Computational Structural Engineering - Automatic Calculation of Mechanical Structures

Table of Contents

CHAPTER 1 Modeling of structures by Finite Elements 1.1 Introduction 1.2 Modeling with 2D elements 1.2.1 Plane stress 1.2.2 Plane strain 1.2.3 Axisymmetric stress 1.3 Modelling with 3D elements 1.4 Modeling with shell elements 1.5 Considerations about connections between parts 1.5.1 Welded connections 1.5.2 Rivet connections 1.5.3 Screw connections 1.5.4 Bonded connections 1.6 One-dimensional elements 1.7 Zero-dimensional elements 1.8 Non structural elements 1.9 Membrane elements 1.10 General comments 1.10.1 Overviews 1.10.2 One-dimensional elements 1.10.3 2D and shell elements 1.10.4 3D elements 1.10.5 Non structural elements 1.11 Conclusions

CHAPTER 2 Modeling of boundary conditions 2.1 Introduction 2.2 Constraint conditions 2.3 Load conditions 2.3.1 Point loads 2.3.2 Distributed loads 2.3.3 Thermal loads 2.3.4 Inertial loads 2.3.5 Volume forces 2.4 Symmetry and antisymmetry 2.4.1 Geometric and load symmetries 2.4.2 Geometric symmetries and load antisymmetries 2.4.3 Modal analysis 2.4.4 Conclusions

CHAPTER 3 Interpreting the results 3.1 Introduction 3.2 Averaged and un-averaged contours 3.3 The reference system 3.4 Shell elements 3.4.1 Top, Bottom e Middle 3.4.2 Intersections among elements located on different planes 3.4.3 Discontinuous joints 3.4.4 Continuous joints 3.5 Solid elements 3.5.1 Discontinuous joints 3.5.2 Continuous joints 3.6 One-dimensional elements 3.7 Non-structural elements 3.8 Reaction forces 3.9 Graphics post-processing considerations 3.9.1 Overview 3.9.2 The flow lines 3.9.3 The funnel effect 3.9.4 Strain energy 3.9.5 Gauss points and nodes

CHAPTER 4 Natural frequencies and mode shapes 4.1 Introduction 4.2 The dynamic problem 4.3 Free-free modal analysis 4.4 Constrained modal analysis 4.5 The importance of discretization 4.6 Effective modal mass and modal participation factor 4.7 Load stiffening 4.8 Conclusions

CHAPTER 5 Instability of elastic equilibrium 5.1 Introduction 5.2 Linear buckling 5.3 The FEM approach 5.4 Some practical examples 5.4.1 Cylinder under external pressure 5.4.2 Cantilevr beam 5.4.3 Thin-walled cylinder subjected to compressive axial loading 5.4.4 Thin-walled cylinder undergoing pure torsion 5.5 Notes on instability in nonlinear domain

CHAPTER 6 Errors in Finite Element calculation 6.1 Introduction 6.2 User errors 6.3 Discretization errors 6.3.1 Introduction 6.3.2 Mesh density 6.3.2.1 A borderline case 6.3.2.2 A practical case 6.3.3 Element type 6.3.3.1 Beam A 6.3.3.2 Beam B 6.3.3.3 Hexahedra versus tetrahedra 6.3.3.4 Quadrangles versus triangles 6.3.3.5 C-section beam 6.3.3.6 Large-curvature beam 6.3.3.7 The skinning technique 6.3.4 Conclusions 6.4 Modeling errors 6.4.1 "Distraction" errors 6.4.2 Conceptual errors 6.4.2.1 Beam modeled exclusively with brick elements 6.4.2.2 Beam modeled with brick/shell - Solution I 6.4.2.3 Beam modeled with brick/shell - Solution II 6.4.2.4 Beam modeled with brick/shell - Solution III 6.4.2.5 Beam modeled with brick/shell - Wrong solution 6.4.2.6 Brick/beam interface 6.4.2.7 Interface between other element types 6.5 Numerical errors 6.5.1 The condition number for the stiffness matrix 6.5.2 Eigenvalues and eigenvectors of the stiffness matrix 6.5.3 Perfectly square plane element 6.5.4 Slightly distorted plane element 6.5.5 Highly distorted element 6.5.6 Unacceptably distorted element 6.6 Pre-processing errors

