MATLAB与Abaqus合作的有限元分析入门指南

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"《有限元分析入门：使用MATLAB®与Abaqus》是一本专门为希望深入了解有限元方法的读者精心编写的英文原版教材。该书由Amar Khennane撰写，并在2013年由Taylor & Francis Group出版。这本书以其详尽的内容和实用性，向初学者展示了如何结合MATLAB®和Abaqus这两个强大的工程计算软件进行有限元分析。有限元素法（Finite Element Method, FEM）是一种数值分析方法，广泛应用于结构力学、流体力学、热传导等领域，用于求解复杂的物理问题。FEM将连续体分解成许多互不重叠的子区域，每个子区域被称为有限元素，通过解决这些小区域的简化模型来近似整个系统的响应。这种方法极大地简化了复杂结构的分析，提高了计算效率。书中涵盖了有限元的基本原理，包括网格划分、节点定义、单元类型及其特性，以及如何建立和求解方程组。MATLAB®在这里起到了关键作用，它是数学计算的强大工具，提供了方便的编程环境和丰富的函数库，使得用户能够快速实现有限元模型的构建和求解。同时，Abaqus作为一个专业级的有限元分析软件，被用来展示如何在实际工程问题中应用这些理论知识。书中还包含了实际案例分析，让读者逐步掌握从问题定义、模型建立到结果解读的全过程。此外，特别强调的是，尽管书中涉及MATLAB®的使用，但并不意味着它在教学中对特定的编程技巧或MATLAB®的特定版本有所偏袒，而是旨在教授有限元分析的核心概念。《有限元分析入门：使用MATLAB®与Abaqus》对于想要进入工程领域，特别是结构工程师、机械工程师和数值模拟分析师的学生和专业人士来说，是一本非常有价值的参考书籍。它不仅提供理论知识，而且通过实践操作，帮助读者建立起扎实的有限元分析基础，并熟练掌握两个实用软件的使用。由于版权原因，书中某些部分可能受到The MathWorks的许可限制，但其价值在于它提供了一个完整的学习路径，助力读者在有限元领域取得成功。"

List of Figures xv

FIGURE 3.1 Beamelement...................................................................... 64

FIGURE 3.2 Differential element of a beam. ................................................... 64

FIGURE 3.3 Nodal degrees of freedom ......................................................... 65

FIGURE 3.4 Statically equivalent nodal loads. ................................................. 68

FIGURE 3.5 Loading, bending moment, and shear force diagrams............................ 68

FIGURE 3.6 Support reactions for individual members. ....................................... 71

FIGURE 3.7 Beam with an internal hinge....................................................... 71

FIGURE 3.8 Beam elements with a hinge....................................................... 73

FIGURE 3.9 Example of a continuous beam. ................................................... 73

FIGURE 3.10 Example 1: Continuous beam results. ............................................ 81

FIGURE 3.11 Problem3.1. ....................................................................... 81

FIGURE 3.12 Problem 3.2 and equivalent nodal loads for elements 3 and 4. .................. 84

FIGURE 3.13 Problem3.3. ....................................................................... 87

FIGURE 3.14 Continuous beam................................................................... 90

FIGURE 3.15 Beam crosssection; dimensionsare in mm....................................... 90

FIGURE 3.16 CreatingtheBeam_Part. .......................................................... 91

FIGURE 3.17 Drawing using the connected line icon. .......................................... 91

FIGURE 3.18 Materialdefinition. ................................................................ 91

FIGURE 3.19 Creating a beam profile. ........................................................... 92

FIGURE 3.20 Enteringthe dimensions ofa profile. ............................................. 92

FIGURE 3.21 Creatinga section. ................................................................. 93

FIGURE 3.22 Editing a beam section............................................................. 93

FIGURE 3.23 Editing section assignments. ...................................................... 94

FIGURE 3.24 Beam orientation................................................................... 94

FIGURE 3.25 Assigning beam orientation. ...................................................... 94

FIGURE 3.26 Rendering beam profile. ........................................................... 95

FIGURE 3.27 Rendered beam..................................................................... 95

FIGURE 3.28 Selecting a beam element.......................................................... 96

FIGURE 3.29 Seeding a mesh by size. ........................................................... 96

FIGURE 3.30 Node and element labels. .......................................................... 97

FIGURE 3.31 Creating a node set................................................................. 97

FIGURE 3.32 Selecting multiple nodes........................................................... 98

FIGURE 3.33 Creatingelement sets. ............................................................. 98

xvi List of Figures

FIGURE 3.34 ImposingBC using createdsets................................................... 98

FIGURE 3.35 Selecting a node set for boundary conditions..................................... 99

FIGURE 3.36 Editing boundary conditions. ..................................................... 99

FIGURE 3.37 ImposingBC using createdsets..................................................100

FIGURE 3.38 Imposing a concentrated load using a node set. ................................. 100

