TR-107453 Stress Indices for Elbows with Trunnion Attachments Technical

Stress Indices for Elbows with Trunnion Attachments

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Technical Report

Stress Indices for Elbows with Trunnion Attachments TR-107453

Final Report December 1998

Prepared for EPRI 3412 Hillview Avenue Palo Alto, California 94304 EPRI Project Manager R. G. Carter

DISCLAIMER OF WARRANTIES AND LIMITATION OF LIABILITIES THIS REPORT WAS PREPARED BY THE ORGANIZATION(S) NAMED BELOW AS AN ACCOUNT OF WORK SPONSORED OR COSPONSORED BY THE ELECTRIC POWER RESEARCH INSTITUTE, INC. (EPRI). NEITHER EPRI, ANY MEMBER OF EPRI, ANY COSPONSOR, THE ORGANIZATION(S) BELOW, NOR ANY PERSON ACTING ON BEHALF OF ANY OF THEM: (A) MAKES ANY WARRANTY OR REPRESENTATION WHATSOEVER, EXPRESS OR IMPLIED, (I) WITH RESPECT TO THE USE OF ANY INFORMATION, APPARATUS, METHOD, PROCESS, OR SIMILAR ITEM DISCLOSED IN THIS REPORT, INCLUDING MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, OR (II) THAT SUCH USE DOES NOT INFRINGE ON OR INTERFERE WITH PRIVATELY OWNED RIGHTS, INCLUDING ANY PARTY'S INTELLECTUAL PROPERTY, OR (III) THAT THIS REPORT IS SUITABLE TO ANY PARTICULAR USER'S CIRCUMSTANCE; OR (B) ASSUMES RESPONSIBILITY FOR ANY DAMAGES OR OTHER LIABILITY WHATSOEVER (INCLUDING ANY CONSEQUENTIAL DAMAGES, EVEN IF EPRI OR ANY EPRI REPRESENTATIVE HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES) RESULTING FROM YOUR SELECTION OR USE OF THIS REPORT OR ANY INFORMATION, APPARATUS, METHOD, PROCESS, OR SIMILAR ITEM DISCLOSED IN THIS REPORT. 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. Box 23205, 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.

CITATIONS

This report was prepared by Wais and Associates, Inc. 3845 Holcomb Bridge Road, Suite 300 Norcross, Georgia 30092 Principal Investigators E. Wais R. Reinecke E. C. Rodabaugh This report describes research sponsored by EPRI. The report is a corporate document that should be cited in the literature in the following manner: Stress Indices for Elbows with Trunnion Attachments, EPRI, Palo Alto, CA: 1998. Report TR-107453.

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REPORT SUMMARY

Trunnions on elbows are generally used as supports and are also used in some applications as anchors. The qualification of trunnions is an important item in the design and fitness-for-service of many piping systems. This report provides equations, based on experimental and test data, for determining the stress indices, B and C, and the flexibility factor, k, for elbows with hollow circular cross-section attachments (trunnions). The report contains explicit modifications to ASME Code Cases 391 and 392 for qualification of trunnions on pipe. It also provides flexibility 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 indices to account for fatigue effects produced by reversing loads and flexibility factors (k) for evaluation of piping configurations. ASME Code Cases 391 and 392 provide procedures for evaluating the design of hollow circular cross-section attachments on Class 1, 2, and 3 pipe. Objectives •

To experimentally derive expressions for B, C, and k factors for analysis of trunnions on elbows.

•

To provide modifications to Code Cases 391 and 392 for improved evaluation of trunnions on elbows.

Approach

A review of the present approach for the evaluation of trunnions on elbows in accordance with the Code provided an understanding of the conservatism in the determination of the fatigue factors. Available data on studies, experiments, and testing were collected and reviewed. Tests and analyses were performed on representative models and the results compared to existing data. v

Results

The present values of A 0, B, and C in Code Cases 391 and 392 were modified as a result of this research and analysis to reduce excess conservatism. Equations, previously unavailable, were derived for flexibility factors for the elbow/trunnion configuration. Equations were derived for both in-plane and out-of-plane bending. Parameter limitations were established for the results to be applicable to short radius and long radius 90o elbows with trunnion attachments. EPRI Perspective

Design for fatigue is a major concern for any power or process facility. Accurate methods of engineering for fatigue are important for cost-effective design, for root cause failures, and for evaluating remaining fatigue life of plant designs. The work being done under EPRI’s SIF optimization program continues to establish the technical justification to allow for reductions in current Code stress indices. The results of this program can provide a basis to reduce the scope of ongoing pressure boundary component testing and inspection programs in operating nuclear power plants. Examples include reductions in the inspection scope of postulated high- and moderateenergy line break locations and reduction of snubber testing. TR-107453 Interest Categories

Piping, reactor vessel, and internals Keywords

ASME Code Fatigue Piping design and analysis Stress intensity factors Stress indices

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EPRI Li censed M at eri al

ABSTRACT

This report was prepared under the auspices of the EPRI project on stress intensification factor optimization. Stress intensification factors and their corresponding stress indices (for ASME Class 1 components) are used in the qualification of piping components to ensure that they have an adequate fatigue life under cyclic loading. Stress intensification factors and stress indices are also used for qualification for other loading conditions. Trunnions on elbows are generally used as supports and are also used in some applications as anchors. The qualification of trunnions is a major concern in the design and qualification of many piping systems. This report presents the results of an investigation of the stress indices and flexibility factors for trunnions on 90 o elbows subject to axial loads and bending and twisting moments. This report reviews existing data and methodologies used for qualification of trunnions. Modified expressions for stress indices are defined. The results of new testing are included. Finally, flexibility factors for accurately modeling the behavior of a trunnion in a piping system are presented. The information presented in this report will significantly improve the qualification of trunnions on elbows.

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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

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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

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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 C L........................................................................... 4-13 Figure 4-2 Comparison of Equations for C N .......................................................................... 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

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LIST OF TABLES Table 3-1 Summary of Test Results........................................................................................ 3-4 Table 3-2 Calculation of C L'..................................................................................................... 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 C L ............................. 3-10 Table 3-5 Trunnion/Elbow-Experimental Evaluation of B L' .................................................... 3-14 Table 4-1 Hankinson FEA Parameters ................................................................................... 4-2 Table 4-2 Comparisons for C L ................................................................................................. 4-5 Table 4-3 Comparisons for C N................................................................................................. 4-7 Table 4-4 Comparisons for C T ................................................................................................. 4-9 Table 4-5 Comparisons for C W .............................................................................................. 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

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1 INTRODUCTION

This report was prepared under the auspices of the EPRI project on stress intensification factor optimization. Stress intensification factors (SIFs) are used to ensure that piping has an adequate fatigue life under cyclic loading. SIFs are not generally used for design of welded attachments such as trunnions; however, the general approach is the same. This report specifically investigates the fatigue behavior of trunnions welded on elbows. 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 to evaluating the trunnion/pipe configuration.

Section 2 of this report provides a summary of the available references regarding trunnions 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 elbows conducted under the auspices of the EPRI research project. The test results are used to derive experimentally based values for the various indices. Section 4 provides an evaluation of the various approaches to evaluating trunnions on elbows. This evaluation includes a comparison to previously published finite element analysis (FEA) data as well as new data. New experimental data is included in the comparison. Specific recommendations regarding proposed analytical approaches are made.

1-1


Introduction

Section 5 compares the experimental data to the results of the analytical approach discussed 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. These conclusions provide new understanding of the behavior of trunnions on elbows. This information 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 Attachments on Classes 2 and 3 Piping, Section 3, Division 1. Appendix B contains the test data and results for this report.