CHAPTER 7 Advanced modeling techniques 7.1 Introduction 7.2 Substructuring 7.2.1 Superelements 7.2.2 A practical example 7.3 Submodeling 7.3.1 A practical example 7.4 The simulation of press fit couplings 7.4.1 Shaft - Flywheel 7.4.2 Wheel - Axle 7.4.3 Gearmotor support 7.4.4 Cage-pin coupling 7.5 Preload in bolted connections 7.6 Conclusions

CHAPTER 8 Linear elastic calculation of composite materials 8.1 Introduction 8.1.1 Historical background 8.2 Element types to be used 8.3 Short and non-oriented fiber composites 8.4 Long and oriented fiber composites 8.4.1 Materials 8.4.2 Stacking of sheets 8.4.3 Orientation of sheets 8.4.4 Orientation of the element normal 8.4.5 The draping 8.5 An example of laminate without core 8.5.1 Bar with symmetrical stacking sequence 8.5.2 Bar with non-symmetrical stacking sequence?8.5.3 Bar with symmetrical stacking and increase of unidirectional sheets 8.5.4 Bar with symmetrical stacking and ifferently oriented sheets 8.5.5 Symmetrically stacked bar subjected to bending - Case 1 8.5.6 Symmetrically stacked bar subjected to bending - Case 2 8.6 Sandwich panels 8.7 The 3D layered elements 8.8 The 3D continuum shell elements 8.9 "Zone based" and "ply based" methods 8.9.1 Introduction 8.9.2 Zone based method 8.9.3 Ply based method 8.9.4 Zone based vs ply based 8.10 More about 3D elements 8.11 3D Composites 8.12 Joining systems 8.13 Notes on metal matrix composites 8.14 Final considerations

CHAPTER 9 Finite element model validation methods 9.1 Introduction 9.2 Numerical validation 9.2.1 Applied loads and reaction forces 9.2.2 Index "EPSILON 9.2.3 Index "MAXRATIO" 9.2.4 Rigid mode check index 9.2.5 Controllo sulla strain energy 9.2.6 Considerations about check indices 9.2.7 Visual checks 9.3 Experimental validation 9.3.1 Load application without stress measurement 9.3.2 Load application with strain gauge measurements 9.3.3 Photoelasticity

CHAPTER 10 Strength assessments 10.1 Introduction 10.2 Static assessment for homogeneous and isotropic materials 10.2.1 Continuous structure parts 10.2.2 Connection systems 10.2.2.1 Introduction 10.2.2.2 Screws 10.2.2.3 Rivets 10.2.2.4 Holes and eyelets 10.2.2.5 Welding 10.3 Fatigue assessment for homogeneous and isotropic materials 10.3.1 Continuous structure parts 10.3.1.1 Classic Method 10.3.1.2 The Gough-Pollard criterion 10.3.1.3 The UIC (Union International des Chemins de Fer) method 10.3.1.4 The Von Mises criterion 10.3.1.5 The b2 coefficient (surface finishing) 10.3.1.6 The b3 coefficient (dimensional effect) 10.3.1.7 Notching coefficients Kt and Kf 10.3.1.8 A practical example 10.3.1.9 Miner's rule 10.3.2 Connection systems 10.4 Failure criteria for composite materials 10.4.1 The maximum stress criterion 10.4.2 The Tsai-Hill criterion 10.4.3 The Tsai-Wu criterion 10.4.4 Interlaminar shear 10.4.5 Considerations 10.4.6 Connection systems 10.4.6.1 Introduction 10.4.6.2 Bonding 10.4.6.3 Insert pull-out 10.5 Fatigue assessment for composite materials 10.6 Assessments beyond the elastic limit 10.7 Conclusions

CHAPTER 11 Geometric nonlinearity 11.1 Introduction 11.2 Geometric nonlinearity 11.3 Considerations 11.4 Post-buckling 11.4.1 Beam in compression 11.4.2 Planar frame 11.4.3 Modeling geometric imperfections 11.4.4 Conclusions

