The Finite Element Method

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1 The Finite Element Method Contents. Introduction 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements The Finite Element Method FEM.-

2 Literature [] Johannes Altenbach, Udo Fischer: Finite-Elemente-Praxis. Fachbuchverlag, Leipzig 99. ISBN [2] Klaus-Jürgen Bathe: Finite-Elemente-Methoden. Matrizen und lineare Algebra, die Methode der finiten Elemente, Lösung von Gleichgewichtsbedingungen und Bewegungsgleichungen. Springer-Verlag, Berlin Heidelberg New York Tokyo 986. ISBN X [3] Josef Betten: Finite Elemente für Ingenieure. Grundlagen, Matrixmethoden, Elastisches Kontinuum. Springer-Verlag, Berlin Heidelberg New York 997. ISBN [4] Josef Betten: Finite Elemente für Ingenieure 2. Variationsrechnung, Energiemethoden, Näherungsverfahren, Nichtlinearitäten. Springer-Verlag, Berlin Heidelberg New York 998. ISBN [5] Richard H. Gallagher: Finite Element Analysis: Fundamentals. Prentice-Hall, Englewood Cliffs, N. J., 975. ISBN Literature FEM.-

3 Literature (cont d) [6] Dietmar Gross, Werner Hauger, Walter Schnell, Peter Wriggers: Technische Mechanik. Band 4: Hydromechanik, Elemente der Höheren Mechanik, Numerische Methoden. Springer- Verlag, Berlin Heidelberg New York 993. ISBN [7] Bernd Klein: FEM. Grundlagen und Anwendungen der Finite-Elemente-Methode. Vieweg, Braunschweig Wiesbaden, dritte, überarbeitete Auflage, 999. ISBN [8] Günther Müller, Clemens Groth: FEM für Praktiker. Die Methode der Finiten Elemente mit dem FE-Programm ANSYS. expert-verlag, Renningen-Malmsheim, dritte, völlig neubearbeitete Auflage, 997. ISBN [9] Douglas H. Norrie, Gerard de Vries: The Finite Element Method. Academic Press, New York 973. ISBN [] J. Tinsley Oden: Finite Elements of Nonlinear Continua. McGraw-Hill, New York 972. Literature (cont d) FEM.-2

4 Literature (cont d) [] Hans Rudolf Schwarz: Methode der finiten Elemente. Eine Einführung unter besonderer Berücksichtigung der Rechenpraxis. B. G. Teubner, Stuttgart 98. ISBN [2] Hans Rudolf Schwarz: FORTRAN-Programme zur Methode der finiten Elemente. B. G. Teubner, Stuttgart 98. ISBN [3] Gilbert Strang and George J. Fix: An Analysis of the Finite Element Method. Prentice-Hall, Englewood-Cliffs, N. J., 973. ISBN [4] Olgierd C. Zienkiewicz: Methode der finiten Elemente. Hanser Verlag, München, zweite, erweiterte und völlig neubearbeitete Auflage, 984. ISBN [5] Olgierd C. Zienkiewicz and Robert L. Taylor: The Finite Element Method. McGraw-Hill, London, fourth edition, 989. Literature (cont d) FEM.-3

5 Contents. Introduction. What is the Finite Element Method.2 Brief History 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements Contents FEM.-

6 Introduction What is the Finite Element Method? The finite element method (FEM) is a numerical method for solving problems of engineering and mathematical physics. Its primary application is in Strength of Materials. The FEM is useful for problems with complicated geometries, loadings, and material properties where analytical solutions cannot be obtained. The model body is divided into a system of small but finite bodies, the finite elements, interconnected at nodal points or nodes. In each of the finite element the unknown fields are approximated by simple functions, which are determined by their nodal values. The discretization by finite elements yields a large system of equations for the unknown nodal values. Introduction FEM.-

7 Brief History A. Hrennikoff (94), Solutions of problems in elasticity by the framework method D. McHenry (943), A lattice analogy for the solution of plane stress problems R. Courant (943), Variational methods for the solutions of problems of equilibrium and vibration J. H. Argyris (954 55), Energy theorems and structural analysis M. J. Turner, R. W. Clough, H. C. Martin, and L. P. Topp (956), Stiffness and deflection analysis of complex structures R. W. Clough (96), The finite element method in plane stress analysis Some Names John H. Argyris, Ivo Babuška, Klaus-Jürgen Bathe, Philipe G. Ciarlet, Richard H. Gallagher, Erwin Stein, Robert L. Taylor, Peter Wriggers, Olek C. Zienkiewicz Brief History FEM.2-

