Stress Indices for Elbows With Trunnion Attachments

Stress Indices for Elbows with TrunnionAttachments

Technical Report

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WARNING:Please read the Export Controland License Agreement on theback cover before removing theWrapping Material.

Stress Indices for Elbows withTrunnion Attachments

TR-107453

Final ReportDecember 1998

Prepared forEPRI3412 Hillview AvenuePalo Alto, California 94304

EPRI Project ManagerR. G. Carter

DISCLAIMER OF WARRANTIES AND LIMITATION OF LIABILITIES

THIS REPORT WAS PREPARED BY THE ORGANIZATION(S) NAMED BELOW AS AN ACCOUNT OF WORKSPONSORED OR COSPONSORED BY THE ELECTRIC POWER RESEARCH INSTITUTE, INC. (EPRI).NEITHER EPRI, ANY MEMBER OF EPRI, ANY COSPONSOR, THE ORGANIZATION(S) BELOW, NOR ANYPERSON ACTING ON BEHALF OF ANY OF THEM:

(A) MAKES ANY WARRANTY OR REPRESENTATION WHATSOEVER, EXPRESS OR IMPLIED, (I) WITHRESPECT TO THE USE OF ANY INFORMATION, APPARATUS, METHOD, PROCESS, OR SIMILAR ITEMDISCLOSED IN THIS REPORT, INCLUDING MERCHANTABILITY AND FITNESS FOR A PARTICULARPURPOSE, OR (II) THAT SUCH USE DOES NOT INFRINGE ON OR INTERFERE WITH PRIVATELY OWNEDRIGHTS, INCLUDING ANY PARTY'S INTELLECTUAL PROPERTY, OR (III) THAT THIS REPORT ISSUITABLE TO ANY PARTICULAR USER'S CIRCUMSTANCE; OR

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ORGANIZATION(S) THAT PREPARED THIS REPORT

Wais and Associates, Inc.

ORDERING INFORMATION

Requests for copies of this report should be directed to the EPRI Distribution Center, 207 Coggins Drive, P.O. Box23205, Pleasant Hill, CA 94523, (925) 934-4212.

Electric Power Research Institute and EPRI are registered service marks of the Electric Power Research Institute, Inc.EPRI. POWERING PROGRESS is a service mark of the Electric Power Research Institute, Inc.

Copyright © 1998 Electric Power Research Institute, Inc. All rights reserved.

iii

CITATIONS

This report was prepared by

Wais and Associates, Inc.3845 Holcomb Bridge Road, Suite 300Norcross, Georgia 30092

Principal Investigators

E. WaisR. ReineckeE. C. Rodabaugh

This report describes research sponsored by EPRI. The report is a corporate documentthat should be cited in the literature in the following manner:

Stress Indices for Elbows with Trunnion Attachments, EPRI, Palo Alto, CA: 1998. ReportTR-107453.

v

REPORT SUMMARY

Trunnions on elbows are generally used as supports and are also used in someapplications as anchors. The qualification of trunnions is an important item in thedesign and fitness-for-service of many piping systems.

This report provides equations, based on experimental and test data, for determiningthe stress indices, B and C, and the flexibility factor, k, for elbows with hollow circularcross-section attachments (trunnions). The report contains explicit modifications toASME Code Cases 391 and 392 for qualification of trunnions on pipe. It also providesflexibility equations for a more accurate evaluation of these configurations.

Background

Fatigue is a significant consideration in the design and engineering of piping systems.The ASME Section III and B31 piping design codes use factors such as B and C indicesto account for fatigue effects produced by reversing loads and flexibility factors (k) forevaluation of piping configurations. ASME Code Cases 391 and 392 provide proceduresfor evaluating the design of hollow circular cross-section attachments on Class 1, 2, and3 pipe.

Objectives

• To experimentally derive expressions for B, C, and k factors for analysis oftrunnions on elbows.

• To provide modifications to Code Cases 391 and 392 for improved evaluation oftrunnions on elbows.

Approach

A review of the present approach for the evaluation of trunnions on elbows inaccordance with the Code provided an understanding of the conservatism in thedetermination of the fatigue factors. Available data on studies, experiments, and testingwere collected and reviewed. Tests and analyses were performed on representativemodels and the results compared to existing data.

vi

Results

The present values of A0, B, and C in Code Cases 391 and 392 were modified as a resultof this research and analysis to reduce excess conservatism. Equations, previouslyunavailable, were derived for flexibility factors for the elbow/trunnion configuration.Equations were derived for both in-plane and out-of-plane bending. Parameterlimitations were established for the results to be applicable to short radius and longradius 90o elbows with trunnion attachments.

EPRI Perspective

Design for fatigue is a major concern for any power or process facility. Accuratemethods of engineering for fatigue are important for cost-effective design, for rootcause failures, and for evaluating remaining fatigue life of plant designs.

The work being done under EPRI’s SIF optimization program continues to establish thetechnical justification to allow for reductions in current Code stress indices. The resultsof this program can provide a basis to reduce the scope of ongoing pressure boundarycomponent testing and inspection programs in operating nuclear power plants.Examples include reductions in the inspection scope of postulated high- and moderate-energy line break locations and reduction of snubber testing.

TR-107453

Interest Categories

Piping, reactor vessel, and internals

Keywords

ASME CodeFatiguePiping design and analysisStress intensity factorsStress indices

EPRI Licensed Material

vii

ABSTRACT

This report was prepared under the auspices of the EPRI project on stressintensification factor optimization. Stress intensification factors and their correspondingstress indices (for ASME Class 1 components) are used in the qualification of pipingcomponents to ensure that they have an adequate fatigue life under cyclic loading.Stress intensification factors and stress indices are also used for qualification for otherloading conditions.

Trunnions on elbows are generally used as supports and are also used in someapplications as anchors. The qualification of trunnions is a major concern in the designand qualification of many piping systems. This report presents the results of aninvestigation of the stress indices and flexibility factors for trunnions on 90o elbowssubject to axial loads and bending and twisting moments. This report reviews existingdata and methodologies used for qualification of trunnions. Modified expressions forstress indices are defined. The results of new testing are included. Finally, flexibilityfactors for accurately modeling the behavior of a trunnion in a piping system arepresented. The information presented in this report will significantly improve thequalification of trunnions on elbows.


ix

CONTENTS

1 INTRODUCTION ................................................................................................................. 1-1

2 BACKGROUND................................................................................................................... 2-1

Nomenclature ..................................................................................................................... 2-1

General ........................................................................................................................... 2-3

ASME Section III and B31.1 Power Piping Code Approach ............................................... 2-3

Review of References......................................................................................................... 2-4

3 TEST PROGRAM................................................................................................................ 3-1

Purpose .............................................................................................................................. 3-1

Design of Test Specimens .................................................................................................. 3-1

Testing Program ................................................................................................................. 3-2

Test Results Summary .................................................................................................... 3-4

Analysis of Test Data ...................................................................................................... 3-4

C2 Indices-Markl Approach .......................................................................................... 3-5

C2 Indices-Class 1 Approach....................................................................................... 3-7

B Indices-from Test Data........................................................................................... 3-11

4 EVALUATIONS OF METHODS TO QUALIFY TRUNNIONS ON ELBOWS ....................... 4-1

Purpose .............................................................................................................................. 4-1

Basic Approach................................................................................................................... 4-1

Potential Methods ............................................................................................................... 4-1

Comparison ........................................................................................................................ 4-4

Results of Comparison...................................................................................................... 4-14

5 COMPARISON OF TEST DATA TO ANALYSIS METHODS.............................................. 5-1

Purpose .............................................................................................................................. 5-1


x

C Indices............................................................................................................................. 5-1

B Indices............................................................................................................................. 5-2

6 INVESTIGATION OF FLEXIBILITY OF TRUNNIONS ON ELBOWS.................................. 6-1

General ............................................................................................................................... 6-1

Discussion: Elbows ............................................................................................................. 6-1

Discussion: Trunnions on Elbows ....................................................................................... 6-3

Finite Element Analysis....................................................................................................... 6-5

FEA Results: Flexibility of Elbows with Trunnions............................................................... 6-9

FEA Results: Flexibility of Trunnions ................................................................................ 6-13

Comparison to Test Data .................................................................................................. 6-21

7 CONCLUSIONS .................................................................................................................. 7-1

8 REFERENCES .................................................................................................................... 8-1

APPENDIX A ASME CODE CASE N-392-3...........................................................................A-1

APPENDIX B TEST DATA AND RESULTS...........................................................................B-1


xi

LIST OF FIGURES

Figure 2-1 Trunnion/Pipe Connection ..................................................................................... 2-1

Figure 3-1 Test Configuration ................................................................................................. 3-2

Figure 3-2 Limit Load Definition ............................................................................................ 3-12

Figure 4-1 Comparison of Equations for CL........................................................................... 4-13

Figure 4-2 Comparison of Equations for CN

.......................................................................... 4-14

Figure 6-1 Configurations ....................................................................................................... 6-3

Figure 6-2 Elbow-Trunnion Model........................................................................................... 6-4

Figure 6-3 Branch Connection Model ..................................................................................... 6-4

Figure 6-4 FEA Model Details................................................................................................. 6-5

Figure 6-5 FEA Model............................................................................................................. 6-6

Figure 6-6 Boundary Conditions ............................................................................................. 6-7

Figure 6-7 Elbow-Trunnion Model......................................................................................... 6-14

Figure 6-8 Beam Model ........................................................................................................ 6-16


xiii

LIST OF TABLES

Table 3-1 Summary of Test Results........................................................................................ 3-4

Table 3-2 Calculation of CL'..................................................................................................... 3-7

Table 3-3 Trunnion/Elbow-Class 1 CUF Evaluation Using Code Case Indices....................... 3-8

Table 3-4 Trunnion/Elbow-Class 1 Minimum CUF = 1.0 Experimental CL............................. 3-10

Table 3-5 Trunnion/Elbow-Experimental Evaluation of BL' .................................................... 3-14

Table 4-1 Hankinson FEA Parameters ................................................................................... 4-2

Table 4-2 Comparisons for CL................................................................................................. 4-5

Table 4-3 Comparisons for CN................................................................................................. 4-7

Table 4-4 Comparisons for CT................................................................................................. 4-9

Table 4-5 Comparisons for CW

.............................................................................................. 4-11

Table 4-6 Comparison of Results for Hankinson [7] Model 12.............................................. 4-16

Table 6-1 FEA Models ............................................................................................................ 6-8

Table 6-2 Summary of Rotations .......................................................................................... 6-10

Table 6-3 Bending of the Pipe-Elbow Flexibility.................................................................... 6-12

Table 6-4 Bending of the Trunnion-Trunnion Flexibility ........................................................ 6-18

Table 6-5 Bending of the Trunnion-Ends Fixed .................................................................... 6-19

Table 6-6 Average Trunnion Flexibility.................................................................................. 6-20


1-1

1 INTRODUCTION

This report was prepared under the auspices of the EPRI project on stressintensification factor optimization. Stress intensification factors (SIFs) are used toensure that piping has an adequate fatigue life under cyclic loading. SIFs are notgenerally used for design of welded attachments such as trunnions; however, thegeneral approach is the same.

This report specifically investigates the fatigue behavior of trunnions welded onelbows. Trunnions are also referred to as “hollow circular cross-section attachments.”

The general approach followed in this report is as follows:

• Review the present approach used for evaluation in accordance with the Code.

• Perform a literature search on the applicable references.

• Perform tests as required and analyze the results.

• In conjunction with analysis, use the test data to develop an updated approach toevaluating the trunnion/pipe configuration.

Section 2 of this report provides a summary of the available references regardingtrunnions on elbows and related references. The limited coverage in the present Codes[1, 2] is also discussed. Potential evaluation methodologies are identified.

Section 3 of this report presents the results of fatigue tests on trunnions on elbowsconducted under the auspices of the EPRI research project. The test results are used toderive experimentally based values for the various indices.

Section 4 provides an evaluation of the various approaches to evaluating trunnions onelbows. This evaluation includes a comparison to previously published finite elementanalysis (FEA) data as well as new data. New experimental data is included in thecomparison. Specific recommendations regarding proposed analytical approaches aremade.


Introduction

1-2

Section 5 compares the experimental data to the results of the analytical approachdiscussed in Section 4.

Section 6 discusses approaches for evaluating flexibility of these configurations.

Section 7 of this report summarizes the conclusions of this research effort. Theseconclusions provide new understanding of the behavior of trunnions on elbows. Thisinformation allows these configurations to be more accurately evaluated.

Appendix A contains American Society of Mechanical Engineers Code Case N-392,Procedure for Evaluation of the Design of Hollow Circular Cross Section Welded Attachmentson Classes 2 and 3 Piping, Section 3, Division 1.

Appendix B contains the test data and results for this report.


2-1

2 BACKGROUND

Nomenclature

Trunnion

MN

W ML

θ

Q1

Q2

MT

Figure 2-1Trunnion/Pipe Connection

Ro = pipe/elbow outside radius, in.

ro = trunnion outside radius, in.

ri = trunnion inside radius, in.

T = nominal pipe/elbow wall thickness, in.

t = nominal trunnion wall thickness, in.

Do = outside diameter of the pipe/elbow, in.

do = outside diameter of the trunnion, in.

D = mean diameter of the pipe/elbow, in.


Background

2-2

d = mean diameter of the trunnion, in.

Rm = mean radius of pipe/elbow, in.

R = nominal bend radius of elbow

h = TR/Rm

2 elbow characteristic

AT = π (ro

2 - ri

2)

ZT = IT/ro

IT = π/4(ro

4 - ri

4)

Am = π/2 (ro

2 - ri

2)

J = lesser of π o

2T or ZT

Z = section modulus of straight pipe section

γ = Ro/T

τ = t/T

β = do/Do

θ = cos-1 (R/(R+Do/2), angle between trunnion and elbow

C = Ao (2γ)n

1 βn

2 τn

, but not less than 1.0

ML = bending moment applied to the trunnion as shown in Figure 2-1, in.-lb.

MN = bending moment applied to the trunnion as shown in Figure 2-1, in.-lb.

MT = torsional moment applied to the trunnion as shown in Figure 2-1, in.-lb.

Q1 = shear load applied to the attachment as shown in Figure 2-1, lb.

Q2 = shear load applied to the attachment as shown in Figure 2-1, lb.

W = thrust load applied to the attachment as shown in Figure 2-1, lb.These moments and loads are determined at the surface of the pipe.

n1, n2, n3 are specified in Code Cases N-391 or N-392 (see Appendix A).


