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أخواني في الله أحضرت لكم كتاب Finite Element Analysis of Solids and Structures Sudip S. Bhattacharjee
![كتاب Finite Element Analysis of Solids and Structures F_e_a_20](https://i.servimg.com/u/f61/15/36/74/14/f_e_a_20.jpg) و المحتوى كما يلي :
Contents Preface .xi Author xv Chapter 1 Introduction to Stress Analysis of Solids and Structures .1 Summary 1 1.1 Introduction – A Brief Summary of Key Historical Developments 2 1.2 Role of Analysis/Simulation in Product Engineering 3 1.3 Static Equilibrium of Structures and Analysis of Forces – Statically Determinate and Indeterminate Systems .5 1.4 Stiffness (Displacement) Method of Structural Analysis 7 1.5 Components of Stresses in a 3D Body 11 1.6 Variation of Stresses and Differential Equations of Equilibrium 13 1.7 Stress Boundary Conditions 15 1.8 Practice Problems on Stress Field Visualization with CAE Tools .16 Chapter 2 Strain–Displacement Relationship and Elasticity of Materials 19 Summary 19 2.1 Measurement of Deformation Intensity (Strain) .20 2.2 General Description of Deformation State in 3D Solids .22 2.3 Compatibility of Strain (Deformation Field) in a Body 23 2.4 Stress-Strain Relationships (Hooke’s Law) .23 2.5 Solution of Elasticity Problems Using Stress Distribution Functions .28 2.6 Solution of Elasticity Problems Using Displacement Variation Functions 30 2.7 Stiffness Method (Finite Element Method) of Structural Analysis .32 2.8 Stiffness Properties of 1-D Truss Element Providing Resistance to Axial Deformation Only 34 2.9 Higher Order Truss Element and Model Refinement 39 2.10 Model Preparation for Computer-Aided Analysis of Structures with Finite Element Simulation Software 41 2.11 Practice Problems – Stress Analysis of a Cable with Finite Element Analysis Software ABAQUS .43vi Contents Chapter 3 Analysis of Solids Represented by 2D Stress Fields 47 Summary 47 3.1 Plane Strain – A Special Form of Elasticity Problem 48 3.2 Plane Stress – A Special Form of Elasticity Problem 50 3.3 Stress Functions for 2D Plane Strain and Plane Stress Elasticity Problems 51 3.4 Example Use of Stress Function Approach to Solve Elasticity Problems 54 3.5 Stiffness Method Analysis of Solids Represented by 2D Stress-Deformation Fields .60 3.6 Iso-Parametric Definition of 2D Solid Finite Elements 63 3.7 Numerical Calculation of Stiffness Matrix for Iso-Parametric 2D Solid Finite Elements 67 3.8 Higher Order Plane-Stress/Plane-Strain Elements 71 3.9 Constant Stress/Strain Triangular Element 73 3.10 Convergence of Finite Element Model Solutions 76 3.11 Selection of Element Types for 2D Stress Analysis with FEA Software Packages 78 3.12 Description of Load Types in General Stress Analysis Problems 82 3.13 Refined Finite Element Model Preparation with a Pre-Processor (HyperMesh) 83 3.14 Practice Problems – Stress Analysis of a Plate Subjected to In-Plane Stresses 84 Chapter 4 FEA Model Preparation and Quality Checks .85 Summary 85 4.1 Adaptation of Finite Element Mesh to Stress Field Variations .85 4.2 Element Type Selection for a Given Stress Analysis Problem 88 4.3 Initial Geometric Design of Structural Components .91 4.4 FEA Model Preparation for Case-Study: Plate with Hole (by Using HyperMesh) .92 4.4.1 CAD Data Preparation for FEA Modeling .92 4.4.2 Finite Element Meshing of the CAD Geometry .94 4.4.3 Finite Element Mesh Quality Checks .95 4.4.4 Material and Part Property Assignment 98 4.4.5 Analysis Parameters – Boundary Constraints, External Loads and Model Outputs 99 4.4.6 Definition of ABAQUS Analysis Step 101 