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 |  موضوع: كتاب Practical Finite Element Analysis - For Mechanical Engineers الأحد 18 أغسطس 2024, 2:03 am | |
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أخواني في الله
أحضرت لكم كتاب
Practical Finite Element Analysis - For Mechanical Engineers
Dominique Madier
و المحتوى كما يلي :
FEA Analyst
TABLE OF CONTENTS
PREFACE 1
Chapter 1 DEFINING FINITE ELEMENT ANALYSIS 5
1.1 OVERVIEW 5
1.2 METHODS FOR SOLVING AN ENGINEERING PROBLEM . 6
1.3 THE DIFFERENT NUMERICAL METHODS . 7
1.4 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS (PDEs) . 8
1.5 WHAT IS FINITE ELEMENT ANALYSIS (FEA)? 10
Chapter 2 WORKING WITH FEA 17
2.1 FROM MATHEMATICS TO COMPUTER SCIENCE 17
2.2 THE MAGIC OF DISCRETIZATION 17
2.3 PRE-PROCESSING 20
2.4 SOLVING 20
2.4.1 DIRECT SOLVER 21
2.4.2 ITERATIVE SOLVER 22
2.5 POST-PROCESSING .22
2.6 FEA PROCESS SUMMARY .23
2.7 CAPABILITIES OF FEA SOFTWARE 27
2.8 HOW ACCURATE IS FEA? .29
2.8.1 CAD SIMPLIFICATION .29
2.8.2 DISCRETIZATION 29
2.8.3 MODELING OF THE JOINTS .30
2.8.4 MATERIAL 30
2.8.5 LOADING 30
2.8.6 BOUNDARY CONDITIONS .31
2.8.7 BEHAVIORS CAPTURED BY FEA 31
2.8.8 CONCLUSION 31
2.9 WHY DO FINITE ELEMENT ANALYSIS? .32
2.10 HOW CAN FEA HELP YOU? 33
2.11 WHAT IS NEEDED TO PERFORM AN FE SIMULATION? .33
Chapter 3 BECOMING AN FEA SPECIALIST 37
3.1 OVERVIEW 37
3.2 WHAT DO YOU NEED TO LEARN IN THE FEA FIELD? .38
3.3 GUIDELINES FOR FEA LEARNING .39
3.4 WHEEL OF STRUCTURAL FEA COMPETENCIES .42
3.5 CONCLUSION 42
Chapter 4 HISTORY OF FEA 45
4.1 THE PIONEERS 45
4.2 FEA TIMELINE .46
Chapter 5 BASIS OF FINITE ELEMENT METHOD THEORY 49
5.1 OVERVIEW 49
5.2 THE EQUILIBRIUM EQUATION .50PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
TABLE OF CONTENTS
5.3 DISPLACEMENT METHOD 51
5.3.1 THREE CONDITIONS .51
5.3.2 STIFFNESS MATRIX 52
5.3.3 LINEAR SPRING MODEL 53
5.3.4 APPLICATION TO THE TWO-SPRING SYSTEM 55
5.3.5 APPLICATION TO THE FOUR-SPRING SYSTEM .58
5.3.6 APPLICATION TO A PARALLEL-SPRING SYSTEM 59
5.4 PRINCIPLE OF MINIMUM POTENTIAL ENERGY 60
5.5 ELEMENT STIFFNESS MATRIX FOR VARIOUS TOPOLOGIES 62
5.5.1 OVERVIEW .62
5.5.2 DEGREES OF FREEDOM 62
5.5.3 SHAPE FUNCTIONS .64
5.5.4 1D TRUSS ELEMENT .64
5.5.5 1D BEAM ELEMENT 73
5.5.6 2D ELEMENTS .75
5.5.7 3D SOLID ELEMENT 89
5.6 HOW IS THE STIFFNESS MATRIX ASSEMBLED? 91
5.6.1 MATRIX ASSEMBLY .91
5.6.2 TAKING ADVANTAGE OF SPARSITY AND SYMMETRY .95
5.6.3 BANDED MATRIX .96
5.6.4 SKYLINE MATRIX STORAGE 97
5.7 HOW ARE FEM EQUATIONS SOLVED? .99
5.7.1 DIRECT SOLUTION .99
5.7.2 ITERATIVE SOLUTION 101
Chapter 6 DEFINING YOUR FEA STRATEGY 105
6.1 OVERVIEW . 105
6.2 TIME . 106
6.3 THE 10 STEPS TO FOLLOW 106
6.4 EXPOSE THE PROBLEM . 107
6.5 DEFINE THE GOALS . 107
6.6 ANALYZE THE HISTORY . 108
6.8 EVALUATE THE BOUNDARIES AND SURROUNDING ENVIRONMENT 108
6.9 UNDERSTAND THE LOADING AND PREDICT THE LOAD PATH .109
6.10 SELECT THE ELEMENT TYPES AND MODEL SIZE .109
6.11 PREDICT THE FINAL RESULTS 109
6.12 REVIEW THE PLAN 110
6.13 14 QUESTIONS YOU SHOULD BE ABLE TO ANSWER BEFORE YOU BEGIN MODELING 110
6.14 LARGE-SCALE MODELING TECHNIQUES .111
6.15 CONCLUSION . 112
Chapter 7 THE LIBRARY OF ELEMENTS 115
7.1 OVERVIEW . 115
7.2 ELEMENT TYPES . 116
7.2.1 OVERVIEW 116
7.2.2 1D ELEMENTS 117
7.2.3 2D ELEMENTS 122
7.2.4 3D ELEMENTS 127
7.2.5 SPECIAL ELEMENTS . 129
7.3 ELEMENT SELECTION CRITERIA .130
7.3.1 ELEMENT TYPE 130PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
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7.3.2 DEGREES OF FREEDOM . 130
7.3.4 COST . 131
7.3.5 ACCURACY 131
7.4 HOW TO CHOOSE THE RIGHT ELEMENT 131
