اسم المؤلف
A.J.M. Ferreira
التاريخ
23 يوليو 2022
المشاهدات
581
التقييم
(لا توجد تقييمات)

MATLAB Codes for Finite Element Analysis – Solids and Structures
A.J.M. Ferreira
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
11.9 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
Index
3D truss problem, 69
2D frame
Stiffness matrix, 89, 93
2D frame problem, 91, 95
2D frames, 89
2D truss
Stiffness matrix, 56
Stresses, 57
2D truss problem, 53, 58
2D truss problem with spring, 63
2D trusses, 51
3D frame
Stiffness matrix, 103
3D frame problem, 104
3D frames, 103
3D truss, 69
Stiffness matrix, 69
Stresses, 77
3D truss problem, 73
Assembly of stiffness matrix, 23, 26
Axes transformation, 104
Axial stresses, 33
B matrix, 40, 146, 164
Bending, 205
Membrane, 205
Shear, 205
Bar element, 19
Bending stiffness, 165
Bending stiffness matrix, 207
Bending stresses, 161
Bernoulli beam, 79
Bernoulli beam problem, 81
Bernoulli beam with spring, 85
Boundary conditions, 171
Buckling analysis of Mindlin plates, 192
Buckling analysis of Timoshenko beams,
136
Coordinate transformation, 104, 113
Cross-ply laminates, 225
Cylindrical bending, 206
Determinant of Jacobian matrix, 164
Distributed forces, 20
Eigenproblem, 136, 193
Equations of motion of Mindlin plates, 182
Essential boundary conditions, 26, 144
Euler-Benoulli beam, 79
External forces, 145
External work, 34, 79
Finite element steps, 21
Force vector
3D frame, 104
Grids, 114
Mindlin plate, 164
Plane stress, 146
Free vibrations of laminated plates, 225
Free vibrations of Mindlin plates, 182
Free vibrations of Timoshenko beams, 130
Fundamental frequency, 183
Generalized eigenproblem, 182, 193
Geometric stiffness matrix
Mindlin plate, 193
Timoshenko beam, 137
Grid example, 116, 119
Grids, 113
Stiffness matrix, 113
233234 Index
Hamilton principle, 182
Hermite shape functions, 80
Hooke’s law, 33
Initially stressed Mindlin plate, 192
Integration points, 149
Interpolation of displacements, 124
Inverse of Jacobian, 148
Jacobian, 35, 148
Kinetic energy, 130
Lagrange shape functions, 147
Laminated plates, 203
Local coordinate system, 53, 89
Mass matrix
Mindlin plate, 182
Timoshenko beam, 130
MATLAB codes
EssentialBC.m, 172
EssentialBC5dof.m, 222
Jacobian.m, 159
Problem1.m, 23
Problem1Structure.m, 30
SrinivasStress.m, 225
forcesInElementGrid.m, 116
formForceVectorMindlinQ4.m, 171
formForceVectorMindlinQ45dof.m, 221
formGeometricStiffnessMindlinQ4.m, 201
formMassMatrixMindlinQ4.m, 192
formMassMatrixMindlinQ4laminated5dof.m,
220
formStiffness2D.m, 157
formStiffness2Dframe.m, 94
formStiffness2Dtruss.m, 57
formStiffness3Dframe.m, 106
formStiffness3Dtruss.m, 76
formStiffnessBernoulliBeam.m, 85
formStiffnessBucklingTimoshenkoBeam.m,
141
formStiffnessGrid.m, 115
formStiffnessMassTimoshenkoBeam.m,
128
formStiffnessMatrixMindlinQ4.m, 170
formStiffnessMatrixMindlinQ45laminated
5dof.m, 218
outputDisplacementsReactionsStructure.m,
31
problem10.m, 91
problem11.m, 95
problem11b.m, 100
problem12.m, 104
problem13.m, 111
problem14.m, 116
problem15.m, 119
problem16.m, 126
problem16Buckling.m, 139
problem16vibrations.m, 133
problem16vibrationsSchultz.m, 135
problem17.m, 149
problem18.m, 152
problem19.m, 165
problem19Buckling.m, 199
problem19Vibrations.m, 189
problem19structure.m, 182
problem2.m, 37
problem20.m, 212
problem21.m, 230
problem3.m, 44
problem3Structure.m, 49
problem3a.m, 46
problem4.m, 55
problem5.m, 58
problem6.m, 64
problem7.m, 69
problem8.m, 75
problem9.m, 84
problem9a.m, 87
shapeFunctionQ4.m, 159
solution.m, 41
solutionStructure.m, 31
srinivasMaterial.m, 215
stresses2D.m, 158
stresses2Dtruss.m, 57
stresses3Dtruss.m, 77
outputDisplacementsReactions.m, 28
Membrane stiffness matrix, 207
Membrane-bending coupling, 207
Membrane-bending stiffness matrix, 207
Mindlin plate theory, 203
Mindlin plates, 161
Modes of vibration, 133, 182
Natural boundary conditions, 144
Natural coordinate system, 34
Natural frequencies, 130, 182
Plane stress, 143
Potential energy, 145
Prescribed degrees of freedom, 28
Problems
Problem 17, 149
Problem 18, 152
Problem 19, 165
Problem16Buckling, 139Index 235
Problem16vibrations, 133
Problem16vibrationsSchultz, 135
problem19Buckling, 199
problem19Vibrations, 189
problem20, 212
Problem21, 230
Problem 1, 23
Problem 10, 91
Problem 11, 95
Problem 11b, 100
Problem 12, 104
Problem 13, 111
Problem 14, 116
Problem 15, 119
Problem 16, 126
Problem 2, 37
Problem 3, 44
Problem 4, 55
Problem 5, 58
Problem 6, 64
Problem 7, 69
Problem 8, 75
Problem 9, 84
Problem 9a, 87
Q4 element, 163
Reactions, 28
Rotation matrix, 104, 113
Shape functions, 34, 35, 80, 124, 145, 147,
163
Shear correction factor, 124, 161, 206
Shear deformations theories
Mindlin theory, 203
Shear locking, 125, 164
Shear stiffness matrix, 207
Spring element, 19
Stiffness matrix, 35
2D frame, 89
2D truss element, 52
3D frame, 103
3D truss, 69
Assembly process, 36
Bar element, 20, 35
Bernoulli beam, 80
Grids, 113
Laminated plate, 207
Mindlin plate, 164
Plane stress, 146
Timoshenko beams, 125
Strain energy, 36, 53
2D frame, 90
Bar element, 33, 36
Bernoulli beam, 79
Laminated plate, 207
Mindlin plate, 161, 164
Plane stress, 145
Timoshenko beams, 123
Stress-strain relations, 163
Stresses
2D truss, 53
Surface tractions, 146
Timoshenko beams, 123
Transformation, 89
Transverse shear stresses, 124, 161
Transverse strains, 161
Two-node bar finite element, 33
Two-node element, 130
Vector of equivalent forces, 36

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