# Practical Introduction to Non-Linear Finite Element Analysis (FEA)

This non-linear Finite Element course is intended for delegates interested in learning how finite elements are used to analyse advanced non-linear problems, difficulties encountered in modelling real-life applications and guidelines for using non-linear finite element technology.

The objectives of this Finite Element course are:

• To provide delegates with an introduction to the fundamental theory of non-linear Finite Element analysis.

• To highlight the possible difficulties that may be encountered in using Finite Element software to analyse non-linear problems.

## Who Should Attend ?

This non-linear finite element course is aimed at engineers and scientists who want to gain an understanding of the fundamental theory of non-linear Finite Element Analysis, solution accuracy, difficulties and application to practical problems.

As this is an advanced Finite Element course, a pre-requisite for this course is a reasonable knowledge of linear Finite Element theory and applications. However, no prior knowledge of non-linear Finite Element theory is required. The course is independent of any finite element software code.

## Technical Content

### Brief Overview of Linear Finite Element analysis

A brief overview of linear Finite Element formulation, numerical algorithms, etc. to provide a foundation for the non-linear formulation.

### General Introduction to Non-linear problems

Classifications of non-linear problems, Comparison of linear and non-linear finite element analysis, Non-linear algorithms and procedures, Difficulties in modelling non-linear problems.

### Plasticity

Basic plasticity theory, Uniaxial and multi-axial plasticity, Work hardening and cycle loading, finite element treatment of plasticity, Solution strategy and accuracy, Discussion of typical practical plasticity applications.

### Creep and Visco-elasticity

Basic theory of creep, uniaxial and multiaxial creep therory, time and strain hardening, Explicit and implicit time integrations, Discussion of typical practical creep applications.

### Contact Problems

Basic theory of contact mechanics, classification of contact configurations, Hertzian and non-Hertzian contact problems, finite element contact algorithms, Penalty methods and Lagrange multipliers, Difficulties in modelling contact problems, Tips and guidelines, Discussion of practical contact problems.

### Geometric Non-linearity

Basic theory of geometric non-linearity, GNL stress-strain definitions, finite element algorithms for geometric non-linearities, buckling problems, Arc-length and line-search methods, Solution strategy and accuracy, Discussion of typical GNL problems.

### Brief introduction to other advanced Finite Element Applications

A brief overview of fracture mechanics, fatigue analysis, thermo-mechnical problems, viscoelastic materials (polymers, plastics, rubbers), explicit finite element codes

