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Same sha e functions are used to inter olate nodal

coordinates and displacements Shape functions are defined for an idealized mapped

e emen e.g. square or any qua r a era e emen

Advantages include more flexible shapes andcom a ibili

We pay the price in complexity and require numericalintegration to calculate stiffness matrices and equivalent

oa s

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Bar element example ree-no e ar examp e use on y or ustrat on

Quadratic variation of both coordinate and displacement

in terms of ideal element coordinate1 4

2 2

2 51 1x a and u a

a a

1

1 1 1

2

2 2 2

1 1 1 1 1 1

1 0 0 1 1 0 0

x a x

x a hence x x

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3 3 31 1 1 1 1 1x a x

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3

1 2 3

T

T

x N x x x

u N u u u

2 2 21 1

12 2

N

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Strains and stiffness matrix

is more complicated

dddu

ddu 1

Jacobian

ddxdx

w ere

u

uN

dxdxx

3

2

2 2

3 3

1 11 2 2 1 2

2 2

dx dJ N x x

d dx x

1 1 1 1

1 2 2 1 22 2

dB N

J d J

And stiffness matrix

TL

T1

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x

10

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6.2: Bilinear quadrilateral We were limited to rectangles because of compatibility

Now both displacement and coordinates are bi-linearfunctions of

H w h r rv m i ili ?

and

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Interpolation Mapping

dNuNucNxNx iiii

&vy iiii

44332211 yxyxyxyxcT

4321

44332211

0000 NNNN

vuvuvuvudT

Interpolation functions

4321 0000 NNNN

11

)1)(1(41)1)(1(

41

21 NN

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4443

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Derivatives Chain rule of differentiation

yx

y

xJoryx

yx,

,

,

,

yx

iiii yNxNyx ,,,, iiii yNxNyx ,,,,

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-

Usin sha e functins

121122

11

)1()1()1()1(1 JJyxyx

J

Then

2221

44

33

yx

1121

12221

2221

1211 1

,

,

,

,

JJ

JJ

JJwhere

y

x

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Strain and stiffness matrices

yx

y

x u,10000001

Where

y

x

xyv,

,0110

,

,

00

00

,

,

2221

1211

u

u

u

u

y

x

,

,

00

00

,

,

2221

1211

v

v

v

v

y

x

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Laboring to obtain B matrix Finally expressed in terms of nodal displacements

1, 2, 3, 4,, 0 0 0 0u N N N N

1, 2, 3, 4,

1*81, 2, 3, 4,

1, 2, 3, 4,

, 0 0 0 0

, 0 0 0 0

, 0 0 0 0

u N N N N

dv N N N N

v N N N N

So B matrix obtained by multiplying these 3x4, 4x4 and4x8 matrices

Stiffness matrix

1

1

1

18*33*33*88*33*33*88*8

ddJtBEBdydxtEEBkTT

University of FloridaEML5526 Finite Element AnalysisR.T. Haftka