n ENGINEERINGlr-----------'-JlIANG-CHUAN (L.C.) PENGliSEN-LOONG (ALVIN) PENG' ~ IIiI! ~! I:1Ij, ); 1\ IPIPE STRESSENGINEERINGbyLiang-Chuan (L.C.) Peng andTsen-Loong (Alvin) PengPeng Engineering, Houston, Texas, USAII, II\~III;\M ~ t :PRESS 2009 by ASME, Three Park Avenue, New York, NY 10016, USA(www.asme.org)All rights reserved. Printed in the United States of America.Except as permitted under the UnitedStates Copyright Act of 1976, no part of this publication may be reproduced or distributed in anyformor byanymeans, or storedin a database or retrieval system, without theprior writtenpermission of the publisher.INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE AMERICANSOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED TO BE RELIABLE.HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARANTEE THE ACCURACYOR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASMENOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS,OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHEDWITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARESUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OROTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICESARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BESOUGHT.ASME shall not be responsible for statements or opinions advanced in papers or ... printed initspublications (B7.1.3). Statement from the Bylaws.For authorization to photocopy material for internal or personal use under those circumstances notfalling 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.copvright.com.This book has been cataloged with the Library of Congress.ISBN: 978-0-7918-0285-4ASME Order No.802854 1'jI, ;jj, ;ueM!ifICONTENTS.Acknowledgments xiPreface xiiiNomenclature xvChapter 1Introduction ;.................................................................................................... 11.1 Scope of Pipe Stress Analysis 21.2 Piping Components and Connecting Equipment.. 41.3 Modes ofFailure ~ 91.3.1 Static Stress Rupture.................................................................................................... 91.3.2 Fatigue Failure........................................................................................................... 121.3.3 Creep Rupture ;....................................... 141.3.4 Stability Failure.......................................................................................................... 171.3.5 Miscellaneous Modes of Failure 181.4 Piping Codes.......................................................................................................................... 191.5 Industry Practice 221.5.1 Load Cases 231.5.2 Local Support Stresses 241.5.3 Local Thermal Stresses 241.5.4 Pressure Effect on Flexibility 251.5.5 Stress Intensification for Sustained Loads 251.5.6 Support Friction .. ~ 251.5.7 Guide and Stop Gaps 261.5.8 Anchor and Restraint Stiffness 261.5.9 Small Piping : 261.6 Design Specification 261.6.1 Owner's Design Specification 261.6.2 Project Specification 281.7 Plant Walk-down 30Chapter 2Strength of Materials Basics 332.1 Tensile Strength 332.1.1 Modulus of Elasticity 342.1.2 Proportional Limit 342.1.3 Yield Strength, Sy 352.1.,4 Ultimate Strength,Su 352.1:5 Stresses at Skewed Plane 352.1.6 Maximum Shear Stress,Ss,max 362.1.7 Principal Stresses , 362.2 Elastic Relationship of Stress and Strain 362.2.1 Poisson's Ratio 372.2.2 Shear Strain and Modulus of Rigidity 38iv Contents2.3 Static Equilibrium 382.3.1 Free-Body Diagram 392.3.2 Static Equilibrium 402.4 Stresses due to Moments...... 402.4.1 Stresses due to Bending Moments 402.4.2 Moment of Inertia............ 422.4.3 Polar Moment of Inertia 422.4.4 Moment ofInertia for Circular Cross-Sections 432.4.5 Stresses due to Torsion Moment 432.5 Stresses in Pipes 452.5.1 Stresses due to Internal Pressure 452.5.2 Stresses due to Forces and Moments 472.6 Evaluation of Multi-Dimensional Stresses 492.6.1 General Two-Dimensional Stress Field 492.6.2 Mohr's Circle for Combined Stresses 512.6.3 Theories of Failure 522.6.4 Stress Intensity (Tresca Stress) 522.6.5 Effective Stress (von Mises Stress) 532.7 Basic Beam Formulas 532.7.1 Guided Cantilever 552.8 Analysis of Piping Assembly 552.8.1 Finite Element 562.8.2 Data Points and Node Points 572.8.3 Piping Assembly 58Chapter 3Thermal Expansion and Piping Flexibility. 613.1 Thermal Expansion Force and Stress 613.1.1 Ideal Anchor Evaluation............................................................................................ 613.1.2 The Real Anchor 623.2 Methods of Providing Flexibility 633.2.1 Estimating Leg Length Required ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2.2 Inherent Flexibility 653.2.3 Caution Regarding Quick Check Formulas 653.2.4 Wall Thickness and Thermal Expansion Stress 663.3 Self-Limiting Stress 663.3.1 Elastic Equivalent Stress 673.4 Stress Intensification and Flexibility Factors 673.4.1 Ovalization of Curved Pipes ~ 683.4.2 Code SIFs 693.5 Allowable Thermal Expansion Stress Range 713.6 Cold Spring 763.6.1 Cold Spring Gap 773.6.2 Location of Cold Spring Gap 783.6.3 Cold Spring Procedure 783.6.4 Multi-Branched System 793.6.5 Analysis of Cold Sprung Piping System 793.7 Pressure Effects on Piping Flexibility 803.7.1 Pressure Elongation 803.7.2 Potential Twisting at Bends 81 1. ,i::--1.U.PIl"II:I'lf'!lIIIIJMIIIIIIII!lilillllliJililiIMHIIBl-.'.'IMMM 'lIfl'lllt11_11_ 11I11,1'1"1'71'11"1"11I,11,,11''I"I.I+lllkI41IIIContents v3.7.3 Pressure Elongation Is Self-Limiting Load 823.7.4 Pressure Effect on Bend Flexibility and SIFs 823.8 General Procedure of Piping Flexibility Analysis 833.8.1 Operating Modes 833.8.2 Anchor Movements 843.8.3 Assignments of Operating Values 843.8.4 Handling of Piping Components 853.8.5 The Analysis 863.9 Problems With Excessive Flexibility 863.9.1 Problems Associated With Excessive Flexibility 883.10 Field Proven Systems 88Chapter 4Code Stress Requirements 914.1 "Design" Chapter of the Piping Codes 914.2 Loadings to be Considered 924.2.1 Pressure 934.2.2 Temperature................................................................................................................ 934.2.3 Weight Effects : 954.2.4 Wind Load 954.2.5 Earthquake................................................................................................................. 964.2.6 Dynamic Fluid Loads 964.2.7 Harmonic Anchor Displacement Loads 984.2.8 Passive Loads............................................................................................................. 984.3 Basic Allowable Stresses 984.3.1 Bases for Establishing Allowable Stresses.................................................................. 984.3.2 Code Allowable Stress Tables.................................................................................... 994.3.3 Weld Strength Reduction Factor 1004.4 Pressure Design 1014.4.1 Straight Pipe 1024.4.2 Curved Segment of Pipe 1044.4.3 Miter Bends ~ . . . . . . . . . . . . . . . . . . . . . . . . 1064.4.4 Branch Connections 1094.4.5 Pressure Design for Other Components 1134.5 Stresses of Piping Components ~ 1134.5.1 Calculations of Component Stresses 1134.5.2 Sustained Stresses 1174.5.3 Occasional Stresses 1194.5.4 Thermal Expansion and Displacement Stress Range 1214.5.5 Code Stress Compliance Report 1244.6 Class 1 Nuclear Piping........................................................................................................ 125Chapter 5Discontinuity Stresses.................................................................................................................... 1335.1 Differential Equation of the Beam Deflection Curve 1335.2 Infu;).ite Beam on Elastic Foundation With Concentrated Load.......................................... 1355.3 Semi-Infinite Beam on Elastic Foundation 1385.4 Application of Beam on Elastic Foundation to Cylindrical Shells 1395.5 Effective Widths.................................................................................................................. 1415.6 Choking Model................................................................................................................... 142viii Contents9.5.2 Basic Piping Support Schemes 3029.5.3 Non-API Pumps 3039.5.4 API Standard 610 Pumps 3049.6 Centrifugal Compressors '" 3089.7 Reciprocating Compressors and Pumps 3139.7.1 Pulsating Flow 3139.7.2 Pulsation Pressure 3159.7.3 Pulsation Dampener for Reciprocating Pumps........................................................ 3169.7.4 Some Notes on Piping Connected to Reciprocating Machine 3199.8 Problems Associated With Some Techniques Used in Reducing Piping Loads 3219.8.1 Excessive Flexibility 3219.8.2 Improper Expansion Joint Installations................................................................... 3229.8.3 Theoretical Restraints.............................................................................................. 3229.9 Example Procedure for Designing Rotation Equipment Piping 324Chapter 10Transportation Pipeline and Buried Piping................................................................................... 32910.1 Governing Codes and General Design Requirements 33010.1.1 B31.4 Liquid Petroleum Pipeline 33110.1.2 B31.8 Gas Transmission Pipeline 33310.2 Behavior of Long Pipeline 33510.2.1 Pressure Elongation 33510.2.2 Anchor Force 33510.2.3 Potential Movement of Free Ends 33610.2.4 Movement of Restrained Ends 33710.2.5 Stresses at Fully Restrained Section....................................................................... 33710.3 Pipeline Bends 33910.4 Basic Elements of Soil Mechanics 34010.4.1 Types of Soils 34010.4.2 Friction Angle 34010.4.3 Shearing Stress 34110.4.4 Soil Resistance Against Axial Pipe Movement 34110.4.5 Lateral Soil Force 34310.4.6 Soil-Pipe Interaction 34410.5 Example Calculations of Basic Pipeline Behaviors 34610.5.1 Basic Calculations 34610.5.2 Soil-Pipe Interaction 34710.6 Simulation of Soil Resistance 34810.7 Behavior of Large Bends 34910.8 Construction of Analytical Model. 35110.9 Anchor and Drag Anchor 353Chapter 11Special Thermal Problems '" 35711.1 Thermal Bowing.................................................................................................................. 35711.1.1 Displacement and Stress Produced by Thermal Bowing........................................ 35711.1.2 Internal Thermal Stresses Generated by Bowing Temperature.............................. 35911.1.3 Occurrences of Thermal Bowing.... 36011.1.4 The Problem Created by a Tiny Line 36411.2 Refractory Lined Pipe 365.. '0. ;J"1II,I_. JIContents ix11.2.1 Equivalent Modulus of Elasticity 36511.2.2 Hot-Cold Pipe Junction 36611.3 Un-Insulated Flange Connections 36911.4 Unmatched Small Branch Connections 36911.5 Socket-Welded Connections 370Chapter 12Dynamic Analysis - Part 1: SDOF Systems and Basics 37312.1 Impact and Dynamic Load Factor ~ 37312.2 SDOF Structures................................................................................................................. 37512.2.1 Working Formula for SDOF Systems 37612.2.2 Un-Damped SDOF Systems 37612.2.3 Damped SDOF Systems 38212.2.4 Summary of the Characteristics ofSDOF Vibration 38612.3 Damping ...............................................................................................................:............ 38712.4 Sonic Velocity Versus Flow Velocity 38912.4.1 Sonic Velocity 39012.4.2 Flow Velocity 39212.5 Shaking Forces due to Fluid Flow 39512.6 Safety Valve Relieving Forces 39712.6.1 Open Discharge System 39712.6.2 Closed Discharge System 40112.7 Steam Turbine Trip Load 403Cbapter 13Dynamic Analysis - Part 2: MDOF Systems and Applications 40913.1 Lumped-Mass Multi-Degree of Freedom Systems 40913.1.1 Mass Lumping 41013.1.2 Free Vibration and Modal Superposition 41213.2 Piping Subject to Ground Motion 41313.2.1 Response Spectra Method 41513.2.2 Combination of Response Spectra Analysis Results 41713.2.3 Comparison of Modal Combination Methods 41913.2.4 Puzzles of Absolute Closely Spaced Modal Combination 42113.2.5 Compensation for the Higher Modes Truncated 42213.2.6 Design Response Spectra 42313.3 Account for Uncertainties 42613.4 Steady-State Vibration and Harmonic Analysis 42813.4.1 Basic Vibration Patterns 42813.4.2 Allowable Vibration Displacement and Velocity 42913.4.3 Formulation of Harmonic Analysis 43713.4.4 Evaluation of Vibration Stress 44413.5 Time-History Analysis 44613.5.1 Treatment of Damping 44613.5.2 Integration Schemes 4481 3 . ~ . 3 Time Step, Stability, and Accuracy 45013.5.4 Example Time-History Analysis 451x ContentsAppendix A .AppendixB .Appendix C .AppendixD .AppendixE .Appendix F - .INDEX .dMe Gwg.r' < ~ ,459 .461462' J464466' ,474 J477, ,, ,. ;nI.1IIJACKNOWLEDGMENTSThis book is essentially the summary of the knowledge accumulated by the authors through 40 yearsof practiceas piping mechanicalengineers. I, thesenior author,would like tousethis opportunityto express my appreciation and gratitude to many friends, colleagues, and supervisors for providingthose learning opportunities and environments. First, I would like to thank Ron Hollmeier of PioneerService & Engineering in Chicago for offering me my first pipe stress job developing a computer pro-gramfor pipe stress analysis in 1967. This job allowed me to stay in the United States and led to a long,interesting career. I am grateful to Bechtel's Bill Doble and Joe Gilchrist, who did not hesitate to sendme toEnglish classes and put me on interesting jobs such as the Trans-Alaskan pipeline and BlackMesacoal slurry pipeline projects. Mymostmemorableworkwasdone at NuclearServicesCor-poration in San Jose. Workingas part of ateam that included BobKeever, Randy Broman, DougMunson and myself, and with help from Professor Gram Powell of University of California-Berkeleyand valuable inputs from Mel Pedell and Dane Shave of Stone and Webster, we created the NUPIPEpipe stress software, which became a very powerful tool in the design of nuclear piping. I am greatlyindebted to Don Mckeehan and Ed Bissaillon of M. W. Kellogg for their encouragement and imple-mentation ofthe SIMFLEX software. As a result of the authors' long association with M. W. Kellogg,this book is noticeably influenced by Kellogg's philosophy and approaches mentioned in the secondedition of the Kellogg's book - Design ofPiping Systems (1956, John Wiley and Sons). SuggestionsfromRay Chaoand David Osage of Exxon Research were very helpful duringthedevelopment ofthe PENGS program. The estimated 1000 engineers, who came to my training classes conducted ina dozen countries, have greatly widened my perspective on piping mechanical work. The authors arevery grateful to ASME Press for valuable comments and excellent editing. We would also like to thankthe twin sisters Lina and Linda for reading the manuscript. The fact that these two non-technical sis-ters have read through the entire draft of the manuscript has greatly encouraged us.Liang-Chuan (L. C.) PengTsen-Loong (Alvin) Pengxi\. J, jI1..-.- ._11I441#11I 91IIIIIIIIIIIIIjnIiI)PREFACEPipe stress analysis calculates the stress in a piping system subject to normal operating loads such aspressure, weight, and thermal expansion, and occasional loads such as wind, earthquake, and waterhammer. Because all piping systems are connected to equipment such as vessels, tanks, pumps, tur-bines, and compressors, the piping stress analysis also involves evaluation of the effect of the pipingforces and moments to the connecting equipment. As the piping stress is controlled by the arrange-ment of the supports and restraints, the scope ofpiping stress includes also pipe supports. The wholescope of this work is generally referred to as piping mechanical.Before the advent of the electronic computer, pipe stress analysis was handled by very specializedengineers. Only large corporations and specialized firms had the personnel to do the job. It normallytook a specialist to use the calculator non-stop for a couple of weeks just to analyze the flexibility ofa moderately complex piping system to absorb the thermal expansion of the pipe. Because only veryfew ofthese engineers knew how to analyze piping stress, most engineers treated it as some type of amysterious subject. Engineers saw there were expansion 1001'S, offsets, and special supports such asspring hangers and constant effort supports, but did not really know why they were there. The limitedscope of pipe stress analysis dealing with the piping flexibility for absorbing thermal expansion wascalled piping flexibility analysis.With the arrival ofthe electronic computer in the 1970s, and especially the personal computer in the1980s, suddenly everybody knew how to analyze pipe stress. This has generated even more mysteryabout the field. Nowadays, we occasionally see an electrical engineer, although discouraged, conduct-ing the analysis just as proficiently as a mechanical or a structural engineer. This partly stems fromthe fact that colleges and universities normally do not offer any course on pipe stress. This leaves theknowledge and skill of pipe stress and piping engineering to be learned by self-study and actual prac-tice. Practitioners who obtain the best computer program and comprehend the manual most will dothe better job.With the rapid advancement in computer technology, a piping flexibility analysis nowadays takesonly few minutes via an appropriate computer software. Therefore, the task ofthe stress engineer hasbeen shifted from the traditional stress calculation to stress engineering. The emphasis is not on howto calculate the stress, but rather on how to utilize the analysis tool to design a better plant. However,just because it is easy to get the stress calculated, engineers often depend too much on the computerand forgetabout thefundamentalsand engineering common sense. Without the fundamentalsandcommon sense, one may not even be aware ofthe unreasonable results produced by the computer, tosay the least about good engineering. This book emphasizes engineering common sense as well as thebasic principles. The following are some examples ofpiping problems that might have been solved byjust good engineering common sense: A plant operated smoothly for the first10 years, and then experienced a leakage at themain process piping about every 4 months after a major revamping. Experts were con-sulted, sophisticated analyses were performed, and expensive modifications were madeto ho avail. Had the engineers used the basic thermal stress common sense, the problemwould have been solved with very little effort (see Section 11.2.2). The pressure thrust force is very critical at a bellow expansion joint. Anchors are oftenneededat bellowexpansion joint installationstoresistthepressurethrustforce, butxiiixiv Prefacesometimes an anchor placed at the wrong location may just be the cause ofthe problem.The wrong anchor had been contributing to severe vibration at some rotating equipment.Had weknown that the pressure thrust forceat an expansion joint isgenerally muchhigher than the force that can be tolerated by the rotating equipment, the problem would have occurred (Chapters 7 and 9). A small steam purge linein a large process pipehadcaused the plant piping totwistwildly, breaking many connecting leads. Several major re-routing of the piping systemswere made, but the problem persisted. Had the involved engineers had some idea aboutthermal bowing, the problem could have been easily corrected (see Section 11.1.4).Providingtheknowledgefor solvingtheproblemssuchastheones listedaboveistheprimarygoal of the book. Chapter 1 summarizesthescopes and requirements related to piping mechanicalactivities, and the subsequent chapters discuss how to deal with them. Chapter I contains some of theauthors' inside views, which we hope will help the readers progress more confidently and comfortablyinto the piping mechanical field.Nowadays, making a calculation with a computer is so fast that we often hear about the "what-if"approach in engineering. What all this "what-if" approach accomplishes is making numerous randomtrials and the wish that one of these trials will hit the mark sooner or later. The problem is that after afew trials, most people lose the ability to make sense ofthe trials. The more they try, the more they getconfused. In contrast, this book puts emphasis on the "what, why, and how" to guide the readers intothis 3-W approach - that is, to be aware ofthe problem, understand the cause ofthe problem, and tosolve the problem or prevent it from happening.The authors will try to explain all the necessary tasks of pipe stress engineering with basic funda-mentals. Althoughonly a fewvery fundamental equations are introduced,theoretical backgroundswill be covered in as much detail as possible. The book is titled Pipe Stress Engineering to distinguish itfrom a regular pipe stress analysis book, which normally lacks the coverage ofthe engineering aspects.Although this book is intended for piping mechanical engineers, it is also a suitable reference bookfor piping designers, plant engineers, and civil-mechanical engineers. The book can be used as thetextbook for a one- or two-semester elective course given at the senior or graduate level., ,, j'''-"l------- ............_...........................illli.IIIII'__.111J14 l!NOMENCLATURE(All are in consistent units. Special usages and non-consistent units are noted in the main text. Ab-breviations are listed at the end)AAAAbAcAmApATaaaBBbbbbCCC[C]C" CzCz,KzCEDCEVCKCwCxIcJccpcvD;D,{D}{D'}DcDe)ICross-section areaThickness allowance including corrosion, thread, etc.Flow areaTotal net cross-section area offlange boltsCorrosion allowance of pipe thicknessManufacturing under-tolerance of pipe thicknessPressure area encircled by the bolt circle of flangeNozzle throat area or valve orifice areaBellow effective pressure thrust areaSonic speedTank nozzle radiusFlange inside diameterTank bottom plate thermal expansion factorFlange effective gasket widthThickness of support saddle, in pipe axial directionWidth of the beam cross-sectionWidth of a small element or stripCold spring factorFlange bolt circle diameterThermal bowing local stress factor, defined in Figure 11.2Damping matrix of the structural systemintegration constants to be determined by boundary conditionsASME Class-l nuclear piping stress indices for displacement loadingEnd coefficient with respect to vibration displacement, defined in Eq. (13.43)End coefficient with respect to vibration velocity, defined in Eq. (13.52)Karman force coefficient on vortex shedding forceRatio oftotal pipe weight, including pipe metal weight, content, and insulation, to pipemetal weightVibration stress amplification factor due to concentrated weightSoil cohesion stressViscous dampingSpecific heat of gas under constant pressure conditionspecific heat of gas under constant volume conditionDiameter of pipe or circular cross-sectionDiameter ofvessel shell or run pipeDisplacement vector in global coordinates, for both element and overall structuresElement displacement vector in local coordinatesCombined equivalent diameter of all the nozzles at rotating equipmentEquivalent diameter of an individual nozzle at rotating equipmentxvxvi Nomenclature"m'. -DLFddddedpEEEeeecomexeyeoFFF{F}{F'}IFIF1{Fc}FKFmaxFnFnsFnTFpFRFRFs{Fs}F(t)FRXFRYFRZFTXFTYFTZffffffi(f3x)Ji(f3x)h(f3x)f4(f3x)fwGG-.-Dynamic load factorDisplacementDiameterDiameter of branch pipeElastic displacement limit of an elastic-plastic restraintPitch diameter of bellowModulus of elasticityLongitudinal joint efficiency, normally expressed with allowable stress as (SE)Equivalent modulus of elasticity for composite pipeStrainCombined total equivalent axial bellow deformation per convolutionAxial bellow deformation per convolutionEquivalent axial bellow deformation per convolution due to lateral displacement yEquivalent axial bellow deformation per convolution due to rotation eForceGas pipeline design factor (defined in Table 10.2)Pipeline anchor force, or potential expansion forceForce vector in global coordinates, for both element and overall structuresElement force vector in local coordinatesAbsolute value of force FForce equivalent to total impulse function =pAV2+ PAVector of cosine components of the harmonic forceKarman force or vortex shedding forceNet maximum peak shaking force of a given piping legNormal forcePressure force at point n due to standing pressure wavePressure force at point n due to traveling pressure waveTank nozzle pressure force defined in Eq. (8.30)Radial forceResultant forceShear forceVector of sine components of the harmonic forceForce as function oftimeRotation spring constant about x axis for a generic flexible jointRotation spring constant about y axis for a generic flexible jointRotation spring constant about z axis for a generic flexible jointTranslation spring constant in x axis for a generic flexible jointTranslation spring constant in y- axis for a generic flexible jointTranslation spring constant in z- axis for a generic flexible jointLine force per unit length of active line in an attachmentNatural frequency of the structural system, or frequency of a vibrationPipeline friction force per unit length of pipeStress range factor for calculating allowable expansion stress rangeSupport friction forceBeam on elastic foundation function defined by Eq. (5.17a)Beam on elastic foundation function defined by Eq. (5.17b)Beam on elastic foundation function defined by Eq. (5.17c)Beam on elastic foundation function defined by Eq. (5.17d)Working (nominal) axial spring constant per bellow convolutionFlange gasket load reaction circle diameterShear modulus of elasticity, modulus of rigidityTil; QW. ) JnI\)IIJglgx, gy, gzHHHhhh1[1]01piiieiLKK[K][K']KAKhKLKNKRKRyKR8KvKvKxkkkkLLLL[L]LMPeeeMM[M];Mmass;,MeMLMRNItNomenclature xviiFlange hub thickness at back of flangeCold spring gap in x, y, z directions, respectivelyDepth of soil cover, from top of pipe to soil surfaceHeight of the beam cross-sectionRing force per unit circumferential breadthFlexibility characteristic of piping component, defined in Table 3.1Safety valve height, defined in Fig. 12.17Soil depth at an arbitrary pointMoment of inertia of a pipe or beam cross-sectionIdentity matrix, or unit diagonal matrix, with 1 on diagonal and 0 at elsewherePolar moment of inertiaStress intensification factorImaginary number, square root of-1Stress intensification factor for circumferential stressStress intensification factor for longitudinal stressBulk modulus ofthe liquid =-dp/(dv/v)Stiffness or spring constant of support structureStiffness matrix in global coordinates, for both element and overall structuresElement stiffness matrix in local coordinatesCoefficient of active lateral soil pressureSpring constant per unit pipe length for horizontal soil resistanceEJMA bellow lateral spring rate =KvNapier constant for steam flowEJMA bellow rotational spring rate = KR8Bellow rotational spring rate due to lateral end deflectionBellowrotational springrate due toendrotation, whenfree lateral deflectionisallowedBellow lateral spring rate due to lateral end deflection. End moment also createdSpring constant per unit pipe length for vertical soil resistanceBellow axial spring rateFlexibility factor ofpiping componentRatio of specific heats =cp/ CvSpring constant and directional stiffnessSpring constant offoundation per unit length of beamDistance from the center of tank nozzle to tank bottomLengthPipeline active lengthSupport spacingTransformation matrix between local coordinates and global coordinatesLarson-Miller parameter defined in Eq. (1.2)ElongationSupport lug lengthLength of elementMassMomentMass matrix of the structural systemConcentrated massCircumferential momentLongitudinal momentResultant momentTorsion momentxviii Nomenclature, 1" ,MwMyMeM(e+tJ.)mmtilNNNNsnPPdPePsPTPp*QQQ[Q]qqqaqcqeqhqyRRRRRR-RRaRajRcRdRyrrrrYmSSSASbMolecular weightMoment produced by bellow lateral displacement without free end rotationMoment produced by bellow rotation with free lateral displacementMoment produced by bellow rotation without free lateral displacementFlange gasket factorMass per unit length of pipeMass flow rateCircumferential force per unit length of pipeNumber of bellow convolutionsNumber of operating cyclesStrouhal number on vortex shedding frequency; Ns=fD/ VNumber of stiffening rings at each support saddlePressureFlange design pressureEquivalent pressure due to force and moment acting on a flangePressure amplitude of a standing wavePressure amplitude of a traveling wavePressureCritical pressure at sonic velocity statePipeline end resistance forceTotal support loadVolumetric flow rateInfluence or relation matrix relating ground motion to every part ofthe structureDistributed external force per unit length of beamPitch of bellow convolutionDeformed pitch at centerline of bellowCompressed pitch at pitch diameter of bellowExtended pitch at pitch diameter of bellowHorizontal soil resistance force per unit length of pipeVertical soil resistance force per unit length of pipeBend radiusGas constant =-R/MwRadius of curvatureRadius of circular cross-sectionRadius ofvessel shellReaction forceUniversal gas constantAcceleration response spectraAcceleration response spectra ofjth independent support motionRadius ofthe crown on a vessel headDisplacement response spectraVelocity response spectraRadius of an arbitrary circular ringRadius of pipe cross-sectionRadius of round attachmentRotationMean radius of pipe cross-sectionBasic allowable stress for pipelineStressBasic allowable thermal expansion stress rangeFlange bolt stressIe.-.J, ," ,i,. ,f 1)JScSESEBSelSelASF{Sg}ShShShpSiSipSMSMYSSpSpwSTSuSySycSyhxTTTT1*ttttttdtoVVVVvVWWWWWWpWsw{X}XAXBNomenclature xixAllowable stress of pipe material at cold ambient conditionExpansion stressBenchmark expansion stress rangeEndurance strength of pipe materialAllowable endurance strengthFla:qge gasket stress due to axial forceVector of independent support motion componentsAllowable stress of pipe material at hot operating conditionPressure hoop stressHoop pressure stressStress intensity (=twice of the maximum shear stress)Longitudinal pressure stressFlange gasket stress due to bending momentSpecified Minimum Yield StrengthFlange gasket stress due to pressureLongitudinal pipe stress due to pressure and weightLocal thermal stress due to thermal bowingUltimate strengthYield strengthYield strength at cold conditionThe lesser of yield strength at hot condition and 160% of the stress producing 0.01 %creep in 1000 hours at the operating temperatureAbsolute temperature, K (=273 + 0c) or R (=460 + OF)Period of vibrationTemperatureVessel thickness or run pipe thicknessCritical absolute temperature at sonic velocity stateThickness of bellowThickness of branch pipeThickness of pipe, genericThickness of tank shell at nozzle locationTimeTime duration of an impulse loadingEffective valve opening timeBellow lateral forceVelocityVolumeTangential shear force per unit circumferential breadthShear forceSpecific volumeTotal flange bolt loadTotal shearforce at pipe cross-sectionWeight load in force unitWeight of the free body, or weight loadWeld strength reduction factorWeight of pipe per unit length of pipeWeight of soil cover per unit length of pipeWeight per unit lengthDisplacement vector of the structural systemDistance between the top of nozzle and tank bottom plateDistance between the bottom of nozzle and tank bottom platexx NomenclatureXc{X}gxx,Y,ZYcYFYLyyyyyZZpZPAzDistance between the center of nozzle and tank bottom plateGround motion displacement vectorAxial displacement of beam or bellowCoordinates in x, y, z directions, respectivelyAllowable coefficient, Fig. 8.25, for circumferential moment on tank nozzleAllowable coefficient, Fig. 8.25, for axial force on tank nozzleAllowable coefficient, Fig. 8.25, for longitudinal moment on tank nozzleAdjustment coefficient (see Table 4.1) for pipe thickness calculationFlange gasket seating stressLocal lateral displacement of beam or bellowPipeline end axial movementRadial displacement of pipe shellSection modulus of a pipe or beam cross-sectionPolar section modulusZero period acceleration of the response spectra curveCompressibility of real gas =pv/(RT)f' Greek symbolsa Constant defining participation ofmass in damping, see Eq. (13.76)a Vibration allowable stress reduction factor, 1.3 for carbon and low alloy steels and 1.0for austenitic stainless and high alloy steelsa Thermal expansion rate, as expansion per unit length per unit temperaturef3 Angle from top of pipe to edge of saddlef3 Angle of branch intersectionf3 Branch/run diameter ratiof3 Characteristic parameter of beam on elastic foundation, defined by Eq. (5.26)f3 Constant defining participation of stiffness in damping, see Eq. (13.76)f3 Frequency ratio of the applied frequency to the natural frequency of the systemr Rotational deformationi1 Clearance between tube and tube sheet holei1 Deflectioni1.. Difference of ..M Amplitude decay due to step-by-step time-history analysisi1T Period elongation due to step-by-step time-history analysisM Integration time step for time-history analysis, Damping ratio, the ratio of damping to critical damping17 Nozzle flow efficiencye Angle of circular wedgee Angle of mitere Angle of the inclining planee Rotatione Support saddle angleA. Tank geometrical parameter defined in Eq. (8.30)p. Friction coefficientv Poisson ratioCoordinate or coefficient of normal mode space, i.e., {X} = Cosine component of modal coordinate vector of harmonic responseSine component of modal coordinate vector of harmonic response1C 3.141592P Angle of the maximum bending moment location at a support ringp Density, mass per unit volumer), 1Nomenclature xxiSummation ofShear stressFlow velocityEigenvector, or natural vibration shapeEigenvector matrix with eigenvectors as columnsAngle from top of pipeBellow rotation per convolutionSoil internal friction angleCircular frequency, or rotational speedCircular natural frequency for damped systemCircular natural frequency for un-damped (and also damped) systemAt origininitial stateAt location 0, I, 2, ... , etc., or at condition 0, I, 2, ... , etc.AllowableAllowableat point a, b, ... , etc.bendingbranch pipecircumferential directioncold or ambient temperaturecritical conditiondischarge sideequivalentfrictionhoop directionhub of flangehot or operating temperaturehoop pressurein-planeinside surface ofthe pipelongitudinal directionlongitudinal directionlongitudinal pressuremean valuemaximum valuenatural vibrationnominalnormalout-planeoutside surface of the pipepressureradial directionresultantrigid body responserun pipering stiffenerrun pipexxii NomenclaturessstTttxyx,y,zYYAbbreviationsABSANSIAPIASMEASTMAWWAB&PVCENcpsDLFftin.HzKksilblbfmMDOFMSSNNRCPapsiRRPMSDOFSIFSRSSWRCsuction sideshearstatictest conditiontangentialtorsionon x plane and in y directioncomponents in x, y, and z directions, respectivelyyield conditionat y distance awayabsoluteAmerican National Standards InstituteAmerican Petroleum InstituteAmerican Society of Mechanical EngineersAmerican Society of Testing and MaterialsAmerican Water Work Works AssociationBoiler and Pressure VesselComite Europeen de Normalisation (European Standard)cycles per seconddynamic load factorfoot or feetinch or inchesHertz =cycles per secondKelvin = 273 + Ckilo pounds per square inch =1000 psipound (weight or force)pound forcemetermulti degrees offreedomManufacturer Standardization Society of the Valve and Fitting IndustryNewtonNuclear Regulatory CommissionPascal =N/m2pounds per squire inchRankin =460 + OFrevolution per minutesingle degree offreedomstress intensification factorsquare root sum of the squaresWelding Research Council-"-............................_ ........... ...iwM ... -------fIliII!lI!-illlllllli lIlIIlIIli'Ilg.IIJI' r ; f : f - - ~-IICHAPTER1INTRODUCTIONA piping system is the most efficient and common means of transporting fluids from one point toanother. Within a petrochemical complex, acres and acres of piping can be seen running in every di-rection and at many different levels. Piping constitutes 25% to 35% ofthe material of a process plant,requires 30% to 40% of the erection labor, and consumes 40% to 48% of the engineering man-hours[1]. The actual importance of piping, however, can far exceed these percentages. An entire piping sys-tem is composed of a large number of components. The failure ofjust one single component has thepotential to shut down the entire plant or, worse yet, cause serious public safety problems. In spite ofthis, piping is generally considered a low-technology subject in the academia. Very few colleges teachthe subject, leaving engineers to gain this knowledge only through actual practice in the field.To find out exactly where pipe stress fits in the piping design process, let us first find out what pro-cedures are involved in designing a piping system. A piping system is designed in the following stepsby different engineering disciplines:(1) Process engineers, basing on process requirements and plant capacity, determine, among otherthings, the flow path, the flow medium and quantity, and operating conditions. They then putall this information into process flow diagrams.(2) Material specification engineers assign suitable categories of specifications for the piping sys-tem based on the process flow and reactivity ofthe contained fluid. Each specification is appli-cable to certain combinations of fluid types, temperature ranges, and pressure ranges. Materialspecifications normally include pipe material, pipe wall thickness for each pipe size, the corro-sion and erosion allowances, flange class, valve types, fitting and branch connection type, boltmaterial, gasket type, etc.(3) System engineers combine process flow diagrams, material specifications, and equipment datasheets to create operational piping diagrams. They select the applicable material specificationand determine the size for each line based on flow quantity, allowable pressure drop, and flowstability. Piping diagramsare generally combined with thenecessary instrument and controlcircuits to become piping and instrument diagrams(P&IDs). Special items such as potentialtwo-phase flow and slug-flow zones are also identified on these diagrams for special consider-ation in design and analysis. In addition to the P&IDs, a line list covering all pipe spools is alsoconstructed. This line list contains most of the design, upset, and operating parameters to beused in the layout, analysis, and fabrication of the piping system.(4) Piping designers, in coordination with other disciplines, conceive an overall plant layout, per-forma piping routingstudy, determinethe pipe rack locations, and place theactual pipingthat connects to designated points. They layout and support the piping by following the rulesand proceduressetupbyeachindividual company. In general, threesetsof drawingsareprepared. The first set istheschematic planning drawings, used asacommunication boardbetween different departments. Actions and comments from related disciplines are all resolvedand recorded in these drawings. Pipe supports are also recorded in this set of drawings. Thesecond set is the composite drawings,consisting of to-scale drawings of all pipes and equip-ment in the area. These drawings, to be used in the construction, are evolved from the planningj1:1. ]2 Chapter 1drawings. The third set of drawings is the isometrics of the piping, used for stress checks andshop fabrications.(5) Piping mechanical engineers check the stresses and supports ofthe systems. Using the P&IDs,they develop operating modes so that all the expected operating conditions are properly ana-lyzed; Proper supports and restraints are selected and placed to optimize the overall cost andperformance of the systems. They also design or specify piping specialty items, such as expan-sion joints, flue heads, special connections, spring hangers, vibration supports, and so forth.It may be a bit surprising that designing a piping systemis so involved. Indeed, on a large project, notonly is every discipline required, but the effort also engages quite a few people. Piping design and pip-ing mechanical are the two disciplines that require the most number ofpersonnel. However, for a smallproject handled by a small outfit, generally only one or two people are assigned to take care of all thework ofvarious disciplines. In such cases, pipe stress activity is often neglected. Due to the resilient na-ture ofductile piping systems, the piping will work most ofthe time, even without going through properstress checks. This may be acceptable for small non-hazardous piping, but not for most of public andindustrial piping systems, which require the piping system to be safe and operational all the time.Item (5) is the main subject ofthis book. The task ofa piping mechanical engineer is generally calledpipe stress and support. The scope ofthe pipe stress and support activity has increased exponentiallyin the past three decades. This is due to the stringent requirements of the modern plant. For instance,in the1960s, the pipe stress and support manpower used for a petrochemical plant was about 4000man-hours. A nuclear power plant in that era would have used about the same amount of pipe stressman-hours. Nowadays, the pipe stress and support manpower required for a petrochemical plant hasincreased by ten times to about 50,000 man-hours. The effort required by a nuclear power plant hasgrown 1000-fold to reach as high as 2 million man-hours [2}. With this exponential growth in the man-hours involved, the probability ofgetting sub-standard output from some ofthe engineers is very high.A time-saving tool, such as an efficient pipe stress computer program, not only significantly reducesthe cost of designing a plant, but also greatly improves the quality of the plant. Good work starts witha good grasp ofthe scope ofthe jobs that need to be done. In this chapter, the non-mathematical con-cepts of pipe stress activities will be discussed.~ J1.1 SCOPE OF PIPE STRESS ANALYSISPipe stress analysis is, of course, the analysis of the stress in the pipe. However, if we ask what is tobe analyzed, many will hesitate to answer. In the 1950s and 1960s, when engineers started analyzingpiping systems, they had only one thing in mind: calculating the stress due to thermal expansion. Inother words, they checked the piping layout to see if the piping system was flexible enough to absorbthe thermal expansion due to temperature change. The analysis was referred toas piping flexibilityanalysis. Books [3-6} and articles [7-9} written during this period dealt mainly with flexibility analysis.Later on, as technology progressed, pipe stress analysis encompassed much more than just checkingflexibility; yet nowadays many engineers still refer to pipe stress analysis as flexibility analysis.Thisslight mix-up in terms is not important. However, the concept that flexibility is the only considerationin pipingstressanalysescan lead toanexpensive, and unsafe, sub-standard design. For instance,many engineers tend to consider that providing additional flexibility in the piping is a conservative ap-proach. In reality, additional flexibility not only increases the material cost and pressure drop, it also'(Oakes the piping prone to vibration, the biggest problem area of the piping in operation. Since thepublication of the 1955 piping code [10) and Kellogg's[3} book, failures due to insufficient flexibilityhavebecomeveryrare. Nowadays, mostfailuresarecaused by vibration, thermal bowing, creep,thermal fatigue not related to flexibility, steam/water hammer, expaJ;lsionjoints, and so forth [11, 12}.These facts should serve as clues toward designing better piping systems.Consider, for example, the piping installed to move the process fluid from the storage tank to theprocess unit asshown in Fig. 1.1. First, we have todeal with tank shell displacement and rotation------------------------IIIIIIJf1'1r'-1Introduction 3Vessel Flexibilityand Local Stresses

EarthquakeSupport andFrictionTank Flexibility andShell RotationTank SettlementTank Nozzle LoadPipe Stress AnalysisIsMuch More than JustA Flexibility CheckFIG. 1.1TASKS OF PIPE STRESS ANALYSISdue to the hydrostatic bulge of theshell. This temperature-independent displacement and rotationwill exert a great influence on the connecting piping. Furthermore, the tank nozzle connection is farfrom rigid. Its flexibility has to be estimated and included in the analysis. Then, after the pipe forcesand moments at the connection are calculated, they have to be evaluated for their acceptance. There4 Chapter 1aremany items likethese, whicharenot normallycalled piping flexibility, butarerequired tobeconsidered in piping stress analysis. These items, using Fig. 1.1, shall be identified one by one startingfrom this tank connection.The next items we come across are flanges and valves. Can they maintain the tightness under pipingforces and.moments? Can valves operate properly under pipe forces and moments? These need to bechecked even though the pipe itself is strong enough for the same forces and moments.We know the support friction can also have a significant effect on piping movement. The situationsand methods to include the friction effect also need to be considered.In addition to the average pipe temperature, the pipe might also have a temperature gradient acrossthe pipe cross-section due to stratified flow or blow off oflow-temperature fluid. Even radiant energyfrom the sun on empty un-insulated pipe can cause this type of temperature difference. This type ofbowing phenomena can create a great problem in the piping and needs to be considered.For piping connecting to the rotating equipment such as pumps, the pipe load has to be maintainedto within the manufacturer's allowable range to prevent the equipment from excessive vibration, wear,and overheating. The piping connected to rotating equipment also needs the consideration of poten-tial water hammer, pulsation, and other dynamic phenomena.Proper spring hangers will need to be selected and placed to ensure that the piping is properly sup-ported under all operating conditions.At the vessel connection, again, the flexibility and displacement at theconnection have to be in-cluded in the analysis. After pipe forces and moments at the connection are calculated, vessel localstresses have to be evaluated to see ifthey are acceptable. Then, of course, if the structure is located inan earthquake or hurricane zone, earthquake and wind loading have to be considered when designingthe piping system.In general, the purpose of pipe stress analysis can be summarized into two broad categories:(a) Ensure structural integrity: This involves the calculation of stresses in the pipe due to all designloads. Necessary procedures are taken to keep the stress within the code allowable limits. Thiscode stress check is the basic assurance that failures from breaks or cracks will not occur in thepiping.(b) Maintain system operability: A piping itself can be very strong, yet the system may not be oper-able due to problems in the connecting equipment. Flange leakage, valve sticking, high stressin the vessel nozzle, and excessive piping load on rotating equipment are some of these prob-lems. The work required in maintaining the system operability is generally much greater thanthatrequiredinensuringthestructuralintegrity. Thisismainlyattributabletothelackofcoordination between engineers of differentdisciplines. Rotatingequipment manufacturers,for instance, design non-pressure parts, such as support and base plate, based mainly on theweight and the torque of theshaft. Then they specify the allowable piping load with that de-sign, disregarding the fact that some practical piping load alwaysexists and needs to be ac-commodated. The allowable loads they provide are generally much too small to be practical.Unfortunately, these allowable values go unchallenged, mainly because the industry as a wholegivesnoincentivestomanufacturerstoproduceequipment thatcanresist theextra pipingload. If more and moreengineers would request theextra strength or give preferential treat-ment to manufacturers that produce stronger equipment, an optimal solution might eventuallybe reached. Until such time comes, piping mechanical engineers should be prepared to spendthree times as many man-hours in stress engineering the piping system connected to rotatingequipment.1.2 PIPING COMPONENTS AND CONNECTINGEQUIPMENTIn this section, we will deal with three categories of hardware: pressure parts of the piping proper,support and restraint elements, and connecting equipment. The functions of support and restraint ele-'-__ I~ - 1-----------.......-------IIIIIIIIIIlIIIIIIIIIIIIlII__IIIIIIIIIiIIIIIDIntroduction 5ments are apparent. And because a separate chapter is dedicated to pipe supports and restraints, weshall not include them in this discussion.In the following, we use Fig. 1.2 as a guide to summarize the types of piping components and arraysof connecting equipment typically encountered in piping stress engineering.Safety ValvePipe Branch Connections:Un-reinforced ReinforceFabricated FabricatedTee TeeRotating Equipment:Steam TurbinePumpCompressorExpanderMotor OperatedPneumatic OperatedHydraulic OperatedControl Valve:Flexible Joint:Bellow JointBall JointFlexible HoseFlexible CouplingExtrudedTeeShell Connections:VesselTankHeat ExchangerWelded-on Branch Connections:Integrally Contoured Half~ ~ E SFlanges:Welding Neck Slip-on Lap-jointciA.Needle ValvePlug ValveCheck Valve5-DBendLong RadiusElbowWeldingTeeValve:Gate ValveGlobe ValveBall ValveButterfly ValveSteam BoilerFire HeaterFurnaceAir CoolerTube Header:(FIG. 1.2PIPING COMPONENTS AND INTERFACE EQUIPMENT-------------------6 Chapter 1(a) Main pipe. Starting with the main pipe, we have to know the pipe material, which is gener-ally given by the American Society of Testing and Materials (ASTM) [13] specification number. Thepipe is generally identified by its nominal diameter and nominal thickness. For pipes 12 in. (300 mm)or smaller, the nominal diameter is very close to the inside diameter of the pipe with standard wallthickness. However, forsizes14 in. (350 mm)and larger, the nominal pipe diameter isexactly thesame as the outside diameter of the pipe. For each pipe size, the outside diameter is fixed for a givennominal diameter. This is done so that all pipes with the same nominal size can use the same pipe sup-port attachments such as clamps, and insulation blocks. Each pipe size also has several commerciallyavailable thicknesses called nominal thickness. The available nominal thicknesses are standardized inrelated standards such as ASTM A-106 [13] and American Society of Mechanical Engineers (ASME)B36.10 [14]. In the United States, these thicknesses are also represented by weight grades as standard(Std), extrastrong(XS), anddoubleextrastrong(XXS), andbyschedulenumbersrangingfromschedule 10 (Sch-10) to schedule 160 (Sch-160). Stainless steel pipes have separate schedule numbersappended with the letter "S" such as Sch-5S, Sch-10S, etc. For instance, a 6-in., Std pipe means thatthepipehasa6.625-in. outsidediameteranda0.280-in. wall thickness, whereasa14-in., Sch-40pipe corresponds to a pipe with a 14-in. outside diameter and 0.438-in. wall thickness. The schedulenumber itself dose not represent any thickness; it has thickness assigned only when tagged with a pipediameter. See Appendix Afor standard nominal pipe wall thicknesses.The pipe material involves different manufacturing processes that might implicate some stress ris-ers and tolerances. The pipe can be broadly classified via the method in which it was manufactured:seamless or welded. Welded pipes, which are somewhat weaker than seamless pipes due to the weld,are further classified into several categories. From the seam position, we have longitudinally weldedand spirally welded (two main types). Fromthe welding process, we have furnace butt welded, electricresistance welded, and electric fusion welded (three methods). Each type of welding has its uniquejoint efficiency that needs to be included in the pressure design.For tolerances, we are mostly con-cerned with thickness under-tolerance, which can reduce the pressure resisting capability of the pipe.Under-tolerance is the allowable amount of thickness in which a manufactured pipe can be made thin-ner than the nominal thickness. In general, seamless pipes and electric resistance welded pipes havea higher under-tolerance of about 12.5% of the nominal wall thickness. Electric fusion welded pipeshave a somewhat different under-tolerancedetermined by the plate used; however, API5L suggest10% to 12.5% for the fusion welded pipe also. This under-tolerance needs to be included in the stressdesign of the pipe.(b) Welds. Toconnect pipingcomponents together, weuseeither circumferential weldorflange connections. The circumferential weld, also called girth weld, is for a permanent connection,whereas theflange is used for locations requiring occasional separation. The girth weld also has ajoint efficiency that works against longitudinal stress. This joint efficiency does not affect pressuredesign, which iscontrolled by circumferential hoopstress. For loadingsother than pressure, thisgirth weld joint efficiency isimplied in thestress intensification factor. Therefore, in most pipingcodes, circumferential weld joint efficiency is seldom mentioned. It is also often overlooked by stressengineers.(c) Weld strength reductionfactor. Inaddition tothe jointefficiency thataffects thegeneralstrength of the piping, the weld also hastens creep failure at creep temperature. The additional reduc-tion o(creepstrength over the non-weld-affected body iscalled the weld strength reduction factor.This is the factor applied,over the joint efficiency, at high temperature ranges. Thesame factor isapplied at both longitudinal welds and circumferential welds. However, longitudinal weld affects onlythe calculation of wall thickness, which is governed by the circumferential hoop stress. On the otherhand, circumferential weld affects only the sustained longitudinal stress due to pressure, weight, andother mechanical loads. The weld strength reduction factor is not applicable to occasional stress dueto the generally short duration of thestress. It also does not affect thermal expansion and displace-ment stress range due to the self-limiting nature ofthe stress. Generally, the temperature that requiresthe application of the weld strength reduction factor starts from 950F (510C). However, B31.1 and_1II'jIntroduction 7B31.3 treat it slightly differently. Details of the application are given in Chapter 4, which deals withcode stress requirements.(d) Flanges. Flangesareavailableinseveraldifferent types. Fromthestructural constructionpoint ofview, it can be classified as welding neck, slip-on, or lap joint (three types). Each type has itslength, weight, and stress intensification factor, all ofwhich have to be identified and considered in thestress analysis. Flanges are also identified by classes. Each class has its set ofpressure-temperature rat-ings, relating allowable pressures with operating temperatures. At a given operating temperature, theselected flange class shall offer an allowable pressure that is either equal to or greater than the designpressure. Table 1.1 shows the pressure-temperature rating for forged A-105 carbon steel flanges. Thetable alsoshows the flangethickness required for each class, for 6-in. and12-in. flanges. The data,taken from ASME B16.5 [16], is presented mainly to show the general trend of the allowable pressureversus temperature and class. It also gives some idea about the magnitude of flange thickness, exclud-ing hub. For Class 2500 flanges, flange thickness equals roughly 2/3 of the pipe diameter.It is also interesting to note that the flange classes were originally called pounds. Class 300 flangeswere called 300-pound flanges, and so forth. This was because, originally, Class 300 flanges were ratedfor 300 pounds per square inch (psi) pressure at a benchmark temperature. Pound (lb) meant psi, andhad nothing to do with the weight of the flange. The benchmark temperatures used were different fordifferentmaterials. For A-105carbonsteel, thebenchmark temperaturewas500F forClass150,and 850F for all other classes. However, as more accurate material data became available and moreaccuratestresscalculations became possible, theoriginal ratingsalsoappeared less accurate. Cur-rently, for A-105 flanges, Class 150 has a rating pressure of 170 psi at 500F benchmark temperature,and Class 300 has a rating pressure of270 psi at 850F benchmark temperature. It is obvious that cur-rent pressure-temperature ratings no longer correspond to the original benchmark idea. Although thepound classification is not very meaningful right now, it is still used by many engineers. See AppendixE for layout dimension and weight of valves and flanges.TABLE 1.1PRESSURE-TEMPERATURE RATING FOR A-105 CARBON STEEL FLANGES [16]Temp.Allowable Pressure, psiClassesOF150 300 400 600 900 1500 2500-20 to 100 285 740 990 1480 2220 3705 6170200 260 675 900 1350 2025 3375 5625300 230 655 875 1315 1970 3280 5470400 200 635 845 1270 1900 3170 5280500 170 600 800 1200 1795 2995 4990600 140 550 730 1095 1640 2735 4560650 125 535 715 1075 1610 2685 4475700 110 535 710 1065 1600 2665 4440750 95 505 670 1010 1510 2520 4200800 80 410 650 825 1235 2060 3430850 65 270 355 535 805 1340 2230900 50 170 230 345 515 860 1430950 35 105 140 205 310 515 8601000 20 50 70 105 155 260 430.S 6-in1.62 1.88 2.19Cl) '"FIg1.00 1.44 3.25 4.25bOa>j ~12-in~ . S : l1.25 2.00 225 2.62 3.12 4.88 7.25PFIg8 Chapter 1----------------- ------------(e) Bends. The turning of the pipe is accomplished by bends. Bends have several general types.Themostcommon bend istheso-called long-radiuselbow,which hasa bend radiusequalto1.5times the nominal pipe diameter. This is a fitting manufactured by the code-approved standard ASMEB16.9 [15] and others. The same standard also includes a short-radius elbow, which has a bend radiusequal to the nominal pipe diameter. Short-radius elbows are used in tight spots where available spaceis not enough for long-radius elbow. The cited factory-madeforged elbows are quite expensive andalso incur high flow friction loss due to the small bend radius. One alternative is to make the bend di-rectly by bending the pipe. To avoid excessive thinning and potential wrinkling, the bend radius ofthistype is generally bigger than three nominal pipe diameters. The one shown in the figure is a 5-D bend.Another alternative, mainly to save cost, is to cut the pipe into angled miters and bring them togetherto form the bend. This type of bend is called a miter bend. All these different bends have different wallthickness requirements,flexibility factors, and stress intensification factorsto beconsidered in thedesign and analysis.(f) Branches. Branch connections are used to form the branches of the piping. The most com-mon rull-size branch connection is the forged welding tee, which is made according to ASME B16.9[15] and other standards. Generally,the welding tee is quiteexpensive, but provides thesmoothestflow passages and least stress intensification factor among all branch connections. Besides the weld-ing tee, the most economical and readily available branch connection is the un-reinforced fabricatedtee. Generally called a stub-in connection, this type of connection is made simply by cuttinga. holeon the run pipe and welding the branch pipe to it. A stub-in connection is cheap and easy to make,but can handleonly about one-half of the pressurethat the pipe can. It also hasa very highstressintensificationfactor. Toimproveboththepressureresistingcapabilityandstressintensification,proper reinforcement is required. When designed properly, the reinforced fabricated branch connec-tion can take the same pressure as a run pipe can, and also substantially reduce stress intensificationfrom the un-reinforced branch connection. Other branch connections include extruded tee, integrallyreinforced weld-on, contoured weld-on, and half coupling. Each of these branch connections has itspressure design requirements and stress intensification that need to be considered in the design andanalysis. See Table 3.1 for the flexibility factors and stress intensification factors ofpiping componentsand connections. See also Appendix B for layout dimension of butt-welding fittings.(g) Valves. The piping also consists of many types of valves. For valves, we have to know thetype, end-to-end length, and weight, for inclusion in the analysis. The valve itself is generally approxi-mated to an equivalent pipe of the same length with three times the stiffness of the connecting pipe.For valves with aheavy operator, such as motor-operated ones, theoperator weight and off-centerlocation has to be included in the design analysis. This is especially important in analyzing dynamiceffects such as earthquakes. For safety-reliefvalves, we also have to consider the dynamic effect due tothe sudden discharge of the fluid when the safety valve pops open. Valves share the same classificationas flanges. They have the same pressure-temperature rating as flanges for a given class and material.(h) Flexible joints. Toincreasetheflexibility of thesystem, the piping may also include sometypesof flexibleconnections. Theoneshown in thefigureisatied bellowexpansion joint. Otherflexible connections that might be used are hinged bellow joints, gimbaled bellow joints, ball joints,flexible couplings, and flexible hoses. Flexible connections are generally considered engineered items,which are not simply picked out of a catalog. Their selection involves some engineering and opera-tional considerations.(i) Terminal connections. In addition to piping components, we also have to consider the effectof the piping on connecting equipment. Because most units of the connecting equipment are weaker;than the piping itself, the piping load satisfactory to piping may not be tolerable to the equipment atall. In fact, to engineer the piping reaction to the acceptable level of the equipment is the most difficulttask for a piping stress engineer. The items listed in the following are some of the most common typesof equipment connected to a piping system.(j) Shell connections. A piping system is often connected to the shell of a pressure vessel, tank,drum, or heat exchanger. Although these shell connections are considered fixed interfaces, they do---:i..'j !--IjI-1Introduction 9impose some movements to the piping at the connection due to expansion of the shell. The bulgingof the shell, as in a low-type tank connection, may also impose some rotation to the piping. The ac-ceptability of the piping load is determined by the local shell stress produced. In some cases, manu-facturers of equipment, such as a heat exchanger, may specify the allowable forces and moments forthe connection on its equipment. Normally, piping stress engineers will work with vessel engineersto decide if the pipi:p.g load is acceptable. However, to expedite the validation process, piping stressengineers will first calculate the local vessel stress, which is integrated in the piping stress analysis, ofthe connection before giving the piping load to vessel engineers for approval. When calculating thepiping load at a shell connection, there is often a disagreement between the vessel engineer and thepiping engineer about the flexibility ofthe shell. Some vessel engineers insist that the flexibility oftheshell should not be included in the piping analysis, so the resulting piping load is artificially increasedto protect the vessel. This is actually very short sighted and may result in a very shaky design of thepiping. Detailed discussions of shellconnectionsaregiven inthechapter dealing withstationaryinterfaces.(k) Tubebundleheader connections. Anothertypeof stationary interfaceistheheadercon-nected to tube bundles. Steam boilers, fired heaters, and air coolers are some examples. This type ofinterface generally has a given set of allowable forces and moments, which are provided either by themanufacturer or by an applicable industry standard. Because this type of interface generally does nothave a precise boundary point, the piping may have to include all or part of the tube bundles in theanalysis. Detailed discussions on this type ofinterface are given in the chapter dealing with stationaryinterfaces.(1) Rotating equipment connections. The biggest challenge to pipe stress engineers is the pipingconnected to rotating equipment such as pumps, compressors, and turbines. Because of its extremelylow permissible force and moment, a rotating equipment piping system is normally stress-engineeredby an experienced piping mechanical engineer. Yet, it still requires considerably more effort to accom-plish the stress engineering than to accomplish the same for a system that is not connected to rotatingequipment. The problem is that rotating equipment cannot endure even a very slight deformation, orelse risk the consequence of shaft misalignment. To maintain smooth operation of the rotating equip-ment,theshaft needsto bekept in perfect alignment withoutcausing binding at thebearings andinterference of the internal parts. Therefore, the acceptance criterion of rotating equipment piping isthe strain rather than the stress. In the case of rotating equipment piping, the allowable strain is onlyequivalent to about one-fifth of the allowable piping stress criterion for a medium-size piping. The al-lowable is even smaller for larger pipes. See Chapter 9 for details.1.3 MODES OF FAILUREThe main purpose of the piping mechanical work is to prevent piping failure. Therefore, it is im-portant to find out how the pipe fails. The pipe can fail in many different modes with many differentmechanisms. Some of the common modes of failure are discussed in the following subsections.1.3.1 Static Stress RuptureThe pipe will fail when it is stressed to beyond its strength, as measured by testing. That is the defini-tion ofthe strength of the material. Static stress means no time element is involved. The failure occursas soon as stress reaches the limit. Because the pipe material will not take any stress higher than thislimit, the limit is also called ultimate strength of the material. Other modes of failure may take con-siderably less stress than the ultimate strength. As the pipe material is statically stressed, it begins todeform (Fig. 1.3). The area under the curve represents the energy required for the failure. This area isalso referred to as the energy absorbing capacity, or the toughness. In general, static rupture can befurther classified into two categories: ductile rupture and brittle rupture.Ductile Fracture10 Chapter 1tBrittle Fracture/ . . . . . . . . . . . . . . . . . . .D J ~:::Flt: -: -:..'Deformation,!:1 ~FIG. 1.3STATIC FRACTUREDuctilerupture. IIiductilerupture, astheload increases, thematerial firstyieldsproducing aconsiderable plastic deformation and failsafter going through fairly largeamount of elongation orcontraction. The material involved is called ductile material. A ductile piping material elongates about25% (one-fourth ofthe original length) before the failure. Therefore, it has a very large energy absorb-ing capacity. The energy absorbing capacity has little effect on the slowly applied static loads, but isvery important in resisting impact loads. Without this large energy absorbing capability, a very smallimpact can translate to a very high damaging stress.Ductility is one of the most important considerations for the material used in piping. In addition toits ability to mitigate the impact load, ductility also redistributes the high concentrated stresses into amore favorablestress distribution through yielding. We often hear the saying that a piping system isvery forgiving. This forgiveness of the piping system is mainly attributed to ductility.The failureof the material mostly starts at thenotches formed by defect or shape discontinuity.Under a given load, the stress produced at the notches is considerably higher than at the bulk of thematerial. The yielding of ductile material will smooth out the stress concentration near the notch, andthus increase the toughness of the material.'Figure 1.4 shows the stress near a circular hole in a plate. The stresses are evenly distributed at thearea remote from the hole. As the area gets closer to the hole, the stress field also changes to followthe passage formed by the material. At the horizontal diametrical direction of the hole, the net areais smallest and the stress flow squeezes near the hole, producing a peak stress much higher than theaverage stress of the net area represented bybod. As the load is increased, the peak stress eventuallyreaches the yield strength ofthe material. At this point, the yielded portion can no longer take any ad-ditional stress, prompting the nearby un-yielded portion to pick up more stress. This essentially limitsthe highest stress to the yield strength until all stresses at the neck reach the yield strength. The yield-ing prevents the stress from reaching the elastic peak stress as shown by the dotted lines. Therefore,ductile material is substantially less notch-sensitive than the non-yielding material.Because ofyielding, a ductile piping systemwill also shift the load to other parts ofthe system, if thestress at a certain portion ofthe systemreaches the yield point. This load-shifting capability effectively. uses the whole systemto resist the load instead of depending on certain local locations. This increases\ the system reliability and the tolerance to abnormalities.It is important to remember that the ductility of the pipe is the presumption of many design rulesand specifications. Without ductility, many ofthe design calculations are meaningless.Brittle rupture. If thepipedoesnot yieldor does not produce plastic deformation, theenergyabsorbing capacity, represented by the area enclosed by the force-deformation curve, is very small asshown in Fig. 1.3. The material that fails without any yielding is called brittle material. JbYieldStressIntroduction 11_J(a) Elastic (b) PlasticFIG. 1.4EFFECT OF NOTCHESBrittle failureoften occurs unexpectedly and suddenly. This is due to two main factors. First, be-cause ofthe low energy absorbing capability, a slight impact translates to a very high stress thus caus-ing the failure. For instance, we can break a glass cup rather easily just by hitting the edge of the cupon a hard object with asmall impact. Another point is that without yielding, the material is unableto relieve the high stress concentration at thecrevices of a notch. Many of us haveobserved how acrack develops on the car windshield. A crack is produced when a tiny hard object, such as a piece ofairborne pebble, hits the glass. The crack forms sharp star-like fissures, whichjust keep growing undernormal driving conditions.Brittleness is the inherent nature of some materials such as glass and gray cast iron. These materials,when used, require strict control of the stress and the nature of the loading. They cannot be used inthe environment with either thermal or mechanical shock loading. The use ofbrittle materials requiresextreme caution and care. A glass cup, for instance, can easily break in a minor incident, such as whenit is dropped on the floor.Somematerials, ontheother hand, become brittle because of temperaturechange. Most pipingmaterials loss their ductility as the temperature drops below acertain limit. For instance, most car-bon steels are susceptible to brittle failure at temperatures lower then -20F (-29C), whereas othermaterials (e.g., austenitic stainless steel, aluminum, copper, and brass) do not become brittle at tem-peratures as low as -425F (-254C). Some materials can also become brittle at high temperature dueto metallurgical change. Mild carbon steel may loss its ductility at temperatures above 800F (427C)due to its susceptibility to graphite formation. For this discussion, however, we are mainly concernedwith the loss of ductility due to cold temperature.All the design philosophy and approaches we commonly use are based on the assumption that thematerial is ductile. Except for inherently brittle materials, we have to be very careful when using a ma-terial at a temperature range that is conducive to brittle failure. Most piping codes list -20F (-29C)as the minimum temperature for which the material is normally suitable without impact testing otherthan required by thematerialspecification. Nevertheless, duetoitswider temperatureapplicationrange, the B31.3 process piping code [17]sets very detailed rules and requirements on low tempera-ture application of the materials.B31.3sets the minimum temperature for each material listed in the allowable stress tables. Again,this minimum temperature for each material is the lowest temperature that the material can withstand12 Chapter 1without impact testing other than that required by the material specification. However, there are someexceptions. For instance, for austenitic stainless materials with carbon content higher than 0.1 %ornotsolution heat treated,impact testing is required fora design temperaturelower than -20F al-though the listed minimum temperature is -425F. For the most commonly used carbon steels, B31.3uses four curves (A, B, C, and D) to determine the impact testing requirements. We will use curve B,which is applicable to mild carbon steels such as A-53, Al06, and A-135, as an example to explain therules and requirements. Curve B is re-plotted in Fig. 1.5. The region above the curve does not requireimpact testing other than that required by the material specification. The figure shows that the higherthe wall thickness, the higher the temperature under which the impact testing is required. This thick-ness correlation is mainly because a thicker wall creates a higher uneven stress distribution and higherprobability of containing bigger size defects. Impact testing, as stipulated by the code, indicates usingthefullallowablestress asgiven in theallowable stress table. Without impact testing, thematerialin question can still be used but at a reduced allowable stress. The rate of allowable stress reductionstartsat 1%per eachFahrenheitdegree(5/9Celsiusdegree)of temperaturelower thanthenon-impact test temperature. The reduction rate maintains the same level for the initial 40F temperaturedifference, and then slows down asymptotically to a maximum total reduction of70% at a temperaturedifference of 217F (120C). In Fig. 1.5, a pipe with a1.5-in. (38.1mm) wall thickness and operat-ing at 32F (OF) would require impact testing in order to use the full allowable stress. At 1.5-in. wallthickness, the temperature not requiring an impact testing is 51F. The operating temperature, in thiscase, is 51-32 =19F lower than the non-impact testing temperature. Therefore, if impact testing isnot performed, the allowable stress has to be reduced by 19%.The loss of ductility is a very serious concern for a piping system. Therefore, the minimum accept-able temperature, the impact test requirements, and the stress reduction provisions as outlined by thecode should be followed closely.1.3.2 Fatigue FailureThe pipe can fail under a stress lower than the ultimate strength ofthe material ifthe stress is cyclic.Bending a paper clip back and forth repeatedly can easily reproduce this effect. This type of failure isdue to fatigue of the material. Fatigue failure is attributed to the combination of the stress amplitude'j'I, jNominal Wall Thickness, mm ~ I1000 1,0 20~ O40 50 6079-40803020>1. 53' Compressive stresses are considered negative stresses in the above comparative quantity. Thisprovides the basis for the comparative damaging effect ofthe stress systems shown in Fig. 2.11.In piping stress analysis, wedeal mostly with two-dimensional stress fieldswith thestress at thethird dimension either zero or insignificant. In this case, it is simpler to calculate the stress intensitydirectlyfromthegeneralstressfieldgiveninFig. 2.12, without calculatingtheprincipal stresses.Again, from Eq. (2.18) we haveSi =2'tinax =2V(5;,;J+ ,,1 =+ Assuming that x-axis is in the axial direction of the pipe, then 5x is the longitudinal stress resultingmainly from bending moments and internal pressure, 5y is the hoop stress mainly from internal pres-sure, and'l'xy ismainly due to the torsion moment. Stresses due to direct axial and shear forcesaregenerally insignificant and are often neglected.There are a couple of things to note in applying Eq. (2.19). One is the nature of the bending stress,which always presents both tension and compression at a given cross-section. The other is the planethat includesthediametrical direction(throughthethicknessdirection) alsohastobeincluded.The normal stress in this diametrical direction is either zeroor equal to the acting pressure, whichis compressive.IL. .J,_J __Jn":';', jStrength of Materials Basics 53(2.21)(2.20)2.7 BASIC BEAM FORMULASThe principal stresses 81 and 82 can be found using Eqs. (2.17a) and (2.17b), respectively, for two-dimensional stress systems. Substituting Eqs. (2.17a) and (2.17b) to the above, the effective stress fortwo-dimensional stress field becomesA piping system is essentially a group of beams connected together to form the shape required fortransporting fluids from one point to another. Therefore, the behavior ofthe beam is the basic compo-nent ofthe pipe stress analysis. Table 2.1 shows some basic beamformulas that stress engineers are allfamiliar with. The following are a few important notes derived from these basic formulas:(1) With a given loading, the bending moment is proportional to the length of the beam, but the. displacement is proportional to the cube ofthe length. A slight increase in length creates a large... increase in displacement, which translates to a large increase in flexibility.(2) For a given configuration, the displacement is inversely proportional to EI. This EI is generallyreferred to as the stiffness coefficient of the cross-section. If a simulation is required for a non-standard cross-section such as refractory lined or concrete lined pipe, the EI of the simulatingpipe has to match the combined EI ofthe pipes being simulated.By comparing Eq. (2.21) with Eq. (2.19), it is difficult to assess whether the effective stress is largeror smaller than thestress intensity. Inother words, itisnot possibletosay which theory ismoreconservative. However,one thing is clear: when either one of the normal stresses is zero, the stressintensity is somewhat larger than the effective stress. The degree of difference depends on the ratio ofthe normal stress and the shear stress. At the extreme condition when both normal stresses are zero,the stress intensity is bigger than the effective stress by a factor of.y4/3 =1.155. However, in mostpractical piping applications, both stress intensity and effective stress can be considered equivalent toeach other.(S1 - sd+(S2 - s3i +(S1 - SJ)2=2 ~ pwhere8yp isthe yield point tensilestress in the tension test. Tomakeastress directly comparableto the yield point stress and other data obtained in the tension test, the effective stress is coined anddefined asFor a two-dimensional stress field, as is commonly the case in practical piping system, 83 can be setto zero, and we have2.6.5 Effective Stress (von Mises Stress)The maximum distortion energy failure theory agrees with the nature of ductile materials the most.This theory is very popular in the European piping community. Based on distortion energy theory, thecondition for yielding is [7]--------------------54 Chapter 2TABLE 2.1BASIC BEAM FORMULAS(a) Simple Beam- Uniform LoadW=wLR~ l ! ! I " I " I I I ~ I ! I " I I I " I ~ RwLR=-.- ,2SwL411 ---max- 384EI'wI!Mmax =--,8(at Center)(at Center)(b) Simple Beam- Concentrated LoadiFF FLRt:R=-, Mmax =-, (at Center)lR2 4LFL311 -- (at Center)max- 48EI'(c) Fixed Beam - Uniform LoadwL4I1max=384EI' (at Center)FL3/). - -- (at Center)max- 192EI'(d) Fixed Beam - Concentrated LoadwLR=-2 'FR=-,2wL2Mmax =--, (atEnds)12FLMmax= -, (at Ends and Center)8l J" 1(3) Thedisplacementformulasincludeonly theterm EI. Thetermsinvolvingshear modulus Gand cross-section area A, associated with shear deformation, arenot included. The formulasare good for practical lengths of beams. However, they are not accurate for short beams whoselength is shorter than ten times the cross-sectional dimension of the beam. The formulas usedby most computer programs include theshear deformation, thus makinga short beam moreflexible than that calculated by the formulas in Table 2.1. This is one ofthe potential discrepan-cies between a computer result and a hand calculation result. In such cases, the computer resultis more accurate.(4) Items (a)and (c) can be used todetermine thestress due to weight under normal supportingspans. Because the actual piping in the plant is supported somewhere between simple supportand fixed support [8], the following average formulas are generally used, instead, for evaluatingweight supports.2.8 ANALYSIS OF PIPING ASSEMBLYFIG. 2.14DIFFERENT TYPES OF CANTILEVER BEAMSl2EIF=-3- 11LFL 6EIM=-=-112 L2I FStrength of Materials Basics 55(b) GuidedCantilever Beame=0,FL311=--,l2EIe>F3EIF=-311;L3EIM=FL=211;L(a) Regular Cantilever BeamR=FFL311=--,3EIFL2e=--,2EIM(jr--........----rThe analysis of a piping assembly is very complicated and is accomplished mostly by using com-puter programs. In general, an engineer is not required to have knowledge of the computing methodsimplemented in the computer programs in order to use the program. However, some common senseregarding general analytical approaches can help analysts to better understand the procedure and bet-ter interpret the results.A pipingsystemconsistsof manycomponentslaidout inall directions. Analysisof the pipingsystem is normally idealized into a combination of straight pipe segments and pipe bend elements.In other words, the analysis is performed using a finite element method consisting of two types of ele-ments: the straight pipe and the pipe bend.2.7.1 Guided CantileverIn piping flexibility analysis, the guided cantilever method is one of the often-mentioned approxi-mateapproaches[8]. Figure2.14showstwodifferent typesof cantilever beamsthat may beusedin piping stress analysis. The regular cantilever beam shown in Figure 2.l4(a) is actually half of thesimple beam subject to a concentrated load as given in Table 2.1(b). The fixed end is equivalent tothe mid-point of thesimple beam. The guidedcantilever beam shown in Figure 2.14(b)isactuallyhalf of the fixed beam subject to a concentrated load as given in Table 2.1(d). These beam models areoften used to calculate the forces and moments, approximately, in a given length of pipe subject to adisplacement resulting from thermal expansion.The regular cantilever model requires an accompanying rotation, e, while absorbing the displace-ment, A. This rotational relief reduces theforcesand moments generated by a given displacement.The problem is that an actual piping system does not freely offer this rotational relief. It does offersome relief, but much less than the amount shown in Fig. 2.l4(a). A more realistic and conservativeapproach is to assume that no rotation is taking place, as shown in Fig. 2.l4(b). This is the so-calledguided cantilever approach. For a given expansion or displacement, the force created by the guidedcantilever is four times as great as that by a regular cantilever, and the moment, thus the stress is twiceas large as that by a regular cantilever.- - - - - - - - - ~-- - - ~ - - - ~ - - - - - ----- ~ - ~--------56 Chapter 22.8.1 Finite ElementThe body of astructure generally hasstresses andstrains that varycontinuously throughout thebody. It is very difficult, if not impossible, to calculate exactly these stresses and strains. However, toobtain a practical result, the body can be divided into many sub-bodies, each with a finite size. Eachsmall body is considered to have a predictable stress and strain distribution over it. These small bod-ies are calied finite elements. In piping analysis, these bodies are actually fairly large compared to thegeneral sense of a finite element. Here, we use straight pipe and curved pipe, two types of beam ele-ments. Each element has two nodes, NI and N2, as shown in Fig. 2.15(a). NIis the beginning node,and N2 is the ending node.The characteristics of the element are expressed in the local coordinates aligned with the elementgeometry. For a straight pipe element, the local x-axis is always in the axial direction pointing from thebeginning node toward the ending node. The local y-axis and z-axis are perpendicular to each otherin the lateral directions. For a curved pipe element, the local axis convention differsslightly amongdifferent investigators. One popular convention, asshown in Fig. 2.15(a), assigns the local x-axis asconnecting the two nodes and pointing from the beginning node to the ending node. The local y-axisis perpendicular to the x-axis and pointing from the mid-cord point toward the bend tangent intersec-tion point.In a general three-dimensional environment, each node has six degrees offreedom, three in transla-tion and three in rotation. At each element, the forces and displacements have a fixed relationship inlocal coordinates and are designated with a prime (') notation. Each node of an element is associatedwith three displacements, Dx', Dy', Dz',and three rotations, R;,Ry', R;.Correspondingly, each nodeis also associated with three forces, F;, Fy', F;, and three moments, M;, My', M;. In general, the termdisplacement is used tocover both displacement and rotation. By the same token, the term force is, Jxzx'y' Nl~x',-Straightz'~ -) y' r.--,8y(a) Local Coordinates (b) Global Coordinates & AssemblyFIG. 2.15COORDINATE SYSTEMSStrength of Materials Basics 572.8.2 Data Points and Node Pointsused to cover both forces and moments. The finite element method is built under the premise that oneach element there is a relationship between these forces and displacements. That is, for each elementwe have the relation in local coordinates as(2.22) . {Fl =[K1{DlBefore a piping system is analyzed,every element in the system has to be identified. Customarily,theseelementsareidentified with point numbers. Thesenumbersserveasthecommunication ad-dresses just like real-life house numbers. Up to three different point numbers may be associated withan analysis. Take,for example, thesimple assembly shown in Fig.2.15(b).The first set of numbersneeded is the set of data point numbers used to describe the geometry ofthe system. The following arethe locations that need to be assigned with data point numbers: ..(1) Terminal points such as anchors, free ends, vessel, and tank connections, etc. Points 10 and 25belong to this category.(2) Bend tangent intersection points (working points). Points 15 and 20 belong to this category.(3) Branch intersection points.(4) Key flange face points.(5) Restraint and loading points.(6) Other points where the response of the system is of interest.The above data points are required for a precise description ofthe system. Fromthese essential datapoints, the computer program may also generate some other data points required for the analysis. Atbend 15, for instance, the required input data point is the tangent intersection point 15, but the pointsrequired for the analysis are the end points15a and 15b of the bend. Point15 is not located at anyphysical part ofthe piping system; therefore, it is not used in the analysis.Fromthe information provided by the data points, the computer program will re-number the entirepiping system toassign analysis node numbers. The analysis node numbers have to beconsecutivestarting from 1. They are generated following the input sequence of the data. In our example system,we have nodes 1 =10,2 =15a, 3 =15b, and so forth. In an actual analysis, these sequentially gener-ated node numbers may be once again re-numbered to achieve the so-called bandwidth optimization.The bandwidth is essentially the difference between the node number of a given point and the nodenumber ofits adjacent connecting point or points. For a multi-branched system, the bandwidth greatly on the re-numbering. The result of this re-numbering is another set of node numbers, whichare called re-numbered nodenumbers. The last re-numbering isintended toachieve the minimumbandwidth possible. Smaller bandwidths require less data storage and also less computing time. Thewhole numbering effort may appear very confusing, but it should not be a concern ofthe analyst. Allanalysis results are given in reference to the original data point numbers assigned.where {F'} is the force vector representing 12 forces and moments at both nodes. That is, {F'} =FYI' Fi\, M;\,My1 , Mi\, F;z, Fyz, Fiz, M;z' Myz, Miz}T. Subscript 1 represents node Nl and subscript 2represents node N2. Superscript T denotes transpose, meaning a column vector. {D'} is the displace-ment vector representing 12 displacements and rotations at both nodes. That is {D'} ={D;\, Dy\, D'z\,R;\,RYI> Ri\, D;z' Dyz, Diz' R;z, Ryz, Riz}T. [K'] is a 12 x 12 symmetric stiffness matrix. The exact termsin [K']are too complex to be included here. Interested readers may find them in related publications[9-12]. Thereareseveral different formsof theseterms, usingsomewhat different localcoordinateconventions.58 Chapter 22.8.3 Piping AssemblyTo mathematically assemble the piping system, all individual elements