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Swiss Federal Institute of Technology Page 1
The Finite Element Method for the Analysis of Linear Systemsy y
Prof. Dr. Michael Havbro Faber Swiss Federal Institute of Technology
ETH Zurich, Switzerland
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 2
C t t f T d ' L tContents of Today's Lecture
• G l Sh ll El t• General Shell Elements
Pure displacement based formulation- Pure displacement based formulation
- Mixed interpolation elements (MITCn)
• Boundary conditions
• Assignment 5
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 3
General Shell ElementsGeneral Shell Elements
t
s
kw0 k
nV
Element midsurface
kβ
k
kv
kukα
02kV
ka
,z w r0
1kV
,y vye
ze
xe 0 0at Gauss integration point at Gauss integration point2 2
kn k n
a aV h V=∑
Method of Finite Elements 1
,x uat Gauss integration point at Gauss integration point2 2n k n
k∑
Swiss Federal Institute of Technology Page 4
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elements
0nV
kβ
r
kv
kukα
02kV
01kV
ka
Top surface
t
s
2a
Gauss pointr
s-coordinate line (r,t constant)r-coordinate line (s,t constant)
re
se
Mid surface
2a
Bottom surface
, , rr s t
××= = =
t es t te e et t t
, , : Tangent vectors to , , coordinate linesr s tr s t
2 2 2r× ×s t t e t
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 5
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elements
• W it th C t i di t f i t i
kβ
r
kv
kukα
02kV
01kV
ka
• We may write the Cartesian coordinates of a point using natural coordinates before and after deformation as:
1 1( , , )
2
( , , )
q ql l k
k k k k nxk kq q
l l kk k k k
tx r s t h x a h V
ty r s t h y a h V
= =
= +
= +
∑ ∑
∑ ∑1 1
1 1
( , , )2
( , , )2
h
k k k k nyk kq q
l l kk k k k nz
k k
y r s t h y a h V
tz r s t h z a h V
= =
= =
+
= +
∑ ∑
∑ ∑where:
, , : Cartesian coordinates of any point in the element, , : Ca
l l l
l l lk k k
x y zx y z rtesian coordinates of -th nodek
: Thickness of the element in t direction at node
, , : Components of unit vector normal to the shell mid
s
kk k k knx ny nz n
a k
V V V V
urface in direction at nodal point . (normal/director t k
Method of Finite Elements 1
vector at node )k
Swiss Federal Institute of Technology Page 6
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elements
• N it th di l t t i t
kβ
r
kv
kukα
02kV
01kV
ka
• Now we can write the displacements at any point as:
1 1
( , , )2k
q qk
k x k k nxk k
tu r s t h u a h V= +∑ ∑1 1
1 1
2
( , , )2
( , , )2
k
k
k kq q
kk y k k ny
k kq q
kk z k k nz
tv r s t h v a h V
tw r s t h w a h V
= =
= =
= +
= +
∑ ∑
∑ ∑1 12
where
kk k= =∑ ∑
1 0k k kn n n= −V V V
We can express the components of efficiently as: knV
0 kV00
1 0
2
, with being the unit vector in the -directionk
y nkyk
y n
y×
=×
e VV e
e V
0 0 0k k kV V V
Method of Finite Elements 1
0 0 02 1k k k
n= ×V V V
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G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
Letting and be the rotations of the director vector around and then we can write (small rotations):
knVkα kβ
01kV 0
2kV
0 0k k kα β+V V V2 1n k kα β= − +V V V
Now we substitute this result into the relations for the displacements and get: p g
0 02 1
1 1
( , , ) ( )2k
q qk k
k x k k x k x kk k
tu r s t h u a h V Vα β= =
= + − +∑ ∑1 1
( , , )2k
q qk
k x k k nxk k
tu r s t h u a h V= =
= +∑ ∑
0 02 1
1 1
0 0
( , , ) ( )2
( ) ( )
k
q qk k
k y k k y k y kk k
q qk k
tv r s t h v a h V V
tw r s t h w a h V V
α β
α β
= =
= + − +
= + − +
∑ ∑
∑ ∑
1 1
( , , )2
( , , )2
k
q qk
k y k k nyk k
q qk
k z k k nz
tv r s t h v a h V
tw r s t h w a h V
= =
= +
= +
∑ ∑
∑ ∑2 11 1
( , , ) ( )2kk z k k z k z k
k k
w r s t h w a h V Vα β= =
= + +∑ ∑1 1
( , , )2kk z k k nz
k k= =∑ ∑
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 8
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
Now we can follow the same procedure as for the beam element in developing the element matrixes
The components of the displacement interpolation matrix H are given in the relation:
0 02 1
1 1( , , ) ( )
2k
q qk k
k x k k x k x kk kq q
tu r s t h u a h V V
t
α β= =
= + − +∑ ∑0 0
2 11 1
0 02 1
( , , ) ( )2
( , , ) ( )2
k
k
q qk k
k y k k y k y kk kq q
k kk z k k z k z k
tv r s t h v a h V V
tw r s t h w a h V V
α β
α β
= =
