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AHMAD SHELL FEM FORMULATION
U n i v e r s i t y o f M i n h o
Department of Civil EngineeringGuimarães, Portugal
[email protected] ∗∗∗∗ www.civil.uminho.pt
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2
1 - Introduction
s3 s3
s2
Top surface
Degenerationprocess of avolume finiteelement into an
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
s2
s1
s3
s2 s1
1
Middle surface of
shell element normal
Degenerated shell
finite element
Volume finite element Bottom surface ma s e n teelement
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3
2 – Basic hypothesis
The deflection orthogonal to the shell middle surface of any point is small
when compared to the shell thickness;
The stress orthogonal to the shell middle surface is negligible;
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
deformed shell middle surface.
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4
3 – Types of nodal points
Structures that have coplanar and kink nodes
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
Kink nodes
Coplanar nodes
Kink edges
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5
ɵnC
D
ɵnC
ɵn A
ɵn D
ɵ ɵn B
ɵn D
Schematic representation of the procedure to distinguish coplanar and kink nodes
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
ɵn A D
C
B
A
element
element
k
A B
C
D
k
k
B
A
(b) The coplanar node k has five
degrees of freedom. (c) The kink nde k has six degrees
of freedom. (a) The k node is common to the
elements A, B, C and D.
B
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6
i1
i3 δ = u
δ = u
s2
n3 ( )nk 3
x3
v2
v3
Degrees of freedom and coordinate systems for coplanar nodes
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
2
δ = u
Superfície s3=0
s1
3
k
nk 2
( )nk 1
( )δ 1 k
( )δ 3 k
( ) ( )δ θ 4 1k nk
=
( ) ( )δ θ 5 2k nk
= x2
x1
n1
( )δ 2 k
k
v1
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7
( )δ 3 k
δ 3 3= u
x3
Degrees of freedom and coordinate system for kink nodes
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
k
( )δ 1 k
( )δ 2 k
( )δ 4 k
( )δ 5 k ( )δ 6 k
δ θ 6 3= x
δ θ 5 2= x
δ θ 4 1= x
δ 2 2= u
x2
δ 1 1= u
x1
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8
4 – Coordinate systems
Middle surface ( s3 0= )
s3 s2
s1
ɵn3 ɵn
n3
v3
Coordinate systems for the Ahamdfinite element formulation: (a)global, (b) natural, (c) nodal and (d)
tangential
(a)(b)
ɵi3
ɵi2
ɵi1
x u x3 3 3, , θ
x u x2 2 2, , θ
x u x1 1 1, , θ
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
Middle surface ( s30= )
θ n2 n2 v2
θ n1
v1
n1
ɵn1
s2
n k 2,'
n k 1,
'
s1
k
k
v3
t 2' t 3 t 2
t 1 s1
π s3
s2
Surface s3 = constant
Integration points
t 3
t 3t 1
t 3
t 1
t 1
(c)
(d)
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9
a ua1 1,
g ug
2 2,
l ul
2 2, β +
Auxiliary coordinate system
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
i
b
( ) ( ) ( )u u ug l a3 3 3≡ ≡
( ) ( ) ( )g l a3 3 3≡ ≡
l ul1 1,
g u g1 1,
a ua
2 2,
α +
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10
4.1 - Nodal coordinate system
n x
s
x
s
x
sk
k
T
1
1
1
2
1
3
1,
'=
∂
∂
∂
∂
∂
∂ n
x
s
x
s
x
sk
k
T
2
1
2
2
2
3
2,
'=
∂
∂
∂
∂
∂
∂ ( )' '
3, 1, 2,' '
1, 2,
1ˆ
k k k
k k
n n nn n
= ××
If is parallel to global axis, the nodal coordinate system is parallel tothe global coordinate system ( = , = and = ). If hasopposite direction to , = , =- and =- .
nk 3, 3 x
1,k n 1 x 2,k n2 x n
k 3, 3 n
k 3,
3 x 1,k n
1 2,k n
2 x nk 3, 3
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
If is not in these special conditionsn k 3,
( )1, 3 3,
3 3,
22, 1 13, 2 3
1 ˆˆ ˆˆ ˆ
ˆ ˆ ˆ0
k k
k
k k
n i ni n
n i n i i
= ××
= − + −
( )ɵ ɵ ɵ, , ,n n nk k k 2 3 1= ×
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11
s1
s3
s2
The vector at k node is defined from the coordinates of the correspondingnodes at top and bottom element surfaces
3n̂
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
k
k θ n2
θ n1
n3
n2
n1
V 2
V 3
V 1
Not necessarily orthogonal to the middle
surface of shell element at node k .