4.4.7 Exporting the Model Input File for ABAQUS Analysis 101Contents vii 4.5 Post-Processing of FEA Results 103 4.6 Practice Problems – Stress Analysis of Plate with Hole 103 Chapter 5 Stress Analysis of Axisymmetric and General 3D Solids 107 Summary 107 5.1 Axisymmetric – A Special Form of 3D Elasticity Problems 107 5.1.1 Equations of Equilibrium in Polar Coordinates 109 5.1.2 Strain–Displacement Relationships 109 5.1.3 Stress–Strain Relationships 110 5.1.4 Compatibility Condition .111 5.2 Stress Analysis of Axisymmetric Example – Thick Wall Cylinder .111 5.3 Finite Element Analysis of Axisymmetric Problems .115 5.4 Stress Analysis of Three-dimensional Bodies – Concentrated Loads .117 5.4.1 Stresses in a Semi-Infinite Solid Subjected to Concentrated Normal Force on the Boundary 117 5.4.2 Stresses in a Solid Beam Subject to Concentrated Lateral Forces .119 5.5 Finite Elements for Stress Analysis of General 3D Solids 121 5.6 Three-Dimensional FEA Model Preparation and Element Quality Checks 123 5.7 Practice Problems – Stress Analysis of Axisymmetric and 3D Solids 124 Chapter 6 Deformation Analysis of Beams for Axial, Bending, Shear, and Torsional Loads 127 Summary 127 6.1 Bending Stresses in a Beam 128 6.2 Stresses Due to Transverse Shear 129 6.3 Transverse Normal Stress in a Beam .132 6.4 Torsional Response of a Beam 133 6.5 Beam Response to Combined Load Effects 142 6.6 Elastic Bending Deflection of Beams 144 6.7 Stress Analysis of Curved Beams 146 6.8 Stiffness Properties of Prismatic Euler–Bernoulli Beam Elements 148 6.9 Stiffness Properties of Beams Including Shear Deformation .154 6.10 Analysis of Beams and Frames with FEA Software Packages 158 6.11 Practice Problems: Load–Deflection Analysis of Beams 162viii Contents Chapter 7 Analysis of 3D Thin-Wall Structures (Plates and Shells) 165 Summary 165 7.1 Bending Stresses and Strains in a Plate .166 7.2 Analytical Solutions for Plate Bending Deflections 169 7.3 In-plane Membrane Stress Resistance of a Shell 172 7.4 Bending Stiffness of Flat Plate Element 174 7.5 Flat Shell Element as a Combination of Plate Bending and Membrane Elements .176 7.6 Shear Deformation in Plates 179 7.7 Curved Shell Elements 181 7.8 Shell Element Mesh Quality and Integration Rules 184 7.9 Analysis of Shells with FEA Software 186 7.10 Practice Problems: Load–Deflection Analysis of Shells .187 Chapter 8 Multi-Component Model Assembly .191 Summary 191 8.1 Element Compatibility and Convergence of Simulation Results .191 8.2 Modeling of Kinematic Joints in Structural Assemblies 194 8.3 Deformable Joint Elements for Part-to-Part Connections .198 8.4 Mesh-Independent Fasteners for Part-to-Part Connections .201 8.5 Simulation of Part-to-Part Interface Contacts 203 8.6 Thin-Layer Interface Elements 207 8.7 Modular Organization of Data in Multi-component Model Assembly 209 8.8 Result Quality Checks .211 8.9 Practice Problems: Analysis with Multi-Component Model Assemblies .213 Chapter 9 Interpretation of Stress Analysis Results for Strength and Durability Assessment 217 Summary 217 9.1 Engineering Properties of Materials 218 9.2 Stress-Strain Results from Linear Elastic Finite Element Analysis of Solids 219 9.3 Strength Assessment of Solids – Use of Material Failure Theories .222 9.4 Post-Processing of Finite Element Stress Analysis Results 229 9.5 Stress Analysis for Durability (Fatigue Life) Assessment of Structures .230Contents ix 9.6 Structural Safety Assessment and Quality of FEA Stress Results .234 9.7 Stresses at Points of Discontinuity: Stress Intensity Factor .235 9.8 Practice Problems: Assessment of Structural Strength and Durability 238 Chapter 10 Vibration Frequency Analysis of Structures with FEA Model .241 Summary 241 10.1 Introduction – Dynamic Response of Structures .242 10.2 Vibration Frequency of a Single Degree of Freedom Spring-Mass System 243 10.3 Forced Vibration Response and Resonance of Structures .245 10.4 Frequency Separation and Design Targets for Structures 249 10.5 Vibration Mode Shape and Frequency of SDOF Structures 250 10.6 Vibration Frequencies of MDOF Systems 253 10.7 Calculation of System Mass Matrix for MDOF Systems 255 10.8 Numerical Calculation of Vibration Mode Shapes and Frequencies of MDOF Systems 259 10.9 Vibration Frequency Analysis with ABAQUS .261 10.10 Practice Problems: Vibration Analysis of Structures .262 Chapter 11 Linear Dynamic Response Analysis of Structures .265 Summary 265 11.1 Linear Elastic Response of SDOF Systems to Impulsive Loading .265 11.2 Response Spectrum of Linear Dynamic Systems 268 11.3 Time-Domain Analysis of Dynamic Structural Response 270 11.4 Numerical Integration Parameters for Time-Domain Analysis of Structures 271 11.5 Time Domain Analysis of MDOF Systems .273 11.6 Mode Superposition Method for Analysis of MDOF Systems 276 11.7 Explicit Time-Domain Analysis of MDOF Systems .277 11.8 Linear Dynamic Response Analysis with ABAQUS .282 11.9 Practice Problems: Dynamic Response Analysis of Structures 286x Contents Chapter 12 Nonlinear Analysis of Structures 289 Summary 289 12.1 Simulation of Nonlinear Force-Deformation Response of Structures .290 12.2 Nonlinear Material Models for Finite Element Simulation of Structures 294 12.3 Simulation of Large Deformation Response – Nonlinear Geometric Problems .301 12.4 Nonlinearity Arising from Changes to Inter-Body Contacts .307 12.5 Nonlinear Dynamic Response Analysis of Structures .308 12.6 Material Failure Simulation in Nonlinear Finite Element Analysis .310 12.7 Computational Methods for Structural Form Simulation 311 12.8 Practice Problems: Nonlinear Response Analysis of Structures .313 References .315 Index 319xi Index A analysis of forces statically determinate, 5 statically indeterminate, 5 aspect ratio, 96 average acceleration method, of dynamic response analysis, 272 axisymmetric problems, 115 B bending deflection of beams, 144 bending stiffness of composite shell, 179 of plate, 174 bending stresses in beams, 128 in dissimilar materials, 129 in plate, 166 bending theory, see Euler–Bernoulli body force, 13, 33 boundary constraints, 9, 34, 99 buckling, thin-wall members, 4 bulk modulus of elasticity, 28 C CAD (computer-aided-design) tools, 16 CAD data exchange formats, 91 CAE (computer aided engineering) tools, 16 central difference theorem of explicit time-domain solution, 278 CFL law (Courant–Friedrichs–Lewy) of time step limit in explicit dynamic analysis, 279 cohesive joint element, 207 combined load effects on beam response, 142 compatibility condition among strains, 23 plane strain, 50 plane stress, 51 in the polar coordinate, 111 completeness requirement, 191 concentrated forces/loads, 33, 45, 117, 119 connector element, 195, 198, 203 constant stress/strain, 73, 122 constitutive models, see stress–strain relationship contact force, 204 contact simulation, 204 contact stiffness, 204 convergence of finite element model solutions, 76, 191 crack-tip, 235 crash strength, 314 crashworthiness simulation, 258, 312 curvature of plate bending, 175 curved beams, 146 cumulative fatigue damage, 231 cut-plane, 212 cyclic load, 231 D damping matrix, 274 damping ratio, 247 damping resistance, 