FIGURE 3.39 Imposinga lineloadon an element set. .........................................101

FIGURE 3.40 Fieldoutput........................................................................101

FIGURE 3.41 Submitting a job in Abaqus CAE. ...............................................101

FIGURE 3.42 Plotting stresses in the bottom fiber. .............................................102

FIGURE 4.1 Beam column element with six degrees of freedom............................. 108

FIGURE 4.2 Example1: Portal frame.......................................................... 110

FIGURE 4.3 Frame with an internal hinge. .................................................... 119

FIGURE 4.4 Finiteelement discretization. .................................................... 119

FIGURE 4.5 Statically equivalent nodal loads. ................................................ 120

FIGURE 4.6 Portalframe. ......................................................................124

FIGURE 4.7 Profiles’sections;dimensionsare in mm........................................ 125

FIGURE 4.8 Creating thePortal_frame part. .................................................. 125

FIGURE 4.9 Material andprofilesdefinitions. ................................................ 126

FIGURE 4.10 Creatingsections..................................................................126

FIGURE 4.11 Editing section assignments. ..................................................... 127

FIGURE 4.12 Assigning beam orientation. ..................................................... 127

FIGURE 4.13 Rendering beam profile. ..........................................................127

FIGURE 4.14 Seeding by number................................................................ 128

FIGURE 4.15 Mesh...............................................................................128

FIGURE 4.16 Creatingthe element setRafters. ................................................129

FIGURE 4.17 ImposingBC using createdsets..................................................129

FIGURE 4.18 Imposing a line load in global coordinates. ..................................... 130

FIGURE 4.19 Imposing a line load in local coordinates. ....................................... 130

FIGURE 4.20 Analyzing a job in Abaqus CAE. ................................................131

FIGURE 4.21 Plotting stresses in the bottom fiber (interactive edition). ......................131

FIGURE 4.22 Plotting stresses in the bottom fiber (keyword edition). ........................ 134

FIGURE 5.1 Internal force components. .......................................................136

FIGURE 5.2 Stress components at a point. ....................................................136

FIGURE 5.3 Stress components on a tetrahedron.............................................. 137

List of Figures xvii

FIGURE 5.4 Equilibrium of an infinitesimal cube. ............................................139

FIGURE 5.5 Principal directionsof a stresstensor. ...........................................141

FIGURE 5.6 Tangential and normal components of the stress vector.........................142

FIGURE 5.7 Mohr’s circles. ....................................................................143

FIGURE 5.8 Schematic representation of the deformation of a solid body...................145

FIGURE 5.9 Reference and current configurations. ........................................... 146

FIGURE 5.10 Deformationsof an infinitesimalelement. ......................................147

FIGURE 5.11 Geometrical representation of the components of strain at a point. ............151

FIGURE 5.12 Monoclinic material. ............................................................. 155

FIGURE 5.13 Symmetryof rotation. ............................................................157

FIGURE 5.14 Astateofplane stress............................................................. 162

FIGURE 5.15 Stateof plane strain...............................................................163

FIGURE 5.16 Change of basis. .................................................................. 165

FIGURE 5.17 Displacement field (Problem 5.3). ...............................................167

FIGURE 5.18 Displacement field (Problem 5.5). ...............................................170

FIGURE 5.19 Strain rosette. .....................................................................171

FIGURE 5.20 Problem5.7. ...................................................................... 172

FIGURE 5.21 Displacements without the rigid walls. .......................................... 173

FIGURE 6.1 Graphical comparison of exact and approximate solution. .....................178

FIGURE 6.2 Integration bypartsin two andthreedimensions. .............................. 180

FIGURE 6.3 Infinitesimal element of the boundary. .......................................... 180

FIGURE 6.4 Graphical comparison of the exact and approximate solutions. ................186

FIGURE 7.1 Thick wall with embedded thermocouples. .....................................192

FIGURE 7.2 Finiteelement discretization. ....................................................193

FIGURE 7.3 Finite element approximation. ...................................................195

FIGURE 7.4 Geometrical illustration of the compatibility principle .........................195

FIGURE 7.5 Linear triangle.....................................................................197

FIGURE 7.6 Geometrical transformation for a triangular element............................ 200

FIGURE 7.7 Three-node triangular element with an arbitrary point O. ......................202

FIGURE 7.8 Three-node triangular reference element. .......................................204

FIGURE 7.9 Geometricaltransformation. ..................................................... 204

FIGURE 7.10 One-dimensional elements. ......................................................207

FIGURE 7.11 Two-dimensional quadrilateral elements. .......................................207

xviii List of Figures

FIGURE 7.12 Two-dimensional triangular elements............................................ 208

FIGURE 7.13 Three-dimensional tetrahedric elements. ........................................209

FIGURE 7.14 Three-dimensional brick elements. .............................................. 210