Background

2-3

CT = 1.0 for β ≤ 0.55

CT = CN for β = 1.0, but not less than 1.0; CT should be linearly interpolated for 0.55<β≤1.0, but not less than 1.0

CL’ = Values of CL based on fatigue test data

BW = 0.5(CW), but not less than 1.0

BL = 0.5(CL), but not less than 1.0

BN = 0.5(CN), but not less than 1.0

BT = 0.5(CT), but not less than 1.0

BL’ = Values of BL based on limit load test data

E = Young’s modulus

G = Bulk modulus

µ = Poisson’s ratio

KT = 1.8 for full penetration welds

W**, MN

**, ML

**, Q1

**, Q2

**, and MT

** are absolute values of maximum loads occurringsimultaneously under all service loading conditions (see Appendix A).

General

Trunnions are often attached to elbows to serve as supports for the piping system.Often these configurations are used to resist water hammer loads. Typically, they willbe subjected to axial and other forces, as well as bending and torsion moments. Thisstudy is limited to trunnions attached to 90o elbows.

ASME Section III and B31.1 Power Piping Code Approach

The present versions of the Codes, Section III [1] and ANSI B31.1 [2], are silent withregard to specific methodologies for qualification of trunnion/elbow configurations.However, Section III, NB-3685.1 does acknowledge that special attention should begiven to this type of configuration: “Stresses in elbows with local discontinuities, suchas longitudinal welds, support lugs, and branch connections in the elbow, shall be


Background

2-4

obtained by appropriate theoretical analysis or by experimental analysis in accordancewith Appendix II.” Unfortunately, this is the extent of the direction.

Review of References

The literature does contain some references that are helpful in evaluation and design oftrunnions on elbows. Slagis [3] and Hankinson et al. [4, 5] provide general discussionsabout the use of trunnions and other attachments, including a discussion regardingjurisdictional boundaries.

Williams and Lewis [6] suggested expressions for B2 and C2 indices for trunnions onelbows that were based on the results of 10 finite element analyses. The expressionssuggested were:

B2 = [-.022(Do/do) + .091] (Do/T) + .973 (Do/T) - .528

C2 = [-.071(Do/do) + .271] (Do/T) + 4.913 (Do/T) - 5.961

Hankinson et al. [7] extended the work performed by Williams and Lewis. They usedthe results from Williams and Lewis and expanded upon them for a total of 26 finiteelement models. They suggested equations for secondary stress indices for moment andforces applied to the trunnion of the form:

C = Ao (Do/T)m1 (do/Do)m2 (t/T)m3

Where Ao, m1, m2, and m3 are constants that vary, depending upon the type of load.

Limits were given for the applicability of the expressions:

0.2 ≤ t/T ≤ 2.0

20 ≤ Do/T ≤ 60 (Eq. 2-1)

0.3 ≤ do/Do ≤ 0.8

These limits were based upon the range of the parameters of the finite elements models.

Hankinson [7] also suggested a modification to the stress indices of the elbow to:


Background

2-5

C2E = 2.55/h.732

where h = TR/R2

m

The Code equation for C2 for an elbow is :

C2 = 1.95/h2/3

There is no available test data in the literature for trunnions on elbows. However,Rawls et al. [8] discuss the results of a series of tests on attachments on elbows andmake a comparison to Code Case N-318 [9]. The attachments tested included bothrectangular and cruciform shapes. Code Case N-318 presents a method ofanalysis/design for rectangular welded attachments on straight pipe that involves useof secondary stress indices. The expression for the C indices is of a simplified form andis a function of geometrical parameters. Rawls used the Code Case methodology toevaluate test data. Based on the test data, the conclusion was that the Code Casemethodology was conservative by a factor of 3.5 to 14.8 when applied to these types ofattachments on elbows.

Code Case N-318 covers rectangular welded attachments. There are two Code Cases(N-391 for Class 1 piping [10] and N-392 for Class 2 and 3 piping [11]) that address theevaluation and design of hollow, circular, cross-section welded attachments (ortrunnions) on straight pipe. The approach followed by these Code Cases is very similarto that of N-318. These Code Cases are important because the attachments are the sameas in this study. Code Case N-392 is included in Appendix A for reference.

N-391 requires the calculation of various stresses:

SMT = BWW/AT + BNMN/ZT + BLML/ZT + Q1/Am + Q2/Am +BTMT/Jm (Eq. 2-2)

SNT = CWW/AT + CNMN/ZT + CLML/ZT + Q1/Am

+ Q2/Am + CTMTJm + 1.7EαTT -TW (Eq. 2-3)

SPT = KT(SNT)

SNT

** = CWW**/AT + CNMN

**/ZT + CLML

**/ZT

+ Q1

**/Am + Q2

**/Am + CTMT

**/Jm


Background

2-6

N-392 has similar expressions except that the 1.7EαTT -TW term in Equation (2-3) isnot included. The stresses calculated by these equations are used in the qualification inmodified standard Code equations by the two Code Cases.

Rodabaugh [12] discusses the background of N-391 and N-392 and is summarizedherein. It should be noted that the original objective in developing these Code Caseswas to provide a simplified and conservative methodology. The approach used toaddress the effects of the various mechanical loads (W, Q1, Q2 , MN, ML, and MT) isdiscussed below.

The original basis for considering the effects of the W, ML, and MN loads was thecorrelation equations given by Potvin et al. [13]. These correlation equations wereconsidered to correspond to the maximum primary-plus-secondary stresses (PL + Pb +Q). Thus, they corresponded to the C-indices of NB-3600 [1] or CW, CL, and CN of theCode Cases. A more generalized form of the correlation equation is given in an earlierwork by Rodabaugh [17]:

C = A(2 γ)n1 (β)n2 (τ)n3 (Do/L)n4(g/Do)n5(sinθ)n6

where:

γ = Ro/Tβ = do/Do

τ = t/T

The constants n1, n2, n3, n4, n5, and n6 vary depending upon the loading. L is thelength of the member corresponding to the trunnion, and g is the distance between thetrunnion and another trunnion. θ is the angle between the trunnion and the straightpipe (see Figure 2-1).

For purposes of the Code Case, this expression was simplified. Note that(sinθ)n6 = 1.0 for θ = 90o. The form of the Code Case expression for the C indices is:

C = Ao(2γ)n1βn2τn3 (Eq. 2-4)

See Appendix A for values of Ao , n1, n2, and n3. This is similar to the form of theequation suggested by Hankinson [7] with different constants.

The range of the applicable parameters in the Code Cases for CW, CL, and CN has beenextended beyond that of Potvin. The applicable range of γ and τ was extended based onWRC Bulletin 198 [7] and WRC Bulletin 297 [8]. The range of β was extended based oncomparison with the equations derived by Wordsworth [16].


Background

2-7

At the time the Code Cases were prepared, data were not available regarding shearloads and torsional moments (Q1, Q2, and MT ). Engineering judgment was used in theevaluation of their effects. For the shear loads (Q1 and Q2), the stress intensity (twice theshear stress) is Q/Am, where Am is one-half the cross-sectional area of the trunnion-pipeinterface (where the load is taken), assumed to be π(ro

2 -ri

2)/2. This is consideredreasonable for small trunnions (small do/Do) but is probably very conservative for largetrunnions (size on size).

The approach used to evaluate the effects of MT was based on comparisons to data onbranch connections [10]. Branch connections are similar to trunnions except that the runpipe has an opening in it. For branch connections with small do/Do, the stress intensityis about Mt/Jm. For do/Do = 1.0, test data [11] indicate that the maximum stress intensityis about the same as for out-of-plane bending (for example, due to MN). Based on thisinformation, the value of CT was taken as 1.0 for β= do/Do ≤ 0.55 and as equal to CN forβ = 1.0. Linear interpolation is used in between. The change at β = 0.55 corresponds toPotvin’s data.

Potvin originally suggested a limit on γ = Ro/T ≥ 8.33. Rodabaugh [12] provides a basisfor extending that to γ = Ro/T ≥ 4.0. This was based on a comparison with Wordsworth[16]. This change was made in Code Case N-392 but not in N-391; however, thisextension is valid for N-391.

The B indices that are in the Code Cases correspond to those of ASME Section III, NB-3600. The B indices are based upon limit load analysis or test. The Code Cases take theB indices as one half the C indices. Based upon data from Rodabaugh [12, 17], it isestimated that the Code Case B indices are conservative by “a factor of at least 1.5” [12].

The approach followed by the Code Case is to calculate the stresses due to the trunnionmechanical loads (W, Q1 , Q2 , MN, ML, and MT) and the thermal stresses (if Class 1piping) and add them to the stresses in the pipe due to loads in the pipe. The stressesare added linearly and then compared to the specific limits dependent upon the pipingclass and the specific requirement. The linear addition of stresses is generally veryconservative. It assumes that all the stresses are maximum at the same point.

Wordsworth’s research [16] warrants further review. This paper reviews the results ofusing acrylic models for determining what were referred to as stress concentrationfactors (SCFs) at tubular joints. The specific application is for offshore steel structures.Test specimens were manufactured from acrylic materials. Data from strain gaugeswere compared to data from the analysis to verify the analysis. Various types of jointswere investigated. The results of what are called “T” joints are of interest for thisinvestigation. The expression given for the SCF for out-of-plane bending is:


Background

2-8

KS = γ τ β (1.6- 1.15 β5) (sin θ)(1.35 + β*β) (Eq. 2-5)

where:

γ = Ro/Tτ = t/Tβ = do/Do

It is assumed that β is based on do and Do.

For in-plane bending, the SCF is given as:

KC = 0.75 γ0.6 τ0.8 (1.6β0.25 - 0.7β2) (sin θ)(1.55 - 1.6β) (Eq. 2-6)

These SCFs are assumed to be equivalent to the indices corresponding to the secondarystresses (that is, CL and CN). Other expressions are provided for other loadingconditions.

Since the connection of the trunnion to the elbow is at an angle θ, it is similar to that ofa lateral. Rodabaugh [18] suggests using Equations 2-5 and 2-6 in the qualification oflaterals connected at an angle of θ to the pipe.

One other reference of interest is Hankinson and Albano’s study of flexibility of elbowswith trunnions [19]. However, this study is limited to the flexibility of the elbows; theflexibility of the trunnion was not investigated.


3-1

3 TEST PROGRAM

Purpose

The purpose of this test program was to obtain some specific data that corresponded tothe test methodology followed by Markl [20]. These tests would provide data thatcould be used as a basis for developing procedures for qualification of trunnions onelbows. All tests were for in-plane bending of the trunnion. This data would be usedfor extrapolation to other loading conditions. As discussed later, the results of thetesting will be expressed in terms of stress intensification factors and stress indices.

Design of Test Specimens

Four specimens were manufactured by Wilson Welding Service, Inc., of Decatur,Georgia. The test specimens consisted of 8-inch NPS schedule 20 A53-B pipe and 8-inchNPS schedule 20 long radius elbows with a 4-inch schedule 40 A53-B trunnion. Thewelds at the interface of the trunnion and pipe were normal full penetration in an as-welded condition. The test specimens were labeled I, J, K, and L.

Figure 3-1 indicates the test configuration.


Test Program

3-2

Load Point4" NPS Sch. 40 Pipe

26-3/4"

Cover Plate

LoadDirection

Flanges

8" Sch. 40 LR Elbow8" NPS Sch. 20 Pipe

63"

L

~ 50"Varies for Test Specimen

8" NPS Sch. 20 Pipe

Flange

Base

Figure 3-1Test Configuration

Testing Program

The testing was performed at the Ohio State University. The fatigue tests wereperformed on a Series 319 dynamically rated axial/torsional load frame made by MTSSystems Corporation. This unit is designed to accommodate either uniaxial ormultiaxial testing. Load frame capacities are 55,000 pounds axial force and 20,000 in.-lb.torsional moment. A computerized control panel provides local, precise operations ofthe cross head, hydraulic grips, and actuator. The maximum actuator displacement is 6inches. The loading pattern applied to an attached sample is controlled byprogrammable servovalves.

Built-in loading programs include sinusoidal and triangular waves with the user ableto select, within machine limits, the desired amplitude and frequency. The actualdisplacement of the actuator is measured by a linear variable differential transformer


Test Program

3-3

(LVDT). The output of either the load cell or the LVDT can be selected for closed loopcontrol of the actuator displacement time history. During a test, the number of cycles ofapplied load is recorded by a digital counter and displayed on the MTS console.

In these tests, the load was sinusoidal at frequencies ranging from 0.3 to 0.5 Hz.Actuator displacement was designated as the test control variable. The selection ofdisplacement as the control parameter meant that actuator movement was used by theMTS system for the feedback in the closed loop controls. This resulted in virtuallyidentical cycles of actuator displacement being recorded throughout the duration ofeach test. The load resulting from the imposition of the specified displacement wasmeasured with a fatigue-rated, 5000-lb. capacity, tension-compression, electronic loadcell manufactured by the Lebow Instrument Company. The output of this load cell wasmonitored continuously throughout the duration of each test.

Both load and actuator displacement were recorded using a computer program writtenat OSU in LabVIEW specifically for that purpose. LabVIEW is a graphical languagedeveloped by National Instruments that allows the user to design in software a testcontrol and data collection system tailored to the requirements of each experimentalprogram. In the LabVIEW application developed for the fatigue tests, the signals fromthe load and displacement transducers were sampled 30 times per second, and the timehistories of each were plotted on the computer screen in real time so that the progressof the test could be readily monitored. By combining the load and displacement timehistories, a plot of load versus displacement at any load cycle could be constructed.This too was done in real time so that changes in the response of the test specimencould be identified while the specimen was still undergoing loading. Any of thesepresentations of the test data could be printed while the test was still in progress.

Figure 3-1 shows the load application point and direction of loading. Note that thedistance from the load point to the surface of the pipe (~50 inches) varies slightly foreach test specimen. The measured distance (L), which is the moment arm for the load,is dependent on the installation and is included in the test data.

The test data, results, and other information are provided in Appendix B. The testswere displacement-controlled cantilever bending tests. The tests followed the standardapproach corresponding to Markl type tests [20, 21]. Each specimen was first tested todetermine the load deflection curve for that particular specimen. The load deflectioncurve was used to determine the stiffness of each specimen and the load applied to thespecimen by a given amount of displacement. The load deflection curves weredetermined for loading in both positive and negative loading directions (down andup). Each specimen was then fatigue tested by cycling the deflection in both directionsof loading by a controlled amount. The cycles to failure were counted to determine thefatigue life. Failure was detected when though wall cracks formed and water leakedthough the cracks.


Test Program

3-4

Test Results Summary

Table 3-1 provides a summary of the test results. This summary includes some the datathat are covered in more detail in Appendix B.

Table 3-1Summary of Test Results

TEST Flb.

Lin.

