eq. �A2� 0.950Eq. �A3� 0.977Eq. �A4� 0.990Eq. �A5� 0.912Eq. �A6� 0.985Eq. �A7� 0.994Eq. �A8� 0.974

FEBRUARY 2006, Vol. 128 / 93


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Sv

0= �− 0.0022 + 4.729� d

D� − 8.674� d

D�2

+ 5.237� d

D�3�

��D

T�0.5260� t

T�0.8120

�A3�

Svm

0= �0.3722 − 0.6740� d

D� + 1.615� d

D�2

− 0.8049� d

D�3�

��D

T�0.5800� t

T�0.8097

�A4�

For out-of-plane moment on the branch:

Sn

0= �− 0.863 + 5.559� d

D� − 5.895� d

D�2

+ 1.780� d

D�3�

��D

T�0.802� t

T�−0.252

�A5�

Snm

0= �− 0.046 + 0.4733� d

D� − 0.4663� d

D�2

+ 0.1542� d

D�3�

��D

T�0.982� t

T�−0.109

�A6�

Sv

0= �0.0947 + 1.099� d

D� − 0.2395� d

D�2

− 0.5410� d

D�3�

��D

T�0.8972� t

T�1.115

�A7�

Svm

0= �− 0.0980 + 0.9376� d

D� − 1.427� d

D�2

+ 0.7837� d

D�3�

��D�0.8566� t �0.7317

�A8�
T T
94 / Vol. 128, FEBRUARY 2006


References�1� Widera, G. E. O., and Wei, Z., 2004, “Large Diameter Ratio Shell Intersec-

tions, Part 3: Parametric Finite Element Analysis of Large Diameter ShellIntersections �External Loadings�,” WRC Bulletin 497.

�2� Rodabaugh, E. C., and Moore, S. E., 1979, “Stress Indices and FlexibilityFactors for Nozzles in Pressure Vessels and Piping,” NUREG/CR-0788,ORNL/Sub-2913.

�3� Rodabaugh, E. C., and Moore, S. E., 1977, “Flexibility Factors for Small�d /D�1/3� Branch Connections with External Loadings,” ORNL/Sub-2913-6.

�4� Rodabaugh, E. C., 1970, “Stress Indices for Small Branch Connections withExternal Loadings,” ORNL-TM-3014.

�5� Steele, C. R., and Steele, M. L., 1983, “Stress Analysis of Nozzles in Cylin-drical Vessels with External Load,” J. Pressure Vessel Technol., 105, pp. 191–200.

�6� ASME Code for Pressure Piping, B31.3, Chemical Plant & Refinery Piping,2004 edition, ASME, New York.

�7� Rodabaugh, E. C., 1987, “Accuracy of Stress Intensity Factors for BranchConnections,” WRC Bulletin 329.

�8� Mershon, J. L., et al., 1984, “Local Stresses in Cylindrical Shells Due toExternal Loadings on Nozzles–Supplement to WRC Bulletin 107,” WRC Bul-letin 297.

�9� Rodabaugh, E. C., and Wais, E. A., 2001, “Standard Flexibility Factor Methodand Piping Burst and Cyclic Moment Tests for Induction Bends and 6061-T6and SS 304 Transition Joints,” Report 1, WRC Bulletin 463.

�10� ASME Boiler and Pressure Vessel Code, Section 3, Division 1, 1992, “Rulesfor Construction of Nuclear Power Plant Components,” ASME, New York.

�11� Fujimoto, T., and Soh, T., 1988, “Flexibility Factors and Stress Indices forPiping Components with D /T� =100 Subjected to In-Plane or Out-of-PlaneMoment,” J. Pressure Vessel Technol., 110, pp. 374–386.

�12� Williams, D., and Clark, J., 1996, “Development of Flexibility Factors forFabricated Tee Branch Connections,” PVP 331.

�13� Wais, E. A., Rodabaugh, E. C., and Carter, R., 1999, “Stress IntensificationFactors and Flexibility Factors for Unreinforced Branch Connections,” PVP383.

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�15� COSMOS/DesignSTAR, Version 3.0, 2000, Structural Research & AnalysisCorporation �SRAC�.

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