Development of reliability-based load and resistance factor design (LRFD) methods for piping : research and development report (LRFD) Methods for Piping (Crtd)

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Research and Development Report

Development of Reliability-Based Load and Resistance Factor Design (LRFD) Methods for Piping

ASME Special Working Group on Probabilistic Methods in Design Endorsed by ASME Boiler & Pressure Vessel Code Committees

ASME Research Committee on Risk Technology

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Research and Development Report

Development of Reliability-Based Load and Resistance Factor Design (LRFD) Methods for Piping

Prepared by ASME Research Task Force on Development of

Reliability-Based Load and Resistance Factor Design (LRFD) Methods for Piping

1828 L St. NW Suite 906, Washington, DC 20036

Prepared for U.S. Nuclear Regulatory Commission

International Institute of Universality, Tokyo, Japan

A S M E


© 2007 by ASME, Three Park Avenue, New York, NY 10016, USA (www.asme.org) ISBN 10: 0-7918-0262-0, ISBN 13: 978-0-7918-0262-5 All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED TO BE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARANTEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT. ASME shall not be responsible for statements or opinions advanced in papers or printed in its publications according to (B7.1.3) statement from the Bylaws: For authorization to photocopy material for internal or personal use under those circumstances not falling within the fair use provisions of the Copyright Act, contact the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, tel: 978-750-8400, www.copyright.com. Disclaimer This research report was prepared as an account of work performed by the ASME Research Task Force on Development of Reliability-Based Load and Resistance Factor Design (LRFD) Methods for Piping through the facilitation of the American Society of Mechanical Engineers (The Society) Center for Research and Technology Development, and for the sponsoring governmental agencies and companies. Neither the Society nor the Sponsors, nor the subcontractors, nor any others involved in the preparation or review of this report nor any of their respective employees, members, or persons acting on their behalf, make any warranty, expressed or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, software or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the Society, the Sponsors, or others involved in the preparation or review of this report, or any agency thereof. The views and opinions of the authors, contributors, and reviewers of the report expressed herein do not necessarily state or reflect those of the Society, the Sponsors, the Sponsorees, financial contributors or others involved in the preparation or review of this report, or any agency thereof.


Summary This research report develops a technical basis for reliability-based load and resistance factor design (LRFD) methods for piping. This document is the work product of the ASME Task Force and Steering Committee on the development of technical basis for reliability-based LRFD methods for piping. This report provides the technical basis for reliability-based load and resistance factor design (LRFD) methods for piping, more specifically for Class 2/3 piping for primary loading that include pressure, deadweight, seismic and accidental loading. The outcomes of the project include design models and equations, and partial safety factors that can be used to compose LRFD guidelines and criteria. It provides a proof of concept of the LRFD for the design of piping. Such design methods should lead to consistent reliability levels. The LRFD guidelines and criteria can initially be used in parallel with currently used procedures. The report provides results based on the following tasks: (1) a state-of-the-art assessment and selection of reliability theories, (2) review and evaluation of existing strength models for piping, (3) selection of strength models and equations that deemed suitable for LRFD development, (4) preliminay analysis of basic random variables to characterize their uncertainties, and (5) development of LRFD guidelines and criteria. The report consists of seven chapters and two appendices. Chapter 1 consists of the introduction and an objective statement. Chapter 2 provides the needed theoretical background for performing reliability-based design and analysis. Chapter 3 gives a summary of the design loads, such as weight, internal pressure, occasional and accidental dynamic loads (e.g., seismic), and also provides a summary of recommended load combinations for this study. Failure modes and limit states for piping systems are provided in Chapter 4. Chapter 5 provides statistical information on basic random variables that are relevant to piping. Chapter 6 calculates the bias of models used for the calculation of the hoop stress and bending moments of pipes. Chapter 7 provides the calculation of the partial load and resistance factors and Chapter 8 recommendations for future work. Appendix A summarizes some of the limit states that are contained in the current ASME code and Appendix B shows the steel that is used for the production of pipes.

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Task Force Organization The task force includes the following members, associates and assistants: Chair Professor Bilal M. Ayyub, PhD, PE, Center for Technology and Systems Management,

University of Maryland, College Park, USA, [email protected], 301-405-1956 Members Dr. Abinav Gupta, North Carolina State University Dr. Nitin Shah, Dominion - Virginia Power Mr. Philip Kotwicki, Westinghouse Electric Company LLC Associates Dr. Ibrahim Assakkaf, Center for Technology and Systems Management, University of

Maryland, College Park Ms. Kleio Avrithi, Center for Technology and Systems Management, University of Maryland,

College Park Steering Committee The steering committee includes the following members: Chair Mr. Ralph S. Hill, Westinghouse Electric Company LLC Members Dr. Syed Ali, U.S. Nuclear Regulatory Commission Mr. Kenneth Balkey, Westinghouse Electric Company LLC Dr. David Harris, Engineering Mechanics Technology, Inc. Mr. Gene Imbro, Chief ME & CE Branch, Office of Nuclear Reactor Regulation, NRC. Mr. John C. Minichiello, Framatome ANP DE&S Dr. Kenzo Miya, International Institute of Universality, Tokyo, Japan Dr. John D. Stevenson, Consultant for DOE, Defense Nuclear Facility Safety Board Mr. James Todd Conner, Exelon Corporation Mr. Edward A. Wais, Wais and Associates, Inc., Atlanta, Georgia

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Peer Reviews

The peer reviews were performed by the following individuals and committee members: Mr. Owen F. Hedden, Representative of Dr. Kenzo Miya, International Institute of Universality,

Tokyo, Japan ASME Working Group on Probabilistic Methods in Design (SG-D)(SCIII) ASME Working Group on Piping

Acknowledgements The financial support of the Nuclear Regulatory Commission and the International Institute of Universality of Japan and the guidance of the Steering Committee, and the comments provided by ASME committees and working groups, particularly the championship and encouragement of Mr. Richard W. Barnes, ANRIC Enterprises, Chairman of ASME Boiler and Pressure Vessel Codes and Standards, Subcommittee Nuclear Power, are greatly appreciated.

Project Administration The project administration includes the following member: Dr. Michael Tinkleman, Director of Research, American Society of Mechanical Engineers

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Table of Contents

Project Administration.................................................................................................................. vii

1. Introduction................................................................................................................................1 1.1. Background.......................................................................................................................1 1.2. History of Reliability-based Design .................................................................................2 1.3. Benefits of Reliability-based Design ................................................................................3 1.4. Challenges in Developing Load and Resistance Factor Design for Piping ......................4 1.5. Piping Pilot Project ...........................................................................................................5 1.6. Objectives .........................................................................................................................5 1.7. Organization......................................................................................................................6

2. Reliability-Based Design and Analysis .....................................................................................8 2.1. Introduction.......................................................................................................................8 2.2. Direct Reliability-Based Design .....................................................................................10 2.3. Load and Resistance Factor Design (LRFD)..................................................................11 2.4. Performance Functions ...................................................................................................11 2.5. First-Order Reliability Method (FORM) ........................................................................13

2.5.1. Algorithm for First-Order Reliability Method....................................................16 2.5.2. Procedure for Calculating Partial Safety Factors (PSF) Using FORM ..............18 2.5.3. Determination of a Strength Factor for a Given Set of Load Factors.................18

2.6. Examples.........................................................................................................................19 2.6.1. Example I ............................................................................................................19 2.6.2. Example II...........................................................................................................21 2.6.3. Example III .........................................................................................................22 2.6.4. Example IV .........................................................................................................24

3. Loads and Load Combinations ................................................................................................26 3.1. Primary Loads.................................................................................................................26

3.1.1. Dead Weight .......................................................................................................27

Disclaimer ...................................................................................................................................... iv

Summary ......................................................................................................................................... v

Task Force Organization................................................................................................................ vi

Steering Committee ....................................................................................................................... vi

Peer Reviews..................................................................................................................................vii

Acknowledgements....................................................................................................................... vii

Notations ....................................................................................................................................... xii

Symbols ....................................................................................................................................... xiv

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3.1.2. Internal Pressure .................................................................................................27 3.1.3. Seismic Loading..................................................................................................28 3.1.4. Nonreversing Dynamic Loads ............................................................................29

3.2. Load Combinations in Non-ASME Structural Codes.....................................................30 3.2.1. American Institute for Steel Construction Code.................................................30 3.2.2. American Society of Civil Engineers Code........................................................31 3.2.3. American Petroleum Institute Code....................................................................32 3.2.4. American Association of State Highway and Transportation Officials

Bridge Design Specifications..............................................................................32 3.2.5. Eurocode 1990 ....................................................................................................33

3.3. Load Combinations for Components of Nuclear Plant...................................................34 3.4. Recommended Load Combinations for Piping...............................................................36

4. Failure Modes and Limit States for Piping..............................................................................37 4.1. Failure Criterion..............................................................................................................37 4.2. Performance Criterion.....................................................................................................39

4.2.1. Limit-Load Capacity...........................................................................................39 4.2.2. Plastic-Instability Collapse Load Using Elastic Slope .......................................41 4.2.3. Plastic-Instability Collapse Load Using Elastic Deformation ............................41 4.2.4. Plastic Instability: Ultimate Moment Definition of Collapse Load....................42

4.3. Existing Code Equations.................................................................................................42 4.3.1. Design Condition ................................................................................................43 4.3.2. Operating Condition (Service Level A)..............................................................44 4.3.3. Upset Loading Condition (Service Level B) ......................................................44 4.3.4. Emergency Loading Condition (Service Level C) .............................................45 4.3.5. Faulted Loading Condition (Service Level D) ...................................................46

4.4. Performance Functions ...................................................................................................46 4.5. Load Combinations for Piping........................................................................................49

4.5.1. Design Condition ................................................................................................49 4.5.2. Operating Condition ...........................................................................................49 4.5.3. Upset Loading Condition....................................................................................49 4.5.4. Emergency Loading Condition ...........................................................................50 4.5.5. Faulted Loading Condition .................................................................................50

5. Basic Random Variables for Piping.........................................................................................52 5.1. Statistical Characteristics of Random Variables.............................................................52 5.2. Strength Variables...........................................................................................................52

5.2.1. Material Properties..............................................................................................52 5.2.1.1. Material Types for Piping ....................................................................52 5.2.1.2. Yield Strength of Steel for Nuclear Piping..........................................53 5.2.1.3. Ultimate Strength of Steel for Nuclear Piping.....................................56

5.2.2. Geometric Properties ..........................................................................................59 5.2.2.1. Pipe Diameter.......................................................................................59 5.2.2.2. Pipe Thickness .....................................................................................60 5.2.2.3. Diameter-to-Thickness Ratio ...............................................................61 5.2.2.4. Summary ..............................................................................................61

5.3. Load variables.................................................................................................................62

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5.3.1. Fluid Pressure in Piping......................................................................................62 5.3.1.1. Operating and Design Pressure............................................................62 5.3.1.2. Peak Pressure .......................................................................................63 5.3.1.3. Testing Pressure ...................................................................................63 5.3.1.4. Accidental Pressure..............................................................................64

5.3.2. Gravity Loads .....................................................................................................64 5.3.2.1. Dead Weight of Pipe............................................................................64 5.3.2.2. Dead Weight of Fittings and Components...........................................65 5.3.2.3. Insulation .............................................................................................65 5.3.2.4. Contents of Pipe...................................................................................66

5.3.3. Non-Reversing Mechanical Loads......................................................................66 5.3.4. Seismic Loads .....................................................................................................68 5.3.5. Summary .............................................................................................................74

6. Modeling Uncertainty ..............................................................................................................75 6.1. Background.....................................................................................................................75 6.2. Hoop Stress .....................................................................................................................75

6.2.1. Lamé or Thick-Wall Theory ...............................................................................76 6.2.2. Thin-Wall Theory ...............................................................................................76

6.2.2.1. The Barlow Formula ............................................................................76 6.2.2.2. The Boardman Equation or Modified Lamé........................................77

6.2.3. Other Models ......................................................................................................78 6.2.4. Experimental Results ..........................................................................................80 6.2.5. Observations and Recommendations..................................................................84

6.3. Bending Moments...........................................................................................................85 6.3.1. Pure Bending.......................................................................................................87 6.3.2. Bending with Internal Pressure...........................................................................88

7. Load and Resistance Factors....................................................................................................90 7.1. Calculation of Partial Safety Factors ..............................................................................90 7.2. General Design Condition ..............................................................................................91

7.2.1. Performance Function g1.....................................................................................91 7.2.2. Performance Function g2.....................................................................................93 7.2.3. Performance Function g3.....................................................................................95

7.3. Operating Condition (Service Level A)..........................................................................98 7.3.1. Performance Function g4.....................................................................................98

7.4. Upset Loading Condition (Service Level B) ................................................................100 7.4.1. Performance Function g5...................................................................................100 7.4.2. Performance Function g6...................................................................................104 7.4.3. Performance Function g7...................................................................................109

7.5. Emergency Loading Condition (Service Level C) .......................................................111 7.5.1. Performance Function g8...................................................................................111 7.5.2. Performance Function g9...................................................................................111 7.5.3. Performance Function g10 .................................................................................115 7.5.4. Performance Function g11 .................................................................................115 7.5.5. Performance Function g12 .................................................................................115

7.6. Faulted Loading Condition (Service Level D) .............................................................115

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7.6.1. Performance Function g13 .................................................................................115 7.6.2. Performance Function g14 .................................................................................117 7.6.3. Performance Function g15 .................................................................................120 7.6.4. Performance Function g16 .................................................................................122 7.6.5. Performance Function g17 .................................................................................125

7.7 Commentary..................................................................................................................129 7.8 Design Example ............................................................................................................130

8. Load and Resistance Factors..................................................................................................135 8.1. Summary .......................................................................................................................135 8.2. Recommendations for Project Completion...................................................................136 8.3. Recommendations for Future Work..............................................................................137

References and Bibliography.......................................................................................................139

Appendix A. Selected Limit States In ASME Code....................................................................151

Appendix B. Steel Used In ASME Code, Part III .......................................................................155

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Notations

Acronyms AASHTO American Association of State Highway and Transportation Officials AFOSM Advanced First-Order Second Moment AISC American Institute of Steel Construction AISI American Iron and Steel Institute ALWR Advanced Light Water Reactor ANSI American National Standards Institute API American Petroleum Institute ASCE American Society of Civil Engineers ASD Allowable Stress Design ASME American Society of Mechanical Engineers AWWA American Water Works Association BOCA Building Officials and Code Administrators BPV Boiler and Pressure Vessel BWR Boiling-Water Reactors CDF Cumulative Distribution Function CEB Comité Européen du Béton CIRIA Construction Industry Research and Information Association CSA Canadian Standard Association EN EuroNorm ESW Essential Service Water FEM Finite Element Method FORM First-Order Reliability Method FOSM First-Order Second Moment FS Factor of Safety HCLPF High-Confidence-Low-Probability-of-Failure IBC International Building Code IIW International Institute of Welding LOCA Loss of Coolant Accident LRFD Load and Resistance Factor Design MPFP Most Probable Failure Point NCHRP National Cooperative Highway Research Program NRC Nuclear Regulatory Commission OBE Operating Basis Earthquake PDF Probability Density Function PRA Probabilistic Risk Assessment PSF Partial Safety Factors PSHA Probabilistic Seismic Hazard Analysis

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PWR Pressurized-Water Reactors SGR Sodium-cooled Fast and Thermal Reactors SMYS Specified Minimum Yield Strength SMTS Specified Minimum Tensile Strength SORM Second Order Reliability Method SRV Safety Relief Valve SSE Safe Shutdown Earthquake PF Performance Function UBS Uniform Building Code WRC Welding Research Council

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Symbols

φ Strength safety factor φfy Safety factor for yield strength of steel φfu Safety factor for ultimate strength of steel β Reliability or safety index β0 Specified target reliability index ψ0,i Reduction partial factor for non-dominant variable actions γΑ Load factor for stress due to sustained weight γΗ Load factor for hoop stress due to internal pressure γEQ Safety factor for earthquake load γG,j,sup Load factor for variable actions γi Partial safety factor or load factor of ith load effect γL Load factor for stress due to LOCA γQ Load factor common for all the variables loads γO Load factor for the stress due to OBE earthquake γPmax Load factor for the stress due to maximum operating pressure γS Load factor for the stress due to SSE earthquake µ Μean Value

NXµ Mean of the equivalent normal distribution

µfy Calculated mean value of fy µfu Calculated mean value of fu µL Mean value of a load variable µnfi Nominal value of load stresses µnR Nominal value of resistance stress µR Mean value of a strength variable µXi Mean value of the basic random variable σ Standard deviation σlong Longitudinal stresses σmax Maximum principal stress σmin Minimum principal stress

NXσ σN

X Standard deviation of the equivalent normal distribution σXi Standard deviation of the basic random variable τmax Maximum shear stress a Peak ground motion acceleration A Additional thickness as allowance for threading, corrosion, etc. Also, ground

acceleration capacity

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~A A Median of random variable A Am Median ground acceleration capacity B Bias factor

~B B Median of random variable B B1 and B2 Primary stress indices for specific product under investigation, defined in Fig.

NC- 3673.2 (b)-1

B1, B2 Primary stress indices equal to 0.5 and 1.0, respectively for straight pipe COV Coefficient of variation d Inside diameter of a pipe D Dead load due to the weight of a structural element Do Outside diameter of a pipe D1 Weight of the structure including the weight of equipment and other objects

permanently mounted on the platform, hydrostatic forces acting on the structure below the waterline including internal pressure

DC Dead load of structural components and nonstructural attachments E Earthquake load, the effect of horizontal and vertical earthquake-induced forces E(X2) Mean square of the variable X Eo Loads generated by the operating basis earthquake (OBE) EQ Earthquake load Ess Loads generated by the safe shutdown or design basis earthquake F Load due to fluids with well-defined pressures and maximum heights, also relative

weight factor or factor of safety Fµ Factor that accounts for additional capacity provided by energy dissipation and

Ductility FC Capacity factor FCm Median capacity factor f Probability density function

fA Normalized stress due to sustained weight fm Flow stress of steel Fhoop Hoop stress or load effect FH Ratio of acceleration levels for HCLPF to the acceleration levels for the SSE ground

motion Fm Median factor of safety ff Stress corresponding to the failure strain εf fM Normalized stress due to mechanical loading fO Normalized stress due to OBE fP Normalized longitudinal stress due to internal pressure

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fPD Normalized longitudinal stress due to internal pressure for service limit D fPO Normalized longitudinal stress due to internal pressure coincident with OBE fPS Normalized longitudinal stress due to internal pressure coincident with SSE fS Normalized stress due to SSE fS(s) Stress probability density function F Relative weight factor FR Structural response factor FRm Median response factor that takes into account differences between the actual

response and the response computed from the designFst Margin of strength over design strength g Performance function G Specific gravity of contents Gk,j,inf Non-favorable permanent loads Gk,j,sup Favorable permanent loads L1 Live load that includes also the weight of fluids in pipes Li Load LL Vehicular live load without wind MA Resultant moment loading on cross section due to weight and other sustained loads,

in-lb = [ ] 2/1222zByBxB MMM ++

MB Resultant moment on a cross-section due to operating basis earthquake MCL Collapse moment MD Maximum moment due to uniform load My Limit-load moment capacity of the cross section, My = SyZp = MLL Mexp Experimental moment capacity Mo Resultant bending moment due to OBE MS Resultant bending due to SSE P Internal pressure, also design pressure Pα Maximum differential pressure load generated by the postulated accident Pf Probability of failure Po Service pressure Pa Allowable pressure PD Pressure occurring coincident with the reversing dynamic load Pmax Peak operating pressure, psi QE Effect of horizontal seismic (earthquake-induced) forces Qk,1 Dominant variable action Qk,i Non-dominant variable actions

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R Strength or resistance of a structural component Rα Pipe and equipment reactions generated by the postulated accident, including Pα

οR R0 pipe reactions during normal operating, start-up, or shutdown conditions, based on the most critical transient or steady-state condition

S Maximum allowable stress for the material at the design temperature SA Stress due to sustained weight SDS Design spectral response acceleration at short periods Sh Material allowable stress at temperature consistent with loading Sm Maximum allowable stress intensity for the material at the design temperature SM Stress due to mechanical loading SO Stress due to OBE SOL Stress due to operating conditions SS Stress due to SSE SSL Stresses due to pressure, weight, and other sustained loads Su Ultimate stress of material SUL Stress due to upset loading conditions Sy Material yield strength at temperature consistent with loading t Specified or actual wall thickness of pipe tm Minimum thickness of pipe tn Nominal wall thickness WA Water load and stream pressure xi Ratio of experimental to nominal pressure calculated by the model i X Vector of basic random variables (X1, X2, ..., Xn) Xf A factor equals 2.326 for 1% probability of failure and 3.090 for 0.1% probability of

failure y Coefficient having a value of 0.4, except that for pipe with Do / tm ratio less than 6,

( )o/ Dddy +=

Z Elastic section modulus Zp Plastic section modulus

ρ Reliability factor Φ(.) Cumulative probability distribution function of the standard normal distribution εf Failure strain εR Random variable with unit medians representing the inherent randomness about the

median εU Random variable with unit medians representing the uncertainty in the median value φ-1 Inverse of the standard Gaussian cumulative distribution function

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Development of Reliability-Based LRFD Methods for Piping – Research and Development Report

1

1. Introduction

Current American Society of Mechanical Engineers (ASME) nuclear codes and standards rely primarily on deterministic and mechanistic approaches to design. The American Institute of Steel Construction and the American Concrete Institute, among other organizations, have incorporated probabilistic methodologies into their design codes. ASME nuclear codes and standards could benefit from developing a probabilistic, reliability-based, design methodology. This report addresses this need of developing such a methodology for piping. This chapter provides general background material, discusses an earlier pilot project to develop probabilistic design procedures for Section III, and identifies current risk-informed regulatory, ASME, and industry initiatives.

1.1. Background All designs involve uncertainty - uncertainty in loading conditions, in material characteristics, in accuracy of analytical models, in geometric properties, in fabrication and installation precision, in examination and inspection results, and in actual usage. Traditionally, engineering design methodology addresses uncertainty through deterministic safety factors, which could lead to inconsistent reliability levels and sometimes overly conservative designs and do not provide insight into the effects of individual uncertainties and the actual margin of safety. Current Section III rules are based on non-probabilistic engineering design methods. In recent years, probabilistic design analysis methods have been developed to address uncertainty and randomness through statistical modeling and probabilistic analysis (Cornell 1969, Hasofer and Lind 1974, Rackwitz and Fiessler 1978, Madsen, Krenk and Lind 1986, Melchers 1987, Ross 1988). Historically, the computational resources to accurately capture uncertainties and estimate probability of failure made application of these methods impractical. Current computing resources and the availability of probabilistic design tools provide an environment for applying probabilistic analysis and optimization effectively to even complex design problems. The ASME Section XI and OM codes have adopted risk-informed methodologies for inservice inspection, preventive maintenance, and repair and replacement decisions. The American Institute of Steel Construction (AISC) and the American Concrete Institute (ACI) have incorporated probabilistic methodologies into their design codes. It is proposed that Section III should undergo a planned evolution integrating it with Section XI that would provide a risk-informed approach across a facility lifecycle - encompassing design, construction, operation, maintenance, and closure.



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As discussed previously, Section III provides rules for component design including pressure vessels, pumps, valves, piping, piping components and supports. These rules involve varying degrees of complexity to account for primary, secondary and peak stresses, as well as a cumulative fatigue usage factor. The deterministic safety factors used by Section III for the loads (pressure, bending moments, etc.) and the allowable stresses are based on many decades of experience as well as supporting test data. The Section III design methodology of deterministic safety factors coupled with allowable stresses is also referred to as working stress design. Section III rules have worked very well for many years, but the reliability of a component designed to Section III can vary considerably. In order to promote consistency and to allow more efficient designs, it would be desirable to design components to expected levels of reliability (or failure probability), with the target level of reliability dependent on specific consequences of failure. This would allow the development of risk-informed design methods, and may result in potential cost savings.

1.2. History of Reliability-based Design The concept of using the probability of failure as a criterion for structural design can be credited to the Russians N. F. Khotsialov and N. S. Streletskii, who presented the idea in the late 1920s. The first exposition of the idea in the United States was made by A. M. Freudenthal in 1947. Considerable interest by many industries and engineering disciplines has evolved in developing reliability-based design codes. Reliability-based design codes using an LRFD format were developed using first-order second-moment reliability methods (Ayyub and McCuen 2003), such as by the American Institute of Steel Construction (AISC 1994, Ravindra and Galambos 1978, Galambos and Ravindra 1978) and by the American Concrete Institute (ACI). An effort was made by the American National Standards Institute (ANSI) to develop probability-based load criteria for buildings (Ellingwood et al 1982a and 1982b) that was published as ASCE 7-93 (ASCE 1993). The American Petroleum Institute (API) extrapolated LRFD technology for its use in fixed offshore platforms (API 1989, Moses 1985 and 1986). Other efforts which provide comprehensive summaries of implementation of modern probabilistic design theory into design codes include those of Siu et al. (1975), Allen (1975) and MacGregor (1976) for the National Building Code of Canada (1977) and the Canadian Standard Association (CSA 1974), Ellingwood, et al (1980) for the National Bureau of Standards, CEB (1976), ASCE (1982), and the CIRIA 63 (1977) report. Ayyub et al. (1995), Ayyub and Atua (1996), Ayyub and Assakkaf (1997), and Ayyub et al. (1998) developed LRFD rules for ship structures for the US Navy. The American Association of State Highway and Transportation Officials (AASHTO) LRFD Bridge Design and Construction Specifications (1994) resulted in part from work in the National Cooperative Highway Research Program (NCHRP) Project 12-33 on bridge girders (Nowak 1993). For example, the procedures for steel structures were developed in the 1980s and 90s and are described in (Manual of Steel Construction 1995, Novak and Lind, White and Ayyub, 1987). Live and dead loads on structures are considered, e.g., snow, seismic, etc., and deterministic multipliers on the loads are specified. These multipliers are based on an underlying probabilistic load definition and analysis of the structural components, and the multipliers are based on a given reliability. In the end, the design process looks much like it did prior to LRFD, but there is



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now an analytical basis for the factors employed where there was none before, and reliability can be factored into the selection of these factors. The need and desirability of LRFD is well summarized in the overview of Modern Steel Construction (Manual of Steel construction 1995). They summarize as follows: “...LRFD is a modern and technologically superior design specification. Its direct representation of ultimate structural behavior is especially relevant for seismic design, design of frames with partially restrained connections, and composite systems design. It offers engineers the opportunity to innovate in the analysis and design of highly reliable and competitive steel structures and serviceability criteria under appropriate combinations of gravity and lateral loads. In this way, LRFD is consistent with the prevailing trend toward limit-states design in all materials, both domestically and internationally.” In addition to the AISC approach mentioned above, the LRFD methodology has been routinely used worldwide in reinforced concrete design since the 1960s. It has also been used in the design of offshore platforms, bridges and aircraft since the 1980s (Freudenthal 1947, Moses and Stevenson 1970, Turkstra 1970, Rowe 1977, Ellingwood, et al 1980, Shinozuka and Yao 1981). While the literature is extensive on the development and use of reliability based structural design in these other structural-mechanical design fields, it has seen very little application to Section III components. References such as Stevenson 1979, Groman, Bergman and Stevenson 1980, Adams and Stevenson 1997, Ghiocel, Wilson and Stevenson 1995 provide examples for such components, but to date essentially no applications to Section III development or requirements have been made.

1.3. Benefits of Reliability-based Design The many advantages and benefits of using reliability-based design methods include the following: • They provide the means for the management of uncertainty in loading, strength, and

degradation mechanisms. • They provide consistency in reliability. • They result in efficient and possibly economical use of materials. • They provide compatibility and reliability consistency across materials, such as steel grades,

aluminum and composites. • They allow for future changes as a result of gained information in prediction models, and

material and load characterization. • They provide directional cosines and sensitivity factors that can be used for defining future

research and development needs. • They allow for performing time-dependent reliability analysis that can form the bases for life

expectancy assessment, life extension, and development of inspection and maintenance strategies.

• They are consistent with other industries as presented by AISC, ASHTO, ACI, API, and ASME.

• They allow for performing system reliability analysis.



4

The basic cost advantage of using LRFD design methodology for piping is the expected reduction in the quantity of lateral seismic restraints. Experience with LRFD in building structures suggests a 3 to 5 percent savings in construction costs. Based on WRC Bulletin 426 (Adams and Stevenson 1997) and the calculation included in Appendix B, this translates into a $1.7 M (1990 Dollars) savings for a 0.6g SSE, 1100 MWe plant. Life cycle cost savings would be even greater due to reduced inspection and maintenance during plant operation. Additional benefits of using LRFD for piping include: • Establishes a more balanced design in load combinations which reduces over-design in one

area (e.g., seismic) which can reduce reliability for other loads (e.g., thermal). • Reduced thermal and seismic displacement stresses • More efficient design of piping, supports and attached components • Major benefit - establishes reliability-based design, which supports the objective of

establishing a risk-informed system design approach. There are a number of risk-informed regulatory, ASME, and industry initiatives, which are establishing a foundation for risk- informed design.

1.4. Challenges in Developing Load and Resistance Factor Design for Piping The primary challenges in developing reliability-based LRFD methods are related to the characterization of failure modes to define limit-states, assessing implicit reliability levels in current practice, and assigning target reliability levels for the identified limit-states. While multiple failure modes may exist, the present practice in accordance with the current code implicitly focuses on designing the piping components against three primary failures that can be referred to as plastic instability, fatigue, and ratcheting. Safeguard against these failures is implicitly built into the various design equations depending upon the type of component and the classification of piping such as Class 1, 2, 3, etc. For example, one of the primary code equations considers primary stresses due to internal pressure, dead load, and inertial seismic loads for ensuring safety against failure due to plastic instability. Selection of target reliability levels for specified limit-states requires calibration of the existing code supplemented by expert opinion. Calibration is the process of evaluating reliability levels in piping components that are designed using the current ASME code criteria and incorporating the insights gained from observed failures, if any. Another challenge relates to the development of a design format that would not only lend itself for transitioning the current design practice to LRFD but also provide flexibility for future enhancements targeted towards a system-based risk-informed design.



5

1.5. Piping Pilot Project In the late 1990s, the ASME Codes and Standards Subcommittee on Nuclear Power and the ASME Center for Research and Technology Development conducted Phases 1 and 2 of a multi-phase pilot project to develop probabilistic design procedures for Section III piping. The objective was to develop probabilistic design procedures that will reduce conservatism, increase safety, reduce life-cycle costs, and provide consistent levels of reliability. This pilot project was successful in calculating probability of piping failures based upon deterministic code rules (Barnes, Harris, Hill, and Stevenson 2000). Phase I investigations found that pipe cross-section failure probabilities are generally below 10-6 per year, when credit is taken for the probability of the seismic event occurring. Phase II analysis used the example from Phase I modified to account for cyclic stresses in hot piping. In this instance, the cumulative leak probabilities were computed to be quite high, as high as 3x10-3 per year. The double-ended pipe break probabilities were several orders of magnitude lower. Results were presented for the location of the design point, which opens the way to a LRFD format for Code evaluations. This provides a basis for design procedures based on a target reliability level. This pilot project demonstrated that adapting LRFD and other risk-informed approaches to the design of piping and other components will put the design procedure on a quantitative analytical foundation and allow the designer to select a design with reliability commensurate with the risk associated with operation of the pressurized component. It will also allow quantification of the reliability that is useful in estimation of lifecycle costs.

1.6. Objectives The primary objective of this study was to develop the technical basis for reliability-based, load and resistance factor design of ASME Section III, Class 2/3 piping for primary loading, i.e., pressure, deadweight, seismic, etc. Achieving this objective should also result in proof of concept in that LRFD can be used in the design of piping, and could achieve consistent reliability levels. Also, the results from this project could form the basis for code cases, and additional research for piping secondary loads. The project produced LRFD methodology, formats and sample partial safety factors - with examples. This report includes a definition of the LRFD methodology, loads and load combinations, failure modes and limit-state formats, and a preliminary characterization of basic random variables. This document provides a development for reliability-based load and resistance factor design methods for piping. The methods are based on structural reliability theory and can be either as direct reliability-based design or in a load and resistance factor design (LRFD) format. The resulting design methods are to be referred to as the LRFD guidelines and criteria for piping. They were developed according to the following requirements: (1) analysis of strength and load effects, (2) building on conventional codes, (3) nominal strength and load values, and (4) achieving target reliability levels. Partial safety factors are evaluated, and examples are provided.



6

1.7. Organization The report consists of eight chapters and two appendices as shown in Figure 1-1. Chapter 1 consists of the introduction and an objective statement. Chapter 2 provides the needed theoretical background for performing reliability-based design and analysis. Chapter 3 gives a summary of the design loads, such as weight, internal pressure, and earthquake, and also provides a summary of recommended load combinations for this study. Failure modes and limit states for piping systems are provided in Chapter 4. Chapter 5 provides statistical information on basic random variables that are relevant to piping. Chapter 6 quantifies the uncertainties for different models that can be used for the calculation of piping burst pressure and bending capacity. Chapter 7 provides the calculation of the partial load and resistance factors and a comparative example of a pipe design according to the present ASME code and the proposed LRFD methodology. Chapter 8 summarizes the conclusions derived from the present study and furthermore gives recommendations for future work. Appendix A summarizes limit states that are contained in the current ASME code and Appendix B provides information for common steels used for nuclear piping.



7

CHAPTER 1INTRODUCTION

BackgroundHistory of Reliability-based DesignBenefits of Reliability-based DesignChallengesPiping Pilot ProjectObjectiveOrganization

CHAPTER 3LOADS AND LOAD COMBINATIONS

Primary LoadsLoad Combinations

CHAPTER 4FAILURE MODES AND LIMIT STATES

FOR PIPESFailure CriterionPerformance CriterionExisting Code EquationsPerformance Functions

CHAPTER 5BASIC RANDOM VARIABLES FOR PIPING

Statistical Characteristics of Random variablesStrength VariablesLoad VariablesReference Statistical Data

CHAPTER 7CALCULATION OF PARTIAL SAFETY

FACTORS FOR FUTURE WORKComparative Example Calculations for a Pipe SegmentCalculations of Partial Safety FactorsSample LRFD Guidelines and Rules for Piping

APPENDIX ASELECTED LIMIT STATES IN ASME

CODE

CHAPTER 2RELIABILITY-BASED DESIGN AND ANALYSIS

IntroductionDirect Reliability-Based DesignLoad and Resistance Factor Design (LRFD)Performance FunctionsFirst-Order Reliability Method (FORM)Example

APPENDIX BSTEEL USED IN

ASME CODE

CHAPTER 6MODELING UNCERTAINTY

Basic Capacity of Pipes Using Different Models and Comparison

CHAPTER 8CONCLUSIONS AND

RECOMMENDATIONS FOR FUTURE WORK Figure 1-1. Organization of the Report



8

2. Reliability-Based Design and Analysis

2.1. Introduction Structural design has been moving toward a more rational and probability-based design procedure referred to as limit states design. Such a design procedure takes into account more information than deterministic methods in the design of structural components. This information includes uncertainties in the strength of various structural elements, in loads, and modeling errors in analysis procedures. Reliability-based design formats are more flexible and rational than working stress formats because they provide consistent levels of safety over various types of structures. The development of a methodology for reliability-based design for piping requires the consideration of the following three components (Ang and Tang 1990, Ang 1984, Ellingwood 1980, Mansour et al. 1996, Ayyub and McCuen 2003): (1) loads, (2) structural strength, and (3) methods of reliability analysis. There are two primary approaches for reliability-based design: (1) direct reliability-based design and (2) load and resistance factor design (LRFD). The two approaches are shown in Figure 2-1. The three components of the methodology are also shown in this figure in the form of several blocks for each. In addition, the figure shows their logical sequence and interaction. The direct reliability-based design approach can include both Level 2 and/or Level 3 reliability methods. Level 2 reliability methods are based on the moments (mean and variance) of random variables, whereas, Level 3 reliability methods use the complete probabilistic characteristics of the random variables. In some cases, Level 3 reliability analysis is not possible because of lack of complete information on the full probabilistic characteristics of the random variables. Also, computational difficulty in Level 3 methods sometimes discourages their uses. The LRFD approach is called a Level 1 reliability method. Level 1 reliability methods utilize partial safety factors (PSF) that are reliability based; but the methods do not require explicit use of the probabilistic description of the variables. The two reliability-based design approaches start with the definition of a structural system or element. Then, the general dimensions and arrangements, structural member sizes, and details need to be assumed. The weight of the structure can then be estimated to ensure its conformance to a specified limit. Using assumed load effects, the two methods can then be used to design or analyze the structural system or element under question. The two approaches, beyond this stage, proceed in two different directions.



9

Both the direct reliability-based design and the LRFD approaches require defining performance functions that correspond to the limit states for significant failure modes (Mansour et al. 1996). They also require the definition of a set of target reliability levels. These levels can be set based on implied levels in the currently used piping design practice with some calibration, or based on cost benefit analysis. Figure 2-2 provides a suggested methodology customized for the needs of this project.

oββ ≥ Is ∑≥ ii LR γφ

Figure 2-1. Reliability-based Design of a Structural Element



10

Definition of PerformanceMeasures

System Definition:(1) Identification of failure modes,(2) Identification of environmentalloads, factors and strength models(3) uncertainty analysis, and(4) Identification of operational andmaintenance factors

Test Results andReported Data

Uncertainty Analysis andModeling

Reliability Analysis

Parametric Analysis

QualitativeRisk Analysis

Selection of Target ReliabilityLevels

Computation of Partial SafetyFactors

LRFD Guidelines for PipingDesign

System Redefinition(if needed)

Definition ofData Needs

Figure 2-2. Methodology for Developing Reliability-Based LRFD Methods

2.2. Direct Reliability-Based Design The direct reliability-based design requires performing spectral analysis and extreme analysis of the loads. Also, linear or nonlinear structural analysis can be used to develop a stress frequency distribution. Then, stochastic load combinations can be performed. Linear or nonlinear structural analysis can then be used to obtain deformation and stress values. Serviceability and strength failure modes need to be considered at different levels of a structural system. The



11

appropriate loads, strength variables, and failure definitions need to be selected for each failure mode. Using reliability assessment methods such as FORM, reliability indices β’s for all modes at all levels need to be computed and compared with target reliability indices '

0β s. The relationship between the reliability index β and the probability of failure is given by

Pf = 1 - Φ(β) (2-1)

where Φ(.) = cumulative probability distribution function of the standard normal distribution, and β = reliability (safety) index. It is to be noted that Eq. 2-1 assumes all the random variables in the limit state equation to have normal probability distribution and the performance function is linear. However, in practice, it is common to deal with nonlinear performance functions with a relatively small level of linearity (Ang and Tang 1990, and Ayyub and McCuen 2003). If this is the case, then the error in estimating the probability of failure Pf is very small, and thus for all practical purposes, Eq. 2-1 can be used to evaluate Pf with sufficient accuracy.

2.3. Load and Resistance Factor Design (LRFD) The second approach (LRFD) consists of the requirement that a factored (reduced) strength of a structural component is larger than a linear combination of factored (magnified) load effects as given by

∑≥n

iii LR

1= γφ (2-2)

In this approach, load effects are increased, and strength is reduced, by multiplying the corresponding characteristic (nominal) values with factors, which are called strength (resistance) and load factors, respectively, or partial safety factors (PSF’s). The characteristic value of some quantity is the value that is used in current design practice, and it is usually equal to a certain percentile of the probability distribution of that quantity. The load and strength factors are different for each type of load and strength. Generally, the higher the uncertainty associated with a load, the higher the corresponding load factor; and the higher the uncertainty associated with strength, the lower the corresponding strength factor. These factors are determined probabilistically so that they correspond to a prescribed level of safety. It is also common to consider two classes of performance function that correspond to strength and serviceability requirements. The difference between the allowable stress design (ASD) and the LRFD format is that the latter use different safety factors for each type of load and strength. This allows for taking into consideration uncertainties in load and strength, and to scale their characteristic values accordingly in the design equation. ASD (or called working stress) formats cannot do that because they use only one safety factor. Piping designers can use the load and resistance factors in limit-state equations to account for uncertainties that might not be considered properly by deterministic methods without explicitly performing probabilistic analysis.

2.4. Performance Functions As stated earlier, reliability-based analysis and design procedures start with defining performance functions that correspond to limit states for significant failure modes. In general,



12

the problem can be considered as one of supply and demand. Failure occurs when the supply (i.e., strength of the system) is less than the demand (i.e., loading on the system). A generalized form for the performance function for a structural system is given by

LRg −=1 (2-3)

where g1 = performance function, R = strength (resistance) and L = loading in the structure. The failure in this case is defined in the region where g1 is less than zero, or R is less than L, that is

LRg << or 0.01 (2-4)

As an alternative approach to Eq. 2-3, the performance function can also be given as

LRg =2 (2-5)

where, in this case, the failure is defined in the region where g2 is less than one, or R is less than L, that is

LRg << or 0.12 (2-6)

If both the strength and load are treated as random variables, then the reliability-based design and analysis can be tackled using probabilistic methods. In order to perform a reliability analysis, a mathematical model that relates the strength and load needs to be derived. This relationship is expressed in the form of a limit state or performance function as given by Eq. 2-3 or Eq. 2-5. Furthermore, the probabilistic characteristics of the basic random variables that define the strength and loads must be quantified. Because the strength R and load L are random variables, there is always a probability of failure that can be defined as

)( Prob)0.0( Prob 1 LRgPf <=<= (2-7)

or

)( Prob)0.1( Prob 2 LRgPf <=<= (2-8)

The probability of failure given by Eqs 2-7 and 2-8 correspond to the performance functions g1 and g2 of Eqs. 2-3 and 2-5, respectively. Figure 2-3 shows these two random variables.



13

Load Effect (L)

Strength (R)

Density Function

Origin 0 Random Value

Area (for g < 0) = Failure probability

Figure 2-3. Reliability Density Functions of Resistance R and Load L (Ayyub 2003)

2.5. First-Order Reliability Method (FORM) The First-Order Reliability Method (FORM) is a convenient tool to assess the reliability of a structural element in piping. It also provides a means for calculating the partial safety factors φ and γi that appear in Eq. 2-2 for a specified target reliability level β0. The simplicity of the first-order reliability method stems from the fact that this method, beside the requirement that the distribution types must be known, requires only the first and second moments; namely the mean values and the standard deviations of the respective random variables. Knowledge of the joint probability density function (PDF) of the design basic variables is not needed as in the case of the direct integration method for calculating the reliability index β. Even if the joint PDF of the basic random variables is known, the computation of β by the direct integration method can be a very difficult task. In design practice, there are usually two types of limit states: the ultimate limit states and the serviceability limit state. Both types can be represented by the following performance function:

) ..., , ,()( 21 nXXXgg =X (2-9)



14

R’

L’

β0

MostProbable

Failure Point

Limit State in ReducedCoordinates

0.0) ,( =LRg

nLLRg

g

−−−=

=

...or

Effect Loads -Strength

1

Failure occurs when g <0.0

Figure 2-4. Space of Reduced Random Variables Showing the Reliability Index and

the Most Probable Failure Point (Ayyub 2003)

in which X is a vector of basic random variables (X1, X2, ..., Xn) for the strengths and the loads. The performance function g(X) is sometimes called the limit state function. It relates the random variables for the limit-state of interest. The limit state is defined when g(X) = 0, and therefore, failure occurs when g(X) < 0 (see Figure 2-4). The reliability index β is defined as the shortest distance from the origin to the failure surface at the most probable failure point (MPFP) as shown in Figure 2-4. The development of FORM over the years resulted in many variations of the method. These variations include such methods as the first-order second moment (FOSM) and the advanced first-order second moment (AFOSM). Both of these methods use the information on first and second moments of the random variables, namely, the mean and standard deviation (or the coefficient of variation, COV) of a random variable. However, the FOSM method ignores the distribution types of the random variables, while AFOSM takes these distributions into account. Clearly, the AFOSM method as the name implies produces more accurate results than FOSM. Nevertheless, FOSM can be used in many situations of preliminary design or analysis stages of a structural component, where the strength and load variables are assumed to follow a normal distribution and the performance function is linear. In these cases, the results of the two methods are essentially the same. The importance of FORM is that it can be used in structural analysis to compute the reliability index β, and also to determine the partial safety factors (PSF’s) in the development of various design codes. The reliability index was defined earlier as shortest distance from the origin to the failure line as shown in Figure 2-4. For normal distributions of the strength and load variables, and linear performance function, β can be computed using the following equation:



15

22LR

LR

σσ

µµβ

+

−= (2-10)

where µR = mean value of strength R, µL = mean value of the load effect L, σR = standard deviation of strength R, and σR = standard deviation of the load effect L. The reliability index according to this definition is commonly referred to as the Hasofer-and-Lind (1974) index. The important relationship between the reliability index β and the probability of failure Pf is given by Eq. 2-1. The nominal values of partial safety factors (PSF’s) according to the linear performance function given by Eq. 2-3, and for normal distributions of the strength and load variables can be calculated using the following two expressions as suggested by Haldar and Mahadevan (2000): For single load case:

RR

R

S δεβδ

φ−−

=11 (2-11)

LL

LL S δ

εβδγ

++

=11 (2-12)

where

LR

LR

σσσσ

ε++

=22

(2-13)

and in which, σR = standard deviation of strength R, σL = standard deviation of the load effect L, δR = coefficient of variation (COV) of the strength R, δL = COV of the load effect L, and SR and SL are parameters used by some classification societies and the industry to approximate the nominal values of the strength and the load effect, respectively. Typical values for SR and SL range from 1 to 3. For multiple load case: The nominal reduction factor φ of strength can still be computed from Equation 2-11. However, the nominal load factors γi’s for the ith load effect become (Haldar and Mahadevan 2000):

i

i

LiL

Lni S δ

βδεγ

+

+=

11 (2-14)

where

222

222

21

21

n

n

LLL

LLLn σσσ

σσσε

+++

+++=

L

L (2-15)



16

in which, ( 222121

,,,nLLL σσσ L ) = standard deviations of the load effects (L1, L2, …, Ln ) and

iLδ = COV of the load effect Li, and

iLS = parameter used to approximate the nominal value of load effect Li. In general, the nominal value of the strength is less than the corresponding mean value, and the nominal value of the load effect is larger than its mean value. For example, if both SR and SL are equal to 2, the nominal value of R would be 2 standard deviations below the mean, and the nominal value for L would be 2 standard deviations above its mean value. If SR and SL have zero values, then Eqs. 2-11 and 12 essentially result in the mean values of the partial safety factors φ and Lγ , respectively. The nominal values of partial safety factors can be used in LRFD design format of the type

nnn LLLR γγγφ +++≥ L2211 (2-16)

For purposes of design, this relationship needs to be satisfied. It is to be noted that Eqs. 2-11 and 12 apply only for linear performance function with two variables (strength and one load effect) having normal distributions, while Eq. 2-14 applies for the multiple linear case. For a general case of a nonlinear function with multiple random variables having different distribution types (i.e., lognormal, Type I, etc.), an advanced version of FORM should be used. Detailed algorithms of the advanced FORM version as well as procedures for calculating and calibrating the partial safety factors using FORM are provided in Section 2.5.1. It is to be noted that the version of FORM given in this section is the advanced first-order second moment (AFOSM). This version of FORM applies for a general case of nonlinear performance function and for any distribution type of the random variables.

2.5.1. Algorithm for First-Order Reliability Method As indicated earlier, the basic approach to develop a reliability-based strength standard is to determine the relative reliability of designs based on current practice. In order to do that, a reliability assessment of existing structural components of piping is needed to estimate a representative value of the reliability index β. The first-order-reliability method is very well suited to perform such a reliability assessment. The following computational steps, as outlined by Ayyub and McCuen (2003), for determining β using the FORM method are provided: 1. Assume a design point ∗

ix and obtain ∗'ix using the following equation:

i

i

X

Xii

xx

σ

µ−=

∗∗' (2-17)

where βα ∗∗ −= i'ix ,

iXµ = mean value of the basic random variable, andiXσ = standard

deviation of the basic random variable. The mean values of the basic random variables can be used as initial values for the design points. The notations ∗x and ∗'x are used respectively for the design point in the regular coordinates and in the reduced coordinates.

2. Evaluate the equivalent normal distributions for the non-normal basic random variables at the design point using the following equations:



17

( ) NXX

NX )(xFx σµ ∗−∗ Φ−= 1 (2-18)

and

( )( )

)(xf

)(xF

X

XNX ∗

∗−Φ=

1σ (2-19)

where =NXµ mean of the equivalent normal distribution, =N

Xσ standard deviation of the

equivalent normal distribution, =∗ )(xFX original cumulative distribution function (CDF) of Xi evaluated at the design point, fX(x∗) = original probability density function (PDF) of Xi evaluated at the design point, Φ(⋅) = CDF of the standard normal distribution, and φ(⋅) = PDF of the standard normal distribution.

3. Compute the directional cosines ( ∗iα , i = 1,2, ..., n) using the following equations:

∑= ∗

∗∗

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=n

i i

ii

x

g

x

g

1

2

'

'

∂

∂

∂

∂

α for i = 1, 2, ..., n (2-20)

where

NX

iiix

gxg σ

∂∂

∂

∂

∗∗⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛' (2-21)

4. With NX

NXi ii

σµα and , ,∗ now known, the following equation can be solved for the root β:

0)( ..., ),(111

=⎥⎦⎤

⎢⎣⎡ −− ∗∗ βσαµβσαµ N

XXNX

NXX

NX nnn

g (2-22)

5. Using the β obtained from step 4, a new design point can be obtained from the following equation:

βσαµ NXi

NXi ii

x ∗∗ −= (2-23)

6. Repeat steps 1 to 5 until a convergence of β is achieved. The reliability index is the shortest distance to the failure surface from the origin in the reduced coordinates as shown in Figure 2-4.

The important relation between the probability of failure and the reliability index is given by Eq. 2-1.



18

2.5.2. Procedure for Calculating Partial Safety Factors (PSF) Using FORM

The first-order reliability method (FORM) can be used to estimate partial safety factors such as those found in the design format of Eq. 2-2. At the failure point ( ∗∗∗

nLLR ..., , , 1 ), the limit state of Eq. 2-2 is given by

0...1 =−−−= ∗∗∗nLLRg (2-24)

or, in a general form

0) ,..., ,()( 21 == ∗∗∗nxxxgg X (2-25)

For given target reliability index β0, probability distributions and statistics (means and standard deviations) of the load effects, and coefficient of variation of the strength, the mean value of the resistance and the partial safety factors can be determined by the iterative solution of Eqs. 2-17 through 2-23. The mean value of the resistance and the design point can be used to compute the mean required partial design safety factors as follows:

R

Rµ

φ∗

= (2-26)

iL

ii

Lµ

γ∗

= (2-27)

2.5.3. Determination of a Strength Factor for a Given Set of Load Factors

In developing design code provisions for piping, it is sometimes necessary to follow the current design practice to insure consistent levels of reliability over various types of pipe structures. Calibrations of existing design codes is needed to make the new design formats as simple as possible and to put them in a form that is familiar to the users or designers. Moreover, the partial safety factors for the new codes should provide consistent levels of safety. For a given reliability index β and probability characteristics for the resistance and the load effects, the partial safety factors determined by the FORM approach might be different for different failure modes for the same structural component. For this reason, calibration of the calculated partial safety factors (PSF’s) is important in order to maintain the same values for all loads at different failure modes. Normally, the calibration is performed on the strength factor φ for a given set of load factors. The following algorithm can be used to accomplish this objective: 1. For a given value of the reliability index β , probability distributions and statistics of the load

variables, and the coefficient of variation for the strength, compute the mean of the strength R using the first-order reliability method as outlined in the Section 2.5.1.

2. With the mean value for R computed in step 1, the partial safety factor φ can be revised as follows:



19

R

n

iLi i

µ

µγφ

∑== 1 (2-28)

where µ Li and µR are the mean values of the loads and strength variables, respectively; and γi, i

= 1, 2, ..., n, are the given set of load factors.

2.6. Examples This section contains four examples to demonstrate the computation of the partial safety factors.

2.6.1. Example I Given: The fully plastic flexural capacity of a beam section can be estimated as Sy Z, where

Sy = yield strength of the material (steel) of the beam and Z = plastic section modulus. If the simply supported beam shown in Figure 2-5 is subjected to mean values of distributed dead and live loads: wD and wL,, respectively; and if Z and L are assumed to be constant, develop the nominal and mean partial safety factors for this beam and the corresponding LRFD-based design formula for a target reliability index of 3. Assume that the nominal values are one standard deviation below the mean for the strength, and one standard deviation above the corresponding mean values for both the dead and live loads. The probabilistic characteristics of the basic random variables are as provided in Table 2-1.

L

wD + wL

Figure 2-5. Beam Design for Example I

Table 2-1. Probabilistic Characteristics of Random Variables for the Beam Problem

Probabilistic Characteristics Variable µ σ Distribution

Sy 248 MPa 12.4 MPa Normal Z 4588 cm3 n/a n/a L 915 cm n/a n/a

wD 0.315 kN/cm 0.044 kN/cm Normal wL 0.438 kN/cm 0.16 kN/cm Normal

n/a = not applicable

Solution: For this analysis, the following linear performance function is considered: LDR MMMg −−=

The plastic moment capacity of the beam Mp can be considered the mean moment capacity, thus



20

LDy MMZSg −−=

m-kN 9.56248

4.128.1137

m-kN 8.1137

)10248)(104588( 36

=⎟⎠⎞

⎜⎝⎛=

=

××=== −

R

yPR ZSMM

σ

05.0

2484.12

==Rδ

36.0438.016.0

14.0315.0044.0

===

===

L

LL

D

DD

µσ

δ

µσ

δ

For simply supported beam, the applied maximum moments at its mid-span can be computed as follows:

m-kN 7.329 )100(8

)915(315.08

22

=

==LwM D

D

m-kN4.458 )100(8

)915(438.08

22

=

==Lw

M LL

Denoting the total moment due to applied dead and live loads as M, its mean, standard deviation, and COV can be estimated: µM = 329.7 + 458.4 = 788.1 kN - m

m-kN 02.165)36.0(4.458and m,-kN 16.46)14.0(7.329

==

==

L

D

M

M

σ

σ

Therefore, ( ) ( ) m-kN 4.17102.16516.46 22 =+=Mσ

22.0

1.7884.171

==Mδ

Using Eqs. 2-13 and 15, the parameters ε and εn are calculated as follows:

( ) ( )79.0

4.1719.564.1719.56 22

=++

=ε

( ) ( )81.0

02.16516.4602.16524.16.46 22

=+

+=nε

According to Eqs. 2-11 and 2-14, and noting that SR = SD = SL = 1 for both the strength and load effects, the nominal partial safety factors (PSF’s) are obtained as follows:



21

( )( )( )( ) 93.0

05.01105.0379.01

=−

−=φ

( ) ( )11.1

)14.0)(1(114.0)3(81.079.01

=+

+=Dγ

( )24.1

)36.0)(1(136.0)3)(81.0(79.01

=+

+=Lγ

Thus, the LRFD-based design formula is given by LDR 24.111.193.0 +≥

The mean values of the partial safety factors can be found using Equations 2-11 and 2-14, with SR = SD = SL = 0. The results are:

88.0=φ

27.1=Dγ

69.1=Lγ

2.6.2. Example II Given: Develop the mean values of partial safety factors for the simply supported beam of

Example I using the probabilistic characteristics for the random variables as provided in Table 2-2. Table 2-2. Probabilistic Characteristics of Random Variables for Example I

Probabilistic Characteristics Variable µ σ Distribution Type

Sy 248 MPa 12.4 MPa Lognormal Z 4588 cm3 n/a n/a L 915 cm n/a n/a

wD 0.315 kN/cm 0.044 kN/cm Normal wL 0.438 kN/cm 0.16 kN/cm Type I


Solution: In this example, we note that the distribution types of the random variables are no longer normal. We have a mixture of distributions for these variables. Therefore, the simplified methods of this section cannot apply directly even though the performance function is the same, that is

LDy MMZSg −−= To compute the mean values of the partial safety factors, the general procedure of FORM, as outlined in Section 2.5.1, should be utilized. The results are as follows:

97.0=φ 05.1=Dγ 63.2=Lγ



22

2.6.3. Example III Given: A Class 2 straight pipe has a constant outside diameter of 6.625 in, constant

thickness of 0.28 in and constant length of 36 in. The sustained weight of pipe wA is uncertain with a mean value of 2.7 lb/in. The service pressure P is also uncertain with a mean value of 1500psig. The pipe is made of austenitic stainless steel with mean yield strength 39.6ksi. As shown in Figure 2-6, the pipe segment is simply supported. Develop the nominal and mean partial safety factors for this pipe and the corresponding LRFD-based design formula for a target reliability index of 2.5. Assume that the nominal values are one standard deviation below the mean for the strength, and one standard deviation above the corresponding mean values for both the dead weight and pressure load. The probabilistic characteristics of the basic random variables are as provided in Table 2-3. Table 2-3. Probabilistic Characteristics of Random Variables for the Pipe Problem


Sy 39602psi 4752.24psi Normal Z 8.50in3 n/a n/a Do 6.625in n/a n/a t 0.28in n/a n/a L 36in n/a n/a

wA 2.7lb/in 0.405b/in Normal P 1500psi 67.5psi Normal


Figure 2-6. Pipe Design for Example III

Solution: The deterministic elastic section modulus, Z of the pipe is equal to:

350.8)625.6(32

)4065.64625.6(32

)44(inπ

oDiDoDπ

Z =−

=−

=

For the analysis, the following linear performance function for Service Limit A is considered:



23

AfpfRfg −−=

fR = 39602psi

12.039602

24.4752==Rδ

psi77.8872)28.0(4

)625.6(15004

===t

PDf o

P

psi27.3991500

5.6777.8872 =⎟⎠⎞

⎜⎝⎛=Pσ

045.01500

5.67==Pδ

psi46.51)5.8(8

)36(7.28

22

===ZLwf A

A

psi72.77.2

405.046.51 =⎟⎠⎞

⎜⎝⎛=Aσ

15.07.2

405.0==Aδ

Denoting the total stress acting on the pipe due to applied service pressure and sustained dead loads as f, its mean, standard deviation, and COV can be estimated:

psi23.892446.5177.8872 =+=fµ

( ) ( ) psi34.399272.7227.399 =+=fσ

045.023.8924

34.399==fδ

Using Eqs. 2-13 and 2-15, the parameters ε and εn are calculated as follows:

926.034.39924.4752

)34.399()24.4752( 22

=++

=ε

981.034.39972.7

)34.399()72.7( 22

=++

=nε

According to Eqs. 2-11 and 2-14, and noting that SR = SP = SA = 1 for both the strength and load effects, the nominal partial safety factors (PSF’s) are obtained as follows:



24

( )( )( )( ) 82.0

12.01112.05.2926.01

=−

−=φ

( )( )( )( ) 06.1

045.011045.05.2981.01

=+

+=Pγ

( )( )( )( ) 19.1

15.01115.05.2981.01

=+

+=Aγ

Thus, the LRFD-based design formula is given by PAR 06.119.182.0 +≥

The mean values of the partial safety factors can be found, using Eqs. 2-11 and 2-14, with SR = SD = SL = 0. The results are:

72.0=φ

11.1=Pγ

37.1=Aγ

2.6.4. Example IV Given: Develop the mean values of partial safety factors for the pipe of Example III, using

the probabilistic characteristics for the random variables as provided in Table 2-4. Table 2-4. Probabilistic Characteristics of Random Variables for Example IV


Sy 39602psi 4752.24psi Lognormal Z 8.50in3 n/a n/a Do 6.625in n/a n/a t 0.28in n/a n/a L 36in n/a n/a

wA 2.7lb/in 0.405b/in Normal P 1500psi 67.5psi Normal

n/a = not applicable Solution: In this example, we note that the distribution types of the random variables are no

longer all normal. We consider a lognormal distribution for the yield strength of pipe. Therefore, the simplified methods of this section cannot apply directly, even though the performance function is the same linear function, that is

ZLw

tPD

Sg Aoy 84

2

−−=

To compute the mean values of the partial safety factors, the general procedure of FORM, as outlined in Sections 2.5.1 and 2.5.2, should be utilized. The results are as follows:

75.0=φ



25

04.1=Pγ 00.1=Aγ



26

3. Loads and Load Combinations

As mentioned earlier, development of reliability-based load and resistance factor design for piping requires the consideration of the following three components: (1) loads, (2) structural strength, and (3) methods of reliability analysis. Obviously, the load and load combinations are of primary importance for such a development. Load and their combinations are to be defined in order to develop the necessary partial safety factor values (PSF’s) for use in an LRFD design equation such as Eq. 2-2. These factors are determined probabilistically so that they correspond to a prescribed level of reliability or safety. LRFD format uses different safety factors for each type of load. This allows for considering uncertainties in loads and scaling their characteristic values accordingly in the design equation. Historically, in the ASD (sometimes called working stress) format, individual load cases such as dead load, internal pressure, and earthquake load were simply added together and compared to an allowable stress value that is reduced by means of a factor of safety (FS). The ASD method cannot properly account for uncertainties in loads and load combinations because it uses only one safety factor. Typical loads that are used individually or in combination with each other include dead load, internal pressure, and seismic loading. Provisions for these loads and their combinations are found in almost all structural and building codes such as AISC, ACI, API, ASSHTO, ASCE-7, UBC, BOCA, and IBC. For comparison purposes, load combinations used in various classification agencies are listed and discussed in Section 3.2 of this chapter. This chapter also provides a definition for the primary loads that are of interest for this study. That is, the weight of the piping, the internal pressure and the seismic loads are presented as described in ASME Boiler and Pressure Vessel (BPV) Code, Section III, Division I (1992). Moreover, it gives the load combinations in an LRFD form that other major organizations utilize for the design of elements. Finally, combinations of primary loads for the design of nuclear piping are recommended.

3.1. Primary Loads Primary loads are the loads that cause primary principal stresses, shear stresses or bending stresses and which must satisfy the laws of equilibrium of external and internal forces and moments. Primary stress is not self-limiting. Therefore, as long as the load is applied, the stress is present and does not reduce with time or as deformation takes place. Primary stresses lead to gross deformations and finally to rupture. This study does not deal with any secondary loads like thermal loads, anchor movements or peak loads like fatigue. Figure 3-1 presents the loads that cause primary stresses, which are the loads that this study is dealing with. Nevertheless, this study does not include the repeating loads e.g. vibrations due to rotating equipment such as compressors, pumps, turbine drivers etc. NB-3622.3 suggests that piping shall be arranged and



27

supported so that vibration will be minimized. The following sections give information on the other loads shown in Figure 3-1.

Figure 3-1. Loads that Cause Primary Stresses

3.1.1. Dead Weight This load category includes the dead weight of piping and its contents. According to the ASME Boiler and Pressure Vessel Code, Section III, Division 1, the dead weight shall consist of the weight of the piping, insulation and other loads permanently imposed upon the piping. Also, the weight of the fluid being handled or of the fluid used for testing (and/or cleaning) should be included in the appropriate load combination. The fluid in this case is a part of the constant load for the operating conditions.

3.1.2. Internal Pressure In ASME Boiler and Pressure Vessel (BPV) Code, Section III, Division I (1992) the following types of pressure are defined: The Internal Design Pressure, P, psi (kPa). This pressure is used to calculate the minimum required wall thickness, tm, for all classes of piping. It is also the pressure that is used in the design conditions for the three classes. As stated in Paragraph NCA-2142.1 (a) of the ASME BPV Code (1992), the Design Pressure shall not be less than the maximum difference in pressure between the inside and outside of the pipe, or between any two chambers of a combination unit, which exists under the most severe loadings for which the Level A Service Limits are applicable.

Primary Loads

Static, (Sustained) Loads Repeating Loads

Weight Internal Pressure

Impact (Dynamic) Loads

Vibration

Occasional Accidental

Dynamic Reversing (e.g. OBE, Mechanical)

Dynamic Non-reversing (e.g. SRV activation)

LOCA SSE



28

Moreover, the Design Pressure shall include allowances for pressure surges, control system error, and system configuration effects such as static pressure heads. The Allowable Pressure, Pa psi (kPa). This pressure is at least equal to the design pressure. It is given in the ASME code by the following equations:

)2/(2 o tyDtSP ma −= , equation NB-3641.1-(3) for class 1 (3-1)

)2/(2 o tyDtSPa −= , equation NC-3641.1-(5) for class 2 (3-2)

)2/(2 o tyDtESPa −= , equation ND-3641.1-(5) for class 3 (3-3)

where Sm = maximum allowable stress intensity for the material at the design temperature (Section II, Part D, Subpart 1, Tables 2A and 2B, psi (kPa); S = maximum allowable stress for the material at the design temperature (Section II, Part D, Subpart 1, Tables 1A and 1B, psi (kPa); E = joint efficiency for the type of longitudinal joint used, as given in Table ND-3613.4-1, or casting quality factor determined in accordance with ND-3613.5; Do = outside diameter of the pipe, in; y = a coefficient having a value of 0.4, except that for pipe with Do / tm ratio less than 6, the value of y shall be taken as ( )o/ Dddy += , where d, in, is the inside diameter of the pipe; and t = the specified or actual wall thickness minus material removed in threading, corrosion or erosion allowance, material manufacturing tolerances, bending allowance, or material to be removed by counterboring, in. The permissible pressure for each class and service limit is expressed as a multiple of Pα.

The Peak Operating Pressure, Pmax, psi (kPa) that can be less than the design pressure. It is used for classes 2 and 3 in the consideration of Level A, B and C Service Limits.

The Pressure, PD, is the pressure occurring coincident with the reversing dynamic load. This pressure is used in the consideration of Level D Service Limits. The Range of Service Pressure, Po, psi (kPa) that is used in Service Limit A for Class 1. Under internal pressure also, the Loss of Coolant Accident (LOCA) Accidental Pressure occurs due to a loss of coolant accident as a result of a pipe break. This pressure is considered only in Service Limit D.

3.1.3. Seismic Loading Seismic loading can be classified as reversing dynamic loading that occur occasionally (called the Operating Basis Earthquake, OBE), Service Limits B and C or under accidental, faulted conditions (called the Safe Shutdown Earthquake, SSE), Service Limit D.



29

The earthquake analysis of a piping system is based on the motion of its supports according to floor spectra. Inelastic effects in the analysis are being ignored. In the ASME code, two types of earthquakes are considered:

1. The Operating Basis Earthquake (OBE) is the earthquake, which considering the regional and local geology and seismology and specific characteristics of local subsurface material, could reasonably be expected to affect the plant site during the operating life of the plant. It is used for Service Limits B and C. This earthquake is typically expected to occur once in one hundred years (Rodabaugh, E. G. 1984). The maximum vibratory ground motion of the OBE should be at least one-half the maximum vibratory ground motion of the SSE unless a lower OBE can be justified on the basis of probability calculations.

2. The Safe Shutdown Earthquake (SSE) is the earthquake, which is based upon an evaluation of the maximum earthquake potential considering the regional and local geology and seismology and specific characteristics of local subsurface material. It is that earthquake which produces the maximum vibratory ground motion for which those structures, systems and components important to safety are designed to remain functional. It is used for Service Limit D. This earthquake is typically expected to occur once in one thousand years (Rodabaugh, E. G. 1984).

The ASME Code also allows for time history analysis for seismic conditions. The Nuclear Regulatory Commission (NRC) has defined the SSE ground motion response spectrum with a specified annual frequency of exceedance (U.S. NRC, 1997). NRC has approved the Regulatory Guide 1.60 spectra with critical damping between 2% and 3% for smaller and larger pipes, respectively. Code case N411 spectra may also be used and are based on 5% damping for the frequencies lower than 10 hertz, tapering damping between 10 and 20 hertz and 2% damping above 20 hertz. This spectrum was being used between 1975 and 1994. The NRC has also permitted the use of the NUREG/CR 0098 shaped median spectra but at a probability level of 10-5/ yr, rather than a 10-4/ yr probability load associated with R.G. 1.60 median plus one standard deviation shaped spectrum. This lateral spectrum is being used mostly for checking existing plants and not for the construction of new plants. The analysis and seismic design of piping is very conservative when pipes are considered to be independent from any of the other structures such as buildings (Gupta and Gupta 1995).

3.1.4. Nonreversing Dynamic Loads Nonreversing dynamic loads are occasional loads that take place under normal operation of the plant (Service Limits B, and C). They do not cycle about a mean value like earthquake loads and include the initial thrust force due to sudden opening or closure of valves and water-hammer resulting from entrapped water in two-phase flow systems.



30

3.2. Load Combinations in Non-ASME Structural Codes Load combinations that are being used in reliability-based design Codes that are based on LRFD format are presented herein. The notation of the loads is not uniform, since the original form of equations is maintained. The loads that are irrelevant to the loads of interest for this study were eliminated from the combinations. Therefore, only loads due to weight, internal pressure, and seismic forces are included. No thermal loading is considered, since this study is limited in scope to cold, primarily straight piping, with some considerations for non-straight pipes. According to Stevenson et al. (1999) cold piping is piping where the differential temperature range in the piping system is less than 150ºF and had an initial installation temperature of 50 to 70ºF and it represents approximately 60 % of the safety-related piping in a nuclear power plant. More specifically load combinations from the American Institute of Steel Construction, the American Society of Civil Engineers, the American Petroleum Institute, the American Association of State Highway and Transportation Officials, and the European Codes are presented. In Section 3.2.4, the recommended combinations for the nuclear piping design are given; while the factor values that are used in the above-mentioned codes are summarized.

3.2.1. American Institute for Steel Construction Code The American Institute for Steel Construction (AISC 2003) provides a framework for developing load combinations. AISC does not include internal pressure loads in its load combinations since they are designated as outside the scope of the AISC code. Therefore, the only pressure mentioned is the maximum differential pressure load, Pα, generated by the postulated accident and it is included in abnormal load combinations. Moreover, every load that has the index α is generated by a postulated accident, which can be caused by a break in high energy piping, or a break in a small line containing high temperature fluids or steam, or by extreme load phenomena which have a probability of occurrence larger than 10-7 per year. The AISC load combinations including these abnormal loads are provided in this section keeping Ro, i.e. reactions resulting from start-up and shutdown conditions. Also, the load factors are retained for the purpose of reference and comparison with the findings of this study. Loads that are provided in underlined text can also be omitted if deemed later on as out of the scope of this study.

1. Normal Load Combinations:

)(4.1 οRD + (AISC NA4-1) (3-4)

LRD o 6.1)(2.1 ++ (AISC NA4-2) (3-5)

2. Severe Environmental Combinations:

oo ELRD 6.18.0)(2.1 +++ (AISC NA4-5) (3-6)

3. Extreme Environmental and Abnormal Load Combinations:

sso ERLD +++ 8.0 (AISC NA4-6) (3-7)



31

αRPLD +++ α2.18.0 (AISC NA4-8) (3-8)

ssαα ERPLD ++++ )(8.0 (AISC NA4-9) (3-9)

where D = dead loads due to the weight of the structural elements; οR = pipe reactions during normal operating, start-up, or shutdown conditions, based on the most critical transient or steady - state condition; L = live load due to occupancy and moveable equipment, including their impact; Eo = loads generated by the operating basis earthquake; Ess = loads generated by the safe shutdown or design basis earthquake; αP = maximum differential pressure load generated by the

postulated accident; and αR = pipe and equipment reactions generated by the postulated

accident, including οR .

3.2.2. American Society of Civil Engineers Code The Structural Engineering Institute (SEI)/ American Society of Civil Engineers (ASCE 7-02) code contain the factored load combinations of the 1982 ANSI Standard for all structural materials. It gives provisions for the earthquake loads for pressure piping systems designed and constructed in accordance with ASME B31. Relevant to the seismic design of pressure piping systems are Sections 9.6.3.11.1, 9.6.1.3 and 9.6.1.4. In Section 2.3 of the code, the following combinations, which are of interest for this study, are presented:

)(4.1 FD + (ASCE 7-02, 2.3.2.1) (3-10)

LFD 6.1)(2.1 ++ (ASCE 7-02, 2.3.2.2) (3-11)

ELD 0.10.12.1 ++ (ASCE 7-02, 2.3.2.5) (3-12)

ED 0.19.0 + (ASCE 7-02, 2.3.2.7.) (3-13)

where D = dead load; F = load due to fluids with well-defined pressures and maximum heights; L = live load; and E = earthquake load The earthquake-induced force effect, E, shall include vertical and horizontal effects as given by the following equations. Eq. 3-14 shall be used with Eq. 3-12, while for Eq. 3-13, E should be calculated using Eq. 3-15:

DSQE DSE 2.0+= ρ (ASCE 7-02, 9.5.2.7-1) (3-14)

DSQE DSE 2.0−= ρ (ASCE 7-02, 9.5.2.7-2) (3-15)

where E = the effect of horizontal and vertical earthquake-induced forces; SDS = the design spectral response acceleration at short periods obtained from Section 9.4.1.2.5; D = the effect of dead load; ρ = the reliability factor; and QE = the effect of horizontal seismic (earthquake-induced) forces. The vertical seismic effect term 0.2 SDS D need not be included where SDS is equal to or less than 0.125.



32

3.2.3. American Petroleum Institute Code This specification, issued by the American Petroleum Institute (API), provides a load and resistance factor design for offshore platforms. Moreover, part of this specification has been developed specifically for fabricated circular tubular shapes which are typical of offshore platform construction. The suggested factored load combinations for all members that are related to this study are as follows:

11 5.13.1 LD + (API, C.2-1) (3-16)

ELD 9.01.11.1 11 ++ (API, C.4-1) (3-17)

ELD 9.08.09.0 11 ++ (API, C.4-2) (3-18)

where D1 = the weight of the structure including the weight of equipment and other objects; permanently mounted on the platform, hydrostatic forces acting on the structure below the waterline including internal pressure; L1 = the live load that includes also the weight of fluids in pipes; and E = the inertially induced load produced by the strength level ground motion. The load combination given by Eq. 3-18 is suggested instead of that given by Eq. 3-17. When inertial forces due to gravity loads oppose the internal forces due to earthquake loads, then gravity load factors should be reduced. In the part of the specification that concerns the circular tubular shapes, the load factors for determining the hydrostatic pressure are as follow: 1.3, for the functional load case that includes pressures that will definitely be encountered during the installation or the life of structure. When interaction with storm loads is considered, this factor reduces to 1.1.

3.2.4. American Association of State Highway and Transportation Officials Bridge Design Specifications

This code refers to the American Association of State Highway and Transportation Officials LRFD Bridge Design Specifications. Although the nature of most of the loads for bridges is very different from the nature of load for piping under consideration, this code is also in LRFD format and involves load factors of interest, specifically those for dead weight and earthquake. The load combination, as described by AASHTO, takes the following general form:

ii qγη Σ (AASHTO, 3.4.1-1) (3-19)

where η = a factor greater than 0.95 relating to ductility, redundancy and operational importance; iγ = a statistically based factor applied to force effects; and qi = the force effect. In Tables 3.4.1-1 and 3.4.1-2 of the code, load combinations, load factors for transient loads, and the load factors for the permanent loads are presented, respectively. The following combinations are selected herein with the factor η omitted:

WALLDC 00.175.125.1 ++ (3-20)



33

WADC 00.15.1 + (3-21)

EQWALLDC EQ 00.100.125.1 +++ γ (3-22)

where DC the dead load of structural components and nonstructural attachments; LL = the vehicular live load without wind; WA = the water load and stream pressure; EQ = earthquake load; and EQγ = this factor shall be determined on a project-specific basis. According to the Commentary of the code in older editions EQγ was zero, but also a value of 0.5 is presented as reasonable.

3.2.5. Eurocode 1990 The EuroNorm 1990 is the head document in the Eurocode suite and provides the basis of structural design. It serves as the reference document for the design of whole structures and component products for the European Union member states. The fundamental combination of persistent and transient situations for the ultimate limit state verification, other than those relating to fatigue, is given in a general form as

∑∑>≥

++1

0111 i

kiiQikQj

kjGj QQG ψγγγ (Clause 6.4.3.2(3)) (3-23)

This combination assumes that besides the permanent actions Gkj a number of variable actions are acting simultaneously but only one is the dominant, Qk1, among the rest of them Qki. When the dominant action is not obvious, each variable action should be considered in turn as the dominant action. In most common cases all factors Qiγ are equal and the permanent actions are divided to favorable and unfavorable actions for the limit state under consideration. Thus, Eq. 3-23 takes the following form:

}⎩⎨⎧

+++ ∑∑∑>

iki

ikQjkjGjkjG QQGG ,1

,01,inf,,inf,,sup,,sup,, ψγγγ (3-24)

where sup,, jkG = favorable permanent loads; inf,, jkG = not favorable permanent loads; sup,, jGγ = load factors for variable actions; Qγ = load factor common for all the variable loads; 1,kQ = dominant variable action; ikQ , = not dominant variable actions; and i,0ψ = reduction partial factors for not dominant variable actions. The seismic design combination in general form is given by the following equation:

∑∑≥≥

++1

21 i

kiiEdj

kj QAG ψ (Clause 6.4.3.4(2)) (3-25)

The range of the reduction partial factors ψ used for the ultimate limit states especially for buildings are presented herein. Their value depends on the Category of the building, with Category E, storage areas, to be the more severe case. Analytically, these values are tabulated in Table A1.1 of EN 1990 as presented by Gulvanessian et al. (2002) with ψ0i = 0.5 to 1.0, and ψ2i = 0.2 to 0.8.



34

The values of the recommended factors γ in Eq. 3-24, which correspond to the EQU limit state, are presented here. The EQU limit refers to the static equilibrium of a structure, or of any part of it, considered as a rigid body. Thus, equation 3-24 takes the form

∑∑∑>≥≥

+++1

011

inf,,1

sup,, 5.150.190.010.1i

kiikj

jkj

jk QQGG ψ (3-26)

In Table A1.2(A) of EN 1990, as mentioned by Gulvanessian et al. (2002), it is noted that in cases where the verification of static equilibrium also involves the resistance of structural members, increased γ factors may be used and Eq. 3-24 can be written as

∑∑∑>≥≥

+++1

011

inf,,1

sup,, 5.150.115.135.1i

kiikj

jkj

jk QQGG ψ (3-27)

3.3. Load Combinations for Components of Nuclear Plant This section summarizes load combinations for structural and mechanical components of nuclear plants based on a literature review. More specifically referring to Table 3-1, the ASCE, SMiRT-4 reference presents combinations of impulsive loads acting on a BWR-Mark I containment. Ravindra, et al. (1981) and Schwartz et al (1981) provide load combinations for the essential service water line (ESW) piping components, which are Class 2 components. The work of Hwang, et al. (1987) develops practical probability-based load and resistance criteria for reinforced concrete containment and shear wall structures in nuclear plants. It can be noticed that in all these combinations, the loads involved are similar to the ones that are applicable to piping design such as the Operating Basis and Safe Shutdown Earthquake, the Safety Relief Discharge Load, the thermal loads etc. Nevertheless only the work of Ravindra, et al. (1981) and Schwartz, et al. (1981) refer exclusively to piping design.



35

Table 3-1 Load Combinations for Nuclear Plant Facilities and Components Load Combinations Loads Reference*

SRVoPFLD 5.10.10.17.14.1 ++++ SRVoRoToPFLD 3.10.10.10.10.13.10.1 ++++++

2)(

2)(25.10.10.10.10.10.10.1 SRVOBEoRoToPOFLD +++++++

LOCAIBASBASRVARATBPFLD 2)/(

2)(25.125.10.10.10.1 +++++++

SRVARATAPFLD 0.10.10.125.10.10.10.1 ++++++

LOCAIBASBASRVOBEARATBPFLD 2)/(

2)(

2)(1.10.10.11.10.10.10.1 ++++++++

SRVA

RA

TA

Po

EFLD 0.10.10.11.11.10.10.10.1 +++++++

2)(

2)(0.10.10.10.10.10.10.1 SRVSSEoRoToPFLD +++++++

2)(2)(0.10.10.10.10.10.10.10.1 SRVSSERRARATBPFLD ++++++++

SRVRRARATAPSSEFLD 0.10.10.10.10.10.10.10.10.1 ++++++++

D=Dead L=Live F=Prestressing To=Operating Temperature Ro=Operating Reactions Po=Operating Pressure SRV=Safety/Relief Valve Eo=OBE Ess=SSE PB=SBA and IBA Pressure TA=Pipe Break Temperature Load RA=Pipe Break Temperature PA=LBA Pressure (including all pool hydrodynamic loadings) RR=Reactions and Jet Forces (Pipe Break)

ASCE, SMiRT-4 (1977)

Design: P+W Service Limit A/B: TRNG Service Limit B: P+W+OBE P+W+HYDTR Service Limit C: P+W+SSE P+W+OBE+HYDRT

WWcWγdPPc

PγRφ +≥111

)(112 TRNGTH

cTHγWWc

WγdPPc

PγRφ ++≥

)(112 NHTR

Hc

HγWWc

WγdPPc

PγRφ ++≥

)(1

112 OBEEo

coE

γWWcWγdPPc

PγRφ ++≥

)(1

)(11

113 NHTR

Hc

HγOBE

Eoc

EγWWc

WγdPPc

PγRφ +++≥

)(12

114 SSESE

cEγWWc

WγdPPc

PγRφ ++≥

)(1222 OBE

oAEcEγRφ ≥ )(2222 SSE

SAEcEγRφ ≥

)(2

)(23

22 NHTR

AHc

HγOBE

oAEcEγRφ +≥ )(3

22 NHTR

AHc

HγRφ ≥

P=Design Pressure W=Weight OBE=Operating Basis Earthquake HYDTR=Hydraulic Transient TRNG=Thermal Range SSE=Safe Shutdown Earthquake HTRN=Hydraulic Transient φij=Resistance Factors γij=Load Factors ci=Influence Coefficients that transform loads into moments

Ravindra, et al (1981) and Schwartz et al (1981)

oRRγoTLLγDDγ 1+++

)(11

SRVSRVγoRRγoTLLγDDγ ++++

oRRγoTLLγDDγ +++1

oRRγoTtWtWγorssEESγLLγDDγ 1

)(1

++++

aPpγαRαTssEEγLLγDDγ +++++11

D=Dead L=Live To=Operating Temperature Ess=SSE Pα=Loca Pressure

Hwang, et al. (1987)

* The notations in this table are uniquely defined in the last column of the table per respective cited references.



36

3.4. Recommended Load Combinations for Piping Based on the reviews provided in the previous sections, the following load combinations are recommended for this study, considering here loads due to sustained weight, the internal pressure, and earthquake loading.:

hoopf1γ (3-28)

PA ff 32 γγ + (3-29)

EPA fff 7,654 γγγ ++ (3-30)

EA ff 10,98 γγ + (3-31)

where fhoop = hoop load effect due to internal pressure; fA = load effect due to weight; fP = load effect due to internal pressure; fE = seismic load effect. When the design earthquake is the OBE then 5γ or 8γ factors should be used. Whereas for SSE earthquake factors 6γ or 9γ should be used; and iγ = load factors. The values of load factors iγ for the various codes are summarized in Table 3-2. It must be emphasized that the definition of pressure and earthquake is different in each of these codes. Table 3-2. Load Factors as Used in Various Structural Codes

Load Factor AISC (2003) ASCE-7 (2003) API (1993) AASHTO

(1994) EN (1990)

1γ NA 1.4 1.3 1.0 NA

2γ 1.4 1.4 1.3 1.5 1.10 or 1.35

3γ NA 1.4 1.3 1.0 **

4γ 1.2 1.2 1.1 1.25 1.0

5γ NA NA 1.1 1.0 **

7,6γ 1.6, 1.0 1.0 0.9 1.0 1.0

8γ NA 0.9 NA NA NA

10,9γ NA 1.0 NA NA NANA = Not Available ** depends on whether favorable or not, variable or not



37

4. Failure Modes and Limit States for Piping

The discussion in this chapter is targeted specifically towards the objective of this project that focuses on the design rules for only the primary stresses in a straight pipe component for Class 2 and Class 3 piping subjected to pressure, deadweight and earthquake loads. Furthermore, the following discussion is focused on cold pipe with no consideration for high temperature related cases. We begin with a discussion on failure criterion that forms the basis of ASME and B31 piping design codes.

4.1. Failure Criterion The equations for evaluating the strength of piping systems are based on the maximum shear stress theory (Tresca Criterion). The criterion states that failure of the material subject to biaxial or triaxial stress occurs when the maximum shear stress at any point reaches the value of the shearing stress at failure in a simple tension or compression test on the same material. The maximum shear stress maxτ is one half the difference between the maximum and minimum principal stresses maxσ and minσ , respectively, i.e.,

2

minmaxmax

σστ −= (4-1)

If the yielding failure in a simple compression or tension test is defined at yield stress Sy, then the maximum shear stress at failure becomes 0.5Sy. Therefore, one can write the Tresca criterion as

22

minmaxmax

yS≤

−=

σστ (4-2)

For a circular straight pipe subjected to internal pressure and bending (neglecting shear forces), the maximum and minimum principal stresses occur along the longitudinal and axial directions, respectively. It should be noted here that the shear stresses are normally much smaller than the moment stresses in piping systems. The shear stresses at the locations, where moment stresses that are maximum are negligible. The longitudinal and hoop stresses due to internal pressure P in a thin-walled pipe of diameter Do and thickness t are given by

t

PD4

olong =σ (4-3)

t

PD2

ohoop =σ (4-4)



38

The longitudinal bending stress due to transverse loading which results in a moment M at any

cross section with elastic section modulus Z has a linear distribution and the maximum value in

the outermost fiber is given by

ZM

long (4-5)

The stress distributions for the two types of longitudinal stresses are shown below which also

shows the combined longitudinal stress distribution. According to this figure, the maximum

longitudinal stress in the outermost fiber of the pipe cross-section can be evaluated simply by

adding the above two expressions.

If the resultant moment (resultant of three orthogonal components at any cross-section) is used,

then ZM is not truly longitudinal but a conservative estimate of the longitudinal stress.

Consequently, the Tresca Criterion, considering the thin wall theory and thus the axial stress

equal to zero, then gives:

ySZM

tPD4

2 omax (4-6)

Some studies have also related the basis of the currently used code equations for Class 2 and 3 to

the Von-Mises criterion of principal stresses (Larson et al. 1974, Larson et al. 1975). Gerdeen et

al. (1979) give a comparison between the two failure theories using theoretical solutions for

plastic collapse of thin cylindrical pipes subjected to bending and internal pressure. They

illustrate that the Tresca criterion is generally the more conservative of the two, and the

maximum difference between the two solutions is 15.4%. Tresca criterion is also simple to use.

Therefore, the Tresca criterion has been considered to serve as the basis for the design of straight

pipes in ASME Section III.

Uniform stress

distribution

due to pressure

Linear stress

distribution due

to bending

Combined

longitudinal stress

distribution

Tension

Tension Tension

Compression Compression

Figure 4-1. Stress distribution in a straight pipe subjected to internal pressure and bending



39

4.2. Performance Criterion

The Section III equations are intended to protect against failures in certain modes, i.e., the code

equations are intended to provide acceptable performance against the following failure modes:

pipe burst,

on-set of yielding,

limit-load capacities (formation of a single hinge or collapse mechanism),

plastic instability or excessive plastic deformation (deformation based criterion), and

low-cycle (high-strain) fatigue.

Primary stresses developed by the applied loads are required to satisfy the equilibrium between

external and internal forces (and moments) of the piping system. Primary stresses are not self

limiting. Therefore, if a primary stress exceeds the yield strength of pipe material through-out

the pipe cross-section then the strain hardening of the material provides resistance to the collapse

(plastic instability). The primary stress limits are thus intended to safeguard against pipe-burst,

on-set of yielding, and plastic instability. These limits do not address fatigue performance.

While the definitions for pipe-burst and on-set of yielding are well defined, the various

definitions for limit-load and plastic instability based criteria have been discussed at length by

researchers and practitioners (Gerdeen et al. 1979, Larson et al. 1974, Matzen and Yang 2002).

The following paragraphs provide a summary of some definitions related to the objective of this

project.

4.2.1. Limit-Load Capacity

In a typical structural engineering design such as that for steel structures (AISC design codes),

the plastic instability at a given cross-section is idealized by the formation of a plastic hinge

assuming that the material is elastic-perfectly plastic as shown below in Figure 4-2.

The stress distributions for three conditions - at the onset of yielding, some plastic deformation,

and complete hinge formation are shown below. The bending stress at the formation of fully

plastic hinge is given by M/Zp where Zp is the plastic section modulus for the cross-section. The

moment calculated as MLL = SyZp is called the “limit-load” moment capacity of the cross section.

The above definition does not utilize the effect of strain hardening in a material beyond yield.

The plastic section modulus for thin pipes is about 1.3Z (Gerdeen et al. 1979). Therefore, the

limit-load can be written as MLL = 1.3SyZ where Z is the elastic section modulus. Mello and

Griffin (1974) relate the 1.2Sy limit in the ASME Section III piping design rules for Service

Level B as a conservative estimate of the 1.3Sy in the definition of limit moment based on the

formation of a single hinge. It is to be noted that the theoretical values for the factors 1.2 and 1.3

is 4/ .

Rodabaugh and Moore (1978) provide a detailed discussion on the evaluation of limit-load

capacities for piping systems. They illustrate that the formation of a single isolated hinge in a

piping system does not result in large (gross) plastic deformation of the piping system.

Formation of hinges at more than one location is needed to develop a “collapse mechanism.”

This study also illustrates the formation of a collapse mechanism in a straight pipe with fixed-



40

fixed end conditions to represent anchors at the two ends. This illustration is used to show that

the first hinge forms at the ends and a total of three hinges (two at the ends and one in the mid-

span) is needed to form a collapse mechanism. The limit-moment for the formation of collapse

mechanism in this case is evaluated at 4/3 times the limit-moment for the formation of a single

hinge. Consequently, one can write MLL = (4/3)1.3SyZ = 1.7 SyZ. This definition of limit load is

then related to the allowable strength factor of 1.5 Sy used in ASME Section III piping design

rules for Service Level C.

Rodabaugh and Moore (1978) also use experimental data to illustrate that the conditions leading

to a limit-moment while necessary to produce collapse are not sufficient. They state that in

actual piping systems development of a collapse mechanism is not likely due to substantial

margins although the degree of margins vary depending upon the piping system and the loading

conditions.

S

Sy

Figure 4-2. Stress-Strain Curve for Idealized Elastic-Perfectly Plastic Material

Onset of Yielding Some yielding

of cross section

Complete yielding

of cross section

Figure 4-3. Stress distribution in a straight pipe subjected to internal pressure and moment at

various stages of yielding



41

4.2.2. Plastic-Instability Collapse Load Using Elastic Slope The ASME Code section NB-3213.25 characterizes collapse moment MCL that is defined based on the load-deformation curve of the type shown below in Figure 4-4. According to this definition, a plastic analysis may be used to develop the load-deformation (load-deflection or load-strain) curve. Then, the angle that the linear part of the curve makes with the ordinate is called θ. A second straight line, called the collapse limit line, is drawn through the origin so that it makes an angle )tan2(tan 1 θφ −= with the ordinate. The collapse load is the load at the intersection of the load-deformation curve and the collapse limit line as shown above in the figure. This definition of collapse moment, typically denoted by Mφ requires a complex inelastic analysis and is related to a large-deformation based criterion for defining the piping performance. This will require nonlinear analysis. Current piping analysis computer programs are based linear analysis.

Figure 4-4. Code Defined Collapse Load

4.2.3. Plastic-Instability Collapse Load Using Elastic Deformation Several researchers (Mello and Griffin 1974, Rawls et al. 1992, Rodabaugh and Moore 1978, Wais 1995) have used an alternative definition of collapse (or plastic) load. According to this definition, typically denoted by M2y, the plastic load can be obtained as that corresponding to a strain ε = 2εy where εy is the strain at yield. Theoretically, one can calculate the collapse moment for pipes as MCL = M2y = 2SyZ. Wais (1995) used the elastic section modulus to obtain a conservative estimate of MCL. Mello and Griffin (1974) used an approximation to the plastic section modulus. It has been shown by several researchers (Gerdeen 1979, Larson et al. 1974, Wais 1995) that experimentally determined value of M2y is higher than the theoretically determined value. Rodabaugh and Moore (1978) use experimental results to conclude that if the moments are restricted to M2y then the restriction in flow area will be small (less than 1%) and the functional capability of the pipe assured. Experimental data from straight pipe tests is used to also conclude that the strains in the piping system will be less than 2% if the moment is



42

restricted to less than M2y. Such strains are considered small and not expected to cause cracking in typical piping materials or pipe welds. Gerdeen et al. (1979) examined several strain and large deformation criteria to conclude that the definition of collapse load (Unlike ASME Code Section NB-3213, Gerdeen et al. (1979) used “plastic” load and did not use the word “collapse” in their work) based on twice-elastic-slope is least subject to error but is impossible to obtain in some cases. Hence, they conclude that twice-elastic-deformation method is more expedient to use. To do so, Gerdeen et al. (1979) recommend that a proper choice of the deformation parameter must be made to properly account for the work done by the loads.

4.2.4. Plastic Instability: Ultimate Moment Definition of Collapse Load

Consistent with their recommendation of choosing an appropriate deformation parameter in determining collapse load, Gerdeen et al. (1979) give a detailed discussion on the choice of an appropriate parameter for pipes subjected to moment loading. According to them, the plastic instability in such cases should be defined with respect to generalized stress and generalized strain approach of Prager (1952). The generalized stress is specified in terms of moment and the generalized strain in terms of the curvature. The plastic instability is then defined as the moment corresponding to the point where the slope of moment-curvature curve becomes zero as shown in Figure 4-5, where MCL = collapse moment. Kennedy (1997) uses a similar approach to define the collapse in terms of an ultimate moment MUD due to reversing dynamic (earthquake type) load.

Curvature

MCL

Mom

ent

Figure 4-5. Moment-Curvature Curve

4.3. Existing Code Equations In this section, we discuss the Section III equations with respect to various limit states and relate them to the performance criteria, and discuss the corresponding limit-states that can be used in LRFD based design equations.



43

4.3.1. Design Condition The minimum thickness tm of pipe wall for Design Pressure is given by Eqs. 3 and 4 of ASME Code Section NC-3641.1 as follows:

APyS

PDtm ++

=)(2

o (4-7)

)(2

22PPySyPASAPdtm −+

++= (4-8)

in which P = design pressure (psi); S = maximum allowable stress for the material (psi); Do = outside diameter of pipe (in); A = additional thickness as allowance for threading,

corrosion, etc.; d = inside diameter of the pipe; and y = 0.4 for 6o ≥mt

D. For 6o <

mtD

, use

Eq. 6 of Section NC-3641.1 as follows:

oDd

dy+

= (4-9)

The allowable pressure is given by Eq. 5 of ASME Code Section NC-3641.1 as follows:

ytD

StPa 22

o −= (4-10)

in which t is the actual wall thickness minus allowance for corrosion, threading, etc. Also note that Pa should be at least equal to the design pressure P.

Equations 4-7 to 4-10 are based on the safeguard against failure due to hoop stress resulting from internal pressure. Excessive internal pressure can cause, among other things, a pipe-burst type of failure which is highly undesirable. Parallels to this situation can be drawn from the building design codes such as the ACI code for concrete structures in which a bending failure of the structural members is ductile and a shear failure is brittle. The brittle failure mode is avoided by over designing in shear relative to bending. Likewise, it would be desirable to avoid a failure due to excessive internal pressure by over designing for it. One way to achieve this would be to use a much higher value of reliability level in the probabilistic analyses. On the contrary, it may be argued that the above equation for allowable pressure which is based on the traditional working stress method with deterministic safety factors has worked well over the years and provides a good basis for over designing against excessive internal pressure. It should also be noted that overpressure relief is required for pressurized piping systems that also limits the potential for failure due to overpressure. Nevertheless, performance functions for these equations are used and included in this report, although a code committee might decide not to change the equations leading to different strength factors for these equations.

Safety against failure due to excessive bending stress resulting from deadweight and

other sustained loads is evaluated by Eq. 8 of Section NC-3652 as follows:



44

hA

nSL S

ZMB

tPDBS 5.12 2

o1 ≤+= (4-11)

in which P = design pressure (psi); MA = resultant moment on a cross-section due to weight and other sustained loads, lb-in.; Z = elastic section modulus, (in3); tn = nominal wall thickness (in); B1,B2 = primary stress indices equal to 0.5 and 1.0, respectively for straight pipe; and Sh = allowable stress for material (psi).

The allowable stress Sh is defined as minimum of 32 yS or 4uS where Sy is the yield stress and Su the ultimate stress.

The maximum allowable stress in the outermost fiber 1.5 Sh reaches Sy when 32 yS is less than 4uS . However, the above design equation limits the stress in outermost fiber to a value even below the yield stress when 4uS is less than 32 yS . The reason behind such a criterion lies in the nature of material and the reserve strength available beyond the yield strength. For materials in which Su is relatively much higher than Sy, the 32 yS limit governs the above equation as sufficient reserve strength is available beyond yield. However, for materials in which Su is relatively closer to Sy, the 4uS limit governs the above design equation as sufficient reserve strength is not available beyond yield. In the current deterministic code based on working stress method of design, the latter performance condition governs for only those materials in which Su is less than 2.67Sy. Therefore, the performance function for design condition to safeguard failure against bending can be defined as on-set of yielding. It is to be noted that the development of load and resistance factors for this limit state should be subjected to the condition that target reliability is consistent with respect to sufficient reserve strength against plastic instability.

4.3.2. Operating Condition (Service Level A) For service level A, the limit-state for safety against failure due to excessive bending stress resulting from deadweight and pressure is evaluated by Eq. 9 of Section NC-3653 as follows:

hA

nOL S

ZMB

tDPBS 8.1

2 2omax

1 ≤+= (4-12)

in which the various terms remain the same as for the Design Condition except for pressure, and SOL is stress resulting from operating condition. In the above equation, Pmax is the Peak Operating Pressure which is less than or equal to Pa. While the right hand side of the above equation permits minor yielding in the outer fibers of a pipe cross section, it is not likely and the Design Condition remains as the governing condition.

4.3.3. Upset Loading Condition (Service Level B) This loading condition corresponds to loads that should be withstood without requiring repair. For service level B, the limit-state for safety against failure due to excessive bending stress



45

resulting from deadweight, pressure, and operating basis earthquake is evaluated by Eq. 9 of Section NC-3653 as follows:

)8.1,5.1min(2 2

omax1 hy

BA

nUL SS

ZMMB

tDPBS ≤⎟

⎠⎞

⎜⎝⎛ +

+= (4-13)

in which MB is the resultant moment on a cross-section due to operating basis earthquake, lb-in, and SUL is stress resulting from upset loading condition. It must be noted that for Service Level B, Pmax can exceed Pa but limited to aPP 1.1max ≤ . The philosophy behind the above equation changes from that in case of Design Condition in the sense that the stress in outer most fiber is permitted to exceed the yield. However, this may not necessarily mean any permanent yielding because of the transient nature of the upset loading. For materials in which 32 yS governs the definition of Sh, the allowable limit 1.8Sh becomes 1.2Sy and for materials in which

4uS governs the definition of Sh, the allowable limit is less than 1.2Sy with sufficient reserve strength against the stress in outermost fiber reaching Su. Mello and Griffin (1974) point out that the 1.2Sy limit is a conservative estimate of the limit moment based on the formation of a single hinge. Therefore, the performance function for Upset Loading Condition to safeguard failure against bending can be defined in accordance with the work of Rodabaugh and Moore (1978) that relates this service loading condition to the formation of a plastic hinge at a single cross-section in a piping system, i.e., limit load based on the formation of single isolated hinge. It is to be noted that the development of load and resistance factors for the above limit state should be subjected to the condition that target reliability is consistent with respect to sufficient reserve strength against plastic instability. As discussed in detail earlier, it must also be noted that formation of a single isolated hinge in a piping system does not result in large (gross) plastic deformation. Formation of hinges at more than one location is needed to develop a “collapse mechanism.” Review of footnote of Table 1-100 of Section II, Appendix 1 reveals that two sets of allowable stress values for Sh are provided in Table 1A for austenitic materials and in Table 1B for specific nonferrous alloys. The higher values are identified by a footnote in those tables.

4.3.4. Emergency Loading Condition (Service Level C) This loading condition corresponds to loads that may require inspection or repair of damaged components. It must be noted that for Service Level C, Pmax is limited to aPP 5.1max ≤ . For service level C, the limit-state for safety against failure due to excessive bending stress resulting from deadweight, pressure, and operating basis earthquake is evaluated by Section NC-3654.

)25.2,8.1min(2 2

omax1 hy

BA

n

SSZ

MMBtDPB ≤⎟

⎠⎞

⎜⎝⎛ +

+ (4-14)

The philosophy behind the above equation is different from that for Service Levels A and B in the sense that the stress in outermost fiber is permitted to exceed much beyond the elastic limit. Based on the work of Rodabaugh and Moore (1978), the limit-state for Emergency Loading Condition to safeguard failure against bending is related to the limit-moment associated with the



46

formation of a mechanism in the piping system due to the formation of more than one plastic hinge. Therefore, the limit-state for this condition can be specified as

• Limit-Load corresponding to the formation of a collapse mechanism as opposed to an isolated single hinge. Rodabaugh and Moore (1978) recommends such a limit-state due to sufficient margins that may be available in a piping system due to cyclic strain hardening, the effect of short-term loading, etc. They also state that this limit-state does not correspond to excessively high strains and that the pipe will not lose a significant part of its flow area.

• Alternatively, one may simply consider the limit-state as that defined by plastic instability with a target reliability that is higher than that in case of Service Level D.

4.3.5. Faulted Loading Condition (Service Level D) This loading condition corresponds to loads that may occur only once during the lifetime and the damage should be contained enough to permit a safe shut down. It must be noted that for Service Level D, Pmax is limited to aPP 0.2max ≤ . For service level D, the limit-state for safety against failure due to excessive bending stress resulting from deadweight, pressure, and safe shutdown earthquake is evaluated by Section NC-3655.

)0.3,0.2min(2 2

omax1 hy

BA

n

SSZ

MMBtDP

B ≤⎟⎠⎞

⎜⎝⎛ +

+ (4-15)

The philosophy behind the above equation is similar to Service Level C in the sense that the stress in outermost fiber is permitted to exceed much beyond the elastic limit closer to the ultimate stress than in case of Service Level C. According to Rodabaugh and Moore (1978), this service level corresponds to a situation in which a pipe may not remain functional but the pressure boundary would be retained. Consequently, the performance function for Emergency Loading Condition to safeguard failure against bending can be described only as plastic instability. Based on the discussion presented in Section 4.2 of this chapter, it must be noted that plastic instability characterizes a large deformation based failure criterion that may be defined either in terms of yield or ultimate stresses.

4.4. Performance Functions The detailed discussion presented above can be summarized in terms of two basic equations. One equation for expressing the limit-states in terms of the yield stress and the other one is for expressing it in terms of the ultimate stress.

yBA

n

SkZ

MMBtDPB 12

omax1 2

≤⎟⎠⎞

⎜⎝⎛ +

+ (4-16)

uBA

n

SkZ

MMBtDPB 22

omax1 2

≤⎟⎠⎞

⎜⎝⎛ +

+ (4-17)



47

where k1 and k2 are constants whose values depend upon the particular service level (or limit-state) under consideration. It must be noted that the premise for using stress indices B1 and B2 lies in simplicity of the code equations especially for non-straight pipe components such as elbows and branch connections. Therefore, these indices case can be considered deterministic for a straight pipe case but not in case of other components. These indices must be represented by respective random variables for other piping components such as elbows and branch connections. The performance functions needed in this study as described in Section 2.4 of this report can be characterized as provided in the following general forms:

APR SSSg −−=1 (4-18)

EAPR SSSSg −−−=2 (4-19)

EAR SSSg −−=3 (4-20)

where SR = strength (stress) random variable equals to Sy for serviceability limit state and equals Su for strength limit state; Sy = yield strength of steel; Su = ultimate strength of steel; SA = random variable representing the stress due to dead weight; SP = random variable representing the stress due to internal pressure in a pipe; and SE = random variable representing the stress due to earthquake loading. A summary of recommended performance functions pertaining to this study is provided in Table 4-1. These functions are based on the load combinations given in Section 3.4, considering moreover mechanical loads and loads due to LOCA, and on limit states of the ASME Code as described earlier in this chapter.



48

Table 4-1. Recommended Performance Functions for Straight Piping Loading Condition Performance Functions Definition of Variables

General Design Condition

HSySg −=1

ASySg −=2

DPSASySg −−=3

t

ytoDP

HS

2

)2( −=

Z

AMAS =

t

DD

P

PDS4

o=

Operating Condition, Service Level A max4 PSASySg −−= t

DP

PS4

max o=

Z

AMAS =

Upset Loading Condition, Service Level B

MSASPSySg −−−= max5

OSMSASoPSySg −−−−=6

OSASySg −−=7

t

Do

P

PoS4

o=

PZ

OMOS =

PZ

AMAS =

PZ

MMMS =

Emergency Loading Condition, Service Level C

MSASPcSySg −−−= max8

OPO SASSySg −−−=9

HSySg −=10

OSASySg −−=11

OSMSASPOSySg −−−−=12

t

DPO

P

oPS

4

o=

PZ

AMAS =

PZ

OMOS =

Faulted Loading Condition, Service Level D

HSSg u −=13

SASPu SSSSg −−−=14

SAu SSSg −−=15

LPAu SSSSg −−−=16

SPLSAu SSSSSg −−−−=17

t

DPS

P

PSS4

o=

PZ

SMSS =

PZ

LM

LS =

Nomenclature

Sy = Yield Strength, Su = Ultimate Strength, SH = Hoop Stress Si, i=P, Pmax,,Po PS = Stress due to Normal Operating Pressure, Maximum Operating Pressure, Pressure Coincident with OBE, Pressure Coincident with SSE, SL = Stress Due to LOCA Accident, y = 0.4, Z = Elastic Section Modulus, ZP = Plastic Section Modulus, SO= Stress Due to OBE, SS = Stress Due to SSE, M = Moment, SM = Stress Due to Mechanical Loading



49

4.5. Load Combinations for Piping In this section, the suggested load combinations having an LRFD format in units of stress are provided. For each load combination, a brief description of the state and service load limit is given according to the ASME BPV Code.

4.5.1. Design Condition The relations presented in this section are used for the preliminary design of piping. Therefore, first the thickness of the piping can be calculated from the following relation:

22

)2(y

St

ytoDPP

φγ ≤− (4-22)

where Do = the outside diameter of pipe, t = thickness of pipe, y = 0.4 and for Do/t<6, y = Di/(Do+Di), P = the allowable pressure. The modified Lamé Equation can be used for the calculation of hoop stress. The following load combination guarantees that the pipe can support its own weight; where the weight includes the weight of the pipe, insulation and any other attachments to the pipe as follows:

yA

A SZ

M31

φγ ≤ (4-23)

where 1Aγ = partial load factor; 2φ = partial strength factor; MA = maximum moment due to

sustained load; Z = elastic section modulus of pipe; and Sy = yield stress of steel. This load combination will not be the critical one, but it is included in order to cover all the possible load combinations.

4.5.2. Operating Condition This combination represents Service Limit A in the ASME BPV Code. The pipe must withstand the loading under normal operation of the plant. Therefore, loadings arising from system startup, operation in the design power range, hot standby and system shutdown by excluding only those loadings covered by Levels B, C, and D Service Loadings or Test Loadings. Moreover, the pipe remains in elastic region under bending stress resulting from deadweight and the peak pressure.

( )uyA

AP SSZ

Mt

PD54

o ,min2 21

φφγγ ≤+ (4-24)

where 2Aγ = partial load factors; φ = strength reduction factor; P = peak internal pressure; Do =

outer diameter of the pipe; and t = thickness of the pipe.

4.5.3. Upset Loading Condition This combination represents Service Limit B in the ASME BPV Code. The loadings in this Service Limit occur often enough and include those transients, which result from any single



50

operator error or control malfunction, transients caused by a fault in a system component requiring its isolation from the system and transients. In this condition we also account for the effects of an operating basis earthquake and a larger pressure.

),min(2 76

o132 uy

BB

AAP SS

ZM

ZM

tPD φφγγγ ≤++ (4-25)

where MB = bending moment due to operating basis earthquake (OBE).

4.5.4. Emergency Loading Condition This combination represents Service Limit C in the ASME BPV Code. For this state the material is permitted to have greater deformations, so that inspection or repair of damaged components is expected. For example if we consider in our example the beam-pipe to be simply supported the formation of one hinge will cause a mechanism. But in piping systems this is not possible, because piping is multi-supported and has high degree of redundancy. In this case the effects of an operating basis earthquake are included and the pressure may have a greater value compared to Service Limits A and B. The conditions for this Service Limit have a low probability of occurrence. The total number of postulated occurrences for these loads may not exceed 25. If more than 25 events are expected then some types of events should be treated under the conditions imposed by Service Limit B.

),min(2 98243 uy

P

BB

P

AA

oP SS

ZM

ZM


where ZP is the plastic section modulus, and MB is the resultant moment on a cross-section due to operating basis earthquake. In this category we can add one more combination that does not include the pressure. This could happen to auxiliary piping that is not being used unless a major problem occurs in the main piping or when the plant shuts down. For primary piping systems in plants with SGR reactors or fast-cooled reactors the pressure can be zero.

),min( 111035 uyP

BB

P

AA SS

ZM

ZM φφγγ ≤+ (4-27)

4.5.5. Faulted Loading Condition This condition corresponds to the Service Level D in the ASME BPV Code. This loading condition may occur only once during the lifetime of the plant and should be contained enough to permit a safe shut down. The conditions of loading are the same as in Service Limit C, only now the earthquake moment can also be the result of the safe-shut down earthquake. A much greater pressure is also allowed, that is twice the pressure that is allowed for the operating condition (Service Level A). For this condition, we have the following load combinations:

),min(2 1312

o164 uy

P

SS

P

AAP SS

ZM

ZM




51

),min( 151427 uyP

SS

P

DA SS

ZM

ZM φφγγ ≤+ (4-29)

where MS is the resultant bending moment due to the safe shutdown earthquake (SSE). The combinations of these loadings are associated with events that have an extremely low probability of occurrence, such as the safe shutdown earthquake or the accidental pressure due to loss of coolant, where core cooling fluid is escaping from a leak or break within the primary cooling system.



52

5. Basic Random Variables for Piping

This chapter summarizes the probabilistic characteristics of strength and load basic random variables that are needed for the development of LRFD design methods for piping. However, it does not cover basic random variables for fatigue.

5.1. Statistical Characteristics of Random Variables The statistical characteristics of random variables of strength and load models are needed for reliability-based design and assessment of piping. The moment methods for calculating partial safety factors (Ang and Tang 1984, Ayyub and McCuen 2003, and Ayyub and White 1987) require full probabilistic characteristics of both strength and load variables in the limit state equation. For example, the relevant strength variables for piping are the basic material’s yield strength (stress) Sy, the ultimate strength Su, pipe thickness t, pipe’s diameter D and the ratio D/t, while the relevant load variables are the dead load the internal pressures P in the pipe, the mechanical loads such as water-hammer and seismic loads.

5.2. Strength Variables

5.2.1. Material Properties Tabulated statistical data on the yield strength Sy and ultimate strength Su are provided in this section. These variables are investigated under separate headings.

5.2.1.1. Material Types for Piping Material designation, class, and size for piping as used in various plants are provided in Table 5-1. Ferritic steel, due to its composition, is not hardened by heat treatment. Austenitic steel also is not hardened by thermal treatment, since its composition has a stable structure at normal (room) temperature. Chromium is a hard crystalline metal used as an alloying element to give resistance to heat, corrosion, and wear and increase strength and hardening.



53

Table 5-1. Typical Steel Piping Material Designations Piping Fitting Type

SA106 Gr. B SA333 Gr. 6

SA105 SA420, WPL6

Ferritic

SA335, P22 SA335, P11

SA182, F11 SA234, WP11

Chrome-Moly

SA312, TP304 SA312, TP304L

SA312, TP304 CL 1 SA312, TP316L SA358, TP304

SA358, TP304 CL 1 SA376, TP316 SA376, TP304

SA182, F316L

SA403, WP304

Austenitic

5.2.1.2. Yield Strength of Steel for Nuclear Piping For piping systems in nuclear plants seamless austenitic stainless steels are mainly used. These are iron-base alloys with chromium, Cr, and nickel, Ni, as primary alloying elements and are specifically intended for high-temperature. More common are AISI Type 304 complying with ASTM A312, A376 or A358 A409, or A813, and Types 304L, 316 and 316L. Carbon steels are also used and mainly the SA 106 Grade B steel and SA 333 Grade 6. The carbon steels are being used mostly for the secondary piping systems, Classes 2 and 3, where no radioactive fluids are contained. Usually the type of reactor specifies the material that the piping system is made of. Piping materials for Pressurized-Water Reactors (PWR), Boiling-Water Reactors (BWR) Sodium-cooled Fast and Thermal Reactors (SGR), are as above. Other less used materials for PWR reactors are ferritic steel ASTM 516 Grade 70 clad with Type 308L austenitic stainless steel (Roberts 1981). In Table 5-2 information on yield stress of carbon and stainless steels is provided. Tables 5-3a and 5-3b present experimental data. The yield strength is specified at room temperature by the offset method of 0.2 per cent. Table 5-4a provides statistical data on yield strength such as the coefficient of variation and the yield strength ratio for carbon steel. Table 5-4b shows statistical properties that are given in ASTM document (ASTM data series DS 5S2, supplement to publication DS5). Ware, A.G. (1995) estimated the margins in the ASME Code for nuclear piping stainless steels, Types 304 (Cast and Wrought) and 316. He gave the best fit curves shown in Table 5-3b for different temperatures, x (oF), for the yield stress, y, (ksi). He assumed a normal distribution for the yield stress in order to estimate the confidence level that the ASME Code yield stress have with respect to the experimental data. He concluded that the yield stress value on the best-fit curve could be used as the mean of the yield stress distribution for a given temperature and the ASME Code value (SMYS) as the 97% lower confidence limit.



54

Table 5-2. Data for the Yield Stress of Carbon and Stainless Steels for Nuclear Piping

Steel Type Nominal Sy (ksi)

Min Sy (ksi)

Max Sy (ksi) Reference

SA-106B NA 35 NA Davis (1996) AISI, Type 304 NA 42 NA Lynch (1989) AISI, Type 304 35 NA NA ASM (1961) AISI, Type 304-L 30 NA NA ASM (1961) AISI, Type 304-L NA 39 NA Lynch (1989) Carbon Steels NA 30 40 Courseware (1982) Stainless Steels NA 40 50 Courseware (1982) AISI, Type 308 35 NA NA ASM (1961) AISI, Type 316 NA 42 NA Benjamin (1983) AISI, Type 316 35 NA NA ASM (1961) AISI, Type 316-L NA 39 NA Benjamin (1983) AISI, Type 316-L 30 NA NA ASM (1961) AISI, Type 304 NA 29.73 110.23 Cardarelli (1999) AISI, Type 304L NA 24.66 44.96 Cardarelli (1999) AISI, Type 304LN NA 29.73 NA Cardarelli (1999) AISI, Type 304N NA 34.81 NA Cardarelli (1999) AISI, Type 347 NA 29.73 44.96 Cardarelli (1999) AISI, Type 304 NA NA 35 Macdonald, et al. (1989),

annealed sheet and strip AISI, Type 304L NA NA 38 Macdonald, et al. (1989),

annealed sheet and strip AISI, Type 316 NA NA 40 Macdonald. et al. (1989),


annealed sheet and strip AISI, Type 316LN NA NA 38 Macdonald, et al. (1989),

annealed sheet and strip AISI, Type 321 NA NA 35 Macdonald, et al. (1989),


annealed sheet and strip NA = Not Available



55

Table 5-3a. Experimental Data for the Yield Stress of Carbon and Stainless Steels for Nuclear Piping

Steel Mean(ksi) Mean/SMYS COV Number of Specimens

Reference

SA106-GR B 45.6 1.30 NA NA Scott, et al. (1994) SA106-GRB 43.65 1.25 NA NA ANL (Chopra, et al.

1996) SA106-GRB 43.51 1.24 NA NA Terrell (Chopra, et al.

1996) SA106-GRB 37.5 1.07 NA NA Simmons, et al. 1955 SA106-GRB 35 1.00 NA NA Simmons, etal. 1955 SA106-GRB 33.1 0.95 NA NA Simmons, et al. 1955

SA333-6 43.8 1.25 NA NA Higuchi (1991) SA333-6 55.55 1.59 NA NA Higuchi (1995) Type 304 38 1.27 +-2ksi** NA Stoner, et al. (1991)

SA312-Type 304 37* 1.23 NA NA Wesley (1993) SA 106-GR B 36* 1.02 NA NA Wesley (1993) SA 106-GR B 41.93 1.20 0.076 3 Marschall, et al. (1993)

SA 376-Type 304 36.05 1.20 NA 2 Marschall, et al. (1993) SA 358-Type 304 42.77 1.43 0.027 3 Marschall, et al. (1993) A-515, Grade 60 39.1 1.22 NA 2 Brust, et al. (1994)

Type 316L 37.5 1.50 NA 2 Brust, et al. (1994) Type 316L 37.71 1.51 NA NA Touboul, et al. (1999) AISI 316 39.16 1.31 NA NA Prost, et al. (1983)

*Median Value, **Confidence Interval SMYS = Specified Minimum Yield Strength (See Appendix B) NA = Not Available Table 5-3b. Best Fit Curves for Yield stress of Nuclear Piping Stainless Steel (Ware 1995)

Stainless Steel

Best Fit Curve (Mean Value) Value at x = 70oF

(ksi)

COV Reference

Type 304, Wrought

9.42)2

10(23.82

)4

10(10.13

)8

10(12.5 +−

−−

+−

−= xxxy R2 = 0.81

37.66 0.108

Type 304, Cast

8.41)2

10(423.72

)4

10(04.13

)8

10(12.5 +−

−−

+−

−= xxxy R2 = 0.877

37.09 0.102

Type 316 8.42)2

10(55.62

)5

10(81.63

)8

10(44.2 +−

−−

+−

−= xxxy R2 = 0.862

38.54 0.118

Ware (1995)

Type 304 09.42)2

10(61.62

)5

10(62.73

)8

10(26.3 +−

−−

+−

−= xxxy R2 = 0.866

37.82 0.069

Type 316 76.42)2

10(98.62

)5

10(76.73

)8

10(88.2 +−

−−

+−

−= xxxy R2 = 0.841

38.24 0.100

Sikka, et al. (1977)

tube specimens



56

Table 5-4a. Statistical Data on Yield Strength for Carbon Steel

Type COV of yS yy SS / Ratio Distribution

Low Strength 0.07 1.1 Normal High strength 0.09 1.2 Lognormal

Table 5-4b. Statistical Properties for Yield Strength of Different Grade Stainless Steel (ASTM DS 5S2, 1969)

Grade of Steel Mean Sy COV Sy No of Data Distribution 304 Pipe 37.7 ksi 0.17 14 Lognormal

304 L 33.8 ksi 0.1065 13 Lognormal 316 38 ksi 0.168 14 Lognormal

316L 33.7 ksi 0.19 6 Lognormal 321 29.7 0.027 4 Lognormal 347 38.2 0.136 18 Lognormal

5.2.1.3. Ultimate Strength of Steel for Nuclear Piping Data on the ultimate strength Su of carbon and stainless steels is provided in Table 5-5. Tables 5-6a and 5-6b summarize experimental results on steel strength and Tables 5-7a and 5-7b provide statistics on ultimate strength such as the coefficient of variation and the ultimate strength ratio for carbon steel. It is to be noted that Stevenson et al. (1999) suggest a lognormal distribution for the ultimate strength while Hill et al. (2000) propose an average value of 67.2 ksi with standard deviation 4.05 ksi. Table 5-7b shows statistical properties that are given in ASTM document (ASTM data series DS 5S2, supplement to publication DS5).



57

Table 5-5. Data for the Ultimate Strength of Carbon and Stainless Steels for Nuclear Piping SA-106B NA 60 NA Davis (1996) AISI, Type 304 NA 84 NA Lynch (1989) AISI, Type 304 85 NA NA ASM (1961) AISI, Type 304-L NA 81 NA Lynch (1989) AISI, Type 304-L 80 NA NA ASM (1961) Carbon Steels NA 55 65 Courseware (1982) Stainless Steels NA 78 100 Courseware (1982) AISI, Type 308 85 NA NA ASM (1961) AISI, Type 316 NA 84 NA Benjamin (1983) AISI, Type 316 85 NA NA ASM (1961) AISI, Type 316-L NA 81 NA Benjamin (1983) AISI, Type 316-L 78 NA NA ASM (1961) AISI, Type 347 NA 85 NA Davis (2000) AISI, Type 347 92 NA NA ASM (1961) Type 304 NA 74.7 NA Ukrainian Industrial Energetic

Company Type 304-L NA 70.34 NA Ukrainian Industrial Energetic

Company Type 316 NA 74.7 NA Ukrainian Industrial Energetic

Company Type 316-L NA 70.34 NA Ukrainian Industrial Energetic

Company AISI, Type 304 87 NA NA www.outokumpu.com/stainless,

2004 AISI, Type 304L 84.1 NA NA www.outokumpu.com/stainless,

2004 AISI, Type 316 82.65 NA NA www.outokumpu.com/stainless,

2004 AISI, Type 316L 82.65 NA NA www.outokumpu.com/stainless,

2004 A 312 TP 304 NA 75 NA Rajdeep Metals, Mumbai A 106 Type B NA 60 NA Rajdeep Metals, Mumbai A 312 TP 304L NA 70 NA Rajdeep Metals, Mumbai AISI, Type 304 NA 74.69 150.11 Cardarelli, F. (1999) AISI, Type 304L NA 65.27 89.92 Cardarelli, F. (1999) AISI, Type 304LN NA 74.69 NA Cardarelli, F. (1999) AISI, Type 304N NA 79.77 NA Cardarelli, F. (1999)



58

Table 5-5 (cont.). Data for the Ultimate Strength of Carbon and Stainless Steels for Nuclear Piping AISI, Type 347 NA 74.69 89.92 Cardarelli, F. (1999) AISI, Type 304 NA NA 85 Macdonald, et al. (1989),




annealed sheet and strip AISI, Type 316LN NA NA 85 Macdonald, et al. (1989),



annealed sheet and strip NA = Not Available Table 5-6a. Experimental Data for the Ultimate Strenght Stress of Carbon and Stainless Steels for Nuclear Piping

Steel Mean(ksi) Mean/SMTS COV Specimens Reference SA106-GR B 75.4 1.26 NA NA Scott, et al. (1994) SA106-GR B 72 1.20 NA NA Simmons, et al. (1955) SA106-GR B 71.4 1.02 NA NA Simmons, et al. (1955) SA106-GR B 67.8 0.97 NA NA Simmons, et al. (1955)

Type 304 91 1.21 +-3ksi** NA Stoner, et al. (1991) SA312-Type 304 86* 1.15 NA NA Wesley (1993)

SA 106-GR B 68* 1.13 NA NA Wesley (1993) SA 106-GR B 75.24 1.25 0.011 3 Marschall, et al. (1993) SA106-GRB 82.96 1.38 NA NA ANL (Chopra, et al.

1996) SA106-GRB 75.85 1.26 NA NA Terrell (Chopra, et al.

1996) SA333-6 70.92 1.18 NA NA Higuchi (1991) SA333-6 79.62 1.33 NA NA Higuchi (1995)

SA 376-Type 304 87.65 1.17 NA 2 Marschall, et al. (1993) SA 358-Type 304 105.23 1.40 0.034 3 Marschall, et al. (1993) A-515, Grade 60 63.9 1.065 NA 2 Brust, et al. (1994)

Type 316L 86.6 1.24 NA 2 Brust, et al. (1994) Type 316L 88.33 1.26 NA NA Touboul, et al. (1999) AISI 316 86.3 1.15 NA NA Prost, et al. (1983)

*Median Value, **Confidence Interval SMTS = Specified Minimum Tensile Strength (See Appendix B) NA = Not Available



59

Table 5-6b. Best Fit Curves for Ultimate Strength of Nuclear Piping Stainless Steel

Stainless Steel

Best Fit Curve (Mean Value) Value at x = 70oF

(ksi)

COV Reference

Type 304, Wrought

95)1

10(46.12

)4

10(38.23

)7

10(27.1 +−

−−

+−

−= xxxy R2 = 0.871 85.90 NA

Type 304, Cast

1.93)1

10(56.12

)4

10(56.23

)7

10(43.1 +−

−−

+−

−= xxxy R2 = 0.831 83.38 NA

Type 316 3.69)

110(31.2

2)

410(46.5

3)

710(75.4

4)

1010(28.1 +

−−

−+

−−

−= xxxxy

R2 = 0.957 82.65 NA

Ware (1995)

Type 304 44.95291.1

2)

410(90.1

3)

810(55.9 +−

−+

−−= xxxy

R2 = 0.969 79.56 0.035

Type 316 58.83)

210(35.6

2)

510(45.9

3)

810(39.5 +

−−

−+

−−= xxxy

R2 = 0.954 79.58 0.05

Sikka, et al. (1977)

tube

specimens

Table 5-7a. Statistical Data on Ultimate Strength Su for Carbon Steel

Type COV of uS uS / uS Ratio Distribution

Low strength 0.06 1.05 Lognormal

Table 5-7b. Statistical Properties for Ultimate Strength of Different Grade Stainless Steel (ASTM DS 5S2, 1969)

Grade of Steel Mean Su COV of Su Specimens Distribution 304 Pipe 84.0 ksi 0.063 14 Lognormal, Weibull

304 L 79.2 ksi 0.034 14 Lognormal, Weibull 316 83.3 ksi 0.077 14 Lognormal, Weibull

316L 78.9 ksi 0.037 9 Lognormal, Weibull 321 81.8 0.089 5 Lognormal, Weibull 347 87.0 0.057 18 Lognormal, Weibull

5.2.2. Geometric Properties Tabulated statistical data on pipe diameter D, pipe thickness t and the ratio D/t are provided in this section. These variables are investigated under separate headings.

5.2.2.1. Pipe Diameter In a nuclear power plant, pipes of different diameters are used (for example, as provided by Zhao 1994, and Crocker and King 1967). Dimensional standards (ASME B36.10M and ANSI/ASME



60

B36.19M) provide the diameter, thickness and weight for all piping schedules for both welded and seamless wrought steel as well as stainless steel piping. Diameter tolerances are quite tight and vary with pipe size. These are governed by the requirements of ASME B16.9 (2003). For example for a 4 in pipe the tolerance is 1/16 in, while for 24 in pipe the tolerance is 1/8. For straight pipes these diameter tolerances are for the entire pipe, but for fittings, they only apply to the ends. The basic idea is that it must be possible to create good welds between pipes and fittings. In typical probabilistic risk assessment studies conducted for existing plants the statistical characteristics for diameter are considered to follow normal distribution with a value of twice standard deviation = ± tolerance values. Table 5-8 provides thickness variations for various pipe sizes. In these studies, the diameter and thickness are assumed to have normal distributions and that the variations correspond to a +2σ to -2σ range. For example, 12.5% thickness variation corresponds to the 4σ range, and 5/32 inch is the 4σ range for pipe diameters of Nominal Pipe Size (NPS) 20 and over. Sotberg, et al. (1994) used for a reliability-based design and code calibration of offshore pipelines the following probabilistic characteristics for the diameter of the pipes normal distribution and COV equal to 0.002. Table 5-8. Properties of Nominal Pipe Sizes

Nominal Pipe Size Diameter Variation (in) Thickness Variation ≤ 1.5 +1/64, -1/32 -12.5%

2.0 thru 4 +1/32, -1/32 -12.5% 5.0 thru 8.0 +1/16, -1/32 -12.5%

10.0 to less than 20.0 +3/32, -1/32 -12.5% ≥ 20.0 +1/8, -1/32 -12.5%

5.2.2.2. Pipe Thickness The thickness of the pipe has historically been indicated by a Schedule number. The higher the Schedule number, the thicker the pipe. The Schedule numbers may be approximated by the following equation:

SPnumbersSchedule /1000= (5-1)

where P = the steam pressure (lb/in2); and S = the working stress of pipe material (usually taken as 10 to 15 percent of ultimate Strength). The minimum thickness for fittings and straight pipes is the nominal thickness minus 12.5%. There is no maximum thickness for fittings, but modern fabrication techniques allow manufacturers to create products that do not exceed nominal values by much except, perhaps at points like the intrados of elbows. There is an average weight tolerance for straight pipes that, in effect, limits the average thickness tolerances to 5%. In typical probabilistic risk assessment studies conducted for existing plants the statistical characteristics for thickness are considered to



61

follow normal distribution with a value of twice standard deviation = ± tolerance values. A minimum coefficient of variation for the thickness of 0.03 can be used. Table 5-9 summarizes the probabilistic characteristics used for the reliability – based design for onshore and offshore pipelines and dented pipes. Table 5-9 Probabilistic Characteristics for the Thickness of Pipes

Distribution Mean COV Reference Normal Nominal value (mm) 0.25 / Mean Zimmerman, et al. (1998) Normal NA 0.025 Sotberg, et al. (1994) Normal NA 0.02 Bai, et al. (1997) Normal 0.925*(Nominal) 0.03 Stewart, et al. (2002)

NA = Not Available

5.2.2.3. Diameter-to-Thickness Ratio The diameter-to-thickness ratio (Do/t) is important, because it is used in the calculation of stresses due to internal pressure. The Do is defined as the outer diameter of the pipe, Di is the internal diameter, and t is its thickness. According to the design conditions presented in the ASME BPV Code, Section III, Division 1, pipes are considered to be thin shells. This is a satisfactory approximation, when Di/t ≥ 20 and it means that the hoop stress does not vary within the thickness of the pipe. Therefore, once the outer fiber of the cross-section reaches the ultimate strength, a plastic hinge is formed. Taking into account the tables for standard commercial pipe sizes, it can be seen that the ratio for pipes made of carbon steel varies between 6 to 100, while for stainless steel has an upper limit of 40. The range of this ratio is needed to perform parametric analysis that is essential for the development of the partial safety factors. Also, simulation is used to assess the probabilistic characteristics of this ratio.

5.2.2.4. Summary Based on the previous sections on geometric properties for pipes, it can be concluded that there are a wide range for the pipe diameter, thickness, and Do/t ratio. For example, a range of 0.5 in. to 42 in. for pipe outer diameter as being manufactured by companies in USA is shown in Table 5-10. The statistical characteristics for thickness are considered to follow normal distribution with a value of twice standard deviation = ± tolerance values. Depth-to-thickness ratio (Do/t) varies between 6 and 100 with an upper limit of 40 for stainless steel. Table 5-10 summarizes these ranges.



62

Table 5-10. Statistical Data on Pipe Thickness, Pipe Outer Diameter, and Diameter-to-Thickness Ratio Geometry Type Do (in.) t (in.) Do/t

Min 0.405 0.068 6

Max 80

(30 for stainless steel)

1.062 (0.312 for

stainless steel)

100 (40 for stainless steel)

Distribution Normal Normal Normal

5.3. Load variables As stated previously statistical data on pressure in pipes, on weight and seismic loads sustained by piping are presented. These load variables are investigated under separate headings.

5.3.1. Fluid Pressure in Piping

5.3.1.1. Operating and Design Pressure Most of the reactors have, as a main function, to generate steam to turn a turbine-generator set to produce electricity. More specifically in each reactor a remarkable concentration of heat is produced. Each of the reactor types and its piping has a specific function or combinations of functions and certain characteristics. Generally the operating pressures in primary piping and auxiliary systems may be as low as 150 psig for liquid-metal coolants to as high as 2,500 psig for light-water-cooled reactors. Primary piping carries radioactive steam and water. In Table 5-11, the representative (average, operating) steam pressure for each type of reactor for primary and secondary (steam) piping is presented (Crocker and King 1967, and Lamit 1981). Statistical information for the fluid pressure in pipes must be found and show how the following types of pressure are related to that pressure:

• Internal Design Pressure, • Peak Operating Pressure, • Pressure that coincides with the earthquake load, and • Accidental pressure.

A minimum coefficient of variation of the pressure can be assumed to be 0.04. Bishop, et al. (1993) used a normal distribution for normal operating pressure of pipes.



63

Table 5-11. Operating Pressure at Operating Temperature (Adapted from Crocker and King 1967, Lamit 1981)

Reactor Type Primary Pressure Average (psig) Steam Pressure (psig)

Pressurized-water (PWR) 2,500 at 514 oF 453 at 460 oF Boiling-water (BWR) 1,000 at 546 oF 500 at 825 oF Sodium-cooled Graphite-moderated Thermal Reactor atmospheric 800 at 825 oF

Sodium cooled Fast-breeder Reactor atmospheric 900 at 780 oF

Gas-cooled Reactor Systems 360 1,450 at 1,000 oF Organic-moderated Reactor-Systems 120 450 at 550 oF

5.3.1.2. Peak Pressure The pressure during operation fluctuates according to the systems’ hydraulic characteristics. According to Stancampiano et al, 1976, the boundaries of pressure can be estimated from the ‘‘required action range’’ for pump head specified in the ASME BPV Code , Section XI, and that is between high (+3%) and low (-10%). Hence, for the peak pressure we can say that it can be approximated by 1.03 of the mean pressure. For the operating pressure Stancampiano et al propose a three-parameter Weibull distribution with the following characteristics.

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−−=≤b

acPPP *exp1*)(Prob (5-2)

where P* = a particular value of P, Po = mean value of operating pressure, α = Weibull scale parameter = 0.10665 Po, b = Weibull shape parameter = 4.2121, and c = Weibul location parameter = 0.9 Po.

5.3.1.3. Testing Pressure The ASME BPV Code requires that any new, modified, or repaired piping system is tested in order to verify its ability to withstand the rated pressure and to remain leak tight. In Table 5-12 example maximum and minimum test pressure, the type of test, and the examination pressure for leaks are provided based on the ASME BPV Code. Hydrostatic testing is the preferred method for testing the piping because it uses water that is safer than the air or nitrogen, which are being used for pneumatic tests. Moreover, the amount of work required to compress water to the desirable test pressure is less than the work required to compress air or any other gas.



64

Table 5-12. Test and Examination Pressure (Nayyar, 2000) Code Section Test type Min Test Pressure Examination Pressure NB Hydrostatic 1.25 (Design Pressure) Greater than D.P. or 0.75(T.P.)* NB Pneumatic 1.2 (Design Pressure) Greater than D.P. or 0.75 (T.P.)* NC Hydrostatic 1.5 (Design Pressure) Greater than D.P. or 0.75 (T.P.)* NC Pneumatic 1.25 (Design Pressure) Greater than D.P. or 0.75 (T.P.)* ND Hydrostatic 1.25 (Design Pressure) Greater than D.P. or 0.75 (T.P.)* ND Pneumatic 1.25 (Design Pressure) Greater than D.P. or 0.75 (T.P.)*

*D.P. stands for Design Pressure and T.P. for Test Pressure

5.3.1.4. Accidental Pressure This pressure occurs under accidental conditions. The statistical characteristics are as provided in Table 5-13. Table 5-13. Statistics on Accidental Pressure

Occurrence Intensity Rate per year Duration Mean/Design COV CDF Reference

NA NA 0.90 0.12 Normal Hwang, et al. (1983) 1.7(10-3) 20 min 0.8 0.20 Type I Ellingwood, et al. (1996)

NA = Not Available

5.3.2. Gravity Loads

5.3.2.1. Dead Weight of Pipe For the computation of the weight of the pipe, the following formula is proposed by King (1967):

)(68.10)/(, tDtFftlbpipeofweight o −= (5-3)

where F = the relative weight factor; t = the wall thickness, in; and Do = the outside diameter, in. This weight of pipe calculation is based on low – carbon steel weighing 0.2833 lb/in3 and is extended to other materials through the factor F, which takes the following values: for austenitic stainless steel F = 1.02, for wrought iron F = 0.98, for carbon steel F = 1.00 and for ferritic stainless steel F = 0.95. Normally, the weight of piping can found in piping catalogs from vendors, and the same weight for carbon steel and austenitic stainless steel is used. The weight per foot of steel pipe is subject to tolerances as illustrated in Table 5-14.



65

Table 5-14. Weight Tolerances of Steel Piping (Crocker and King 1967) Specification Size Tolerance

ASTM A376 and A312 12 in and under +6.5% -3.5% ASTM A106 Sch. 10-120

Sch. 140-160 +6.5% -3.5% +10% -3.5%

ASTM A53 Std wt and XS wt +5% -5%

5.3.2.2. Dead Weight of Fittings and Components Except the weight of the pipe itself the dead weight for calculations should include also all the fittings and components that are part of the pipe run, including valves, meters and other special equipment. Because of the relative large diameters and wall thickness of the nuclear pipes the accompanying equipment is usually of greater weight than on a conventional pipeline, especially for items such as nuclear valves with their motorized or hydraulic actuators (Lamit 1981).

5.3.2.3. Insulation All the nuclear pipes connecting the nuclear vessel with the steam generator, turbines and condenser require high quality insulation in order to protect surrounding equipment and instrumentation. For these pipes the insulation is never permanently bonded, it can be easily removed and stored during the regular inspection of the pipes. Block insulation, metal insulation and blanket insulation have been used in such cases. Most protective insulation must be clad with stainless steel or aluminum sheets. For pipes that inspection is not necessary, common commercial and industrial piping insulation materials that have resistance in high temperatures and reduce heat losses are used. The most common of these materials are provided in ASTM Specifications. The weight of pipe insulation W is given by the relation (Helguero, 1983):

)(0218.0)/( TDTIftlbW o += (5-4)

where T = insulation thickness, Do = outside diameter of pipe (in), I = insulation density (lb/ft3). Table 5-15 gives values for I for some insulation materials.



66

Table 5-15. Insulation Density I, (Helguero V.M., 1983 and Kannappan, S.*, 1986) Material Density, I (lb/ft3)

Calcium Silicate 11, 12.25* 85% Magnesium 10 to 11 Thermobestos 11.53 Kalo 19 to 21 High Temperature 24 Super -X 25 Poly-Urethane 2.3, 2.0* Mineral Wool 8 Fiber Glass 3.25* Foam Glass 8.50* Polystyrene 2.00*

5.3.2.4. Contents of Pipe Crocker and King (1967) propose for the weight (lb/ft) of contents of a pipe as follows:

2)2(3405.0 tDGpipeofcontentsofweight −= (5-5)

where G = the specific gravity of contents; t = the pipe thickness, in; and D = the pipe outside diameter, in. Different coolants are used for different types of reactor. The density of these coolant materials varies considerably. Gas cooled reactors have coolant densities below 0.1 lb/ft3 and water cooled reactors may have coolant densities over 60 lb/ft3. According to NCIG-05 Revision-1, the uniformly distributed weight (including the insulation and fluid weights) as well as the concentrated weight due to valves, flanges etc. may vary by ± 20% from the as-analyzed weight. In typical probabilistic risk assessment studies conducted for existing plants the statistical characteristics for dead weight are considered as follows to follow normal distribution with a value of twice standard deviation = ± tolerance values.

5.3.3. Non-Reversing Mechanical Loads Such loads include water hammer and pressure surge. Water hammer refers to shocks sounding like hammer blows produced by a rapid change of fluid flow velocity in a closed pipeline. It can happen due to rapid closure of the valve where the fluid stops suddenly. The kinetic energy is then converted into pressure energy. The pressure rise causes in turn elastic waves that travel upstream and downstream from the point of origin. These elastic waves cause increases or decreases in pressure that are called water hammer surge or transient pressure. In most cases water hammer occurs during a plant startup or during return of an isolated plant system into service or when safety valves are actuated. The discharge piping is more susceptible to transient hydrodynamic loads, since the opening time of safety valves is very fast and can induce large hydrodynamic loads on the downstream piping, especially when loop seals are present. The pressure rise, P, for instantaneous valve closure is given in the following equation (AWWA Manual, 2004).



67

gVWaP

144= (5-6)

where W is the weight of fluid (lb/ft3), V is the velocity of flow (fps), g = 32.2 fps/sec, and α is the magnitude of the surge wave velocity, which is independent of the length of the pipe and for steel is equal to:

fps))100/(1(

4660tD

ai+

= (5-7)

where, Di is the inside diameter of the pipe and t its thickness. The pressure rise exert a force to the pipe which is equal to the pressure times the cross sectional area of the pipe. The water hammer mechanism can be also generally described by Joukowsky’s Law as follows:

uαρp ∆=∆ (5-8)

where ∆p = the dynamic pressure change resulting from the change of the flow velocity in the pipe by an amount ∆u, ρ = the density of the fluid, α = the speed of a pressure wave in the fluid flowing in the pipe. Table 5-16 presents the statistics on Safety Relief Valve (SRV) discharge loads as presented in the consensus estimation studies of Hwang, et al. (1983), and Table 5-17 shows the statistical properties of Safety Relief Valve (SRV) discharge loads, presented in other papers. SRV loads occur mostly in BWR plants. Table 5-16. Statistics on Safety Relief Valve (SRV) Discharge Loads, (Hwang, et al. 1983)

Load Case A B C D

Design Value (psi d) 13.49 17.40 16.46 28.23 Design Value (bar d) 0.93 1.20 1.14 1.95 Predicted Value (bar d) 0.60 0.79 0.74 1.11 Mean/Design Value 0.65 0.66 0.65 0.57 Variance 0.00357 0.00407 0.00363 0.0154 Standard Deviation 0.0597 0.0638 0.0602 0.124 COV 0.10 0.08 0.08 0.11 No of Occurrence in 40 years

271 1313 NA 1620

Occurrence Rate per year 6.775 32.825 NA 40.5 A: First actuation of one or two valves (100oF suppression pool) B: First actuation of three or more adjacent valves (100oF suppression pool) C: First actuation of an ADS valve (120oF suppression pool) D: Subsequent actuation of a single valve (120oF suppression pool)



68

Table 5-17. Statistics on Safety Relief Valve (SRV)

Occurrence Intensity Rate per year Duration Mean COV CDF Reference

N/A 1sec 0.8PSRV 0.14 Normal Ellingwood, et al. (1996) N/A = not available

5.3.4. Seismic Loads The seismic design of piping involves many complexities and uncertainties that the designers try to overcome by adding a great number of rigid supports and snubbers to the nuclear piping systems in order to lower the seismic stresses. Parameters that add uncertainties are:

• The calculation of soil and building response and their interaction with the piping system. • The damping of the piping system. • The method of analysis that is linear elastic and therefore does not take into account the

material hardening, the load redistribution. Existing nuclear power plants define their plant seismic margins in terms of the High-Confidence-Low-Probability-of-Failure (HCLPF) seismic capacity (USNRC, 1991). The ratio FH of acceleration levels for HCLPF to the acceleration levels for the SSE ground motion give the seismic margin for the SSE design basis for new plants, i.e.,

SSEHCLPFH AAF /= (5-9)

The USNRC (1997) has also required that the SSE ground motion response spectrum be defined with a specified annual frequency of exceedance. The seismic margins are calculated using a conventional seismic probabilistic risk assessment (PRA). Key elements of a conventional seismic PRA are: (1) Probabilistic seismic hazard analysis (PSHA) that specifies seismic hazard curves, (2) seismic fragility evaluation of key structural systems such as buildings, equipment, piping, etc., (3) plant systems and accident analysis including development of fault/event trees and events, and (4) estimation of core damage frequencies and the frequencies of serious releases. A seismic hazard analysis is performed by experts in ground motion and is based on studies related to seismic sources, seismic activity rates, attenuation of seismic ground motion, etc. One can think of the hazard curves as the information on the earthquake input to be used in the evaluation on seismic fragility for structural systems. Seismic fragility is thus the conditional probability of failure for a structural component or system for a given level of seismic input that is typically defined in terms of peak ground acceleration. Seismic fragilities are convolved with seismic hazard to quantify the fault trees and used to estimate the failure frequency of initiating events. Evaluation of seismic fragility requires information on the best estimates of design parameters, and design and qualification procedures for structural components and systems. A probabilistic response analysis of the structural components and systems should then be conducted to evaluate the fragility. To do so, it is essential to identify the critical failure modes of components as well as the system and to identify the interdependencies of failure modes in interacting components or subsystems. Consequently, evaluation of seismic fragility is a critical



69

step in seismic PRA. In the current practice, a family of fragility curves is generated for a particular component in terms of the best estimate of the median ground acceleration capacity Am. The HCLPF capacity, A, is given by

URmAA εε= (5-10)

in which εR and εU are random variables with unit medians representing the inherent randomness about the median and the uncertainty in the median value, respectively. Both these random variables are assumed to be lognormally distributed with logarithmic standard deviations βR and βU, respectively. These assumptions and the above model are used to facilitate the development of fragility curves such that fragility at each acceleration value a is represented using a probability density function such as

⎥⎥⎦

⎤

⎢⎢⎣

⎡

βφβ+

φ=′−

R

1Um )Q()A/aln(

f (5-11)

where ]/[ affPQ ′<= , that is, the probability (confidence) that the conditional probability of failure f is less than f ′ for a peak ground acceleration a, and φ-1 is the inverse of the standard Gaussian cumulative distribution function. For convenience in developing an estimate of the fragility parameters, the current practice uses an intermediate random variable called factor of safety F on ground acceleration capacity above the safe shutdown earthquake (SSE) level ASSE specified for design, i.e. A = F ASSE. The median factor of safety is then directly related to the median ground acceleration capacity Am as

SSE

mm A

AF = (5-12)

The logarithmic standard deviations of Fm, representing the inherent randomness and uncertainty, are identical to those for the HCLPF capacity A. The factor of safety is further simplified by expressing it as a product of factors corresponding to the capacity FC and the structural response FR. The structural response factor FR in itself is modeled as a product of factors influencing the variability in the response of structural components or systems due to an uncertainty in the structural parameters of the building, equipment or piping. These structural parameters represent an uncertainty in the deterministically evaluated frequencies, damping ratios etc. As the seismic design calculations for components (mechanical, electrical or structural) in a nuclear plant is conservative Ellingwood (1994) proposes a model of capacity in earthquake loads in order to obtain best estimates of in-situ median capacity and variability in terms of the following parameters:

µFFF stmC == factorCapacityMedian (5-13)

where Fst = the margin of strength over design strength; and Fµ = the factor that accounts for additional capacity provided by energy dissipation and ductility.

...ssmdrRm FFFFF = (5-14)



70

FRm is the median response factor that takes into account differences between the actual response and the response computed from the design spectrum assumptions taking into account damping, conservatism in modeling soil-structure interaction, duration of earthquake, spectrum shape, and any other factor of this kind. The median capacity, Am, of the component in a nuclear power plant can be modeled in the following form:

SSEmRCSSEmm AFFAFA m== (5-15)

where mm RC FF , = median capacity and response factors, respectively, defined as above; and ASSE = the safe-shutdown earthquake acceleration. In the initial studies, the HCLPF seismic capacity as described above was defined at 95% confidence of less than about 5% probability of failure (in the existing plants, 95% and 5% are still being used). However, USNRC (1991a) recommended that the HCLPF seismic capacity can be approximated as the 1% composite (mean) probability of failure. Such an approximation simplifies the process described above in the sense that the estimates of variability do not need to be separated into “uncertainty” and “randomness”. Instead, one can specify a mean fragility curve that gives the variation of mean probability of failure with respect to the ground motion acceleration levels. In addition, seismic PRA studies conducted on existing nuclear power plants reported in ERI/NRC 94-502 (ERI 1994) have shown that for existing plants 25.1≥HF . Also, the target goal for the standardized Advanced Light Water Reactor (ALWR) has been set at 67.1≥HF . Seismic PRA studies based upon the pre-1994 ASME Code seismic design criteria have concluded that piping does not control the plant HCLPF seismic capacity which is a desirable goal for future designs. Piping is estimated to not control the plant HCLPF capacity when the probability of piping failure Pf at plant HCLPF capacity is substantially less than 1% (Kennedy, 1996). To achieve such a goal, it was suggested that %1.0<fP at Plant HCLPF Seismic Capacity Level. To achieve the desired seismic margins in piping, let us consider the median factor of safety defined above in Eqs. 5-12 and 5-15. This factor is conventionally assumed to be log-normally distributed such that its logarithmic standard deviation β can be obtained from the corresponding βC and βR values associated with the median capacity and response factors mCF and mRF , respectively.

222RC βββ += (5-16)

It must be noted that specification of variability in terms of “randomness” and “uncertainty” is not needed while specifying βC and βR. As discussed earlier, the plant HCLPF is defined at the 1% probability of failure. For a log-normal distribution, one can then write

βfXmH eFF −= (5-17)

Note that Xf = 2.326 for 1% probability of failure and Xf = 3.090 for 0.1% probability of failure. Kennedy and Short (1994) evaluated capacities using Code specified minimum ultimate strength,



71

Code specified limit-state, and Code specified Service Level D limits. This study specifies a capacity factor %1CF greater than unity on the 1% probability. Therefore, the median capacity factor is expressed as

β326.2%1 eFF CCm = (5-18)

Standard Review Plan (USNRC 1991b) specifies the guidelines for seismic response analyses which are aimed at evaluating the seismic response at 84% probability. Most response analyses are excessively conservative (Gupta and Gupta 1994, and Zhao 1994). Therefore, the response factor %84RF is greater than unity when compared to the probabilistically computed response at 84% probability level. Consequently, one can write

Rm eFF RR

β%84= (5-19)

The above equations can be combined to express the capacity factor %1CF corresponding to 1% plant HCLPF as

1)(%84

%1−

⎟⎟⎠

⎞⎜⎜⎝

⎛= βf

FFFR

HC (5-20)

in which,

ββββ

fCR Xef −+= 326.2 (5-21)

Kennedy et al. (1997) consider a range of βC from 0.3 to 0.6 and βR from 0.2 to 0.4 for piping systems to tabulate values of 1)( −

βf as given in Table 5-18. Kennedy et al. (1997) calculated the mean of various values in the above table as 1.42 and the COV as 0.07 indicating that the variation in 1)( −

βf values is small and that it can be considered

nearly constant. They recommend that 1)( −βf be defined at 84% probability which gives

5.1)1()()( 11 =+= −− COVff mββ (5-22)

While Kennedy, et al. (1997) recognize the excessive conservatism in %84RF , they recommend

that credit cannot be taken for such conservatism as situations may exist wherein 0.1%84 ≈RF .

They recommend 0.1%84 ≥RF for existing plants and 25.1%84 ≥RF standardized ALWR. The higher number for standardized ALWR plants is due to additional conservatism introduced in the design by consideration of seismic responses that are calculated from an envelope of the responses computed over many sites. As described earlier, 25.1≥HF for existing plants and

67.1≥HF for the standardized ALWR. Consequently, Kennedy, et al. (1997) evaluate %1CF as 1.875 for existing plants and 2.0 for ALWR. Kennedy, et al. (1997) also calculated the sensitivity of this factor to the selected probability of failure value of 0.1% at plant HCLPF level by varying the probability values between a lower bound of 0.02% and an upper bound of 0.5%.



72

The corresponding values of %1CF were found to lie between 1.4 and 2.7. The authors

recommend a range for %1CF between 1.5 and 2.5 centered on 2.0. Finally, it must be noted that the above discussion based on various existing studies focuses on specification of plant HCLPF at 1% probability that corresponds to the occurrence of a safe shut down earthquake (SSE). Therefore, these discussions relate to Service Level D for piping design. For other service levels, the required capacity margin needs to be defined at some higher probability failure – say Y%. Kennedy, et al. (1997) illustrate that for Y = 5%, the COV for

1)( −βf is 0.12 compared to 0.07 for Y = 1%. Furthermore, they illustrate that for Y > 1%, the

largest value of 1)( −βf always occurs at βC = 0.6. Therefore, they conservatively define %YCF as

6.0)326.2( %%1%

YY

XCC eFF −= (5-23)

Table 5-19 shows recommended cumulative distribution functions and statistical properties for the intensity of earthquake loads used in different studies. Hwang, et al. (1987) used a probability distribution function FA(α) of the annual peak ground acceleration, A, is a Type II distribution of extreme values. Therefore, the seismic hazard curves are given by the equation:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=−=

−a

AA µααFαG exp1)(1)( (5-24)

in which µ, α are the two distribution parameters. The parameter, α, is related to the coefficient of variation in the peak ground acceleration. In the study only one value of α = 2.7 is selected for the whole country, in order for the results not to be tied to a specific site. The coefficient of variation in A is implied to be 0.85. The parameter µ is computed by assuming that the annual probability of exceeding the SSE at any site is approximately 4x10-4/ year. During the earthquake the ground acceleration is idealized as a segment of a zero mean stationary Gaussian process with the following power spectral density function:

2

2

22

22

41

41

)(

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

gg

g

gg

ogg

ωωζ

ωω

ωωζ

SωS (5-25)

Parameters ωg and ζg depend on the soil conditions of the plant area. If it is rock ωg= 8π (rad/sec) and if it is deep cohesionless soil it is ωg= 5π (rad/sec). For both cases ζg = 0.6. The parameter So is related to intensity (peak ground acceleration of earthquake) and its distribution can be derived from the distribution of A.



73

Ellingwood, 1995 gives the following hazard curve that is related to the plant site.

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=−=

−β

AA µααFαG exp1)(1)( (5-26)

where α = ground acceleration and µ, β the parameters of the distribution. β is related to the COV of the annual effective peak acceleration. If an overall value for β is used, as Hwang et al (1987) did, it is estimated to be 2.3 with COV = 1.38. This is considered enormous and therefore different values of β should be considered for each location, whereas some example values are given for Western US, namely β = 5.5 with COV = 0.28 and for Eastern US, β = 2.3 with COV = 1.38. The coefficient of variation for Eastern US remains high, since there is uncertainty due to absence of historical events in this area during the period of modern instrumentation.

The pulse intensity and the distribution of annual extreme effective peak ground acceleration can be related with the following equation:

EAAE λaGαFλαF /)(1)(ln)/1(1)( −≈+= (5-27)

From this relation the minimum acceleration for a structurally damaging earthquake can be estimated to be 0.05g. The parameter λE varies from 0.01/yr for Eastern US to 0.5/yr for sites in Western US. Moreover, the CDF of the annual extreme earthquake structural action can be considered also Type II with a coefficient of variation between 0.8 and 1.3.

Table 5-18. Values of 1)( −

βf as provided by Kennedy, et al. (1997) 1)( −

βf βC

βR = 0.2 βR = 0.3 βR = 0.4 0.3 1.24 1.37 1.56 0.4 1.29 1.37 1.52 0.5 1.35 1.40 1.52 0.6 1.43 1.46 1.54



74

Table 5-19. Probabilistic Characteristics for the Seismic Load Occurrence Intensity

Rate per year Duration Mean COV CDF Reference 0.02 30sec Site-dep. 0.85 Type II Ellingwood (1995) 0.05 30sec (0.08Es) 0.90 Type II Ellingwood, et al. (1996)

lnFA(0.05)/year 10-20sec Eqs 5-23 & 5-24 0.85 Type II* Hwang & Ellingwood, et al. (1987)

2/year 60sec NA 0.70 Type I Casciati (1983) 1 to 4/year 60sec NA 0.35-0.70 Type I or II Casciati, et al. (1982)

NA = Not Available, *the ground acceleration is described by normal distribution. Intensity refers to peak ground acceleration

5.3.5. Summary Table 5-20 summarizes the statistical data on load variables for piping that include operating or design pressure, static loading, and earthquake loads. The coefficient of variation for the intensity of an earthquake was based on the work of Ellingwood (1994 and 1995). Table 5-20. Statistical Data on Load Variables for Piping

Load Type Design Pressure (psig)

Weight Load (mean/design)

OBE Seismic Load

SSE Seismic Load

Min 50 0.84 Eq. 5-22 Eq. 5-22 Max 2,500 1.00 Eq. 5-22 Eq. 5-22 COV 0.05 0.10 0.85 0.85

Distribution Lognormal Normal Type II Type II NA = not available



75

6. Modeling Uncertainty

6.1. Background This chapter analyzes the modeling uncertainty associated with prediction models used to obtain nominal values in the design of pipes compared to experimental results. The objective herein is to evaluate the bias, which is the ratio of the true (mean) value of a variable based on experimental results to its nominal value calculated according to a model. First the bias (Experimental Value / Nominal Value) of models used for the prediction of the hoop stress and in consequence of the pipe’s thickness is presented and then the bias in the prediction of pipes bending moments. This bias can be used to compare the conservatism (bias>1) or non conservatism (bias<1) of different models. The COV of the bias should be small and combined with the COV that reflects the basic, ‘‘inherent’’, statistical variability of the load and resistance variables.

6.2. Hoop Stress In the ASME Codes the calculation of the pipe thickness is based on the concept that the pipe should resist the tension exerted by the circumferential or hoop stress due to internal pressure of gas or liquid. Hoop stress is greater (twice when the cylinder is considered thin) than the longitudinal stress and therefore the critical stress for the selection of the thickness of the pipe. When the pressure in a pipe is raised to a bursting point, the pipe splits along a longitudinal line. Therefore, generally in pipes, any existing longitudinal joints should also be stronger than the girth joints. The way the thickness is calculated depends on the calculation of the hoop stress, which in the ASME BVP Code is called allowable stress Pα. Once the pressure design thickness, t, is calculated, in order to find the minimum wall thickness, tm, an additional thickness A should be added, which accounts for corrosion or erosion allowance, for the material that is removed in threading, and for the material required for structural strength of the pipe during erection. Therefore, the wall thickness is

Attm += (6-1)

The subsequent sections provide information on the theories (models) for the calculation of hoop stress and pipe thickness. After the introduction of theory, the comparison is performed and summarized.



76

6.2.1. Lamé or Thick-Wall Theory A precise calculation procedure of wall thickness of a pipe subjected to an internal pressure (P) is the one developed by Lamé (ca 1833) for thick-wall cylinders. It is the most general formula for any Di/t ratio depending on the radial distance, r, within the thickness of the pipe. The hoop stress obtains its largest value at r = Di/2 = r1 and the smallest at r = Do/2 = r2, where Di is the inside diameter of the cylinder and Do the outside diameter. The hoop stress is

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−= 2

22

21

22

21 1

rr

rrrPσH (6-2)

For r = r1, the stress is

)()(

21

22

22

21

max rrrrPσH

−

+= (6-3)

For r2 = r1 + t and Di = 2r1, the above equation becomes

( )

PtDt

Dσi

iH

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+= 1

)1)/(2 2

2

max (6-4)

6.2.2. Thin-Wall Theory In the thin-wall cylinder theory, the hoop stress is considered to be constant within the thickness of the pipe, and the hoop stress is given simply by the relation:

t

DPσ i

2=Η (6-5)

or

H

i

σDPt

2= (6-6)

6.2.2.1. The Barlow Formula When the inside diameter, from the thin theory is substituted by the outside one, one gets the formula also known as the Barlow formula, which is:

t

DPσ o

2=Η (6-7)

And solving for t, it is

H

o

σDPt

2= (6-8)

This equation according to Becht (2004) is always conservative.



77

A model similar to Eqs. 6-5 and 6-7 can also be used to express the hoop stress, having the mean diameter, Dm rather than the inside or the outside diameter. These approaches are nevertheless similar, when the pipe is really thin, that is if t<<<, then Do ≈ Di ≈ Dm. Since in many cases the assumptions of thin theory are not satisfied, these formulae give different results in comparison to experimental analysis. As the ASME Code should be applicable to a wide variety of pipes considering not only their functionality but also their geometry, all these formulae need to be examined.

6.2.2.2. The Boardman Equation or Modified Lamé The Boardman equation is an empirical approximation with no theoretical basis. It is used in the ASME BVP Code, and is like calculating the hoop stress in an intermediate diameter between the inside and the outside values. Hence, the hoop stress is given by the relation:

t

tyDPσ oH 2

2−= (6-9)

where y = 0.4 and for Do/t<6, y = Di/(Do+Di) Solving the above equation for t, the pressure design thickness is obtained that also is used in the Code. Thus,

)(2 Pyσ

PDtH

o

+= (6-10)

Substituting in the previous equations σH with the allowable stresses of the material as defined in the ASME BPV Code, that is Sm for Class 1, S for Class 2 and SE for Class 3, where E defines the joint efficiency for the type of longitudinal joint used as given in Table ND-3613.4-1 of the ASME BPV Code, 2002, we get some of the formulas that exist in the ASME Piping Codes. For example, from Eq. 6-10 we get Eqs. NB-3641.1-(1), NC-3641.1-(3) and ND-3641.1-(3) of the ASME BPV Code, 2002. In this Code under section NB-3649 exists also the Barlow formula that is used for calculating the pressure of the weakest pipe that is attached to a piping product, whose design pressure needs to be calculated, and for the calculation of longitudinal stress due internal pressure. It is necessary to clarify that all the above models are based on the Tresca Criterion, while some equations derived from the application of the Von Mises Criterion and others, considering the strain hardening of the material obeying different laws, are presented in Section 6.2.3. Moreover, these models predict burst pressure which is related to the ultimate strength of the steel. If the ultimate strength is substituted by the yield strength, the pressure that yields the pipe is obtained. As shown in subsequent section, experimental results that provide yield pressure are limited.



78

6.2.3. Other Models Table 6-1 summarizes models for the calculation of burst pressure, considering a strain hardening material, obeying different power laws, while either the Tresca or Von Mises Yield Criterion is used. These equations though are not compared with experimental results. The comparison is limited only for the models presented in Sections 6.2.1 and 6.2.2 that are models related to those used in ASME codes.



79

Table 6-1 Burst Pressure for Strain Hardening Material Criterion Linear-Elastic, Perfectly-Plastic Reference

Tres

ca

iDoD

yfyP ln= Zhu, et al. 2005, Steward, et al. 1994

Strain Hardening Material Reference

Tres

ca

ufmD

t

nuP2

2

1=

with corresponding ultimate strain

2

nuε =

Based on pure power-law curve

Zhu, et al. 2005, Steward et al. 1994

Closed-End Cylinders

( ) ufmD

tnuP

41

3

1+=

where the strengthening coefficient n is 596.0

1239.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

yfufn


3

nuε =

Zhu, et al. 2005, Steward, et al. 1994, Cooper, 1957,

Svensson, 1958

Closed-End Cylinders n

oDuft

uP ⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

=3

12

3

2


2

nuε =

Open-End Cylinders

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

=oDuft

uP2

3

2


3

2 nuε =

Gerdeen, 1976

Von

Mis

es

Closed-End Cylinders

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

−

+=

toD

tyfufuP

075.23

4

Material follows the Ramberg - Osgood power law

Kirkemo, 2001



80

6.2.4. Experimental Results This section presents a comparative evaluation of the pipe stress models provided by Eqs. 6-4, 6-6, 6-8 and 6-10 with experimental burst results of pipes without defects that were found in literature. As shown in Table 6-1 these tests provide the burst pressures of pipes. Only few tests exist, which measure the yield stress that precedes the burst of the pipe. That is because is difficult to measure exactly this stress or define exactly when it happens. Nevertheless, Table 6-2 gives also some calculations, using the yield stress of the material fy instead of the ultimate strength fu for the ratios of experimental yield pressure to nominal value in Eqs. 6-11 to 6-14. where,

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+

===

1)1)/((2 2

2

burstburstlame

tDtD

fP

PP

xxx

i

i

unn (6-11)

i

unD

ftP

PPxx 2

burstbursttheorythinint_thin === (6-12)

tyDft

PP

Pxxx

o

unn2

2burstburst

board

−

=== (6-13)

o

unD

ftP

PP

xx2

burstburstbarlowthin === (6-14)


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

81

Tabl

e 6-

1. C

ompa

rativ

e Ev

alua

tion

of P

ipe

Bur

st S

tress

for a

ny D

i/t

Do

(mm

) D

i (m

m)

t (m

m)

Di/t

f y

(MPa

) f m

(M

Pa)

f u

(MPa

) p y

(exp

) p b

urst

la

me

(u)

boar

d (u

) th

in (u

) x l

ame

(6-1

2)

x boa

rd

(6-1

4)

x thi

n (6

-15)

x i

nt_t

hi

(6-1

3)

Ref

eren

ce

88.9

80

.9

4 20

.2

336

411

486

25.5

44

.8

45.6

9 45

.37

43.7

3 0.

98

0.99

1.

02

0.93

W

ellin

ger a

nd S

turm

(1

971)

88

.9

71.3

8.

8 8.

1 32

4 39

0.5

457

57.8

97

.4

99.2

2 98

.26

90.4

7 0.

98

0.99

1.

08

0.86

101.

6 81

.6

10

8.2

284

346

408

97

.5

88.0

3 87

.18

80.3

1 1.

11

1.12

1.

21

0.98

139.

7 11

4.7

12.5

9.

2 26

6 33

3 40

0

74.7

5 77

.86

77.1

0 71

.58

0.96

0.

97

1.04

0.

86

88

.9

80.9

4

20.2

51

2 57

7 64

2

61.8

60

.36

59.9

3 57

.77

1.02

1.

03

1.07

0.

97

88

.9

71.3

8.

8 8.

1 50

6 57

0 63

4

153.

1 13

7.64

13

6.31

12

5.52

1.

11

1.12

1.

22

0.98

101.

6 81

.6

10

8.2

689

714.

5 74

0

183

159.

67

158.

12

145.

67

1.15

1.

16

1.26

1.

01

13

9.7

114.

7 12

.5

9.2

648

675

702

15

2 13

6.65

13

5.31

12

5.63

1.

11

1.12

1.

21

0.99

610

573.

6 18

.2

31.5

63

6 64

0.5

645

41

.76

39.6

3 39

.43

38.4

9 1.

05

1.06

1.

08

1.02

M

axey

(198

6)

610.

1 57

2.3

18.9

30

.3

563

576

589

37

.86

37.6

2 37

.42

36.4

9 1.

01

1.01

1.

04

0.97

610.

1 57

2.3

18.9

30

.3

607

618.

5 63

0

40.7

9 40

.24

40.0

2 39

.03

1.01

1.

02

1.05

0.

98

411.

1 38

4.1

13.5

28

.5

364

443.

5 52

3

36.5

35

.47

35.2

8 34

.35

1.03

1.

03

1.06

0.

99

Roy

er a

nd R

olfe

(1

974)

40

3.6

378

12.8

29

.5

807

838

869

59

.6

56.8

6 56

.55

55.1

2 1.

05

1.05

1.

08

1.01

38.1

35

.7

1.2

29.8

48

4.2

33

31

.46

31.2

9 30

.50

1.05

1.

05

1.08

1.

01

38

.1

34.9

1.

6 21

.8

484.

2

43

42.3

7 42

.08

40.6

7 1.

01

1.02

1.

06

0.97

50.8

48

.4

1.2

40.3

48

4.2

23

23

.41

23.3

2 22

.88

0.98

0.

99

1.01

0.

96

Pret

oriu

s, e

t al.

(199

6)

50.8

47

.6

1.6

29.8

48

4.2

22

31

.46

31.2

9 30

.50

0.70

0.

70

0.72

0.

68

Sta

inle

ss s

teel

Typ

e 30

4L

63.5

61

.1

1.2

50.9

48

4.2

26

.5

18.6

5 18

.58

18.3

0 1.

42

1.43

1.

45

1.39

pi

pes

fixed

at t

he

ends

63

.5

60.3

1.

6 37

.7

484.

2

27

25.0

1 24

.90

24.4

0 1.

08

1.08

1.

11

1.05

76.2

73

.8

1.2

61.5

48

4.2

15

.9

15.4

9 15

.44

15.2

5 1.

03

1.03

1.

04

1.01

76.2

73

1.

6 45

.6

484.

2

22

20.7

6 20

.68

20.3

3 1.

06

1.06

1.

08

1.04

916.

1 87

1.1

22.5

38

.7

526

567

608

27

.93

30.6

0 30

.46

29.8

7 0.

91

0.92

0.

94

0.89

G

raha

m, e

t al.

(199

4)

172

152.

4 9.

8 15

.6

602

689

776

86

.6

93.4

3 92

.65

88.4

3 0.

93

0.93

0.

98

0.87


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

82

Tabl

e 6-

1 (C

ont.)

. Com

para

tive

Eval

uatio

n of

Pip

e B

urst

Stre

ss fo

r any

Di/t

D

o (m

m)

Di

(mm

) t

(mm

) D

i/t

f y (M

Pa)

f m

(MPa

) f u

(M

Pa)

p y(e

xp)

p bur

st

lam

e (u

) bo

ard

(u)

thin

(u)

x lam

e (6

-12)

x b

oard

(6

-14)

x t

hin

(6-1

5)

x int

_thi

(6

-13)

R

efer

ence

508

492.

15

7.92

5 62

.1

43

50.5

58

1.

54

1.91

1.

84

1.83

1.

81

1.04

1.

04

1.06

1.

02

Nak

ai, e

t al.

(198

2)

508

493.

73

7.13

7 69

.2

43.2

49

.35

55.5

1.

43

1.73

1.

58

1.58

1.

56

1.09

1.

10

1.11

1.

08

Pre

ssur

e: k

g/m

m2

609.

6 59

6.9

6.35

94

.0

43.9

49

.65

55.4

1.

04

1.2

1.17

1.

16

1.15

1.

03

1.03

1.

04

1.02

609.

6 59

5.33

7.

137

83.4

54

.3

56.7

5 59

.2

1.32

1.

54

1.40

1.

40

1.39

1.

10

1.10

1.

11

1.08

609.

6 59

0.55

9.

525

62.0

44

.6

51.2

5 57

.9

1.59

1.

83

1.84

1.

83

1.81

1.

00

1.00

1.

01

0.98

38.1

35

.7

1.2

29.8

48

4.2

41

.41

31.4

6 31

.29

30.5

0 1.

32

1.32

1.

36

1.27

38.1

34

.9

1.6

21.8

48

4.2

55

.86

42.3

7 42

.08

40.6

7 1.

32

1.33

1.

37

1.26

50.8

48

.4

1.2

40.3

48

4.2

29

.5

23.4

1 23

.32

22.8

8 1.

26

1.27

1.

29

1.23

Pr

etor

ius,

et a

l. (1

996)

50.8

47

.6

1.6

29.8

48

4.2

39

31

.46

31.2

9 30

.50

1.24

1.

25

1.28

1.

20

Sta

inle

ss s

teel

Typ

e 30

4L

63.5

61

.1

1.2

50.9

48

4.2

23

.5

18.6

5 18

.58

18.3

0 1.

26

1.26

1.

28

1.24

63.5

60

.3

1.6

37.7

48

4.2

33

.4

25.0

1 24

.90

24.4

0 1.

34

1.34

1.

37

1.30

76.2

73

.8

1.2

61.5

48

4.2

18

.9

15.4

9 15

.44

15.2

5 1.

22

1.22

1.

24

1.20

76.2

73

1.

6 45

.6

484.

2

26.7

20

.76

20.6

8 20

.33

1.29

1.

29

1.31

1.

26

M

ean

1.09

0 1.

096

1.13

1 1.

043

Varia

tion

0.02

2 0.

021

0.02

2 0.

022

σ 0.

148

0.14

5 0.

148

0.14

9

C

OV

0.13

6 0.

132

0.13

1 0.

143


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

83

Tabl

e 6-

2. C

ompa

rativ

e Ev

alua

tion

of P

ipe

Yie

ld S

tress

bef

ore

Bur

st fo

r any

Di/t

D

o (m

m)

D

i(mm

) t

(mm

) D

i/t

f y(M

Pa)

p y(e

xp)

lam

e (u

) bo

ard

(u)

thin

(u

) th

in_i

nt

(u)

x lam

e (6

-12)

x b

oard

(6

-14)

x t

hin

(6-1

5)

x int

_thi

(6

-13)

R

efer

ence

88.9

80

.9

4 20

.2

336

25.5

31

.59

31.3

7 33

.23

30.2

4 0.

81

0.81

0.

77

0.84

88.9

71

.3

8.8

8.1

324

57.8

70

.34

69.6

6 79

.98

64.1

4 0.

82

0.83

0.

72

0.90

W

ellin

ger a

nd S

turm

(197

1)

508

492.

2 7.

925

62.1

43

1.

54

1.36

1.

36

1.38

1.

34

1.13

1.

13

1.11

1.

15

50

8 49

3.7

7.13

7 69

.2

43.2

1.

43

1.23

1.

23

1.25

1.

21

1.16

1.

16

1.14

1.

18

Nak

ai, e

t al.

(198

2)

609.

6 59

6.9

6.35

94

.0

43.9

1.

04

0.92

0.

92

0.93

0.

91

1.13

1.

13

1.11

1.

14

Pre

ssur

e: k

g/m

m2

609.

6 59

5.3

7.13

7 83

.4

54.3

1.

32

1.29

1.

28

1.30

1.

27

1.03

1.

03

1.01

1.

04

60

9.6

590.

6 9.

525

62.0

44

.6

1.59

1.

42

1.41

1.

44

1.39

1.

12

1.13

1.

11

1.14

M

ean

1.02

8 1.

032

0.99

7 1.

055

Varia

tion

0.02

3 0.

022

0.03

1 0.

018

σ 0.

152

0.15

0 0.

177

0.13

3

C

OV

0.14

8 0.

145

0.17

8 0.

126

N

ote:

In E

qs 6

-12

to 6

-15

f y w

as u

sed

inst

ead

of f u



84

6.2.5. Observations and Recommendations Tables 6-3 and 6-4 summarize the probabilistic characteristics of the ratio of burst pressure to nominal value calculated by different models and the ratio of yield pressure to nominal value, respectively. Results are categorized according to different ratios of Di/t for pipes. The first category includes all the results, the second only thick pipes (Di/t < 20) and the third thin pipes (Di/t >20). Some researchers consider thin pipes those having Di/t less than 10. Nevertheless, here the definition as given in the theory of strength of materials is used (Hearn, 1985). Values of the ratio obtained from Eqs. 6-11 to 6-14 greater than one, show the conservatism of the model, whereas the coefficient of variation, which shows the uncertainty included in the obtained mean value, should be as low as possible. From the results presented the following conclusions can be made:

- Thin theory is inadequate for thick pipes. Referring to Table 6-3, the thin theory model is the only one that gives a ratio less than one. In all other cases it gives a relatively high COV.

- Referring to Table 6-3, the Barlow model gives the more conservative results and a high COV. It should be also noticed that the Barlow formula is used in the ASME Code, when the longitudinal stress due to internal pressure is taken into account.

- The Boardman model, which is used in the Code for the calculation of hoop stress and thickness, gives in almost all cases mean ratios near unity and a relatively smaller COV, and moreover has a simpler form than the Lame equation. Therefore, it is maintained in the LRFD Equations, when hoop stress is calculated.

- Results in Table 6-4 should be seen with reservations since the experimental results are few, and the determination of yield pressure may differ from a researcher to another.

In the LRFD methodology, two limit states may be used for securing failure due to burst of the pipe as follows:

- Prevent Yielding. Although it is assumed that prevention against failure (burst) must be satisfied, yielding proceeds the burst of the pipe, and this failure mode is usually used in the codes. Since the material is regularly in elevated temperatures, even brittle metals (depends on their chemical composition, carbon content, etc.) can perform as ductile materials. This limit is also suitable when any deformation before yielding is undesirable, although deformation might not be significant before bursting. A performance function for reliability analysis can be defined as

g = fhy - σΗ (6-15)

where fhy is the yield strength of the metal of the pipe in the hoop direction, and σΗ the hoop stress. It is important to mention that when making the assumption of a homogenous, isotropic material fhy = fy, where fy is the yield strength obtained from specimens tested in the longitudinal direction.

- Prevent Burst. When clearly burst of the pipe should be prevented as illustrated in

Figure 6-1, the following performance function may be used:

g = fhu- σΗ ≈ fu - σΗ (6-16)



85

where fu is the ultimate strength of the metal of the pipe, and σΗ the hoop stress. This limit state can also be used especially when a higher reliability index, β is required.

Table 6-3. Summary of Probabilistic Characteristics of the Ratio of Experimental Burst Pressure to Nominal Value

Di/t Any 20≤ >20 Model

Mean COV Mean COV Mean COV Lamé 1.090 0.136 1.049 0.085 1.100 0.140 Boardman 1.096 0.132 1.060 0.086 1.105 0.140 Barlow 1.131 0.131 1.143 0.094 1.128 0.139 Thin-Theory 1.043 0.143 0.935 0.073 1.069 0.142 Table 6-4. Summary of Probabilistic Characteristics of the Ratio of Experimental Yield Pressure to Nominal Value

Di/t

Any 20≤ >20 Model Mean COV Mean COV Mean COV

Lamé 1.028 0.148 NA NA 1.062 0.125 Boardman 1.032 0.145 NA NA 1.066 0.124 Barlow 0.997 0.178 NA NA 1.043 0.136 Thin-Theory 1.055 0.126 NA NA 1.081 0.116 NA = enough specimens are not available

Figure 6-1. Pipe Failure due to Internal Pressure by a Burst (Woods 1999)

6.3. Bending Moments As mentioned in previous chapters, the steel of the pipes is considered ductile, linear elastic perfectly plastic material and limit theory is considered. In such case, the failure moment equals:

pypyf ZfMMM === (6-17)



86

where fy = yield strength and Zp the plastic section modulus given by any of the following equivalent equations:

ttDDD

Z oio

P2

33

)(6

−≈−

= (6-18)

or

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

33 2116 o

oP D

tDZ (6-19)

where, Do, Di and t are the outside diameter, the inside diameter and thickness of pipe, respectively. If also work-hardening material is considered, the failure moment has an upper bound as follows:

puuf ZfMM == (6-20)

where fu is the ultimate strength of the material. If now a bilinear approximation of work hardening material is considered, the failure moment is given as:

ZffZfMM yfpynf )( −+== (6-21)

where Zp is defined as above and Z is the elastic cross-section modulus expressed as:

( )

o

io

DDDπ

Z32

44 −= (6-22)

In Eq. 6-21 ff is the stress that corresponds to a given ‘‘failure’’ strain, εf, different from the ultimate, as shown in Figure 6-2. If the stress – strain curve is not available, ff is usually considered equal to the ultimate strength, fu.

Figure 6-2. Real Stress-Strain Diagram with Perfectly Plastic and Bilinear Approximations



87

The following sections present experimental results of bending tests on straight pipes without and with internal pressure. The interpretation of these tests is a difficult task, since usually there is no detailed description of the experiments or of the characteristics of the materials used. Therefore, only the conclusions drawn by respective researchers are summarized.

6.3.1. Pure Bending Figure 6-3 presents a summary of experimental failure moments of pipes, according to different researchers, compared to the yield moment calculated by Eq. 6-17. The results are dependent not only on the ratio δ = Do/t but also on the ratio of the ultimate to yield strength. The tests were performed on pipes made of both carbon and stainless steel. Belke (1983) extracts the following bounding relations for the failure moment: For fu / fy ≥ 2

3/1

)3.4(−

⎟⎠

⎞⎜⎝

⎛=t

DMM o

yf (9) or (6-23)

yf MM 2= Do/t ≤ 10

For fu / fy<2

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

−

=y

uo

y

u

yf ff

tDff

MM80

)/()1( (6-24)

In Figure 6-3, Eq. (6-23) (noted as Eq. (9) in the figure) is plotted, which bounds all the experimental results. The dashed line is in a 10% range. From all these equations is evident that the failure moment in most cases exceeds the limit moment My, whereas for the cases that Do/t>80 the experimental failure moment is smaller than the limit moment My. Pipes with Do/t > 80 are susceptible to failure due to buckling and ovalization of the cross-section and therefore are expected to fail earlier. Belke (1983) does not define what experimental failure moment Mexp is, since as shown in Sections 4.2.2 to 4.2.4 the failure (collapse moment) can be calculated by different methods. He though states that the results of Gerber, Jirsa, et al., Sorenson, et al. and Franzen and Stockey are obtained from the Rodabaugh, et al. (1978) report, where the experimental collapse moment is calculated according to Section 4.2.3, and therefore is the moment that corresponds to a displacement two times the extrapolated elastic displacement. Table 6-5 and 6-6 summarize results of the ratio of experimental data to values obtained by Eqs. 6-17 and 6-20 for pure bending, respectively. For Table 6-20 the results are obtained from fewer data, since Rodabaugh, et al. (1978) does not give the ultimate strength of steel. The data in Table 6-5 are part of that presented in Figure 6-3.



88

Table 6-5 reveals that pipes with ratio of diameter to thickness greater than 80 are susceptible to buckling, and therefore the theoretical nominal moment is not reached.

Figure 6-3. Experimental Results of Bending Failure Moments (Belke, 1983) Table 6-5. Statistics of Bias (Eq.6-17) for Pure Bending

Bias Properties Mean COV Do/t Steel 1.18 0.23 Stainless 1.06 0.13

<80 Carbon

0.76 0.11 >80 Carbon Table 6-6. Statistics of Bias (Eq. 6-20) for Pure Bending

Bias Properties Mean COV Do/t Steel 0.78 0.25 30-50 Stainless

6.3.2. Bending with Internal Pressure As stated by Weiner, et al. (1976), the lack of internal pressure leads to moments lower than those predicted by theory for pipes with Do/t > 80, because the ovalization of these pipes cross-



89

section is high. Axial tension has the same results as internal pressure; i.e., compression, as internal pressure limits the ovalization of the cross section that leads to lower bending capacity. On the other hand Rodabaugh, et al. (1976) commend that the internal pressure generally reduces the pipe’s moment capacity. In most cases bending moments due to different loads are combined with internal pressure. Table 6-7 gives the reduction of moment capacity obtained by Franzen, et al. (1972) as described by Rodabaugh, et al. (1976). Table 6-8 shows the total bias of these data, when nominal value is obtained again according to Eq. 6-17. Table 6-7. Reduction of Moment Capacity Due to Internal Pressure, Rodabaugh et al (1976).

Test No t

PDo

2 Moment Capacity as a Fraction of Zero

Pressure Moment Capacity 2 14.8 0.84 3 22.3 0.76 4 29.7 0.56 6 22.2 0.88 7 28.7 0.78

Table 6-8. Statistics of Bias (Eq.6-20) under Bending and Internal Pressure

Bias Properties Mean COV Do/t Steel 0.78 0.17 Stainless 0.93 NA 30-50 Carbon

NA: Not Available



90

7. Load and Resistance Factors

In this chapter, the load and resistance partial factors are calculated for the performance functions presented in Table 4-1 for a range of target reliability indices β. A summary of the probabilistic characteristics of strength and loads is given for each performance function based on the data provided in Chapter 5 under different subsections. A sensitivity analysis is also performed in some cases of the factors to changes of the probabilistic characteristics of variables. This chapter provides also an example of design according to the current practice but also to the equations derived in this report.

7.1. Calculation of Partial Safety Factors The procedure for calculating the partial safety factors can be summarized into three steps: a) First, the probabilistic characteristics of the random variables (stresses), which are

normalized with respect to the stress due to sustained weight SA, are summarized and using the AFOSM, as described in sections 2.5.1 and 2.5.2, the load and resistance factors applicable to the mean values of variables are calculated. For the normalized stresses the

symbol f is used and for example 1==A

AA S

Sf always.

b) Then, using the total bias as described in Chapter 6, the factors are converted to factors applicable to nominal values for the variables under consideration. This step is necessary, since in common practice for the piping design the nominal values of loads and resistance are used.

c) Finally, for a given set of load factors the strength factor, φ, is determined for each performance function. This step provides uniform partial safety factors applied to nominal loads and thus equations that are used in the design are obtained. These factors are obtained from step b, using the relation:

nR

n

i nfii

µ

µγφ

∑== 1 (7-1)

where µnfi and µnR are the nominal values of load stresses and the resistance, respectively, and γi the predefined desirable nominal load factors.

Steps b and c can be reversed, and hence given the desirable factors for the mean values of loads, the mean strength factor can be calculated, using Eq. 2-28, and then obtain the nominal factors by using the bias. In this chapter, steps a, b and c are followed, because this procedure is more flexible in order to obtain the desirable factors.



91

7.2. General Design Condition For loading condition, a designer selects the appropriate geometry (Do, t) for the pipe under consideration. Afterwards, the pipe is checked for one or more operating conditions (Service Limits). The following sections present the load and resistance factors for performance functions g1, g2, and g3.

7.2.1. Performance Function g1 This performance function, as in the present ASME BPV Code, can be used to calculate the pipe thickness without taking into consideration the allowance accounting for corrosion, erosion, etc. The allowance can be estimated according to the present practice. The performance function lacks the term of stress due to sustained weight, fA and therefore can not be normalized with respect to it.

2

)2(

1 t

ytoDD

P

yfg

−−= (7-2a)

where y has a deterministic value equal to 0.4 for Do/t>6, and otherwise it is y = Di/(Do+Di). Considering first y equal to 0.4, Eq. 7-2a can be written as

)4.05.0(1 −−= αPfg Dy (7-2b)

For the case of Do/t<6, Eq.7-2a takes the form:

⎟⎠⎞

⎜⎝⎛

−−

−−=1

15.05.01 aaaPfg Dy (7-1c)

where in all equations α = Do/t and is considered constant. The probabilistic characteristics of the variables fy and P are given in Table 7-1. Table 7-3a shows the calculated factors for the mean values of variables shown in Table 7-2. Eqs 7-2b and 7-2c yield the same results in all combinations of pressure and resistance considered. Figure 7-1a gives the pressure factors and 7-1b the resistance factors also shown in Table 7-3a, considering normal and lognormal distribution for pressure. Nevertheless, for pressure the lognormal distribution is suggested and therefore the safety factors applied to nominal values of variables are calculated accordingly as Tables 7-3 b shows.



92

Table 7-1. Ranges of Parameters

Parameter Ranges β 2, 3, 3.5, 4.5, 5.5

COV (fy)

Carbon 0.07 to 0.09

Stainless 0.03 to 0.19

COV(PD) 0.10 to 0.20 Table 7-2. Selected Probabilistic Characteristics of Random Variables Random Variabl

e

Recommended (COV) Mean Value Distribution Type Total Bias

fy 0.08 carbon 0.14 stainless

30 to 50 ksi

Lognormal 1.13 carbon 1.35 stainless

P 0.15 50 to 3000 psi Normal/Lognormal 1.15 Table 7-3a. Load and Resistance Factors Applied to Mean Values of Variables Considering Different Distribution Types for Internal Pressure, g1

Carbon Steel Stainless Steel Distributions Β

φfy γH φfy γH 2.0 0.91 1.25 0.80 1.20 3.0 0.87 1.36 0.72 1.29 3.5 0.84 1.42 0.68 1.33 4.5 0.79 1.52 0.60 1.41

fy: Lognormal

P: Normal 5.5 0.75 1.62 0.53 1.49 2.0 0.92 1.29 0.82 1.23 3.0 0.89 1.47 0.74 1.37 3.5 0.87 1.57 0.71 1.45 4.5 0.84 1.79 0.65 1.61

fy: Lognormal

P: Lognormal

5.5 0.81 2.04 0.59 1.80 Table 7-3b. Load and Resistance Factors Applied to Nominal Values of Variables, g1

Carbon Steel Stainless Steel Distributions Β

φfy γH φfy γH 2.0 1.04 1.48 1.11 1.41 3.0 1.00 1.69 1.00 1.58 3.5 0.98 1.81 0.96 1.67 4.5 0.95 2.06 0.88 1.85

fy: Lognormal

P: Lognormal 5.5 0.92 2.35 0.80 2.07



93

Figure 7-1a. Pressure Factors for Normal and Lognormal Distribution for Pressure

Figure 7-1b. Resistance Factors for Normal and Lognormal Distribution for Pressure

7.2.2. Performance Function g2

As mentioned in this report, this performance function is applicable to pipes of all safety categories and service limits. A higher reliability index is expected to be applicable to this performance function, which simply guarantees that the pipe can hold its sustained weight.

Afyfg −=2 (7-3)

Table 7-6a gives the partial safety factors (φfy, γΑ) applied to mean values of variables shown in Table 7-5. The variable µfy refers to the calculated mean value of the strength resistance. Table 7-6b presents the factors applied to nominal values, while Table 7-6c gives the recommended values for the partial resistance and load factors applied to nominal values.

Carbon Steel, g 1

0.5

0.6

0.7

0.8

0.9

1

2 3 4 5 6Target Reliability Index, β

φ fy

P,normal P,lognormal

Stainless Steel, g 1

0.5

0.6

0.7

0.8

0.9

1


φfy


Carbon Steel, g 1

11.21.41.61.8

22.2

2 3 4 5 6

Target Reliability Index, β

γ H


Stainless Steel, g 1

1

1.2

1.4

1.6

1.8

2

2.2


γ H




94

Table 7-4. Ranges of Parameters Parameter Ranges

β 2, 3, 3.5, 4.5, 5.5, 6.5

COV (fy) Carbon

0.07 to 0.09 Stainless

0.03 to 0.19 COV (fA) 0.05 to 0.10

Table 7-5. Probabilistic Characteristics of Random Variables Random Variable Mean Recommended (COV) Distribution Type Total Bias

fy NA 0.08 for carbon steel 0.14 for stainless steel Lognormal 1.13 for carbon

1.35 for stainless fA 1 0.10 Normal 1.00

Table 7-6a. Load and Resistance Factors Applied to Mean Values of Variables, g2

Carbon Steel Stainless Steel Distributions β

µfy φfy γΑ µfy φfy γΑ 2.0 1.147 0.895 1.147 1.415 0.784 1.109 3.0 1.441 0.844 1.215 1.668 0.694 1.158 3.5 1.525 0.818 1.248 1.810 0.653 1.182 4.5 1.705 0.769 1.311 2.129 0.576 1.227 5.5 1.902 0.721 1.370 2.500 0.508 1.271

fy: Lognormal

fA: Normal

6.5 2.117 0.675 1.428 2.932 0.448 1.310 Table 7-6b. Load and Resistance Factors Applied to Nominal Values of Variables, g2

Carbon Steel Stainless Steel Distributions β

φfy γΑ φfy γΑ 2.0 1.01 1.15 1.06 1.11 3.0 0.95 1.22 0.94 1.16 3.5 0.92 1.25 0.88 1.18 4.5 0.87 1.31 0.78 1.23 5.5 0.81 1.37 0.69 1.27

fy: Lognormal

fA: Normal

6.5 0.77 1.43 0.60 1.31



95

Table 7-6c. Suggested Nominal Load and Resistance Factors, g2 Carbon Steel Stainless Steel

Distributions β φfy γΑ φfy γΑ

2.0 0.98 1.0 0.95 1.0 3.0 0.94 1.2 0.97 1.2 3.5 0.89 1.2 0.90 1.2 4.5 0.80 1.2 0.76 1.2 5.5 0.71 1.2 0.65 1.2

fy: Lognormal

fA: Normal

6.5 0.64 1.2 0.55 1.2


This performance function as the previous ones is used for the preliminary selection of pipe geometry. The stress fPD corresponds to the longitudinal stress due to the expected mean pressure (design).

DPfAfyfg −−=3 (7-4)

Table 7-7 gives the ranges of parameters, while Table 7-8 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-9a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-8. The recommended distribution for the design pressure is the lognormal. Tables 7-9b and 7-9c give the factors applicable to nominal values of variables and the suggested ones, respectively. Table 7-7. Ranges of Parameters


COV (fy) Carbon


0.03 to 0.19 COV (fA) 0.05 to 0.10 COV (fPD) 0.05 to 0.10

Table 7-8. Probabilistic Characteristics of Random Variables

Random Variable

Recommended (COV)

Mean Range Distribution Type Total Bias

fy 0.08 for carbon 0.14 for stainless

N.A. Lognormal 1.13 for carbon 1.35 for stainless

fA 0.10 1 Normal 1.00 fPD 0.10 1, 10, 100, (1000*) Normal /Lognormal 1.00

N.A. = Not Available


Dev

elop

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t of R

elia

bilit

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ased

LR

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r Pip

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– R

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rch

and

Dev

elop

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t Rep

ort

96

Tabl

e 7-

9a. L

oad

and

Res

ista

nce

Fact

ors A

pplie

d to

Mea

n V

alue

s of V

aria

bles

C

arbo

n St

eel

Stai

nles

s Ste

el

D

istri

butio

ns

β µ f

y φ f

y γ Α

γ P

D

µ fy

φ fy

γ Α

γ PD

2.0

2.47

4 0.

881

1.08

9 1.

061

2.76

0.

770

1.06

1 1.

061

3.0

2.53

3 0.

825

1.13

1.

13

3.21

0.

678

1.09

0 1.

090

3.5

2.88

2 0.

799

1.15

1 1.

151

3.47

0.

636

1.10

3 1.

103

4.5

3.18

5 0.

747

1.19

1.

19

4.04

0.

559

1.13

0 1.

130

f y: L

ogno

rmal

f A: N

orm

al

f P

D: N

orm

al

5.5

3.51

7 0.

698

1.22

7 1.

227

4.71

0.

491

1.15

6 1.

156

2.0

2.47

0.

883

1.08

8 1.

094

2.75

0.

771

1.06

1 1.

061

3.0

2.74

0.

829

1.12

6 1.

15

3.21

0.

68

1.08

9 1.

096

3.5

2.89

0.

804

1.14

5 1.

18

3.47

0.

638

1.10

2 1.

114

4.5

3.21

0.

755

1.18

1.

24

4.05

0.

563

1.12

8 1.

152

fPD = 1 f y:

Log

norm

al

f A

: Nor

mal

f PD: L

ogno

rmal

5.

5 3.

56

0.70

9 1.

211

1.31

1 4.

73

0.49

6 1.

152

1.19

2 2.

0 13

.95

0.89

1 1.

014

1.14

15

.43

0.78

0 1.

01

1.10

2 3.

0 15

.61

0.83

9 1.

021

1.20

7 18

.14

0.69

1.

015

1.14

9 3.

5 16

.49

0.81

3 1.

024

1.24

19

.65

0.64

8 1.

017

1.17

2 4.

5 18

.38

0.76

3 1.

03

1.30

23

.05

0.57

1 1.

021

1.21

5

f y: L

ogno

rmal

f A: N

orm

al

f P

D: N

orm

al

5.5

20.4

4 0.

715

1.03

6 1.

357

27

0.50

3 1.

026

1.25

7 2.

0 14

.01

0.89

8 1.

013

1.15

6 15

.43

0.78

5 1.

00

1.11

3.

0 15

.83

0.85

2 1.

018

1.24

7 18

.23

0.69

9 1.

014

1.17

3 3.

5 16

.82

0.83

1 1.

020

1.29

5 19

.82

0.66

0 1.

016

1.20

6 4.

5 19

.02

0.78

9 1.

024

1.39

8 23

.43

0.58

8 1.

02

1.27

5

fPD = 10

f y: L

ogno

rmal

f A: N

orm

al

f P

D: L

ogno

rmal

5.

5 21

.50

0.75

1.

028

1.50

9 27

.70

0.52

4 1.

023

1.34

8 2.

0 12

9.35

0.

894

1.00

1.

147

142.

73

0.78

3 1.

00

1.10

8 3.

0 14

5.24

0.

843

1.00

1.

215

168.

25

0.69

4 1.

00

1.15

7 3.

5 15

3.72

0.

818

1.00

1.

247

182.

55

0.65

2 1.

00

1.18

1 4.

5 17

1.84

0.

768

1.00

1.

310

214.

63

0.57

6 1.

00

1.22

6

f y: L

ogno

rmal

f A: N

orm

al

f P

D: N

orm

al

5.5

191.

62

0.72

1.

00

1.31

0 25

1.97

0.

508

1.00

1.

269

2.0

130

0.90

2 1.

00

1.16

2

0.78

9 1.

00

1.11

7 3.

0 14

7.63

0.

857

1.00

1.

256

169.

42

0.70

4 1.

00

1.18

3 3.

5 15

7.32

0.

836

1.00

1.

306

184.

53

0.66

5 1.

00

1.21

8 4.

5 17

8.66

0.

795

1.00

1.

411

218.

92

0.59

4 1.

00

1.29

1

fPD = >100

f y: L

ogno

rmal

f A:N

orm

al

f P

D: L

ogno

rmal

5.

5 20

2.91

0.

757

1.00

1.

525

259.

72

0.53

0 1.

00

1.36

7



97

Table 7-9b. Load and Resistance Factors Applied to Nominal Values of Variables, g3 fp<10* fp ≥ 10 Distributions Steel β

φfy γΑ γPD φfy γΑ γPD 2.0 1.00 1.09 1.09 1.04 1.01 1.16 3.0 0.94 1.13 1.15 0.92 1.02 1.26 3.5 0.91 1.15 1.18 0.86 1.02 1.31 4.5 0.85 1.18 1.24 0.76 1.02 1.41

fy: Lognormal

fA: Normal

fPD: Lognormal

Carbon

5.5 0.80 1.21 1.31 0.67 1.03 1.52 2.0 1.01 1.06 1.06 1.06 1.00 1.12 3.0 0.96 1.09 1.10 0.95 1.01 1.18 3.5 0.94 1.10 1.11 0.89 1.02 1.22 4.5 0.89 1.13 1.15 0.80 1.02 1.28

fy: Lognormal

fA: Normal

fPD: Lognormal

Stainless

5.5 0.85 1.15 1.19 0.70 1.02 1.36 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi. Table 7-9c. Suggested Nominal Load and Resistance Factors, g3

fp<10* fp ≥ 10 Distributions Steel β φfy γΑ γPD φfy γΑ

2.0 1.0 1.0 1.2 0.97 1.0 1.1 3.0 0.95 1.0 1.3 0.93 1.0 1.2 3.5 0.98 1.2 1.3 0.95 1.2 1.3 4.5 0.88 1.2 1.3 0.84 1.2 1.3

fy: Lognormal

fA: Normal

fPD: Lognormal

Carbon

5.5 0.79 1.2 1.3 0.75 1.2 1.3 2.0 0.98 1.0 1.0 0.96 1.0 1.0 3.0 0.97 1.0 1.3 0.96 1.0 1.2 3.5 0.97 1.2 1.3 0.97 1.2 1.3 4.5 0.83 1.2 1.3 0.82 1.2 1.3

fy: Lognormal

fA: Normal

fPD: Lognormal

Stainless

5.5 0.71 1.2 1.3 0.6 1.2 1.3 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200



98

7.3. Operating Condition (Service Level A) This condition is described by only the performance function g4, which takes into consideration the maximum pressure during normal operation of a plant.


This performance function checks the pipe during normal operation, considering the maximum pressure that can occur and the sustained weight. Therefore, stress fPmax corresponds to the normalized longitudinal stress due to the maximum values of pressure during normal operation.

max4 PfAfyfg −−= (7-5)

Table 7-10 gives the ranges of parameters, while Table 7-11 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-12a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-11. The recommended distribution for the maximum operating pressure is the extreme Type I of largest values. Tables 7-12b and 7-12c give the factors applicable to nominal values of variables and the suggested ones, respectively. Table 7-10. Ranges of Parameters


COV (fy) Carbon


0.03 to 0.19 COV (fA) 0.05 to 0.10

COV (fPmax) 0.10 to 0.13 Table 7-11. Probabilistic Characteristics of Random Variables

Random Variable

Recommended (COV)

Mean Range Distribution Type Total Bias



fA 0.10 1 Normal 1.00 fPmax 0.12 1, 10, 100, (1000*) Lognormal/ Type I

(Largest) 1.10 N.A. = Not Available


Dev

elop

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elia

bilit

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ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

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elop

men

t Rep

ort

99

Tabl

e 7-

12a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es

Car

bon

Stee

l St

ainl

ess S

teel

Dis

tribu

tions

β

µ fy

φ fy

γ Α

γ Pm

ax

µ fy

φ fy

γ Α

γ Pm

ax

2.0

2.50

0.

888

1.08

2 1.

134

2.77

0.

776

1.05

9 1.

088

3.0

2.79

0.

838

1.11

7 1.

217

3.24

0.

686

1.08

5 1.

140

3.5

2.94

0.

814

1.13

3 1.

262

3.51

0.

645

1.09

7 1.

168

4.5

3.29

0.

768

1.16

2 1.

361

4.11

0.

571

1.12

1 1.

228

f y: L

ogno

rmal

f A: N

orm

al

f P

max

: Log

norm

al

5.5

3.67

0.

725

1.18

7 1.

473

4.82

0.

505

1.14

3 1.

292

2.0

2.49

0.

896

1.07

4 1.

159

2.75

0.

779

1.05

8 1.

087

3.0

2.83

0.

864

1.09

2 1.

351

3.24

0.

699

1.07

9 1.

189

3.5

3.03

0.

851

1.09

6 1.

483

3.53

0.

666

1.08

7 1.

266

4.5

3.51

0.

827

1.10

0 1.

807

4.21

0.

612

1.09

6 1.

483

fPmax = 1

f y: L

ogno

rmal

f A: N

orm

al

f P

max

: Typ

e I

5.5

4.12

0.

803

1.10

3 2.

203

5.08

0.

566

1.10

1.

775

2.0

14.3

8 0.

908

1.01

1 1.

205

15.7

3 0.

796

1.00

9 1.

151

3.0

16.4

9 0.

867

1.01

5 1.

328

18.7

9 0.

714

1.01

3 1.

241

3.5

17.6

6 0.

847

1.01

7 1.

395

20.5

4 0.

676

1.01

4 1.

288

4.5

20.2

7 0.

809

1.02

0 1.

538

24.5

5 0.

607

1.01

7 1.

389

f y: L

ogno

rmal

f A: N

orm

al

f P

max

: Log

norm

al

5.5

23.2

7 0.

773

1.02

2 1.

697

29.3

6 0.

545

1.01

9 1.

499

2.0

14.5

9 0.

926

1.00

9 1.

250

15.7

5 0.

815

1.00

8 1.

183

3.0

17.5

6 0.

904

1.01

1.

487

19.4

2 0.

759

1.00

9 1.

374

3.5

19.4

1 0.

894

1.04

1 1.

634

21.7

1 0.

735

1.01

0 1.

495

4.5

23.9

3 0.

871

1.01

1.

983

27.4

3 0.

687

1.01

1.

783

fPmax = 10

f y: L

ogno

rmal

f A: N

orm

al

f P

max

: Typ

e I

5.5

29.6

2 0.

845

1.01

2.

403

34.9

0 0.

637

1.01

2.

12

2.0

133.

89

0.91

2 1.

00

1.21

1 14

5.99

0.

801

1.00

1 1.

159

3.0

154.

48

0.87

2 1.

00

1.34

17

5.29

0.

72

1.00

1 1.

253

3.5

165.

93

0.85

3 1.

00

1.40

5 19

2.09

0.

683

1.00

1 1.

302

4.5

191.

47

0.81

6 1.

00

1.55

2 23

0.67

0.

615

1.00

2 1.

408

f y: L

ogno

rmal

f A: N

orm

al

f P

max

: Log

norm

al

5.5

220.

94

0.78

1.

00

1.71

4 27

7.01

0.

553

1.00

2 1.

521

2.0

136.

11

0.93

1.

00

1.25

6 14

6.01

0.

822

1.00

1 1.

194

3.0

165.

62

0.91

1.

00

1.49

5 18

2.32

0.

768

1.00

1 1.

391

3.5

184.

04

0.89

8 1.

00

1.64

3 20

4.91

0.

744

1.00

1 1.

514

4.5

228.

98

0.87

5 1.

00

1.99

4 26

1.24

0.

695

1.00

1 1.

806

fPmax = >100

f y: L

ogno

rmal

f A: N

orm

al

f P

max

: Typ

e I

5.5

285.

55

0.85

1.

00

2.41

6 33

4.88

0.

645

1.00

1 2.

15



100

Table 7-10b. Load and Resistance Factors Applied to Nominal Values of Variables, g4 fp<10* fp ≥ 10 Distributions Steel β

φfy γΑ γPmax φfy γΑ γPmax 2.0 1.01 1.07 1.27 1.05 1.00 1.38 3.0 0.98 1.09 1.49 1.02 1.00 1.64 3.5 0.96 1.10 1.63 1.01 1.00 1.79 4.5 0.93 1.10 1.99 0.98 1.00 2.19

fy: Lognormal

fA: Normal

fPmax: Type I

Carbon

5.5 0.91 1.10 2.42 0.95 1.00 2.65 2.0 1.05 1.16 1.20 1.11 1.01 1.31 3.0 0.94 1.19 1.31 1.03 1.01 1.52 3.5 0.90 1.20 1.39 1.00 1.01 1.65 4.5 0.83 1.21 1.63 0.93 1.01 1.97

fy: Lognormal

fA: Normal

fPmax: Type I

Stainless

5.5 0.76 1.21 1.95 0.86 1.01 2.34 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi. Table 7-10c. Load Suggested Nominal Load and Resistance Factors, g4

fp<10* fp ≥ 10 Distributions Steel β φfy γΑ γPmax φfy γΑ γPmax

2.0 0.95 1.1 1.1 0.93 1.1 1.2 3.0 0.95 1.2 1.3 0.84 1.2 1.3 3.5 0.89 1.2 1.3 0.76 1.2 1.3 4.5 0.77 1.2 1.3 0.61 1.2 1.3

fy: Lognormal

fA: Normal

fPmax: Type I

Carbon

5.5 0.65 1.2 1.3 0.50 1.2 1.3 2.0 0.98 1.2 1.1 0.96 1.2 1.1 3.0 0.99 1.2 1.3 0.90 1.2 1.3 3.5 0.91 1.2 1.3 0.81 1.2 1.3 4.5 0.76 1.2 1.3 0.64 1.2 1.3

fy: Lognormal

fA: Normal

fPmax: Type I

Stainless

5.5 0.63 1.2 1.3 0.50 1.2 1.3 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi.

7.4. Upset Loading Condition (Service Level B) This section provides the performance functions for operating loads such as non reversing mechanical and the OBE loading, as defined in Chapter 3


This state limit refers to Service Limit B and considers loads that occur occasionally during the operation of the plant.

MfAfPfyfg −−−=max

5 (7-6)



101

Table 7-11 gives the ranges of parameters, while Table 7-12 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-13a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-12. The recommended distribution for the maximum operating pressure is the extreme Type I of largest values. Tables 7-13b to 7-13d give the factors applicable to nominal values of variables and the suggested ones, respectively. Table 7-11. Ranges of Parameters


COV (fy) Carbon


0.03 to 0.19 COV (fA) 0.05 to 0.10

COV (fPmax) 0.10 to 0.13 COV (fM) 0.07 to 0.15


Random Variable

Recommended (COV)

Mean Range Distribution Type Total Bias (Recommended)



fA 0.10 1 Normal 1.00 fPmax 0.12 1, 10, 100, (1000*) Type I (Largest) 1.10 fM 0.14 0.5, 1, 2 Lognormal 0.60 to 1.00 (Use

0.90) N.A. = Not Available


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

102

Tabl

e 7-

13a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es fo

r Car

bon

Stee

l C

arbo

n St

eel

f M

0.5

1.0

2.0

D

istri

butio

ns

β µ f

y φ f

y γ Α

γ P

max

γ M

µ f

y φ f

y γ Α

γ P

max

γ M

µ f

y φ f

y γ Α

γ P

max

γ M

2

3.06

7 0.

887

1.06

7 1.

123

1.06

1 3.

684

0.88

6 1.

057

1.08

5 1.

121

4987

0.

892

1.03

9 1.

040

1.18

5 3

3.43

4 0.

847

1.08

8 1.

278

1.08

6 4.

110

0.83

9 1.

078

1.18

4 1.

186

5.60

0 0.

846

1.05

4 1.

077

1.30

3 3.

5 3.

648

0.83

2 1.

093

1.39

4 1.

094

4.34

7 0.

819

1.08

6 1.

261

1.21

4 5.

938

0.82

4 1.

061

1.09

8 1.

368

4.5

4.15

2 0.

806

1.09

9 1.

697

1.10

1 4.

889

0.78

7 1.

096

1.49

9 1.

252

6.68

4 0.

784

1.07

2 1.

145

1.51

1

fPmax = 1

f y: L

ogno

rmal

f A

: Nor

mal

f P

max

: Typ

e I

f M: L

ogno

rmal

5.

5 4.

776

0.78

2 1.

102

2.07

9 1.

105

5.54

3 0.

760

1.10

1 1.

845

1.26

9 7.

539

0.74

6 1.

081

1.20

2 1.

671

2 15

.123

0.

924

1.00

9 1.

246

1.00

15

.666

0.

922

1.00

8 1.

243

1.00

7 16

.770

0.

918

1.00

8 1.

234

1.02

4 3

18.1

1 0.

902

1.01

0 1.

482

1.00

18

.668

0.

900

1.01

0 1.

477

1.00

9 19

.804

0.

895

1.01

0 1.

465

1.02

9 3.

5 19

.971

0.

891

1.01

0 1.

629

1.00

20

.536

0.

889

1.01

0 1.

623

1.00

9 21

.686

0.

884

1.01

0 1.

610

1.03

0 4.

5 24

.507

0.

868

1.01

0 1.

977

1.00

25

.086

0.

866

1.01

0 1.

970

1.01

0 26

.268

0.

861

1.01

0 1.

955

1.03

1

fPmax = 10 f y:

Log

norm

al

f A: N

orm

al

f Pm

ax: T

ype

I f M

: Log

norm

al I

5.5

30.2

12

0.84

3 1.

010

2.39

6 1.

00

30.8

09

0.84

1 1.

010

2.38

8 1.

011

32.0

28

0.83

6 1.

010

2.37

0 1.

032

2 13

6.64

2 0.

930

1.00

1 1.

255

1.00

13

7.17

5 0.

930

1.00

1 1.

255

0.99

2 13

8.24

4 0.

929

1.00

1 1.

255

0.99

4 3

166.

160

0.90

9 1.

001

1.49

5 1.

00

166.

706

0.90

8 1.

001

1.49

4 0.

992

167.

800

0.90

8 1.

001

1.49

3 0.

994

3.5

184.

592

0.89

8 1.

001

1.64

3 1.

00

185.

144

0.89

8 1.

001

1.64

2 0.

992

186.

251

0.89

7 1.

001

1.64

1 0.

994

4.5

229.

546

0.87

5 1.

001

1.99

4 1.

00

230.

113

0.87

5 1.

001

1.99

3 0.

992

231.

249

0.87

4 1.

001

1.99

2 0.

994

fPmax = 100

f y: L

ogno

rmal

f A

: Nor

mal

f P

max

: Typ

e I

f M: L

ogno

rmal

5.

5 28

6.13

0 0.

849

1.00

1 2.

415

1.00

28

6.11

4 0.

849

1.00

1 2.

415

0.99

2 28

7.88

4 0.

849

1.00

1 2.

413

0.99

4

Stai

nles

s Ste

el

2 3.

404

0.77

2 1.

049

1.06

2 1.

04

4.09

4 0.

772

1.04

1 1.

043

1.07

8 5.

516

0.77

9 1.

029

1.02

1 1.

123

3 3.

986

0.68

6 1.

069

1.13

2 1.

064

4.78

7 0.

684

1.05

8 1.

090

1.12

5 6.

486

0.69

2 1.

041

1.04

5 1.

200

3.5

4.31

7 0.

648

1.07

8 1.

183

1.07

4 5.

180

0.64

4 1.

067

1.12

0 1.

149

7.03

4 0.

652

1.04

7 1.

058

1.24

2 4.

5 5.

081

0.58

5 1.

091

1.33

7 1.

091

6.07

0 0.

573

1.08

1 1.

205

1.19

5 8.

280

0.58

0 1.

057

1.08

6 1.

331

fPmax = >1

f y: L

ogno

rmal

f A

: Nor

mal

f P

max

: Typ

e I

f M: L

ogno

rmal

5.

5 6.

022

0.53

6 1.

098

1.58

2 1.

099

7.13

3 0.

515

1.09

2 1.

349

1.23

3 9.

755

0.51

7 1.

066

1.11

9 1.

431

2 16

.357

0.

812

1.00

8 1.

177

0.99

8 16

.975

0.

808

1.00

8 1.

171

1.00

5 18

.233

0.

802

1.00

7 1.

158

1.02

0 3

20.0

73

0.75

5 1.

009

1.36

4 0.

999

20.7

40

0.75

0 1.

009

1.35

4 1.

008

22.1

05

0.74

2 1.

009

1.33

3 1.

027

3.5

22.3

94

0.73

0 1.

010

1.48

4 1.

000

23.0

84

0.72

5 1.

010

1.47

3 1.

009

24.4

98

0.71

6 1.

010

1.44

8 1.

029

4.5

28.1

61

0.68

2 1.

010

1.76

9 1.

000

28.9

01

0.67

7 1.

010

1.75

6 1.

010

30.4

17

0.66

8 1.

010

1.72

6 1.

031

fPmax = >10

f y: L

ogno

rmal

f A

: Nor

mal

f P

max

: Typ

e I

f M: L

ogno

rmal

5.

5 35

.684

0.

633

1.01

0 2.

107

1.00

0 36

.481

0.

629

1.01

0 2.

091

1.01

0 38

.115

0.

620

1.01

0 2.

057

1.03

2 2

147.

021

0.82

2 1.

001

1.19

3 0.

991

147.

624

0.82

1 1.

001

1.19

3 0.

992

148.

833

0.82

1 1.

001

1.19

1 0.

993

3 18

2.96

8 0.

768

1.00

1 1.

390

0.99

1 18

3.61

4 0.

767

1.00

1 1.

389

0.99

2 18

4.90

8 0.

766

1.00

1 1.

387

0.99

4 3.

5 20

5.57

9 0.

743

1.00

1 1.

513

0.99

1 20

6.24

6 0.

743

1.00

1 1.

512

0.99

2 20

7.58

4 0.

742

1.00

1 1.

510

0.99

4 4.

5 26

1.95

6 0.

695

1.00

1 1.

805

0.99

1 26

2.67

0 0.

694

1.00

1 1.

804

0.99

2 26

4.10

2 0.

693

1.00

1 1.

801

0.99

4

fPmax = >100

f y: L

ogno

rmal

f A

: Nor

mal

f P

max

: Typ

e I

f M: L

ogno

rmal

5.

5 33

5.64

7 0.

644

1.00

1 2.

148

0.99

1 33

6.41

7 0.

644

1.00

1 2.

147

0.99

2 33

7.96

0 0.

643

1.00

1 2.

144

0.99

4



103

Table 7-13b. Suggested Resistance and Load and Resistance Factors Applied to Mean Values of Variables, g5

fp<10* fp ≥ 10 Steel β µfy φfy γΑ γPmax γM µfy φfy γΑ γPmax γM 2.0 4.027 0.889 1.053 1.081 1.123 15.946 0.924 1.006 1.245 1.010 3.0 4.517 0.843 1.082 1.177 1.194 18.957 0.902 1.006 1.480 1.012 3.5 4.793 0.825 1.077 1.327 1.231 20.828 0.884 1.006 1.627 1.012 4.5 5.418 0.796 1.090 1.421 1.306 25.387 0.868 1.006 1.975 1.013

Carbon

5.5 6.157 0.764 1.091 1.640 1.388 31.120 0.843 1.006 2.393 1.013 2.0 4.460 0.775 1.039 1.042 1.081 17.295 0.812 1.006 1.176 1.007 3.0 5.236 0.688 1.055 1.089 1.132 21.089 0.755 1.006 1.361 1.009 3.5 5.676 0.648 1.063 1.121 1.158 23.446 0.729 1.006 1.480 1.010 4.5 6.681 0.579 1.074 1.211 1.211 29.289 0.681 1.006 1.765 1.011

Stainless

5.5 7.889 0.526 1.082 1.350 1.265 36.900 0.632 1.006 2.102 1.012 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi. Table 7-13c. Resistance and Load Factors Applied to Nominal Values of Variables

fp<10* fp ≥ 10 Steel Β φfy γΑ γPmax γM φfy γΑ γPmax γM

2.0 1.00 1.053 1.081 1.123 1.04 1.006 1.37 0.91 3.0 0.95 1.082 1.177 1.194 1.02 1.006 1.63 0.91 3.5 0.93 1.077 1.327 1.231 1.00 1.006 1.79 0.91 4.5 0.90 1.090 1.421 1.306 0.98 1.006 2.17 0.91

Carbon

5.5 0.86 1.091 1.640 1.388 0.95 1.006 2.63 0.91 2.0 1.05 1.039 1.042 1.081 1.10 1.006 1.29 0.91 3.0 0.93 1.055 1.089 1.132 1.02 1.006 1.50 0.91 3.5 0.87 1.063 1.121 1.158 0.98 1.006 1.63 0.91 4.5 0.78 1.074 1.211 1.211 0.92 1.006 1.94 0.91

Stainless

5.5 0.71 1.082 1.350 1.265 0.85 1.006 2.31 0.91 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi.



104

Table 7-13d. Suggested Resistance and Load Factors Applied to Nominal Values of Variables fp<10* fp ≥ 10 Steel β

φfy γΑ γPmax γM φfy γΑ γPmax γM 2.0 0.96 1.0 1.1 1.0 0.94 1.0 1.2 1.0 3.0 0.97 1.1 1.3 1.1 0.87 1.2 1.3 1.1 3.5 0.92 1.2 1.3 1.1 0.78 1.2 1.3 1.1 4.5 0.85 1.2 1.3 1.1 0.65 1.2 1.3 1.1

Carbon

5.5 0.75 1.2 1.3 1.1 0.53 1.2 1.3 1.1 2.0 1.00 1.0 1.0 0.95 0.98 1.0 1.1 1.0 3.0 0.98 1.2 1.3 1.0 0.93 1.2 1.3 1.1 3.5 0.94 1.2 1.3 1.1 0.83 1.2 1.3 1.1 4.5 0.80 1.2 1.3 1.1 0.67 1.2 1.3 1.1

Stainless

5.5 0.69 1.2 1.3 1.1 0.53 1.2 1.3 1.1 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi.


This function also refers to Service Limit B, where loads that occur occasionally during operation of a plant are encountered.

O

fMfPOfAfyfg −−−−=5 (7-5)

Table 7-14 gives the ranges of parameters, while Table 7-15 presents the probabilistic characteristics used for the calculation of the partial safety factors. As explained in performance function g7, the coefficient of 0.83 is selected for the load due to OBE. Results are shown in Tables 7-16a to 7-16d. Table 7-14. Ranges of Parameters

Parameter Ranges β 1.5, 2, 3

COV (fy) Carbon


0.03 to 0.19 COV (fA) 0.05 to 0.10 COV (fPO) 0.15 to 0.25 COV (fM) 0.07 to 0.15 COV (fo) 0.40 to 1.00



105

Table 7-15. Probabilistic Characteristics of Random Variables Random Variable

Recommended (COV)




fA 0.10 1 Normal 1.00 fPo 0.20 1, 10, 100, (1000*) Type I (Largest) 0.90 fM 0.14 0.5, 1, 2 Lognormal 0.60 to 1.00 (Use

0.90) fo 0.50, 0.83 1, 2, 3

Type II (Largest) 0.50 to 0.90 (Use 0.65)



Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

106

Tabl

e 7-

16a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es fo

r Car

bon

Stee

l C

arbo

n St

eel

f M

β =

1.5

0.5

1.0

2.0

fo

µ fy

φ fy

γ Α

γ Po

γ M

γ o

µ fy

φ fy

γ Α

γ Po

γ M

γ o

µ fy

φ fy

γ Α

γ Po

γ M

γ o

1 4.

452

0.96

7 1.

011

1.00

7 1.

001

1.78

5 4.

973

0.96

3 1.

011

1.00

8 1.

012

1.75

9 6.

041

0.95

6 1.

011

1.00

9 1.

037

1.67

8 2

6.32

6 0.

976

1.00

5 0.

986

0.99

6 1.

841

6.83

8 0.

974

1.00

5 0.

986

1.00

1 1.

833

7.87

3 0.

970

1.00

6 0.

986

1.01

2 1.

812

1

3 8.

217

0.97

8 1.

004

0.97

9 0.

994

1.85

3 8.

726

0.97

7 1.

004

0.97

9 0.

997

1.84

9 9.

751

0.97

5 1.

004

0.97

9 1.

005

1.83

8 1

16.0

18

0.95

3 1.

005

1.28

9 0.

995

0.87

3 16

.542

0.

952

1.00

5 1.

287

0.99

9 0.

872

17.5

97

0.94

9 1.

005

1.28

2 1.

008

0.87

2 2

16.9

88

0.95

2 1.

004

1.26

9 0.

995

0.98

6 17

.572

0.

951

1.00

4 1.

267

0.99

9 0.

985

18.5

69

0.94

8 1.

004

1.26

2 1.

008

0.98

3

10

3 18

.142

0.

954

1.00

4 1.

203

0.99

4 1.

258

18.6

64

0.95

2 1.

004

1.20

2 0.

998

1.25

2 19

.718

0.

950

1.00

4 1.

200

1.00

6 1.

239

1 13

8.15

2 0.

958

1.00

0 1.

301

0.99

0.

812

138.

67

0.95

8 1.

000

1.30

1 0.

991

0.81

2 13

9.70

5 0.

958

1.00

0 1.

300

0.99

2 0.

812

2 13

9.00

2 0.

958

1.00

0 1.

300

0.99

1 0.

817

139.

52

0.95

8 1.

000

1.30

0 0.

991

0.81

7 14

0.55

5 0.

958

1.00

0 1.

300

0.99

2 0.

817

fPo

100

3 13

9.85

9 0.

958

1.00

0 1.

300

0.99

1 0.

823

140.

38

0.95

8 1.

000

1.30

0 0.

991

0.82

3 14

1.41

2 0.

957

1.00

0 1.

299

0.99

2 0.

823

β

= 2.

0

β

= 3.

0

1 5.

392

0.97

0 1.

008

0.99

6 0.

998

2.72

5 5.

909

0.96

7 1.

008

0.99

6 1.

007

2.70

3 6.

962

0.96

1 1.

008

0.99

7 1.

024

2.63

9

2 8.

221

0.97

6 1.

004

0.98

1 0.

994

2.77

2 8.

732

0.97

5 1.

004

0.98

1 0.

998

2.76

5 9.

761

0.97

2 1.

004

0.98

1 1.

006

2.74

7

1

3 11

.064

0.

978

1.00

3 0.

976

0.99

3 2.

783

11.5

72

0.97

8 1.

003

0.97

6 0.

996

2.77

9 12

.594

0.

976

1.00

3 0.

976

1..0

01

2.77

0 1

17.8

85

0.94

2 1.

005

1.44

7 0.

995

0.88

4 18

.415

0.

941

1.00

5 1.

444

1.00

0 0.

884

19.4

85

0.93

8 1.

005

1.43

7 1.

011

0.88

4 2

18.8

97

0.94

0 1.

005

1.41

8 0.

995

1.04

6 19

.428

0.

939

1.00

5 1.

415

1.00

0 1.

045

20.5

00

0.93

6 1.

005

1.40

8 1.

010

1.04

3

10

3 20

.635

0.

959

1.00

3 1.

113

0.99

3 2.

387

21.1

54

0.95

8 1.

003

1.11

4 0.

996

2.37

6 22

.198

0.

956

1.00

3 1.

116

1.00

2 2.

352

1 15

6.53

0 0.

948

1.00

1 1.

461

0.99

1 0.

812

157.

05

0.94

8 1.

001

1.46

1 0.

991

0.81

2 15

8.09

9 0.

948

1.00

1 1.

461

0.99

2 0.

812

2 15

7.39

1 0.

948

1.00

1 1.

461

0.99

1 0.

819

157.

91

0.94

8 1.

001

1.46

1 0.

991

0.81

9 15

8.96

0 0.

948

1.00

1 1.

460

0.99

2 0.

819

fPo

100

3 15

8.25

8 0.

948

1.00

1 1.

460

0.99

1 0.

826

158.

78

0.94

8 1.

001

1.46

0 0.

991

0.82

6 15

9.82

7 0.

948

1.00

1 1.

460

0.99

2 0.

826

1 10

.608

0.

976

1.00

3 0.

978

0.99

3 7.

879

11.1

18

0.97

5 1.

003

0.97

8 0.

996

7.86

6 12

.144

0.

973

1.00

3 0.

978

1.00

3 7.

833

2 18

.685

0.

979

1.00

2 0.

972

0.99

2 7.

909

19.1

92

0.97

8 1.

002

0.97

2 0.

993

7.90

4 20

.210

0.

977

1.00

2 0.

972

0.99

6 7.

892

1

3 26

.767

0.

980

1.00

1 0.

971

0.99

1 7.

917

27.2

74

0.97

9 1.

001

0.97

1 0.

992

7.91

4 28

.288

0.

979

1.00

1 0.

971

0.99

4 7.

907

1 22

.701

0.

923

1.00

6 1.

854

0.99

6 0.

896

23.2

43

0.92

1 1.

006

1.85

0 1.

002

0.89

6 24

.339

0.

918

1.00

6 1.

841

1.01

3 0.

896

2 23

.775

0.

920

1.00

6 1.

806

0.99

6 1.

151

24.3

18

0.91

8 1.

006

1.80

2 1.

001

1.15

0 24

.417

0.

915

1.00

6 1.

793

1.01

3 1.

148

10

3 35

.866

0.

974

1.00

1 1.

005

0.99

1 7.

789

36.3

75

0.97

3 1.

001

1.00

5 0.

992

7.78

4 37

.396

0.

973

1.00

1 1.

005

0.99

4 7.

775

1 20

4.16

2 0.

929

1.00

1 1.

874

0.99

1 0.

813

204.

70

0.92

9 1.

001

1.87

3 0.

991

0.81

3 20

5.76

3 0.

929

1.00

1 1.

873

0.99

3 0.

813

2 20

5.04

1 0.

929

1.00

1 1.

873

0.99

1 0.

820

205.

57

0.92

9 1.

001

1.87

3 0.

991

0.82

0 20

6.64

3 0.

928

1.00

1 1.

872

0.99

3 0.

820

fPo

100

3 20

5.92

8 0.

929

1.00

1 1.

872

0.99

1 0.

828

206.

46

0.92

8 1.

001

1.87

2 0.

991

0.82

8 20

7.53

0 0.

928

1.00

1 1.

871

0.99

3 0.

828


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

107

Tabl

e 7-

16a.

(Con

t.) L

oad

and

Res

ista

nce

Fact

ors A

pplie

d to

Mea

n V

alue

s of V

aria

bles

for C

arbo

n St

eel

Stai

nles

s Ste

el

f M

β =

1.5

0.5

1.0

2.0

fo

µ fy

φ fy

γ Α

γ Po

γ M

γ o

µ fy

φ fy

γ Α

γ Po

γ M

γ o

µ fy

φ fy

γ Α

γ Po

γ M

γ o

1 4.

046

0.91

1 1.

012

1.00

9 1.

002

1.16

5 5.

185

0.89

1 1.

012

1.01

0 1.

014

1.58

6 6.

352

0.87

0 1.

012

1.01

1 1.

040

1.42

0 2

6.50

8 0.

928

1.00

6 0.

986

0.99

6 1.

774

7.04

8 0.

923

1.00

6 0.

986

1.00

1 1.

755

8.14

6 0.

912

1.00

6 0.

987

1.01

3 1.

706

1

3 8.

429

0.93

6 1.

004

0.97

9 0.

994

1.80

3 8.

962

0.93

3 1.

004

0.98

0 0.

997

1.79

3 10

.040

0.

926

1.00

4 0.

980

1.00

5 1.

767

1 16

.838

0.

876

1.00

4 1.

238

0.99

4 0.

867

17.4

08

0.87

3 1.

004

1.23

4 0.

999

0.86

6 18

.559

0.

869

1.00

4 1.

224

1.00

7 0.

865

2 17

.880

0.

873

1.00

4 1.

219

0.99

4 0.

962

18.4

51

0.71

1.

004

1.21

5 0.

998

0.96

0 19

.604

0.

867

1.00

4 1.

206

1.00

6 0.

955

10

3 19

.069

0.

876

1.00

4 1.

178

0.99

4 1.

137

19.6

38

0.87

3 1.

004

1.17

6 0.

998

1.12

9 20

.788

0.

869

1.00

4 1.

171

1.00

5 1.

114

1 14

4.52

2 0.

887

1.00

0 1.

259

0.99

1 0.

811

145.

80

0.88

7 1.

000

1.25

8 0.

991

0.81

1 14

6.20

0 0.

886

1.00

0 1.

258

0.99

2 0.

811

2 14

5.44

0 0.

887

1.00

0 1.

258

0.99

1 0.

817

139.

52

0.95

8 1.

000

1.30

0 0.

991

0.81

7 14

7.11

9 0.

886

1.00

0 1.

257

0.99

2 0.

817

fPo

100 3

146.

365

0.88

6 1.

000

1.25

7 0.

991

0.82

2 14

6.92

0.

886

1.00

0 1.

257

0.99

1 0.

822

148.

044

0.88

5 1.

000

1.25

6 0.

992

0.82

2

β =

2.0

β

= 3.

0

1 5.

583

0.90

9 1.

009

0.99

8 0.

999

2.57

1 6.

137

0.90

1 1.

009

0.99

8 1.

008

2.51

5 7.

278

0.88

2 1.

009

1.00

0 1.

028

2.35

7 2

8.45

4 0.

929

1.00

4 0.

981

0.99

4 2.

686

8.99

1 0.

925

1.00

4 0.

981

0.99

8 2.

668

10.0

79

0.91

7 1.

004

0.98

1 1.

007

2.62

4

1

3 11

.351

0.

936

1.00

3 0.

976

0.99

3 2.

714

11.8

83

0.93

3 1.

003

0.97

6 0.

996

2.70

4 12

.956

0.

923

1.00

3 0.

976

1.00

1 2.

680

1 19

.015

0.

847

1.00

5 1.

373

0.99

5 0.

880

19.6

04

0.84

4 1.

005

1.36

7 1.

000

0.87

9 20

.798

0.

839

1.00

5 1.

354

1.01

0 0.

878

2 20

.127

0.

844

1.00

5 1.

344

0.99

5 1.

018

20.7

19

0.84

1 1.

005

1.33

8 1.

000

1.01

6 21

.917

0.

835

1.00

5 1.

326

1.00

9 1.

011

10

3 21

.644

0.

870

1.00

4 1.

153

0.99

4 1.

935

22.2

17

0.86

6 1.

004

1.15

8 0.

997

1.88

6 23

.380

0.

857

1.00

4 1.

171

1.00

5 1.

769

1 16

5.45

3 0.

861

1.00

1 1.

401

0.99

1 0.

812

166.

03

0.86

0 1.

001

1.40

0 0.

991

0.81

2 16

7.18

2 0.

860

1.00

1 1.

399

0.99

2 0.

812

2 16

6.40

0 0.

860

1.00

1 1.

400

0.99

1 0.

818

166.

98

0.86

0 1.

001

1.39

9 0.

991

0.81

8 16

8.13

1 0.

859

1.00

1 1.

398

0.99

2 0.

818

fPo

100

3 16

7.35

6 0.

860

1.00

1 1.

399

0.99

1 0.

825

167.

93

0.85

9 1.

001

1.39

8 0.

991

0.82

5 16

9.08

7 0.

859

1.00

1 1.

397

0.99

2 0.

825

1 10

.907

0.

929

1.00

3 0.

978

0.99

3 7.

658

11.4

44

0.92

6 1.

003

0.97

8 0.

997

7.62

2 12

.527

0.

920

1.00

3 0.

978

1.00

3 7.

539

2 19

.163

0.

937

1.00

2 0.

972

0.99

2 7.

739

19.6

93

0.93

5 1.

002

0.97

2 0.

993

7.72

4 20

.759

0.

932

1.00

2 0.

972

0.99

7 7.

692

1

3 27

.428

0.

939

1.00

1 0.

971

0.99

1 7.

762

27.9

56

0.93

8 1.

001

0.97

1 0.

992

7.75

3 29

.017

0.

936

1.00

1 0.

971

0.99

4 7.

734

1 24

.650

0.

797

1.00

6 1.

725

0.99

6 0.

894

25.2

78

0.79

4 1.

006

1.71

7 1.

001

0.89

4 26

.551

0.

787

1.00

6 1.

698

1.01

3 0.

893

2 25

.884

0.

792

1.00

6 1.

674

0.99

6 1.

121

26.5

16

0.78

8 1.

006

1.66

6 1.

001

1.11

9 27

.798

0.

782

1.00

6 1.

648

1.01

2 1.

115

10

3 36

.988

0.

921

1.00

1 1.

007

0.99

1 7.

498

37.5

27

0.92

0 1.

001

1.00

7 0.

992

7.48

4 36

.608

0.

918

1.00

1 1.

007

0.99

5 7.

457

1 22

0.21

1 0.

812

1.00

1 1.

765

0.99

1 0.

813

220.

82

0.81

2 1.

001

1.76

4 0.

991

0.81

3 22

2.04

5 0.

811

1.00

1 1.

762

0.99

3 0.

813

2 22

1.21

7 0.

811

1.00

1 1.

763

0.99

1 0.

820

221.

83

0.81

1 1.

001

1.76

3 0.

991

0.82

0 22

3.05

2 0.

810

1.00

1 1.

761

0.99

3 0.

820

fPo

100

3 22

2.23

3 0.

811

1.00

1 1.

762

0.99

1 0.

828

222.

84

0.81

0 1.

001

1.76

1 0.

991

0.82

8 22

4.06

9 0.

810

1.00

1 1.

760

0.99

3 0.

828



108


fp<10* fp ≥ 10 Steel β

µfy φfy γΑ γPo γM γO µfy φfy γΑ γPo γM γO 1.5 7.10 0.97 1.01 0.99 1.02 1.77 17.87 0.95 1.00 1.25 1.00 0.82

2.0 8.99 0.97 1.01 0.99 1.00 2.74 20.04 0.95 1.00 1.30 1.00 1.60

Car

bon

3.0 19.45 0.98 1.00 0.97 1.00 7.88 30.05 0.94 1.01 1.44 1.00 4.30 1.5 7.04 0.90 1.01 1.00 1.02 1.61 18.81 0.91 1.00 1.22 1.00 0.97 2.0 9.27 0.91 1.01 0.99 1.01 2.53 21.20 0.85 1.00 1.28 1.00 1.37

Stai

nles

s

3.0 19.56 0.93 1.00 0.98 1.00 7.65 31.09 0.85 1.00 1.39 1.00 4.15 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi. Table 7-16c. Resistance and Load Factors Applied to Nominal Values of Variables


φfy γΑ γPo γM γO φfy γΑ γPo γM γO 1.5 1.10 1.01 0.89 0.92 1.15 1.07 1.00 1.13 0.90 0.53

2.0 1.10 1.01 0.89 0.90 1.78 1.07 1.00 1.17 0.90 1.04

Car

bon

3.0 1.11 1.00 0.87 0.90 5.12 1.06 1.01 1.30 0.90 2.80 1.5 1.21 1.01 0.90 0.92 1.05 1.23 1.00 1.10 0.90 0.63 2.0 1.23 1.01 0.89 0.91 1.64 1.15 1.00 1.15 0.90 0.89

Stai

nles

s

3.0 1.25 1.00 0.88 0.90 4.97 1.15 1.00 1.25 0.90 2.70 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi. Table 7-16d. Suggested Resistance and Load Factors Applied to Nominal Values of Variables


φfy γΑ γPo γM γO φfy γΑ γPo γM γO 1.5 0.95 0.90 0.90 0.90 0.90 1.00 0.90 0.90 0.90 0.90

2.0 0.97 1.00 1.20 1.20 1.2 0.96 1.00 1.00 1.00 1.00

Car

bon

3.0 0.50 1.20 1.30 1.20 1.4 0.87 1.20 1.30 1.20 1.40 1.5 0.98 0.80 0.80 0.80 0.80 0.99 0.80 0.80 0.80 0.70 2.0 0.97 1.00 1.00 1.00 1.00 0.99 0.90 0.90 0.90 1.00

Stai

nles

s

3.0 0.59 1.20 1.30 1.20 1.40 0.98 1.20 1.30 1.20 1.40 *Such cases are usual, for example, when L>300in & P<2000 psi or when L<300in & P<200 psi.



109


This function refers to Service Limit B. The stress fO is the result of the OBE earthquake and it is applicable to auxiliarypiping used only for emergency conditions. Table 7-17 gives the ranges of parameters, while Table 7-18 presents the probabilistic characteristics used for the calculation of the partial safety factors.

OfAfyfg −−=7 (7-6)

For this performance function, the earthquake load is selected to have a coefficient of variation equal to 0.83, although the value of COV = 0.50 is also reasonable, since the return period of the OBE is only 100 years (Rodabaugh, 1984). Nevertheless, the following suggested factors consider the case of COV = 0.83 for the earthquake load. The values of target reliability index are also smaller, since from experience of buildings the reliability index, when earthquake load is present, is between 1.75 and 2.50. Results are shown in Tables 7-19a to 7-19d. Table 7-17. Ranges of Parameters

Parameter Ranges β 1.5, 2, 3, 3.5

COV (fy) Carbon


0.03 to 0.19 COV (fA) 0.05 to 0.10 COV (fo) 0.40 to 1.00


Random Variable

Recommended (COV)




fA 0.10 1 Normal 1.00 fo 0.50, 0.83 1, 2, 3





110

Table 7-19a. Load and Resistance Factors Applied to Mean Values of Variables Carbon Steel

fO 1 2 3 Distributions β

µfy φfy γΑ γo µfy φfy γΑ γo µfy φfy γΑ γo 1.5 2.715 0.970 1.016 1.617 4.389 0.975 1.008 1.635 6.068 0.977 1.005 1.640 2 3.318 0.970 1.014 2.203 5.596 0.974 1.007 2.221 7.879 0.975 1.004 2.226 3 6.056 0.972 1.007 4.879 11.078 0.974 1.003 4.894 16.103 0.975 1.002 4.898

fy: Lognormal fA: Normal fo: Type II, with COV=0.50 3.5 9.222 0.973 1.004 7.969 17.414 0.974 1.002 7.983 25.606 0.975 1.001 7.987

1.5 2.925 0.977 1.011 1.847 4.820 0.981 1.005 1.860 6.718 0.982 1.004 1.864 2 3.873 0.977 1.008 2.778 6.718 0.980 1.004 2.790 9.566 0.981 1.003 2.793 3 9.106 0.979 1.003 7.914 17.189 0.980 1.002 7.923 25.273 0.980 1.001 7.926

fy: Lognormal fA: Normal fo: Type II, with COV=0.83 3.5 16.458 0.980 1.002 15.123 31.894 0.980 1.001 15.13 47.331 0.980 1.001 15.13 Stainless Steel

1.5 2.811 0.912 1.017 1.545 4.519 0.926 1.008 1.589 6.237 0.931 1.005 1.60 2 3.436 0.910 1.014 2.112 5.769 0.922 1.007 2.157 8.110 0.927 1.005 2.17 3 6.255 0.917 1.007 4.727 11.417 0.923 1.003 4.768 16.583 0.925 1.002 4.78


1.5 3.004 0.932 1.011 1.790 4.933 0.942 1.006 1.822 6.868 0.946 1.004 1.83 2 3.978 0.933 1.008 2.702 6.881 0.94 1.004 2.733 9.788 0.943 1.003 2.74 3 9.334 0.938 1.0003 7.752 17.602 0.941 1.002 7.780 25.871 0.942 1.001 7.79


Table 7-19b. Suggested Load and Resistance Factors Applied to Mean Values of Variables

Distributions Steel β µfy φfy γΑ γo 1.5 4.821 0.979 1.007 1.855 2.0 6.719 0.938 1.005 2.785 3.0 17.189 0.980 1.002 7.920

fy: Lognormal fA: Normal fO: Type II

Carbon

3.5 31.894 0.980 1.000 31.50 1.5 4.936 0.939 1.007 1.810 2.0 6.833 0.938 1.005 2.722 3.0 17.602 0.939 1.002 7.770

fy: Lognormal fA: Normal fO: Type II

Stainless

3.5 32.656 0.941 1.001 31.646 Table 7.19c. Load and Resistance Factors Applied to Nominal Values of Variables

Distributions Steel β φfy γΑ γo 1.5 1.11 1.01 1.21 2.0 1.06 1.01 1.81 3.0 1.11 1.00 5.15

fy: Lognormal fA: Normal fO: Extr. II

Carbon

3.5 1.11 1.00 20.5 1.5 1.27 1.01 1.18 2.0 1.27 1.01 1.77 3.0 1.27 1.00 5.05

fy: Lognormal fA: Normal fO: Extr II

Stainless

3.5 1.27 1.00 20.57



111

Table 7-19d. Suggested Nominal Values for the Variables, g7

Distributions Steel β φfy γΑ γo 1.5 0.96 1.0 1.0 2.0 0.94 1.2 1.5 3.0 0.38 1.2 1.5


Carbon

3.5 0.12 1.2 1.5 1.5 1.00 0.9 0.9 2.0 0.94 1.1 1.2 3.0 0.45 1.2 1.5


Stainless

3.5 0.17 1.2 2.0

7.5. Emergency Loading Condition (Service Level C) This section provides the performance functions for emergency loading including mechanical loads that now can be not only non-reversing but also reversing. Moreover, the loading due to OBE is considered. In the following analysis, the reversing mechanical loads are not taken into consideration since information about their statistical properties is nonexistent.


This performance function is the same as g5 with the maximum pressure referring to that of service limit C. The target reliability index β might be different.

7.5.2. Performance Function g9 This performance function can be applied either to Service Limit B or Service Limit C expecting that the reliability index, β, for the first case will be higher.

OfPofAfyfg −−−=9 (7-7)

Table 7-20 gives the ranges of parameters, while Table 7-21 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-22a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-21. Table 7-22b shows the suggested factors applicable to mean values of variables and Tables 7-22c and d the nominal and the suggested nominal factors, respectively. Table 7-22d shows recommended values.



112


β 1.5, 2, 3, 3.5

COV (fy) Carbon


0.03 to 0.19 COV (fA) 0.05 to 0.10 COV (fP0) 0.15 to 0.25 COV (f0) 0.40 to 1.00


Random Variable

Recommended (COV)




fA 0.10 1 Normal 1.00 fPo 0.20 1, 10, 100, (1000*) Type I (Largest) 0.90

fo 0.83 1, 2, 3 Type II (Largest) 0.50-0.90 (Use

0.65) N.A. = Not Available


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

113

Tabl

e 7-

22a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es fo

r Car

bon

Stee

l C

arbo

n St

eel

f 0 1

2 3

D

istri

butio

ns

β µ f

y φ f

y γ Α

γ P

O

γ O

µ fy

φ fy

γ Α

γ PO

γ O

µ fy

φ fy

γ Α

γ PO

γ O

1.5

3.93

8 0.

970

1.01

1 1.

007

1.80

2 5.

817

0.97

7 1.

005

0.98

6 1.

847

7.71

0 0.

980

1.00

4 0.

979

1.85

7 2

4.88

0 0.

972

1.00

8 0.

996

2.74

0 7.

713

0.97

7 1.

004

0.98

1 2.

777

10.5

57

0.97

9 1.

003

0.97

6 2.

786

3.0

10.1

00

0.97

7 1.

003

0.97

8 7.

890

18.1

79

0.97

9 1.

002

0.97

2 7.

914

26.2

62

0.98

1.

001

0.97

1 7.

920

fPmax = 1

f y: L

ogno

rmal

f A

: Nor

mal

f P

o: Ty

pe I

f o: T

ype

II

3.5

17.4

48

0.97

9 1.

002

0.97

3 15

.103

32

.882

0.

980

1.00

1 0.

970

15.1

23

48.3

18

0.98

1.

001

0.96

9 15

.129

1.

5 15

.498

0.

954

1.00

5 1.

291

0.87

3 16

.467

0.

953

1.00

5 1.

271

0.98

7 17

.622

0.

955

1.00

4 1.

203

1.26

3 2

17.3

59

0.94

4 1.

005

1.44

9 0.

884

18.3

69

0.94

2 1.

005

1.42

0 1.

047

20.1

18

0.96

0 1.

003

1.11

2 2.

396

3.0

22.1

64

0.92

4 1.

006

1.85

8 0.

896

23.2

36

0.92

1 1.

006

1.81

0 1.

152

35.3

57

0.97

4 1.

001

1.00

5 7.

793

fPmax = 10

f y: L

ogno

rmal

f A

: Nor

mal

f P

o: Ty

pe I

f o: T

ype

II

3.5

25.1

73

0.91

4 1.

006

2.10

9 0.

899

41.9

33

0.97

5 1.

001

0.99

7 14

.958

57

.308

0.

977

1.00

1 0.

986

15.0

42

1.5

137.

636

0.95

8 1.

000

1.30

1 0.

812

138.

486

0.95

8 1.

000

1.30

1 0.

817

139.

342

0.95

8 1.

000

1.30

0 0.

823

2 15

6.00

8 0.

949

1.00

1 1.

462

0.81

2 15

6.86

8 0.

948

1.00

1 1.

461

0.81

9 15

7.73

5 0.

948

1.00

1 1.

461

0.82

6 3.

0 20

3.62

8 0.

929

1.00

1 1.

874

0.81

3 20

4.50

7 0.

929

1.00

1 1.

873

0.82

0 20

5.39

5 0.

9219

1.

001

1.87

3 0.

828

fPmax = 100

f y: L

ogno

rmal

f A

: Nor

mal

f P

o: Ty

pe I

f o: T

ype

II

3.5

233.

502

0.91

9 1.

001

2.12

7 0.

813

234.

391

0.91

9 1.

001

2.12

7 0.

821

235.

289

0.91

8 1.

001

2.12

6 0.

829

St

ainl

ess S

teel

1.

5 4.

075

0.91

2 1.

011

1.00

9 1.

696

5.97

5 0.

933

1.00

6 0.

986

1.79

0 7.

900

0.93

9 1.

004

0.97

9 1.

812

2 5.

039

0.91

7 1.

009

0.99

7 2.

616

7.92

1 0.

933

1.00

4 0.

981

2.70

2 10

.822

0.

938

1.00

3 0.

976

2.72

4 3.

0 10

.374

0.

932

1.00

3 0.

978

7.69

0 18

.634

0.

938

1.00

2 0.

972

7.75

3 26

.901

0.

940

1.00

1 0.

971

7.77

1

fPmax = >1

f y: L

ogno

rmal

f A

: Nor

mal

f P

o: Ty

pe I

f o: T

ype

II

3.5

17.8

94

0.93

7 1.

002

0.97

3 14

.785

33

.685

0.

940

1.00

1 0.

970

14.8

42

49.4

80

0.94

1 1.

001

0.96

9 14

.859

1.

5 16

.273

0.

878

1.00

4 1.

242

0.86

8 17

.312

0.

876

1.00

4 1.

223

0.96

4 18

.503

0.

878

1.00

4 1.

181

1.14

4 2

18.4

30

0.85

0 1.

005

1.37

9 0.

880

19.5

39

0.84

6 1.

005

1.34

9 1.

021

21.0

75

0.87

4 1.

003

1.14

8 1.

979

3.0

24.0

28

0.80

1 1.

006

1.73

4 0.

894

25.2

58

0.79

5 1.

006

1.68

3 1.

123

36.4

50

0.92

2 1.

001

1.00

7 7.

510

fPmax = >10

f y: L

ogno

rmal

f A

: Nor

mal

f P

o: Ty

pe I

f o: T

ype

II

3.5

27.5

92

0.77

6 1.

006

1.95

1 0.

898

43.1

76

0.92

5 1.

001

0.99

8 14

.487

58

.880

0.

931

1.00

1 0.

987

14.6

57

1.5

143.

964

0.88

7 1.

000

1.25

9 0.

811

144.

882

0.88

7 1.

000

1.25

9 0.

817

145.

806

0.88

6 1.

000

1.25

8 0.

822

2 16

4.87

7 0.

861

1.00

0 1.

401

0.81

2 16

5.82

5 0.

860

1.00

1 1.

400

0.81

8 16

6.78

0 0.

860

1.00

1 1.

399

0.82

5 3.

0 21

9.60

1 0.

812

1.00

1 1.

766

0.81

3 22

0.60

7 0.

812

1.00

1 1.

764

0.82

0 22

1.62

2 0.

811

1.00

1 1.

763

0.82

8

fPmax = >100

f y: L

ogno

rmal

f A

: Nor

mal

f P

o: Ty

pe I

f o: T

ype

II

3.5

254.

568

0.78

8 1.

001

1.98

7 0.

813

255.

605

0.78

7 1.

001

1.98

5 0.

821

256.

653

0.78

7 1.

001

1.98

4 0.

828



114


fp<10 fp ≥ 10 Steel β µfy φfy γΑ γPO γO µfy φfy γΑ γPO γO 1.5 5.82 0.98 1.01 0.99 1.83 16.56 0.95 1.00 1.25 1.04 2 7.72 0.97 1.01 0.99 2.76 18.74 0.95 1.00 1.29 1.60 3 18.18 0.98 1.01 0.97 7.91 28.76 0.95 1.00 1.44 4.30 Carbon

3.5 32.88 0.98 1.00 0.97 15.12 41.24 0.94 1.00 1.56 7.93 1.5 5.99 0.92 1.01 1.01 1.75 17.39 0.88 1.00 1.22 0.98 2 7.93 0.93 1.01 0.99 2.67 19.75 0.86 1.00 1.27 1.40 3 18.64 0.94 1.00 0.97 7.73 30.24 0.86 1.00 1.39 4.16 Stainless

3.5 33.69 0.94 1.00 0.97 14.82 43.24 0.85 1.00 1.49 7.73 Table 7-22c. Resistance and Load Factors Applied to Nominal Values of Variables

fp<10 fp ≥ 10 Steel β φfy γΑ γPO γO φfy γΑ γPO γO

1.5 1.11 1.01 0.89 1.19 1.07 1.00 1.13 0.68 2 1.10 1.01 0.89 1.79 1.07 1.00 1.16 1.04 3 1.11 1.01 0.87 5.14 1.07 1.00 1.30 2.80

Carbon

3.5 1.11 1.00 0.87 9.80 1.06 1.00 1.40 5.20 1.5 1.24 1.01 0.91 1.14 1.19 1.00 1.10 0.64 2 1.26 1.01 0.89 1.74 1.16 1.00 1.14 0.91 3 1.27 1.00 0.87 5.02 1.16 1.00 1.25 2.70

Stainless

3.5 1.27 1.00 0.87 9.60 1.06 1.00 1.34 5.00 Table 7-22d. Suggested Resistance and Load Factors Applied to Nominal Values of Variables

fp<10 fp ≥ 10 Steel β φfy γΑ γPO γO φfy γΑ γPO γO

1.5 1.00 1.0 1.0 1.0 0.96 1.0 0.9 1.0 2.0 0.97 1.0 1.2 1.4 0.95 1.0 1.0 1.0 3.0 0.41 1.0 1.2 1.4 0.79 1.0 1.2 1.4

Carbon

3.5 0.23 1.0 1.2 1.4 0.56 1.0 1.2 1.4 1.5 1.00 1.0 0.9 1.0 0.99 0.9 0.8 0.9 2.0 0.91 1.0 1.2 1.2 0.94 1.0 1.0 1.0 3.0 0.43 1.0 1.2 1.4 0.85 1.0 1.2 1.4

Stainless

3.5 0.24 1.0 1.2 1.4 0.61 1.0 1.2 1.4



115


This performance function is similar to the design performance function g1 where the pressure is checked and not the thickness. This performance function aims to be equivalent to the pressure limit of 1.5Ρα that the ASME BPV Code requires. Therefore,

)2(

2P

ytoDyft

−≤ (7-8)

The factors obtained from performance function g1 can be also used here with possibly different value of β.


This performance is similar with the performance function g7 of service limit B. Similarly, here the calibration should lead to a different value for β or in other words a different probability of failure.


This performance function is similar to the performance function g6 of service limit B with the reversing mechanical loads. Since the reversing mechanical loading is not considered as out of scope of this work, the factors obtained in Section 7.4.2 for the performance function g6 can also be used herein.

7.6. Faulted Loading Condition (Service Level D) This service limit includes performance functions g13 to g17, where accidental loads are combined.


This performance function works again as limit to the maximum pressure occurring during service limit D loading. It comes to substitute the 2Ρα limit available in the ASME BPV Code. Again herein the Boardman equation is used as follows:

2

)2(13 t

ytoDPufg

−−= (7-9a)

where y has a deterministic value equal to 0.4 for Do/t>6, and otherwise it is y = Di/(Do+Di). Considering first y equal to 0.4, Eq. 7-9a can be written as

)4.05.0(12 −−= αPfg u (7-9b)

For the case Do/t<6, Eq.7-9a takes the form:

⎟⎠⎞

⎜⎝⎛

−−

−−=1

15.05.012 aaaPfg u (7-9c)



116

where in all equations α = Do/t and is considered constant. The probabilistic characteristics of the variables fu and P are given in Tables 7-23 and 7-24. Table 7-25a shows the calculated factors for the mean values of variables shown in Table 7-23. Eqs 7-9b and 7-9c yield the same results in all considered combinations of pressure and resistance. Table 7-25c provides suggested factors. Table 7-23. Ranges of Parameters


COV (fu) 0.03 to 0.09 COV(P) 0.10 to 0.20

Table 7-24. Selected Probabilistic Characteristics of Random Variables Random Variabl

e

Recommended (COV) Mean Value Distribution Type Total Bias

fu 0.06 for carbon & stainless

45 to 80 ksi

Lognormal 1.18 carbon & stainless

P 0.15 50 to 3000 psi Lognormal 1.15 Table 7-25a. Factors Applied to Mean Values of Variables

Carbon & Stainless Steel Distributions β

φfu γp 2.0 0.95 1.30 3.0 0.93 1.50 3.5 0.92 1.61 4.5 0.90 1.87

fu: Lognormal

P: Lognormal 5.5 0.88 2.19

Table 7-25b. Factors Applied to Nominal Values of Variables


φfu γp 2.0 1.12 1.50 3.0 1.10 1.72 3.5 1.08 1.85 4.5 1.06 2.15

fu: Lognormal

P: Lognormal 5.5 1.04 2.52



117

Table 7-25c. Suggested Factors Applied to Nominal Values of Variables


φfu γp 2.0 0.95 1.30 3.0 0.93 1.50 3.5 0.92 1.61 4.5 0.90 1.87

fu: Lognormal

P: Lognormal 5.5 0.88 2.19


This performance function considers the loading due to the SSE as follows:

SfPSfAfufg −−−=14 (7-10)

Table 7-26 gives the ranges of parameters, while Table 7-27 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-28a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-27. Table 7-28b gives recommended factors applied to mean values of variables and Tables 7-28c and 7-28d give the factors applicable to nominal values of variables and the suggested ones, respectively. Table 7-26. Ranges of Parameters

Parameter Ranges β 1.5, 2, 3, 3.5

COV (fu) 0.03 to 0.09 COV (fA) 0.05 to 0.10 COV (fPS) 0.15 to 0.35 COV (fS) 0.40 to 1.10



118


Recommended (COV)


fu 0.06 for carbon

& stainless N.A. Lognormal 1.18 carbon & stainless

fA 0.10 1 Normal 1.00 fPS 0.25 1, 10, 100, (1000) Type I (Largest) 0.90

fS 0.83 1.5, 2, 3, 4, 5 Type II (Largest) 0.50 to 0.90 (Use 0.65)



Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

119

Tabl

e 7-

28a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es fo

r Car

bon

Stee

l C

arbo

n &

Sta

inle

ss S

teel

f S

1.

5 2

3

D

istri

butio

ns

β µ f

u φ f

u γ Α

γ P

S γ S

µ f

u φ f

u γ Α

γ P

S γ S

µ f

u φ f

u γ Α

γ P

S γ S

1.

5 4.

843

0.98

6 1.

007

0.99

9 1.

845

5.78

1 0.

987

1.00

5 0.

988

1.85

6 7.

665

0.98

8 1.

004

0.97

8 1.

865

2 6.

253

0.98

6 1.

005

0.98

8 2.

782

7.66

5 0.

987

1.00

4 0.

980

2.79

2 10

.496

0.

988

1.00

3 0.

973

2.79

9 3.

0 14

.056

0.

988

1.00

2 0.

970

7.94

2 18

.076

0.

988

1.00

2 0.

967

7.94

8 26

.120

0.

989

1.00

1 0.

964

7.95

2

fPS= 1

f u: L

ogno

rmal

f A

: Nor

mal

f P

S: Ty

pe I

f S: T

ype

II

3.5

25.0

26

0.98

8 1.

001

0.96

5 15

.180

32

.706

0.

989

1.00

1 0.

963

15.1

84

48.0

66

0.98

9 1.

001

0.96

2 15

.188

1.

5 16

.565

0.

977

1.00

4 1.

384

0.89

3 17

.033

0.

976

1.00

4 1.

375

0.93

6 18

.057

0.

976

1.00

4 1.

337

1.08

1 2

18.8

02

0.97

1 1.

004

1.58

9 0.

908

19.2

83

0.97

0 1.

004

1..5

78

0.96

5 20

.392

0.

970

1.00

4 1.

490

1.29

2 3.

0 24

.572

0.

960

1.00

5 2.

119

0.92

4 25

.072

0.

959

1.00

5 2.

104

0.99

9 35

.155

0.

985

1.00

1 1.

020

7.81

0

fPS = 10

f u: L

ogno

rmal

f A

: Nor

mal

f P

S: Ty

pe I

f S: T

ype

II II

3.

5 28

.172

0.

953

1.00

5 2.

446

0.92

8 28

.678

0.

953

1.00

5 2.

430

1.00

9 56

.949

0.

987

1.00

1 0.

989

15.1

05

1.5

144.

700

0.97

9 1.

000

1.39

5 0.

813

145.

116

0.97

9 1.

000

1.39

4 0.

815

145.

951

0.97

9 1.

000

1.39

4 0.

820

2 16

6.85

3 0.

974

1.00

0 1.

603

0.81

4 16

7.27

2 0.

974

1.00

0 1.

602

0.81

6 16

8.11

3 0.

974

1.00

0 1.

602

0.82

2 3.

0 22

4.21

3 0.

963

1.00

0 2.

136

0.81

5 22

4.63

7 0.

963

1.00

0 2.

136

0.81

7 22

5.48

9 0.

963

1.00

0 2.

136

0.82

3

fPS = 100 f u:

Log

norm

al

f A: N

orm

al

f PS:

Type

I f S:

Typ

e II

3.

5 26

0.04

7 0.

957

1.00

0 2.

465

0.81

5 26

0.47

4 0.

956

1.00

0 2.

465

0.81

8 26

1.33

2 0.

956

1.00

0 2.

464

0.82

4

4 5

1.5

9.53

3 0.

989

1.00

3 0.

973

1.86

8 11

.443

0.

989

1.00

2 0.

970

1.87

0 2

13.3

29

0.98

9 1.

002

0.96

9 2.

802

16.1

63

0.98

9 1.

002

0.96

7 2.

803

3.0

34.1

64

0.98

9 1.

001

0.96

3 7.

954

42.2

09

0.98

9 1.

001

0.96

2 7.

955

fPS = >1

f u: L

ogno

rmal

f A

: Nor

mal

f P

S: Ty

pe I

f S: T

ype

II

3.5

63.4

26

0.98

9 1.

000

0.96

1 15

.189

78

.787

0.

989

1.00

0 0.

961

15.1

90

1.5

19.3

19

0.97

8 1.

003

1.22

1 1.

420

20.9

06

0.98

1 1.

002

1.12

7 1.

650

2 22

.740

0.

981

1.00

2 1.

113

2.54

4 25

.400

0.

983

1.00

2 1.

067

2.66

0 3.

0 43

.115

0.

986

1.00

1 1.

003

7.87

3 51

.111

0.

987

1.00

1 0.

993

7.90

2

fPS= 10

f u: L

ogno

rmal

f A

: Nor

mal

f P

S: Ty

pe I

f S: T

ype

II

3.5

72.2

69

0.98

8 1.

000

0.98

1 15

.139

87

.605

0.

988

1.00

0 0.

976

15.1

55

1.5

146.

791

0.97

9 1.

000

1.39

4 0.

825

14.6

37

0.97

9 1.

000

1.39

3 0.

830

2 16

8.96

0 0.

973

1.00

0 1.

602

0.82

7 16

9.81

3 0.

973

1.00

0 1.

601

0.83

3 3.

0 22

6.34

8 0.

962

1.00

0 2.

135

0.83

0 22

7.21

3 0.

962

1.00

0 2.

135

0.83

6

fPS= >100

f u: L

ogno

rmal

f A

: Nor

mal

f P

S: Ty

pe I

f S: T

ype

II

3.5

262.

197

0.95

6 1.

000

2.46

4 0.

830

263.

068

0.95

6 1.

000

2.46

3 0.

837



120


fp<10 fp ≥ 10 Steel β µfu φfu γΑ γPS γS µfu φfu γΑ γPS γS 1.5 8.143 0.99 1.00 0.98 1.86 18.735 0.98 1.00 1.26 1.23 2 11.209 0.99 1.00 0.97 2.79 16.860 0.99 1.00 1.33 1.75 3 28.132 0.99 1.00 0.97 7.949 33.39 0.97 1.00 1.56 4.36

Carbon &

Stainless 3.5 51.906 0.99 1.00 0.96 15.18 53.48 0.97 1.00 1.72 7.98 Table 7-28c. Resistance and Load Factors Applied to Nominal Values of Variables

fp<10 fp ≥ 10 Steel β φfu γΑ γPS γS φfu γΑ γPS γS

1.5 1.17 1.00 0.88 1.21 1.16 1.00 1.13 0.80 2 1.17 1.00 0.87 1.81 1.17 1.00 1.20 1.14 3 1.17 1.00 0.87 5.17 1.14 1.00 1.40 2.83

Carbon &

Stainless 3.5 1.17 1.00 0.86 9.90 1.14 1.00 1.55 5.20

Table 7-28d Suggested Resistance and Load Factors Applied to Nominal Values of Variables

fp<10 fp ≥ 10 Steel β φfu γΑ γPS γS φfu γΑ γPS γS

1.5 0.95 1.0 0.9 0.9 0.96 0.9 0.8 0.9 2.0 0.94 1.0 1.2 1.3 0.99 1.0 1.0 1.0 3.0 0.43 1.2 1.3 1.5 0.85 1.2 1.3 1.5

Carbon &

Stainless 3.5 0.34 1.2 1.3 2.5 0.56 1.2 1.3 1.5


In this performance function, the pipe is not pressurized and is susceptible to sustained weight and SSE loading.

SfAfufg −−=15 (7-11)

Table 7-29 gives the ranges of parameters, while Table 7-30 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-31a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-30. Table 7-31b gives recommended factors applied to mean values of variables and Tables 7-31c and 7-31d give the factors applicable to nominal values of variables and the suggested ones, respectively.



121


β 1.5, 2, 3, 3.5 COV (fu) 0.03 to 0.09 COV (fA) 0.05 to 0.10 COV (fs) 0.40 to 1.10


Random Variable

Recommended (COV)



N.A. Lognormal 1.18 for carbon & stainless

fA 0.10 1 Normal 1.00 fs 0.83 1.5, 2, 3, 4, 5 Type II (Largest) 0.50 to 0.90 (Use

0.65) N.A. = Not Available Table 7-31a. Calculated Load and Resistance Factors Applied to Mean Values of Variables

Carbon & Stainless Steel fs

1.5 2 3 Distributions β µfu φfu γΑ γs µfu φfu γΑ γs µfu φfu γΑ γs

1.5 3.851 0.988 1.007 1.866 4.796 0.989 1.005 1.869 6.686 0.99 1.004 1.871 2 5.267 0.988 1.005 2.800 6.684 0.989 1.004 2.802 9.519 0.989 1.003 2.804 3 13.080 0.989 1.002 7.953 17.102 0.989 1.002 7.954 25.147 0.989 1.001 7.956

fu: Lognormal fA: Normal fs: Type II

3.5 24.053 0.989 1.001 15.188 31.734 0.989 1.001 15.19 47.094 0.989 1.001 15.19 4 5

1.5 8.577 0.990 1.003 1.872 10.468 0.990 1.002 1.873 2 12.354 0.989 1.002 2.805 15.189 0.990 1.002 2.806 3 33.192 0.989 1.001 7.957 41.237 0.989 1.001 7.958

fu: Lognormal fA: Normal fs: Type II

3.5 62.455 0.989 1.000 15.19 77.816 0.989 1.000 15.19

Table 7-31b. Suggested Load and Resistance Factors Applied to Mean Values of Variables

Distributions Steel β µfu φfu γΑ γs 1.5 7.16 0.99 1.00 1.87 2.0 10.23 0.99 1.00 2.80 3.0 27.16 0.99 1.00 7.96

fu: Lognormal fA: Normal fs: Extr. II

Carbon &

Stainless 3.5 50.94 0.99 1.00 15.19



122

Table 7-31c. Suggested Load and Resistance Factors Applied to Nominal Values of Variables Distributions Steel β φfu γΑ γs

1.5 1.17 1.00 1.22 2.0 1.17 1.00 1.82 3.0 1.17 1.00 5.17

fu: Lognormal fA: Normal fs: Extr. II

Carbon &

Stainless 3.5 1.17 1.00 9.88

Table 7-31d. Suggested Nominal Values for the Variables, g15

Distributions Steel β φfu γΑ γs 1.5 0.99 1.0 1.0 2.0 0.95 1.2 1.4 3.0 0.49 1.2 2.0

fu: Lognormal fA: Normal fs Extr. II

Carbon &

Stainless 3.5 0.26 1.2 2.0


This performance function adds loading due to LOCA.

Lu fPfAffg −−−=16 (7-12)

Table 7-32 gives the ranges of parameters, while Table 7-33 presents the probabilistic characteristics used for the calculation of the partial safety factors. Table 7-34a shows the calculation of partial factors for all combinations of the probabilistic characteristics shown in Table 7-33. Tables 7-34b and 7-34c give the suggested mean factors and factors applicable to nominal values, respectively, whereas Table 7-34d presents the suggested factors applied to nominal values of variables. Table 7-32. Ranges of Parameters


COV (fu) 0.03 to 0.09 COV (fA) 0.05 to 0.10 COV (fP) 0.10 to 0.13 COV (fL) 0.07 to 0.15



123


Recommended (COV)




fA 0.10 1 Normal 1.00 fP 0.12 1, 10, 100, (1000) Lognormal 1.10 fL 0.20 0.5, 1, 2 Type I (Largest) 0.80



Dev

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124

Tabl

e 7-

34a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es

Car

bon

& S

tain

less

Ste

el

f L 0.

5 1.

0 2.

0

D

istri

butio

ns

β µ f

u φ f

u γ Α

γ P

γ L

µ f

u φ f

u γ Α

γ P

γ L

µ f

u φ f

u γ Α

γ P

γ L

2

2.99

9 0.

927

1.07

5 1.

119

1.17

1 3.

678

0.94

0 1.

048

1.06

9 1.

339

5.15

0 0.

954

1.02

6 1.

031

1.42

7 3

3.30

9 0.

897

1.10

0 1.

175

1.38

3 4.

181

0.92

3 1.

057

1.08

4 1.

716

6.10

3 0.

940

1.02

9 1.

036

1.83

7 3.

5 3.

483

0.85

5 1.

109

1.19

6 1.

552

4.48

6 0.

915

1.05

9 1.

088

1.96

0 6.

694

0.93

4 1.

030

1.03

8 2.

092

4.5

3.88

2 0.

865

1.11

8 1.

220

2.04

3 5.

222

0.90

1 1.

061

1.09

2 2.

552

8.12

5 0.

920

1.03

1 1.

039

2.70

1

fP = 1

f u: L

ogno

rmal

f A

: Nor

mal

f P

: Log

norm

al

f L: T

ype

I 5.

5 4.

367

0.84

8 1.

122

1.23

1 2.

704

6.13

5 0.

885

1.06

2 1.

094

3.27

3 9.

907

0.90

3 1.

031

1.04

0 3.

440

2 14

.581

0.

941

1.01

2 1.

222

0.98

7 15

.112

0.

940

1.01

2 1.

218

1.01

0 16

.215

0.

938

1.01

1 1.

208

1.06

2 3

16.4

90

0.91

5 1.

016

1.35

7 0.

995

17.0

44

0.91

3 1.

016

1.35

2 1.

028

18.2

17

0.91

0 1.

015

1.33

4 1.

116

3.5

17.5

42

0.90

2 1.

018

1.43

0 0.

999

18.1

07

0.90

0 1.

018

1.42

4 1.

037

19.3

16

0.89

7 1.

017

1.40

2 1.

144

4.5

19.8

65

0.87

7 1.

021

1.59

0 1.

004

20.4

51

0.87

5 1.

021

1.58

1 1.

051

21.7

33

0.87

1 1.

020

1.56

0 1.

203

fP = 10 f u:

Log

norm

al

f A: N

orm

al

f P: L

ogno

rmal

f L:

Typ

e I

5.5

22.5

14

0.85

3 1.

023

1.76

7 1.

009

23.1

21

0.85

0 1.

023

1.75

7 1.

063

24.4

74

0.84

6 1.

022

1.71

5 1.

264

2 13

1.54

3 0.

945

1.00

1 1.

229

0.96

9 13

2.05

7 0.

945

1.00

1 1.

229

0.97

1 13

3.08

6 0.

945

1.00

1 1.

228

0.97

5 3

150.

188

0.92

0 1.

002

1.36

7 0.

970

150.

716

0.92

0 1.

002

1.36

7 0.

972

151.

776

0.92

0 1.

002

1.36

6 0.

978

3.5

160.

486

0.90

8 1.

002

1.44

2 0.

970

161.

021

0.90

8 1.

002

1.44

2 0.

973

162.

097

0.90

7 1.

002

1.44

1 0.

979

4.5

183.

265

0.88

4 1.

002

1.60

5 0.

971

183.

815

0.88

3 1.

002

1.60

4 0.

974

184.

922

0.88

3 1.

002

1.60

3 0.

981

fP = 100

f u: L

ogno

rmal

f A

: Nor

mal

f P

: Log

norm

al

f L: T

ype

I 5.

5 20

9.29

9 0.

860

1.00

2 1.

786

0.97

1 20

9.86

4 0.

860

1.00

2 1.

785

0.97

5 21

1.00

2 0.

859

1.00

2 1.

784

0.98

3



125


fp<10 fp ≥ 10 Steel β µfu φfu γΑ γP γL µfu φfu γΑ γP γL 2.0 4.074 0.94 1.05 1.08 1.30 15.398 0.94 1.01 1.22 1.01 3.0 4.706 0.92 1.06 1.10 1.61 17.353 0.91 1.01 1.35 1.04 3.5 5.088 0.89 1.07 1.12 1.82 18.429 0.90 1.01 1.42 1.06 4.5 6.003 0.89 1.07 1.13 2.37 20.799 0.88 1.01 1.58 1.09

Carbon &

Stainless 5.5 7.137 0.87 1.08 1.13 3.07 23.494 0.85 1.01 1.77 1.11

Table 7-34c. Resistance and Load Factors Applied to Nominal Values of Variables

fp<10 fp ≥ 10 Steel β φfu γΑ γP γL φfu γΑ γP γL

2.0 1.11 1.05 1.19 1.04 1.11 1.01 1.34 0.81 3.0 1.09 1.06 1.21 1.29 1.07 1.01 1.49 0.83 3.5 1.05 1.07 1.23 1.46 1.06 1.01 1.56 0.85 4.5 1.05 1.07 1.24 1.96 1.04 1.01 1.74 0.87

Carbon &

Stainless 5.5 1.03 1.08 1.24 2.46 1.00 1.01 1.95 0.89

Table 7-34d. Suggested Resistance and Load Factors Applied to Nominal Values of Variables

fp<10 fp ≥ 10 Steel β φfu γΑ γPx γL φfu γΑ γPx γL

2.0 0.98 1.0 1.0 0.90 0.90 1.0 1.0 1.0 3.0 0.97 1.0 1.1 1.1 0.93 1.0 1.2 1.2 3.5 0.97 1.0 1.2 1.3 0.95 1.0 1.3 1.3 4.5 0.90 1.0 1.3 1.5 0.89 1.2 1.3 1.5

Carbon &

Stainless 5.5 0.80 1.2 1.3 1.5 0.75 1.2 1.3 1.5


This performance function combines earthquake load due to SSE with load due to LOCA.

S

fLfPSfAfu

fg −−−−=17

(7-13)

Table 7-35 gives the ranges of parameters, while Table 7-36 presents the probabilistic characteristics used for the calculation of the partial safety factors. Tables 7-37a to 7-37d show the results.



126


β 1.5, 2, 3 COV (fu) 0.03 to 0.09 COV (fA) 0.05 to 0.10 COV (fPS) 0.15 to 0.35 COV (fL) 0.20 COV (fS) 0.40 to 1.10


Random Variable

Recommended (COV)




fA 0.10 1 Normal 1.00 fPS 0.25 1, 10, 100, (1000) Type I (Largest) 0.90 fL 0.20 0.5, 1, 2 Type I (Largest) 0.80 fS 0.83 1.5, 2, 3, 4, 5




Dev

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Dev

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t Rep

ort

127

Tabl

e 7-

37a.

Loa

d an

d R

esis

tanc

e Fa

ctor

s App

lied

to M

ean

Val

ues o

f Var

iabl

es

Car

bon

&St

ainl

ess S

teel

f L

β =

1.5

0.5

1.0

2.0

fs

µ fu

φ fu

γ Α

γ PS

γ L

γ S

µ fu

φ fu

γ Α

γ PS

γ L

γ S

µ fu

φ fu

γ Α

γ PS

γ L

γ S

1.5

5.33

6 0.

985

1.00

7 0.

999

0.97

9 1.

838

5.83

7 0.

983

1.00

7 0.

999

0.99

3 1.

827

6.86

3 0.

980

1.00

7 1.

000

1.02

3 1.

783

2 6.

273

0.98

6 1.

005

0.98

8 0.

976

1.85

2 6.

771

0.98

5 1.

005

0.98

8 0.

986

1.84

6 7.

783

0.98

3 1.

006

0.98

9 1.

007

1.82

2 3

8.15

6 0.

988

1.00

4 0.

978

0.97

3 1.

863

8.65

0 0.

987

1.00

4 0.

978

0.97

9 1.

859

9.65

0 0.

986

1.00

4 0.

978

0.99

2 1.

849

4 10

.044

0.

989

1.00

3 0.

973

0.97

2 1.

867

10.5

36

0.98

8 1.

003

0.97

3 0.

976

1.86

5 11

.529

0.

987

1.00

3 0.

973

0.98

6 1.

859

1

5 11

.933

0.

989

1.00

2 0.

970

0.97

1 1.

869

12.4

24

0.98

9 1.

002

0.97

0 0.

974

1.86

7 13

.414

0.

988

1.00

2 0.

970

0.98

2 1.

863

1.5

17.0

62

0.97

6 1.

004

1.38

2 0.

973

0.89

3 17

.562

0.

975

1.00

4 1.

381

0.98

0 0.

893

18.5

74

0.97

4 1.

004

1.37

7 0.

993

0.89

2 2

17.5

30

0.97

6 1.

004

1.37

4 0.

973

0.93

6 18

.031

0.

975

1.00

4 1.

372

0.98

0 0.

936

19.0

43

0.97

4 1.

004

1.36

8 0.

993

0.93

5 3

18.5

54

0.97

5 1.

004

1.33

6 0.

973

1.08

0 19

.055

0.

975

1.00

4 1.

335

0.97

9 1.

079

20.0

67

0.97

3 1.

004

1.33

2 0.

992

1.07

6 4

19.8

15

0.97

7 1.

003

1.22

1 0.

972

1.41

6 20

.313

0.

977

1.00

3 1.

221

0.97

7 1.

412

21.3

20

0.97

6 1.

003

1.22

2 0.

988

1.40

2 10

5 21

.400

0.

981

1.00

2 1.

127

0.97

1 1.

647

21.8

96

0.98

1 1.

002

1.12

7 0.

975

1.64

5 22

.895

0.

980

1.00

2 1.

127

0.98

3 1.

638

1.5

145.

194

0.97

9 1.

000

1.39

4 0.

968

0.81

3 14

5.69

0.

979

1.00

0 1.

394

0.96

8 0.

813

146.

678

0.97

9 1.

000

1.39

4 0.

970

0.81

3 2

145.

610

0.97

9 1.

000

1.39

4 0.

968

0.81

5 14

6.10

0.

979

1.00

0 1.

394

0.96

8 0.

815

147.

094

0.97

9 1.

000

1.39

4 0.

970

0.81

5 3

146.

445

0.97

9 1.

000

1.39

4 0.

968

0.82

0 14

6.94

0.

979

1.00

0 1.

394

0.96

8 0.

820

147.

930

0.97

9 1.

000

1.39

4 0.

970

0.82

0 4

147.

285

0.97

9 1.

000

1.39

4 0.

968

0.82

5 14

7.78

0.

979

1.00

0 1.

394

0.96

8 0.

825

148.

770

0.97

9 1.

000

1.39

4 0.

970

0.82

5

fPS

100

5 14

8.13

1 0.

979

1.00

0 1.

393

0.96

8 0.

830

148.

62

0.97

9 1.

000

1.39

3 0.

968

0.83

0 14

9.61

0 0.

979

1.00

0 1.

394

0.97

0 0.

830

β

= 2.

0

1.5

6.74

6 0.

985

1.00

5 0.

988

0.97

6 2.

777

7.24

4 0.

984

1.00

5 0.

988

0.98

6 2.

767

8.25

6 0.

982

1.00

5 0.

989

1.00

7 2.

735

2 8.

157

0.98

7 1.

004

0.98

0 0.

974

2.78

8 8.

652

0.98

6 1.

004

0.98

0 0.

981

2.78

2 9.

656

0.98

4 1.

004

0.98

1 0.

996

2.76

4 3

10.9

86

0.98

8 1.

003

0.97

3 0.

972

1.79

7 11

.479

0.

987

1.00

3 0.

973

0.97

6 2.

794

12.4

73

0.98

6 1.

003

0.97

3 0.

986

2.78

5 4

13.8

19

0.98

8 1.

002

0.96

9 0.

970

2.80

0 14

.311

0.

988

1.00

2 0.

969

0.97

4 2.

798

15.3

00

0.98

7 1.

002

0.96

9 0.

981

2.79

3

1

5 16

.653

0.

989

1.00

2 0.

967

0.97

0 2.

802

17.1

44

0.98

8 1.

002

0.96

7 0.

972

2.80

1 18

.131

0.

988

1.00

2 0.

967

0.97

8 2.

797

1.5

19.3

02

0.97

0 1.

004

1.58

7 0.

974

0.90

8 19

.806

0.

970

1.00

4 1.

586

0.98

1 0.

908

20.8

26

0.96

8 1.

004

1.58

1 0.

997

0.90

8 2

19.7

84

0.97

0 1.

004

1.57

6 0.

974

0.96

5 20

.288

0.

969

1.00

4 1.

574

0.98

1 0.

965

21.3

09

0.96

8 1.

004

1.56

9 0.

995

0.96

5 3

20.8

93

0.96

9 1.

004

1.48

9 0.

974

1.28

9 21

.397

0.

969

1.00

4 1.

488

0.98

1 1.

286

22.4

17

0.96

7 1.

004

1.48

6 0.

995

1.27

7 4

23.2

34

0.98

1 1.

002

1.11

3 0.

971

2.54

1 23

.730

0.

980

1.00

2 1.

113

0.97

4 2.

538

24.7

29

0.97

9 1.

002

1.11

4 0.

982

2.52

8

10

5 25

.892

0.

983

1.00

2 1.

067

0.97

0 2.

659

26.3

86

0.98

3 1.

002

1.06

7 0.

973

2.65

7 27

.379

0.

982

1.00

2 1.

067

0.97

8 2.

651

1.5

167.

350

0.97

4 1.

000

1.60

2 0.

968

0.81

4 16

7.85

0.

974

1.00

0 1.

602

0.96

9 0.

814

168.

843

0.97

3 1.

000

1.60

2 0.

970

0.81

4 2

167.

769

0.97

4 1.

000

1.60

2 0.

968

0.81

6 16

8.27

0.

974

1.00

0 1.

602

0.96

9 0.

816

169.

262

0.97

3 1.

000

1.60

2 0.

970

0.81

6 3

168.

610

0.97

4 1.

000

1.60

2 0.

968

0.82

2 16

9.11

0.

973

1.00

0 1.

602

0.96

9 0.

822

170.

103

0.97

3 1.

000

1.60

2 0.

970

0.82

2 4

169.

457

0.97

3 1.

000

1.60

2 0.

968

0.82

7 16

9.95

0.

973

1.00

0 1.

601

0.96

9 0.

827

170.

950

0.97

3 1.

000

1.60

1 0.

970

0.82

7

fPS

100

5 17

0.31

0 0.

973

1.00

0 1.

601

0.96

8 0.

833

170.

81

0.97

3 1.

000

1.60

1 0.

969

0.83

3 17

1.80

3 0.

973

1.00

0 1.

601

0.97

0 0.

833


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

128

Tabl

e 7-

37a.

(Con

t.) L

oad

and

Res

ista

nce

Fact

ors A

pplie

d to

Mea

n V

alue

s of V

aria

bles

β =

3.0

1.

5 14

.546

0.

988

1.00

2 0.

970

0.97

1 7.

938

15.0

39

0.98

7 1.

002

0.97

0 0.

974

7.93

3 16

.029

0.

986

1.00

2 0.

970

0.98

1 7.

918

2 18

.566

0.

988

1.00

2 0.

967

0.97

0 7.

945

19.0

58

0.98

8 1.

002

0.96

7 0.

972

7.94

1 20

.045

0.

987

1.00

2 0.

967

0.97

8 7.

932

3 26

.609

0.

988

1.00

1 0.

964

0.96

9 7.

951

27.1

00

0.98

8 1.

001

0.96

4 0.

971

7.94

9 28

.084

0.

988

1.00

1 0.

964

0.97

4 7.

944

4 34

.653

0.

989

1.00

1 0.

963

0.96

8 7.

953

35.1

44

0.98

9 1.

001

0.96

3 0.

970

7.95

2 36

.126

0.

988

1.00

1 0.

963

0.97

2 7.

948

1

5 42

.698

0.

989

1.00

1 0.

962

0.96

8 7.

955

43.1

88

0.98

9 1.

001

0.96

2 0.

969

7.95

3 44

.169

0.

988

1.00

1 0.

962

0.97

1 7.

951

1.5

25.0

78

0.95

9 1.

005

2.11

7 0.

975

0.92

4 25

.589

0.

958

1.00

5 2.

114

0.98

3 0.

924

26.6

24

0.95

7 1.

005

2.10

8 1.

000

0.92

4 2

25.5

78

0.95

8 1.

005

2.10

2 0.

975

0.99

9 26

.089

0.

957

1.00

5 2.

099

0.98

3 0.

999

27.1

25

0.95

6 1.

005

2.09

3 1.

000

0.99

9 3

35.6

47

0.98

5 1.

001

1.02

0 0.

969

7.80

7 36

.139

0.

985

1.00

1 1.

020

0.97

1 7.

805

37.1

27

0.98

4 1.

001

1.02

0 0.

974

7.79

3 4

43.6

06

0.98

6 1.

001

1.00

3 0.

968

7.87

2 44

.097

0.

986

1.00

1 1.

003

0.97

0 7.

870

45.0

82

0.98

6 1.

001

1.00

3 0.

972

7.86

6

10

5 51

.602

0.

987

1.00

1 0.

993

0.96

8 7.

901

52.0

92

0.98

7 1.

001

0.99

3 0.

969

7.89

9 53

.076

0.

986

1.00

1 0.

993

0.97

1 7.

896

1.5

224.

715

0.96

3 1.

000

2.13

6 0.

968

0.81

5 22

5.22

0.

963

1.00

0 2.

136

0.96

9 0.

815

226.

225

0.96

2 1.

000

2.13

6 0.

970

0.81

5 2

225.

139

0.96

3 1.

000

2.13

6 0.

968

0.81

7 22

5.64

0.

962

1.00

0 2.

136

0.96

9 0.

817

226.

649

0.96

2 1.

000

2.13

5 0.

970

0.81

7 3

225.

992

0.96

2 1.

000

2.13

5 0.

968

0.82

3 22

6.49

0.

962

1.00

0 2.

135

0.96

9 0.

823

227.

502

0.96

2 1.

000

2.13

5 0.

970

0.82

3 4

226.

850

0.96

2 1.

000

2.13

5 0.

968

0.83

0 22

7.35

0.

962

1.00

0 2.

135

0.96

9 0.

830

228.

361

0.96

2 1.

000

2.13

4 0.

970

0.83

0

fPS

100

5 22

7.71

6 0.

962

1.00

0 2.

134

0.96

8 0.

836

228.

22

0.96

2 1.

000

2.13

4 0.

969

0.83

6 22

9.22

7 0.

962

1.00

0 2.

134

0.97

0 0.

836



129


fp<10 fp ≥ 10 Steel β

µfu φfu γΑ γPS γL γS µfu φfu γΑ γPS γL γS 1.5 9.375 0.98 1.00 0.98 1.00 1.83 19.98 0.98 1.00 1.31 0.98 1.23

2.0 12.45 0.99 1.00 0.98 0.99 2.77 23.34 0.97 1.00 1.33 0.98 1.74

Car

bon

&

Stai

nles

s

3.0 29.36 0.99 1.00 0.97 0.97 7.94 39.08 0.97 1.00 1.56 0.98 4.36

Table 7-37c. Resistance and Load Factors Applied to Nominal Values of Variables


φfu γΑ γPS γL γS φfu γΑ γPS γL γS 1.5 1.16 1.00 0.88 0.80 1.19 1.16 1.00 1.18 0.78 0.80

2.0 1.17 1.00 0.88 0.79 1.80 1.14 1.00 1.20 0.78 1.13

Car

bon

&

Stai

nles

s

3.0 1.17 1.00 0.87 0.78 5.16 1.14 1.00 1.40 0.78 2.83

Table 7-37d. Suggested Resistance and Load Factors Applied to Nominal Values of Variables


φfu γΑ γPS γL γS φfu γΑ γPS γL γS 1.5 1.00 1.00 0.9 0.9 0.9 0.95 1.00 0.8 0.9 0.9

2.0 0.98 1.00 1.2 1.2 1.2 0.91 1.00 0.9 0.9 0.9

Car

bon

&

Stai

nles

s

3.0 0.49 1.00 1.3 1.3 1.5 0.85 1.00 1.3 1.3 1.5

7.7 Commentary Hereby, recommendations are provided for further improvement of the derived design equations, according to LRFD. As mentioned in previous chapters for the combination of stresses the Tresca criterion is used, which is considered conservative and simplifications are made, such as neglecting shear stresses in obtaining the principal stresses, or the combination of moments in three directions and the use of one single value. Ravindra, et al. (1973) on the derivation of load and resistance factors for steel building suggested a variable E that multiplies the loads and includes uncertainties induced by simplified structural analysis or by simplified modeling of complex, real three dimensional structures. In the case of pipes such a variable can be used, for example, as:

)( PPAAyy SγSγESφ +≥ (7-14)



130

Of course, the derivation of the probabilistic characteristics of the variable E is a very difficult task and in the case of piping can be just estimated by 3D analysis of piping systems, using for example the finite element method etc. In Service Limit C mechanical reversing loading is to be combined with the OBE loads. The ASME Code requires that the moments should first be combined in each direction before calculating a resultant moment as:

222zyx MMMM ++= (7-15)

In the way the LRFD equations are formulated such a procedure is not possible and Eq. 7-15 is used for each load separately, resulting in a conservative result. A variable C can be used to account for such uncertainties and therefore the equation can then have the form:

)( 00 SγSγCSγSγSφ MMPPAAyy +++≥ (7-16)

Again with analytical work, the probabilistic characteristics of variable C can be estimated.

7.8 Design Example This example aims to illustrate calculations made, using the derived equations. Figure 7-1 shows a Class 2 straight pipe segment anchored at both ends (e.g., pipe between two tanks) that should be designed to withstand the loads given in Table 7-38. Table 7-39 shows the calculations made also by using equations in the ASME BPV Code, Section III-NC, 2001. Loading is given readily in the form of moments. For the use of the LRFD equations the values of the reliability index β are arbitrary chosen for each performance function.

(a)

(b)

Figure 7-1. A Piping Segment (a) and Cross-section (b) with Anchored Ends Used for Sample Calculations



131

Table 7-38. Assumed Data for Illustrative Example Material SA312, Type 304 Yield Strength, Sy

* 30 ksi Ultimate Strength, Su

* 75 ksi Sh

* = Sc* = S* 18.8 ksi

Do = 12.75 in L = 236 in Z = 43.82in3 ZP = 57.46in3 Design Temperature 80 oF PD = 820 psi Pmax = 845 psi PO = 1080 psi PS = 1350 psi Pcmax = 984 psi Mx

A = 46,420 lb.in Mx

M = 36000 lb.in Mx

O = 75,000 lb.in My

O = 80,000 lb.in Mx

S = 140,000 lb.in My

S = 150,000 lb.in *Values obtained from ASME BPV Code, Part II Loading Nomenclature Mx

A = Moment due to sustained weight Mx

O, MyO = Moments due to OBE

MxS, My

S = Moments due to SSE Mx

M = Moment due to an instantaneous valve closure PD = Design Pressure Pmax = Maximum Operating Pressure Pcmax = Maximum Operating Pressure for Service Limit C PO = Pressure coincident with OBE PS = Pressure coincident with SSE


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

132

Tabl

e 7-

39. E

xam

ple

Cal

cula

tions

for t

he P

ipe

Segm

ent o

f Fig

ure

7-1

†ASM

E B

VP

Cod

e, S

ectio

n II

I - N

C, 2

001

†LR

FD

NC

-36

41.1

(3)

Des

ign

[]

in37

5.0

1016

.027

34.0

1016

.0)

820

(4.0

1880

02

)75.

12(82

0

)(2

o=

+=

++

=+

+=

APy

SPDmt

A

= 0.

1016

in is

allo

wan

ce fo

r cor

rosi

on, e

rosi

on o

r thr

eate

ning

C

hoos

e a

NPS

12

stan

dard

sche

dule

pip

e

t

yto

DDP

yS

2

)2

(58.1

−≥

or

()

int

2706

.082

0()4.0(

58.130

000

2

)75.

12()82

0(

58.1=

+=

t is t

he th

ickn

ess w

ithou

t the

allo

wan

ce, y

= 0

.4 fo

r Do/t

>6

A =

0.10

44in

. Cho

ose

a N

PS 1

2 st

anda

rd sc

hedu

le p

ipe

β =

3 an

d P f

= 1

.35E

-03

g 1

NC

-36

41.1

(5)

Des

ign

psi

820

)27

3.0()4.0(2

75.12

)27

3.0()

1880

0(2

2

2=

−=

−=

tyo

D

tS

αP

ZAM

yS

2.165.0

≥or

2.

1271

82.43

4642

0)2.1(

1950

0=

≥

β =

5.5

and

P f =

1.9

0E-0

8

g 2

NC

-36

52(8

) D

esig

n

hSZA

M

nt

PDSLS

5.14

o≤

+=

or

)18

800

(5.182.

434642

0

)37

5.0(4

)75.

12(82

0≤

+

or

2820

080

30≤

and

Saf

ety

Fac

tor =

3.5

1

f p>10

to

DDP

ZAM

yf4

3.12.1

97.0+

≥ o

r )

375

.0(4

)75.

12)(82

0(3.1

82.43

)46

420

(2.129

100

+≥

or 2

9100

9061

≥

β =

3.5

and

P f =

2.3

3E-0

4

g 3

f p>10

to

DP

ZA

My

S4

max

3.12.1

81.0+

≥ o

r 1060

8)

375

.0(4

)75.

12)(84

5(3.1

82.43

)46

420

(2.124

300

=+

≥

β =

3.5

and

P f =

2.3

3E-0

4

g 4

NC

-36

53.1

(9)

S. L

.: A

&

B

)5.1,

8.1m

in(

)(

4

om

axy

ShS

ZB

MA

M

nt

DP

OL

S≤

++

= o

r

3384

082.

43

)36

000

4642

0(

)37

5.0(

4

)75.

12(84

5≤

++

or

3384

090

64≤

and

Saf

ety

Fac

tor =

3.7

3

f p>10

PZMM

to

DP

PZ

AM

yS

1.14

max

3.12.1

83.0+

+≥

or

1099

646.

57)

3600

0(1.1

)37

5.0(4

)75.

12)(84

5(3.1

46.57

)46

420

(2.124

900

=+

+≥

β

= 3.

5 an

d P f

= 2

.33E

-04

g 5


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

133

†ASM

E B

VP

Cod

e, S

ectio

n II

I - N

C, 2

001

†LR

FD

f p>10

inlb

1096

602 )

(2 )

(=

+=

o yM

o xM

OM

PZO

M

PZMM

toD

OP

PZ

AM

yS

4.12.1

43.1

2.198.0

++

+≥

1632

746.

57

)10

9660

(4.1

46.57

)36

000

(2.1

)37

5.0(4

)75.

12)(10

80(3.1

46.57

4642

0)2.1(

2940

0=

++

+≥

β =

3 an

d P f

= 1

.35E

-03

g 6 & g 12

NC

-36

53.1

(11

a)

S. L

.:

A &

B

()

hScS

fA

SZR

iMRS

25.025.1

0.20.2

+=

≤=

. Fo

r f =

i =1

and

for M

R

cons

ider

ing

just

the

OB

E an

d no

t ran

ges a

s the

Cod

e st

ates

, (th

is d

esig

n do

es

not i

nclu

de a

lso

anch

or m

ovem

ents

due

to re

vers

ing

dyna

mic

load

s) a

nd th

is

way

it y

ield

s

5640

0)

1880

0(3

2503

82.4310

9660

=≤

==

RS a

nd S

afet

y F

acto

r = 2

2.5!

P

ZO

M

PZ

AM

yf5.1

2.145.0

+≥

or 38

3346.

57

)10

9660

(5.1

46.57

)46

420

(2.113

500

=+

≥

β =

3 an

d P f

= 1

.35E

-03

g 7

&

g 11

NC

-36

54.2

(a)

S. L

.: C

)8.1,

25.2m

in(

)(

4

om

axy

ShS

ZB

MA

M

nt

DP

OL

S≤

++

=

or

4230

082.

43

)36

000

4642

0(

)37

5.0(

4

)75.

12(98

4≤

++

or

4230

010

245

≤an

d Sa

fety

Fac

tor =

4.12

PZM

M

t

oD

CP

PZ

AM

yS1.1

4

max

3.12.1

83.0+

+≥

or

1253

346.

57)

3600

0(1.1

)37

5.0(4

)75.

12)(98

4(3.1

46.57

)46

420

(2.124

900

=+

+≥

β =

3.5

and

P f =

2.3

3E-0

4

g 8

NC

-36

54.2

(b)

S. L

.: C

mSmS

EM

IoD

Bnt

oD

DPB

10.2)

70.0(32' 2

21

=≤

+ o

r

4200

082.

43

)10

9660

4642

0(

)37

5.0(

4

)75.

12(10

80≤

++

or

4200

012

742

≤an

d Sa

fety

Fac

tor =

3.3

PZ

OM

toD

OP

PZ

AM

yS4.1

42.1

61.0+

+≥

1449

646.

57)

1096

60(4.1

)37

5.0(4

)75.

12)(10

80(2.1

46.57

)46

420

(18

300

=+

+≥

β

= 3.

5 an

d P f

= 2

.33E

-04

g 9


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

134

†ASM

E B

VP

Cod

e, S

ectio

n II

I - N

C, 2

001

†LR

FD

NC

-36

54.1

S.

L.:

C

αPP

5.1≤

or

12

30≤

Pps

i

)2

)(58.1(

2P

yto

DySt

−≤

or

1144

)45.

12()58.1(

)30

000

()37

5.0(2

=≤

P p

si

β =

3 an

d P f

= 1

.35E

-03

g 10

NC

-36

55(α

)(1)

S. L

.: D

αPP

2≤

or

16

40≤

Pps

i

)2

(

2P

yto

DuSt −

≤or

1816

)40

2.0(

)45.

12(

)75

000

()37

5.0(2

=≤

P p

si

β =5

.5 a

nd P

f = 1

.90E

-08

g 13

NC

-36

56(a

) S.

L.:

D

)2,

0.3m

in(

)(

4max

yS

hSZ

BM

AM

nt

DP

OL

S≤

++

=o

or

5640

082.

434642

0

)37

5.0(

4

)75.

12(13

50≤

+

or

5640

012

534

≤ a

nd S

afet

y F

acto

r =4.

56

f p>10

inlb

2051

802 )

(2 )

(=

+=

o yM

o xM

SM

PZ

SM

toD sP

PZ

AM

S u5.1

43.1

2.185.0

++

≥

2124

346.

57

)20

5180

(5.1

)37

5.0(4

)75.

12)(13

50(3.1

46.57

4642

0)2.1(

6375

0=

++

≥

β =

3 an

d P f

= 1

.35E

-03

g 14

NC

-36

56(b

) S.

L.:

D

mSE

MIo

DB

nto

DDP

B3

2'

22

1≤

+ o

r

6000

082.

43

)20

5180

4642

0(

)37

5.0(

4

)75.

12(13

50≤

++

or

6000

017

217

≤ a

nd S

afet

y F

acto

r =3

.48

PZM

PZ

AM

uSS

0.22.1

49.0+

≥

or

8111

46.57

)20

5180

(0.2

46.57

)46

420

(2.136

750

=+

≥

β =

3 an

d P f

= 1

.35E

-03

g 15

Not

e: S

.L.=

Ser

vice

Lim

it, P

f = P

roba

bilit

y of

failu

re, β

= R

elia

bilit

y In

dex;

†In

all

equa

tions

the

resu

lt of

cal

cula

tions

is in

psi



135

8. Load and Resistance Factors

8.1. Summary The main objective of this report was to develop reliability-based Load and Resistance Factor Design (LRFD) methods for piping, more specifically for Class 2/3 piping for primary loading that include pressure, deadweight, seismic and mechanical loading. The LRFD methods were developed based on structural reliability theories, on previous practices in the design of pipes, and on LRFD rules and specifications adopted by other related structural codes. In these developments, commonly used design equations for piping were collected from various sources and investigated in terms of their limitations, applicability, uncertainties, and biases. The probabilistic characteristics of the basic random variables for both the strength and the load were quantified based on statistical analyses of data collected, on values recommended in other studies, and occasionally on sound engineering judgment. The report provides results based on the following tasks: (1) a state-of-the-art assessment and selection of reliability theories, (2) review and evaluation of existing strength models for piping, (3) selection of strength models and equations deemed suitable for LRFD development, and (4) preliminary analysis of basic random variables to characterize their uncertainties. The report also presented the detailed development work for LRFD methods of structural design for piping. Based on the LRFD development of this report, the following conclusions can be drawn:

1. The reliability-based design and assessment can easily be performed for the design of various piping components, such as straight pipes, elbows, and supports.

2. Selection of strength models and equations deemed suitable for LRFD development can

be done by performing reliability assessment, uncertainty analyses, and validation. 3. The probabilistic characteristic of both the strength and load variables play a vital role in

reliability assessment and reliability-based design for piping structural systems. Quantification of the probabilistic characteristics of these variables is an essential element for developing LRFD rules. For example, determination of partial safety factors (PSF’s) for the design format in the limit state function depends on these characteristics. Therefore, PSF’s for both the strength and load variables are as good as the probabilistic characteristics from which they were derived.



136

4. In formulating a strength design model for piping, a balance must be achieved between the model accuracy, applicability, and simplicity, all of which are desired features. The assessment of a model uncertainty and bias can be performed by comparing its strength prediction with ones that are more accurate or real values from experimental test results.

5. In developing design code provisions for piping, it is sometimes necessary to follow the

current design practices to ensure consistent levels of reliability among various types of piping components. Therefore, calibration is often needed on the strength factor to maintain the same values for all load factors.

8.2. Recommendations for Project Completion Based on the LRFD development presented in this study for structural design of piping components, the following recommendations should be considered:

1. The reliability-based LRFD methods for piping were developed in this study based primarily on stresses in a pipe due to internal pressure, sustained weight, seismic and mechanical loading and their combinations. It is recommended that other types of loads in combinations with the primary loads be included in the LRFD formats. The other types of loads such as fatigue and thermal effect in combinations with primary loads may have an effect on the overall behavior of a pipe.

2. The probabilistic characteristics of the basic random variables for both the load and

strength play a vital role in reliability assessment and reliability-based design for piping components. Quantification of the probabilistic characteristics of these variables is an essential element for developing LRFD rules. The characteristics can be developed from laboratory testing of mechanical properties of materials and measurements in plants for dimensions and tolerances.

3. Sensitivity analysis and the effect of correlation between the load and strength random

variables.

4. Identification of a target range for reliability levels to compute the partial safety factors (also known as load and strength factors).

5. Computations of partial safety factors for use in LRFD design formats.

6. Sample LRFD guidelines and rules for piping summarizing target reliability levels,

strength factors, and load factors (see Table 8-1). The values in Table 8-1 are for illustration purposes and to provide an example of how the final results will be presented.

7. Comparative design examples from actual nuclear plant configurations to illustrate the

use of load and strength factors in selected LRFD-based design equations.



137

8.3. Recommendations for Future Work The following recommended tasks were developed and submitted to the Department of Energy for support that are directly related to this project: 1. Develop technical basis for incorporation of reliability-based design methodology (e.g., LRFD) into codes and standards and more specifically:

• Develop the technical basis for code rules to permit the use of reliability based load – resistance factor design (LRFD) methods for pressure equipment in nuclear power stations; hydrogen storage and transport; oil and gas production and transport; refineries and chemical plants; and fossil fuel-fired power plants. LRFD methods will be developed for the passive pressure-retaining function of piping, vessels, pumps and valves under primary, secondary, and dynamic loadings, including other design considerations. The technical basis will include: o Methods for determining the probability distribution of the parameters that are used

in the design calculations, such as material static and fatigue strength (initially and after deterioration in service), flaw distributions (initial and due to in-service deterioration and damage), fabrication tolerances and loading variations. The use of “guaranteed material properties” that are negotiated with the materials supplier, rather than the minimum or typical properties that are currently published in the codes will also be considered.

o Development of limit states to address failure modes in pressure containing equipment and use of these limit states with probability distributions of basic random variables to derive “partial safety factors”, which are design margins applied to individual design parameters to achieve target reliability levels.

• Develop methods for determining an acceptable probability of failure (reliability level) considering the failure mode and consequences. This task includes gathering historical failure data and demonstration of the methodology.

• Provide technical information and background to resolve concerns and assist codes & standards committees and jurisdictional authorities in adopting codes and standards based on LRFD methods.

2. Improve fatigue analysis rules by developing an improved approach to the determination of design life under cyclic loading conditions (fatigue rules) for temperatures up to about 700oF. This is an important component of LRFD approaches. Extensive work has been done in this area in support of the re-write of ASME Section VIII, Division 2, but more work must be done to optimize the rules. The project will focus primarily on welded joints, evaluate the “structural stress” method, include both traditional life vs. peak stress range approaches (S-N approaches) and fracture mechanics approaches, consider the effects of residual stress, include approaches for handling complex loading sequences, and provide technical information and background to resolve concerns and assist codes & standards committees and jurisdictional authorities in adopting codes and standards based on these new methods and data.


Dev

elop

men

t of R

elia

bilit

y-B

ased

LR

FD M

etho

ds fo

r Pip

ing

– R

esea

rch

and

Dev

elop

men

t Rep

ort

138

Tabl

e 8-

1. S

ampl

e Ta

rget

Rel

iabi

lity

Leve

ls a

nd P

artia

l Saf

ety

fact

ors f

or D

emon

stra

tion

Purp

oses

Ta

rget

Rel

iabi

lity

Inde

x, β

2

3 Lo

adin

g C

ondi

tion

Des

ign

Equa

tion

γ 1

γ 2

γ 3

φ 1

or

φ 2

γ 1

γ 2

γ 3

φ 1

or

φ 2

Des

ign

Con

ditio

n (h

oop

stre

ss)

APy

SPDt m

++

=)

(2o

, or

)(2

22

PPy

SyP

ASA

Pdt m

−+

++

=

NA

N

A

NA

N

A

NA

N

A

NA

N

A

Gen

eral

Con

ditio

n )

,m

in(

21

1u

yA

SS

ZMφ

φγ

≤

1.14

N

A

NA

0.

82

1.20

N

A

NA

0.

73

Ope

ratin

g C

ondi

tion,

Se

rvic

e Le

vel A

)

,m

in(

22

11

o2

uy

AS

SZM

tPD

φφ

γγ

≤+

1.

05

1.22

N

A

0.87

1.

06

1.36

N

A

0.81

Ups

et L

oadi

ng C

ondi

tion,

Se

rvic

e Le

vel B

)

,m

in(

22

13

1o

2u

yB

AS

SZM

ZMt

PDφ

φγ

γγ

≤+

+

1.01

1.

04

2.23

0.

94

1.01

1.

03

3.55

0.

92

Emer

genc

y Lo

adin

g C

ondi

tion,

Ser

vice

Lev

el C

),

min

(2

21

31

o2

uy

PB

PAS

SZM

ZMt

PDφ

φγ

γγ

≤+

+

),

min

(2

12

1u

yPB

PAS

SZM

ZMφ

φγ

γ≤

+

1.01

1.

01

1.27

2.

26

NA

N

A

0.94

0.

95

1.01

1.

01

1.38

3.

59

NA

N

A

0.92

0.

93

Faul

ted

Load

ing

Con

ditio

n,

Serv

ice

Leve

l D

),

min

(2

21

31

o2

uy

PS

PAS

SZM

ZMt

PDφ

φγ

γγ

≤+

+

),

min

(2

12

1u

yPS

PAS

SZM

ZMφ

φγ

γ≤

+

1.18

1.

01

1.18

2.

26

2.21

N

A

0.94

0.

96

1.20

1.

00

1.20

3.

59

3.53

n/

a

0.92

0.

94

NA

= N

ot a

pplic

able



139

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151

Appendix A. Selected Limit States In ASME Code

The following tables summarize the limit states equations according to ASME Code Section III for dead loads, sustained loads, internal pressure and seismic loads, without thermal loads:

Design Condition

Class 2 (NC3600, 1992 edition)

Class 3 (ND3600, 1992 edition)

B31.1 (1992 Edition)

Design Condition

NC3652 Eq. 8 (NC-3652): Load combination: SSL = B1 (PDo)/(2tn) + B2 (MA)/Z Strength Limit: 1.5Sh

Same as Class 2 104.8.1 Eq. 11A Effects of pressure, weight and sustained loads: Load combination: SL = (PDo)/(4tn)+0.75i(MA/Z) Strength Limit: 1.0Sh



152

Design Condition




Level A & B Service Limits

NC3653 Eq. 9 (NC-3653.1) with: Load combination: SOL = B1(PmaxDo)/(2tn)+B2(MA+MB)/Z Strength Limit: Smaller of 1.8Sh or 1.5Sy Eq. 10 (NC-3653.2 (a)) for thermal expansion with Load combination: SE = iMC/Z Strength Limit: SA = f(1.25Sc +Sh) Eq. 10a (NC-3653.2(b)) for nonrepeated anchor movement with Load combination: iMD/Z Strength Limit: 3.0Sc Eq. 11 (NC-3653.2 (c)) with pressure, weight and sustained loads: Load combination: STE = (PDo)/(4tn)+0.75i(MA/Z) +i(MC/Z) Strength Limit: Sh + SA

Same as Class 2 Except Eq. 11a allowable value = 3. 0SA ( This can be an error)

104.8.2 Eq. 12A Effects of pressure, weight, sustained and occasional loads: Load combination: (PDo)/(4tn)+0.75i(MA/Z) + 0.75i(MB/Z) Strength Limit: kSh k = 1.15 for occasional loads acting 10% of any 24 hr operating period. (See Para 102.2.4) k = 1.2 for occasional loads acting 1% of any 24 hr operating period. (See Para 102.2.4) 104.8.3 Eq. 13Afor thermal expansion with Load combination: SE = iMC/Z Strength Limit: SA + f(Sh - SL) SA = f(1.25Sc +Sh)



153

Design Condition




Level C NC3654 - (Condition of Eq. (9) for Service Loadings for Level C) Eq. 9 (NC-3652) with: Load combination: S = B1(PmaxDo)/(2tn)+B2(MA+MB)/Z Strength Limit: Smaller of 2.25Sh or 1.8Sy

Same as Class 2 Emergency condition? (Cannot find a reference) Load combination: (PDo)/(4tn)+0.75i(MA/Z) + 0.75i(MB/Z) Strength Limit: 1.8Sh

Level D NC3655 (Condition of Eq. (9) for Service Loadings for Level D) Eq. 9 (NC-3653.1) with: Load combination: S = B1(PmaxDo)/(2tn)+B2(MA+MB)/Z Strength Limit: Smaller of 3.0Sh or 2.0Sy

Same as Class 2



154

Design Condition




Definition of Variables

SSL= Stress due to effects of pressure, weight and other sustained mechanical loads SOL= Stress due to effects of pressure, weight, other sustained and occasional loads, including earthquake B1, B2 = Primary stress indices for the specific product under investigation P = internal Design Pressure, psi Do = outside diameter of pipe, in. tn = nominal wall thickness, in. MA = resultant moment loading on cross section due to weight and other sustained loads, in-lb (NC-3653.3) Z = section of modulus of pipe, in3 Sh = basic material allowable stress at Design Temperature, psi Pmax = peak pressure, psi MB = resultant moment loading on cross section due to occasional loads, such as thrusts from relief and safety valve loads from pressure and flow transients and earthquake. For earthquake, use only one-half the range. Effects of anchor displacement due to earthquake may be excluded from Eq. (9) if they are included in Eq.(10) and EQ (11) (NC-3653.2) Sy = material yield strength at temperature consistent with the loading under consideration, psi Sh = material allowable stress at temperature consistent with the loading under consideration, psi Sc = material allowable stress at minimum (cold) temperature, psi MC = range of resultant moments due to thermal expansion, in-lb.; also include moment effects of anchor displacements due to earthquake if anchor displacement effects were omitted from Eq. (9) (NC-3653.1) SE = expansion stress SA = allowable stress range for expansion stresses (NC-3611.2) psi i = stress intensification factor (NC-3673.2) MD = resultant moment due to any single non-repeated anchor movement (e. g. predicted building settlement), in-lb. STE = stress due to pressure, weight, other sustained loads and thermal expansion f = stress range reduction factor for cyclic conditions for total number N of full temperature cycles over total number of years during which system is expected to be in operation, from Table NC-3611.2(e)-1.

Same as Class 2

P = internal Design Pressure, psi Do = outside diameter of pipe, in. tn = nominal wall thickness, in. MA = resultant moment loading on cross section due to weight and other sustained loads, in-lb (Para. 104.8.4)) Z = section of modulus of pipe, in3 i = stress intensification factor (See Appendix in B31.1 code) the product 0.75i shall never be taken as less than 1.0. SL = sum of longitudial stresses due to pressure, weight, and other sustained loads MB = resultant moment loading on cross section due to occasional loads [see Para. 102.3.3(A)], such as thrusts from relief and safety valve loads from pressure and flow transients and earthquake. For earthquake, use only one-half the range. Effects of anchor displacement due to earthquake may be excluded from Eq. (12) if they are included in Eq.(13) (see Para. 104.8.4) MC = range of resultant moments due to thermal expansion. Also include moment effects of anchor displacements due to earthquake if anchor displacement effects were omitted from Eq. (12) (see Para. 104.8.4) f = stress range reduction factor for cyclic conditions for total number N of full temperature cycles over total number of years during which system is expected to be in operation, from Table 102.3.2 (C). SA = allowable stress range for expansion stresses



155

Appendix B. Steel Used In ASME Code, Part III

The following Table presents the Specified Minimum Yield Strength (SMYS) and the Specified Minimum Tensile Strength (SMTS) of steels used in the ASME Code, Part III, for the design of piping.

SPEC # Gr.,Cl., Ty. Nominal Product UNS # SMYS SMTS Notes Common

Composition (ksi) (ksi) Name

SA-53 Ty S-Gr

A C Stl W&SP K02504 48 30 black & hot-

dipped

Ty S-Gr

B C-Mn Stl K03005 60 35 zinc coated

Ty E-Gr

A C Stl K02504 48 30

Ty E-Gr

B C-Mn Stl K03005 60 35 SA-106 Gr A C-Si Stl SP K02501 48 30

carbon steel pipe

Gr B C-Si Stl K03006 60 35 for high-

temperature Gr C C-Si Stl K03501 70 40 service

SA-134 C Stl WP >=NPS 16

A36, A283, A285, A570

SA-312 Gr TP304 18 Cr-8 Ni Sm&WP S30400 75 30

Gr

TP304H 18 Cr-8 Ni S30409 75 30 Austentic stainless

Gr

TP304L 18 Cr-8 Ni S30403 70 25 steel

Gr

TP304N 18 Cr-8 Ni-N S30451 80 35

Gr

TP304LN 18 Cr-8 Ni-N S30453 75 30

Gr

TP309S 23 Cr-12 Ni S30908 75 30

Gr

TP309Cb 23 Cr-12 Ni-

Cb S30940 75 30

Gr

TP310S 25 Cr-20 Ni S31008 75 30

Gr

TP310Cb 25 Cr-20 Ni-

Cb S31040 75 30

Gr TP316 16Cr-12Ni-

2Mo S31600 75 30

Gr

TP316H 16Cr-12Ni-

2Mo S31609 75 30

Gr

TP316L 16Cr-12Ni-

2Mo S31603 70 25

Gr

TP316N 16Cr-12Ni-

2Mo-N S31651 80 35

Gr

TP316LN 16Cr-12Ni-

2Mo-N S31653 75 30



156



Gr TP317 18Cr-13Ni-

3Mo S31700 75 30

Gr TP321 18 Cr-10 Ni-

Ti S32100 75 30 Sm <3/8in

Gr

TP321H 18 Cr-10 Ni-

Ti S32109 75 30 Sm <3/8in

Gr TP347 18 Cr-10 Ni-

Cb S34700 75 30

Gr TP347

H 18 Cr-10 Ni-

Cb S34709 75 30

Gr TP348 18 Cr-10 Ni-

Cb S34800 75 30

Gr TP348

H 18 Cr-10 Ni-

Cb S34809 75 30

Gr TP XM19

22Cr-13Ni-Mn S20910 100 55

nitronic 50 or 22-13-5

SA-333 Gr 1 C- Mn Stl Sm&WP K03008 55 30

low-temperature

Gr 6 C- Mn-Ci Stl K03006 60 35 service Gr 8 9Ni K81340 100 75 Gr9 2Ni-1Cu K22035 63 46

SA-335 Gr P1 C- 1/2Mo SP K11522 55 30

Gr P2 1/2Cr- 1/2Mo K11547 55 30 ferritic alloy

steel

Gr P5 5Cr- 1/2Mo K41545 60 30 for high-

temperature Gr P9 9Cr- 1Mo K81590 60 30 service

Gr P11 11/4Cr-

1/2Mo-Si K11597 60 30 Gr P12 1Cr- 1/2Mo K11562 60 30 Gr P21 3Cr- 1/2Mo K31545 60 30 Gr P22 21/4Cr- 1Mo K21590 60 30

SA-358 Gr 304 18Cr- 8Ni WP S30400 75 30

Gr 304L 18Cr- 8Ni S30403 70 25 electric-fusion

Gr 304N 18Cr- 8Ni-N S30451 80 35 welded

austentic

Gr

304LN 18Cr- 8Ni-N S30453 75 30 chromium-

nickel Gr 304H 18Cr- 8Ni S30409 75 30 alloy steel pipe

Gr 309 23Cr- 12Ni S30900 75 30 low-

temperature Gr 310 25Cr- 20Ni S31000 75 30 service

Gr 316 16Cr- 12Ni-

2Mo S31600 75 30

Gr 316L 16Cr- 12Ni-

2Mo S31603 70 25

Gr 316H 16Cr- 12Ni-

2Mo S31609 75 30

Gr 316N 16Cr- 12Ni-

2Mo-N S31651 80 35

Gr 316N 16Cr- 12Ni-

2Mo-N S31653 75 30 Gr 321 18Cr-10Ni-Ti S32100 75 30

Gr 347 18Cr-10Ni-

Cb S34700 75 30

Gr 348 18Cr-10Ni-

Cb S34800 75 30



157



Gr XM-

19 22Cr-13Ni-

5Mn S22100 100 55 SA-369 Gr FP1 C-1/2Mo FBP K11522 55 30

Gr FP2 1/2Cr-1/2Mo K11547 55 30 carbon and

ferretic Gr FP5 5Cr-1/2Mo K41545 60 30 alloy steel

Gr FP9 9Cr-1Mo K90941 60 30 for high-

temperature Gr FP11

11/4Cr-1/2Mo-Si K11597 60 30 service

Gr FP12 1Cr-1/2Mo K11562 60 30 Gr FP21 3Cr-1Mo K31545 60 30 Gr FP22 21/4Cr-1Mo K21590 60 30

SA-376

Gr TP304 18Cr- 8Ni SP S30400 75 30

Gr

TP304H 18Cr-8Ni S30409 75 30 austentic steel

pipe

Gr

TP304N 18Cr-8Ni-N S30451 80 35 for high

temperature

Gr

TP304LN 18Cr-8Ni-N S30453 75 30 central station

Gr

TP316 16Cr- 12Ni-

2Mo S31600 75 30 service

Gr

TP316H 16Cr- 12Ni-

2Mo S31609 75 30

Gr

TP316N 16Cr- 12Ni-

2Mo-N S31651 80 35

Gr

TP316LN 16Cr- 12Ni-

2Mo-N S31653 75 30

Gr

TP321 18Cr-10Ni-Ti S32100 75 30 <3/8in

Gr

TP321 18Cr-10Ni-Ti S32100 70 25 >3/8in

Gr

TP321H 18Cr-10Ni-Ti S32109 75 30 <3/8in

Gr

TP321H 18Cr-10Ni-Ti S32109 70 25 >3/8in

Gr

TP347 18Cr-10Ni-

Cb S34700 75 30

Gr

TP347H 18Cr-10Ni-

Cb S34709 75 30

Gr TP 348

18Cr-10Ni-Cb S34800 75 30

SA-409

Gr TP304 18Cr- 8Ni WP S30400 75 30

Gr

TP304L 18Cr-8Ni S30403 70 25 large diameter

Gr

TP316 16Cr-12Ni-

2Mo S31600 75 30 austentic steel

Gr

TP316L 16Cr-12Ni-

2Mo S31603 70 25 for corrosive or

Gr

TP321 18Cr-10Ni-Ti S32100 75 30 high-

temperature

Gr

TP347 18Cr-10Ni-

Cb S34700 75 30 service Gr 18Cr-10Ni-Ti S34800 75 30



158



TP348 SA-426 Gr CP1 C-1/2Mo CCP J12521 65 35 CP1

Gr CP2 1/2Cr-1/2Mo J11547 60 30 Centrfugally

cast CP2

Gr CP5 5Cr-1/2Mo J42045 90 60 ferritic alloy

steel CP5

Gr CP9 9Cr-1Mo J82090 90 60 for high-

temperature CP9

Gr CP11 11/4Cr-1/2Mo J12072 70 40 service CP11

Gr CP12 1Cr-1/2Mo J11562 60 30 CP12 Gr CP21 3Cr-1Mo J31545 60 30 CP21 Gr CP22 21/4Cr-1Mo J21890 70 40 CP22

Gr

CPCA15 13Cr J91150 90 65 CPCA15 SA-430

Gr FP304 18Cr-8Ni FBP S30400 70 30

Gr

FP304H 18Cr-8Ni S30409 70 30 austentic steel

Gr

FP304N 18Cr-8Ni-N S03451 75 35 for high-

temperature

Gr

FP316 16Cr-12Ni-

2Mo S31600 70 30 service

Gr

FP316H 16Cr-12Ni-

2Mo S31609 70 30

Gr

FP316N 16Cr-12Ni-

2Mo-N S31651 75 35

Gr

FP321 18Cr-10Ni-Ti S32100 70 30

Gr

FP321H 18Cr-10Ni-Ti S32109 70 30

Gr

FP347 18Cr-10Ni-

Cb S34700 70 30

Gr

FP347H 18Cr-10Ni-

Cb S34709 70 30 SA-451 Gr CPF3 18Cr-8Ni CCP J92500 70 30 CPF3

Gr

CPF3A 18Cr-8Ni J92500 77 25 Centrifugally

cast CPF3A

Gr

CPF3M 16Cr-12Ni-

2Mo J92800 70 30 austentic steel CPF3M

Gr CPF8 18Cr-8Ni J92600 70 30 for high-

temperature CPF8

Gr

CPF8A 18Cr-8Ni J92600 77 35 service CPF8A

Gr

CPF8M 16Cr-12Ni-

2Mo J92900 70 30 CPF8M

Gr

CPF8C 18Cr-10Ni-

Cb J92700 70 30 CPF8C

Gr

CPH8 25Cr-12Ni J93400 65 28 CPH8

Gr

CPK20 25Cr-20Ni J94202 65 28 CPK20

Gr

CPH20 25Cr-12Ni J93402 70 30 CPH20 SA- Gr 18Cr-8Ni CWP S30409 75 30 Centrifugally



159



452 TP304H cast

Gr

TP347H 18Cr-10Ni-

Cb S34709 75 30 austentic steel

Gr

TP316H 16Cr- 12Ni-

2Mo S31609 75 30 for high-

temperature SA-660 Gr WCA C-Si Stl CCP J02504 60 30

Centrifugally cast WCA

Gr WCB C-Si Stl J03003 70 36 carbon steel WCB

Gr WCC C-Mn-Si Stl J02505 70 40 for high-

temperature WCC SA-671

Gr CA55 C Stl WP K02801 55 30 SA-285, Gr C

Gr CB60 C-Si Stl K02401 60 32 electric -fusion SA-515, Gr60 Gr CB65 C-Si Stl K02800 65 35 welded pipe SA-515, Gr65

Gr CB70 C-Si Stl KO310

1 70 38 for atmospheric

and SA-515, Gr70

Gr

CC60 C-Mn-Si Stl K02100 60 32 lower

temperatures SA-516, Gr60

Gr

CC65 C-Mn-Si Stl K02403 65 35 SA-516, Gr65

Gr

CC70 C-Mn-Si Stl K02700 70 38 SA-516, Gr70

Gr

CD70 C-Mn-Si Stl K02400 70 50 SA-537, Cl 1

Gr

CD80 C-Mn-Si Stl K02400 80 60 SA-537, Cl 2

Gr

CE55 C-Mn-Si Stl KO220

2 55 30 SA-442, Cr 55

Gr

CE60 C-Mn-Si Stl K02402 60 32 SA-442, Cr 60

Gr

CK75 C-Mn-Si Stl K02803 75 40 SA-299 SA-672 Gr A45 C Stl WP K01700 45 24 SA-285, Gr A

Gr A50 C Stl K02200 50 27 electric -fusion SA-285, Gr B Gr A55 C Stl K02801 55 30 welded pipe SA-285, Gr C

Gr B55 C-Si Stl K02001 55 30 for high-pressure SA-515, Gr55

Gr B60 C-Si Stl K02401 60 32 service at moderate SA-515,Gr60

Gr B65 C-Si Stl K02800 65 35 temperature SA-515,Gr65 Gr B70 C-Si Stl K03101 70 38 SA-515,Gr70 Gr C55 C-Si Stl K01800 55 30 SA-515,Gr55 Gr C60 C-Mn-Si Stl K02100 60 32 SA-516,Gr60 Gr C65 C-Mn-Si Stl K02403 65 35 SA-516,Gr65 Gr C70 C-Mn-Si Stl K02700 70 38 SA-516,Gr70 Gr D70 C-Mn-Si Stl K02400 70 50 SA-537, Cl 1 Gr D80 C-Mn-Si Stl K02400 80 60 SA-537, Cl 2 Gr E55 C-Mn-Si Stl K02202 55 30 SA-442, Gr55 Gr E60 C-Mn-Si Stl K02402 60 32 SA-442, Gr60 Gr H75 Mn-1/2Mo K12021 75 45 SA-302, Gr A



160



Gr J80 Mn-1/2Mo-

1/2Ni K12539 80 50 SA-533, Gr B, Cl

1

Gr J90 Mn-1/2Mo-

1/2Ni K12539 90 70 SA-533, Gr B, Cl

2

Gr J100 Mn-1/2Mo-

1/2Ni K12539 100 83 SA-533, Gr B, Cl3 Gr L65 C-1/2Mo K11820 65 37 SA-204, Gr A Gr L70 C-1/2Mo K12020 70 40 SA-204, Gr B Gr L75 C-1/2Mo K12320 75 43 SA-204, Gr C Gr N75 C-Mn-Si Stl K02803 75 40 SA-299

SA-691

Gr CM65 C-1/2Mo WP K11820 65 37

carbon and alloy A204, Gr A

Gr

CM70 C-1/2Mo K12020 70 40 electric-fusion A204, Gr B

Gr

CM75 C-1/2Mo K12320 75 43 welded for high A204, Gr C

Gr CMSH-

70 C-Mn-Si Stl K02400 70 50 pressure and A537, Cl1

Gr CMS-

75 C-Mn-Si Stl K02803 75 40 temperature SA-731

Gr TPXM-33 27Cr-1Mo-Ti Sm&WP S44626 65 40 martensitic

Gr TPXM-33 27Cr-1Mo S44627 65 40 stainless steel

SA-813

Gr TP304 18Cr-8Ni WP S30400 75 30

Gr

TP304H 18Cr-8Ni S30409 75 30 single or double

Gr TP304

L 18Cr-8Ni S30403 70 25 welded

Gr

TP304N 18Cr-8Ni-N S30451 80 32 austentic

Gr

TP304LN 18Cr-8Ni-N S30453 75 30 stainless steel

Gr

TP309S 23Cr-12Ni S30908 75 30

Gr

TP316 16Cr-12Ni-

2Mo S31600 75 30

Gr

TP316H 16Cr-12Ni-

2Mo S31609 75 30

Gr

TP316L 16Cr-12Ni-

2Mo S31603 70 25

Gr TP316N

16Cr-12Ni-2Mo-N S31651 80 32

Gr

TP321 18Cr-10Ni-Ti S32100 75 30

Gr

TP321H 18Cr-10Ni-Ti S32109 75 30

Gr

TP347 18Cr-10Ni-

Cb S34700 75 30

Gr

TP347H 18Cr-10Ni-

Cb S34709 75 30

Gr

TP348 18Cr-10Ni-

Cb S34800 75 30 Gr 18Cr-10Ni- S34809 75 30



161



TP348H Cb

SA-814

Gr TP304 18Cr-8Ni CWWP S30400 75 30

Gr

TP304H 18Cr-8Ni S30409 75 30 cold-worked

Gr TP304

L 18Cr-8Ni S30403 70 25 welded

austentic

Gr

TP304N 18Cr-8Ni-N S30451 80 35 stainless steel

Gr

TP304LN 18Cr-8Ni-N S30453 75 30

Gr

TP316 16Cr-12Ni-

2Mo S31600 75 30

Gr

TP316H 16Cr-12Ni-

2Mo S31609 75 30

Gr

TP316L 16Cr-12Ni-

2Mo S31603 70 25

Gr TP316N

16Cr-12Ni-2Mo-N S31651 80 35

Gr

TP321 18Cr-10Ni-Ti S32100 75 30

Gr

TP321H 18Cr-10Ni-Ti S32109 75 30

Gr

TP347H 18Cr-10Ni-

Cb S34709 75 30

Gr

TP348 18Cr-10Ni-

Cb S34800 75 30

Gr

TP348H 18Cr-10Ni-

Cb S34809 75 30 Sm = Seamless Pipe W = Welded Pipe







Documents

Development of reliability-based load and resistance factor design (LRFD) methods for piping : research and development report (LRFD) Methods for Piping (Crtd)