7.4.1 PREDICT YOUR STRUCTURE’S BEHAVIOR 131
7.4.2 EXPERIMENT YOUR LIBRARY OF ELEMENTS 131
7.4.3 GEOMETRY SIZE AND SHAPE .131
7.4.4 ELEMENT ORDER: LINEAR OR QUADRATIC? .132
7.4.5 INTEGRATION SCHEME 135
7.4.6 CHOOSE THE ELEMENTS IN RELATION TO THE SOLUTION .140
7.4.7 RULES FOR SELECTING THE RIGHT ELEMENTS 140
7.5 SHEAR LOCKING . 141
7.5.1 WHAT IS SHEAR LOCKING? . 141
7.5.2 HOW TO PREVENT SHEAR LOCKING .142
7.6 HOURGLASSING . 143
7.6.1 WHAT IS HOURGLASSING? . 143
7.6.2 HOW TO PREVENT HOURGLASSING .144
7.7 EXAMPLES . 144
7.7.1 QUADRILATERAL ELEMENTS VS TRIANGULAR ELEMENTS 144
7.7.2 HIGHER ORDER TETRAHEDRAL ELEMENTS VS LOWER ORDER ELEMENTS (TET10 VS TET4) .146
7.7.3 EFFECT OF THE INTEGRATION SCHEME ON SHEAR LOCKING AND HOURGLASSING 150
Chapter 8 MESHING 153
8.1 OVERVIEW . 153
8.2 UNDERSTANDING ELEMENT BEHAVIOR .155
8.3 PLANNING THE MESHING . 155
8.3.1 STUDY THE GEOMETRY IN DETAIL .156
8.3.2 CLEAN UP THE GEOMETRY . 156
8.3.3 SELECT THE ELEMENT TYPES .156
8.4 SELECTING THE ELEMENT SIZE 157
8.4.1 FACTORS THAT INFLUENCE MESH SIZE 157
8.4.2 DEFLECTION, STIFFNESS, OR STRESS? 157
8.4.3 PREDICT AND MATCH THE DEFORMED SHAPE .157
8.4.4 MESHING OF CRITICAL REGIONS .158
8.4.5 KEEP IT SIMPLE WHEN THE DESIGN IS NOT MATURE .158
8.5 HOW TO DO MESH REFINEMENT .159
8.5.1 WHY DO MESH REFINEMENT? 159
8.5.2 THE MESH REFINEMENT PROCESS 159
8.5.3 ADVANTAGES AND DISADVANTAGES OF MESH REFINEMENT 159
8.5.4 EXAMPLES OF MESH REFINEMENT TECHNIQUES 160
8.5.5 CONVERGENCE STUDY METHODOLOGY .163
8.5.6 OVER WHAT DISTANCE IS THE MESH REFINED? 164
8.5.7 CAN YOU USE AN EXISTING CONVERGENCE STUDY IN OTHER MODELS? 165
8.5.8 THE DIFFERENT MESH REFINEMENT METRICS .165
8.5.9 CONVERGENCE STUDY GUIDELINES 166
8.5.10 EXAMPLE OF A CONVERGENCE STUDY 166
8.6 WHAT IS A PHYSICAL INTERFACE? 169
8.7 WHAT ARE THE PREFERRED SHAPES FOR 2D AND 3D MODELS? .169
8.8 HOW TO DO A MESH TRANSITION 170
8.8.1 MESH TRANSITION USING VARIOUS ELEMENT TYPES .170
8.8.2 MESH TRANSITION USING HIGHER ORDER ELEMENTS .171PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
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8.8.3 MESH TRANSITION BETWEEN DISSIMILAR ELEMENT TYPES .171
8.9 1D MESHING RULES . 174
8.10 2D MESHING RULES . 174
8.10.1 WHY MESH IN 2D INSTEAD OF 3D? .174
8.10.2 THE MID-PLANE CONCEPT 175
8.10.3 THE TWO RULES OF MID-PLANE CREATION .176
8.10.4 VARIABLE THICKNESS . 176
8.10.5 COMPARISON BETWEEN LINEAR AND QUADRATIC ELEMENTS 177
8.10.6 RULES FOR MODELING HOLES AND FILLETS 179
8.10.7 HOW TO CHECK A 2D MESH 180
8.10.8 THE FOUR MOST COMMON 2D MESHING ERRORS 182
8.10.9 HOW TO IMPROVE YOUR 2D MESH QUALITY .183
8.10.10 OTHER RECOMMENDATIONS FOR 2D MESHING 183
8.11 3D MESHING RULES . 184
8.11.1 TETRAHEDRAL MESHING TECHNIQUES .184
8.11.2 RECOMMENDATIONS FOR TETRAHEDRAL MESHING .188
8.11.3 LINEAR VS QUADRATIC TETRAHEDRAL ELEMENTS 189
8.11.4 HOW TO CHECK A TETRAHEDRAL MESHING .189
8.11.5 HEXAHEDRAL MESHING TECHNIQUES .190
8.11.6 HOW TO CHECK HEXAHEDRAL MESHING .191
8.11.7 ARE YOU ACTUALLY FACED WITH A 3D PROBLEM? .192
Chapter 9 SETTING YOUR UNITS 195
9.1 CONSISTENT SYSTEMS OF UNITS 195
9.2 THE MASS PROBLEM . 196
9.3 WEIGHT AND MASS DENSITY OF COMMON MATERIALS 198
9.4 ENGINEERING UNITS FOR COMMON VARIABLES 199
Chapter 10 MATERIAL MODELING 201
10.1 OVERVIEW . 201
10.2 ISOTROPIC MATERIAL 201
10.2.1 DEFINING AN ISOTROPIC MATERIAL .201
10.2.2 STRESS AND STRAIN 202
10.2.3 STRESS-STRAIN CURVE . 202
10.2.4 PLASTIC AND ELASTIC STRAIN .203
10.2.5 STRAIN HARDENING . 204
10.2.6 STRESS-STRAIN CURVE USING THE RAMBERG-OSGOOD MODEL 206
10.2.7 STRESS-STRAIN CURVE USING THE HOLLOMON MODEL 207
