MATLAB Codes for Finite Element Analysis

1 Short introduction to MATLAB . 1 1.1 Introduction . . . . . . . . 1 1.2 Matrices . 1 1.3 Operating with matrices . . . . . . . . 2 1.4 Statements . . . . . . . . . 3 1.5 Matrix functions . . . . 3 1.6 Conditionals, if and switch. . . . . . 4 1.7 Loops: for and while . 5 1.8 Relations . 6 1.9 Scalar functions . . . . . 7 1.10 Vector functions . . . . . 8 1.11 Matrix functions . . . . 9 1.12 Submatrix 10 1.13 Logical indexing . . . . . 12 1.14 M-files, scripts and functions . . . . 13 1.15 Graphics . 14 1.15.1 2D plots . . . . . 14 1.15.2 3D plots . . . . . 16 1.16 Linear Algebra . . . . . . 16 2 Discrete systems . . . . . . . 21 2.1 Introduction . . . . . . . . 21 2.2 Springs and bars . . . . 21 2.3 Equilibrium at nodes . 22 2.4 Some basic steps . . . . 23 2.5 First problem and first MATLAB code . . . . . 23 2.6 New code using MATLAB structures . . . . . . . 30 3 Analysis of bars . . . . . . . 35 3.1 A bar element . . . . . . . 35 3.2 Numerical integration 39 3.3 An example of isoparametric bar 40 3.4 Problem 2, using MATLAB struct . . . . . . . . . 43 3.5 Problem 3 47 4 Analysis of 2D trusses . 53 4.1 Introduction . . . . . . . . 53 4.2 2D trusses 53 4.3 Stiffness matrix . . . . . 54 4.4 Stresses at the element . . . . . . . . . 55 4.5 First 2D truss problem . . . . . . . . . 55 4.6 A second truss problem . . . . . . . . 61 4.7 An example of 2D truss with spring . . . . . . . . 66 5 Trusses in 3D space . . . . 71 5.1 Basic formulation . . . . 71 5.2 A 3D truss problem. . 72 5.3 A second 3D truss example . . . . . 75 6 Bernoulli Beams . . . . . . . 81 6.1 Introduction . . . . . . . . 81 6.2 Bernoulli beam problem . . . . . . . . 83 6.3 Bernoulli beam with spring . . . . . 88 7 2D frames . . . 7.1 An example of 2D frame . . . . . . . 93 7.2 An example of 2D frame . . . . . . . 97 8 Analysis of 3D frames . 107 8.1 Introduction . . . . . . . . 107 8.2 Stiffness matrix and vector of equivalent nodal forces . . . 107 8.3 First 3D frame example . . . . . . . . 108 8.4 Second 3D frame example . . . . . . 113 9 Analysis of grids . . . . . . . 117 9.1 Introduction . . . . . . . . 117 9.2 A first grid example . 120 9.3 A second grid example . . . . . . . . . 123 10 Analysis of Timoshenko beams . . 127 10.1 Introduction . . . . . . . . 127 10.2 Formulation for static analysis . . 127 10.3 Free vibrations of Timoshenko beams . . . . . . 134 10.4 Buckling analysis of Timoshenko beams . . . . 142 11 Plane stress . 11.1 Introduction . . . . . . . . 149 11.2 Displacements, strains and stresses . . . . . . . . 150 11.3 Boundary conditions . 150 11.4 Potential energy . . . . . 151 11.5 Finite element discretization . . . . 151 11.6 Interpolation of displacements . . . 151 11.7 Element energy . . . . . 152 11.8 Quadrilateral element Q4 . . . . . . . 153 11.9 Example: plate in traction . . . . . . 156 11.10Example: Beam in bending . . . . . 159 12 Analysis of Mindlin plates . . . . . . . 169 12.1 Introduction . . . . . . . . 169 12.2 The Mindlin plate theory . . . . . . . 169 12.2.1 Strains . . . . . . . 170 12.2.2 Stresses . . . . . . 171 12.3 Finite element discretization . . . . 172 12.4 Example: a square Mindlin plate in bending 173 12.5 Free vibrations of Mindlin plates 190 12.6