ZT

in. 3M

in.-lb.N

Cycles toFailure

it

Note (1)

I 1816 51.500 3.21 93,524 2,125 1.820

J 2319 50.625 3.21 117,399 1,231 1.617

K 2221 51.375 3.21 114,104 968 1.746

L 2168 51.500 3.21 111,652 1,405 1.658

Notes:

1. The value of it is calculated from it = 245,000 N-0.2/S, where N ≡ cycles to failure, andS = M/ZT. ZT is based on nominal dimensions for the trunnion. If there was morethan one loading condition with different deflections, then N is an equivalent valuecalculated from:

Neq = Σ(δi/δmax)5 * Ni

where δi is the deflection for the ith loading condition, Ni is the number of cycles forthe ith loading condition, and δmax is the maximum deflection.

Analysis of Test Data

There are several methods available to analyze the data. In general, for this type ofloading condition, the purpose of analysis is to be able to express the results in terms ofSIFs (i-factors), B2 indices, C2 indices, and K2 indices. The literature that is available usesSCFs (or C indices, etc.) and not SIFs in the qualification of trunnion/pipeconfigurations; therefore, the focus will be on B2, C2, and K2. Because the welds were as-welded, full penetration welds, it is believed that K2 = 1.8 is reasonable. Hence, thefocus will be on B2 and C2.


Test Program

3-5

C2 Indices-Markl Approach

As discussed earlier, the tests that were performed as a part of this investigation werefatigue tests. There are two methods that can be used to evaluate the results. The firstwill be referred to as the C2 Indices-Markl approach. The second is the Class 1 fatigueapproach used in Class 1 analysis per NB-3600 [1]. The Markl approach is presentedfirst.

The fatigue tests followed the Markl approach [20, 21]. Markl used the followingexpression for Grade B Carbon steel:

iS = 245,000 N-0.2 (Eq. 3-1)

where S is the nominal stress in the component and N is the number of cycles whenthrough wall cracks occur and water leaks. This is used as the definition of the SIF (or ifactor) and is used in the design for fatigue for B31.1 piping [2] and ASME Section IIIClasses 2 and 3 piping [1].

The C-indices correspond to primary-plus-secondary stresses. The C2 indices, which areapplicable to moment loading in piping, are related to the SIFs. Section NC-3672.2 [1]provides the following equation:

i = C2K2/2 (Eq. 3-2)

This expression will be used to evaluate the value of C2 (or equivalent) that is used inthe Code Cases. The approach follows that developed in Rawls et al. [8]. For theloading condition used in the tests, C2 corresponds to CL which is associated with thein-plane moment in Equation 2-3 (Code Case N-391). We will use the nomenclature CL

in this evaluation.

In N-392, the following equation is provided (neglecting the shear stress term, Q2/A,which is negligible when compared to the bending stress):

SE = iMc/Z + SPT/2

and also:

SPT = KT SNT


Test Program

3-6

For this application,

SNT = CLML/ZT

Therefore:

SPT = KT CLML/ZT

and:

SE = iMc/Z + KT CLML/(2ZT ) (Eq. 3-3)

Substituting equation (3-2) into equation (3-3) with K2 = KT yields:

SE = (KT/2) C2M/Z + (KT/2) CLML/ZT (Eq. 3-4)

SE in Equation 3-4 is equivalent to the iS term in Equation 3-1. Substituting andrearranging yields:

CL = {245,000 N-0.2(2/KT) - C2 M/Z} ZT/M

Because this CL is derived from fatigue tests, to distinguish it from the CL from the CodeCase, it will be called CL’. Now assume (conservatively) that this trunnion is attached toa straight pipe. Since C2 = 1.0 for straight pipe, Equation 3-4 becomes:

CL’ = {245,000 N-0.2(2/KT) - M/Z} ZT/M (Eq. 3-5)

Thus CL’ is a fatigue-based value that can be compared to the values calculated byvarious methods. Table 3-2 summarizes this calculation for the four tests.


Test Program

3-7

Table 3-2Calculation of C L'

TEST ZT

in. 3M

in.-lb.N

Cycles toFailure

it Zin. 3

KT

Note (1)CL'

Note (2)

I 3.21 93524 2125 1.820 13.39 1.8 1.78

J 3.21 117399 1231 1.617 13.39 1.8 1.55

K 3.21 114104 968 1.746 13.39 1.8 1.70

L 3.21 111652 1405 1.658 13.39 1.8 1.60

Average= 1.66

Notes:

1. K2 = 1.8 for full penetration welds.2. Calculated using Equation 3-5.

Table 3-2 shows that the value of CL’ , on the average is 1.66. This is with theassumption that C2 was equal to that of a straight pipe. The actual value of C2 for theelbow is about 6.3. If this were to be used, the value of C2 would be less than presentedin Table 3-2 (approximately 1.03). However, FEA analysis indicates that the stresspeaks on the elbow are on the sides of the elbows and there is little interaction with thelocal stresses near the trunnion. Thus, a value of C2 =1.0 is reasonable.

C2 Indices-Class 1 Approach

The second method of evaluating the data follows the fatigue evaluation approach usedfor Class 1 analysis (NB-3600). This method is referred to as the C2 Indices-Class 1Approach. The methodology of Code Case N-392 is used as the basis for calculating andcombining stresses as well as evaluating those stresses. Table 3-3 provides a summaryof a fatigue analysis of the data using the values for the various stress indices calculatedin accordance with the Code Case. The tacit assumption is that the trunnion on anelbow acts like a trunnion on a pipe. CL calculated for the trunnion using code Case N-392 yields CL = 3.77.


Table 3-3Trunnion/Elbow-Class 1 CUF Evaluation Using Code Case Indices

Case Mkips

SNT

ksiC2 Mi/Z

ksiSn

ksi3Sm

ksiKe SPT

ksiSP

ksiSalt

ksiN

AllowableN

FailureCUF

I 93.5 219.6 7.0 226.6 60.0 5.00 395.3 402.3 1005.8 77 2125 27.44

J 117.4 275.8 8.8 284.5 60.0 5.00 496.4 505.1 1262.8 49 1231 25.29

K 114.1 268.0 8.5 276.5 60.0 5.00 482.4 490.9 1227.4 52 968 18.76

L 111.7 262.4 8.3 270.7 60.0 5.00 472.3 480.6 1201.5 54 1405 26.08

Average= 24.39

Notes:

1. C2 (pipe)=1.02. K2 (pipe)=1.03. KT (trun)=1.84. CL(trun)=3.775. Z (pipe) =13.39 in.3

6. ZT(trun)=3.21 in.3

7. SNT = CLMT/ZT

8. Sn = C2 Mi/Z + SNT

9. SPT = KT SNT

10. SP = K2C2Do/2IMi + SPT

11. Calculations are based on nominal dimensions.


Test Program

3-9

The test cumulative usage factor, or CUF, is based on an allowable number of cyclesfrom the expression:

Nallowable = (8,664,000/(Salt-21,645))2

This expression does not include the factors of two on stress and 20 on cycles that arepart of the Section III Class 1, Appendix I, S-N design curves [1]. If these were included,the calculated CUF would be much greater.

The value of Sm used was 20 ksi. as specified by the Code. As indicated in Table 3-3, thecumulative usage factor (CUF) for the tests is, on the average, 24.39 versus a coderequirement of 1.0. This indicates that the Code is very conservative. Contributors tothe conservatism could be the value of the indices and/or the value of Ke.

Table 3-4 presents the results of a fatigue analysis in which the value of CL’ was varieduntil the average CUF = 1.00. The corresponding value of CL’ was 1.496. In this case, thevalue of Ke varies from 2.14 to 2.75, which reduces the potential of contribution to theoverall conservatism.


Table 3-4Trunnion/Elbow-Class 1 Minimum CUF = 1.0 Experimental C L

Case Mkips.

SNT

ksi.C2 Mi/Z

ksi.Sn

ksi.3Sm

ksi.Ke SPT

ksi.SP

ksi.Salt

ksi.N

AllowableN

FailureCUF

I 93.5 94.2 7.0 101.2 60.0 2.37 169.6 176.5 209.4 2126 2125 1.00

J 117.4 118.3 8.8 127.0 60.0 3.23 212.9 221.7 358.5 661 1231 1.86

K 114.1 115.0 8.5 123.5 60.0 3.12 206.9 215.4 335.6 761 968 1.27

L 111.7 112.5 8.3 120.9 60.0 3.03 202.6 210.9 319.4 846 1405 1.66

Average= 1.45

Notes:

1. C2 (pipe)= 1.02. K2 (pipe)= 1.03. KT (trun)= 1.84. CL(trun)= 1.6175. Z (pipe) = 13.39 in.3

6. ZT(trun)= 3.21 in.3

7. SNT = CLMT/ZT

8. Sn = C2 Mi/Z + SNT

9. SPT = KT SNT

10. SP = K2C2Do/2IMi + SPT

11. Calculations are based on nominal dimensions.


Test Program

3-11

If Table 3-4 were modified such that the minimum CUF was 1.0, the correspondingvalue of CL would be 1.617. The average CUF would be 1.45.

It should be noted that in this approach the stresses were all assumed to be based onML/ZT. In the actual qualification, the stresses would also include a contribution due tothe elbow, that is, iM/Z + 1/2 KTML/ZT. Thus, this method is conservative.

B Indices-from Test Data

Code Case N-392 specifies that the value of BL be taken as one-half the value of CL., butnot less than 1.0. It is worthwhile to determine if the data obtained in this study can beused for evaluating these indices.

The ASME Code [1] uses limits on the primary stress intensity to limit gross plasticdeformation of piping. The Code has specific limits that it applies to stresses calculatedusing B-indices. The basic equations of the Code are modified to include the effects ofthe trunnions in the Code case. See Equation 2-2. The terms in Equation 2-2 can beneglected except for the term with BL because of the loading.

Therefore the equation reduces to:

SMT = BL ML/ZT

Using SY as the allowable stress and solving for ML (the limit moment) yields:

ML = SYZT/BL

Or rearranging:

BL =SYZT/ ML

Because this value of BL is based on test data, it will be referred to as BL’ to distinguish itfrom the value of BL calculated from the Code case. Hence:

BL’ =SYZT/ ML

To determine the limit moment experimentally, a load-deflection curve must bedeveloped. The limit moment (or limit load) is defined as when the deflection is equalto twice that predicted assuming linear behavior (Article II-1000, Section II-1430,reference 1). This is shown in Figure 3-2.


Test Program

3-12

Load

Deflection

Limit Load

Linear Action

Test Data(See Appendix A)

Twice DeflectionAssuming Linear Action

δ 2δ

Figure 3-2Limit Load Definition

The tests performed for this report were directed toward obtaining fatigue data ratherthan limit load (or moment) data. However, the data that was taken during the initialphase of the testing can be used to obtain an estimate of the limit loads.

The first phase of the tests involved determining the stiffness of the test specimen. Thatwas determined by obtaining a load deflection curve. (See Appendix B for curves.) Theloads in these tests were taken slightly into the plastic region. As such, the maximumloads can be used to estimate the limit load. This would be a lower limit because thedeflection was not allowed to go to twice that based on elastic behavior. This maximumload is used to investigate the value of BL’.

A review of the curves in Appendix B shows that the specimens were loaded in boththe positive and negative direction. Thus BL’ can be estimated for both directions ofloading.


Test Program

3-13

Table 3-5 shows the calculation of BL’ based on the maximum force used in determiningthe load deflection curve. As noted earlier, this force is less than the limit load and,hence, it will underpredict BL’. Column 5 lists the calculated values of BL’. The averagevalue of BL’ is 1.56.

Note that the value of SY is based on the material certification data provided by the testspecimen manufacturer.

In order to obtain more insight into the actual value of BL’, the values of the limit loadswere estimated from the load deflection curves. This was performed by extrapolatingthe load deflection curves, assuming that they would continue to follow the shape ofthe curves beyond the point where the loading was stopped. In other words, it wasassumed that there would be no sudden change in the behavior. This is believed to be areasonable assumption.

Column 6 of Table 3-5 lists the estimated limit loads (FLIM-EST). Column 7 lists theassociated value of BL’. The average value of BL’ is 1.07 using FLIM-EST.

The conclusions to be made from these results are discussed later.


Table 3-5Trunnion/Elbow-Experimental Evaluation of B L'

TESTSPECIMEN

COLUMNLOADING

DIRECTION

mlb./in.

Lin.

(1)Fmax

lb.

(2)M=F*Lin.-lb.

(3)Z

in. 3

(4)Sy

ksi

(5)BL

' =SyZ/M

Using (2)

(6)FLIM-EST

(7)BL

' = SyZ/M

Using (6)

I POSITIVE 1817 51.5 2145 110,468 3.215 63.3 1.84 3000 1.32

NEGATIVE 1630 51.5 2065 106,348 3.215 63.3 1.91 3510 1.13

Average for specimen I = 1.88 1.22

J POSITIVE 2009 50.625 2724 137,903 3.215 63.3 1.48 4100 0.98

NEGATIVE 1840 50.625 2923 147,977 3.215 63.3 1.38 4900 0.82

Average for specimen J = 1.43 0.90

K POSITIVE 1887 51.375 2754 141,487 3.215 63.3 1.44 3300 1.20

NEGATIVE 1789 51.375 2754 141,487 3.215 63.3 1.44 4000 0.99

Average for specimen K = 1.44 1.10

L POSITIVE 1888 51.5 2629 135,394 3.215 63.3 1.50 3450 1.15

NEGATIVE 1756 51.5 2659 136,939 3.215 63.3 1.49 3950 1.00

Average for specimen L = 1.49 1.07

Average for all specimens, bothloading directions =

1.56 1.07


4-1

4 EVALUATIONS OF METHODS TO QUALIFY

TRUNNIONS ON ELBOWS

Purpose

The purpose of this section is to review and assess different methods of evaluatingtrunnions on elbows. The overall objective is to select a general approach that isconservative but not unreasonably so.

Basic Approach

The basic approach utilized was to follow the methodology used by Code Case N-392(or N-391). This methodology was modified as required to satisfy any specialrequirements for the trunnion/elbow configuration. The general approach used byCode Case 392 (or 391) is clearly conservative and can be extended to cover trunnionson elbows. This is the same general approach that was used by Rodabaugh [22]. Asdiscussed earlier, the objective of Code Case 392 is to present a simplified conservativemethod. The methodology to determine the indices is presented next.

Potential Methods

There are several equations available for calculating the various C stress indices. Theseinclude:

• Hankinson [7]

• Wordsworth [16]

• Code Case 392 [11] (based on Potvin [13])

• Code Case 392 modified to include the effect of the angle θ (see Figure 2-1)

• Hankinson [7] modified to represent a “best fit” of the data


Evaluations of Methods to Qualify Trunnions on Elbows

4-2

In order to make a comparison of these equations, the finite element database used byHankinson [7] served as the basis of comparison. Table 4-1 lists the characteristics of themodels. There are 26 models in this data base for CT, CL, and CN. For CW, there are 16models. The first step in the evaluation is a comparison of the indices calculated by thefive methods for these models.