247 deep beam, 146 deformation axial, 10 bending, 88 distortional, 20 shear, 20 torsional, 134 degree of freedom master, 195 multi-degree of freedom (MDOF) system, 9 single degree of freedom (SDOF) spring, 7 slave, 195 direction cosines, 137, 148, 181 displacement control method of structural analysis, 293 field, 32, 34 method of structural analysis, 7 variation functions, 30 virtual, 32 distortional stress, 227 distributed loads, 45 ductile material, 3, 219 Duhamel integral, 266 durability, 230 dynamic amplification factor damped system, 247 to impact force, 267 un-damped system, 246 dynamic equilibrium MDOF, 273 SDOF, 247, 270 dynamic response nonlinear, 308 of flexible spring-mass, 243, 271 of free rigid body mass, 242 to impact force, 266320 Index dynamic response analysis average acceleration method, 272 modal superposition method, 276 time domain analysis, 270 E earthquake load, 249 eigenpair, 255 eigenvalue, 254 eigenvalue shift, 259 eigenvector, 254 elasticity property matrix, 27, 294 tangential, 296 elastic modulus, 24, 219 element characteristic length, for explicit dynamic analysis, 281 element compatibility, 191 element types axisymmetric, 117 beam element, 154, 159 mass, 261 mixed, 192 shell elements, 186 3D elements, 122 truss elements, 34, 39 2D solids, 78, 88, 94 energy error, 102 engineering strain, 24, 218 engineering stress, 24, 218 equations of dynamic equilibrium, 274 equilibrium in polar coordinates, 109 static equilibrium, 7, 11 strain compatibility, 23 stress compatibility, 28 stress equilibrium, 14 stress–strain relationship, 23 Euler–Bernoulli beam theory, 128, 146, 148, 151, 154, 160, 251 F fatigue crack growth, 237 fatigue life, 230 FEA solvers, 16 finite elements axisymmetric, 117 beam elements, 154, 160 plane-strain, 80 plane-stress, 78 shell, 176, 181, 186 3D solid, 122 truss, 34, 42 first moment of the beam section, 130 flexibility method of structural analysis, 6 flexural rigidity, 144, 169 force method pf structural analysis, 6 form-finding, 312 fracture modes, 236 fracture response simulation, 310 fracture toughness, 236 free-vibration response, 243 frequency response function, 284 frequency separation, 249 fundamental frequency of vibration, 252 G gage length, 20 Gauss quadrature, 69 Green-Lagrange strains, 301 Guyan reduction, 259 H hardening plasticity, 295 Hooke’s law, 23, 110, 168, 294 hourglass deformation, 71, 90, 184 householder-QR method, 260 high cycle fatigue, 230 hinge connection, 200 hydrostatic pressure, 28 hysteresis, 248 I implicit time integration, 272 impulse, 266 in-compatible element deformation, 73, 80, 193 initial penetration, 206 interface contact, 203 interface element, 207 interpolation functions beam deformation, 151 coordinate, 34 displacement, 36 general 3D solids, 122 3-node truss elements, 40 2D deformation fields, 60 2-nodetruss element, 35 intersection, 206 J Jacobian definition of, 66 determinant, 96 distorted 2D solid element, 66, 70 inverse of, 68 physical interpretation, 65 3D system, 122 triangular elements, 74Index 321 J-integral, 236 Johnson–Cook plasticity model, 295, 299, 309 joint elements, deformable, 198 K kinematic constraint, 194 kinematic coupling, 196, 201 kinematic joints, 194 kinematic relations, 270 Kronecker delta, 254 L Lagrangian element, 72, 80 Lagrangian method of nonlinear structural analysis total, 305 updated, 306 Lagrangian multiplier, 195, 204 Lame’s constants, 28 Lame’s equations, 13 Lanczos transformation method, 260 large deformation response, 301 linear acceleration method, of dynamic response analysis, 272 