A. J. M. Ferreira 
MATLAB Codes for Finite Element Analysis 
Solids and Structures

This book intend to supply readers with some MATLAB codes for ?nite element analysis of solids and structures. After a short introduction to MATLAB, the book illustrates the ?nite element implementation of some problems by simple scripts and functions. The following problems are discussed: • Discrete systems, such as springs and bars • Beams and frames in bending in 2D and 3D • Plane stress problems • Plates in bending • Free vibration of Timoshenko beams and Mindlin plates, including laminated composites • Buckling of Timoshenko beams and Mindlin plates The book does not intends to give a deep insight into the ?nite element details, just the basic equations so that the user can modify the codes. The book was prepared for undergraduate science and engineering students, although it may be useful for graduate students. The MATLABcodesofthisbookareincludedinthedisk.Readersarewelcomed to use them freely. The author does not guarantee that the codes are error-free, although a major e?ort was taken to verify all of them. Users should use MATLAB 7.0 or greater when running these codes. Any suggestions or corrections are welcomed by an email to ferreira@fe.up.pt.

Jadual kandungan

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