1-2


2 BACKGROUND

Nomenclature Trunnion

Q2 MN

W

ML

Q1

MT

θ

Figure 2-1 Trunnion/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.

2-1


Background

d = mean diameter of the trunnion, in. Rm = mean radius of pipe/elbow, in. R = nominal bend radius of elbow h = TR/Rm2 elbow characteristic AT = π (ro2 - ri2) ZT = IT/ro IT = π/4(ro4 - ri4) Am = π/2 (ro2 - ri2) J = lesser of π o2T 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 )n1 βn2 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). 2-2


Background

CT = 1.0 for β ≤ 0.55 CT = CN for β = 1.0, but not less than 1.0; C T should be linearly interpolated for 0.55<β≤1.0, but not less than 1.0 CL’ = Values of C L 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 occurring simultaneously 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 will be subjected to axial and other forces, as well as bending and torsion moments. This study is limited to trunnions attached to 90 o 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 with regard to specific methodologies for qualification of trunnion/elbow configurations. However, Section III, NB-3685.1 does acknowledge that special attention should be given to this type of configuration: “Stresses in elbows with local discontinuities, such as longitudinal welds, support lugs, and branch connections in the elbow, shall be 2-3


Background

obtained by appropriate theoretical analysis or by experimental analysis in accordance with 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 of trunnions on elbows. Slagis [3] and Hankinson et al. [4, 5] provide general discussions about the use of trunnions and other attachments, including a discussion regarding jurisdictional boundaries. Williams and Lewis [6] suggested expressions for B 2 and C2 indices for trunnions on elbows that were based on the results of 10 finite element analyses. The expressions suggested 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 used the results from Williams and Lewis and expanded upon them for a total of 26 finite element models. They suggested equations for secondary stress indices for moment and forces 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:

2-4


Background

C2E = 2.55/h.732 where h = TR/R2m The Code equation for C 2 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 and make a comparison to Code Case N-318 [9]. The attachments tested included both rectangular and cruciform shapes. Code Case N-318 presents a method of analysis/design for rectangular welded attachments on straight pipe that involves use of secondary stress indices. The expression for the C indices is of a simplified form and is a function of geometrical parameters. Rawls used the Code Case methodology to evaluate test data. Based on the test data, the conclusion was that the Code Case methodology was conservative by a factor of 3.5 to 14.8 when applied to these types of attachments 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 the evaluation and design of hollow, circular, cross-section welded attachments (or trunnions) on straight pipe. The approach followed by these Code Cases is very similar to that of N-318. These Code Cases are important because the attachments are the same as 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 + CTMT Jm + 1.7EαTT -TW

(Eq. 2-3)

SPT = KT(SNT) SNT** = CWW**/AT + CNMN**/ZT + CLML**/ZT + Q1**/Am + Q2**/Am + CTMT**/Jm

2-5


Background

N-392 has similar expressions except that the 1.7EαTT -TW term in Equation (2-3) is not included. The stresses calculated by these equations are used in the qualification in modified standard Code equations by the two Code Cases. Rodabaugh [12] discusses the background of N-391 and N-392 and is summarized herein. It should be noted that the original objective in developing these Code Cases was to provide a simplified and conservative methodology. The approach used to address the effects of the various mechanical loads (W, Q1, Q2 , MN, ML, and MT) is discussed below. The original basis for considering the effects of the W, M L, and MN loads was the correlation equations given by Potvin et al. [13]. These correlation equations were considered to correspond to the maximum primary-plus-secondary stresses (P L + P b + Q). Thus, they corresponded to the C-indices of NB-3600 [1] or C W, CL, and CN of the Code Cases. A more generalized form of the correlation equation is given in an earlier work by Rodabaugh [17]: C = A(2 γ )n1 (β)n2 (τ)n3 (Do/L)n4(g/Do)n5(sinθ)n6 where:

γ = Ro/T β = do/Do τ = t/T The constants n1, n2, n3, n4, n5, and n6 vary depending upon the loading. L is the length of the member corresponding to the trunnion, and g is the distance between the trunnion and another trunnion. θ is the angle between the trunnion and the straight pipe (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 A o , n1, n2, and n3. This is similar to the form of the equation suggested by Hankinson [7] with different constants. The range of the applicable parameters in the Code Cases for C W, CL, and CN has been extended beyond that of Potvin. The applicable range of γ and τ was extended based on WRC Bulletin 198 [7] and WRC Bulletin 297 [8]. The range of β was extended based on comparison with the equations derived by Wordsworth [16]. 2-6


Background

At the time the Code Cases were prepared, data were not available regarding shear loads and torsional moments (Q 1, Q2, and MT ). Engineering judgment was used in the evaluation of their effects. For the shear loads (Q 1 and Q2), the stress intensity (twice the shear stress) is Q/A m, where Am is one-half the cross-sectional area of the trunnion-pipe interface (where the load is taken), assumed to be π(ro2 -ri2)/2. This is considered reasonable for small trunnions (small d o/Do) but is probably very conservative for large trunnions (size on size). The approach used to evaluate the effects of M T was based on comparisons to data on branch connections [10]. Branch connections are similar to trunnions except that the run pipe has an opening in it. For branch connections with small d o/Do, the stress intensity is about Mt/Jm. For do/Do = 1.0, test data [11] indicate that the maximum stress intensity is about the same as for out-of-plane bending (for example, due to M N). Based on this information, the value of CT was taken as 1.0 for β= do/Do ≤ 0.55 and as equal to CN for β = 1.0. Linear interpolation is used in between. The change at β = 0.55 corresponds to Potvin’s data. Potvin originally suggested a limit on γ = Ro/T ≥ 8.33. Rodabaugh [12] provides a basis for 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, this extension is valid for N-391. The B indices that are in the Code Cases correspond to those of ASME Section III, NB3600. The B indices are based upon limit load analysis or test. The Code Cases take the B indices as one half the C indices. Based upon data from Rodabaugh [12, 17], it is estimated 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 trunnion mechanical loads (W, Q1 , Q2 , MN, ML, and MT) and the thermal stresses (if Class 1 piping) and add them to the stresses in the pipe due to loads in the pipe. The stresses are added linearly and then compared to the specific limits dependent upon the piping class and the specific requirement. The linear addition of stresses is generally very conservative. It assumes that all the stresses are maximum at the same point. Wordsworth’s research [16] warrants further review. This paper reviews the results of using acrylic models for determining what were referred to as stress concentration factors (SCFs) at tubular joints. The specific application is for offshore steel structures. Test specimens were manufactured from acrylic materials. Data from strain gauges were compared to data from the analysis to verify the analysis. Various types of joints were investigated. The results of what are called “T” joints are of interest for this investigation. The expression given for the SCF for out-of-plane bending is:

2-7


Background

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

(Eq. 2-5)

where:

γ = Ro/T τ = t/T β = do/Do It is assumed that β is based on d o and Do. For in-plane bending, the SCF is given as: KC = 0.75 γ 0.6 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 secondary stresses (that is, CL and CN). Other expressions are provided for other loading conditions. Since the connection of the trunnion to the elbow is at an angle θ, it is similar to that of a lateral. Rodabaugh [18] suggests using Equations 2-5 and 2-6 in the qualification of laterals connected at an angle of θ to the pipe. One other reference of interest is Hankinson and Albano’s study of flexibility of elbows with trunnions [19]. However, this study is limited to the flexibility of the elbows; the flexibility of the trunnion was not investigated.

