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MECh300H Introduction to Finite Element Methods

Finite Element Analysis (F.E.A.) of 1-D Problems

Historical Background

- Hrenikoff, 1941 frame work method
- Courant, 1943 piecewise polynomial

interpolation - Turner, 1956 derived stiffness matrice for

truss, beam, etc - Clough, 1960 coined the term finite element

Key Ideas - frame work method

piecewise polynomial approximation

Axially Loaded Bar

Review

Stress

Stress

Strain

Strain

Deformation

Deformation

Axially Loaded Bar

Review

Stress

Strain

Deformation

Axially Loaded Bar Governing Equations and

Boundary Conditions

- Differential Equation
- Boundary Condition Types
- prescribed displacement (essential BC)
- prescribed force/derivative of displacement

(natural BC)

Axially Loaded Bar Boundary Conditions

- Examples
- fixed end
- simple support
- free end

Potential Energy

- Elastic Potential Energy (PE)

- Spring case

Unstretched spring

Stretched bar

x

- Axially loaded bar

undeformed

deformed

- Elastic body

Potential Energy

- Work Potential (WE)

f

P

f distributed force over a line P point

force u displacement

B

A

- Total Potential Energy

- Principle of Minimum Potential Energy

For conservative systems, of all the

kinematically admissible displacement

fields, those corresponding to equilibrium

extremize the total potential energy. If the

extremum condition is a minimum, the equilibrium

state is stable.

Potential Energy Rayleigh-Ritz Approach

Example

f

P

A

B

Step 1 assume a displacement field

f is shape function / basis function n is the

order of approximation

Step 2 calculate total potential energy

Potential Energy Rayleigh-Ritz Approach

Example

f

P

A

B

Step 3select ai so that the total potential

energy is minimum

Galerkins Method

Example

f

P

A

B

Seek an approximation so

In the Galerkins method, the weight function is

chosen to be the same as the shape function.

Galerkins Method

Example

f

P

A

B

3

2

1

1

2

3

Finite Element Method Piecewise Approximation

u

x

u

x

FEM Formulation of Axially Loaded Bar Governing

Equations

- Differential Equation
- Weighted-Integral Formulation
- Weak Form

Approximation Methods Finite Element Method

Example

Step 1 Discretization

Step 2 Weak form of one element

P2

P1

x1

x2

Approximation Methods Finite Element Method

Example (cont)

Step 3 Choosing shape functions - linear

shape functions

x

x

x-1

x0

x1

x2

x1

l

Approximation Methods Finite Element Method

Example (cont)

Step 4 Forming element equation

E,A are constant

Let , weak form becomes

Let , weak form becomes

Approximation Methods Finite Element Method

Example (cont)

Step 5 Assembling to form system equation

Approach 1

Element 1

Element 2

Element 3

Approximation Methods Finite Element Method

Example (cont)

Step 5 Assembling to form system equation

Assembled System

Approximation Methods Finite Element Method

Example (cont)

Step 5 Assembling to form system equation

Approach 2 Element connectivity table

global node index (I,J)

local node (i,j)

Approximation Methods Finite Element Method

Example (cont)

Step 6 Imposing boundary conditions and forming

condense system

Condensed system

Approximation Methods Finite Element Method

Example (cont)

Step 7 solution

Step 8 post calculation

Summary - Major Steps in FEM

- Discretization
- Derivation of element equation
- weak form
- construct form of approximation solution over

one element - derive finite element model
- Assembling putting elements together
- Imposing boundary conditions
- Solving equations
- Postcomputation

Exercises Linear Element

Example 1

E 100 GPa, A 1 cm2

Linear Formulation for Bar Element

xx1

Higher Order Formulation for Bar Element

Natural Coordinates and Interpolation Functions

x1

x-1

x

xx1

x x2

Natural (or Normal) Coordinate

Quadratic Formulation for Bar Element

Quadratic Formulation for Bar Element

Exercises Quadratic Element

Example 2

E 100 GPa, A1 1 cm2 A1 2 cm2

Some Issues

Non-constant cross section

Interior load point

Mixed boundary condition

k

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