load-deformation response monotonic, 293 snap-back, 293 load types, 82 low cycle fatigue, 230 M mass matrix calculation, 255 consistent, 256 lumped, 257 master–slave concept, 204 master surface, 204 material failure simulation, 310 material failure theories, 222 matrix equilibrium equations, 9 measurement of deformation intensity, 20 mechanics of materials, 2 membrane analogy, 138 membrane element, 183 membrane stress resistance of shell, 172 mesh-dependent localization of failure, 311 mesh-independent fasteners, 201 mesh quality checks, 95, 123, 184 mesh refinement effect on predicted results, 76 h-adaptivity, 86 p-adaptivity, 86 selective, 90 Mindlin’s plate deformation theory, 179 Miner’s rule, 231 Modak’s T-method, for dynamic response analysis, 272 modal damping, 274, 277 modal superposition method of dynamic response analysis, 269, 276 model preparation a-to-g key process steps, 102 axisymmetric problem, 116 for contact simulation, 206 multi-component, 209 plate with hole, 92 three-dimensional FEA, 123 modular organization of data, 209 Mohr circle of stress, 226 moment-curvature relationship of plates, 176 moment of inertia, 153 moment release, 194 momentum, 242 MPC (multi-point constraint), 195 multi-component model, 209 N neutral axis, 10, 56, 128, 130, 147 Newmark integration, 272 Newton–Raphson method, 291 Newton’s second law, 242 nodes, 33 nonlinear analysis contact problems, 307 dynamic response, 308 force-deformation response, 290 geometric problems, 301, 306 material models, 294 numerical damping, in dynamic response analysis, 272 numerical integration examples, 68 Gauss quadrature, 69 of higher order elements, 72 of mass matrix, 256 parameters for time-domain dynamic response analysis, 271 reduced, 90 of stiffness function, 70, 122 P parasitic shear, 89 Paris law, 237 part property, 98 part-to-part connections, 198, 201 part-to-part contact, 203 patch test, 76 penalty method, 204 penalty stiffness, 204322 Index period of vibration, 244 perturbation of stress flow, 86 Piola–Kirchoff stresses, 305 plane strain, 48 plasticity models associated, 298 non-associated, 298 plastic potential function, 298 plastic strain, 295 plate bending theory Kirchoff’s, 166 Mindlin’s, 179 Poisson’s ratio, 25 polar coordinates, 107 polar moment of inertia, 141 post-processors, 16 pre-processors, 16, 83 principal strain, 225 principal stress, 223 principle of superposition, of stresses, 25 prismatic member, 148 product development, automotive, 4 R radius of curvature, 147 Rayleigh damping coefficient, 274 Rayleigh’s quotient, 255 reference node, 196 resonance, 245 response spectrum, 268 result quality checks, 77, 103, 211, 234 rigid body, 196 rigid body motion, 9, 191 S SDOF (single degree of freedom) system, 7 self-contact, 206 semi-infinite solid, 119 shape functions beam deflection, 151 general 3D solids, 121 higher order elements, 71 iso-parametric definition, 64 2D deformation fields, 60 shear center, 143 shear correction factor, 146, 156, 158, 183 shear deformation of cantilever beam, 59 in plates, 179 shear flow through beam, 143 shear locking of beam, 158 shear modulus, 27 shear stress distribution in beam, 131 in plate, 168 simulation model limitations, 4 singular element, 236 size effect, 220 skeletal beam, 10, 148 structural members, 10 skew, 96 slave node, 204 slender beam, 154, 158 S–N curve, 231 softening response of structures, 293 software CAD, 16, 91–94, 102, 123, 186, 214 CAE tools, 16 FEA solvers, 16, 211, 212 finite element simulation, 78, 83 post-processor, 16 pre-processor, 16, 48, 83, 84, 90, 92, 