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3 TEST PROGRAM

Purpose The purpose of this test program was to obtain some specific data that corresponded to the test methodology followed by Markl [20]. These tests would provide data that could be used as a basis for developing procedures for qualification of trunnions on elbows. All tests were for in-plane bending of the trunnion. This data would be used for extrapolation to other loading conditions. As discussed later, the results of the testing 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-inch NPS schedule 20 long radius elbows with a 4-inch schedule 40 A53-B trunnion. The welds at the interface of the trunnion and pipe were normal full penetration in an aswelded condition. The test specimens were labeled I, J, K, and L. Figure 3-1 indicates the test configuration.

3-1


Test Program Load Point

4" NPS Sch. 40 Pipe

Cover Plate 26-3/4"

Load Direction

Flanges 8" Sch. 40 LR Elbow

8" NPS Sch. 20 Pipe

L

~ 50" Varies for Test Specimen 63" 8" NPS Sch. 20 Pipe

Flange

Base

Figure 3-1 Test Configuration

Testing Program The testing was performed at the Ohio State University. The fatigue tests were performed on a Series 319 dynamically rated axial/torsional load frame made by MTS Systems Corporation. This unit is designed to accommodate either uniaxial or multiaxial 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 of the cross head, hydraulic grips, and actuator. The maximum actuator displacement is 6 inches. The loading pattern applied to an attached sample is controlled by programmable servovalves. Built-in loading programs include sinusoidal and triangular waves with the user able to select, within machine limits, the desired amplitude and frequency. The actual displacement of the actuator is measured by a linear variable differential transformer 3-2


Test Program

(LVDT). The output of either the load cell or the LVDT can be selected for closed loop control of the actuator displacement time history. During a test, the number of cycles of applied 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 of displacement as the control parameter meant that actuator movement was used by the MTS system for the feedback in the closed loop controls. This resulted in virtually identical cycles of actuator displacement being recorded throughout the duration of each test. The load resulting from the imposition of the specified displacement was measured with a fatigue-rated, 5000-lb. capacity, tension-compression, electronic load cell manufactured by the Lebow Instrument Company. The output of this load cell was monitored continuously throughout the duration of each test. Both load and actuator displacement were recorded using a computer program written at OSU in LabVIEW specifically for that purpose. LabVIEW is a graphical language developed by National Instruments that allows the user to design in software a test control and data collection system tailored to the requirements of each experimental program. In the LabVIEW application developed for the fatigue tests, the signals from the load and displacement transducers were sampled 30 times per second, and the time histories of each were plotted on the computer screen in real time so that the progress of the test could be readily monitored. By combining the load and displacement time histories, 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 specimen could be identified while the specimen was still undergoing loading. Any of these presentations 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 the distance from the load point to the surface of the pipe (~50 inches) varies slightly for each 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 tests were displacement-controlled cantilever bending tests. The tests followed the standard approach corresponding to Markl type tests [20, 21]. Each specimen was first tested to determine the load deflection curve for that particular specimen. The load deflection curve was used to determine the stiffness of each specimen and the load applied to the specimen by a given amount of displacement. The load deflection curves were determined for loading in both positive and negative loading directions (down and up). Each specimen was then fatigue tested by cycling the deflection in both directions of loading by a controlled amount. The cycles to failure were counted to determine the fatigue life. Failure was detected when though wall cracks formed and water leaked though the cracks. 3-3


Test Program

Test Results Summary

Table 3-1 provides a summary of the test results. This summary includes some the data that are covered in more detail in Appendix B. Table 3-1 Summary of Test Results TEST

F lb.

L in.

ZT 3 in.

M in.-lb.

N Cycles to Failure

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 i t = 245,000 N-0.2/S, where N ≡ cycles to failure, and S = M/ZT. ZT is based on nominal dimensions for the trunnion. If there was more than one loading condition with different deflections, then N is an equivalent value calculated from: Neq = Σ(δi/δmax)5 * Ni where δi is the deflection for the ith loading condition, N i is the number of cycles for the 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 of loading condition, the purpose of analysis is to be able to express the results in terms of SIFs (i-factors), B 2 indices, C2 indices, and K2 indices. The literature that is available uses SCFs (or C indices, etc.) and not SIFs in the qualification of trunnion/pipe configurations; therefore, the focus will be on B 2, C2, and K2. Because the welds were aswelded, full penetration welds, it is believed that K 2 = 1.8 is reasonable. Hence, the focus will be on B 2 and C2.

3-4


Test Program

C2 Indices-Markl Approach

As discussed earlier, the tests that were performed as a part of this investigation were fatigue tests. There are two methods that can be used to evaluate the results. The first will be referred to as the C 2 Indices-Markl approach. The second is the Class 1 fatigue approach used in Class 1 analysis per NB-3600 [1]. The Markl approach is presented first. The fatigue tests followed the Markl approach [20, 21]. Markl used the following expression 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 when through wall cracks occur and water leaks. This is used as the definition of the SIF (or i factor) and is used in the design for fatigue for B31.1 piping [2] and ASME Section III Classes 2 and 3 piping [1]. The C-indices correspond to primary-plus-secondary stresses. The C 2 indices, which are applicable 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 C 2 (or equivalent) that is used in the Code Cases. The approach follows that developed in Rawls et al. [8]. For the loading condition used in the tests, C 2 corresponds to CL which is associated with the in-plane moment in Equation 2-3 (Code Case N-391). We will use the nomenclature C L in this evaluation. In N-392, the following equation is provided (neglecting the shear stress term, Q 2/A, which is negligible when compared to the bending stress): SE = iMc/Z + SPT/2 and also: SPT = KT SNT

3-5


Test Program

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 K 2 = 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 and rearranging 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 C L from the Code Case, it will be called CL’. Now assume (conservatively) that this trunnion is attached to a 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 by various methods. Table 3-2 summarizes this calculation for the four tests.

3-6


Test Program

Table 3-2 Calculation of C L' TEST

ZT 3 in.

M in.-lb.

N Cycles to Failure

it

Z 3 in.

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 the assumption that C2 was equal to that of a straight pipe. The actual value of C 2 for the elbow is about 6.3. If this were to be used, the value of C 2 would be less than presented in Table 3-2 (approximately 1.03). However, FEA analysis indicates that the stress peaks on the elbow are on the sides of the elbows and there is little interaction with the local stresses near the trunnion. Thus, a value of C 2 =1.0 is reasonable. C2 Indices-Class 1 Approach

The second method of evaluating the data follows the fatigue evaluation approach used for Class 1 analysis (NB-3600). This method is referred to as the C 2 Indices-Class 1 Approach. The methodology of Code Case N-392 is used as the basis for calculating and combining stresses as well as evaluating those stresses. Table 3-3 provides a summary of a fatigue analysis of the data using the values for the various stress indices calculated in accordance with the Code Case. The tacit assumption is that the trunnion on an elbow acts like a trunnion on a pipe. C L calculated for the trunnion using code Case N392 yields CL = 3.77.

3-7


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

M kips

SNT ksi

C2 Mi/Z ksi

Sn ksi

3Sm ksi

Ke

SPT ksi

SP ksi

Salt ksi

N Allowable

N Failure

CUF

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)=1.0 2. K2 (pipe)=1.0 3. KT (trun)=1.8 4. CL(trun)=3.77 5. 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

The test cumulative usage factor, or CUF, is based on an allowable number of cycles from 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 are part 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, the cumulative usage factor (CUF) for the tests is, on the average, 24.39 versus a code requirement of 1.0. This indicates that the Code is very conservative. Contributors to the conservatism could be the value of the indices and/or the value of K e. Table 3-4 presents the results of a fatigue analysis in which the value of C L’ was varied until the average CUF = 1.00. The corresponding value of C L’ was 1.496. In this case, the value of Ke varies from 2.14 to 2.75, which reduces the potential of contribution to the overall conservatism.

