1、Introduction to Finite Element Method 1The Finite Element Method(FEM)in Engineering工程中的有限元法Introduction to Finite Element Method 2 Virtual 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 Ana
2、lysis (结构分析结构分析)Key WordsIntroduction to Finite Element Method 3How can we achieve“understanding”？!Chinese Idiom:Practice makes progress.Review leads to deeper understanding.(1)based on knowledge(2)processed by thinkingIntroduction to Finite Element Method 4What can we learn in class?What should we
3、learn from engineering application?Introduction to Finite Element Method 5Major Reference BooksReading is important for you!Introduction to Finite Element Method 6Brief History of the FEMIntroduction to Finite Element Method 7Introduction to Finite Element Method 88 The purpose of this subject is to
4、 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 the
5、oretical but applicable.Subject ObjectiveIntroduction to Finite Element Method 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)Introduction to Finite Element Method 11Mesh 1Mesh 2Continuous Element Assemblag
6、e (连续体）连续体）(单元集合体单元集合体)Introduction to Finite Element Method 12FEM Modelling for an AircraftIntroduction to Finite Element Method 13FEM Modelling for an AircraftIntroduction to Finite Element Method 14FEM Modelling for the Vertical Tail in DesigningIntroduction to Finite Element Method 15FEM modelli
7、ng for rudderIntroduction to Finite Element Method 16Shell IntersectionsFEM ModellingIntroduction to Finite Element Method 17 The 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
8、（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.Introduction to Finite Element Method 18 Although 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 mode
9、lling the structure,and spotting wrong answers due to using inadequate elements.Introduction to Finite Element Method 19The Second Thing The second part of the finite element process is the assembly of the elements（单元集成单元集成）and then solving the complete structure.The commercial FEM packages will do
10、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.Introduction to Finite Element Method 20Modelling with 3 node triangular elementsTriangular Elements三
11、节点三角形单元三节点三角形单元Introduction 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 nodesIntroduction to Finite Element Method 22Example of a mesh with elements and node properly numberedIntroduction to Finite Element Method 232341657
12、823143-D Brick and Tetrahedral Elements （空间六面体和四面体元）（空间六面体和四面体元）Introduction to Finite Element Method 24An Automatically Generated Mesh of Tetrahedra for the Exterior Region of a AircraftIntroduction to Finite Element Method 25Axisymmetric Element（轴对称单元）（轴对称单元）Introduction to Finite Element Method 2
13、626FEM Modelling of Nozzle Exit Cone(喷管出口整流锥喷管出口整流锥)Introduction to Finite Element Method 27 Plate and Shell Elements（板壳单元）（板壳单元）Introduction to Finite Element Method 28The fuselage of an aircraft to be considered as a typical shell structureIntroduction to Finite Element Method 29The Accuracy of FE
14、M ResultsIntroduction to Finite Element Method 30Mesh Divisions for thick cylinder under Internal Pressure Introduction 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）;
15、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).Introduction to Finite Element Method 32 Mathematically,the finite
16、element method represents an approximate solution of a boundary-value problem（边值问题）（边值问题）described by the differential equations.Introduction to Finite Element Method 33Introduction to Finite Element Method 34Introduction to Finite Element Method 35Introduction to Finite Element Method 36Fig.1.1 The
17、 Route of Structure Analysis The route followed in displacement method is Assume the pattern of displacements.Introduction to Finite Element Method 37Numerical Approach in FEMIntroduction to Finite Element Method 38Beam and Truss SystemIntroduction to Finite Element Method 39Introduction to Finite E
18、lement Method 40Piping systemand idealization Introduction to Finite Element Method 41Introduction to Finite Element Method 42Degrees of Freedom（DOF，自由度），自由度）of a Truss Element=2Example121121If1 with0,thus the axial force developed at node 1 and node 2 denote and respectively.uukkIntroduction to Fin