FIGURE 8.1 Positions of the sampling points for a triangle: Orders 1, 2, and 3. ............ 219

FIGURE 8.2 Gauss quadrature over an arbitrary area. ........................................219

FIGURE 8.3 Double change of variables.......................................................220

FIGURE 8.4 Coarse mesh of two 8-nodded elements. ........................................ 221

FIGURE 8.5 Eight elements finite element approximation with two 8-nodded elements.... 222

FIGURE 8.6 Estimation of rainfall using finite element approximation. .....................226

FIGURE 9.1 Discretization error involving overlapping. .....................................234

FIGURE 9.2 Discretization errorinvolving holesbetween elements. ........................235

FIGURE 9.3 Plane elements with shape distortions. ..........................................235

FIGURE 9.4 Geometricaldiscretizationerror..................................................235

FIGURE 9.5 Linear triangular element. ........................................................ 236

FIGURE 9.6 Element nodal forces. ............................................................ 239

FIGURE 9.7 Analysis of a cantilever beam in plane stress. ................................... 240

FIGURE 9.8 Finite element discretization with linear triangular elements. ..................241

FIGURE 9.9 Deflection of the cantilever beam. ............................................... 248

FIGURE 9.10 Stresses along the x-axis.......................................................... 249

FIGURE 9.11 Automatic mesh generation with the CST element. ............................252

FIGURE 9.12 Deflection of the cantilever beam obtained with the fine mesh.................253

FIGURE 9.13 Stresses along the x-axisobtained withthefine mesh. .........................253

FIGURE 9.14 CreatingtheBeam_CST Part.....................................................254

FIGURE 9.15 Drawing using the create-lines rectangle icon. ..................................254

FIGURE 9.16 Creating a partition. ..............................................................255

FIGURE 9.17 Creatinga planestress section. .................................................. 255

FIGURE 9.18 Editing section assignments. ..................................................... 255

FIGURE 9.19 Mesh controls. .................................................................... 256

FIGURE 9.20 Selecting element type. ...........................................................256

FIGURE 9.21 Seeding part by size...............................................................256

FIGURE 9.22 Mesh...............................................................................257

FIGURE 9.23 Imposing BC using geometry. ...................................................257

FIGURE 9.24 Imposing a concentrated force using geometry. .................................257

FIGURE 9.25 Analyzing a job in Abaqus CAE. ................................................258

List of Figures xix

FIGURE 9.26 Plotting displacements on deformed and undeformed shapes. .................258

FIGURE 9.27 Generating a mesh manually in Abaqus. ........................................ 261

FIGURE 9.28 Displacement contour. ............................................................ 263

FIGURE 9.29 Linear strain triangular element. .................................................263

FIGURE 9.30 Automatic mesh generation with the LST element.............................. 271

FIGURE 9.31 Deflection of the cantilever beam obtained with the LST element. ............272

FIGURE 9.32 Stresses along the x-direction obtained withtheLST element. ................ 273

FIGURE 9.33 Aluminum plate witha hole......................................................273

FIGURE 9.34 Making useof symmetry. ........................................................ 273

FIGURE 9.35 CreatingthePlate_LSTPart......................................................274

FIGURE 9.36 Creatinga planestress section. ..................................................274

FIGURE 9.37 Editing section assignments. .....................................................275

FIGURE 9.38 Mesh controls. ....................................................................275

FIGURE 9.39 Seeding edge by size and simple bias. ........................................... 276

FIGURE 9.40 Creating a node set................................................................276

FIGURE 9.41 Creating a surface. ................................................................ 277

FIGURE 9.42 Imposing BC using node sets..................................................... 277

FIGURE 9.43 Imposing a pressure load on a surface. .......................................... 278

FIGURE 9.44 Plotting the maximum in-plane principal stress (under tension)................ 279

FIGURE 9.45 Plotting the maximum in-plane principal stress (under compression)..........279

FIGURE 9.46 Linear quadrilateral element......................................................280

FIGURE 9.47 Element loading. ..................................................................283

FIGURE 9.48 Equivalent nodal loading. ........................................................ 284

FIGURE 9.49 Finite element discretization with 4-nodded quadrilateral elements. ..........285

FIGURE 9.50 Contour of the vertical displacement v

.......................................... 290

FIGURE 9.51 Contour of the stress σ

.......................................................... 291

FIGURE 9.52 Automatic mesh generation with the Q4 element. ..............................295

FIGURE 9.53 Contour of the vertical displacement v

.......................................... 295

FIGURE 9.54 Contour of the stresses along the x-axis σ

. .................................... 295

FIGURE 9.55 Creatingthe Beam_Q4 Part. .....................................................296

FIGURE 9.56 Creating a partition. ..............................................................296

FIGURE 9.57 Creatinga planestress section. ..................................................297

FIGURE 9.58 Editing section assignments. .....................................................297

FIGURE 9.59 Mesh controls. ....................................................................297

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