3-9


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

M kips.

SNT ksi.

C2 Mi/Z ksi.

Sn ksi.

3Sm ksi.

Ke

SPT ksi.

SP ksi.

Salt ksi.

N Allowable

N Failure

CUF

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)= 1.0 2. K2 (pipe)= 1.0 3. KT (trun)= 1.8 4. CL(trun)= 1.617 5. 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

If Table 3-4 were modified such that the minimum CUF was 1.0, the corresponding value 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 on ML/ZT. In the actual qualification, the stresses would also include a contribution due to the 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 B L be taken as one-half the value of C L., but not less than 1.0. It is worthwhile to determine if the data obtained in this study can be used for evaluating these indices. The ASME Code [1] uses limits on the primary stress intensity to limit gross plastic deformation of piping. The Code has specific limits that it applies to stresses calculated using B-indices. The basic equations of the Code are modified to include the effects of the trunnions in the Code case. See Equation 2-2. The terms in Equation 2-2 can be neglected except for the term with B L because of the loading. Therefore the equation reduces to: SMT = BL ML/ZT Using SY as the allowable stress and solving for M L (the limit moment) yields: ML = SYZT/BL Or rearranging: BL =SYZT/ ML Because this value of B L is based on test data, it will be referred to as B L’ to distinguish it from the value of B L calculated from the Code case. Hence: BL’ =SYZT/ ML To determine the limit moment experimentally, a load-deflection curve must be developed. The limit moment (or limit load) is defined as when the deflection is equal to twice that predicted assuming linear behavior (Article II-1000, Section II-1430, reference 1). This is shown in Figure 3-2. 3-11


Test Program

Linear Action

Limit Load

d a o L

Test Data (See Appendix A)

Twice Deflection Assuming Linear Action

δ

2δ

Deflection

Figure 3-2 Limit Load Definition

The tests performed for this report were directed toward obtaining fatigue data rather than limit load (or moment) data. However, the data that was taken during the initial phase 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. That was determined by obtaining a load deflection curve. (See Appendix B for curves.) The loads in these tests were taken slightly into the plastic region. As such, the maximum loads can be used to estimate the limit load. This would be a lower limit because the deflection was not allowed to go to twice that based on elastic behavior. This maximum load is used to investigate the value of B L’. A review of the curves in Appendix B shows that the specimens were loaded in both the positive and negative direction. Thus B L’ can be estimated for both directions of loading.

3-12


Test Program

Table 3-5 shows the calculation of B L’ based on the maximum force used in determining the load deflection curve. As noted earlier, this force is less than the limit load and, hence, it will underpredict B L’. Column 5 lists the calculated values of B L’. The average value of BL’ is 1.56. Note that the value of S Y is based on the material certification data provided by the test specimen manufacturer. In order to obtain more insight into the actual value of B L’, the values of the limit loads were estimated from the load deflection curves. This was performed by extrapolating the load deflection curves, assuming that they would continue to follow the shape of the curves beyond the point where the loading was stopped. In other words, it was assumed that there would be no sudden change in the behavior. This is believed to be a reasonable assumption. Column 6 of Table 3-5 lists the estimated limit loads (F LIM-EST). Column 7 lists the associated value of B L’. The average value of B L’ is 1.07 using F LIM-EST. The conclusions to be made from these results are discussed later.

3-13


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

I

COLUMN LOADING DIRECTION

m lb./in.

POSITIVE

1817

NEGATIVE

1630

L in.

(1) Fmax lb.

(2) M=F*L in.-lb.

(3) Z 3 in.

(4) Sy

(5) ' BL =SyZ/M

ksi

Using (2)

51.5

2145

110,468

3.215

63.3

1.84

3000

1.32

51.5

2065

106,348

3.215

63.3

1.91

3510

1.13

Average for specimen I = J

Using (6)

1.88

1.22

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

1.43

0.90

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 = L

(7) ' BL = SyZ/M

POSITIVE

Average for specimen J = K

(6) FLIM-EST

1.44

1.10

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, both loading directions =

1.56

1.07


4 EVALUATIONS OF METHODS TO QUALIFY TRUNNIONS ON ELBOWS

Purpose The purpose of this section is to review and assess different methods of evaluating trunnions on elbows. The overall objective is to select a general approach that is conservative 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 special requirements for the trunnion/elbow configuration. The general approach used by Code Case 392 (or 391) is clearly conservative and can be extended to cover trunnions on elbows. This is the same general approach that was used by Rodabaugh [22]. As discussed earlier, the objective of Code Case 392 is to present a simplified conservative method. The methodology to determine the indices is presented next.

Potential Methods There are several equations available for calculating the various C stress indices. These include: •

Hankinson [7]

•

Wordsworth [16]

•

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

•

Code Case 392 modified to include the effect of the angle

•

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

θ (see Figure 2-1)

4-1


Evaluations of Methods to Qualify Trunnions on Elbows

In order to make a comparison of these equations, the finite element database used by Hankinson [7] served as the basis of comparison. Table 4-1 lists the characteristics of the models. There are 26 models in this data base for C T, CL, and CN. For CW, there are 16 models. The first step in the evaluation is a comparison of the indices calculated by the five methods for these models. The first three methods (Hankinson [7], Wordsworth, and Code Case 392 (Potvin) have been discussed in detail earlier. The other two methods are discussed below. Table 4-1 Hankinson FEA Parameters

4-2

Model No.

D in.

T in.

d in.

t in.

2γ=

β=

D/T

d/D

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


Evaluations of Methods to Qualify Trunnions on Elbows Table 4-1 (cont.) Hankinson FEA Parameters Model No.

D in.

T in.

d in.

t in.

2γ=

β=

D/T

d/D

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 straight pipe where the angle θ is 90o. WRC Bulletin 256 [17] includes a modified equation which includes a θ term that is (sin θ)n6, where n6 is a constant that depends on the type of loading. To include the effects of θ, the basic formulation for C indices of Code Case N-392 is modified to: C = Ao(2γ )n1βn2τn3 (sin θ)n6 The last method to be investigated listed above is a modification of the expressions developed by Hankinson [7] based on what could be called a “sequential” regression analysis. Hankinson indicated that the approach used was to first find a relationship between the FEA and one variable, for example, (t/T). A logarithmic regression analysis would provide the exponent m. The FEA results would then be normalized by dividing by (t/T)m. The process would be continued until all the exponents (and A o) 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-indices can be correlated by equations of the form of Equation 2-4: C = Ao(2γ )n1βn2τn3. 4-3



The constants Ao, n1, n2 and n3 were determined by a multiple regression analysis using 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 best fit 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 to 5-4, 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 C L and CN. Regression equations were also made for data where t/T ≤ 1.0 and additional equations for data where t/T > 1.0. These are called the “t/T ≤ 1.0 regression.” The equations 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 C L, CN, CT, and CW. The values of C are calculated using the various methods. The calculated values are referred to as CL’, CN’, CT’, and CW’. These are then compared to the results of the finite element analysis (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 values are presented for comparison. It should be noted that Wordsworth did not develop indices for torsional moments. 4-4


Table 4-2 Comparisons for C L

Model

FEA

2

No.

