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Introduction to Finite Element Method 1The Finite Element Method(FEM)in Engineering工程中的有限元法1The Finite Element Method(FEIntroduction to Finite Element Method 2Virtual Work and Variational Principles (虚功和变分原理虚功和变分原理)Discretization of Problems (问题离散化问题离散化)Formulation of Finite Element Matrices (有限元矩阵公式的建立有限元矩阵公式的建立)Structure Analysis (结构分析结构分析)Key Words2Virtual Work and Variational Introduction to Finite Element Method 3How can we achieve“understanding”？UNDERSTANDINGUNDERSTANDING!Chinese Idiom:Practice makes progress.Review leads to deeper understanding.(1)based on knowledge(2)processed by thinking3How can we achieve“understanIntroduction to Finite Element Method 4What can we learn in class?What should we learn from engineering application?4What can we learn in class?WhIntroduction to Finite Element Method 5Major Reference BooksReading is important for you!5Major Reference BooksReading Introduction to Finite Element Method 6Brief History of the FEM6Brief History of the FEMIntroduction to Finite Element Method 77Introduction to Finite Element Method 88The purpose of this subject is to present the necessary concepts,fundamental principles and effective techniques of the FEM，so that you can use any well known commercial FEM software package comfortably to solve engineering problems.The general philosophy is to introduce the FEM insightful but simple;informative but concise;and theoretical but applicable.Subject Objective88The purpose of this subject Introduction to Finite Element Method 9COURSE ASSESSMENT 9COURSE ASSESSMENT Introduction to Finite Element Method 101.1 Basic Concept of Finite Element MethodDiscretization of Structure (结构离散化结构离散化)101.1 Basic Concept of Finite Introduction to Finite Element Method 11Mesh 1Mesh 2Continuous Element Assemblage (连续体）体）(单元集合体单元集合体)11Mesh 1Mesh 2Continuous EleIntroduction to Finite Element Method 12FEM Modelling for an Aircraft12FEM Modelling for an AircrafIntroduction to Finite Element Method 13FEM Modelling for an Aircraft13FEM Modelling for an AircrafIntroduction to Finite Element Method 14FEM Modelling for the Vertical Tail in Designing14FEM Modelling for the VerticIntroduction to Finite Element Method 15FEM modelling for rudder15FEM modelling for rudderIntroduction to Finite Element Method 16Shell IntersectionsFEM Modelling16Shell IntersectionsFEM ModelIntroduction to Finite Element Method 17The first part of the finite element process is fundamental and involves choosing the correct and appropriate types of elements and describing and evaluating their properties（单元选择单元选择）.The First ThingIn the FEM analysis,we can say that there are two separate things to be done.17The First ThingIn the FEM anIntroduction to Finite Element Method 18Although commercial FEM software packages will do this,evaluating element properties does help to be able to understand exercise judgment(用于判断用于判断)when modelling the structure,and spotting wrong answers due to using inadequate elements.18Although commercial FEM softIntroduction to Finite Element Method 19The Second ThingThe second part of the finite element process is the assembly of the elements（单单元元集集成成）and then solving the complete structure.The commercial FEM packages will do this of course,but again it pays to know what is involved since sometimes this process breaks down,or it simply becomes an inefficient process because the structure has been described inconveniently.19The Second ThingThe second pIntroduction to Finite Element Method 20Modelling with 3 node triangular elementsTriangular Elements三节点三角形单元三节点三角形单元20Modelling with 3 node triIntroduction to Finite Element Method 212D Isoparametric Quadrilateral elements（平面四边形等参单元）（平面四边形等参单元）n=4n=8n=12n the number of element nodes212D Isoparametric QuadrilaterIntroduction to Finite Element Method 22Example of a mesh with elements and node properly numbered22Example of a mesh with elemeIntroduction to Finite Element Method 232341657823143-D Brick and Tetrahedral Elements （空（空间六面体和四面体元）六面体和四面体元）232341657823143-D Brick Introduction to Finite Element Method 24An Automatically Generated Mesh of Tetrahedra for the Exterior Region of a Aircraft24An Automatically Generated MIntroduction to Finite Element Method 25Axisymmetric Element（轴对称单元）（轴对称单元）25Axisymmetric Element（轴对称单元）Introduction to Finite Element Method 2626FEM Modelling of Nozzle Exit Cone(喷管出口整流锥喷管出口整流锥)2626FEM Modelling of Nozzle ExIntroduction to Finite Element Method 27 Plate and Shell Elements（板壳单元）（板壳单元）27 Plate and Shell Elements（板Introduction to Finite Element Method 28The fuselage of an aircraft to be considered as a typical shell structure28The fuselage of an aircraft Introduction to Finite Element Method 29The Accuracy of FEM Results29The Accuracy of FEM ResultsIntroduction to Finite Element Method 30Mesh Divisions for thick cylinder under Internal Pressure 30Mesh Divisions for thick cylIntroduction to Finite Element Method 31 1.2 General Description of Finite Element Method In structure analysis,the governing equations are usually derived from equilibrium equations（平衡方程平衡方程）;constitutive laws(stress-strain relations)(本构或物理方程本构或物理方程);strain-displacement relations(几何方程或应变几何方程或应变-位移关系位移关系);prescribed boundary conditions(给定边界条件给定边界条件).They can be characterized by a set of differential equations (微分方程组微分方程组).31 1.2 General Description Introduction to Finite Element Method 32Mathematically,the finite element method represents an approximate solution of a boundary-value problem（边边值值问问题题）described by the differential equations.32Introduction to Finite Element Method 3333Introduction to Finite Element Method 3434Introduction to Finite Element Method 3535Displacement Method Introduction to Finite Element Method 36Fig.1.1 The Route of Structure Analysis The route followed in displacement method is Assume the pattern of displacements.36Fig.1.1 The Route of StructuIntroduction to Finite Element Method 37Numerical Approach in FEM37Numerical Approach in FEMIntroduction to Finite Element Method 38Beam and Truss System38Beam and Truss SystemIntroduction to Finite Element Method 3939Introduction to Finite Element Method 40Piping systemand idealization 40Piping systemIntroduction to Finite Element Method 4141Introduction to Finite Element Method 42Degrees of Freedom（DOF，自，自由度）由度）of a Truss Element=2Example42Degrees of Freedom（DOF，自由度）Introduction to Finite Element Method 43DOF of a Beam Element=4EI43DOF of a Beam Element=4EIIntroduction to Finite Element Method 44ExampleThe Third DOF44ExampleThe Third DOFIntroduction to Finite Element Method 4545Introduction to Finite Element Method 46Element 1-global degrees of freedom46Element 1-global degrees oIntroduction to Finite Element Method 47Element 2Element 3ks-spring stiffness Element Stiffness Matrices Global Degrees of Freedom47Element 2Element 3ks-spriIntroduction to Finite Element Method 48Element 4Element Stiffness Matrix Global Degrees of Freedom48Element 4Element Stiffness MIntroduction to Finite Element Method 49(3)Structure Stiffness Matrix Direct Stiffness Method（直接刚度法（直接刚度法)49(3)Structure Stiffness MatrIntroduction to Finite Element Method 5050Direct Stiffness Method Introduction to Finite Element Method 51Expanded Matrices51Expanded MatricesIntroduction to Finite Element Method 52Structure Stiffness Matrix52Structure Stiffness MatrixIntroduction to Finite Element Method 5353Introduction to Finite Element Method 54Boundary Conditions:U1=0,U7=054Boundary Conditions:Introduction to Finite Element Method 5555Boundary ConditionsIntroduction to Finite Element Method 5656Introduction to Finite Element Method 5757Governing EquationsIntroduction to Finite Element Method 58Accuracy of Results58Accuracy of ResultsIntroduction to Finite Element