= + − +
= + − +
∑ ∑
∑ ∑ 2 11 1
( , , ) ( )2kk z k k z k z k
k kβ
= =∑ ∑
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 9
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
In evaluating the strain-displacement matrix B we first need the derivatives of the displacements in regard to the natural coordinates
[( , , ) 1 k kkhu v w tg tg∂∂ ⎡ ⎤⎡ ⎤ ⎤⎦⎢ ⎥⎢ ⎥ [
[
1( , , ) 2( , , )
1( , , ) 2( , , )1
1
( , , ) 1
x y z x y z
kqk kkx y z x y z k
k
tg tgrr uhu v w tg tg
s sαβ=
⎤⎦⎢ ⎥⎢ ⎥ ∂∂ ⎡ ⎤⎢ ⎥⎢ ⎥∂∂ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎤= ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥∂ ∂
⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦
∑0
1 2
0
121
k kk
k k
a
a
= −
= −
g V
g V[ 1( , , ) 2( , , )
( , , ) 0 kk kk x y z x y z
u v w h tg gt
β⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦∂ ⎤⎢ ⎥⎢ ⎥ ⎦⎢ ⎥ ⎢ ⎥∂⎣ ⎦ ⎣ ⎦
2 1 2 ka= −g V
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 10
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
As usual we must now transform the derivatives to achieve their Cartesian components (global coordinates):
1−∂ ∂=
∂ ∂J
x r
then we may write:
( ) h⎡ ⎤∂∂⎡ ⎤1( , , ) 2( , , )
1( ) 2( )
( , , )
( , , )
k k k kkx y z x x y z x
kqk k k kk
k
hu v w g G g Gxr uhu v w g G g G α
⎡ ⎤∂∂⎡ ⎤⎢ ⎥⎢ ⎥ ∂∂ ⎢ ⎥ ⎡ ⎤⎢ ⎥∂∂ ⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ∑ where:
1( , , ) 2( , , )1
1( , , ) 2( , , )
( , , )
x y z y x y z y kk
kk k k kkx y z z x y z z
g G g Gs y
u v w h g G g Gt z
αβ=
⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦∂ ∂⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥∂⎣ ⎦ ∂⎣ ⎦
∑1 1
11 12
1 1 1
k k k
k k k
h h hJ Jx r s
h hG t J J J h
− −
− − −
∂ ∂ ∂= +
∂ ∂ ∂∂ ∂⎛ ⎞+ +⎜ ⎟
Method of Finite Elements 1
⎣ ⎦ 1 1 111 12 13
k k kx kG t J J J h
r s⎛ ⎞= + +⎜ ⎟∂ ∂⎝ ⎠
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G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
Having established the displacement derivatives we can now assemble the strain-displacement matrix B for the shell element.
In order to establish the strain-stress matrix (constitutive law) we must impose the shell assumption (constitutive law) we must impose the shell assumption that the stress in the direction of the normal to the shell surface is zero
with:sh=τ C ε
Txx yy zz xy yz zx
Txx yy zz xy yz zx
τ τ τ τ τ τ
γ γ γ
⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦
τ
ε ε ε ε
Method of Finite Elements 1
yy y y⎣ ⎦
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G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
Where
sh=τ C ε
T ⎡ ⎤⎣ ⎦T
xx yy zz xy yz zx
Txx yy zz xy yz zx
τ τ τ τ τ τ
γ γ γ
⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦
τ
ε ε ε ε
⎛ ⎞⎡ ⎤1 0 0 0 01 0 0 0 0
0 0 0 0
ν⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥
2
0 0 0 0 (1 ) 0 0 1 2 (1 )
Tsh sh
E νν
ν
⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥−⎜ ⎟⎢ ⎥=−⎜ ⎟⎢ ⎥
⎜ ⎢ ⎥
C Q sh
⎟
Q(1 )symmetrical 0
2(1 )
2
k
k
ν
ν
−⎜ ⎢ ⎥⎜ ⎢ ⎥⎜ −⎢ ⎥⎜ ⎢ ⎥⎣ ⎦⎝ ⎠
⎟⎟⎟⎟
Method of Finite Elements 1
2⎢ ⎥⎣ ⎦⎝ ⎠
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G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
The transformation from the Cartesian shell aligned coordinate system to the global coordinate system is made through the 4th order tensor transformation
, ,r s t
2 2 21 1 1 1 1 1 1 1 12 2 2
l m n l m m n n ll l l
⎡ ⎤⎢ ⎥2 2 2
2 2 2 2 2 2 2 2 22 2 23 3 3 3 3 3 3 3 3
1 2 1 2 1 2 1 2 2 1 1 2
2 2 2sh
l m n l m m n n ll m n l m m n n ll l m m n n l m l m m n m
=+ +
Q2 1 1 2 2 1n n l n l
⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
+⎢ ⎥2 3 2 3 2 3 2 3 3 2
3 1 3 1 3 1 3 1 1 3
2 2 22 2 2l l m m n n l m l ml l m m n n l m l m
++
2 3 3 2 2 3 3 2
3 1 1 3 3 1 1 3
where
m n m n n l n lm n m n n l n l
⎢ ⎥⎢ ⎥+ +⎢ ⎥
+ +⎢ ⎥⎣ ⎦
1 1 1
2 2 2
wherecos( , ), cos( , ), cos( , )
cos( , ), cos( , ), cos( , )x r y r z r
x s y s z s
l m n
l m n
= = =
= = =
e e e e e e
e e e e e e
Method of Finite Elements 1
3 cos( , ),y
x tl = e e 3 3 cos( , ), cos( , )y t z tm n= =e e e e
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G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
In general this transformation has to evaluated at each integration point during the integration of the stiffness matrix
– there are exceptions for e.g. flat plates