D/T

1

20.0

d/D

t/T

CL

Wordsworth Kc~CL

Kc/CL

Hankinson

Full Regression

t/T 1 Regression

Code Case N-392

CC N-392 With adj.

CL'

CL'

CL'

CL'

CL'

CL'/CL

CL'/CL

CL'/CL

CL'/CL

CL'/CL

.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

Model No.

FEA

2 D/T

d/D

t/T

CL

Wordsworth Kc~CL

Kc/CL

Hankinson CL'

CL'/CL

Full Regression CL'

CL'/CL

t/T 1 Regression

Code Case N-392

CL'

CL'

CL'/CL

CL'/CL

CC N-392 With adj.

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

0.45

0.80

0.73

0.80

MIN=

0.75

MIN=

0.59

MIN=

MIN=

MIN=

MIN=


Table 4-3 Comparisons for C N

Model 2 No.

FEA

D/T

d/D

t/T

CN

Wordsworth Ks~CN

Ks/CN

Hankinson

Full Regression

Regression

Code Case N-392

CC N-392 With adj.

CN'

CN'

CN'

CN'

CN'

CN'/CN

CN'/CN

t/

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

EPRI Li cense censed d M at eri al

Table 4-3 (cont.) Comparisons for CN

Model 2 No.

FEA

D/T

d/D

t/T

CN

Wordsworth Ks~CN

Ks/CN

Hankinson

Full Regression

Regression

Code Case N-392

CC N-392 With adj.

CN'

CN'

CN'

CN'

CN'

CN'/CN

CN'/CN

t/

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

0.69

0.64

1.86

0.84

MIN=

MIN=

MIN=

MIN=


Table 4-4 Comparisons for C T

Model No.

FEA D/T

d /D

t/T

CT

Hankinson CT'

Code Case N-392

Regression

CT'/CT

CT'

CT'/CT

CT'

CC N-392 With adj.

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 No.

FEA D/T

d /D

t/T

CT

Hankinson CT'

Code Case N-392

Regression

CT'/CT

CT'

CT'/CT

CT'

CC N-392 With adj.

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 4-5 Comparisons for C W

Model No.

FEA

2 D/T

d/D

t/T

CW

Wordsworth

Hankinson

Regression

CW'

CW'

CW'

CW'/CW

CW'/CW

CW'/CW

Code Case N-392 CW'

CW'/CW

CC N-392 With adj. 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 No.

FEA

2 D/T

d/D

t/T

16

40.3

.843

2.00

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

CW 6.90

Wordsworth

Hankinson

Regression

CW'

CW'

CW'

CW'/CW

CW'/CW

CW'/CW

Code Case N-392 CW'

CW'/CW

CC N-392 With adj. CW'

CW'/CW

12.53

1.82

8.28

1.20

6.05

0.88

34.27

4.97

17.01

2.47

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



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

Wordsworth y c n e u q e r F

y c n e u q e r F

10 Frequency

5 0

y c n e u q e r F

4 . 0

6 . 0

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

4 . 0

6 . 0

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

CL´/CL

Full Regression

t/T < 1.0 Regression y c n e u q e r F

Frequency 6 . 0

0

Frequency

CL´/CL

5 4 . 0

5

8 . 1

10

0

10

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

8 . 1

8 . 1

10 5 0

Frequency 4 . 0

6 . 0

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

8 . 1

CL´/CL

CL´/CL

Code Case N-392 y c n e u q e r F

4 2 0

Frequency 4 . 0

6 . 0

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

8 . 1

CL´/CL

Figure 4-1 Comparison of Equations for C L

4-13



Hankinson

Wordsworth y c n e u q e r F

6 4 2 0

y c n e u q e r F

Frequency 5 . 0

5 . 1

5 . 2

5 . 3

5 . 4

5 . 5

5 . 6

4 2 0

5 . 7

Frequency 4 . 0

6 . 0

8 . 0

CN´/CN

5

y c n e u q e r F

Frequency 4 . 0

6 . 0

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

8 . 1

t/T < 1.0 Regression

10

0

2 . 1

CN´/CN

Full Regression y c n e u q e r F

0 . 1

4 . 1

6 . 1

8 . 1

10 5 0

CN´/CN

Frequency 4 . 0

6 . 0

8 . 0

0 . 1

2 . 1

4 . 1

6 . 1

8 . 1

CN´/CN

Code Case N-392 y c n e u q e r F

6 4 2 0

Frequency 5 . 1

5 . 2

5 . 3

5 . 4

5 . 5

5 . 6

5 . 7

5 . 8

CN´/CN

Figure 4-2 Comparison of Equations for C N

Results of Comparison First, consider in-plane bending or C L. A review of Table 4-2 shows that all the methods provide results that are reasonably close to the FEA results. Hankinson’s [7] equation provides results that, on the average, are about 25% higher than the FEA. This is to be expected since Hankinson’s [7] equation was a “curve fit” and it was intended to be conservative. The other expressions were “best fit” regression expressions and were expected to be a closer fit. For comparison, the values of C L that correspond to the parameters of the test specimens are: CL = 4.37 for Hankinson [7] CL = 3.53 for Wordsworth CL = 3.77 for Code Case N-392 4-14



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 4-3 indicates, as expected, the various regression expressions and Hankinson’s [7] equations produce results close to the FEA results. However, Wordsworth, Code Case N-392, and Code Case N-392 modified to include the θ effects are all consistently higher. The results of Tables 4-2 and 4-3 indicated that there was little difference in the full regression and the t/T ≤ 1.0 Regression. For torsion of the trunnion, C T, Hankinson [7] provided results about 30% higher than the FEA. The regression analysis yielded the closest fit and the Code Case N-392 results were very high. For the axial force, C W, Hankinson[ 7] provided results about 12% higher, and again the regression 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 was selected because the variation was high. The FEA was conducted using COSMOS version 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 elements were used in the model. The models were fixed at one end, and moments in the three orthogonal directions were applied at the trunnion or the end of the straight pipe. Load combinations that simultaneously applied the six moments were also run. The stresses used by Hankinson [7] in the development of the stress indices are based on the stress intensity located at the centroid of each element. The stress intensities are evaluated at mid-thickness and at the top and bottom shell surfaces. The general methodology used in the various EPRI studies that are a part of this project is at the juncture of the trunnion (or in other cases, the branch) to project the stress intensities from the adjacent elements to the juncture. This is similar to experimental stress measurement techniques where the stress at the juncture is projected from strain gages located on both sides of the juncture. FEA nodal results at the juncture, which are an averaging of the elbow and trunnion side results, are not used. However, for comparison, they will be considered. Table 4-6 contains a comparison of the results for CL ,CN and CT. The FEA results are with loads on the end of the trunnion.

4-15


Evaluations of Methods to Qualify Trunnions on Elbows Table 4-6 Comparison of Results for Hankinson [7] Model 12 This Study

This Study

CC

CC

Hankinson

FEA

FEA

N-392

N-392

FEA

Element

Projected

Wordsworth

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

with Effect

Hankinson [7] also reported a value of C 2e, which represented the C index for the elbow. The value was C 2e = 8.1 for Model 12. The FEA for this study indicates that the C for loading the elbow is 5.19 for torsion or out-of-plane loading and 8.19 for in-plane loading. For comparison, an FEA was run for an elbow without a trunnion. This produced C values of 5.24 for torsion or out-of-plane bending and 8.55 for in-plane bending. For reference, the Code value of C 2 = 1.95/h2/3= 8.85. Hankinson’s [7] models covered only long radius elbows (R = 1.5D where D is the nominal pipe diameter). However, it is assumed that trunnions would also be used on short radius elbows (R=1.0D where D is the nominal pipe diameter). The major impact is that the angle θ changes from 41.4° to 48.2°. Based on the conservatism of the overall process, it is assumed that the conclusions derived from Hankinson’s studies are also applicable to trunnions attached to short radius elbows. The following observations can be made from comparing Table 4-2 to Table 4-5 and from Table 4-6: •

The FEA from this study verifies the results from Hankinson [7] when the stress intensity 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 for CL, 19% lower for C N, and 14% lower for C T. This corresponds to the results from other studies prepared under the EPRI SIF Optimization Project when the FEA results for element stresses are compared to those that are projected.

•

The results from Wordsworth and Code Case N-392 yield reasonable results for inplane loading (CL) but are overly conservative for other loadings. These methodologies were based on trunnions on straight pipe; apparently, the elbow does have an effect. When compared to the FEA from this study, Wordsworth is conservative by a factor of about 3 for C N. Code Case N-392 is conservative by a

4-16



factor of 4.6 . When Code Case N-392 is adjusted for the factor of 2.39.

θ effect, this drops to a

•

The FEA from this study indicates that the presence of the trunnion decreases the stress in the elbow when the loading is on the elbow. This is confirmed by Hankinson [7] in that the value of C 2 determined is less than that of a plain elbow.

•

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

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

4-17


5 COMPARISON OF TEST DATA TO ANALYSIS METHODS

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

C Indices The experimental value of C L was, on the average, 1.66 using the C T Indices -Markl Approach and about 1.5 using the C T Indices-Class 1 Approach. Using Equation 4-1, a value of CL = 3.76 would be calculated. The tests indicate that the indices developed from Equation 4-1 can be reduced by a factor of .43–.40. Considering the spread of the data 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 theoretical approach (finite element analysis), it is reasonable to assume that the same degree of conservatism exists for these indices. Multiplying Equations 4-1, 4-3 and 4-4 by 0.56 leads 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)

5-1


Comparison of Test Data to Analysis Methods

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 of testing 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 B L was determined to be B L = 1.07 on the average. Based on this value, the value of B L is .64–.71 times the experimentally derived value of CL. The present version of Code Case N-392 specifies that the B index be taken as 0.5 times the C index. It is recognized that the method of estimating the value of B L from the test data is very conservative. Recognizing also that the equations for the C indices include a factor of 1.3 (to cover the statistical distribution), it is believed that it is reasonable to maintain the relationship that the B indices are 0.5 times the C indices. It is also believed to be reasonable to assume that the same relationship exists for the other B indices, that is, BW, BN, and BT.

5-2


6 INVESTIGATION OF FLEXIBILITY OF TRUNNIONS ON ELBOWS

General There are two parts to the evaluation of the flexibility of the trunnion-elbow configuration. 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. With large trunnions, the ovalization will be restricted. This is to be investigated. The second part is the flexibility of the trunnion. This is important because trunnion-elbows are often used as anchors in piping systems. This flexibility study is based on the results from FEA. The test data (Section 3) do not envelope all the boundary conditions that are used to verify the results. As discussed later, the flexibility is a function of the boundary conditions of the trunnion-elbow. The test data are used to confirm the results of the study.

Discussion: Elbows In piping analysis, the various components are modeled as one dimensional beam elements. In order to accurately represent the load displacement (flexibility) action of the components, flexibility factors are used. From Rodabaugh [23], when bending a straight pipe of length L, the rotation, φ, of one end with respect to the other is:

φ= 1/EI ∫ oL M dx where M is the bending moment. For a torsional moment, the rotation is given by:

φ = (1/GJ) ∫ oL M dx = 1.3/EI ∫ oL M dx

(Eq. 6-1) 6-1


Investigation of Flexibility of Trunnions on Elbows

where M is the torsional moment. This is based on the relationship between J and I in addition to G= E/(2(1+µ)). For curved pipe or an elbow of 90 o, the length of the one-dimensional segment used to represent it is R π/2 where R is the bend radius. The rotation for in-plane bending is given 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 inplane bending moment. The evaluation will be performed by comparing the rotations from FEA to those from closed form solutions for the beam model considering both bending and torsional rotations due to in-plane and out-of-plane bending. The flexibility factor for an elbow is defined by the Codes as k = 1.65/h, h = tR/r 2. Rodabaugh [22] provides the following expressions for evaluating the flexibility factors for 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 corresponds to the rotation, φ, indicated in Figure 6-1(a). The rotation, φfea, for out-of-plane bending is similarly defined.

6-2



M

α

R M (a) Elbow

M

α

R

M

(b) Elbow with Pipe Attached

Figure 6-1 Configurations

Discussion: Trunnions on Elbows The flexibility of the trunnion connection will also be investigated. For configurations such 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 the area of intersection. 6-3



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

A

Figure 6-2 Elbow-Trunnion Model

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

Rigid Link

Figure 6-3 Branch Connection Model

6-4

Point Spring



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 of pipe (or trunnion) diameters that would be added to represent the local flexibility. The flexibility factor for the spring is calculated by: k =(φfea - φ b)/(Mdo/It)

(Eq. 6-4)

where φfea is the rotation from the FEA and φ b is the rotation from the beam model. This is discussed in more detail later.

Finite Element Analysis The general dimensions for the FEA models are shown in Figure 6-4. Typically, straight sections of pipe (or trunnion) equal to four diameters are used in the models. COSMOS version 1.75 from Structural Research and Analysis Corporation was used. Stresses were not calculated in this evaluation, only deflections. L1

L2

L3

3

2

θ

L4 R L1 = 4 D o θ = arccos(R/(R+Do /2)) L2 = (R+Do /2)sin θ L3 = R+Do /2+4do-L2

L5

L4 = 4 d o L5 = 4 D o 1

Figure 6-4 FEA Model Details

6-5



Shell elements were used in the models, which typically consisted of approximately 7000 elements. A typical model is shown in Figure 6-5. The elements at the ends of the pipe and trunnion sections are connected to elements that are “rigid bars” or “rigid links,” that is, the stiffness of the rigid links is infinite (or very large).

Figure 6-5 FEA Model

The material properties used in the analyses are E = 30E6, G = 12E6, and µ = 0.28. The beam equations (Equations 6-1 and the others) assumed a value of µ = 0.3. This difference 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. The local rotation at the juncture of the elbow and the trunnion is somewhat dependent on the boundary conditions at the ends of the pipes (points 1 and 2). It is important to recognize that in evaluating flexibility factors, there is no “conservative” value that would be applicable for all piping layouts. As an example, a high value might mean that the loads are lower in other components in a piping system than the “true” values.

6-6



Consequently, the “best” value to use is the one that is most representative of the actual value. This is complicated because the flexibility is a function of the end conditions at the ends of the elbow (or attached straight pipe). This will, of course, be a function of the layout. As a result of this, two sets of boundary conditions were used in the evaluation. The first 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 the second case, both ends were fixed and the loads were applied at the end of the trunnion. The flexibility factors are based on the average of the results. This is depicted in Figure 6-6. (2)

(3)

(1)

(a) Moment on Elbow

(2)

(3)

(1)

(b) Moment on Trunnion

(2)

(3)

(1)

(c) Moment on Trunnion

Figure 6-6 Boundary Conditions

The 22 models are listed in Table 6-1 along with the dimensions and other pertinent data. The cases listed are representative of actual usage. As an example, it is not expected that a very small trunnion would be used on a large elbow (small d/D). Table 6-1 includes the moments used in the FEA, which are based on a nominal bending stress of 10 ksi in the pipe or trunnion (that is, M/Z or M/ZT where Z is the section modulus for the pipe and Z T is the section modulus for the trunnion). Note that only the deflections were calculated in the FEA. The bend radius, R, is also listed in Table 6-1. For models of actual pipe, R = 1.5 D where D is the nominal pipe size. For the other cases, R = 1.5 D o. These values of R correspond to long radius elbows. Based on the conservatism of the overall process, it is assumed that the conclusions derived from the FEA will also be applicable to short radius elbows (R = 1.0 D).

6-7


Table 6-1 FEA Models Model

Do

T

do

t (in.)

R

do/D

t/T

Do/T do/t TR/r

(in.)

2

D (in.)

d

D/T

d/D

d/t

(in.)

Ma

(in.)

(in.)

(in.)

T1

8.625

0.250

4.5

0.237 1 2.0 0 .522 0.948

34.5

19.0 0 .171 8.375 4.263 33.5

0.51 18.0

33827

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

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 33.0

Mb

(in.-lb.) (in.-lb.)

Ie 4

It

L3 4

(in )

(in )

(degrees) (in.)

137721

57.7

7.2

42.6

22.6

451036

279.3

7.2

42.4

24.6

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

0.82 17.1 481195

451036

279.3

245.2

42.4

50.6

TE1

10.00

0.500

5.0

0.250 1 5.0

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 1 9.0

0.75 19.0 1 49517

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 1 9.0

0.85 19.0 2 17652

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 1 5.0

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 1 9.0

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

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 2 9.0

30580

244637

118.4

7.4

41.4

25.9

0.75 29.0 1 03206

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 2 9.0

0.85 29.0 1 50238

244637

118.4

61.8

41.4

40.5

TE10

10.00

0.333

5.0

0.333 1 5.0

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 2 9.0

0.72 14.0 1 92422

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 1 5.0

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 1 5.0

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 1 5.0

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 1 5.0

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 1 5.0

0.75

1.500

50.0

25.0 0 .125 9.800 7.200 49.0

0.73 24.0 1 22145

150859

74.0

44.0

41.4

36.3

TE18

10.00

0.200

8.5

0.340 1 5.0

0.85

1.700

50.0

25.0 0 .125 9.800 8.160 49.0

0.83 24.0 1 77807

150859

74.0

72.7

41.4

40.5

Notes:

1. See Figure 6-4.



FEA Results: Flexibility of Elbows with Trunnions Table 6-2 lists the rotations at the indicated points for the specific load cases. These rotations were taken directly from the FEA output. The results are discussed in the following sections.

6-9



Table 6-2 Summary 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

Model

PT-3 Y

PT-2 Y

PT-3

PT-2

PT-3

PT-2

Y

Y

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 (See Figure 6-4). 3.

6-10

PT-1 is fixed.



Table 6-3 lists the flexibility factors for elbows derived from the FEA rotations using Equations 6-2 and 6-3. The loads considered are the in-plane and out-of-plane loading at the end of the pipe (point 2). Rodabaugh [22] provides an evaluation of end effects on elbows subjected to moment loading. It concludes that for in-plane bending for configurations of 90 o elbows with straight sections of pipe on the ends: k = 1.3/h

(Eq. 6-5)

where h is the elbow characteristic TR/Rm2. Consequently, for this study, a comparison is 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 significantly affected by the trunnion. The percentage difference from k = 1.3/h (a maximum difference of 7.7%, an average difference of -1.6%, and a standard deviation of the difference 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 for the flexibility of 90 o elbows with straight sections of pipe on the ends. The percentage difference from k = 1.25/h is slightly higher (a maximum difference of 9.0%, an average difference 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 of the 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%.

6-11


Table 6-3 Bending of the Pipe-Elbow Flexibility IN-PLANE LOAD Model

D/T

d/D

d/t

Z

k

1.3/h

OUT-OF-PLANE LOAD

% DIFF

Reg Eq

% DIFF

(1)

(2)

(3)

7.6

7.6

6.5

T1

33.5

0.51

18.0

1.774E-02

8.17

7.60

T2

33.0

0.34

18.0

1.787E-02

8.14

7.37

10.3

9.0

T3

33.0

0.84

28.5

1.467E-02

6.03

7.37

-18.2

6.1

T4 TE1

33.0 19.0

0.82 0.50

17.1 19.0

1.399E-02 1.230E-02

5.59 4.06

7.37 3.91

-24.2 3.9

TE2 TE3

19.0 19.0

0.75 0.85

19.0 19.0

1.147E-02 1.085E-02

3.56 3.18

3.91 3.91

-8.9 -18.6

TE4

19.0

0.47

9.0

1.217E-02

3.99

3.91

TE5

19.0

0.71

9.0

1.110E-02

3.34

3.91

TE6 TE7

19.0 29.0

0.81 0.50

9.0 29.0

1.037E-02 1.620E-02

2.89 6.59

3.91 6.07

TE8 TE9 TE10

29.0 29.0 29.0

0.75 0.85 0.48

29.0 29.0 14.0

1.483E-02 1.368E-02 1.605E-02

5.74 5.03 6.49

TE11

29.0

0.72

14.0

1.423E-02

TE12

29.0

0.82

14.0

1.281E-02

TE13 TE14

49.0 49.0

0.50 0.75

TE15 TE16

49.0 49.0

TE17 TE18

49.0 49.0

Y

k

1.3/h

% DIFF (4)

1.302E-02

7.65

-10.6

1.297E-02

7.45

7.37

1.0

-1.7

1.284E-02

7.28

7.37

-1.3

5.8 4.3

-4.5 -5.1

1.280E-02 1.056E-02

7.22 3.70

7.37 3.91

-2.0 -5.4

3.5 3.3

1.7 -3.3

1.049E-02 1.043E-02

3.61 3.54

3.91 3.91

-7.6 -9.4

1.9

4.0

-0.7

1.053E-02

3.66

3.91

-6.3

-14.7

3.3

1.5

1.045E-02

3.57

3.91

-8.8

-26.0 8.4

3.1 6.6

-6.8 -0.8

1.036E-02 1.238E-02

3.46 6.15

3.91 6.07

-11.6 1.2

6.07 6.07 6.07

-5.5 -17.1 6.9

5.4 5.1 6.2

5.3 -1.6 4.7

1.230E-02 1.223E-02 1.234E-02

6.05 5.96 6.10

6.07 6.07 6.07

-0.4 -1.9 0.4

5.37

6.07

-11.6

5.1

5.7

1.226E-02

6.00

6.07

-1.2

4.50

6.07

-26.0

4.8

-6.0

1.215E-02

5.86

6.07

-3.5

49.0 49.0

2.404E-02 11.61 2.179E-02 10.20

10.40 10.40

11.6 -1.9

11.4 9.3

2.1 8.7

1.603E-02 1.593E-02

10.90 10.78

10.40 10.40

4.8 3.6

0.85 0.49

49.0 24.0

1.940E-02 8.71 2.389E-02 11.52

10.40 10.40

-16.3 10.7

8.8 10.5

-0.5 8.5

1.587E-02 1.598E-02

10.70 10.84

10.40 10.40

2.9 4.2

0.73 0.83

24.0 24.0

2.081E-02 1.782E-02

10.40 10.40

-7.8 -25.7

8.6 8.1

10.0 -5.1

1.590E-02 1.580E-02

10.74 10.61

10.40 10.40

3.2 2.0

9.59 7.73

7.60

0.7



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/h 3. 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 with attached tangent piping. Consequently, it was decided to curve fit the data of Table 6-3 to 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 the average 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 as the 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 the model from the center line of the run pipe to its surface. At that juncture, a point spring is used to represent the local flexibility of the connection. For the elbow-trunnion model, there are several possibilities. The trunnion will be modeled as a beam with a point spring located at point A (see Figure 6-7). A rigid link can be connected from point A to either points B, C, or D. Then the appropriate sections of the elbows would be modeled as beams. The lengths of the sections would depend upon the points selected.

6-13


Investigation of Flexibility of Trunnions on Elbows A

B

C D

Figure 6-7 Elbow-Trunnion Model

The first evaluation performed for this study assumed the rigid link was from point A to B. The link from A to B was selected because the trunnion’s centerline passed through point B. The model was fixed at the bottom end, and the moments were applied at the end of the trunnion. However, this model resulted in negative values for the flexibility factor for the point spring. The values were negative because the flexibility of the elbow (including the modification to account for the trunnion) was greater than the flexibility in the FEA model. Next, the model was changed so that the rigid link went from point A to C. This corresponds to the model for branch connections in that the rigid link is perpendicular to the centerline of the elbow. In the branch connection model, the rigid link is perpendicular to the centerline of the run pipe. However, the flexibility factors were still negative. Finally, the model was changed so that the rigid link was from point A to D. This resulted 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 for loads on the trunnion. The rotation at the end of the trunnion (Point 6) with respect to the fixed end (Point 1) is given by:

φ = φ6-5 + φ5-4 + φ4-3 + φ3-2 +φ2-1 6-14



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]

6-15


Investigation of Flexibility of Trunnions on Elbows L1

L3

L2

6

5

7

4 L6

3 ψ

L7

2

ψ = cos-1(L2 /R) L6 = πR/2 (90-ψ )/90 L7 = πR/2 ψ /90 L5 Points 4 and 5 are at the trunnion/elbow interface. All other points are on the centerline of the elbow, trunnion, or pipe.

1

Figure 6-8 Beam Model

For out-of-plane bending of the trunnion, k e of the elbow is replaced by the regression Equation 6-6 for out-of-plane bending of the elbow. M is the out-of-plane bending moment. 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 pipe ends, Equations 6-7 and 6-8 can be combined and modified for in-plane moments:

φ3-1 =φ3-2 + φ2-1 = 1/4 M L b/(EIe)[4(La/Lc) - 9(La/Lc)2 + 6(La/Lc)3 -1] where: La = L 1 + ke L6 L b = L5 + ke L7 Lc = La + L b 6-16



For out of plane moments, k e, determined from Equation 6-6, is used. Also since segment 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 inplane and out-of-plane bending for the case with only one end fixed. For in-plane bending, the values of k are reasonably small with a maximum of about 3. For out of plane 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-plane moments, there is little difference from the values in Table 6-4 for the case with one end fixed. The flexibility is lower, on the average, by a value of .6. This is within the analysis methodology tolerance. For out-of-plane flexibility, the situation is different. The average difference between the results for the two sets of boundary conditions is about 3 with a maximum difference of over 8. The boundary conditions clearly affect the flexibility factors. It is assumed that the average of the two conditions is representative of actual applications. Table 6-6 lists the average values of k and also provides a comparison to equations 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 a maximum 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 a maximum of about 11%.

6-17



Table 6-4 Bending of the Trunnion-Trunnion Flexibility In-Plane Moment Model

D/T

d/D

d/t

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.

6-18

Point 1 is fixed.

Z

k

Out-of-Plane Moment Y

k



Table 6-5 Bending of the Trunnion-Ends Fixed Trunnion Flexibility In-Plane Moment Model

D/T

d/D

d/t

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

Z

k

Out-of-Plane Moment Y

k

Notes:

1.

Points 1 and 2 are fixed.

6-19



Table 6-6 Average 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

6-20



Comparison to Test Data While the tests discussed in Section 3 were not specifically for determining flexibility factors, they can be used to evaluate the equations derived above. Deflections and loads at point 6 in Figure 6-8 for in-plane bending were recorded and are included in Appendix 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 per Figure 6-8, the calculated deflection was .43 inches. This calculation used Equation 6-6 for the elbow and Equation 6-9 for the trunnion. Considering that the calculation did not include the flexibility of the testing frame or bolted connections and also that Equation 6-9 is based on the results from the average of the two cases with different boundary conditions, this is considered as verification of the methodology.

6-21


7 CONCLUSIONS

The conclusions arrived at from the analyses and tests described in this report are enumerated below: 1.

The basic approach used by Code Cases 391 and 392, with modification of the indices, can be used in the design and qualification of trunnions on 90 o 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 models that 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 for a straight pipe. For evaluation of elbows with trunnions, these indices and i factors shall be based on the elbow configuration. The evaluation of the 7-1


Conclusions

experimental data either conservatively assumed indices for a straight pipe or conservatively neglected the contribution of the elbow. 3.

The values of KT, shall be as specified in Code Cases 391 and 392 for the corresponding 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. (Equation 6-6)

(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 the trunnion. (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 models that were used in the development of the constants. The use of this methodology will allow for a more accurate evaluation of trunnions on elbows.

7-2


8 REFERENCES

1.

ASME Boiler and Pressure Vessel Code, Section III, “Nuclear Power Plant Components.” 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,” Pressure Vessels and Piping. Vol. 169, ASME (1989).

4.

R. F. Hankinson, D. A. Van Duyne, D. H. Stout, and Y. K. Tang, “Design Guidance 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 Circular Trunnion Attachments to Piping, Pressure Vessels and Piping. Vol. 218, ASME (1991).

6.

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

7.

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

8.

G. B. Rawls, E. A. Wais, and E. C. Rodabaugh, “Evaluation of the Capacity of Welded Attachments to Elbows as Compared to the Methodology of ASME Code 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 Section Attachments on Class 2 or 3 Piping, Section III, Division 1, American Society of Mechanical Engineers, New York.

10

Case N-391-2, Procedure for Evaluation of the Design of Hollow Circular Cross Section Welded Attachments on Class 1 Piping, Section III, Division 1, American Society of Mechanical Engineers, New York. 8-1


References

11.

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

12.

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

13.

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

14.

W. G. Dodge, “Secondary Stress Indices for Integral Structural Attachments to Straight Pipe,” and E. C. Rodabaugh, W. G. Dodge, and S. E. Moore, “Stress Indices 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, “Local Stresses in Cylindrical Shells due to External Loadings on Nozzles-Supplement to WRC Bulletin No. 107,” Welding Research Council Bulletin, No. 297 (August 1984).

16.

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

17.

E. C. Rodabaugh, “Review of Data Relevant to the Design of Tubular Joints in Fixed Offshore Platforms,” Welding Research Council Bulletin, No. 256 (January 1980).

18.

E. C. Rodabaugh, “Stress Indices, Pressure Design, and Stress Intensification Factors for Lateral in Piping,” Welding Research Council Bulletin, No. 360, (January 1991).

19.

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

20.

A. R. C. Markl, “Fatigue Tests of Piping Components,” ASME paper no. 51-PET21, May 21, 1951.

21.

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

8-2


References

22.

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

23.

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

8-3


A ASME CODE CASE N-392-3

Reprinted with the permission of The American Society of Mechanical Engineers from ASME BPVC, Section XI-1998 Edition A-1


ASME Code Case N-392-3












B TEST DATA AND RESULTS

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Test Data and Results

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TR-107453 Stress Indices for Elbows with Trunnion Attachments Technical

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