Finite Element Method

ME438

Introduction

Dr Aamir Mubashar

PhD Mechanical Engineering Loughborough University, Loughborough, United Kingdom

MSC Advanced Manufacturing Technology and Systems Management The University of Manchester, Manchester, United Kingdom

Course Contents

Introduction

Stiffness (Displacement) Method

Development of Truss Equations

Development of Beam Equations

Plane Stress / Strain Stiffness Equations

Practical Considerations in Modelling

Usage of Commercial Finite Element Software (Abaqus)

Term Project (4 weeks)

Marks Distribution

Quizzes 15%

Projects / Assignments 10%

OHT-1 15%

OHT-2 15%

End Semester Exam 45%

5% Class Quizzes20% Term Project

INTRODUCTION TO FEM

Finite Element Method

FEM is a numerical method for solving problems or engineering and mathematical physics

Typical classes of problems

Structural analysis

Heat transfer

Fluid flow

Mass transport

Electromagnetic potential

etc.

Can solve problems involving complicated geometries, loadings and material properties for which analytical solutions are not possible

GENERAL STEPS OF FEM

Introduction to FEM

General Steps of FEM

For simplicity, we will consider the structural problem for now

Engineers seeks to determine displacements and stresses throughout a structure, which is in equilibrium and subjected to applied loads

Two general approaches traditionally associated with FEM

Force or flexibility method: uses internal forces as unknowns, result is a set of algebraic equations for determining unknown forces

Displacement or stiffness method: uses displacements of nodes as unknowns

For computational purposes, displacement or stiffness method is simpler to formulate for most of the problems and will be discussed further

How FEM Works?

How FEM Works?

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 1 & 8 are generally decided by AnalystSteps 2-7 are carried out automatically by computer program

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 1: Discretise and Select the Element Types

Dividing the body into an equivalent system of finite elements with associated nodes

Choose most appropriate element type to model most closely the actual physical behaviour

Primary Engineering Judgements

Total number of elements used

Variation in size and type of elements within a given body

Elements

Small enough to give useable results

Large enough to reduce computational effort

Small element (and possibly higher order elements) are generally desirable where the results are changing rapidly such as where changes in geometry occurs

Large elements can be used where results are relatively constant

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 2: Select a Displacement Function

Involves choosing a displacement function within each element

Defined using nodal values of the element

Linear, quadratic and cubic polynomials are frequently used functions as they are simple to use in FEM formulation

Trigonometric series can also be used

For a two dimensional element, displacement function is a function of the coordinates in its plane (e.g. x-y plane)

Step 2: Select a Displacement Function

The functions are expressed in terms of nodal unknowns (in two dimensional problem, in terms of x-y components)

Same general displacement function can be used repeatedly for each element

Hence, in FEM a continuous quantity such as displacementthroughout the body is approximated by a discrete modelcomposed of a set of piece-wise continuous functions defined within each finite domain or finite element

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 3: Define the Strain / Displacement and

Stress / Strain Relationships

Necessary for deriving the equations for each finite element

In case of one dimensional deformation, say x-direction

=

Stresses must be related to strains by a constitutive law. Simplest relationship is given by Hookes law

=

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 4: Derive the Element Stiffness Matrix

and Equations

Stiffness matrix and element equations relating nodal forces to nodal displacements are obtained using force equilibrium conditions

Most easily adaptable to line or one dimensional elements

Can be used to illustrate development for spring, bar, and beam elements

Several methods are used for determining the element stiffness matrix

Direct Equilibrium or Stiffness Method

Step 4: Derive the Element Stiffness Matrix

and Equations

Using any of the above methods, we get the equations to describe the behaviour of an element

123

=

11 12 13 121 22 23 231 32 33 3 1

123

Vector of element

nodal forces

Element Stiffness Matrix

Vector of unknown element

nodal dofs

= In compact form

Generalised displacements may include such quantities as actual displacements, slopes or even curvatures

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 5: Assemble to Obtain the Global

Equations and Introduce Boundary Conditions

The individual element nodal equilibrium equations generated in Step 4 are assembled into the global nodal equilibrium equations

The direct method of superposition (called the direct stiffness method), based on nodal force equilibrium, can be used

Implicit in the direct stiffness method is the concept of continuity or compatibility

Compatibility means that the structure remains together and no tears occur anywhere inside the structure

The final assembled equation can be written as

=

Step 5: Assemble to Obtain the Global

Equations and Introduce Boundary Conditions

For most problems, global stiffness matrix is square and symmetric

It can be shown that global stiffness matrix is singular i.e. its determinant is equal to zero

To remove singularity, boundary conditions (constraints or supports) are applied so structure remains in place and no rigid body motion occurs

Applied known loads are accounted for in the global force matrix

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 6: Solve for Unknown Degrees of

Freedom (or Generalised Displacements)

After modification to account for boundary conditions, a set of simultaneous algebraic equations is obtained

Where is the structure total number of unknown nodal degrees of freedom

Can be solved for s using elimination method (such as Gausss method) or an iterative method (such as Gauss-Seidel method)

123

=

11 12 13 121 22 23 231 32 33 3

1

123

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 7: Solve for the Element Strains and Stresses

For structural stress analysis, important secondary quantities of strain and stress can be obtained from displacements

Typical relationships between strain and displacements and between stress and strain can be used

General Steps of FEM

Step 1

Discretise and select the element types

Step 2

Select a displacement function

Step 3

Define the strain / displacement and stress / strain relationships

Step 4

Derive the element stiffness matrix and equations

Step 5

Assemble to obtain global equations and apply boundary conditions

Step 6

Solve for unknown degree of freedom (displacements)

Step 7

Solve for element strains and stresses

Step 8

Interpret the results

Step 8: Interpret the Results

Final goal is to interpret and analyse the results for use in the design / analysis process

Determination of locations in structures where large deformationsand large stresses occur is generally important

Post processor computer programs help the user to interpret the results by displaying them in graphical form