The FEM Handbook
This text combines the previous two, Computer Aided Structural Design and Advanced Techniques in Structural Design, into a single volume. This operation was not a simple “Copy & Paste“, but a total revision, which required many additions and eliminated almost nothing.
There has also been a reorganization of the subjects, so that some topics have been moved to Part I, dedicated to the linear elastic calculations, making Part II a little less “meaty”, but enriched with some practical examples. All the examples, old and new, have been redone using ABAQUS in its version 6.14-2, but they can be replicated using any other Finite Element code; indeed, the encouragement is to try to compare the results obtained with other programs.
Claudio Gianini
Computational Structural Engineering
Automatic calculation of mechanical structures
Introduction to Part I: Rory Byrne
Introduction to Part II: Luca Marmorini
CG CAE Sagl
Index
Modeling of structures by Finite 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
Modeling of boundary conditions
2.1 Introduction
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
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
3.9.4 Strain energy
3.9.5 Gauss points and nodes
Instability of elastic equilibrium
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
4.8 Conclusions
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 Cantilever 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
Errors in Finite Element calculation
6.1 Introduction
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
6.3.3.7 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
Advanced modeling techniques
7.1 Introduction
7.2 Substructuring
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
Linear elastic calculation of composite materials
8.1.1 Historical background
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 differently 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
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
9.3 Experimental validation
9.3.1 Load application without stress measurement
9.3.2 Load application with strain gauge measurements
9.3.3 Photoelasticity
Strength assessments
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
Geometric nonlinearity
11.1 Introduction
11.2 Geometric nonlinearity
11.3 Considerations
11.4.1 Beam in compression
11.4.2 Planar frame
11.4.3 Modeling geometric imperfections
11.4.4 Conclusions
Contact nonlinearity
12.1 Introduction
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
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)
13.7.1 Introduction
13.7.2 Uniaxial tensile-compression test
13.7.3 Study of an O-Ring
13.8 Conclusions
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
14.6.1 Introduction
14.6.2 Comparison with the implicit method
14.6.3 Some considerations about the explicit approach
14.6.4 ConclusionS
Structural optimization
15.1.1 Size optimization
15.1.2 Shape optimization
15.1.3 Topological optimization
15.2 A case study
15.3 Conclusions
Damage simulation
16.1 Introduction
16.2 Damage in ductile materials
16.3 Damage in composite materials
16.4 Damage in bonding
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
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
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
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
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