CHAPTER 12 Contact nonlinearity 12.1 Introduction 12.2 GAP elements 12.2.1 Sphere on infinitely rigid plane 12.2.2 The finite element model 12.3 Contact surfaces 12.3.1 Sphere on deformable plane 12.3.2 The finite element model 12.3.3 Gearmotor support (Chapter 7) 12.3.4 Cage - pin coupling (Chapter 7) 12.3.5 Self-contact 12.4 Some suggestions 12.5 Conclusions

CHAPTER 13 Material nonlinearity 13.1 Introduction 13.2 Beam in bending at elastic limit 13.3 Beam in bending beyond the elastic limit 13.4 Beam in torsion beyond the elastic limit 13.5 Industry practice 13.5.1 Drive shafts for racing car 13.5.2 Spacer for flange connection 13.6 True stresses and true strains 13.6.1 Flanged pipe (Chapter 7) 13.7 Elastomeric materials 13.7.1 Introduction 13.7.2??Uniaxial tensile-compression test 13.7.3??Study of an O-Ring 13.8 Conclusions

CHAPTER 14 Dynamic analyses 14.1 Introduction 14.2 Frequency response 14.2.1??Structural damping 14.2.2??Solution techniques 14.2.3??Direct integration 14.2.4??Modal superposition 14.2.5??Comparison between the two methods 14.2.6??Modal superposition with an insufficient number of modes 14.2.7??Conclusions 14.3 Transient dynamic analysis 14.3.1??Direct integration 14.3.2??Modal superposition 14.3.3??Comparison with the static case 14.3.4??Material nonlinearity 14.3.5??Conclusions 14.4 Spectrum analysis 14.5 Random vibration analysis 14.6 Explicit methods 14.6.1??Introduction 14.6.2??Comparison with the implicit method 14.6.3??Some considerations about the explicit approach 14.6.4??Conclusionsi

CHAPTER 15 Structural optimization 15.1 Introduction 15.1.1 Size optimization 15.1.2 Shape optimization 15.1.3 Topological optimization 15.2 A case study 15.3 Conclusions

CHAPTER 16 Damage simulation 16.1 Introduction 16.2 Damage in ductile materials 16.3 Damage in composite materials 16.4 Damage in bonding

CHAPTER 17 Examples of advanced calculations 17.1 Introduction 17.2 Modeling of ball bearings 17.2.1 Introduction 17.2.2 The wheel group 17.2.3 The FE model of the bearing 17.2.4 Validation of the calculation model 17.2.5 Wheel group optimization 17.2.6 Conclusions 17.3 Modeling of wheel rim and tire 17.3.1 Introduction 17.3.2 The finite element model 17.4 Bolted connections 17.4.1 Introduction 17.4.2 The preload 17.4.3 Preload + symmetrical orthogonal external load 17.4.4 Preload + nonsymmetrical orthogonal external load 17.4.5 Preload + external tangential load 17.4.6 Conclusions 17.5 T-bracket (Chapters 3 and 10) 17.5.1 Introduction 17.5.2 Without preload 17.5.3 With preload 17.5.4 Conclusions 17.6 The calculation of lugs 17.6.1 Introduction 17.6.2 Classical calculation 17.6.3 Finite element calculation 17.7 Conclusions

CHAPTER 18 State of the art and future developments 18.1 Introduction 18.1.1 When to use traditional methods 18.1.2 When to use numerical methods 18.1.3 When to use a "hybrid" method 18.1.4 The principle of the least necessary mass 18.1.5 Conclusions 18.2 Classical FE methods 18.3 Multibody methods and FE 18.4 Multiphysics methods 18.5 The process simulation 18.6 CAE in CAD 18.7 Conclusions

APPENDIX A Notes on structural calculations in the linear elastic domain A.1 Introduction A.2 The stress-strain relationship A.3 The strain-displacement equations A.4 The indefinite equilibrium equations A.5 The plane stress state A.6 The plane strain state A.7 The axisymmetric stress state

APPENDIX B The stiffness matrix for the plane stress 3-node element B.1 Introduction B.2 Finite Elements B.3 Shape functions for the plane stress triangular element B.4 The stiffness matrix for the CST element B.5 A practical example

APPENDIX C The numerical solution of linear algebraic equation systems C.1 Introduction C.2 The system of equations C.3 Direct methods C.4 Iterative methods C.5 Comparison between direct and iterative methods C.6 Conclusions