10.2.8 TRUE STRESS AND STRAIN 208
10.2.9 SUMMARY OF THE TYPICAL BEHAVIORS OF METALLIC MATERIALS 209
10.3 TWO-DIMENSIONAL ORTHOTROPIC MATERIAL .210
10.4 TWO-DIMENSIONAL ANISOTROPIC MATERIAL .210
10.5 THREE-DIMENSIONAL ANISOTROPIC MATERIAL 211
10.6 THREE-DIMENSIONAL ORTHOTROPIC MATERIAL 211
Chapter 11 DEFINING LOADS AND BOUNDARY CONDITIONS 215
11.1 OVERVIEW . 215
11.2 WHAT IS A BOUNDARY CONDITION? .215
11.3 WHY DO WE NEED BOUNDARY CONDITIONS? .215
11.4 WHAT ROLE DO BOUNDARY CONDITIONS PLAY? 216
11.5 THE DIFFERENT TYPES OF BOUNDARY CONDITIONS .216PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
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11.6 USING BOUNDARY CONDITIONS TO CONSTRAIN A MODEL .217
11.6.1 WHAT IS RIGID BODY MODE? 217
11.6.2 WHAT IS A MECHANISM? 218
11.6.3 HOW TO DETECT MECHANISMS IN AN FEA .219
11.6.4 CONSTRAINT TYPES . 219
11.6.5 WHAT ARE SINGLE-POINT CONSTRAINTS? .219
11.6.6 EXAMPLES OF CONSTRAINTS FOR 2D AND 3D PROBLEMS 221
11.6.7 COMPATIBILITY OF BOUNDARY CONDITIONS WITH ELEMENTS .222
11.6.8 CONSTRAINTS AND ENFORCED DISPLACEMENT 225
11.6.9 HOW TO USE BOUNDARY CONDITIONS TO MODEL SYMMETRY AND ANTI-SYMMETRY .225
11.7 INFLUENCE OF BOUNDARY CONDITIONS ON A SIMPLE PLATE MODEL 226
11.8 USING BOUNDARY CONDITIONS TO SIMPLIFY A PROBLEM 227
11.9 STRATEGY FOR PROPERLY DEFINING BOUNDARY CONDITIONS 229
11.9.1 BOUNDARY CONDITIONS ARE NEVER PERFECT 229
11.9.2 THE SEVEN QUESTIONS YOU SHOULD ANSWER TO SUCCESSFULLY DEFINE BOUNDARY CONDITIONS 229
11.9.3 STRATEGY 229
11.10 HOW TO CREATE ISOSTATIC RESTRAINTS .231
11.11 THE OVER-STIFFENING AND UNDER-STIFFENING PROBLEM 232
11.11.1 OVER-STIFFENING 232
11.11.2 UNDER-STIFFENING 235
11.12 HOW TO AVOID SINGULARITIES 237
11.12.1 WHAT IS A SINGULARITY? 237
11.12.2 RULES FOR AVOIDING SINGULARITIES .237
11.13 ABOUT SUPPORT STIFFNESS . 238
11.14 HOW TO LOAD A MODEL 238
11.14.1 LOADING TYPES 238
Chapter 12 RIGID BODY ELEMENTS AND MULTI-POINT CONSTRAINTS 241
12.1 OVERVIEW . 241
12.2 TERMINOLOGY 242
12.3 R-TYPE ELEMENTS . 243
12.3.1 INTRODUCTION TO R-TYPE ELEMENTS 243
12.3.2 SMALL DISPLACEMENT THEORY .244
12.3.3 TWO-NODE RIGID ELEMENT 244
12.3.4 N-NODE RIGID ELEMENT 248
12.3.5 INTERPOLATION ELEMENT 249
12.3.6 R-TYPE ELEMENT SUMMARY .264
12.4 MULTI-POINT CONSTRAINTS . 265
12.4.1 DEFINITION 265
12.4.2 SET UP AN MPC 265
12.4.3 EXAMPLE 1: CREATE A DISPLACEMENT EQUALITY RELATIONSHIP ON A PER DEGREE OF FREEDOM LEVEL .266
12.4.4 EXAMPLE 2: COMPUTE RELATIVE DISPLACEMENT 266
12.4.5 EXAMPLE 3: ENFORCE A SEPARATION BETWEEN NODES .267
12.4.6 EXAMPLE 4: AVERAGE MOTION .269
12.4.7 EXAMPLE 5: CREATE A LINEAR CONTACT BETWEEN NODES 269
12.4.8 EXAMPLE 6: CREATE A PRELOAD IN A 3D BOLT 269
12.4.9 KEY POINTS OF THE MPC 270
Chapter 13 MODELING BOLTED JOINTS 273
13.1 OVERVIEW . 273
13.2 DO YOU REALLY NEED TO MODEL THE BOLTS? 274PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
TABLE OF CONTENTS
13.3 THE VARIOUS FINITE ELEMENT MODELING APPROACHES FOR BOLTED JOINTS .275
13.3.1 FASTENERS MODELED WITH RIGID ELEMENTS .275
13.3.2 FASTENERS MODELED WITH DISCRETE SPRING ELEMENTS .276
13.3.3 FASTENERS MODELED WITH BEAM ELEMENTS 277
13.3.4 FASTENERS MODELED WITH CONNECTORS .277
13.3.5 FASTENERS MODELED WITH THE RUTMAN METHOD 278
13.4 HOW TO CALCULATE THE SPRING FASTENER STIFFNESS 285
13.4.1 WHY CALCULATE THE FASTENER STIFFNESS? 285
13.4.2 AXIAL STIFFNESS . 286
13.4.3 SHEAR STIFFNESS . 286
13.4.4 BENDING STIFFNESS . 288
13.4.5 TORSIONAL STIFFNESS . 288
13.5 HOW TO CONNECT THE FASTENER ELEMENTS TO THE SURROUNDING MESH .289
13.5.1 CONNECT THE FASTENER WHEN THE HOLE IS MODELED 289
