Finite Element Analysis for Mechanical and Aerospace Design

Finite Element Analysis for Mechanical and Aerospace

Design

Prof. Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering

101 Rhodes Hall Cornell University Ithaca, NY 14853-3801 zabaras@cornell.edu

CORNELL

UNIVERSITY

MAE 4700 ? FE Analysis for Mechanical & Aerospace Design

N. Zabaras (02/20/2014)

1

Development of discrete equations

? Consider the elastic bar problem with arbitrary BCs.

dw T

dx

EA

du dx

dx

-

This

wT

is a scalar

At

|

t

- wT bdx =

This is a scalar

0, w(x) in 0 <

x < L, with w U0

This is a scalar

? Note that we added the transpost ()T on w terms for allowing us to do easy matrix operations at a later time.

At this time, this has no effects as w is a scalar!

? Having selected the FE mesh and introducing smooth

approximation functions over each element, we can

write:

e

e

dwe dx

T

Ee Ae

du e dx

dx

-

e

weT Ae t | t -

e

weT bedx = 0, w U0

e

CORNELL

UNIVERSITY

MAE 4700 ? FE Analysis for Mechanical & Aerospace Design

N. Zabaras (02/20/2014)

2

Development of discrete equations

e

e

dwe dx

T

Ee Ae

due dx dx

-

e

we T

Ae t

| t

-

scalar

e

weT bedx =0, wF U0

e scalar

scalar

scalar

? We use the same approximations for we and ue :

= u e

scalar

{ } N e

row vector

u= e , due

column vector

dx

scalar

{ } = Be ue

row vector

{ } = we w= eT

Scalars

we T N e T ,

dwe dx

T

=

{ } Be d e

row vector Nodal DOF

{ }we T Be T

? Substitution into the weak form gives:

{ } { } we

T

Be T AeEe Be dx d e -

T

N e

T

bedx + N e

Ae t | t

= 0, {wF }

e

e

Element stiffness

e e

Element Load Distributed

Element load Concentrated

CORNELL

UNIVERSITY

MAE 4700 ? FE Analysis for Mechanical & Aerospace Design

N. Zabaras (02/20/2014)

3

Development of discrete equations

{ } { } [ ] { } e

we

T

e

Be

T

Ae Ee

B e dx

de

-

e

T

N e

e

T

bedx - N e

Ae t | t

= 0, wF

K e = Be T AeEe Be dx

e

T

T

{ } = f e N e bedx + (N e Ae t) |t

Column vector

e

{ }fe

{ fe}

{we} = Le {w}

? The element and global matrices are related: {d e} = Le {d}

? The global stiffness and load now take the form:

{ } {w}T

Le T K e Le {d}

e

[K]

-

Le T f e

e

{F}

= 0,{wF }

CORNELL

UNIVERSITY

MAE 4700 ? FE Analysis for Mechanical & Aerospace Design

N. Zabaras (02/20/2014)

4

Partitioning of the global solution

? As we did in earlier lectures, we partition the solution and weight function to account for essential boundary conditions:

=d

d

E= , w

dF

= wwEF

0

wF

? So in our discretized weak form the

statement for every wU0 needs to be

translated to for every {wF } .

CORNELL

UNIVERSITY

MAE 4700 ? FE Analysis for Mechanical & Aerospace Design

N. Zabaras (02/20/2014)

5

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