8 FEM.2-2

9 Contents. Introduction 2. A Simple Example 2. Statement of Problem and Exact Solution 2.2 Approximate Solution Using Finite Elements 2.3 New Approach: Strain Energy 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements Contents FEM 2.-

10 Elastic Rod Loaded by Self-Weight and End Load ρ, E, A Elongation or strain ε = du dx x u(x) g Stress σ = Eε l Tensile force F = Aσ = AE du dx x F l Equilibrium condition df dx = ρga Elastic Rod Loaded by Self-Weight and End Load FEM 2.-

11 Boundary Value Problem and Solution Differential equation d dx ( AE du ) + ρga = dx x ρ, E, A u(x) g Boundary conditions u() =, du dx = F l x=l AE l x F l Assumption: Constant tensile stiffness, AE = const Closed-form solution of the boundary-value problem [ ρg ( u = l x ) + F ] l x E 2 EA Boundary Value Problem and Solution FEM 2.-2

12 Exact Solution Gl 2EA Gl EA u G 2G F l x F l = G / 2 G u = Gx ( + F l EA G x ) 2l l x G / 2 G F = F l + G ( x ) l Exact Solution FEM 2.-3

13 Discretization by Finite Elements Total system element node h l 4 x Single element ξ node coordinate transformation x = x + hξ Displacement ansatz Nodal displacements Interpolation functions Strain in element Stress resultant u = ( ξ)u + ξu u, u N = ξ, N = ξ ε = du dx = h du dξ = u u h F = EA h (u u ) Discretization by Finite Elements FEM 2.2-

14 Collecting the Elements Overall system Single element Return to global numbering within the overall system element h node node F i = EA h (u i u i ) i =, 2, 3, l x ξ coordinate transformation x = x + hξ Global vector of stress resultants F F 2 F 3 = EA h F 4 u u u 2 u 3 u 4 Collecting the Elements FEM 2.2-2

15 F 2 ρgah F Equilibrium Conditions Node with adjacent half elements F = F + 2 ρgah F k F k = F k+ + ρgah < k < 4 k 2 ρgah 2 ρgah F 4 = F l + 2 ρgah 4 F k+ F 4 2 ρgah F l Equilibrium conditions in matrix form F F 2 F 3 = ρgah F 4 / 2 / 2 + F F l Equilibrium Conditions FEM 2.2-3

16 Resulting System of Equations Equilibrium conditions F F 2 F 3 F 4 = ρgah / 2 / 2 + F F l Stress resultants F F 2 F 3 F 4 = EA h u u u 2 u 3 u 4 System of equations EA h u u u 2 u 3 u 4 = ρgah / 2 / 2 + F F l unknown Resulting System of Equations FEM 2.2-4

17 Comparison Between Exact and Approximate Solutions Displacement Stress resultant Gl 2EA Gl EA u G 2G F h h 2h 2h 3h 3h 4h 4h x x Comparison Between Exact and Approximate Solutions FEM 2.2-5

18 Strain Energy Displacement within single element Strain u = ( ξ)u + ξu ε = h (u u ) Strain energy of single element Π elem = 2 ξ= EAε 2 h dξ = EA 2h (u 2 2u u + u 2 ) Strain energy of total system Π = EA [ u 2 2h 2u u + u u 2 2u u 2 + u u2 2 2u 2 u 3 + u u3 2 2u 3 u 4 + u ] 4 2 Strain Energy FEM 2.3-

19 Strain Energy (contd.) Matrix representation of strain energy Π = 2 ut Ku Global nodal displacement vector u = u u u 2 u 3 u 4 Global stiffness matrix K = EA h Strain Energy (contd.) FEM 2.3-2

20 Principle of Virtual Work Virtual work of the external forces (δw ) = p T δu Global load vector, vector of virtual displacements / 2 F p = ρgah + F l / 2, δu = δu δu δu 2 δu 3 δu 4 Principle of virtual work δπ = (δw ) for arbitrary virtual displacements δu Principle of Virtual Work FEM 2.3-3

21 Principle of Virtual Work (contd.) Strain energy Π = 2 ut Ku Variation of strain energy Consequence of the Principle of Virtual Work δπ = 2 ut ( K + K T) δu = u T K δu due to symmetry of the global stiffness matrix K (Ku p) T δu = for arbitrary δu Linear system of equations Ku = p Principle of Virtual Work (contd.) FEM 2.3-4