13.5.2 CONNECT THE FASTENER WHEN THE HOLE IS NOT MODELED 291
13.6 HOW TO CAPTURE THE PRYING EFFECT IN A BOLTED JOINT MODELED WITH A 1D SPRING .293
13.7 PIN JOINT MODELING APPROACH .299
13.8 BOLT PRELOAD 301
13.8.1 PRELOAD IN A 1D BOLT 301
13.8.2 PRELOAD IN A 3D BOLT 302
13.9 DISCUSSION 305
Chapter 14 MODELING CONTACT 307
14.1 OVERVIEW . 307
14.2 WHAT IS A CONTACT? . 308
14.2.1 INTRODUCTION 308
14.2.2 DEFINITIONS 309
14.2.3 CONTACT STRATEGY 309
14.2.4 CONTACT FORCE . 310
14.2.5 FRICTION FORCE . 311
14.2.6 LINEAR OR NONLINEAR? . 312
14.3 CONTACT TYPES . 312
14.3.1 POINT-TO-POINT LINEAR CONTACT .312
14.3.2 POINT-TO-POINT NONLINEAR CONTACT .313
14.3.3 GENERAL CONTACT . 314
14.4 CONTACT ANALYSIS PROCEDURE .315
14.4.1 THE TWO TYPES OF CONTACT INTERACTION 315
14.4.2 THE TWO TYPES OF CONTACT BODY 316
14.4.3 THE MASTER-SLAVE CONCEPT 316
14.4.4 CONTACT DETECTION . 317
14.4.5 CONTACT TOLERANCE AND DETECTION ALGORITHMS .319
14.4.7 SPECIFY THE CONTACT BETWEEN BODIES 322
14.4.6 INFLUENCE OF THE LOAD INCREMENT ON CONTACT DETECTION 322
14.5 GUIDELINES FOR DEFINING CONTACT .323
14.5.1 KEEP IT SIMPLE IN THE BEGINNING 323
14.5.2 DO NOT VARY THE MESH DENSITY VERY MUCH 323
14.5.3 PAY ATTENTION TO THE RIGID-DEFORMABLE CONTACT .324
14.5.4 MESH REQUIREMENTS 324
14.5.5 PENALTY-BASED CONTACT METHOD 325
14.5.6 PREVENTING RIGID BODY MOTION IN CONTACT SIMULATIONS 325
14.5.7 ISOLATE THE PROBLEMS . 326PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
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14.5.8 INITIAL CONTACT . 326
14.5.9 AVOID CRACKS IN THE CONTACT SURFACES 329
14.5.10 CONTACT AT CORNERS . 329
14.5.11 MPCS INVOLVED IN CONTACT SURFACES 330
14.5.12 SELF-CONTACT 330
14.6 DO YOU REALLY NEED TO REPRESENT CONTACT IN YOUR SIMULATION? .330
14.6.1 ARE THERE BODIES IN CONTACT IN YOUR MODEL? 331
14.6.2 CAN A BODY TOUCH A RIGID SUPPORT IN THE MODEL? 331
14.6.3 IS THERE AN INITIAL CONTACT? .331
14.6.4 CAN YOU PREDICT WHERE THE CONTACT WILL BE? 332
14.7 EXAMPLES . 334
14.7.1 POINT-TO-POINT LINEAR CONTACT BETWEEN TWO NODES .334
14.7.2 POINT-TO-POINT LINEAR CONTACT ON A GROUNDED SURFACE .337
14.7.3 POINT-TO-POINT NONLINEAR CONTACT .339
14.7.4 GLUED CONTACT 341
14.7.5 TOUCHING CONTACT 342
14.7.6 CONTACT BETWEEN DEFORMABLE BODIES .344
14.7.7 DEFORMABLE-RIGID CONTACT 346
Chapter 15 SUBMODELING 349
15.1 WHAT IS SUBMODELING? . 349
15.2 WHY DO SUBMODELING? 350
15.3 HOW TO DO SUBMODELING 350
15.3.1 SUBMODEL A GLOBAL FEM 350
15.3.2 EXTRACT A PART OF THE GLOBAL FEM 350
15.4 TIPS AND HINTS FOR SUBMODELING 350
15.5 DISPLACEMENT-BASED SUBMODELING VS FORCE-BASED SUBMODELING .351
15.6 STATIC CONDENSATION . 353
15.6.1 FROM FEM TO MATRIX 353
15.6.2 TERMINOLOGY AND STATIC CONDENSATION CONCEPT .354
15.6.3 THE STATIC CONDENSATION PROCESS .355
15.6.4 STATIC CONDENSATION VALIDATION 358
15.6.5 LIMITATIONS OF THE STATIC CONDENSATION PROCESS 359
15.7 EXAMPLES OF SUBMODELING .360
15.7.1 SUBMODELING A GLOBAL FEM 360
15.7.2 SUBMODELING BY EXTRACTING A COMPONENT FROM THE GLOBAL FEM 364
15.7.3 SUBMODELING BY STATIC CONDENSATION .365
Chapter 16 VALIDATING AND CORRELATING YOUR FEA 373
16.1 OVERVIEW . 373
16.2 ACCURACY CHECKS 374
16.3 MATHEMATICAL VALIDITY CHECKS 376
16.3.1 BASIC CONCEPTS FOR UNDERSTANDING MATHEMATICAL CHECKS 377
16.3.2 MATHEMATICAL VALIDITY CHECK 1: FREE-FREE MODAL CHECK 381
16.3.3 MATHEMATICAL VALIDITY CHECK 2: UNIT GRAVITY CHECK .382
16.3.4 MATHEMATICAL VALIDITY CHECK 3: UNIT ENFORCED DISPLACEMENT CHECK 383
16.3.5 MATHEMATICAL VALIDITY CHECK 4: THERMAL EQUILIBRIUM CHECK .384
16.4 DEFORMATION CHECK . 385
16.5 HOW ACCURATE ARE THE HOT SPOTS? 385
