Matrix Analysis of Framed Structures Matrix Analysis of Framed Structures Third Edition William Weaver, Jr. Professor Emeritus of Structural Engineering, Stanford University James M. Gere Professor Emeritus of Structural Engineering, Stanford University ~ VANNOSTRAND REINHOLD ~ _______ New York Copyright1990byVanNostrandReinhold Softcover reprint of the hardcover 1st edition1990 Libraryof CongressCatalogCardNumber 89-21458 ISBN-13:978-1-4684-7489-3 Allrightsreserved.Certainportionsof thiswork1965,1980by VanNostrandReinhold.Nopartof thisworkcoveredbythecopyright hereonmaybereproducedor usedinanyformor byanymeans-graphic, electronic,or mechanical,includingphotocopying,recording,taping, or informationstorageandretrievalsystems-withoutwrittenpermission of thepublisher. VanNostrandReinhold 115FifthAvenue NewYork,NewYork10003 VanNostrandReinholdInternationalCompanyLimited 11NewFetter Lane LondonEC4P 4EE,England VanNostrandReinhold 480 La TrobeStreet Melbourne,Victoria3000,Australia NelsonCanada 1120Birchmount Road Scarborough,OntarioM1K 5G4,Canada 1615141312111098765432 Libraryof Congress Cataloging-in-Publication Data Weaver,William,1929-Matrixanalysisof framedstructures/William Weaver,Jr.,James M.Gere.-3rd ed. p.cm. Includesbibliographicalreferences. ISBN-13:978-1-4684-7489-3e-ISBN-13:978-1-4684-7487-9 DOl:10.1007/978-1-4684-7487-9 1.Structural analysis(Engineering) JamesM.II.Title. T A645. W361990 624.1 '773-dc20 2.Matrices.I.Gere, 89-21458 C1P Preface Matrixanalysisofstructuresisavitalsubjecttoeverystructuralanalyst, whether workinginaero-astro,civil,or mechanicalengineering.It provides acomprehensiveapproachtotheanalysisofawidevarietyofstructural types,and therefore offers a major advantage over traditionalm t h o ~ which often differ for each typeof structure.The matrix approach also provides an efficientmeansofdescribingvariousstepsintheanalysisandiseasily programmedfordigitalcomputers.Useofmatricesisnaturalwhen performingcalculationswithadigitalcomputer,becausematricespermit large groups of numbers to be manipulated in a simple and effective manner. This book,nowinitsthird edition,waswritten for both college students andengineersinindustry.It servesasatextbookforcoursesateitherthe senior or first-year graduate level,and it also provides a permanent reference for practicing engineers.The book explains both the theoryand the practical implementation of matrix methods of structural analysis.Emphasis isplaced ondevelopingaphysicalunderstandingof thetheoryandtheabilitytouse computer programsfor performingstructuralcalculations. Inpreparing thisnewedition,we havetriedtomaintain thestrengthsof theearliereditionswhilealsoaddingnewmaterialtoallowpersonal computers to be used in the solution of problems.The direct stiffness method ispresentedindetailbecauseitisthebestandmostgeneralapproachfor theanalysisof structuresbydigitalcomputation.Theflexibilitymethodis includedasasupplementarymethod,partlyforcompletenessandpartly becauseitoftenisnecessarytoobtainstiffnessesandfixed-endactionsby flexibilitytechniques. Throughoutthebook,newexamplesandproblemshavebeenaddedto aidinteachingthesubject.Twonewtopics,repeatedsubstructuresandthe omissionof axialstrainsinframes,arenowincludedinChapter 6.Anew chapter,"Finite-ElementMethodforFramedStructures,"hasbeenadded at the end of the text toshow how the analysis of framed structures fitswithin thescopeofthemoregeneralfinite-elementmethod.Thischapteralso providesanintroductiontothefinite-elementmethod,usingonlyone-dimensionalelementstomodeltheslender membersof framedstructures. Prerequisitesforthestudyof matrixanalysisof structuresarestatics, mechanicsof materials,algebra,andintroductorycalculus.Inaddition,a previouscourseinelementarystructuralanalysisiscertainlybeneficial, althoughnotessential.Elementarymatrixalgebraisusedthroughoutthe v viPreface book,and the reader must be familiar with basicmatrix operations,suchas addition,multiplication,andinversion.Becausethesetopicsarenotdiffi-cult,thereadercanacquirethenecessaryknowledgeof matricesthrough self-studyduringa period of onlytwoor threeweeks. Computer programs for the six basic types of framed structures are given in Chapter 5 in the form of FORTRAN-oriented flowcharts. These programs are available on a diskette that also contains the data for all examples shown in that chapter.The diskettemaybe purchasedfromDr.PaulR.Johnston, Manager,StructuralAnalysis,FailureAnalysisAssociates,149Common-wealthDrive,MenloPark,CA94025[phone(415)688-7210].Forthis purpose,a tear-out order formisincluded at theback of the book. Thefirstchapter of thebook coversthefundamentalconceptsof struc-turalanalysisthatareneededfortheremainingchapters.Thosewhohave previouslystudiedstructuraltheorywillfindthatthismaterialismostly review.However,anyoneencounteringthissubjectforthefirsttimewill needtobecome thoroughlyfamiliar with thebasictopicspresentedhere. The flexibilityandstiffnessmethods are introducedin Chapters 2 and3, respectively.Eachofthesechaptersexplainsthetheoryindetail,with examplesandproblems,andconcludeswithasectiononformalizingthe method (Sections 2.7 and 3.6). Although these last sections show the general mathematicalapproach,theyareindirectandarenotneededforimple-mentingthemethods.In Chapter 4thedirectstiffnessmethodisdeveloped furtherinacomputer-orientedmanner,aspreparationforprogramming. TheninChapter5thestiffnessmethodisappliedinFORTRAN-oriented flowchartsof programsfortheanalysisof thesixbasictypesof framed structures. Because the emphasis in the first five chapters is on fundamental concepts, the treatment of manyspecial topicsispostponedtoChapter 6.Included in that chapter are such matters as symmetric structures, nonprismatic members, elasticconnections,andsoon.Thesetopicscanbeconsideredasmodifi-cationsof thebasicproceduresdescribedinearlierchapters.Asalready explained,a new Chapter 7 describeshowthe finite-elementmethod can be applied toframedstructures. Problemsforhandsolutionaregivenattheendsof thechapters,and theyaregenerallyplacedinorderof increasingdifficulty.Newproblems have been added to allproblem sets,and those of Chapter 6 are completely new.The examples and problems in Chapters1,2,3, and 6 (and the Appen-dixes)areinliteralform.Numericalexamplesandproblemsof Chapters4 and 5 arein bothUSand SI units. Referencesforfurtherstudy,listsofnotations,fiveappendixes,and answerstoproblemsareattheendof thebook.Theappendixescontain tables of useful information,a description of the unit-load method for calcu-latingdisplacements(AppendixA),andcomputerroutinesforsolving equations(AppendixD). Prefacevii TheauthorswishtothankthegraduatestudentsatStanfordandthe teachers at other colleges and universitieswho contributed ideas to this book. SpecialappreciationisduePaulR.Johnston,whomodifiedtheprograms forusewithpersonalcomputersandproducednewoutputsforChapter5. JeffreyE.Jonessolvedandorganizedtheproblemsetsinanexcellent manner,usingatexteditor onhisowncomputer.Finally,JudithC.Clark did an outstanding job of word processing for parts of the revised manuscript. WilliamWeaver,Jr. JamesM.Gere Contents Preface/v 1BasicConcepts of Structural Analysis /1 1. 1Introduction /1 1.2Typesof FramedStructures/1 1. 3DefonnationsinFramedStructures /4 1.4ActionsandDisplacements / 7 1. 5Equilibrium /13 1.6Compatibility /14 1.7StaticandKinematicIndetenninacy /14 1. 8StructuralMobilities /21 1.9Principleof Superposition/22 1.10ActionandDisplacementEquations /25 1.11FlexibilityandStiffnessMatrices /31 1.12Equivalent JointLoads/36 1.13EnergyConcepts /38 1.14VirtualWork/44 References /48 Problems /48 2Fundamentals of the Flexibility Method / 55 2.1Introduction / 55 2.2FlexibilityMethod/55 2.3Examples /63 2.4TemperatureChanges,Prestrains,andSupportDisplacements /75 2.5JointDisplacements,Member End-Actions,andSupport Reactions / 80 2.6Flexibilitiesof PrismaticMembers/86 2.7Fonnalization of theFlexibilityMethod /92 Problems /107 3Fundamentals of the Stiffness Method / 117 3.1Introduction /117 3.2StiffnessMethod /117 3.3Examples /129 ix x Contents 3.4Temperature Changes,PrestrainsandSupportDisplacements/149 3.5Stiffnessof PrismaticMembers/153 3.6Formalization of theStiffnessMethod /157 Problems /168 4Computer-Oriented Direct StiffnessMethod/181 4.1Introduction /181 4.2Direct StiffnessMethod /181 4.3CompleteMember StiffnessMatrices /184 4.4Formationof JointStiffnessMatrix /192 4.5Formationof LoadVector /193 4.6Rearrangement of StiffnessandLoadArrays /196 4.7Calculation of Results/197 4.8Analysisof ContinuousBeams /199 4.9Example/209 4.10PlaneTrussMember Stiffnesses /215 4.11Analysis of PlaneTrusses/219 4.12Example /228 4.13Rotationof AxesinTwoDimensions /233 4.14Application toPlaneTrussMembers/235 4.15Rotation of AxesinThreeDimensions/239 4.16PlaneFrameMember Stiffnesses/24I 4.17Analysisof PlaneFrames /245 4.18Example/250 4.19GridMember Stiffnesses /254 4.20Analysis of Grids/258 4.21SpaceTrussMember Stiffnesses/26I 4.22Selection of SpaceTrussMember Axes/263 4.23Analysisof SpaceTrusses/267 4.24SpaceFrameMember Stiffnesses /270 4.25Analysisof SpaceFrames/277 Problems /281 5Computer Programs forFramed Structures/291 5. 1Introduction /291 5.2FORTRANProgramming andFlowCharts/291 5.3ProgramNotation/298 5.4Preparation of Data / 300 5.5Description of Programs/306 5.6ContinuousBeamProgram /309 5.7PlaneTrussProgram/325 5.8PlaneFrameProgram/330 5.9GridProgram / 340 5.10SpaceTrussProgram/345 Contents 5.11SpaceFrameProgram/353 5.12CombinedProgramforFramedStructures/368 References /372 6Additional Topics for the Stiffness Method/373 6.1Introduction/373 6.2Rectangular Framing/374 6.3Symmetric andRepeatedStructures /378 6.4LoadsBetween Joints/383 6.5AutomaticDeadLoadAnalysis/386 6.6TemperatureChangesandPrestrains/387 6.7Support Displacements /388 6.8ObliqueSupports/389 6.9ElasticSupports /391 6.10TranslationofAxes/394 6.11Member StiffnessesandFixed-EndActionsfromFlexibilities /396 6.12NonprismaticMembers /402 6. 13CurvedMembers /409 6. 14ReleasesinMembers /414 6. 15ElasticConnections /419 6.16ShearingDeformations/422 6.17OffsetConnections /424 6.18Axial-FlexuralInteractions /428 6.19AxialConstraintsinFrames /432 References /441 Problems /442 7Finite-Element Method forFramed Structures /447 7. 1Introduction /447 7.2StressesandStrainsinContinua /448 7.3Virtual-WorkBasisof Finite-ElementMethod/449 7.4One-DimensionalElements /453 7.5Application toFramedStructures /464 References /467 GeneralReferences /468 Notation /470 AppendixA.Displacementsof FramedStructures /476 A.lStressesandDeformationsinSlender Members /476 A.2DisplacementsbytheUnit-LoadMethod/485 A.3Displacementsof Beams /493 A.4Integralsof ProductsforComputingDisplacements /493 References /495 xi xii Appendix B.End-ActionsforRestrainedMembers/498 AppendixC.Properties of Sections /505 AppendixD.Computer RoutinesforSolvingEquations/506 D.IFactorizationMethodforSymmetricMatrices/506 D.2SubprogramFACTOR/512 D.3SubprogramSOLVER/514 D.4SubprogramBANFAC/514 D.5SubprogramBANSOL/518 References /518 AppendixE.SolutionwithoutRearrangement / 520 AnswerstoProblems /523 Index/541 Order FormforDiskette /547 Contents