22 FEM 2.3-5

23 . Introduction 2. A Simple Example 3. Trusses 3. Data of a Truss Contents 3.2 Element Stiffness Matrix 3.3 Global Stiffness Matrix 3.4 Supports and Reactive Forces 3.5 How to Develop a Truss Program 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements Contents FEM 3.-

24 Trusses Characteristics of a truss Assembly of pin-jointed members External forces applied to nodes Members loaded in axial direction Prescribed: Geometry Material data Support Load Demanded: Nodal displacements Member forces Reactive forces Trusses FEM 3.-

25 Elongation of a Rod Rod in undeformed and deformed configuration e u 2 l u l + l Elongation l = e. (u 2 u ) Matrix representation l = [ e T +e ] T u u 2 = [ u T u T 2 ] e +e Elongation of a Rod FEM 3.2-

26 Element Stiffness Matrix of a Single Member Strain energy stored in a single member Π = 2 EA l ( l)2 Strain energy represented in terms of nodal displacements Π = EA [ ] ee T ee T u 2 l u 2 ee T ee T u u 2 = 2 ut Ku Nodal displacement vector and element stiffness matrix u = u u 2 K = EA l ee T ee T ee T ee T Element Stiffness Matrix of a Single Member FEM 3.2-2

27 Contribution of a Single Rod to the Global Stiffness Matrix (2) 4 3 Element stiffness matrix of rod (2 4) 2 () K (2 4) = K K 2 K 2 K 22 Global nodal displacement vector and contribution of rod (2 4) to the global stiffness matrix u = u u 2 u 3 K K 2 K = u 4 K 2 K 22 Contribution of a Single Rod to the Global Stiffness Matrix FEM 3.3-

28 Support and Reaction Forces Types of support u x = u y = u x = u y = F x F x F y F y Decomposition of the nodal displacement and force vectors u = u e p = p e u x p x u e free displacements p e given forces u x fixed displacements p x reactive forces Support and Reaction Forces FEM 3.4-

29 Decomposition of the System of Equations System of equations K ee K ex u e = p e K xe K xx u x p x unknown quantity Decomposition. K ee u e = p e K ex u x = u e free displacements 2. p x = K xe u e + K xx u x = p x reactive forces Decomposition of the System of Equations FEM 3.4-2

30 How to Develop a Truss Program. Input nodal coordinates. Reserve memory for Nodal displacement vector u Nodal force vector p Global stiffness matrix K (clear matrix to ) 2. Input and process member data: Compute element stiffness matrices K Elem = accumulate them in the global stiffness matrix K K K 2 K 2 K 22, split them up, and 3. Allow for supports: Split up the nodal vectors u = u e and p = p e u x p x Enter the fixed nodal displacements u x How to Develop a Truss Program FEM 3.5-

31 How to Develop a Truss Program (contd.) 4. Input loads Enter the vector p e of given loads 5. Solve the reduced system of equations for the unknown free displacements K ee u e = p e K ex u x = u e 6. Compute the reaction forces p x = K xe u e + K xx u x = p x 7. Compute member forces F = EA l e. (u 2 u ) How to Develop a Truss Program (contd.) FEM 3.5-2

32 FEM 3.5-3

33 Contents. Introduction 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 4. Some Mathematical Foundations 4.2 Cholesky Decomposition 4.3 How to Store Sparse Matrices 4.4 Other Methods 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements Contents FEM 4.-

34 Linear Systems of Equations Ax = b det A det A = Solution exists, is unique. y T b = for all y satisfying y T A = Solution exists, is not unique. y T b for some y satisfying y T A = Solution does not exist. Linear Systems of Equations FEM 4.-

35 Cholesky s Method Solution of the linear system Ax = b with a symmetric, positiv definite matrix A.. Factorization Decompose the symmetric matrix A into the product A = U T U where U is an upper triangular matrix. 2. Forward Substitution Solve the lower triangular system U T y = b for the auxiliary vector y. 3. Backward Substitution Solve the upper triangular system Ux = y for the requested vector x. André-Louis Cholesky (875 98): French military officer involved in geodesy and surveying in Crete and North Africa Cholesky s Method FEM 4.2-

36 Cholesky s Method: Factorization. Factorization s = A ij i k= U ki U kj A = U T U U ij = s U ii if i < j s if i = j i =,..., j j =,..., n Cholesky s Method: Factorization FEM 4.2-2