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 15
G l Sh ll El t
s
t
kw0 k
nV
Element midsurface
General Shell Elementskβ
r
kv
kukα
02kV
01kV
ka
We can compare the present formulation with a shell element constructed through superposition of a plate bending and a membrane stress behavior
Let us assume that we apply the general shell element as a flat element in the modeling of a shellflat element in the modeling of a shell
- in this case we can construct the stiffness matrix by superposition of a plate stiffness matrix and a plane stress superposition of a plate stiffness matrix and a plane stress stiffness matrix
the effective difference lies in the fact that when we the effective difference lies in the fact that when we establish the stiffness matrix for the general shell element we integrate in all directions whereas for the plate element and the plane stress elements we do not integrate in the
Method of Finite Elements 1
and the plane stress elements we do not integrate in the t-direction.
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G l Sh ll El tGeneral Shell Elements
Let us consider an example
z 100 2
nV0 3nV
y1.20.8 2
t 45o23
15
s
15
0 8 2
r0 1
nV
0 4nV
0.80.8 2
45o 14
Method of Finite Elements 1
x 10
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G l Sh ll El t
y
z
1.20.8 2
10
t 45o23
0 2nV0 3
nV
General Shell Elements
Let us consider an example
15
0.80.8 2
s
r
4
0 1nV
0 4nV
x
0.845o 1
4
10For this element the interpolation functions are those of the 4-node two dimensionalfunctions are those of the 4 node two dimensionalelement (solid element)
The directional vectors are:The directional vectors are:
0 1 0 2 0 3 0 4
0 0 0 01/ 2 , 1/ 2 , 0 , 0
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − = − = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥V V V V1/ 2 , 1/ 2 , 0 , 0
1 11/ 2 1/ 2n n n n⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
V V V V
0 1 0 2 0 3 0 41 1 1 1
hence10⎡ ⎤⎢ ⎥= = = = ⎢ ⎥V V V V
Method of Finite Elements 1
1 1 1 1
0⎢ ⎥⎢ ⎥⎣ ⎦
Swiss Federal Institute of Technology Page 18
G l Sh ll El t
y
z
1.20.8 2
10
t 45o23
0 2nV0 3
nV
General Shell Elements
Further we have
15
0.80.8 2
s
r
4
0 1nV
0 4nV
x
0.845o 1
4
10
0 1 0 2 0 3 0 42 2 2 2
0 01/ 2 ; 1
0
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
V V V V01/ 2
and
0 8 2 1 2 0 8a a a a
⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦
= = = =1 2 3 40.8 2, 1.2, 0.8a a a a= = = =
The thickness and the director vector are determinedas:
40 0
midpoint , 012 2
kkn k r s n
k
aa h ==
⎛ ⎞ =⎜ ⎟⎝ ⎠⇓
∑V V
0 0
0 0 0 0 0.00.8 2 1.2 0.81/ 2 0 0 0.2 0.406 , 0.985n n
a a
⇓
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎛ ⎞ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − + + = − ⇒ = − =⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠
V V
Method of Finite Elements 1
,2 4 8 8
1 1 0.45 0.9141/ 2n n⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
Swiss Federal Institute of Technology Page 19
G l Sh ll El tGeneral Shell Elements
So far we have considered a pure displacement based element – however, as for the beam and the plate element shear locking (and membrane locking) is a problem (at least cubic displacement interpolation p ( p pfunctions are required)
In order to solve this problem we use again mixed In order to solve this problem we use again mixed interpolation functions (Dvorkin & Bathe)
( ) ( ) ( )r r s s r s s r r t t r s t t s( ) ( ) ( )r r s s r s s r r t t r s t t srr ss rs rt st= + + + + + + +ε ε g g ε g g ε g g g g ε g g g g ε g g g g
In-layer strains Transverse shear strainsIn-layer strains Transverse shear strains
[ ]; ; , Tr s t x y zd d d∂ ∂ ∂
= = = =x x xg g g x
Method of Finite Elements 1
[ ]r s t ydr ds dt
g g g
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G l Sh ll El tGeneral Shell Elements
The Green-Lagrange covariant strain tensor components are determined from :
1 1 1 0 0 0 11 ( )( )ε ∂ ∂ += − = =
x x ug g g g g gi i0 ( ), , 2ij i j i j i i
i ir rε = − = =
∂ ∂g g g g g gi i
The objective is to interpolate the in-layer and The objective is to interpolate the in layer and transverse shear strains independently and then to express these in terms of the usual displacement interpolation functionsinterpolation functions.
We then get a formulation of the stiffness matrixas usual in terms of the usual nodal point as usual in terms of the usual nodal point displacement (and rotations) .
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 21
G l Sh ll El tGeneral Shell Elements
The four-node shell element (MITC4) proposed by Dvorkin and Bathe is attractive –
The in-layer strains are computed from the displacementy p pinterpolations
The covariant shear strains are computed from the The covariant shear strains are computed from the displacement interpolation functions at discrete locationsas for the plate element.
1
ˆijn
ij DIij k ij k
khε
=
=∑ B u
Better general performance is achieved from the MITC9 and MITC16 elements by Bucalem and Bathe.
Method of Finite Elements 1
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G l Sh ll El tGeneral Shell Elements
In the MITC9 element the discrete locations in which themixed interpolations are fixed with the displacement interpolation functions are:
ijn
For the strain components,rr rtε ε
: are the 6-node displacement ijkh
1
ˆij
ij DIij k ij k
khε
=
=∑ B u
1 1s
interpolation functionsk
3 3
r
35
r
35
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 23
G l Sh ll El tGeneral Shell Elements
In the MITC9 element the discrete locations in which themixed interpolations are fixed with the displacement interpolation functions are:
ijn
∑For the strain componentrsε
: are the 4-node displacement ijkh
1
ˆij DIij k ij k
khε
=
=∑ B u
1 1s
interpolation functionsk
3 3
r
13r13
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 24
B d C ditiBoundary Conditions
The plate elements we considered in the previous lecture were based on the Reissner-Midlin plate theory
- transverse displacements and section rotationsp
The Kirchhoff plate theoryThe Kirchhoff plate theory
- transverse displacements
Boundary conditions are prescribed accordingly
therefore the Reissner-Midlin formulation providesmore accurate representation of boundary conditions
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 25
B d C ditiBoundary Conditions
Let us consider an example:
,z w
,y v,x u L
h Rigid support
L
1hL
rigid
Method of Finite Elements 1
Swiss Federal Institute of Technology Page 26
B d C diti
,z w
,y v,x u L
Boundary Conditions
Let us consider an example:L
1hL
h Rigid support
rigid
,z w ,z w
,y v,x u 1i +
i
2i +
1i +i
2i +,y v,x u yθ
i ixθ
0, , 1, 2,..k k ku v w k i i i= = = = + + 0, , 1, 2,..kw k i i i= = + +
Reissner-Midlinsoft – only w
3-D solidDi l t i ll
Method of Finite Elements 1
soft – only whard - also θ
Displacement in all directions
Swiss Federal Institute of Technology Page 27
B d C diti
,z w
,y v,x u L
Boundary Conditions
Let us consider an example:L
1hL
h Rigid support
rigid
,z w ,z w
,y v,x u 1i +
i
2i +
1i +i
2i +,y v,x u yθ
i ixθ
0, , 1, 2,..k k ku v w k i i i= = = = + + 0, , 1, 2,..kw k i i i= = + +
3-D solidDi l t i ll
Kirschhoffw and the derivatives i e
w∂
Method of Finite Elements 1
Displacement in all directions
w and the derivatives i.e. dx
Swiss Federal Institute of Technology Page 28
B d C diti
,z w
,y v,x u L
Boundary ConditionsL
1hL
h Rigid support
rigid
The main message here is that we have to specify boundary conditions in accordance with the characteristics of elements we are using !
Method of Finite Elements 1