## PSE Competencies addressed by this training course

IDCompetence Statement
Plasticity
PLASkn1 For a beam under pure bending sketch the developing stress distribution from first yield, to collapse.
PLASkn2 For a simple steel thick cylinder or sphere under internal pressure, state the location of first yield.
PLASkn7 Sketch a stress-strain curve for an elastic-perfectly plastic and bi-linear hardening material showing elastic
and plastic modulii.
PLASco1 Discuss salient features of the inelastic response of metals.
PLASco2 Explain the terms Isotropic Hardening, Kinematic Hardening and Rate Independency.
PLASco3 Discuss the role of the Hydrostatic and Deviatoric Stress Components in yield criteria for isotropic,
polycrystalline solids.
PLASco7 Explain the phenomenon of Shakedown and define the term Shakedown Load.
PLASco8 Contrast the terms Ratchetting and Low Cycle Fatigue.
PLASco11 Explain how plastic effects in a Finite Element system are commonly handled as a series of incremental
iterative linear analyses
PLASco12 Explain, in general terms, the function of the Mises Flow Rule or Prandtl - Reuss Equations, used in a
finite element solver.
PLASco13 Outline how the cumulative and incremental displacements, total strains, elastic strains, elastic stresses
and plastic strains are related.
PLASco14 Illustrate typical examples of Local Plastic Deformation and Gross Plastic Deformation.
PLASco16 Explain the significance of a Hysteresis Loop in a load/deflection test.
PLASco23 Describe the Bauschinger Effect.
PLASco27 Explain the process of Stress Redistribution.
PLASco28 Describe the process and common purpose of Autofrettage.
PLASap4 Use FEA to illustrate Shakedown for a range of components/structures and actions.
PLASap5 Use FEA to determine the presence of ratchetting for a range of components and actions.
PLASap7 Using standard material data, derive a true stress vs true strain curve to be used for nonlinear analysis.
PLASsy2 Plan a series of simple benchmarks in support of a more complex plasticity analysis.
PLASsy4 Prepare an analysis specification for a nonlinear material analysis, including modelling strategy, highlighting
any assumptions relating to geometry, loads, etc.
PLASev1 Select appropriate solution schemes for non-linear material problems.
PLASev4 Assess the significance of simplifying geometry, material models, mass, loads or boundary conditions, on
a non linear material analysis.
Creep and Time Dependency
CTDkn2 State the Time Hardening and Strain Hardening Laws, based on Norton s Power Law, for primary creep.
CTDkn3 State how typical creep laws depend on temperature.
CTDkn4 List the range of creep and time-dependent constitutive models available in any finite element used.
CTDkn5 Identify the extent to which your application software allows modification of creep solution parameters.
CTDkn6 State the basic definitions of stress relaxation and creep.
CTDco1 Describe and illustrate a standard creep curve for steels, highlighting the steady state regime.
CTDco2 Using the standard creep curve, describe the effects of (i) increasing stress level and (ii) removing the
stress.
CTDco3 Describe different ways of presenting creep data.
CTDco4 Explain the term Stress Redistribution in a structure subject to creep under load.
CTDco9 Contrast the creep solution procedure with the procedure commonly employed for plasticity.
CTDco10 Discuss the complexities arising from a multiaxial stress state and illustrate how these are normally handled.
CTDco11 Discuss the advantage and validity of using a stiffness matrix that doesn t vary during the creep solution.
CTDco14 Explain why it is important to carefully consider the output required from a finite element system for this
type of analysis.
CTDco17 Contrast Explicit and Implicit Creep Integration.
CTDco19 Describe why a creep analysis is necessary for relevant components in your organisation or sector.
CTDap1 Define creep constitutive data as appropriate.
CTDap2 Use FEA to obtain creep solutions for a range of typical components.
CTDsy3 Prepare an analysis specification for a time dependent analysis, including modelling strategy, highlighting any
assumptions relating to geometry, loads, boundary...
CTDev3 Assess the significance of simplifying geometry, material models, mass, loads or boundary conditions, on
a time dependent analysis.
CTDev4 Select appropriate solution schemes for time dependent problems.
Nonlinear Geometric Effects and Contact
NGECkn1 Identify the contact facilities available in a finite element system, including friction models.
NGECkn2 Identify the algorithm used to follow highly non-linear equilibrium paths in a finite element system.
NGECkn3 List common categories of geometric non-linearity and contact.
NGECco1 Discuss the terms Geometric Strengthening and Geometric Weakening.
NGECco2 Explain why load sequencing can give rise to different end results and identify relevant examples.
NGECco3 Explain how large displacement effects can be handled as a series of linear analyses.
NGECo5 Discuss the term Load Following.
NGECo7 Contrast the terms Large Displacement and Large Strains.
NGECo8 Discuss the meshing requirements for accurate contact area and contact pressure.
NGECo9 Discuss the limitations of contact algorithms available in a finite element system.
NGECo10 Discuss the theoretical basis of the contact algorithms available in a finite element system.
NGECo11 Explain the challenges of following a highly non-linear equilibrium path with both load control and
displacement control.
NGECo12 Contrast the Newton-Raphson method and the Riks arc-length method.
NGECap1 Identify whether a system has automatic re-meshing and implement a re-meshing strategy as appropriate, due
to significant distortion of a mesh.
NGECap2 Conduct large displacement analyses.
NGECap3 Carry out large strain analyses.
NGECap4 Use an analysis system to carry out contact analyses.
NGECap6 Carry out analyses with load following.
NGECan1 Analyse the results from geometrically nonlinear analyses (including contact) and determine whether
they satisfy inherent assumptions.
NGECsy1 Plan a series of simple benchmarks in support of a more complex nonlinear analysis.
NGECsy2 Plan modelling strategies for geometrically nonlinear problems, including contact.
NGECev1 Assess whether Load Following is likely to be required in any analysis.
NGECev2 Select appropriate solution schemes for geometrically non-linear problems
Buckling and Instability
BINco3 Explain why theoretical Buckling Loads (including those calculated using FEA) often vary significantly from
test values.
BINco5 Discuss the snap-through buckling of a shallow spherical shell subjected to a lateral load and explain why a
linear buckling analysis is not appropriate.
BINco13 Explain the meaning of Stable Buckling and provide examples.
BINco14 Explain the meaning of Unstable Buckling and provide examples.
BINco18 Explain when geometric non-linear analysis should be used in a buckling analyses.
BINsy3 Plan a series of simple benchmarks in support of a more complex instability analysis.
BINsy4 Plan modelling strategies for buckling and instability problems.
BINev2 Select appropriate idealisation(s) for a buckling analysis.
BINev3 Assess whether a non-linear buckling analysis is necessary.
BINev4 Select appropriate solution schemes for buckling problems.

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