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12
5 – Geometry definition
1 1 31
2 2 3 32
1 1
3 3 33
2
n nk
k k
k k
k k
x x nh
x N x N s n
x x n
= =
= +
∑ ∑ 1,1
2,1
3,1
31,1
132,1
1 1 3 3
33,11
1
0 0 0 0 ... 0 0 0 02 2
nn n
n
x
x
x
n
hh n N N s N N s
n xhh
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
k h
k _2
2 1 1 3 3
1,31
1 1 3 32,
3,
31,
32,
33,
......2 2
0 0 0 0 ... 0 0 0 02 2
n n
nn
n nn
n
n
n
n
x x hh N N s N N s x
x
n
n
n
=
,11
,22
,33
x
x
x
x
x N
x N X
x N
N X
=
=
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13
6 – Displacement field
θ n k 1,
n k 3,
n k 2,
1 1,
2 2,
13 3,
k n
k k
k k
u u
u N u
u u=
=
∑
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
n k 2 ,
θ n k 1,
n k 1,
uk
hk
k
k
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14
k
s3
hk V 1
θ n k 2,
θ n k 1,
ɵ,n k 1 n k 1,
n k 2,
v1,k P
n k 3,
v2, k
s3
hk k V 2
θ n k 1,
θ n k 2,
ɵ,n k 2n k 1, n k 2,
P
n k 3,
Contribution of the rotations for the displacement field
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1, 3 2,v
2
nk k k
hs θ =
1 , 1, 1,
3 2, 1,
ˆv v .
ˆ.2
n k k k
nk k k
n
hs nθ
=
=
2, 3 1,v
2
nk k k
hs θ = −
2 , 2, 2,
3 1, 2,
ˆv v .
ˆ.2
n k k k
nk k k
n
hs nθ
=
= −1,
3 2, 1,
2,
ˆ ˆv U2
n
g k k k k k k n
k
hs n n
θ
θ
= = −
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15
1
1,
2 3 2, 1,
1 2,
3
ˆ ˆ2
nng k k k k k k n
k k
uh
U u N s n n
u
θ
θ =
= = −
∑
n ggnk k k U T U =
1
2
3
n
n
k
k
u
U u
u
=
1
2
3
g
g
k
k
u
U u
u
=
1 11 12 13
2 21 22 23
3 31 32 33
ˆ
ˆ
ˆ
T
gn T
k
T
k k
n n n n
T n n n n
n n n n
= =
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1,
11 12 131,
2,
21 22 232,
3,
1,
1,
2,
2,
3,
ˆ
ˆ
g
k n
k g
k n
k gk k
g
k T
k g
k T
k g
k
n n n
n n n
n
n
θ θ
θ θ
θ
θ
θ
θ
=
=
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1 1,
1,
2 3 2, 1, 2,
1 2,
3 3,
ˆˆ ˆ
ˆ2
g
k T nk gk
k k k k T k k g
k
unh
u N s n nn
u
θ
θ
θ =
= −
∑
1 21 11 1,
2 3 22 12 2,
1
3 23 13 3,
0
02
0
n
k nnk
k k
k n
k k
u n nh
u N s n n
u n n
θ
θ
θ =
−
= − −
∑
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
11 12 13 1,
3 21 22 23 2,
1
31 32 33 3,
2
n
k nnk
k k
k n
k k
P P Ph N s P P P
P P P
θ θ
θ =
=
∑
11 21 12 11 13
21 22 22 12 23
31 23 32 13 33
0
0
0
P n P n P
P n P n P
P n P n P
= − = =
= − = =
= − = =
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1 21 11 1,
11 12 13
2 3 22 12 2,
1 21 22 23
3 23 13 3,
11 12 13 1,
3 21 22 23 2,
1
31 32 33 3,
2
2
g
k ngk
k k
k gk k k
g
k ngk
k k
k g
k k
u n nn n nh
u N s n nn n n
u n n
P P Ph N s P P P
P P P
θ
θ
θ
θ θ
θ
=
=
−
= − −
=
∑
∑
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
11 12 21 12 11 22 13 21 13 11 23
21 22 11 12 21 22 23 22 13 12 23
31 23 11 13 21 32 23 12 13 22 33 23 13 13 23
0
0
P P n n n n P n n n n
P n n n n P P n n n n
P n n n n P n n n n P n n n n
= = − + = − +
= − + = = − +
= − + = − + = − +
1 1 11 12 13 1
2 2 3 21 22 23 2
1 1
3 3 31 32 33 3
2
g g
n nk
k k
k k
k k k
u u P P Ph
u N u N s P P P
u u P P P
θ
θ
θ = =
= +
∑ ∑
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1 1
2 2
3 3
n
n
n
k k
θ θ
θ θ
θ θ
=
1 1
2 2
3 3
g
g
g
k k
θ θ
θ θ
θ θ
=
if k is a coplanar node: if k is a kink node:
1 1 11 1 3 11,1 1 3 12,1 1 3 13,1 3 11, 3 12, 3 13,
1
0 0 ... 0 02 2 2 2 2 2
n n nn n n n n n n
g
h h hh h h N N s P N s P N s P N N s P N s P N s P
u
1,1
2,1
3,1
1,1
2,1
3,1
u
uu
θ
θ
θ
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1 1 1
2 1 1 3 21,1 1 3 22,1 1 3 23,1 3 21, 3 22, 3 23,
31
1 1 3 31,1
0 0 ... 0 02 2 2 2 2 2
0 02
n n n
n n n n n n n
u N N s P N s P N s P N N s P N s P N s P
uh
N N s P
=
1,1 1
1 3 32,1 1 3 33,1 3 31, 3 32, 3 33,2,
3,
1,
2,
3,
...
... 0 02 2 2 2 2
nn n n
n n n n n n nn
n
n
n
n
uh h hh h
N s P N s P N N s P N s P N s P u
u
θ
θ
θ
,11
,22
3 ,3
g
u
u
u
u
N u
u N U
u N
N U
=
=
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19
7 – Strain components
1
1
21
2
2
1 2
12
2 1
t
t
w
t
wt
w w
t t
∂
∂
∂ ε ∂
ε ∂ ∂
ε γ ∂ ∂
+ = =
1 1 1
1 2 3
2 2 2
1 2 3
t
w w w
t t t
w w w
t t t
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂ ε
∂ ∂ ∂
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
23
3213
3 2
31
3 1
wwt t
ww
t t
∂ ∂ γ ∂ ∂
∂ ∂
∂ ∂
+
+
3 3 3
1 2 3
w w w
t t t ∂ ∂ ∂
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 2 3
g
u u u
x x
u u u x x
u u u
x x
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂ ε ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
=
T gt gt t gT T ε ε =
1 11 12 13
2 21 22 23
3 31 32 33
ˆ
ˆ
ˆ
gt
t t t t
T t t t t
t t t t
= =
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20
1g s J ε ε −=
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
s
u u u
s s s
u u u
s s s
u u u
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂ ε
∂ ∂ ∂
∂ ∂ ∂
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1 2 3s s s∂ ∂ ∂
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 2 3
x x x
s s s
x x J s s s
x x x
s s s
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
=
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21
1
1 1,1 1, 1 31,1 31, 1
2 2,1 2, 3 32,1 32,
3 3,1 3, 33,1 33,
02
0 2
n n
n n
n n n n n
h
x x x N n n N
x x x s n n
x x x N n n h N
= +
⋯ ⋯
⋯ ⋮ ⋯ ⋱ ⋮
⋯ ⋯
1 x
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
2 3 3
3
T T
V n V
x X N s L H N
x
= +
1,1 1,
2,1 2,
3,1 3,
n
T
n
n
x x
X x x x x
=
⋯
⋯
⋯
1
V
n
N
N N
=
⋮
31,1 31,
3 32,1 32,
33,1 33,
n
T
n n
n
n n
L n n
n n
=
⋯
⋯
⋯
1 02
02
n
h
H
h
=
⋱
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22
1 1 1 11,1 1, 3 31,1 3 31,
1 1 1 1 1
= + + + +2 2
n n nn n
N N h x N N h x x s n s n
s s s s s
∂ ∂∂ ∂ ∂+
∂ ∂ ∂ ∂ ