6.1-6.3
Transcript of 6.1-6.3
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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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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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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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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