37 Cholesky s Method: Forward and Backward Substitution 2. Forward Substitution ( U T y = b y i = b U i ii i k= U ki y k ) i =,..., n 3. Backward Substitution ( U x = y x i = y U i ii n k=i+ U ik x k ) i = n,..., Cholesky s Method: Forward and Backward Substitution FEM 4.2-3

38 . Introduction 2. A Simple Example 3. Trusses Contents 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 5. Displacements 5.2 Strain 5.3 Stress 5.4 Equilibrium 5.5 Strain Energy 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements Contents FEM 5.-

39 Strain y y u x dy dy y dx ( + u ) y dy u x u y ( + u ) x dx u y dx x x x Elongations Shear ε x = u x ε y = u y γ xy = u x + u y Strain FEM 5.2-

40 Hooke s Law Uniaxial stress σ ε = E σ ε t = ν E σ E Young s modulus ν Poisson s ratio Shear stress τ γ = G τ G Shear modulus Hooke s Law FEM 5.3-

41 Simple Shear: Principal Stresses τ xy τ xy τ xy σ = τ xy σ 2 = τ xy π / 2 σ σ = τ xy π / 4 σ 2 = τ xy τ τ xy Simple Shear: Principal Stresses FEM 5.3-2

42 Simple Shear: Deformation a π / 4 a σ 2 = τ σ = τ π / 4 γ / 2 a( + ε) a( ε) τ τ Simple Shear: Deformation FEM 5.3-3

43 Relation Among Material Constants Geometry ε ( π + ε = tan 4 γ ) 2 = sin γ + sin γ Approximation for small strain: γ 2ε Hooke s Law ε = E (σ νσ 2 ) = + ν E τ γ = G τ Shear modulus expressed in terms of Young s modulus and Poisson s ratio G = E 2( + ν) Relation Among Material Constants FEM 5.3-4

44 Governing Equations of Plane Stress Problems Definition of strain ε x ε y γ xy = [ ux u y ] Stress-strain relation (Hooke s law) σ x σ y τ xy = E ν 2 ν ν ν 2 ε x ε y γ xy Specific strain energy W = [ ] εx ε 2 y γ xy σ x σ y τ xy Governing Equations of Plane Stress Problems FEM 5.3-5

45 Contents. Introduction 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 6. Linear Triangular Elements 6.2 Bilinear Quadrilateral Elements 6.3 Higher Elements 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements Contents FEM 6.-

46 Linear Interpolation η 3 Problem: Determine the linear function u = u(ξ, η) satisfying u(, ) = u u(, ) = u 2 u(, ) = u 3 Solution: 2 ξ u(ξ, η) = ( ξ η)u + ξu 2 + ηu 3 Interpolation or shape functions N (ξ, η) = ξ η N 2 (ξ, η) = ξ N 3 (ξ, η) = η Characteristic property: N i (ξ k, η k ) = i = k if i k Linear Interpolation FEM 6.-

47 Affine Transformation y η y y 2 2 y x x 3 x 2 x = 2 ξ Coordinate transformation [ x y ] = [ x x 2 x 3 y y 2 y 3 ] ξ η ξ η Affine Transformation FEM 6.-2

48 Derivatives Jacobi matrix ξ ξ η η = [ x x 2 x 3 y y 2 y 3 ] = [ x2 x x 3 x y 2 y y 3 y ] Determinant of Jacobi matrix, Jacobian J = (x 2 x )(y 3 y ) (x 3 x )(y 2 y ) Inverse Jacobi matrix ξ η ξ η = J [ y3 y x x 3 y y 2 x 2 x ] Transformation of derivatives [ ϕ ϕ ] = [ ϕ ξ ϕ η ] ξ η ξ η Derivatives FEM 6.-3

49 Linear Triangular Element y y 3 3 Shape functions y 2 y 2 N (ξ, η) N 2 (ξ, η) N 3 (ξ, η) = ξ η ξ η x x 3 x 2 x Derivatives of the shape functions with respect to the global coordinates (x, y) N N N N ξ η ξ ξ N 2 N 2 = N 2 N 2 ξ η = [ ] y3 y x x 3 = J y y 2 x 2 x J N 3 N 3 N 3 ξ N 3 η η η y 2 y 3 x 3 x 2 y 3 y x x 3 y y 2 x 2 x Jacobian J = (x 2 x )(y 3 y ) (x 3 x )(y 2 y ) Linear Triangular Element FEM 6.-4