16.6 CORRELATION 386
16.6.1 OBJECTIVE 386PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
TABLE OF CONTENTS
16.6.2 STRAIN GAUGE MEASUREMENTS 386
16.6.3 TAP TESTING 388
16.6.4 VALIDATION FACTORS AND CORRELATION PLOT 388
16.7 MODEL CHECKOUT DOCUMENTATION .390
16.8 MATHEMATICAL VALIDITY CHECK EXAMPLE 395
16.8.1 EXAMPLE INTRODUCTION 395
16.8.2 FREE-FREE MODAL CHECK 397
16.8.3 UNIT GRAVITY CHECK . 402
16.8.4 UNIT ENFORCED DISPLACEMENT CHECK 404
Chapter 17 UNDERSTANDING FEA OUTPUTS 409
17.1 OVERVIEW . 409
17.2 STANDARD OUTPUTS 409
17.2.1 DEFORMED SHAPES 409
17.2.2 ELEMENT FORCE 410
17.2.3 STRESSES IN ELEMENTS . 422
17.2.4 PRINCIPAL STRESS OR VON MISES STRESS? .429
17.2.5 FORCES AT BOUNDARY CONDITIONS .430
17.2.6 FREE BODY DIAGRAM 431
17.3 THE BASIC RULES OF POST-PROCESSING .436
17.3.1 ANIMATE THE DISPLACEMENT FIRST 436
17.3.2 CONTOUR PLOTS 437
17.3.3 SELECT THE APPROPRIATE STRESS PLOT .437
17.3.4 EXTRAPOLATION 438
17.3.5 SELECT THE APPROPRIATE TYPE OF STRESS .441
17.3.6 DO NOT NEGLECT THE CONVERGENCE TEST .441
17.3.7 VALIDATE THE LINEAR ASSUMPTION 441
17.3.8 DO NOT CONFUSE FORCES AND FLOWS FOR 2D SHELL ELEMENTS 442
17.3.9 PAY ATTENTION TO COORDINATE SYSTEMS .442
17.3.10 ADJUSTING THE SCALE OF THE COLOR BAR .442
17.3.11 REPORT THE MAXIMUM STRESS LOCATION 443
17.3.12 TOP AND BOTTOM STRESSES FOR 2D SHELL ELEMENTS .443
17.3.13 GRAPH THE RESULTS . 444
17.3.14 INTERPRETATION OF RESULTS AND DESIGN MODIFICATIONS 445
17.3.15 EXPORT THE RESULTS IN REPORTS 445
17.3.16 USE THE READING ELEMENTS .445
17.3.17 VECTOR PLOT . 446
17.4 HOW TO DEAL WITH SINGULARITIES 447
17.4.1 ARE YOU INTERESTED IN RESULTS AROUND A SINGULARITY? 447
17.4.2 IMPACT OF A SINGULARITY 447
17.4.3 CAN I IGNORE SINGULARITIES? 447
17.4.4 HOW DO I AVOID A SINGULARITY DUE TO A POINT LOADING? .448
Chapter 18 IMPROVING YOUR PERFORMANCE COMPUTING 451
18.1 OVERVIEW . 451
18.2 CPU POWER AND CLOCK SPEED .452
18.3 MEMORY SIZE . 453
18.4 CACHE SIZE 453
18.5 HARD DRIVE SPEED 454
18.6 PARALLEL COMPUTING 454
18.6.1 OVERVIEW 454PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
TABLE OF CONTENTS
18.6.2 PARALLEL COMPUTER ARCHITECTURES: SMP VS DMP 455
18.6.3 THE BASICS OF HIGH-PERFORMANCE COMPUTING (HPC) .456
18.7 WAYS TO SPEED UP YOUR SIMULATIONS .457
18.7.1 SYSTEM OPTIMIZATION . 457
18.7.2 MANAGE MEMORY . 458
18.7.3 OPTIMIZE THE OUTPUT REQUESTS .459
18.7.4 MAKE USE OF MULTIPLE CORES (SMP) 459
18.7.5 ABOUT HYPER-THREADING 459
Chapter 19 DOCUMENTING YOUR FEA 461
19.1 OVERVIEW . 461
19.2 MODEL DESCRIPTION . 462
19.3 GEOMETRY SOURCE 462
19.4 MODEL ASSUMPTIONS 463
19.5 SIMULATION PARAMETERS 464
19.6 VERIFICATION AND VALIDATION .464
Chapter 20 LINEAR STATIC ANALYSIS 467
20.1 OVERVIEW . 467
20.2 WHAT IS LINEAR STATIC ANALYSIS? 467
20.3 HOW TO SOLVE A LINEAR STATIC PROBLEM .468
20.4 CHARACTERISTICS OF A LINEAR ANALYSIS 469
20.4.1 LOAD-DISPLACEMENT RELATION .469
20.4.2 STRESS-STRAIN RELATION . 469
20.4.3 SCALABILITY . 469
20.4.4 SUPERPOSITION . 470
20.4.5 REVERSIBILITY AND LOAD HISTORY .470
20.4.6 SOLUTION SETTINGS . 470
20.5 EXAMPLES OF LINEAR STATIC ANALYSIS .470
20.5.1 CHARACTERISTICS OF A LINEAR STATIC ANALYSIS .470
20.5.2 HOW DOES MATERIAL AFFECT STRESS IN A LINEAR STATIC SOLUTION? .474
Chapter 21 NONLINEAR STATIC ANALYSIS 477
21.1 OVERVIEW . 477
21.2 WHAT IS A NONLINEAR SYSTEM? .478
21.3 CHARACTERISTICS OF A NONLINEAR ANALYSIS .479
21.3.1 LOAD-DISPLACEMENT RELATION .479
21.3.2 STRESS-STRAIN RELATION . 479
21.3.3 SCALABILITY . 479
21.3.4 SUPERPOSITION . 479
21.3.5 INITIAL STATE OF STRESS . 479
21.3.6 LOAD HISTORY . 479
21.3.7 REVERSIBILITY . 480
21.3.8 SOLUTION SETTINGS . 480
