有限单元法——基本原理及其在土木、水利、机械和航天工程中的应用

Preface 
The.niteelementmethod(FEM)issopowerfulthatmanyverycomplicatedengineer-ing problems can be solved by it. This book is primarily written for engineers. It intro-duces the basic principles and applications of FEM. It may also be used as textbook inuniversitiesandcolleges.
The .rst purpose of this book is to make an easy read for engineers, so the physicalideas are enhanced and the basic principlesand computing methods are introduced in aneasybutaccurateway.
Thesecondpurposeofthisbookistobeofpracticalvaluetoengineers,sotheformulasthatcanbeusedtoanalyzeproblemsinpracticalengineeringaregivenindetail.
Thus, therearethree distinguishingcharacteristics of this book: (1) itis easyto read; (2)thetheoryandcomputingformulasof.niteelementmethodarecomplete;(3)itisofpracticalusetoreaders,especiallytoengineersandprofessorsandengineeringstudents.
Beforethepublicationofthe.rsteditionofthisbookinChinesein1979,thepredictedreaderswereengineers,butafterpublicationitwasnoticedthatitwaswellreceivednotonly by engineers but also by professors and students in universities and colleges. It is now not only a widely accepted reference book for engineers but also widely used astextbookforprofessorsandstudentsinuniversitiesandcollegesinChina.
According to the Information Center of Chinese Academy of Science The Finite Ele-mentMethod,Theory,andApplications(inChinese)isoneofthemostwell-received10booksinChinainwaterresourcesandhydropowerdomain.
Now the new book in English will be published, I hope it will be well received notonly by engineers working in practical engineering project but also by professors andstudents. 

About the Author 
BofangZhu,anacademicianoftheChineseAcademyofEngineeringandafamoussci-entist of hydraulic structures and solid mechanics in China, was born on October 17, 1928, in Yujiang County, Jiangxi Province. In 1951, he graduated in civil engineeringfromShanghaiJiaoTongUniversityandthenparticipatedinthedesignofthe.rstthree concretedamsinChina(Fozilingdam,Meishandam,andXianghongdiandam).In1957,hewastransferredtotheChinaInstituteofWaterResourcesandHydropowerResearch where he was engaged in the research work of high concrete dams. He was awarded China National Outstanding Young Scientist in 1984 and was elected the academician oftheChineseAcademyofEngineeringin1995.Heisnowtheconsultantofthetechni-calcommitteeoftheMinistryofWaterResourcesofChina,amemberoftheconsultant group of the three very high dams in the world: the Xiaowan dam, the Longtan dam, and the Baihetan dam. He was a member of the eighth and the ninth Chinese Peo-ple’sPoliticalConsultativeConference,theboardchairmanoftheInstituteofComputer ApplicationofChinaCivilEngineeringSociety,andamemberofthestandingcommit-tee of the China Civil Engineering Society and the standing committee of the China HydropowerEngineeringSociety.
He is the founder of the theory of thermal stresses of mass concrete, the shape opti-mizationofarch dams,thesimulatingcomputationofconcretedam,andthetheoryofcreepofmassconcreteintheword. 
He has established a perfect system of the theory of thermal stress and temperaturecontrol of mass concrete, including two basic theorems of creep of nonhomogeneous concrete structures; the law of variation and the methods of computation of the ther-mal stresses of arch dams, gravity dams, docks, sluices, tunnels, and various massive concrete structures; the method of computation of temperature in reservoirs and pipe cooling,thermalstressinbeamsonfoundation,coldwave,heighteningofgravitydam;andthemethodsandcriteriaforcontroloftemperatures.Heproposedtheideaof“long-time thermal insulation as well as comprehensive temperature control” that ended the historyof“noconcretedamwithoutcrack”andsomeconcretedamswithoutcrackthat had been .rst constructed in China in recent years, including the Sanjianghe concretearchdamandthethirdstageofthefamousThreeGorgesconcretegravitydam. 
Heproposedthemathematicalmodelandmethodsofsolutionforshapeoptimizationofarchdams,whichwasrealizedforthe.rsttimeintheworldanduptonowhadbeen applied to more than 100 practical dams, resulting in 10–30% saving of dam concrete,andthee.ciencyofdesignwasraisedagreatdeal. 
He had a series of contributions to the theory and applications of the .nite elementmethod(FEM).
He proposed a lot of new methods for .nite element analysis, including the com-pound element, di.erent time incrementsin di.erent regions, the equivalent equation of heat conduction for pipe cooling, and the implicit method for computing elastocre-epingstressesbyFEM.
HedevelopedthemethodofsimulatingcomputationofhighconcretedamsbyFEM.Allfactors,includingthecourseofconstruction,thevariationofambienttemperatures,theheathydrationofcement,thechangeofmechanicalandthermalpropertieswithageofconcrete,andthepipecooling,precooling,andsurfaceinsulationcanbeconsideredintheanalysisofthestressstate.Ifthetensilestressislargerthantheallowablevalue,themethods of temperature control must be changed until the maximum tensile stress isnotbiggerthantheallowablevalue.Thuscrackswillnotappearinthedam.Experienceshowsthatthisisanimportantcontributionindamtechnology.
He proposed the equivalent stress for FEM and its allowable values that had been
adoptedinthe design speci.cations of archdamsinChina; thustheconditionforsub-
stitutingthetrialloadmethodbyFEMisprovided.
Theinstrumentalmonitoringcangiveonlythedisplacementofsomeparticularpointsbutcannotgivethestress.eldandthecoe.cientofsafetyofconcretedams.Inorderto overcomethisdefect,heproposedanewmethodofnumericalmonitoringbyFEMthat can give thestress .eld andthe coe.cientofsafetyand raisethelevelof safety control ofconcretedamandthathadbeguntobeappliedinpracticalprojectsinChina.
Anewideaforsemimatureageofconcretehasbeenproposedbyhim.Thecrackresis-tanceofconcretemaybepromotedbychangingitssemimatureage.
Avastamountofscienti.cresearchworkshadbeenconductedunderhisdirectionfor aseriesofimportantconcretedamsinChina,suchasThreeGorges,Xiaowan,Longtan,Xiluodu,Sanmenxia,Liujiaxia,Xing’anjiang,andsoon.Fourteenresultsofhisscienti.cresearch were adopted in the design speci.cations of gravity dams, arch dams, docks,andhydraulicconcretestructures.
He has published 10 books: Theory and Applications of the Finite Element Method(1st ed. in 1979, 2nd ed. in 1998, 3rd ed. in 2009), Thermal Stresses and Temperature Control of Mass Concrete (1st ed. in 1999, 2nd ed. in 2012), Thermal Stresses and Tem-perature Control of Hydraulic Concrete Structures (1976), Theory and Applications of Structural Optimization (1984), Design and Research of Arch Dams (2002), Collected WorksonHydraulicStructuresandSolidMechanics(1988),SelectedPapersofAcademi-cianBofangZhu(1997),NewDevelopmentsinTheoryandTechnologyofConcreteDams (2009), and Thermal Stresses and Temperature Control of Mass Concrete (in English)(2014).Hehaspublishedmorethan200scienti.cpapers.
HewasawardedthetitleofChinaNationalOutstandingYoungScientistin1984,theChinaNationalPrizeofNaturalSciencein1982forhisresearchworkinthermalstresses inmassconcrete,theChinaNationalPrizeofScienti.cProgressin1988forhisresearchwork in the optimum design of arch dams, and the China National Prize of Scienti.cProgressin2001forhisresearchworksinsimulatingcomputationandthermalstresses.HebecameICOLD(InternationalCommissiononLargeDams)HonoraryMember. 

