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Finite Element Method

- Brian Hammond
- Ivan Lopez
- Ingrid Sarvis

Fundamental Concept of FEM

- A continuous field of a certain domain having

infinite degrees of freedom is approximated by a

set of piecewise continuous models with a number

of finite regions called elements. The number of

unknowns defined as nodes are determined using a

given relationship i.e.FKd.

Fundamental Concept of FEM

- Red line-Continuous field over the entire domain.
- Blue line-Finite number of linear approximations

with the finite number of elements

General Steps

- Discretize the domain
- a) Divide domain into finite elements using

appropriate element types (1-D, 2-D, 3-D, or

Axisymmetric) - 2) Select a Displacement Function
- a) Define a function within each element using

the nodal values - Define the Strain/Displacement and Stress/strain

Relationships - 4) Derive the Element Stiffness Matrix and

Equations - a)Derive the equations within each element

General Steps

- 5) Assemble the Element Equations to Obtain the

Global or Total Equations and Introduce Boundary

Conditions - a)Add element equations by method of

superposition to obtain global equation - 6) Solve for the Unknown Degrees of Freedom (i.e

primary unknowns) - 7) Solve for the Element Strains and Stresses
- 8) Interpret the Results

Applications

- Stress Analysis
- Truss and frame analysis
- Stress concentration
- Buckling
- Vibration analysis
- Heat transfer
- Fluid flow

Advantages of FEM

- Model irregularly shaped bodies
- Compute General load conditions
- Model bodies composed of different materials
- Solve unlimited numbers and kinds of boundary

conditions - Able to use different element sizes in places

where loads or stresses are concentrated - Handle non-linear behavior using linear

approximations - Reduce System Cost

FEM Packages

- Large Commercial Programs
- Designed to solve many types of problems
- Can be upgraded fairly easily
- Initial Cost is high
- Less efficient
- Special-purpose programs
- Relatively short, low development costs
- Additions can be made quickly
- Efficient in solving their specific types of

problems - Cant solve different types of problems

FEM Packages

- Algor
- ANSYS
- COSMOS/M
- STARDYNE

- IMAGES-3D
- MSC/NASTRAN
- SAP90
- GT-STRUDL

Note on Stiffness matrix

For a 1-D bar, the stiffness matrix is derived

from the stress/strain relationship in Hookes

law and the definitions of stress and strain. sx

Eex. sx P/A ex

du/dx (d2x d1x)/L By substitution -f1x

EA (d2x d1x)

L f1x EA (d1x d2x)

L Similarly for f2x f2x EA (d2x

d1x)

L Combining into matrix form, the stiffness

matrix is defined as

L

E

A

f2

f1

d1x

d2x

- -1
- -1 1

k EA

L

Note on the displacement function

- For a given set of nodes there exists a function

that approximates - the displacement at any position along the

bar. - This function, called the displacement function,

is derived from - Pascals Triangle.
- A new constant is introduced into the function

for every node in - the discretized domain.

For 1-D

u(x) a1 a2x a3x2

Note on the displacement function

If a 1-D bar is broken into 2 elements, the

displacement function would be

u(x) a1 a2x a3x2.

a1 a2 a3

Putting it into matrix notation u(x) 1 x

x2

By knowing the distances to the nodes and the

displacements at those nodes, the equation

becomes

u1 u2 u3

- 0 0
- 1 x2 x22
- 1 x3 x32

a1 a2 a3

, where

x1 0, x2 and x3 are the distances to the nodes

and u1, u2, and u3 are the displacements.

The coefficients are found by solving the

equation.

Example

E

2E

A A

P

a

b

L

L

Determine displacements of materials a and b if

the load P is applied to the end of the bar

given the above information.

Example cont.

1) Discretize the domain with appropriate

elements.

Element a

Element b

f3 P

f1

2

3

1

u1

u2

u3

f21

f22

2

2

3

f1

f3 P

1

u1

u2

u3

u2

Example cont.

2) Select a displacement function

There will be new term for each element, and the

terms are

derived from Pascals triangle.

1

2

3

u1

u2

u3

u(x) a1 a2x a3x2

Example cont.

3) Define stress/displacement and stress/strain

relationships

sx Eex

ex du/dx

4) Derive the element stiffness matrix and

element equations

F kd k stiffness

matrix

- -1
- -1 1

k EA

F Force

L

d displacement

a

b

1 -1 -1 1

EA

f1 f21

u1 u2

2 -2 -2 2

EA

f22 f3

u2 u3

,

L

L

Example cont.

- Construct Global equation and introduce boundary
- conditions and known variables.

- -1 0
- -1 3 -2
- 0 -2 2

f1 f21f22 f3

EA

u1 u2 u3

Global Equation

L

B.C. (x 0) u1 0 Known variables f3

P and f2 f21f22 0

Example cont.

6) Solve for unknowns.

- -1 0
- -1 3 -2
- 0 -2 2

f1 0 P

EA

0 u2 u3

L

f1 -EAu2 L

0 EA(3u2-2u3) L

P EA(2u3-2u2) L

u2 PL EA

u3 3PL 2EA

f1 -P

Example cont.

7) Solve for the element strains and stresses.

ea P u2 EA L

sa Eea P A

eb 3P u3 2EA L

sa 2E eb P 3 A

8) Interpret the Results

- After solving for the displacements, the

coefficients of the - displacement function can be determined.

Bibliography

- Logan, Daryl L. A First Course in the Finite

Element Method Using Algor. Brooks/Cole,

Pacific Grove, CA. 2001 - http//www.finite-element-method.info
- http//www.eng.fsu.edu/chandra/courses/eml4536/
- http//urbana.mie.uc.edu/yliu/FEM-525/FEM_Lecture_

Notes_Liu_UC.pdf

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