21.4 GEOMETRIC NONLINEARITY 480
21.4.1 SOURCES OF GEOMETRICAL NONLINEARITY .480
21.4.2 HOW DOES NONLINEAR GEOMETRY WORK? 481
21.4.3 DO YOU REALLY NEED A NONLINEAR GEOMETRIC ANALYSIS? .483
21.4.4 THE FOLLOWER LOAD CONCEPT 484
21.4.5 SMALL OR LARGE STRAIN? . 485
21.4.6 EXAMPLE OF GEOMETRIC NONLINEARITY .485PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
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21.5 MATERIAL NONLINEARITY 487
21.5.1 YIELD CRITERIA . 487
21.5.2 HARDENING RULES . 488
21.5.3 MATERIAL MODELS . 489
21.5.4 ENGINEERING STRESS-STRAIN OR TRUE STRESS-STRAIN? .492
21.5.5 HOW DOES NONLINEAR MATERIAL WORK? 493
21.5.6 DO YOU REALLY NEED A NONLINEAR MATERIAL ANALYSIS? .495
21.6 BOUNDARY NONLINEARITY . 496
21.6.1 LOAD VARIATION 496
21.6.2 CONSTRAINT VARIATION . 496
21.6.3 CONTACTS . 497
21.7 CHOOSING THE RIGHT ELEMENTS FOR A NONLINEAR ANALYSIS 497
21.8 HOW DO FEA SOFTWARE COMPUTE NONLINEAR PROBLEMS? 498
21.8.1 CHARACTERIZATION AND FORMULATION OF A NONLINEAR PROBLEM .498
21.8.2 NEWTON-RAPHSON METHOD 498
21.8.3 MODIFIED NEWTON-RAPHSON METHOD 501
21.8.4 NEWTON-RAPHSON METHOD EXAMPLES 502
21.8.5 COMPUTATIONAL METHODS IN NONLINEAR ANALYSIS 506
21.8.6 EQUILIBRIUM PATH AND CRITICAL POINTS 511
21.8.7 ADAPTIVE SOLUTION STRATEGIES .511
21.8.8 STIFFNESS MATRIX UPDATE STRATEGIES .512
21.8.9 CHOOSING THE INCREMENTAL LOAD STEP 514
21.8.10 ARC-LENGTH METHODS 515
21.8.11 LINE SEARCH PROCEDURES 518
21.8.12 CONVERGENCE CRITERIA 519
21.8.13 HOW TO DEAL WITH CONVERGENCE ISSUES .519
21.8.14 SUMMARY OF ITERATIVE SOLUTION SCHEMES 520
21.8.15 HOW TO SELECT THE RIGHT ITERATIVE SOLUTION SCHEME .521
21.8.16 SUMMARY OF THE NONLINEAR SOLUTION STRATEGY 522
21.9 GENERAL RECOMMENDATIONS FOR NONLINEAR ANALYSIS 523
21.9.1 UNDERSTAND THE NONLINEAR FEATURES .523
21.9.2 UNDERSTAND YOUR PROBLEM AND STRUCTURAL BEHAVIOR .523
21.9.3 UNDERSTAND THE DIFFERENCE BETWEEN A LINEAR SUBCASE AND A NONLINEAR SUBCASE 524
21.9.4 SIMPLIFY YOUR MODEL . 524
21.9.5 USE AN ADEQUATE MESH AND ELEMENT TYPES .524
21.9.6 APPLY LOADING GRADUALLY .525
21.9.7 READ THE OUTPUT 525
21.9.8 NUMBER OF INCREMENTS . 525
21.9.9 CONVERGENCE PROBLEMS 525
21.9.10 KEEP AN EYE ON YOUR MATERIAL DEFINITION 526
21.10 COMMON MISTAKES IN NONLINEAR ANALYSIS .526
21.11 EXAMPLES OF NONLINEAR STATIC ANALYSIS .528
21.11.1 GEOMETRIC NONLINEARITY AND HISTORY PATH 528
21.11.2 CUMULATIVE EFFECT OF A NONLINEAR ANALYSIS 531
21.11.3 INFLUENCE OF THE INCREMENTAL LOAD STEP ON RESULTS .536
21.11.4 MATERIAL NONLINEARITY: ELASTOPLASTIC PLATE 540
21.11.5 HIGHLY NONLINEAR PROBLEM .546
Chapter 22 LINEAR BUCKLING ANALYSIS 553
22.1 WHAT IS LINEAR BUCKLING ANALYSIS? .553
22.2 ASSUMPTIONS AND LIMITATIONS OF LINEAR BUCKLING ANALYSIS .554PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
TABLE OF CONTENTS
22.3 LINEAR BUCKLING ANALYSIS OUTCOMES 555
22.4 HOW DO SOLVERS COMPUTE LINEAR BUCKLING PROBLEMS? 556
22.4.1 THE EQUATION OF MOTION WITH DIFFERENTIAL STIFFNESS MATRIX 556
22.4.2 HOW TO COMPUTE THE EIGEN EQUATION .557
22.4.3 SOLUTION OF THE BUCKLING PROBLEM 557
22.5 THE LINEAR BUCKLING STRATEGY 558
22.5.1 EVERYTHING STARTS WITH A LINEAR STATIC ANALYSIS .558
22.5.2 SELECT YOUR BUCKLING CASES 558
22.5.3 MESHING HINTS . 558
22.6 EXAMPLES OF LINEAR BUCKLING ANALYSIS .558
22.6.1 EULER BEAM BUCKLING 558
22.6.2 PANEL BUCKLING . 560
22.6.3 STIFFENED PANEL BUCKLING .564
22.6.4 INFLUENCE OF MESHING DENSITY ON BUCKLING PREDICTIONS 565
Chapter 23 NONLINEAR BUCKLING ANALYSIS 569
23.1 OVERVIEW . 569
23.2 WHY PERFORM A NONLINEAR BUCKLING ANALYSIS? 569
23.3 THE STABILITY PATH AND THE CONVERGED SOLUTION .571
