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AN INTRODUCTION TO THE FINITE ELEMENT METHOD

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J.P. PONTHOTAerospace & Mechanical Laboratory/LTAS-MN2L

University of Liège, Belgium

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ACKNOWLEDGEMENTS

The present course material is based on the lecture notes developed for the introductory Finite Element Method (FEM) course taught at the University of Liège.

This course has been handled for over fifteen years by late Prof. G. SANDER, to whom large tribute has to be paid for having launched new and synthetic ideas on teaching FEM to undergraduate students and engineers.

It was then handled over more than fifteen years by Prof. M. HOGGE. I am very indebted to these two people for influencing quite deeply my “thinking” over FEM and for their seminal contribution to these notes.

I am also very thankful to Prof. C. FELIPPA, from the University of Boulder, Colorado, for posting his own material notes on the web with free access.

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An introduction to Finite Element Method

CONTENT OF THE COURSE:

1.INTRODUCTION

Overview

Brief History

Basic Concepts

Idealization process

Applications in aeronautics, space & automotive engineering

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2. THE DIRECT STIFFNESS METHOD (PART I) OR A FIRST STEP INTO STRUCTURAL MECHANICS

Goal of the Direct Stiffness Method

The linear spring

Energy considerations

A two spring assembly

Automatic assembly process

Boundary conditions

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3. THE DIRECT STIFFNESS METHOD (PART II) OR A FIRST STEP INTO STRUCTURAL MECHANICS

Goal of the method

A simple 1D truss/bar element

Truss structures

Spatial element formulation

Stiffness matrix

Assembly and Global stiffness matrix

Stress computation

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4. CONSTRUCTING MoM MEMBERS, OR MATRIX STRUCTURAL ANALYSES

What are Mom members? (MoM = Mechanics Of Materials)

The Bar element

The Spar element

The Shaft element

The Shear panel

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5. PRINCIPLE OF VIRTUAL WORK AND VARIATIONAL PRINCIPLES (1)

Calculus of Variations

Exact solutions for 3D linear elastic problems

Approximations

Finite differences solutions

Strong and weak forms

Variational Principle on Displacements

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5. PRINCIPLE OF VIRTUAL WORK AND VARIATIONAL PRINCIPLES (2)

The Minimum Potential Energy Principle

The Hu-Washizu principle

The Hellinger-Reissner principle

The Fraeijs de Veubeke principle

Variational principle on stresses

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6. RAYLEIGH-RITZ METHOD

Approximations methods

Rayleigh-Ritz approximation method

1D shearing problem

3D case

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7. APPROXIMATION METHODS (1)

The tightly stretched rope

Analytical solution

Approximations to the analytical solution

Collocation method

Least square method

Galerkin’s method

Rayleigh-Ritz Method

Finite element approximation in 1D

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7. APPROXIMATION METHODS (2)

Piecewise Rayleigh-Ritz approximation

Characteristics of the approximation

Solution procedure

Shape function properties

Summary

Common features between Rayleigh-Ritz and FEM

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8. VARIATIONAL PROCEDURE FOR FINITE ELEMENTS: APPLICATION TO THE BAR ELEMENT

General procedure

Concept of kinematically admissible variation

Natural coordinates

Strain interpolation matrix

Stiffness matrix

Energy conjugated quantities

Example

Rigid body modes and assumed displacements

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8. SHAPE FUNCTIONS AND CONSISTENT LOADING IN 1D

Introduction

Lagrange interpolation and Lagrange polynomial

Element interpolation and local coordinates

Consistent nodal forces

Hermitian interpolation

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10. KINEMATICALLY ADMISSIBLE FINITE ELEMENTS (1)

Introduction

Examples of applications

Variational formulation of FEM in 3D

Plane stress problems

Finite element models

Displacement approximation

Connectors

External work and consistent nodal loading

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Assembly of the element

Principle of virtual work

Galerkin method

Constant strain triangle

Stress calculation

Linear strain triangle

Quadratic strain triangle

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Linear and quadratic quadrilateral

Bilinear quadrilateral

Quadrilateral in bending

Wilson-Q6 element

Biquadratic element

Bubble modes

Patch test

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11. ISOPARAMETRIC ELEMENTS (1)

Introduction

Geometry interpolation

Isoparametric triangles

Completeness of isoparametric elements

Construction of shape functions for Lagrange elements

Quadrangular elements

Serendipity elements

Transition elements

3D Lagrange elements

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11. ISOPARAMETRIC ELEMENTS (2)

3D Serendipity elements

Strains

Stiffness matrix

Gauss integration rule

Required order of integration

Reduced integration

Hourglass modes

Convergence requirements

Stress recovery

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12. BEAM ELEMENTS AND FRAME STRUCTURES

Mechanics of beam

Variational Principle

Discretization & shape functions

Stiffness matrix

Generalized loading

Examples

Beam-column element

Assembly of trusses frame structures

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13. FEM PRACTISE AND RECOMMANDATION

Basic steps of FEA

+ and – of FEA

General FEM modeling rule

Minimum support conditions

Using symmetries

Where finer mesh should be used

Element choice

Good practice of FEM

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14. STATICALLY ADMISSIBLE FINITE ELEMENTS

Principle of Variation on Stresses (PVS) – Weak Form

Discretization

Relations between connectors and internal parameters

Locking of kinematical modes in a patch of elements

Example 1: The triangular element

Example 2: Modeling of stiffened pannels

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15. PLATE AND SHELL ELEMENTS

Plate theory

Thin plate theory (Kirchhoff plate theory)

Governing equations

Boundary conditions

Thick plate theory (Mindlin plate theory)

Plate elements: Kirchoff plate element

Plate elements: Mindlin plate element

Shell theory

Shell elements

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16. SUPERELEMENTS

Substructures

Static condensation

Global-local analysis

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17. EXTENSION TO OTHER PROBLEMS (SURVEY)

Dynamics

Field Problems (Heat Transfer)

Extensions for nonlinear problems

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