Introduction to FEM
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Transcript of Introduction to FEM
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Simulation Tool
Response of a structure or system to the loads imposed
Structural Analysis Tool
Measuring critical loads or failure criteria
Structural Design Tool
Modifying the structure to improve performance
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Number of Edges
Calculated
circumference
2r
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Node
Element
Mesh
Load
Boundary Condition
Material Property
TrueGrid
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Compatibility
Equilibrium
Constitutive Law
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Things fit together with no gaps
Each node and element boundary matches the one beside it.
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F1 F2
Fk Fn
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E
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P
P
A
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P
P
A
A A Pn
0lim nA
P
A
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Ps A
0lim sA
P
A
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Megson
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Megson
xy yx
yz zy
zx xz
, ,x y zNormal components
, , ,
, ,
xy yx yz
zy zx xz
Shear components
Perpendicular to
this axis
Parallel to this
axis
xy
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x xy xz
yx y yz
zx zy z
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x
y
z
yz
zx
xy
Careful with the
shear terms,
different sources
use different order
x
y
z
yz
zx
xy
Sigma
Tau
Epsilon
Gamma
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E
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1 0 0 01 1
1 0 0 01 1
1 0 0 01 1(1 )
(1 2 )(1 )(1 2 ) 0 0 0 0 02(1 )
(1 2 )0 0 0 0 0
2(1 )
(1 2 )0 0 0 0 0
2(1 )
x x
y y
z z
yz yz
zx zx
xy x
E
E
y
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Continuum
(Displacement only)
Structural
(Displacement and Slope)
1D
Bar (u)
Truss (u,v,w)
Beam (u,v,w,
x,y,z)
2D
Plane stress
(u,v)
Plane Strain
(u,v)
Plate (u,v,w,
x,y) Shell
(u,v,w,
x,y,z)
3D
Bricks (u,v,w)
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Real Structure
Model of Structure
Discretised Model
Different Model
of Structure
Discretised Model
Ma
gn
itude o
f E
rrors
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Remember:
Refinement does
not make your
closer to REALITY.
Refinement makes
your results closer
to your MODEL!
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(a) Von mises stress (b) Fore-aft load paths showing detail
around mast step
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11 12 1 1
21 22 2 2
k k u P
k k u P
P1 P2
u1 u2
E, A, L
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P1 P2
u1=0 u2=1
N N
Node 1 Node 2
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What if we have multiple elements or loads?
We can assemble multiple elements using equilibrium at the nodes (+ compatibility)
I II III
P1 P2 P3 P4
1
IP2
IP2
IIP3
IIP3
IIIP4
IIIP
I II III
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Garth Pearce 2012
At Node 1:
At Node 2:
1 1
1 2 1
2 2 2
1 2 3 2
0
0
I
I I I I
I I
I II
I I I I II II II II
I I II II
P P
A E A Eu u P
L L
P P P
A E A E A E A Eu u u P
L L L L
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Garth Pearce 2012
In matrix form
1 1
2 2
3 3
4 4
0 0
0
0
0 0
I I I I
I I
I I I I II II II II
I I II II
II II II II III III III III
II II III III
III III III III
III III
A E A E
L L
u PA E A E A E A E
L L L L u P
u PA E A E A E A E
L L L L u P
A E A E
L L
u1 u3 u2 u4
I II III
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Ku P
K u P
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Steel Bar, A = 100mm2, L = 1m Al Bar, A = 50mm2, L = 600mm
F1 = 3kN F2 = 1kN
X=0 X=1000mm X=1600mm
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ij j iij
EAF A u u
L
j iij
ij
u u
L
ij j iij
EE u u
L
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1 kN
2 kN
1000 1000 1000
A = 50 mm2
E = 70 GPa
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P1
P2
u1
u2
Q2
Q1
v1
v2
2 2
1 1
2 2
1 1
2 2
2 2
2 2
2 2
cos
where
sin
j i
j i
u Pl lm l lm
v Qlm m lm mAE
u PL l lm l lm
v Qlm m lm m
x xl
L
y ym
L
j iijj iij
u uEAF l m
v vL
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2 2
2 2
2 2
2 2
2 2
2 2
2 2 2
[ ]
T
i i i j j j
T
i i i j j j
ij
j i j i j i
ij j i j i j i
ij ij
u v w u v w
P Q R P Q R
l lm ln l lm ln
lm m mn lm m mn
ln mn n ln mn nEA
L l lm ln l lm ln
lm m mn lm m mn
ln mn n ln mn n
x x y y z zL x x y y z z l m n
L L
u
P
K
ij
j i
ij j i
ij
j i
L
u uEA
F l m n v vL
w w
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http://bendingmomentdiagram.com/
v3 v2
u1 u3 u2
I II v1
Try with truss elements
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3 2 3 2
1 1
2 21 1
2 2
3 2 3 2
2 2
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
EI EI EI EI
L L L Lv PEI EI EI EI
ML L L L
v PEI EI EI EI
L L L L M
EI EI EI EI
L L L L
P2 P1 v2
M1 M2
2
v1
1