23.4 NONLINEAR BUCKLING PROCEDURE 571
23.5 POST-BUCKLING . 571
23.6 ESSENTIAL STEPS IN NONLINEAR BUCKLING ANALYSIS .573
23.7 EXAMPLES OF NONLINEAR BUCKLING ANALYSIS 573
23.7.1 NONLINEAR BUCKLING OF A CURVED PANEL 573
23.7.2 SNAP-THROUGH: NEWTON-RAPHSON VS ARC-LENGTH .576
Chapter 24 NORMAL MODE ANALYSIS 583
24.1 OVERVIEW . 583
24.2 HOW TO SOLVE THE REAL EIGENVALUE PROBLEM 584
24.2.1 THE EQUATION OF MOTION 584
24.2.2 HOW TO COMPUTE THE EIGEN EQUATION .584
24.2.3 SOLUTION OF THE EIGEN EQUATION .587
24.2.4 EIGENVALUE EXTRACTION METHOD .587
24.3 WHAT A MODE IS AND WHAT IT IS NOT 588
24.3.1 NATURAL FREQUENCIES 588
24.3.2 WHAT A MODE IS . 588
24.3.3 WHAT A MODE IS NOT . 589
24.4 HOW ARE NATURAL FREQUENCIES AND MODE SHAPES INFLUENCED? .589
24.5 WHY COMPUTE A MODAL ANALYSIS? .592
24.5.1 FINDING WEAKNESSES IN A MODEL .592
24.5.2 AVOID RESONANCE . 593
24.6 EXAMPLES OF MODAL ANALYSIS 594
24.6.1 MODEL CHECKS 594
24.6.2 FIND THE NATURAL FREQUENCIES TO AVOID RESONANCE .594
24.6.3 EVALUATE THE MODAL EFFECTIVE MASS .596
24.6.4 INFLUENCE OF THE PRE-STIFFNESS ON THE NATURAL FREQUENCIES .598
Chapter 25 GOOD MODELING PRACTICES 603
25.1 OVERVIEW . 603
25.2 GOOD MODELING PRACTICES APPROACH 604
25.3 IT ALL STARTS WITH A GOOD PLAN .605PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
TABLE OF CONTENTS
25.4 UNDERSTAND THE PROBLEM TO ANALYZE IN DETAIL .605
25.5 DEFINE YOUR DESIGN OBJECTIVE 605
25.6 BE SURE OF THE INPUTS AND REQUIREMENTS .606
25.7 SELECT THE RIGHT TYPE OF ANALYSIS .606
25.8 CLEAN UP THE GEOMETRY . 607
25.9 CHECK THE GEOMETRY 607
25.10 SELECT THE PROPER ELEMENTS .607
25.11 CREATE AN INTELLIGIBLE MESH 608
25.12 DEFINE THE RIGHT BOUNDARY CONDITIONS .609
25.13 VALIDATE THE INPUT DATA 610
25.14 DEFINE CONTACT PROPERLY . 610
25.15 MODEL THE RIGHT MATERIAL BEHAVIOR .610
25.16 MANAGE THE UNITS 611
25.17 SHOULD YOU MODEL THE ENTIRE STRUCTURE? .611
25.18 MANAGE THE SINGULARITIES 611
25.19 SHOULD YOU MODEL THE BOLTS? 611
25.20 MANAGE INCOMPATIBLE DEGREES OF FREEDOM 612
25.21 KEEP AN EYE ON THE SOLUTION’S PARAMETERS 612
25.22 VERIFY AND VALIDATE YOUR MODEL .612
25.23 READ THE SOLVER’S MESSAGES 613
25.24 KEEP A CRITICAL EYE ON THE RESULTS .613
25.25 DOCUMENT EVERYTHING 614
25.26 ASK FOR HELP 614
25.27 THE MOST COMMON MISTAKES IN FEA .615
25.28 THE 10 COMMANDMENTS OF THE FEA ANALYST .617
GLOSSARY AND ABBREVIATIONS 619
REFERENCES 629
IMAGE CREDITS 633
INDEX 635 PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
INDEX
A
Accuracy 29
Accuracy Checks 374
Adaptive Solution Strategies .511
Anisotropic Material 210
Anti-Symmetry 225
Arc-Length .515
Assembly Phase .358
B
Banded Matrix .96
Beam Element .118
Bolted Joints 273
Axial Stiffness 286
Beam .277
Bending Stiffness 288
Bolt Preload 301
Connectors 277
Pin Joint 299
Prying Effect 293
Rigid Elements 275
Rutman Method .278
Shear Stiffness 286
Spring 276
Stiffness Calculation 285
Bolt Preload .301
Boundary Conditions .215
Anti-Symmetry 225
Isostatic Restraints 231
Loads 238
Mechanism .218
Over-Stiffening 232
Rigid Body Mode 217
Role .216
Single-Point Constraints 219
Singularity .237
Strategy .229
Symmetry .225
Type 216
Under-Stiffening .232
Boundary Element Method .7
Boundary Nonlinearity 496
Boundary Simmetry .225
BRICK Element .128
C
Checkout 373
Accuracy Checks .374
Applied Loads .379
Correlation 386
Correlation Plot 388
Documentation .390
Free-Free Modal Check .381
Gauges Measurement 386
Load Path 381
Mathematical Checks .376
Mechanism .377
Post-Processor Checks 380
Reacted Loads .380
Singularity .377
Thermal Equilibrium Check .384
Unit Enforced Displacement Check .
