MATLAB Codes for Finite Element Analysis

MATLAB Codes for Finite Element Analysis – Solids and Structures
A.J.M. Ferreira
Universidade do Porto – Portugal
Contents
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 15
1.16 Linear algebra 16
2 Discrete systems 19
2.1 Introduction 19
2.2 Springs and bars 19
2.3 Equilibrium at nodes 20
2.4 Some basic steps 21
2.5 First problem and first MATLAB code 21
2.6 New code using MATLAB structures 28
3 Analysis of bars 33
3.1 A bar element 33
3.2 Numerical integration 36
viiviii Contents
3.3 An example of isoparametric bar 37
3.4 Problem 2, using MATLAB struct 41
3.5 Problem 3 44
4 Analysis of 2D trusses 51
4.1 Introduction 51
4.2 2D trusses 51
4.3 Stiffness matrix 52
4.4 Stresses at the element 53
4.5 First 2D truss problem 53
4.6 A second truss problem 58
4.7 An example of 2D truss with spring 63
5 Trusses in 3D space. 69
5.1 Basic formulation 69
5.2 A 3D truss problem 69
5.3 A second 3D truss example 73
6 Bernoulli beams 79
6.1 Introduction 79
6.2 Bernoulli beam problem 81
6.3 Bernoulli beam with spring 85
7 2D frames 89
7.1 Introduction 89
7.2 An example of 2D frame 91
7.3 Another example of 2D frame 95
8 Analysis of 3D frames 103
8.1 Introduction 103
8.2 Stiffness matrix and vector of equivalent nodal forces 103
8.3 First 3D frame example 104
8.4 Second 3D frame example 108
9 Analysis of grids 113
9.1 Introduction 113
9.2 A first grid example 116
9.3 A second grid example 119
10 Analysis of Timoshenko beams 123
10.1 Introduction 123
10.2 Formulation for static analysis 123
10.3 Free vibrations of Timoshenko beams 130
10.4 Buckling analysis of Timoshenko beams 136Contents ix
11 Plane stress 143
11.1 Introduction 143
11.2 Displacements, strains and stresses 143
11.3 Boundary conditions 144
11.4 Potential energy 145
11.5 Finite element discretization 145
11.6 Interpolation of displacements 145
11.7 Element energy 146
11.8 Quadrilateral element Q4. 147
11.9 Example: plate in traction 149
11.10 Example: beam in bending 152
12 Analysis of Mindlin plates 161
12.1 Introduction 161
12.2 The Mindlin plate theory. 161
12.2.1 Strains. 162
12.2.2 Stresses 163
12.3 Finite element discretization 163
12.4 Example: a square Mindlin plate in bending 165
12.5 Free vibrations of Mindlin plates 182
12.6 Buckling analysis of Mindlin plates 192
13 Laminated plates 203
13.1 Introduction 203
13.2 Displacement field 203
13.3 Strains 203
13.4 Strain-displacement matrix B 205
13.5 Stresses 205
13.6 Stiffness matrix 207
13.7 Numerical example 208
13.8 Free vibrations of laminated plates 225
References 231
Index
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