50 Linear Triangular Element: Stiffness Matrix Matrix of derivatives of shape functions DN = J y 2 y 3 y 3 y y y 2 x 3 x 2 x x 3 x 2 x x 3 x 2 y 2 y 3 x x 3 y 3 y x 2 x y y 2 Matrix of material constants E = E ν 2 ν ν ν 2 Element stiffness matrix of the linear triangular element K = 2 Jh (DN)T E (DN) Linear Triangular Element: Stiffness Matrix FEM 6.-5

51 Bilinear Interpolation η 3 4 Problem: Determine the bilinear function u = u(ξ, η) satisfying u(, ) = u u(, ) = u 2 u(, ) = u 3 u(, ) = u 4 Solution: 2 ξ u(ξ, η) = ( ξ)( η)u + ξ( η)u 2 + ( ξ)ηu 3 + ξηu 4 Interpolation or shape functions N (ξ, η) = ( ξ)( η) N 2 (ξ, η) = ξ( η) N 3 (ξ, η) = ( ξ)η Characteristic property: N i (ξ k, η k ) = i = k if i k N 4 (ξ, η) = ξη Bilinear Interpolation FEM 6.2-

52 Bilinear Parallelogram Element y y 3 y 2 y Shape functions N (ξ, η) N 2 (ξ, η) N 3 (ξ, η) N 4 (ξ, η) = ( ξ)( η) ξ( η) ( ξ)η ξη x x 3 x 2 x Derivatives of the shape functions with respect to the global coordinates (x, y) N N 2 N 3 N 4 N N 2 N 3 N 4 = N ξ N 2 ξ N 3 ξ N 4 ξ N η N 2 η N 3 η N 4 η [ ξ η ξ η ( η) ( ξ) ] = η ξ η ξ η ξ J [ y3 y x x 3 y y 2 x 2 x Jacobian J = (x 2 x )(y 3 y ) (x 3 x )(y 2 y ) ] Bilinear Parallelogram Element FEM 6.2-2

53 Bilinear Transformation y y 4 y η 3 4 y 2 2 y x x 3 x 2 x 4 x = 2 ξ Coordinate transformation [ x y ] = [ x x 2 x 3 x 4 y y 2 y 3 y 4 ] ( ξ)( η) ξ ( η) ( ξ) η ξ η Bilinear Transformation FEM 6.2-3

54 Derivatives Jacobi matrix ξ ξ η η = [ x x 2 x 3 x 4 y y 2 y 3 y 4 ] ( η) ( ξ) η ξ η ξ η ξ Determinant of Jacobi matrix, Jacobian J = ξ η ξ η Inverse Jacobi matrix ξ η ξ η = ξ ξ η η = J η ξ η ξ matrix elements are rational functions of ξ, η Transformation of derivatives [ ϕ ϕ ] = [ ϕ ξ ϕ η ] ξ η ξ η Derivatives FEM 6.2-4

55 Bilinear Quadrilateral Element: Stiffness Matrix Derivatives of shape functions [ Nk N k ] = [ Nk ξ N k η ] ξ η ξ η... rational functions of ξ, η Matrix of derivatives of shape functions DN = N N N N N 2 N 2 N 2 N 2 N 3 N 3 N 3 N 3 N 4 N 4 N 4 N 4 Element stiffness matrix of the bilinear quadrilateral element K = η= ξ= (DN) T E (DN) Jh dξ dη numerical integration (Gauss) Bilinear Quadrilateral Element: Stiffness Matrix FEM 6.2-5

56 Quadratic Triangular Elements η ξ N = ( ξ η)( 2ξ 2η) N 2 = ξ(2ξ ) N 3 = η(2η ) N 4 = 4ξη N 5 = 4η( ξ η) N 6 = 4ξ( ξ η) y Affine Transformation 3 2 y Isoparametric Transformation x x Quadratic Triangular Elements FEM 6.3-

57 Quadrilateral Elements of Lagrange Type 7 η 6 5 N = 4 ξ( ξ)η( η) N 5 = 4 ξ( + ξ)η( + η) ξ N 2 = 2 ( ξ2 )η( η) N 3 = ξ( + ξ)η( η) 4 N 6 = 2 ( ξ2 )η( + η) N 7 = ξ( ξ)η( + η) N 4 = 2 ξ( + ξ)( η2 ) N 8 = 2 ξ( ξ)( η2 ) N 9 = ( ξ 2 )( η 2 ) y Affine Transformation 7 3 y Isoparametric Transformation x x Quadrilateral Elements of Lagrange Type FEM 6.3-2

58 7 η 6 Quadrilateral Elements of Serendipity Type 5 N = ( ξ)( η)( + ξ + η) N 4 2 = 2 ( ξ2 )( η) ξ N 3 = 4 N 5 = 4 N 7 = 4 ( + ξ)( η)( ξ + η) ( + ξ)( + η)( ξ η) ( ξ)( + η)( + ξ η) N 4 = 2 ( + ξ)( η2 ) N 6 = 2 ( ξ2 )( + η) N 8 = 2 ( ξ)( η2 ) y Affine Transformation 7 y Isoparametric Transformation x x Quadrilateral Elements of Serendipity Type FEM 6.3-3

59 Deriving the Stiffness Matrix of a Finite Element Displacement vector u(ξ, η) = n N i (ξ, η) u i i= Strain ɛ = D u = n (DN i ) u i i= Stress Strain energy σ = E ɛ = n E (DN k ) u k k= Π = 2 ɛ T σ dv = 2 n n u T i (DN i ) T E (DN k ) dv u k V i= k= V Deriving the Stiffness Matrix of a Finite Element FEM 6.3-4

60 Element Stiffness Matrices of Finite Elements for Plane Stress Problems General structure Submatrices K = K K 2... K n K 2 K K 2n K ik = A (DN i ) T E (DN k ) h da K n K n2... K nn K ik = E ν 2 A N i ν N i N k + ν 2 N k + ν 2 N i N i N k N k ν N i N i N k + ν 2 N k + ν 2 N i N i N k N k h da Element Stiffness Matrices of Finite Elements for Plane Stress Problems FEM 6.3-5

61 Contents. Introduction 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 7. Linear Tetrahedral Elements 7.2 Trilinear Hexahedral Elements 7.3 Higher Elements 8. Dynamical Problems 9. Beam Elements Contents FEM 7.-

62 Contents. Introduction 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 8. Natural and Forced Vibrations 8.2 Mass Matrices 8.3 Natural frequencies and modes 9. Beam Elements Contents FEM 8.-

63 Contents. Introduction 2. A Simple Example 3. Trusses 4. Linear Systems of Equations 5. Basic Equations of Elasticity Theory 6. Finite Elements for Plane Stress Problems 7. Finite Elements for Three-Dimensional Problems 8. Dynamical Problems 9. Beam Elements 9. Hermite Interpolation 9.2 Mass and Stiffness Matrices of Beam Elements 9.3 Cubic Splines Contents FEM 9.-

64 Beam Elements: Hermite Interpolation w x h x w w w w x = x + hξ w = N (ξ)w + hn (ξ)w + N (ξ)w + hn (ξ)w Beam Elements: Hermite Interpolation FEM 9.-

65 Shape Functions N = ( ξ) 2 ( + 2ξ) N = ξ 2 (3 2ξ) N = ξ( ξ) 2 N = ξ 2 ( ξ) Shape Functions FEM 9.-2

66 Element Stiffness Matrix Strain energy of beam element Π = 2 x +h x=x EI ( d 2 w dx 2 ) 2 dx = 2h 3 ξ= EI ( d 2 ) 2 w dξ dξ 2 EI bending stiffness Displacement w = N (ξ)w + hn (ξ)w + N (ξ)w + hn (ξ)w Nodal displacement and load vectors, element stiffness matrix u = w w w w p = F M F M K = 2EI h 3 6 3h 6 3h 3h 2h 2 3h h 2 6 3h 6 3h 3h h 2 3h 2h 2 Element Stiffness Matrix FEM 9.2-

67 Element Mass Matrix Kinetic energy of beam element Π = 2 x +h x=x ρaẇ 2 dx = h 2 ξ= ρaẇ 2 dξ ρa mass per unit length Displacement velocity ẇ = N (ξ)ẇ + hn (ξ)ẇ + N (ξ)ẇ + hn (ξ)ẇ Nodal velocity vector and element mass matrix u = ẇ ẇ ẇ ẇ M = m h 54 3h 22h 4h 2 3h 3h h 56 22h 3h 3h 2 22h 4h 2 Element Mass Matrix FEM 9.2-2

68 FEM 9.2-3

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps