# Finite Element Primer for Engineers: Part 2

Mechanics

Oct 29, 2013 (4 years and 5 months ago)

108 views

Finite Element Primer for Engineers: Part 2

Mike Barton & S. D. Rajan

©, 2000, Barton & Rajan

2

Contents

Introduction to the Finite Element Method (FEM)

Steps in Using the FEM (an Example from Solid
Mechanics)

Examples

Commercial FEM Software

Competing Technologies

Future Trends

Internet Resources

References

©, 2000, Barton & Rajan

3

FEM Applied to Solid Mechanics Problems

Create elements

of the beam

d
xi 1

d
xi 2

d
yi 1

d
yi 2

1

2

4

3

Nodal displacement and forces

A

FEM

model

in

solid

mechanics

can

be

thought

of

as

a

system

of

assembled

springs
.

When

a

is

applied,

all

elements

deform

until

all

forces

balance
.

F = Kd

K is dependant upon Young’s
modulus and Poisson’s ratio,
as well as the geometry.

Equations from discrete elements
are assembled together to form
the global stiffness matrix.

Deflections are obtained by
solving the assembled set of
linear equations.

Stresses and strains are
calculated from the deflections.

©, 2000, Barton & Rajan

4

Classification of Solid
-
Mechanics Problems

Analysis of solids

Static

Dynamics

Behavior of Solids

Linear

Nonlinear

Material

Fracture

Geometric

Large Displacement

Instability

Plasticity

Viscoplasticity

Geometric

Classification of solids

Skeletal Systems

1D Elements

Plates and Shells

2D Elements

Solid Blocks

3D Elements

Trusses

Cables

Pipes

Plane Stress

Plane Strain

Axisymmetric

Plate Bending

Shells with flat elements

Shells with curved elements

Brick Elements

Tetrahedral Elements

General Elements

Elementary

Stress Stiffening

©, 2000, Barton & Rajan

5

Governing Equation for Solid Mechanics Problems

[K]

{u}

=

{F
app
}

+

{F
th
}

+

{F
pr
}

+

{F
ma
}

+

{F
pl
}

+

{F
cr
}

+

{F
sw
}

+

{F
ld
}

[K]

=

total

stiffness

matrix

{u}

=

nodal

displacement

{F
app
}

=

applied

nodal

force

vector

{F
th
}

=

applied

element

thermal

vector

{F
pr
}

=

applied

element

pressure

vector

{F
ma
}

=

applied

element

body

force

vector

{F
pl
}

=

element

plastic

strain

vector

{F
cr
}

=

element

creep

strain

vector

{F
sw
}

=

element

swelling

strain

vector

{F
ld
}

=

element

large

deflection

vector

Basic equation for a static analysis is as follows:

©, 2000, Barton & Rajan

6

Six Steps in the Finite Element Method

Step 1
-

Discretization
: The problem domain is discretized
into a collection of simple shapes, or elements.

Step 2
-

Develop Element Equations
: Developed using the
physics of the problem, and typically Galerkin’s Method or
variational principles.

Step 3
-

Assembly
: The element equations for each element
in the FEM mesh are assembled into a set of global equations
that model the properties of the entire system.

Step 4
-

Application of Boundary Conditions
: Solution
cannot be obtained unless boundary conditions are applied.
They reflect the known values for certain primary unknowns.
Imposing the boundary conditions modifies the global
equations.

Step 5
-

Solve for Primary Unknowns
: The modified global
equations are solved for the primary unknowns at the nodes.

Step 6
-

Calculate Derived Variables
: Calculated using the
nodal values of the primary variables.

©, 2000, Barton & Rajan

7

Process Flow in a Typical FEM Analysis

Start

Problem

Definition

Pre
-
processor

nodes and elements
(ex: ANSYS)

material property data.

boundary conditions
constraints.)

Processor

Generates
element shape
functions

Calculates master
element equations

Calculates
transformation
matrices

Maps element
equations into
global system

Assembles
element equations

Introduces
boundary
conditions

Performs solution
procedures

Post
-
processor

Prints or plots
contours of stress
components.

Prints or plots
contours of
displacements.

Evaluates and
prints error
bounds.

Analysis and

design decisions

Stop

Step 1, Step 4

Step 6

Steps 2, 3, 5

©, 2000, Barton & Rajan

8

Step 1: Discretization
-

Mesh Generation

airfoil geometry

mesh

generator

surface model

ET,1,SOLID45

N, 1, 183.894081 ,
-
.770218637 , 5.30522740

N, 2, 183.893935 ,
-
.838009645 , 5.29452965

.

.

TYPE, 1

E, 1, 2, 80, 79, 4, 5, 83, 82

E, 2, 3, 81, 80, 5, 6, 84, 83

.

.

.

meshed model

©, 2000, Barton & Rajan

9

Step 4: Boundary Conditions for a Solid Mechanics Problem

Displacements

DOF constraints usually
specified at model boundaries to define rigid
supports.

Forces and Moments

nodes usually specified on the model exterior.

Pressures

Surface loads usually specified on
the model exterior.

Temperatures

Input at nodes to study the
effect of thermal expansion or contraction.

Loads that affect the entire
structure (ex: acceleration, rotation).

©, 2000, Barton & Rajan

10

Step 4: Applying Boundary Conditions (Thermal Loads)

Temp

mapper

Nodes from

FE Modeler

Thermal
Soln Files

bf, 1,temp, 149.77

bf, 2,temp, 149.78

.

.

.

bf, 1637,temp, 303.64

bf, 1638,temp, 303.63

©, 2000, Barton & Rajan

11

Step 4: Applying Boundary Conditions (Other Loads)

Speed,

temperature

and

hub

fixity

applied

to

sample

problem
.

FE

Modeler

used

to

apply

speed

and

hub

constraint
.

antype,static

omega,10400*3.1416/30

d,1,all,0,0,57,1