本书系统地阐述了有限单元法的基本原理及其在工程问题中的应用，包括弹性力学平面问题和空间问题、薄板、薄壳、厚板、厚壳、弹性稳定、塑性力学、大位移、断裂、动力反应、徐变、岩土力学、混凝土与钢筋混凝土、流体力学、热传导、工程反分析、仿真计算、网络自动生成、误差估计及自适应技术。该书内容丰富、取材新颖，概念清晰，可供土木、水利、机械、航空、力学专业的设计、科研人员使用，并可用做高等院校有关专业的教材。

朱伯芳，中国工程院院士，1951年毕业于上海交通大学土木系后，参加我国首批三座大坝设计，1957年调至中国水科院，1995年当选院士，发表论文210篇出版著作10本，曾获国家自然科学奖1项，国家科技进步奖2项。国际大坝会议奖1项，部奖8项，建立了混凝土温度应力，拱坝优化等新学科。首次把有限元引入混凝土温度应力和水坝仿真计算。

Preface i 
About the Author iii 

1  Introduction to Finite Element Method and Matrix Analysis of Truss 1 
1.1  IntroductiontoFiniteElementMethod 1 
1.2  TrussAnalysisOverview 5 
1.3  Sti.nessMatrixofHorizontalBarElement 8 
1.4  Sti.nessMatrixofInclinedBarElement 10 
1.5  CoordinateTransformation 11 
1.6  NodalEquilibriumEquationandGlobalSti.nessMatrix 14 
1.7  TreatmentofBoundaryConditions 15 Bibliography 23 
2  Plane Problems in Theory of Elasticity 25 
2.1  DiscretizationofContinuousMedium 25 
2.2  DisplacementFunction 28 
2.3  ElementStrain 30 
2.4  InitialStrain 31 
2.5  ElementStress 32 
2.5.1  IsotropicBody:PlaneStress 32 
2.5.2  IsotropicBody:PlaneStrain 33 
2.5.3  AnisotropicBody 34 
2.6  EquivalentNodalForceandElementSti.nessMatrix 35 
2.7  NodalLoads 40 
2.7.1  EquivalentNodalLoadsofDistributedBoundaryForces 41 
2.7.2  NodalLoadsofUniformVolumeForce 41 
2.7.3  NodalLoadsDuetoPotentialofVolumeForce 42 
2.7.4  NodalLoadsCausedbyInitialStrain 43 
2.8  NodalEquilibriumEquationandGlobalSti.nessMatrix 43 
2.9  EstablishtheGlobalSti.nessMatrixbytheCodingMethod 48 
2.10  CalculationExample 51 
2.10.1  StressConcentrationneartheCircularHole 51 
2.10.2  StressAnalysisofIBeamwithaHoleinWeb 51 
2.10.3  StressAnalysisoftheConcreteGravityDam 51 Bibliography 51 
3  Element Analysis 53 
3.1  PrincipleofVirtualDisplacement 53 
3.2  ElementDisplacement 56 
3.3  ElementStrainandStress 57 
3.4  NodalForceandElementSti.nessMatrix 57 
3.5  NodalLoad 59 
3.5.1  DistributedVolumeForce 60 
3.5.2  Distr