The first three methods (Hankinson [7], Wordsworth, and Code Case 392 (Potvin) havebeen discussed in detail earlier. The other two methods are discussed below.

Table 4-1Hankinson FEA Parameters

ModelNo.

Din.

Tin.

din.

tin.

2 γ =2 γ =D/T

β =β =d/D

τ =t/T

1 5 .25 1.98 .050 20.0 .396 .20

2 5 .25 2.03 .100 20.0 .406 .40

3 5 .25 2.33 .400 20.0 .466 1.60

4 5 .25 2.98 .075 20.0 .595 .30

5 5 .25 3.05 .150 20.0 .610 .60

6 5 .25 3.90 .100 20.0 .780 .40

7 5 .25 4.00 .200 20.0 .800 .80

8 30 .50 12.10 .100 60.0 .403 .20

9 30 .50 12.20 .200 60.0 .407 .40

10 30 .50 12.80 .800 60.0 .427 1.60

11 30 .50 18.15 .150 60.0 .605 .30

12 30 .50 18.30 .300 60.0 .610 .60

13 30 .50 24.20 .200 60.0 .807 .40

14 30 .50 24.40 .400 60.0 .813 .80

15 30.25 .75 19.35 1.350 40.3 .640 1.80

16 30.25 .75 25.50 1.500 40.3 .843 2.00

17 6.625 .28 3.50 .300 23.7 .528 1.07



4-3

Table 4-1 (cont.)Hankinson FEA Parameters

ModelNo.

Din.

Tin.

din.

tin.

2 γ =2 γ =D/T

β =β =d/D

τ =t/T

18 6.625 .28 4.50 .337 23.7 .679 1.20

19 12.75 .38 6.63 .432 34.0 .520 1.15

20 12.75 .50 10.75 .500 25.5 .843 1.00

21 16 .38 8.63 .500 42.7 .539 1.33

22 16 .50 12.75 1.000 32.0 .797 2.00

23 24 .38 12.75 .375 64.0 .531 1.00

24 24 .50 16.00 .625 48.0 .667 1.25

25 36 .63 20.00 .625 57.6 .556 1.00

26 36 .73 24.00 1.218 49.7 .667 1.68

As discussed earlier, Code Case N-392 specifies stress indices for trunnions on straightpipe where the angle θ is 90o. WRC Bulletin 256 [17] includes a modified equationwhich includes a θ term that is (sin θ)n6, where n6 is a constant that depends on the typeof loading. To include the effects of θ, the basic formulation for C indices of Code CaseN-392 is modified to:

C = Ao(2γ)n1βn2τn3 (sin θ)n6

The last method to be investigated listed above is a modification of the expressionsdeveloped by Hankinson [7] based on what could be called a “sequential” regressionanalysis. Hankinson indicated that the approach used was to first find a relationshipbetween the FEA and one variable, for example, (t/T). A logarithmic regressionanalysis would provide the exponent m. The FEA results would then be normalized bydividing by (t/T)m. The process would be continued until all the exponents (and Ao)were determined. They would then be adjusted “to better envelope the data” [7].

The approach selected for use in this study starts with the assumption that C-indicescan be correlated by equations of the form of Equation 2-4:

C = Ao(2γ)n1βn2τn3

.



4-4

The constants Ao, n1, n2 and n3 were determined by a multiple regression analysisusing Hankinson [7] FEA results, that is, those under the FEA column in Tables 4-1, 4-2,4-3, and 4-4. In the following comparison, this is referred to a “full regression.” The bestfit equations so obtained are:

CL = 3.0(2γ)0.0734β-0.01τ0.769 (Eq. 4-1)

CN = 1.287(2γ)0.21β-0.355τ0.84 (Eq. 4-2)

CT = 2.24 (2γ)0.158β-0.06τ0.717 (Eq. 4-3)

CW = 1.34 (2γ)0.229β-0.42τ0.85 (Eq. 4-4)

In Section 5, Equations 4-1 to 4-4 are adjusted for test results to obtain Equations 5-1 to5-4, which then become the first part of Item 1 of Section 7, Conclusions.

As part of this study, for comparison, an additional modification was made for CL andCN. Regression equations were also made for data where t/T ≤ 1.0 and additionalequations for data where t/T> 1.0. These are called the “t/T ≤ 1.0 regression.” Theequations are:

CL = 2.85(2γ)0.055βn0.116τ0.707

for t/T ≤ 1.0

CL = 2.87(2γ)0.15β0.375τ0.756 for t/T > 1.0

CN = 1.34(2γ)0.173β-0.39τ0.75for t/T ≤ 1.0

CN = .918(2γ)0.299β-0.233τ1.52 for t/T > 1.0

Comparison

Tables 4-2 through 4-5 list the results of the comparison for CL, CN, CT, and CW. Thevalues of C are calculated using the various methods. The calculated values are referredto as CL’, CN’, CT’, and CW’. These are then compared to the results of the finite elementanalysis (FEA), which is used as the base line. These values are referred as C L, CN, CT,and CW. The ratio of CL’/ CL is calculated. The average, maximum, and minimum valuesare presented for comparison.

It should be noted that Wordsworth did not develop indices for torsional moments.


Table 4-2Comparisons for C L

Model 2 γ = β = τ = FEA Wordsworth HankinsonFull

Regressiont/T≤1

RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T C L Kc~C L Kc/C L CL' CL'/CL CL' CL'/CL CL' CL'/CL CL' CL'/CL CL' CL'/CL

1 20.0 .396 .20 1.50 0.67 0.45 1.24 0.80 1.09 0.73 1.20 0.80 1.69 1.13 1.55 1.03

2 20.0 .406 .40 1.80 1.17 0.65 2.13 1.18 1.86 1.04 1.95 1.08 2.18 1.21 2.00 1.11

3 20.0 .466 1.60 4.30 3.73 0.87 7.47 1.74 5.41 1.26 4.82 1.12 4.29 1.00 3.39 0.79

4 20.0 .595 .30 1.70 1.05 0.62 1.71 1.00 1.49 0.88 1.52 0.90 1.69 0.99 1.55 0.91

5 20.0 .610 .60 2.00 1.84 0.92 2.93 1.46 2.54 1.27 2.48 1.24 2.18 1.09 2.00 1.00

6 20.0 .780 .40 1.90 1.39 0.73 2.14 1.13 1.85 0.97 1.81 0.95 1.70 0.89 1.56 0.82

7 20.0 .800 .80 2.60 2.43 0.93 3.67 1.41 3.16 1.21 2.95 1.13 2.31 0.89 2.01 0.77

8 60.0 .403 .20 1.30 1.30 1.00 1.36 1.05 1.19 0.91 1.27 0.98 2.16 1.66 1.98 1.52

9 60.0 .407 .40 1.70 2.27 1.33 2.33 1.37 2.02 1.19 2.07 1.22 2.80 1.65 2.57 1.51

10 60.0 .427 1.60 5.60 6.99 1.25 8.16 1.46 5.87 1.05 5.50 0.98 8.32 1.49 6.57 1.17

11 60.0 .605 .30 1.50 2.04 1.36 1.87 1.24 1.61 1.08 1.62 1.08 2.16 1.44 1.98 1.32

12 60.0 .610 .60 2.20 3.56 1.62 3.20 1.46 2.75 1.25 2.63 1.20 3.53 1.60 2.79 1.27

13 60.0 .807 .40 1.90 2.70 1.42 2.34 1.23 2.01 1.06 1.91 1.01 2.46 1.30 1.98 1.04

14 60.0 .813 .80 3.50 4.70 1.34 4.01 1.15 3.42 0.98 3.12 0.89 4.47 1.28 3.53 1.01

15 40.3 .640 1.80 5.90 6.84 1.16 9.07 1.54 6.21 1.05 6.59 1.12 7.14 1.21 5.64 0.96


Table 4-2 (cont.)Comparisons for C L


Regressiont/T≤1

RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T C L Kc~C L Kc/C L CL' CL'/CL CL' CL'/CL CL' CL'/CL CL' CL'/CL CL' CL'/CL

16 40.3 .843 2.00 6.90 7.70 1.12 10.26 1.49 6.72 0.97 7.92 1.15 7.73 1.12 6.10 0.88

17 23.7 .528 1.07 3.80 3.11 0.82 4.78 1.26 4.02 1.06 3.83 1.01 3.34 0.88 2.73 0.72

18 23.7 .679 1.20 4.90 3.65 0.74 5.47 1.12 4.38 0.89 4.59 0.94 3.66 0.75 2.89 0.59

19 34.0 .520 1.15 4.50 4.08 0.91 5.35 1.19 4.36 0.97 4.24 0.94 4.42 0.98 3.50 0.78

20 25.5 .843 1.00 3.70 3.36 0.91 4.45 1.20 3.81 1.03 3.47 0.94 3.23 0.87 2.55 0.69

21 42.7 .539 1.33 5.60 5.31 0.95 6.45 1.15 4.96 0.89 4.97 0.89 5.74 1.03 4.54 0.81

22 32.0 .797 2.00 9.00 6.70 0.74 10.06 1.12 6.61 0.73 7.49 0.83 6.74 0.75 5.32 0.59

23 64.0 .531 1.00 4.40 5.36 1.22 4.79 1.09 4.10 0.93 3.86 0.88 5.72 1.30 4.52 1.03

24 48.0 .667 1.25 4.20 5.73 1.36 6.05 1.44 4.75 1.13 5.22 1.24 5.78 1.38 4.57 1.09

25 57.6 .556 1.00 4.60 5.10 1.11 4.75 1.03 4.06 0.88 3.81 0.83 5.36 1.17 4.23 0.92

26 49.7 .667 1.68 7.50 7.40 0.99 8.53 1.14 5.98 0.80 6.56 0.87 7.61 1.01 6.01 0.80

AVE= 1.02 AVE= 1.25 AVE= 1.01 AVE= 1.01 AVE= 1.16 AVE= 0.97

MAX= 1.62 MAX= 1.74 MAX= 1.27 MAX= 1.24 MAX= 1.66 MAX= 1.52

MIN= 0.45 MIN= 0.80 MIN= 0.73 MIN= 0.80 MIN= 0.75 MIN= 0.59


Table 4-3Comparisons for C N


Regressiont/ Τ ≤ 1

RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T C N Ks~C N Ks/C N CN' CN'/CN CN' CN'/CN CN' CN'/CN CN' CN'/CN CN' CN'/CN

1 20.0 .396 .20 1.10 0.67 0.61 0.58 0.53 0.87 0.79 0.97 0.88 2.14 1.95 0.92 0.84

2 20.0 .406 .40 1.30 1.38 1.06 1.03 0.79 1.54 1.18 1.61 1.24 3.18 2.45 1.37 1.05

3 20.0 .466 1.60 4.10 6.15 1.50 6.38 1.56 4.70 1.15 5.49 1.34 8.74 2.13 4.59 1.12

4 20.0 .595 .30 1.20 1.34 1.11 0.92 0.77 1.06 0.88 1.12 0.93 3.39 2.82 1.46 1.22

5 20.0 .610 .60 1.70 2.70 1.59 1.63 0.96 1.87 1.10 1.86 1.09 4.89 2.88 2.15 1.26

6 20.0 .780 .40 1.20 1.76 1.47 1.27 1.06 1.22 1.02 1.25 1.04 3.67 3.06 1.58 1.32

7 20.0 .800 .80 1.70 3.44 2.02 2.25 1.33 2.17 1.27 2.08 1.22 5.30 3.12 2.34 1.38

8 60.0 .403 .20 1.30 2.06 1.58 0.76 0.58 1.09 0.84 1.16 0.89 5.53 4.25 2.38 1.83

9 60.0 .407 .40 2.00 4.14 2.07 1.34 0.67 1.94 0.97 1.94 0.97 8.09 4.05 3.64 1.82

10 60.0 .427 1.60 7.60 17.22 2.27 8.53 1.12 6.11 0.80 7.78 1.02 24.71 3.25 12.97 1.71

11 60.0 .605 .30 1.30 4.04 3.10 1.20 0.93 1.32 1.02 1.34 1.03 8.58 6.60 3.70 2.85

12 60.0 .610 .60 2.20 8.10 3.68 2.12 0.97 2.36 1.07 2.25 1.02 12.45 5.66 6.51 2.96

13 60.0 .807 .40 1.30 5.11 3.93 1.67 1.28 1.52 1.17 1.49 1.14 9.25 7.12 3.99 3.07

14 60.0 .813 .80 2.40 10.12 4.22 2.95 1.23 2.71 1.13 2.49 1.04 13.42 5.59 7.03 2.93


Table 4-3 (cont.)Comparisons for CN


Regressiont/ Τ ≤ 1

RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T C N Ks~C N Ks/C N CN' CN'/CN CN' CN'/CN CN' CN'/CN CN' CN'/CN CN' CN'/CN

15 40.3 .640 1.80 6.60 16.57 2.51 7.80 1.18 5.37 0.81 7.52 1.14 21.43 3.25 11.25 1.70

16 40.3 .843 2.00 5.90 16.11 2.73 8.06 1.37 5.32 0.90 8.28 1.40 19.83 3.36 10.41 1.76

17 23.7 .528 1.07 2.70 5.30 1.96 4.02 1.49 3.32 1.23 3.05 1.13 8.00 2.96 4.20 1.56

18 23.7 .679 1.20 3.50 6.56 1.87 4.24 1.21 3.35 0.96 3.43 0.98 8.42 2.41 4.42 1.26

19 34.0 .520 1.15 3.30 8.11 2.46 4.79 1.45 3.83 1.16 3.81 1.15 12.15 3.68 6.38 1.93

20 25.5 .843 1.00 3.90 5.09 1.31 2.91 0.75 2.70 0.69 2.51 0.64 7.24 1.86 3.54 0.91

21 42.7 .539 1.33 4.20 12.04 2.87 5.92 1.41 4.49 1.07 5.04 1.20 17.91 4.26 9.40 2.24

22 32.0 .797 2.00 7.00 13.81 1.97 7.76 1.11 5.17 0.74 7.82 1.12 16.25 2.32 8.53 1.22

23 64.0 .531 1.00 3.50 13.43 3.84 3.13 0.89 3.86 1.10 3.52 1.01 20.64 5.90 10.84 3.10

24 48.0 .667 1.25 3.30 13.80 4.18 5.28 1.60 4.04 1.22 4.51 1.37 18.00 5.45 9.45 2.86

25 57.6 .556 1.00 3.40 12.41 3.65 3.10 0.91 3.71 1.09 3.40 1.00 19.86 5.84 10.43 3.07

26 49.7 .667 1.68 5.40 19.19 3.55 7.47 1.38 5.22 0.97 7.14 1.32 24.23 4.49 12.72 2.36

AVE= 2.43 AVE= 1.10 AVE= 1.01 AVE= 1.09 AVE= 3.87 AVE= 1.90

MAX= 4.22 MAX= 1.60 MAX= 1.27 MAX= 1.40 MAX= 7.12 MAX= 3.10

MIN= 0.61 MIN= 0.53 MIN= 0.69 MIN= 0.64 MIN= 1.86 MIN= 0.84


Table 4-4Comparisons for C T

Model 2 γ = β = τ = FEA Hankinson RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T C T CT' CT'/CT CT' CT'/CT CT' CT'/CT CT' CT'/CT

1 20.0 .396 .20 1.40 1.44 0.75 1.20 0.86 1.00 0.71 1.00 0.71

2 20.0 .406 .40 1.90 2.38 1.25 1.97 1.04 1.00 0.53 1.00 0.53

3 20.0 .466 1.60 4.80 7.86 1.64 5.27 1.10 1.00 0.21 1.00 0.21

4 20.0 .595 .30 1.90 1.88 0.99 1.56 0.82 1.24 0.65 1.05 0.55

5 20.0 .610 .60 2.50 3.10 1.24 2.57 1.03 1.52 0.61 1.15 0.46

6 20.0 .780 .40 2.10 2.27 1.08 1.89 0.90 2.36 1.13 1.30 0.62

7 20.0 .800 .80 2.60 3.74 1.44 3.11 1.19 3.39 1.30 1.75 0.67

8 60.0 .403 .20 1.40 1.73 1.24 1.42 1.02 1.00 0.71 1.00 0.71

9 60.0 .407 .40 2.50 2.86 1.14 2.34 0.94 1.00 0.40 1.00 0.40

10 60.0 .427 1.60 8.80 9.51 1.08 6.31 0.72 1.00 0.11 1.00 0.11

11 60.0 .605 .30 1.90 2.25 1.19 1.86 0.98 1.93 1.01 1.33 0.70

12 60.0 .610 .60 3.00 3.72 1.24 3.05 1.02 2.53 0.84 1.73 0.58

13 60.0 .807 .40 2.20 2.72 1.23 2.25 1.02 5.71 2.59 2.70 1.23

14 60.0 .813 .80 3.20 4.49 1.40 3.69 1.15 8.27 2.58 4.53 1.42

15 40.3 .640 1.80 8.00 9.88 1.24 6.29 0.79 5.07 0.63 3.04 0.38


Table 4-4 (cont.)Comparisons for C T

Model 2 γ = β = τ = FEA Hankinson RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T C T CT' CT'/CT CT' CT'/CT CT' CT'/CT CT' CT'/CT

16 40.3 .843 2.00 8.30 10.92 1.32 6.67 0.80 13.26 1.60 7.13 0.86

17 23.7 .528 1.07 3.10 5.05 1.63 4.03 1.30 1.00 0.32 1.00 0.32

18 23.7 .679 1.20 4.00 5.66 1.42 4.32 1.08 3.13 0.78 1.98 0.50

19 34.0 .520 1.15 3.80 5.84 1.54 4.50 1.18 1.00 0.26 1.00 0.26

20 25.5 .843 1.00 4.00 4.56 1.14 3.78 0.94 5.07 1.27 2.65 0.66

21 42.7 .539 1.33 5.00 7.15 1.43 5.17 1.03 1.00 0.20 1.00 0.20

22 32.0 .797 2.00 8.10 10.55 1.30 6.45 0.80 9.37 1.16 5.13 0.63

23 64.0 .531 1.00 3.90 5.51 1.41 4.49 1.15 1.00 0.26 1.00 0.26

24 48.0 .667 1.25 3.90 6.67 1.71 4.97 1.27 5.41 1.39 3.19 0.82

25 57.6 .556 1.00 3.90 5.39 1.38 4.40 1.13 1.23 0.32 1.12 0.29

26 49.7 .667 1.68 6.50 9.42 1.45 6.17 0.95 7.02 1.08 4.04 0.62

AVE= 1.30 AVE= 1.01 AVE= 0.87 AVE= 0.57

MAX= 1.71 MAX= 1.30 MAX= 2.59 MAX= 1.42

MIN= 0.75 MIN= 0.72 MIN= 0.11 MIN= 0.11


Table 4-5Comparisons for C W

Model 2 γ = β = τ = FEA Wordsworth Hankinson RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T CW CW' CW'/CW CW' CW'/CW CW' CW'/CW CW' CW'/CW CW' CW'/CW

1 20.0 .396 .20 1.33 1.06 0.79 1.05 0.79 1.00 0.75 3.82 2.87 1.90 1.43

2 20.0 .406 .40 1.62 2.12 1.31 1.84 1.13 1.78 1.10 7.59 4.69 3.77 2.33

3 20.0 .466 1.60 5.52 8.59 1.56 6.50 1.18 5.47 0.99 28.99 5.25 14.39 2.61

4 20.0 .595 .30 1.30 1.52 1.17 1.29 0.99 1.19 0.91 4.69 3.61 2.33 1.79

5 20.0 .610 .60 1.75 3.00 1.71 2.25 1.28 2.12 1.21 9.18 5.24 4.55 2.60

6 20.0 .780 .40 1.37 1.49 1.09 1.49 1.09 1.36 0.99 4.38 3.20 2.17 1.59

7 20.0 .800 .80 2.03 2.84 1.40 2.60 1.28 2.42 1.19 8.33 4.10 4.13 2.04

8 60.0 .403 .20 1.38 3.18 2.31 1.36 0.99 1.28 0.93 6.96 5.05 3.45 2.51

9 60.0 .407 .40 1.76 6.36 3.62 2.39 1.36 2.29 1.30 13.89 7.90 6.89 3.92

10 60.0 .427 1.60 7.71 25.64 3.33 8.70 1.13 7.30 0.95 65.68 8.52 32.60 4.23

11 60.0 .605 .30 1.49 4.52 3.03 1.66 1.12 1.52 1.02 8.46 5.67 4.20 2.82

12 60.0 .610 .60 2.55 8.99 3.53 2.92 1.15 2.73 1.07 16.79 6.59 8.33 3.27

13 60.0 .807 .40 1.72 4.17 2.43 1.92 1.12 1.72 1.00 7.49 4.36 3.72 2.16

14 60.0 .813 .80 3.40 8.19 2.41 3.37 0.99 3.09 0.91 15.04 4.43 7.46 2.20

15 40.3 .640 1.80 6.80 17.53 2.58 7.99 1.17 6.21 0.91 44.65 6.57 22.16 3.26


Table 4-5 (cont.)Comparisons for C W

Model 2 γ = β = τ = FEA Wordsworth Hankinson RegressionCode Case

N-392CC N-392

With θ adj.

No. D/T d/D t/T CW CW' CW'/CW CW' CW'/CW CW' CW'/CW CW' CW'/CW CW' CW'/CW

16 40.3 .843 2.00 6.90 12.53 1.82 8.28 1.20 6.05 0.88 34.27 4.97 17.01 2.47

17 23.7 .528 1.07

18 23.7 .679 1.20

19 34.0 .520 1.15

20 25.5 .843 1.00

21 42.7 .539 1.33

22 32.0 .797 2.00

23 64.0 .531 1.00

24 48.0 .667 1.25

25 57.6 .556 1.00

26 49.7 .667 1.68

AVE= 2.13 AVE= 1.12 AVE= 1.01 AVE= 5.19 AVE= 2.58

MAX= 3.62 MAX= 1.36 MAX= 1.30 MAX= 8.52 MAX= 4.23

MIN= 0.79 MIN= 0.79 MIN= 0.75 MIN= 2.87 MIN= 1.43



4-13

In addition, histograms of the ratios of the calculated values of the indices to thecorresponding FEA results are presented in Figures 4-1 and 4-2.

Full Regression

Fre

quen

cy 10

5

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Hankinson

Fre

quen

cy 10

5

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

t/T < 1.0 Regression

Fre

quen

cy 10

5

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Code Case N-392

Fre

quen

cy 4

2

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Wordsworth0.

4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Fre

quen

cy

Frequency

10

5

0

CL´/CL

CL´/CL

CL´/CL

CL´/CL

CL´/CL

Figure 4-1Comparison of Equations for C L



4-14

Full Regression

Fre

quen

cy 10

5

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Hankinson

Fre

quen

cy 4

2

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

t/T < 1.0 Regression

Fre

quen

cy 10

5

0

Frequency

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Code Case N-392

Fre

quen

cy

420

Frequency

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

Wordsworth

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

Fre

quen

cyFrequency

0

246

6

CN´/CN

CN´/CN

CN´/CN

CN´/CN

CN´/CN

Figure 4-2Comparison of Equations for C N

Results of Comparison

First, consider in-plane bending or CL. A review of Table 4-2 shows that all the methodsprovide results that are reasonably close to the FEA results. Hankinson’s [7] equationprovides results that, on the average, are about 25% higher than the FEA. This is to beexpected since Hankinson’s [7] equation was a “curve fit” and it was intended to beconservative. The other expressions were “best fit” regression expressions and wereexpected to be a closer fit.

For comparison, the values of CL that correspond to the parameters of the testspecimens are:

CL = 4.37 for Hankinson [7]

CL = 3.53 for Wordsworth

CL = 3.77 for Code Case N-392



4-15

CL = 3.76 for Full Regression

CL = 3.60 for t/T ≤ 1.0 Regression

The average value is 3.8. Except for Hankinson [7], these results are very close.

For out-of-plane bending, the results are somewhat different. A review of Table 4-3indicates, as expected, the various regression expressions and Hankinson’s [7]equations produce results close to the FEA results. However, Wordsworth, Code CaseN-392, and Code Case N-392 modified to include the θ effects are all consistentlyhigher.

The results of Tables 4-2 and 4-3 indicated that there was little difference in the fullregression and the t/T ≤ 1.0 Regression.

For torsion of the trunnion, CT, Hankinson [7] provided results about 30% higher thanthe FEA. The regression analysis yielded the closest fit and the Code Case N-392 resultswere very high.

For the axial force, CW, Hankinson[ 7] provided results about 12% higher, and again theregression analysis was the closest fit. Wordsworth and the Code Case were very high.

As a result of the differences among the FEA results, Wordsworth, and the Code Case,an FEA analysis was performed to verify the Hankinson [7] results. Model 12 wasselected because the variation was high. The FEA was conducted using COSMOSversion 1.75. Figure 4-1 shows the model that was developed. The FEA used four node,quadrilateral shell elements built up of four triangular elements. About 6300 elementswere used in the model. The models were fixed at one end, and moments in the threeorthogonal directions were applied at the trunnion or the end of the straight pipe. Loadcombinations that simultaneously applied the six moments were also run.

The stresses used by Hankinson [7] in the development of the stress indices are basedon the stress intensity located at the centroid of each element. The stress intensities areevaluated at mid-thickness and at the top and bottom shell surfaces. The generalmethodology used in the various EPRI studies that are a part of this project is at thejuncture of the trunnion (or in other cases, the branch) to project the stress intensitiesfrom the adjacent elements to the juncture. This is similar to experimental stressmeasurement techniques where the stress at the juncture is projected from strain gageslocated on both sides of the juncture. FEA nodal results at the juncture, which are anaveraging of the elbow and trunnion side results, are not used. However, forcomparison, they will be considered. Table 4-6 contains a comparison of the results forCL ,CN and CT. The FEA results are with loads on the end of the trunnion.



4-16

Table 4-6Comparison of Results for Hankinson [7] Model 12

This Study This Study CC CC

Hankinson FEA FEA N-392 N-392

FEA Element Projected Wordsworth with θEffect

CL 2.2 2.47 3.18 3.56 3.53 2.79

CN 2.2 2.46 2.72 8.10 12.45 6.51

CT 3.0 3.1 3.5 N/A 2.53 1.73

Hankinson [7] also reported a value of C2e, which represented the C index for theelbow. The value was C2e = 8.1 for Model 12. The FEA for this study indicates that the Cfor loading the elbow is 5.19 for torsion or out-of-plane loading and 8.19 for in-planeloading. For comparison, an FEA was run for an elbow without a trunnion. Thisproduced C values of 5.24 for torsion or out-of-plane bending and 8.55 for in-planebending. For reference, the Code value of C2 = 1.95/h2/3= 8.85.

Hankinson’s [7] models covered only long radius elbows (R = 1.5D where D is thenominal pipe diameter). However, it is assumed that trunnions would also be used onshort radius elbows (R=1.0D where D is the nominal pipe diameter). The major impactis that the angle θ changes from 41.4° to 48.2°. Based on the conservatism of the overallprocess, it is assumed that the conclusions derived from Hankinson’s studies are alsoapplicable to trunnions attached to short radius elbows.

The following observations can be made from comparing Table 4-2 to Table 4-5 andfrom Table 4-6:

• The FEA from this study verifies the results from Hankinson [7] when the stressintensity used in the comparison is based on the element stress (within about 12%).

• The [7] FEA for Model 12 results are 31% lower than the results from this study forCL, 19% lower for CN, and 14% lower for CT. This corresponds to the results fromother studies prepared under the EPRI SIF Optimization Project when the FEAresults for element stresses are compared to those that are projected.

• The results from Wordsworth and Code Case N-392 yield reasonable results for in-plane loading (CL) but are overly conservative for other loadings. Thesemethodologies were based on trunnions on straight pipe; apparently, the elbowdoes have an effect. When compared to the FEA from this study, Wordsworth isconservative by a factor of about 3 for CN. Code Case N-392 is conservative by a



4-17

factor of 4.6 . When Code Case N-392 is adjusted for the θ effect, this drops to afactor of 2.39.

• The FEA from this study indicates that the presence of the trunnion decreases thestress in the elbow when the loading is on the elbow. This is confirmed byHankinson [7] in that the value of C2 determined is less than that of a plain elbow.

• The results from the full regression and the t/T ≤ 1.0 regression produce similarresults. There appears to be no advantage in using the more complicated t/T ≤ 1.0regression equations.

It is concluded that the Hankinson [7] FEA data can serve as the basis of expressions forvalues of the secondary stress indices. It is also concluded that the modified correlationequations for full regression should serve as the basis of this analysis. Theseexpressions are used as a basis of this study for developing the indices.


5-1

5 COMPARISON OF TEST DATA TO ANALYSIS

METHODS

Purpose

The purpose of this section is to summarize the test results and compare them to thosedetermined by analysis methods.

C Indices

The experimental value of CL was, on the average, 1.66 using the CT Indices -MarklApproach and about 1.5 using the CT Indices-Class 1 Approach. Using Equation 4-1, avalue of CL = 3.76 would be calculated. The tests indicate that the indices developedfrom Equation 4-1 can be reduced by a factor of .43–.40. Considering the spread of thedata in Tables 4-2 through 4-5 (the maximum ratio of C’/C was 1.3), a factor of 1.3*0.43= .56 is suggested. Multiplying Equation 4-1 by 0.56 leads to:

CL = 1.68(2γ)0.0734β-0.01τ0.769 (Eq. 5-1)

The tests performed for this investigation, were for in-plane bending of the trunnion.Because the indices for other loading conditions were based on the same theoreticalapproach (finite element analysis), it is reasonable to assume that the same degree ofconservatism exists for these indices. Multiplying Equations 4-1, 4-3 and 4-4 by 0.56leads to:

CN = 0.721(2γ)0.21β-0.355τ0.84 (Eq. 5-2)

CT = 1.25 (2γ)0.158β-0.06τ0.717 (Eq. 5-3)

CW = 0.75 (2γ)0.229β-0.42τ0.85 (Eq. 5-4)


Comparison of Test Data to Analysis Methods

5-2

Equations 5-1 through 5-4 are the first part of item 1 of Section 7, Conclusions.

B Indices

The tests performed as a part of this study were not specifically focused on the type oftesting required to provide data for experimental determination of the B indices.However, an upper bound estimate was made based on the data. An upper bound of BL

was determined to be BL = 1.07 on the average.

Based on this value, the value of BL is .64–.71 times the experimentally derived value ofCL. The present version of Code Case N-392 specifies that the B index be taken as 0.5times the C index.

It is recognized that the method of estimating the value of BL from the test data is veryconservative. Recognizing also that the equations for the C indices include a factor of1.3 (to cover the statistical distribution), it is believed that it is reasonable to maintainthe relationship that the B indices are 0.5 times the C indices. It is also believed to bereasonable to assume that the same relationship exists for the other B indices, that is,BW, BN, and BT.


6-1

6 INVESTIGATION OF FLEXIBILITY OF TRUNNIONS ON

ELBOWS

General

There are two parts to the evaluation of the flexibility of the trunnion-elbowconfiguration. The first part is the impact of the trunnion on the flexibility of the elbow.The flexibility of elbows is directly related to the ability of the elbow to ovalize. Withlarge trunnions, the ovalization will be restricted. This is to be investigated. The secondpart is the flexibility of the trunnion. This is important because trunnion-elbows areoften used as anchors in piping systems.

This flexibility study is based on the results from FEA. The test data (Section 3) do notenvelope all the boundary conditions that are used to verify the results. As discussedlater, the flexibility is a function of the boundary conditions of the trunnion-elbow. Thetest data are used to confirm the results of the study.

Discussion: Elbows

In piping analysis, the various components are modeled as one dimensional beamelements. In order to accurately represent the load displacement (flexibility) action ofthe components, flexibility factors are used.

From Rodabaugh [23], when bending a straight pipe of length L, the rotation, φ, of oneend with respect to the other is:

φ= 1/EI ∫o

L M dx

where M is the bending moment.

For a torsional moment, the rotation is given by:

φ = (1/GJ) ∫o

L M dx = 1.3/EI ∫o

L M dx (Eq. 6-1)


Investigation of Flexibility of Trunnions on Elbows

6-2

where M is the torsional moment. This is based on the relationship between J and I inaddition to G= E/(2(1+µ)).

For curved pipe or an elbow of 90o, the length of the one-dimensional segment used torepresent it is Rπ/2 where R is the bend radius. The rotation for in-plane bending isgiven by:

φ =(k/EIe )∫oπ/2 M Rdα = kπMR/(2EIe )

where α is shown in Figure 6-1, k is the flexibility factor for the elbow, and M is the in-plane bending moment.

The evaluation will be performed by comparing the rotations from FEA to those fromclosed form solutions for the beam model considering both bending and torsionalrotations due to in-plane and out-of-plane bending. The flexibility factor for an elbow isdefined by the Codes as k = 1.65/h, h = tR/r2.

Rodabaugh [22] provides the following expressions for evaluating the flexibility factorsfor elbows with attached pipe lengths of length L.

In-plane moment:

k = [φfea/(M/EIe) - 2 L]/(Rπ/2) (Eq. 6-2)

Out-of-plane moment:

k = [φfea/(M/EIe) - 2.3 L - 1.021 R]/(Rπ/4) (Eq. 6-3)

where L is the length of the attached pipe and φfea represents the rotation from the FEA.This assumes that G = E/(2(1+µ). The rotation, φfea, for in-plane moments correspondsto the rotation, φ, indicated in Figure 6-1(a). The rotation, φfea, for out-of-plane bendingis similarly defined.



6-3

M

R

M

α

M

R

M

α

(a) Elbow

(b) Elbow with Pipe Attached

Figure 6-1Configurations

Discussion: Trunnions on Elbows

The flexibility of the trunnion connection will also be investigated. For configurationssuch as branch connections or trunnions on elbows, there is no well-defined length,such as L or Rα, to integrate over. The rotations are due to local deformations in thearea of intersection.



6-4

There are several possible ways to model the trunnion/elbow connection. Figure 6-2indicates one possible model. A rigid link is used to connect point A to B. At point B, apoint spring is used to represent the local flexibility of the connection.

BA

Figure 6-2Elbow-Trunnion Model

For comparison, Figure 6-3 indicates a typical model used for branch connections. It canbe seen that this is equivalent to the trunnion/elbow model, assuming the elbow ischanged from a 90o elbow to 0o and the trunnion is rotated.

Rigid LinkPoint Spring

Figure 6-3Branch Connection Model



6-5

It is convenient to define the flexibility of the spring by:

φ = k M do/EIt

where It is the section modulus of the trunnion. Then k is equivalent to the number ofpipe (or trunnion) diameters that would be added to represent the local flexibility.

The flexibility factor for the spring is calculated by:

k =(φfea - φb)/(Mdo/It) (Eq. 6-4)

where φfea is the rotation from the FEA and φb is the rotation from the beam model. Thisis discussed in more detail later.

Finite Element Analysis

The general dimensions for the FEA models are shown in Figure 6-4. Typically, straightsections of pipe (or trunnion) equal to four diameters are used in the models. COSMOSversion 1.75 from Structural Research and Analysis Corporation was used. Stresseswere not calculated in this evaluation, only deflections.

R

2

1

3

L1 L3

L4

L2

L5

θ

L1 = 4 Do

θ = arccos(R/(R+Do/2))

L2 = (R+Do/2)sin θ

L3 = R+Do/2+4do-L2

L4 = 4 do

L5 = 4 Do

Figure 6-4FEA Model Details



6-6

Shell elements were used in the models, which typically consisted of approximately7000 elements. A typical model is shown in Figure 6-5. The elements at the ends of thepipe and trunnion sections are connected to elements that are “rigid bars” or “rigidlinks,” that is, the stiffness of the rigid links is infinite (or very large).

Figure 6-5FEA Model

The material properties used in the analyses are E = 30E6, G = 12E6, and µ = 0.28. Thebeam equations (Equations 6-1 and the others) assumed a value of µ = 0.3. Thisdifference is considered insignificant.

For evaluation of the flexibility of the elbow, one end of the model (the lower end),point 1 in Figure 6-4, was fixed and moments were applied to the upper end, point 2.The trunnion end (point 3) was free.

For evaluation of the flexibility of the trunnion, a different approach was used. Thelocal rotation at the juncture of the elbow and the trunnion is somewhat dependent onthe boundary conditions at the ends of the pipes (points 1 and 2). It is important torecognize that in evaluating flexibility factors, there is no “conservative” value thatwould be applicable for all piping layouts. As an example, a high value might meanthat the loads are lower in other components in a piping system than the “true” values.



6-7

Consequently, the “best” value to use is the one that is most representative of the actualvalue. This is complicated because the flexibility is a function of the end conditions atthe ends of the elbow (or attached straight pipe). This will, of course, be a function ofthe layout.

As a result of this, two sets of boundary conditions were used in the evaluation. Thefirst was where the lower end of the model (point 1) was fixed and the upper end(point 2) was free; the loads were applied at the end of the trunnion (point 3). In thesecond case, both ends were fixed and the loads were applied at the end of thetrunnion. The flexibility factors are based on the average of the results. This is depictedin Figure 6-6.

(2)(3)

(1)

(b) Moment on Trunnion

(2)(3)

(1)

(c) Moment on Trunnion

(2)(3)

(1)

(a) Moment on Elbow

Figure 6-6Boundary Conditions

The 22 models are listed in Table 6-1 along with the dimensions and other pertinentdata. The cases listed are representative of actual usage. As an example, it is notexpected that a very small trunnion would be used on a large elbow (small d/D). Table6-1 includes the moments used in the FEA, which are based on a nominal bendingstress of 10 ksi in the pipe or trunnion (that is, M/Z or M/ZT where Z is the sectionmodulus for the pipe and ZT is the section modulus for the trunnion). Note that only thedeflections were calculated in the FEA.

The bend radius, R, is also listed in Table 6-1. For models of actual pipe, R = 1.5 Dwhere D is the nominal pipe size. For the other cases, R = 1.5 Do. These values of Rcorrespond to long radius elbows. Based on the conservatism of the overall process, it isassumed that the conclusions derived from the FEA will also be applicable to shortradius elbows (R = 1.0 D).


Table 6-1FEA Models

Model D o T do t R do/D t/T Do/T do/t TR/r 2 D d D/T d/D d/t Ma Mb Ie It θ L3

(in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.-lb.) (in.-lb.) (in 4) (in 4) (degrees) (in.)

T1 8.625 0.250 4.5 0.237 12.0 0.522 0.948 34.5 19.0 0.171 8.375 4.263 33.5 0.51 18.0 33827 137721 57.7 7.2 42.6 22.6

T2 12.75 0.375 4.5 0.237 18.0 0.353 0.632 34.0 19.0 0.176 12.37 4.263 33.0 0.34 18.0 33827 451036 279.3 7.2 42.4 24.6

T3 12.75 0.375 10.75 0.365 18.0 0.843 0.973 34.0 29.5 0.176 12.37 10.38 33.0 0.84 28.5 309169 451036 279.3 160.7 42.4 50.6

T4 12.75 0.375 10.75 0.594 18.0 0.843 1.584 34.0 18.1 0.176 12.37 10.15 33.0 0.82 17.1 481195 451036 279.3 245.2 42.4 50.6

TE1 10.00 0.500 5.0 0.250 15.0 0.50 0.500 20.0 20.0 0.332 9.500 4.750 19.0 0.50 19.0 44301 354411 168.8 10.6 41.4 25.9

TE2 10.00 0.500 7.5 0.375 15.0 0.75 0.750 20.0 20.0 0.332 9.500 7.125 19.0 0.75 19.0 149517 354411 168.8 53.4 41.4 36.3

TE3 10.00 0.500 8.5 0.425 15.0 0.85 0.850 20.0 20.0 0.332 9.500 8.075 19.0 0.85 19.0 217652 354411 168.8 88.1 41.4 40.5

TE4 10.00 0.500 5.0 0.500 15.0 0.50 1.000 20.0 10.0 0.332 9.500 4.500 19.0 0.47 9.0 79521 354411 168.8 18.1 41.4 25.9

TE5 10.00 0.500 7.5 0.750 15.0 0.75 1.500 20.0 10.0 0.332 9.500 6.750 19.0 0.71 9.0 268385 354411 168.8 91.7 41.4 36.3

TE6 10.00 0.500 8.5 0.850 15.0 0.85 1.700 20.0 10.0 0.332 9.500 7.650 19.0 0.81 9.0 390689 354411 168.8 151.3 41.4 40.5

TE7 10.00 0.333 5.0 0.167 15.0 0.50 0.500 30.0 30.0 0.214 9.667 4.833 29.0 0.50 29.0 30580 244637 118.4 7.4 41.4 25.9

TE8 10.00 0.333 7.5 0.250 15.0 0.75 0.750 30.0 30.0 0.214 9.667 7.250 29.0 0.75 29.0 103206 244637 118.4 37.5 41.4 36.3

TE9 10.00 0.333 8.5 0.283 15.0 0.85 0.850 30.0 30.0 0.214 9.667 8.217 29.0 0.85 29.0 150238 244637 118.4 61.8 41.4 40.5

TE10 10.00 0.333 5.0 0.333 15.0 0.50 1.000 30.0 15.0 0.214 9.667 4.667 29.0 0.48 14.0 57014 244637 118.4 13.4 41.4 25.9

TE11 10.00 0.333 7.5 0.500 15.0 0.75 1.500 30.0 15.0 0.214 9.667 7.000 29.0 0.72 14.0 192422 244637 118.4 67.7 41.4 36.3

TE12 10.00 0.333 8.5 0.567 15.0 0.85 1.700 30.0 15.0 0.214 9.667 7.933 29.0 0.82 14.0 280110 244637 118.4 111.7 41.4 40.5

TE13 10.00 0.200 5.0 0.100 15.0 0.50 0.500 50.0 50.0 0.125 9.800 4.900 49.0 0.50 49.0 18857 150859 74.0 4.6 41.4 25.9

TE14 10.00 0.200 7.5 0.150 15.0 0.75 0.750 50.0 50.0 0.125 9.800 7.350 49.0 0.75 49.0 63644 150859 74.0 23.4 41.4 36.3

TE15 10.00 0.200 8.5 0.170 15.0 0.85 0.850 50.0 50.0 0.125 9.800 8.330 49.0 0.85 49.0 92646 150859 74.0 38.6 41.4 40.5

TE16 10.00 0.200 5.0 0.200 15.0 0.50 1.000 50.0 25.0 0.125 9.800 4.800 49.0 0.49 24.0 36191 150859 74.0 8.7 41.4 25.9

TE17 10.00 0.200 7.5 0.300 15.0 0.75 1.500 50.0 25.0 0.125 9.800 7.200 49.0 0.73 24.0 122145 150859 74.0 44.0 41.4 36.3

TE18 10.00 0.200 8.5 0.340 15.0 0.85 1.700 50.0 25.0 0.125 9.800 8.160 49.0 0.83 24.0 177807 150859 74.0 72.7 41.4 40.5

Notes:

1. See Figure 6-4.



6-9

FEA Results: Flexibility of Elbows with Trunnions

Table 6-2 lists the rotations at the indicated points for the specific load cases. Theserotations were taken directly from the FEA output. The results are discussed in thefollowing sections.



6-10

Table 6-2Summary of Rotations

In-Plane Out-of-Plane In-Plane Out-of-Plane

Moment on Moment on Moment on Moment on

Trunnion Trunnion Pipe Pipe

PT-3 PT-2 PT-3 PT-2 PT-3 PT-2 PT-3 PT-2

Model φΖ φΖ φY φY φΖ φΖ φY φY

T1 6.337E-03 2.257E-03 7.994E-03 2.552E-03 9.190E-03 1.774E-02 1.039E-02 1.302E-02

T2 5.538E-03 6.863E-04 6.335E-03 7.807E-04 9.150E-03 1.787E-02 1.041E-02 1.297E-02

T3 8.025E-03 5.349E-03 1.091E-02 6.771E-03 7.803E-03 1.467E-02 9.878E-03 1.284E-02

T4 1.044E-02 7.981E-03 1.500E-02 1.047E-02 7.481E-03 1.399E-02 9.816E-03 1.280E-02

TE1 4.902E-03 7.937E-04 5.556E-03 1.004E-03 6.350E-03 1.230E-02 8.035E-03 1.056E-02

TE2 5.993E-03 2.537E-03 7.288E-03 3.187E-03 6.014E-03 1.147E-02 7.554E-03 1.049E-02

TE3 6.783E-03 3.530E-03 8.372E-03 4.509E-03 5.748E-03 1.085E-02 7.341E-03 1.043E-02

TE4 5.786E-03 1.413E-03 6.804E-03 1.760E-03 6.295E-03 1.217E-02 7.843E-03 1.053E-02

TE5 8.024E-03 4.435E-03 1.031E-02 5.673E-03 5.857E-03 1.110E-02 7.491E-03 1.045E-02

TE6 9.399E-03 6.081E-03 1.230E-02 8.017E-03 5.517E-03 1.037E-02 7.273E-03 1.036E-02

TE7 5.204E-03 1.052E-03 6.175E-03 1.257E-03 8.401E-03 1.620E-02 1.003E-02 1.238E-02

TE8 6.515E-03 3.311E-03 8.340E-03 4.010E-03 7.849E-03 1.483E-02 9.504E-03 1.230E-02

TE9 7.401E-03 4.485E-03 9.691E-03 5.664E-03 7.311E-03 1.368E-02 9.233E-03 1.223E-02

TE10 6.305E-03 1.952E-03 7.814E-03 2.279E-03 8.377E-03 1.605E-02 9.781E-03 1.234E-02

TE11 9.079E-03 5.971E-03 1.242E-03 7.412E-03 7.592E-03 1.423E-02 9.424E-03 1.226E-02

TE12 1.063E-02 7.902E-03 1.498E-02 1.045E-02 6.898E-03 1.281E-02 9.135E-03 1.215E-02

TE13 5.725E-03 1.588E-03 7.091E-03 1.676E-03 1.270E-02 2.404E-02 1.341E-02 1.603E-02

TE14 7.570E-03 4.952E-03 1.014E-02 5.544E-03 1.174E-02 2.179E-02 1.314E-02 1.593E-02

TE15 8.659E-03 6.482E-03 1.217E-02 7.928E-03 1.056E-02 1.940E-02 1.291E-02 1.587E-02

TE16 7.236E-03 3.044E-03 9.326E-03 3.154E-03 1.269E-02 2.389E-02 1.315E-02 1.598E-02

TE17 1.115E-02 9.162E-03 1.603E-02 1.058E-02 1.132E-02 2.081E-02 1.307E-02 1.590E-02

TE18 1.297E-02 1.154E-02 1.993E-02 1.510E-02 9.791E-03 1.782E-02 1.281E-02 1.580E-02

Notes:1. All rotations are in radians.2. PT-3 refers to the end of the trunnion and PT-2 refers to the end of the pipe (SeeFigure 6-4).

3. PT-1 is fixed.



6-11

Table 6-3 lists the flexibility factors for elbows derived from the FEA rotations usingEquations 6-2 and 6-3. The loads considered are the in-plane and out-of-plane loadingat the end of the pipe (point 2). Rodabaugh [22] provides an evaluation of end effectson elbows subjected to moment loading. It concludes that for in-plane bending forconfigurations of 90o elbows with straight sections of pipe on the ends:

k = 1.3/h (Eq. 6-5)

where h is the elbow characteristic TR/Rm

2. Consequently, for this study, a comparisonis made to the value of k = 1.3/h versus the value of 1.65/h listed in the Code.

It is seen from Table 6-3 that for out-of-plane loads, the flexibility is not significantlyaffected by the trunnion. The percentage difference from k = 1.3/h (a maximumdifference of 7.7%, an average difference of -1.6%, and a standard deviation of thedifference of 4.5%) is well within the methodology tolerance.

It is noted that Rodabaugh [22] suggests for out-of-plane bending a value of 1.25/h forthe flexibility of 90o elbows with straight sections of pipe on the ends. The percentagedifference from k = 1.25/h is slightly higher (a maximum difference of 9.0%, an averagedifference of 2.3%, and a standard deviation of the difference of 4.9%).

For in-plane loads, Table 6-3 indicates that the flexibility is affected by large trunnions.The flexibility is decreased by up to 26% for large trunnions. A regression analysis ofthe data yields the following expression:

k= .116 (D/T).99 (d/D)1.51 (d/t).11 (r2 = .98)

Comparing this equation to the results of the FEA analysis, the maximum difference is-10.2% and the average is .2%.


Table 6-3Bending of the Pipe-Elbow Flexibility

IN-PLANE LOAD OUT-OF-PLANE LOAD

Model D/T d/D d/t φZ k 1.3/h % DIFF Reg Eq % DIFF φY k 1.3/h % DIFF

(1) (2) (3) (4)

T1 33.5 0.51 18.0 1.774E-02 8.17 7.60 7.6 7.6 6.5 1.302E-02 7.65 7.60 0.7

T2 33.0 0.34 18.0 1.787E-02 8.14 7.37 10.3 9.0 -10.6 1.297E-02 7.45 7.37 1.0

T3 33.0 0.84 28.5 1.467E-02 6.03 7.37 -18.2 6.1 -1.7 1.284E-02 7.28 7.37 -1.3

T4 33.0 0.82 17.1 1.399E-02 5.59 7.37 -24.2 5.8 -4.5 1.280E-02 7.22 7.37 -2.0

TE1 19.0 0.50 19.0 1.230E-02 4.06 3.91 3.9 4.3 -5.1 1.056E-02 3.70 3.91 -5.4

TE2 19.0 0.75 19.0 1.147E-02 3.56 3.91 -8.9 3.5 1.7 1.049E-02 3.61 3.91 -7.6

TE3 19.0 0.85 19.0 1.085E-02 3.18 3.91 -18.6 3.3 -3.3 1.043E-02 3.54 3.91 -9.4

TE4 19.0 0.47 9.0 1.217E-02 3.99 3.91 1.9 4.0 -0.7 1.053E-02 3.66 3.91 -6.3

TE5 19.0 0.71 9.0 1.110E-02 3.34 3.91 -14.7 3.3 1.5 1.045E-02 3.57 3.91 -8.8

TE6 19.0 0.81 9.0 1.037E-02 2.89 3.91 -26.0 3.1 -6.8 1.036E-02 3.46 3.91 -11.6

TE7 29.0 0.50 29.0 1.620E-02 6.59 6.07 8.4 6.6 -0.8 1.238E-02 6.15 6.07 1.2

TE8 29.0 0.75 29.0 1.483E-02 5.74 6.07 -5.5 5.4 5.3 1.230E-02 6.05 6.07 -0.4

TE9 29.0 0.85 29.0 1.368E-02 5.03 6.07 -17.1 5.1 -1.6 1.223E-02 5.96 6.07 -1.9

TE10 29.0 0.48 14.0 1.605E-02 6.49 6.07 6.9 6.2 4.7 1.234E-02 6.10 6.07 0.4

TE11 29.0 0.72 14.0 1.423E-02 5.37 6.07 -11.6 5.1 5.7 1.226E-02 6.00 6.07 -1.2

TE12 29.0 0.82 14.0 1.281E-02 4.50 6.07 -26.0 4.8 -6.0 1.215E-02 5.86 6.07 -3.5

TE13 49.0 0.50 49.0 2.404E-02 11.61 10.40 11.6 11.4 2.1 1.603E-02 10.90 10.40 4.8

TE14 49.0 0.75 49.0 2.179E-02 10.20 10.40 -1.9 9.3 8.7 1.593E-02 10.78 10.40 3.6

TE15 49.0 0.85 49.0 1.940E-02 8.71 10.40 -16.3 8.8 -0.5 1.587E-02 10.70 10.40 2.9

TE16 49.0 0.49 24.0 2.389E-02 11.52 10.40 10.7 10.5 8.5 1.598E-02 10.84 10.40 4.2

TE17 49.0 0.73 24.0 2.081E-02 9.59 10.40 -7.8 8.6 10.0 1.590E-02 10.74 10.40 3.2

TE18 49.0 0.83 24.0 1.782E-02 7.73 10.40 -25.7 8.1 -5.1 1.580E-02 10.61 10.40 2.0



6-13

Notes:1. Percentage difference between k and 1.3/h.2. Regression equation (6-11). k = 1.01 (d/D)-.372 (t/T)-0.12/h3. Percentage difference between regression equation and k.4. Percentage difference between k and 1.3/h.5. Point 1 is fixed.

As discussed earlier, the expression k = 1.3/h has been suggested for elbows withattached tangent piping. Consequently, it was decided to curve fit the data of Table 6-3to an equation of the form:

k = A (d/D)B (t/T)C /h

where A, B, and C are constants.

Curve fitting this expression, using regression analysis, yields:

k = 1.01 (d/D)-0.372 (t/T)-0.12/h (r2 = .84) (Eq. 6-6)

Comparing this equation to the FEA results, the maximum difference is -11.4% and theaverage is 0.33 %. There is very little difference in the results of the two equations.Equation 6-6 appears to be the most desirable because it has the same general form asthe flexibility factor for an elbow.

FEA Results: Flexibility of Trunnions

Figure 6-3 shows a typical model for branch connections. A rigid link is included in themodel from the center line of the run pipe to its surface. At that juncture, a point springis used to represent the local flexibility of the connection. For the elbow-trunnionmodel, there are several possibilities. The trunnion will be modeled as a beam with apoint spring located at point A (see Figure 6-7). A rigid link can be connected frompoint A to either points B, C, or D. Then the appropriate sections of the elbows wouldbe modeled as beams. The lengths of the sections would depend upon the pointsselected.



6-14

B

C

D

A

Figure 6-7Elbow-Trunnion Model

The first evaluation performed for this study assumed the rigid link was from point Ato B. The link from A to B was selected because the trunnion’s centerline passedthrough point B. The model was fixed at the bottom end, and the moments wereapplied at the end of the trunnion. However, this model resulted in negative values forthe flexibility factor for the point spring. The values were negative because theflexibility of the elbow (including the modification to account for the trunnion) wasgreater than the flexibility in the FEA model.

Next, the model was changed so that the rigid link went from point A to C. Thiscorresponds to the model for branch connections in that the rigid link is perpendicularto the centerline of the elbow. In the branch connection model, the rigid link isperpendicular to the centerline of the run pipe. However, the flexibility factors werestill negative.

Finally, the model was changed so that the rigid link was from point A to D. Thisresulted in flexibility factors that were positive and of reasonable magnitude.

The model depicted in Figure 6-8 is used as the basis of evaluating the FEA results forloads on the trunnion. The rotation at the end of the trunnion (Point 6) with respect tothe fixed end (Point 1) is given by:

φ = φ6-5 + φ5-4 + φ4-3 + φ3-2 +φ2-1



6-15

Where φi-j is the rotation of point i with respect to point j. Note that for in-plane bending(where M is the in-plane bending moment):

φ6-5 = ML3/EIt

φ5-4 = k Mdo/EIt point spring, from Equation 6-4

φ4-3 = 0, since this is the rotation over a rigid link

φ3-2 = ke ML7/EIe (Eq. 6-7)

where ke is determined from Equation 6-6 for in-plane bending of the elbow,

and

φ2-1 = ML5/EIe (Eq. 6-8)

Replacing φ with φfea , and rearranging yields:

k = 1/(Mdo/EIt) [φfea - φ6-5 - φ3-2 - φ2-1]



6-16

L1 L2 L3

5 6

4

3

2

1

L5

L6

L7ψ

7

ψ = cos-1(L2/R)

L6 = πR/2 (90-ψ)/90

L7 = πR/2 ψ/90

Points 4 and 5 are at thetrunnion/elbow interface.All other points are on thecenterline of the elbow,trunnion, or pipe.

Figure 6-8Beam Model

For out-of-plane bending of the trunnion, ke of the elbow is replaced by the regressionEquation 6-6 for out-of-plane bending of the elbow. M is the out-of-plane bendingmoment. Also since the segment of the beam model from point 1 to point 2 is in torsion,Equation 6-8 is replaced by

φ2-1 = 1.3 ML5/EIe

As discussed earlier, the FEA was performed with two sets of boundary conditions,fixed at the bottom pipe end and fixed at both ends. When fixed at both of the pipeends, Equations 6-7 and 6-8 can be combined and modified for in-plane moments:

φ3-1 =φ3-2 + φ2-1 = 1/4 M Lb/(EIe)[4(La/Lc) - 9(La/Lc)2 + 6(La/Lc)

3 -1]

where:

La = L1 + ke L6

Lb = L5 + ke L7

Lc = La + Lb



6-17

For out of plane moments, ke, determined from Equation 6-6, is used. Also sincesegment L5 is in torsion, L5 can be replaced by 1.3 L5.

The results are presented in Tables 6-4 through 6-6. Table 6-4 lists the values of k for in-plane and out-of-plane bending for the case with only one end fixed. For in-planebending, the values of k are reasonably small with a maximum of about 3. For out ofplane bending, the values are larger with a maximum of about 9.

Table 6-5 lists the values of k for the condition with both ends fixed. For in-planemoments, there is little difference from the values in Table 6-4 for the case with one endfixed. The flexibility is lower, on the average, by a value of .6. This is within theanalysis methodology tolerance.

For out-of-plane flexibility, the situation is different. The average difference betweenthe results for the two sets of boundary conditions is about 3 with a maximumdifference of over 8. The boundary conditions clearly affect the flexibility factors.

It is assumed that the average of the two conditions is representative of actualapplications. Table 6-6 lists the average values of k and also provides a comparison toequations developed from regression analysis for the two loading conditions:

In-plane bending:

k = .142 (D/T)1.11 (d/D)-0.22 (d/t)-0.55 (r2 = .94) (Eq. 6-9)

The average difference between Equation 6-9 and the FEA results is 1.5% with amaximum of about 13%.

Out-of-plane bending:

k = .146 (D/T)1.41 (d/D)0.36 (d/t)-0.61 (r2 = .98) (Eq. 6-10)

The average difference between Equation 6-10 and the FEA results is 0.8% with amaximum of about 11%.



6-18

Table 6-4Bending of the Trunnion-Trunnion Flexibility

In-Plane Moment Out-of-Plane Moment

Model D/T d/D d/t φZ k φY k

T1 33.5 0.509 18.0 6.337E-03 2.03 7.994E-03 4.11

T2 33.0 0.344 18.0 5.538E-03 1.74 6.335E-03 2.86

T3 33.0 0.839 28.5 8.025E-03 1.72 1.091E-02 4.59

T4 33.0 0.821 17.1 1.044E-02 2.37 1.500E-02 6.66

TE1 19.0 0.500 19.0 4.902E-03 0.93 5.556E-03 1.74

TE2 19.0 0.750 19.0 5.993E-03 0.94 7.288E-03 2.16

TE3 19.0 0.850 19.0 6.783E-03 0.98 8.372E-03 2.23

TE4 19.0 0.474 9.0 5.786E-03 1.23 6.804E-03 2.38

TE5 19.0 0.711 9.0 8.024E-03 1.47 1.031E-02 3.39

TE6 19.0 0.805 9.0 9.399E-03 1.46 1.230E-02 3.52

TE7 29.0 0.500 29.0 5.204E-03 1.26 6.175E-03 2.57

TE8 29.0 0.750 29.0 6.515E-03 1.24 8.340E-03 3.18

TE9 29.0 0.850 29.0 7.401E-03 1.21 9.691E-03 3.36

TE10 29.0 0.483 14.0 6.305E-03 1.75 7.814E-03 3.63

TE11 29.0 0.724 14.0 9.079E-03 2.03 1.242E-02 5.26

TE12 29.0 0.821 14.0 1.063E-02 1.86 1.498E-02 5.58

TE13 49.0 0.500 49.0 5.725E-03 1.69 7.091E-03 3.64

TE14 49.0 0.750 49.0 7.570E-03 1.71 1.014E-02 4.64

TE15 49.0 0.850 49.0 8.659E-03 1.55 1.217E-02 5.23

TE16 49.0 0.490 24.0 7.236E-03 2.49 9.326E-03 5.24

TE17 49.0 0.735 24.0 1.115E-02 3.01 1.603E-02 8.06

TE18 49.0 0.833 24.0 1.297E-02 2.41 1.993E-02 9.11

Notes:1. Point 1 is fixed.



6-19

Table 6-5Bending of the Trunnion-Ends Fixed

Trunnion Flexibility

In-Plane Moment Out-of-Plane Moment

Model D/T d/D d/t φZ k φY k

T1 33.5 0.509 18.0 4.867E-3 1.64 5.7100E-3 1.60

T2 33.0 0.344 18.0 5.095E-3 1.70 5.6310E-3 1.70

T3 33.0 0.839 28.5 4.325E-3 0.84 5.1400E-3 0.67

T4 33.0 0.821 17.1 4.835E-3 1.09 6.0910E-3 0.79

TE1 19.0 0.500 19.0 4.334E-3 0.89 4.6860E-3 0.87

TE2 19.0 0.750 19.0 4.118E-3 0.66 4.6700E-3 0.59

TE3 19.0 0.850 19.0 4.115E-3 0.58 4.7500E-3 0.46

TE4 19.0 0.474 9.0 4.770E-3 1.13 5.3020E-3 1.10

TE5 19.0 0.711 9.0 4.707E-3 0.96 5.6690E-3 0.81

TE6 19.0 0.805 9.0 4.729E-3 0.80 5.8720E-3 0.57

TE7 29.0 0.500 29.0 4.385E-3 1.03 4.8600E-3 1.02

TE8 29.0 0.750 29.0 4.227E-3 0.83 4.8770E-3 0.74

TE9 29.0 0.850 29.0 4.217E-3 0.70 4.9230E-3 0.55

TE10 29.0 0.483 14.0 4.994E-3 1.58 5.7790E-3 1.55

TE11 29.0 0.724 14.0 4.892E-3 1.23 6.0480E-3 1.04

TE12 29.0 0.821 14.0 4.902E-3 0.99 6.2040E-3 0.68

TE13 49.0 0.500 49.0 4.565E-3 1.31 5.2530E-3 1.31

TE14 49.0 0.750 49.0 4.372E-3 0.95 5.1400E-3 0.84

TE15 49.0 0.850 49.0 4.353E-3 0.75 5.1500E-3 0.55

TE16 49.0 0.490 24.0 5.318E-3 2.10 6.4640E-3 2.07

TE17 49.0 0.735 24.0 5.141E-3 1.43 6.5280E-3 1.17

TE18 49.0 0.833 24.0 5.131E-3 1.04 6.6180E-3 0.59

Notes:1. Points 1 and 2 are fixed.



6-20

Table 6-6Average Trunnion Flexibility

In-Plane Out-of-Plane

REG REG

Model D/T d/D d/t AVE k EQ % DIF AVE k EQ % DIF

T1 33.5 0.51 18.0 1.83 1.66 9.6 2.85 2.78 2.7

T2 33.0 0.34 18.0 1.72 1.78 -3.1 2.28 2.36 -3.6

T3 33.0 0.84 28.5 1.28 1.13 11.6 2.63 2.46 6.4

T4 33.0 0.82 17.1 1.73 1.51 12.9 3.73 3.33 10.6

TE1 19.0 0.50 19.0 0.91 0.86 5.1 1.31 1.20 8.4

TE2 19.0 0.75 19.0 0.80 0.79 1.9 1.37 1.39 -1.1

TE3 19.0 0.85 19.0 0.78 0.77 1.9 1.35 1.45 -7.9

TE4 19.0 0.47 9.0 1.18 1.31 -11.2 1.74 1.86 -6.5

TE5 19.0 0.71 9.0 1.21 1.20 1.0 2.10 2.15 -2.1

TE6 19.0 0.81 9.0 1.13 1.17 -3.2 2.05 2.25 -9.7

TE7 29.0 0.50 29.0 1.14 1.09 4.7 1.79 1.68 6.2

TE8 29.0 0.75 29.0 1.03 1.00 3.3 1.96 1.95 0.7

TE9 29.0 0.85 29.0 0.96 0.97 -1.5 1.96 2.04 -4.0

TE10 29.0 0.48 14.0 1.67 1.64 1.5 2.59 2.59 -0.1

TE11 29.0 0.72 14.0 1.63 1.50 8.1 3.15 3.00 4.8

TE12 29.0 0.82 14.0 1.42 1.46 -2.4 3.13 3.14 -0.2

TE13 49.0 0.50 49.0 1.50 1.46 2.6 2.47 2.56 -3.6

TE14 49.0 0.75 49.0 1.33 1.34 -0.7 2.74 2.96 -8.1

TE15 49.0 0.85 49.0 1.15 1.30 -13.2 2.89 3.10 -7.2

TE16 49.0 0.49 24.0 2.29 2.18 5.2 3.66 3.93 -7.4

TE17 49.0 0.73 24.0 2.22 1.99 10.5 4.61 4.54 1.5

TE18 49.0 0.83 24.0 1.72 1.94 -12.3 4.85 4.75 2.1



6-21

Comparison to Test Data

While the tests discussed in Section 3 were not specifically for determining flexibilityfactors, they can be used to evaluate the equations derived above. Deflections and loadsat point 6 in Figure 6-8 for in-plane bending were recorded and are included inAppendix A. The average deflection of the four tests at a load of 1000 lb. was .52 inches.Using an average value of L3, other dimensions from Figure 3-1, and the model as perFigure 6-8, the calculated deflection was .43 inches. This calculation used Equation 6-6for the elbow and Equation 6-9 for the trunnion. Considering that the calculation didnot include the flexibility of the testing frame or bolted connections and also thatEquation 6-9 is based on the results from the average of the two cases with differentboundary conditions, this is considered as verification of the methodology.


7-1

7 CONCLUSIONS

The conclusions arrived at from the analyses and tests described in this report areenumerated below:

1. The basic approach used by Code Cases 391 and 392, with modification of theindices, can be used in the design and qualification of trunnions on 90o elbows.The equation for C shall be maintained as

C= Ao (2γ)n1βn2τn3 but not less than 1.0

The values of the constants are modified to:

Ao n1 n2 n3

CW 0.75 0.229 -0.42 0.85

CL 1.68 0.0734 -0.01 0.769

CN 0.72 0.210 -0.355 0.84

CT 1.25 0.158 0.158 0.717

These constants shall be applicable for the following range of parameters:

(a) 10 ≤ γ = Ro/T ≤ 30

(b) 0.2 ≤ τ = t/T ≤ 2.0

(c) 0.3 ≤ β = do/Do ≤ 0.8

These limits are based upon the range of parameters used in the FEA modelsthat were used in the development of the constants [7].

2. The equations in Code Cases 391 and 392 include B or C indices and i factors fora straight pipe. For evaluation of elbows with trunnions, these indices and ifactors shall be based on the elbow configuration. The evaluation of the


Conclusions

7-2

experimental data either conservatively assumed indices for a straight pipe orconservatively neglected the contribution of the elbow.

3. The values of KT, shall be as specified in Code Cases 391 and 392 for thecorresponding type of weld.

4. Items 1–3 shall be applicable for long and short radius 90o elbows only, that is,R= 1.0 D to 1.5 D where D is the nominal pipe size.

5. The flexibility model is as indicated in Figure 6-7. It is applicable only to 90o

elbows.

(a) k = 1.3/h for out-of-plane moments on the elbow. (Equation 6-5)

(b) k= 1.01 (d/D)-.372(t/T)-0.12/h for in-plane bending on the elbow. (Equation6-6)

(c) k = .142 (D/T)1.11 (d/D)-0.22 (d/t)-0.55 for in-plane bending on the trunnion.(Equation 6-9)

(d) k= .146 (D/T)1.41 (d/D)0.36 (d/t)-0.61 for out-of-plane bending on thetrunnion. (Equation 6-10)

These equations are applicable for the following range of parameters:

(a) 19 ≤ d/D ≤ 33

(b) 0.34 ≤d/D ≤ 0.85

(c) 9 ≤ d/t ≤ 49

These limits are based upon the range of parameters used in the FEA modelsthat were used in the development of the constants.

The use of this methodology will allow for a more accurate evaluation of trunnions onelbows.


8-1

8 REFERENCES

1. ASME Boiler and Pressure Vessel Code, Section III, “Nuclear Power PlantComponents.” American Society of Mechanical Engineers, New York.

2. American National Standards Institute (ANSI), Code for Pressure Piping, B31.1,Power Piping. American Society of Mechanical Engineers, New York.

3. G. Slagis, “Commentary on the 1987 Section III Attachment Rules,” PressureVessels and Piping. Vol. 169, ASME (1989).

4. R. F. Hankinson, D. A. Van Duyne, D. H. Stout, and Y. K. Tang, “DesignGuidance for Integral Welded Attachments,” Pressure Vessels and Piping. Vol.237-2, ASME (1992).

5. R. F. Hankinson and R. A. Weiler, A Review, Discussion, and Comparison of CircularTrunnion Attachments to Piping, Pressure Vessels and Piping. Vol. 218, ASME(1991).

6. D. K. Williams and G. D. Lewis, “Development of Primary and SecondaryMoment Loading Stress Indices for Trunnion Elbows,” 84-PVP-98 (1984).

7. R. F. Hankinson, L. A. Budlong, and L. D. Albano, “Stress Indices for PipingElbows with Trunnion Attachments for Moment and Axial Loads,” presented atthe ASME Pressure Vessels and Piping Conference (October 1987).

8. G. B. Rawls, E. A. Wais, and E. C. Rodabaugh, “Evaluation of the Capacity ofWelded Attachments to Elbows as Compared to the Methodology of ASMECode Case N-318,” Pressure Vessels and Piping. Vol. 237-2, ASME (1992).

9. Code Case N-318, Procedure for Evaluation of the Design of Rectangular Cross SectionAttachments on Class 2 or 3 Piping, Section III, Division 1, American Society ofMechanical Engineers, New York.

10 Case N-391-2, Procedure for Evaluation of the Design of Hollow Circular Cross SectionWelded Attachments on Class 1 Piping, Section III, Division 1, American Society ofMechanical Engineers, New York.


References

8-2

11. Case N-392-3, Procedure for Evaluation of the Design of Hollow Circular Cross SectionWelded Attachments on Classes 2 and 3 Piping, Section III, Division 1, AmericanSociety of Mechanical Engineers, New York.

12. E. C. Rodabaugh, Background of ASME Code Cases N-391 and N-392 Trunnions ofStraight Pipe, September 1990. Attachment to ASME Code Committee, WorkingGroup on Piping Design (WGPD) meeting minutes (April 1991).

13. A. B. Potvin, J. G. Kuang, R. D. Leick, and J. L. Kablich, “Stress Concentration inTubular Joints,” Society of Petroleum Engineers Journal, pp. 287–299 (August 1977).

14. W. G. Dodge, “Secondary Stress Indices for Integral Structural Attachments toStraight Pipe,” and E. C. Rodabaugh, W. G. Dodge, and S. E. Moore, “StressIndices at Lug Supports on Piping Systems,” Welding Research Council Bulletin,No. 198 (September 1974).

15. J. L. Mershon, K. Mokhtarian, G. V. Ranjan, and E. C. Rodabaugh, “LocalStresses in Cylindrical Shells due to External Loadings on Nozzles-Supplementto WRC Bulletin No. 107,” Welding Research Council Bulletin, No. 297 (August1984).

16. A. C. Wordsworth and G. P. Smedley, Stress Concentrations of Unstiffened TubularJoints, European Offshore Steels Research Seminar, Cambridge (1978).

17. E. C. Rodabaugh, “Review of Data Relevant to the Design of Tubular Joints inFixed Offshore Platforms,” Welding Research Council Bulletin, No. 256 (January1980).

18. E. C. Rodabaugh, “Stress Indices, Pressure Design, and Stress IntensificationFactors for Lateral in Piping,” Welding Research Council Bulletin, No. 360, (January1991).

19. R. F. Hankinson and L. D. Albano. An Investigation of Elbow Flexibility for Elbowswith Circular Attachments, presented at the ASME/JSME Pressure Vessels andPiping Conference, July 1989.

20. A. R. C. Markl, “Fatigue Tests of Piping Components,” ASME paper no. 51-PET-21, May 21, 1951.

21. E. C. Rodabaugh, “Developing Stress Intensification Factors: (1) StandardizedMethod for Developing Stress Intensification Factors for Piping Components,”Welding Research Council Bulletin, Number 392 (June 1994).


References

8-3

22. E. C. Rodabaugh, S. K. Iskander, and S. E. Moore, End Effects on Elbows Subjectedto Moment Loadings, ORNL/Sub-2913/7 (March 1978).

23. E. C. Rodabaugh and S. E. Moore, Stress Indices and Flexibility Factors for Nozzlesin Pressure Vessels and Piping, NUREG/CR-0718 (June 1979).


Reprinted with the permission of The American Society of Mechanical Engineers fromASME BPVC, Section XI-1998 Edition

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Stress Indices for Elbows With Trunnion Attachments