93, 95, 98 spot-welds, 197, 233 steady state dynamics, 285 stiffness matrix of beam element, 153 of finite element, 33 of flat shell element, 180 fully integrated, 70 of general 3D solids, 122 membrane resistance, 177 positive definite, 9, 40 reduced integration, 71 spring elements, 7 system assembly, 8, 34, 39 tangent, 292, 305 of 2-node truss element, 37 stiffness method of structural analysis, 7, 32, 60 stiffness property beam shear deformation, 156 calculation of finite elements, 32 Euler–Bernoulli beam element, 148 flat shell element, 177 joint element, 200 spring, 7 truss element, 34 strain bi-directional, 21 compatibility, 23 components, 22 energy, 9, 34, 219 energy release rate, 237 failure, 3 Green-Lagrange, 301 hardening parameter, 296 measurement of, 20 normal, 20 plastic, 295 rosette, 21 shear, 20Index 323 thermal, 83 virtual, 32 strain–displacement relationship axisymmetric problem, 115 beam element, 152 in differential element, 22 general 3D solid, 121 Green-Lagrange, 301 iso-parametric 2D solid, 67 nonlinear, 302 plane-strain, 48, 61 plane-stress, 50 plate bending, 167 in polar coordinates, 110 shell element, 181 two-node truss element, 36 triangular element, 76 stress at the crack tip, 235 beam section, 10 boundary conditions, 15, 50 components in 3D solids, 11 concentration factor, 86, 234 equilibrium equations, 14, 49 initial, 33 intensity factor, 235 normal, 12, 129 Piola–Kirchoff, 305 shear, 12 singularity, 78, 88 tangential, 114 thermal, 83 torsional, 134 transverse shear, 129 vector, 13 stress function in polar coordinates, 109 prismatic bar, 29 thin cantilever beam, 54 torsional problem, 136 for 2D stress field, 52, 84 stress flow in a plate with hole, 86 torsion, 139 stress–strain relationship axisymmetric, 110, 116 for bending deformation of beam, 152 for curved shell elements, 182 incremental, 299 nonlinear, 295 plane-strain, 49, 63 plane-stress, 51, 63 for plate bending, 168 strain-rate effect, 309 truss element, 36 structural adhesive, 233 structural analysis, step-by-step load control, 293 structural form simulation, 311 structural instability, 40 strum sequence check, 261 stamping of metal blank, 311 stress wave propagation, 280 subspace iteration method, 260 surface force, 33 T tangent modulus, 297 Taylor’s series for strain definition, 13 for time domain response, 272 thin-layer joint, 207 thin shell, 179 time domain analysis error in predicted response, 273, 279 explicit method, 277 implicit method, 272 of MDOF system, 273 of SDOF system, 271 time step limit, 273, 276, 279 Timoshenko beam theory, 146, 156 torsional response of beam, 133 torsional rigidity, 135 torsional stiffness of member, 141 transformation matrix of beam element stiffness matrix, 148 of global and local DOF of truss element, 37 transverse normal stress in beam, 132 true stress, 24, 219 true strain, 24, 219 U ultimate strength analysis of beam, 291 V variable amplitude stress cycle, 232 vector of applied loads, 11, 33 nodal displacements, 11, 33 vector iteration method, 260 velocity of stress wave, 280, 282 vibration frequency MDOF, 253 numerical calculation, 259 SDOF, 244 vibration mode shape MDOF, 259 numerical calculation, 259 SDOF, 250 simply supported beam, 250 simply supported plate, 252324 Index virtual cut-plane, 212 displacement, 32 work, 32 von Mises stress, 228, 297 W warpage, 96 warpage-induced deformation, 135 Wilson θ method of dynamic response analysis, 272 Winker formula, 147 Y yield function, 297 yield strength, 3, 219 Young’s modulus, 24
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