383
Unit Gravity Check 382
Validation Factor .388
Weight 379
Compatibility of deformation 52
Contact . 307, 497
Analysis Procedure .315
Definition 309
Deformable Bodies .309
Deformable-Rigid Contact 344
Detection 317
Force .310
Friction 311
General Contact 314
Glued Contact 315, 341
Guidelines .323
Master 316
Master-Slave Concept .316
Node-to-Segment .317
Point-to-Point Linear Contact
312, 334, 337
Point-to-Point Nonlinear Contact
313, 339
Segment-to-Segment 318
Slave .316
Strategy .309
Tolerance 319
Touching Contact . 315, 342
Types of Contact .312
Convergence 163
Convergence Criteria .519
Correlation 373, 386
Critical Points .511
D
Deformable Bodies 344
Deformable Contact Body .316
Degrees of Freedom . 62, 130
Direct Solver 21
Discretization .17
Displacement Method 51, 64
Documentation 461
E
Elastic Strain 203
Element Types .116
Engineering Stress-Strain .492
Equilibrium Conditions 51
Equilibrium Equation .50
Equilibrium Path 511
F
FEA Capabilities .27
FEA Concept 23
FEA History 45
FEA Process .23
FEA Strategy 17
FEA Timeline 46
Finite Difference Method 7
Finite Element Method 7
Finite Volume Method .7
Follower Load 484
Free-Free Modal Check .381
G
Gauss Integration 136
Geometric Nonlinearity .480
Glued Contact 341
Good Modeling Practices 603
Guyan Reduction .353636
PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
INDEX
H
Hardening Rules 488
HEX Element 128
Hollomon Model .207
Hourglassing 143
Huth .287
I
Incremental Load Step .514
Integration Scheme .135
Gauss Integration 136
Interpolation Element .241
Isostatic Restraints .231
Isotropic Material 201
Iterative Schemes 520
Iterative Solver 22
L
Learning FEA 37
Library of Elements 115
1D Elements 117
2D Elements 122
3D Elements 127
Beam Element 118
BRICK Element 127
Element Order 132
Element Selection .130
Element Type 116
HEX Element .127
Integration Scheme 135
Membrane 124
PENTA Element .127
Plane Strain .123
Plane Stress 123
Plate 124
Shell 125
Special Elements .129
Spring Element .122
TET Element 127
Truss Element .117
Linear Buckling Analysis 553
Assumptions .554
Definition 553
Differential Stiffness .556
Eigen Equation 557
Limitations 554
Outcomes .555
Solving 556
Strategy .558
Linear Element 132
Linear Static Analysis .467
Characteristics 469
Definition 467
Solving 468
Line Search 518
Loads .238
M
Material .201
Anisotropic .210
Elastic Strain .203
Hollomon 207
Isotropic 201
Orthotropic 210, 211
Plastic Strain .203
Ramberg-Osgood 206
Strain Hardening .204
Stress-Strain Curve .202
Stress Strain Relationship .202
True Strain 208
True Stress 208
Material Models 489
Material Nonlinearity 487
Mathematical Checks 376
Matrix Assembly 91
Matrix Sparsity 95
Mechanism . 218, 377
Membrane .124
Membrane CST 76
Membrane LST 81
Meshing .153
1D Meshing Rules .174
2D Meshing Rules .174
3D Meshing Rules .184
BRICK Element 190
Check 3D Hexa 191
Check 3D Tetra 189
Ckeck 2D Mesh .180
Convergence .163
Element Selection .156
Element Size .157
HEX Element .190
Interface .169
Mid-Plane .175
Plan .155
Refinement .159
TET Element 184
Transition 170
Meshing Rules 174, 184
Meshing Size 157
Mesh Refinement 159
Methods 6
Mid-Plane 175
Modal Analysis See Normal Mode
Analysis
Modeling Process 23
Mode Shape 589
MPC 241, 265
Multi-Point Constraints 241, 265
N
Natural Frequency .588
Nonlinear Buckling Analysis 569
Post-Buckling 571
Stability Path .571
Steps .573
Nonlinear Static Analysis .477
Adaptive Solution Strategies .511
Arc-Length 515
Boundary 496
Characteristics 479
Common Mistakes 526
Computational Methods .506
Convergence Criteria 519
Convergence Issues 519
Critical Points 511
Definition 478
Elements .497
Equilibrium Path .511
Follower Load .484
Geometric .480
Incremental Load Step 514
Iterative Schemes .520
Line Search .518
Material 487
Modified Newton-Raphson 501
Newton-Raphson 498
Recommendations 523
Solving 498
Stiffness Matrix Update Strategy512
Normal Mode Analysis 583
Application 592
Eigen Equation 584
Mode 588
Mode Shape .589
Natural Frequency . 588, 589
Solving 584
Numerical Methods .7INDEX
637
PRACTICAL FINITE ELEMENT ANALYSIS FOR MECHANICAL ENGINEERS
O
Orthotropic Material 210, 211
Outputs .409
Basic Rules 436
Force .410
Freebody Diagram 431
Singularity .447
Stress 422
Over-Stiffening .232
P
Partial Differential Equations .8
Performance Computing .451
Cache 453
Clock Speed 452
CPU Power 452
DMP 455
Hard Drive .454
HPC .456
Memory 453
Parallel Computing .454
Recommendations 457
SMP .455
Pin Joint .299
Pioneers .45
Plane Strain .123
Plane Stress .123
Plastic Strain 203
Plate 124
Post-Buckling .571
Post-Processing 22
Pre-Processing .20
Principle of Minimum Potential
Energy 60
Prying Effect .293
Q
Quadratic Elements
129, 132, 177, 189
R
Ramberg-Osgood Model .206
Reduction Phase 357
Rigid Body Elements 241
Rigid Elements .241
R-Type Elements . 241, 243
Interpolation Element .249
N-Node Rigid Element 248
Small Displacement Theory 244
Summary 264
Two-Node Rigid Element 244
Typical Elements .242
Rutman Fastener .278
Behavior 279
Compatibility of Displacement .282
Example 283
Modeling 280
Stiffness 279
S
Saint-Venant's Principle .349
Shape Functions 64
Truss Element 65, 67
Shear Locking 141
Shell 86, 125
Single-Point Constraints 219
Singularity . 237, 377
Skyline Matrix Storage .97
Solid Element .89
Solving .20
Solving the FEM Equations 99
Direct Solution 99
Iterative Solution 101
Sparsity 95
Spring Element 122
Static Condensation .353
Assembly Phase 358
Concept 354
Guyan Reduction 353
Limitations 359
Process 355
Reduction Phase .357
Validation 358
Stiffness Matrix . 52, 62
2D Element .75
Membrane CST .76
Membrane LST . 76, 81
Plate 83
Shell 86
Solid Element 89
Truss Element .64
Stiffness Matrix Update Strategy .512
Strain Hardening 204
Stress-Strain Curve 202
Stress-Strain Relationship 202
Submodeling .349
Swift .287
Symmetry 225
T
Tate & Rosenfeld 287
TET Element .128
Theory .49
Thermal Equilibrium Check .384
Touching Contact .342
True Strain .208
True Stress .208
True Stress-Strain 492
Truss Element 117
U
Under-Stiffening 232
Unit Enforced Displacement Check
383
Unit Gravity Check .382
Units 195
V
Validation 373
Y
Yield Criteria 487
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