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*

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

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