Method 59The Fundamental Steps in FE Analysis59The Fundamental Steps in FIntroduction to Finite Element Method 60The Fundamental Steps in FE Analysis60The Fundamental Steps in FIntroduction to Finite Element Method 6161Introduction to Finite Element Method 62 Short Cantilever Beam Analysis62 Short Cantilever Beam AnalyIntroduction to Finite Element Method 631.3 Engineering Applications of the FEM631.3 Engineering ApplicationsIntroduction to Finite Element Method 64Typical equilibrium problems suitable for finite element analysisAerospace Engineering 64Typical equilibrium problemsIntroduction to Finite Element Method 65Biomedical Engineering (生物医学工程生物医学工程)Typical equilibrium problems suitable for finite element analysis65Biomedical EngineeringTypicaIntroduction to Finite Element Method 66Civil Engineering (土木工程土木工程)Typical equilibrium problems suitable for finite element analysis66Civil Engineering (土木工程Introduction to Finite Element Method 67Mechanical Engineering (机械工程机械工程)Typical equilibrium problems suitable for finite element analysis67Mechanical EngineeringTypicaIntroduction to Finite Element Method 68Nuclear Engineering (核工程核工程)Typical equilibrium problems suitable for finite element analysis68Nuclear Engineering Typical Introduction to Finite Element Method 69Principle of Virtual Displacements (虚位移原理）虚位移原理）69Principle of Virtual DisplacIntroduction to Finite Element Method 70(1)Concept of virtual work(虚功概念）虚功概念）1.4.1 Principle of Virtual Displacements（虚位移原理）（虚位移原理）70(1)Concept of virtual work Introduction to Finite Element Method 71Force System in Equilibrium Compatible Deformation SystemExternal Virtual WorkInternal Virtual Work71Force System in Equilibrium Introduction to Finite Element Method 72Principle of Virtual Work72Principle of Virtual WorkIntroduction to Finite Element Method 73“Virtual”work73“Virtual”workIntroduction to Finite Element Method 7474Introduction to Finite Element Method 75(2)Principle of Virtual Displacements(PVD)75(2)Principle of Virtual DisIntroduction to Finite Element Method 76Real Force System in Equilibrium Virtual System of Displacements and StrainsPrinciple of Virtual DisplacementsInternal Virtual WorkExternal Virtual Work76Real Force System in EquilibIntroduction to Finite Element Method 77Internal Virtual WorkExternal Virtual Work77Internal Virtual WorkExternaIntroduction to Finite Element Method 7878Introduction to Finite Element Method 79Principle of Virtual Displacements79Principle of Virtual DisplacIntroduction to Finite Element Method 80Virtual Field Quantities （虚场变量虚场变量）80Virtual Field QuantitiesIntroduction to Finite Element Method 81Internal Virtual Work 81Internal Virtual Work Introduction to Finite Element Method 82External Virtual Work ST82External Virtual Work STIntroduction to Finite Element Method 83Principle of Virtual Displacements 83Principle of Virtual DisplacIntroduction to Finite Element Method 84Equations of Compatibility(变形协调方程或几何方程变形协调方程或几何方程)84Equations of Compatibility(Introduction to Finite Element Method 85Equations of Compatibility 85Equations of Compatibility Introduction to Finite Element Method 86For virtual displacements and virtual strains,the compatibility equations have the form Since86For virtual displacements anIntroduction to Finite Element Method 87Gauss Ostrogradski Identity（奥高公式）（奥高公式）Su87Gauss Ostrogradski IdentiIntroduction to Finite Element Method 88GaussOstrogradski Identity88GaussOstrogradski IdentityIntroduction to Finite Element Method 89Su89SuIntroduction to Finite Element Method 9090Introduction to Finite Element Method 91 Static Equilibrium Conditions STV91 Static Equilibrium ConditioIntroduction to Finite Element Method 9292Introduction to Finite Element Method 9393Introduction to Finite Element Method 94Fig.194Fig.1Introduction to Finite Element Method 9595
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