19、ite Element Method 43DOF of a Beam Element=4EI11213242,UvUUvUIntroduction to Finite Element Method 44Example2112131231332432If1 with0,thus,.yyvvkPkMkPkMThe Third DOFIntroduction to Finite Element Method 45Introduction to Finite Element Method 46Element 1-global degrees of freedomIntroduction to Fini
20、te Element Method 47Element 2Element 3ks-spring stiffness Element Stiffness Matrices Global Degrees of FreedomIntroduction to Finite Element Method 48Element 4Element Stiffness Matrix Global Degrees of FreedomIntroduction to Finite Element Method 49(3)Structure Stiffness Matrix Direct Stiffness Meth
21、od（直接刚度法（直接刚度法)Introduction to Finite Element Method 50Introduction to Finite Element Method 51Expanded MatricesIntroduction to Finite Element Method 52Structure Stiffness Matrix 123456713232223323242325671261260004620002461212020128012212216.02sUUUUUUUUEIEIEIEILLLLUEIEIEILLLEIEIEIEIULLLLEIEIEIKULLL
22、EIEAEIEAULLLLEIsymkULEALU ()ieiKKIntroduction to Finite Element Method 53Introduction to Finite Element Method 54Boundary Conditions:U1=0,U7=0Introduction to Finite Element Method 55Introduction to Finite Element Method 56Introduction to Finite Element Method 57Introduction to Finite Element Method
23、58Accuracy of ResultsIntroduction to Finite Element Method 59The Fundamental Steps in FE AnalysisIntroduction to Finite Element Method 60The Fundamental Steps in FE AnalysisIntroduction to Finite Element Method 61Introduction to Finite Element Method 62 Short Cantilever Beam AnalysisIntroduction to
24、Finite Element Method 631.3 Engineering Applications of the FEMIntroduction to Finite Element Method 64Typical equilibrium problems suitable for finite element analysisAerospace Engineering Introduction to Finite Element Method 65Biomedical Engineering (生物医学工程生物医学工程)Typical equilibrium problems suit
25、able for finite element analysisIntroduction to Finite Element Method 66Civil Engineering (土木工程土木工程)Typical equilibrium problems suitable for finite element analysisIntroduction to Finite Element Method 67Mechanical Engineering (机械工程机械工程)Typical equilibrium problems suitable for finite element analy
26、sisIntroduction to Finite Element Method 68Nuclear Engineering (核工程核工程)Typical equilibrium problems suitable for finite element analysisIntroduction to Finite Element Method 69Principle of Virtual Displacements (虚位移原理）虚位移原理）Introduction to Finite Element Method 70(1)Concept of virtual work(虚功概念）虚功概念
27、）1.4.1 Principle of Virtual Displacements（虚位移原理）（虚位移原理）Introduction to Finite Element Method 71Force System in Equilibrium Compatible Deformation SystemExternal Virtual WorkInternal Virtual WorkIntroduction to Finite Element Method 72Principle of Virtual WorkIntroduction to Finite Element Method 73“
28、Virtual”workIntroduction to Finite Element Method 74Introduction to Finite Element Method 75(2)Principle of Virtual Displacements(PVD)Introduction to Finite Element Method 76Real Force System in Equilibrium Virtual System of Displacements and StrainsPrinciple of Virtual DisplacementsInternal Virtual
29、 WorkExternal Virtual WorkIntroduction to Finite Element Method 77Internal Virtual WorkExternal Virtual WorkIntroduction to Finite Element Method 78Introduction to Finite Element Method 79Principle of Virtual DisplacementsIntroduction to Finite Element Method 80Virtual Field Quantities （虚场变量虚场变量）Int
30、roduction to Finite Element Method 81Internal Virtual Work Introduction to Finite Element Method 82External Virtual Work STIntroduction to Finite Element Method 83Principle of Virtual Displacements Introduction 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)Introduction
31、 to Finite Element Method 85Equations of Compatibility Introduction to Finite Element Method 86For virtual displacements and virtual strains,the compatibility equations have the form SinceIntroduction to Finite Element Method 87Gauss Ostrogradski Identity（奥高公式）（奥高公式）SuIntroduction to Finite Element
32、Method 88GaussOstrogradski Identity TTTTTddVSuuvunSIntroduction to Finite Element Method 89 TTTddd0TvsVVSuVPuVPuSSuIntroduction to Finite Element Method 90Introduction to Finite Element Method 91 Static Equilibrium Conditions BBB00in0yxxzxxxyyzyyyzxzzzfxyzfVxyzfxyzSTVIntroduction to Finite Element Method 92Introduction to Finite Element Method 93Introduction to Finite Element Method 94Problem Set 1 1.1 Calculate the deflection and rotation under the load in the statically indeterminate beam shown in Fig.1.Fig.1Introduction to Finite Element Method 95
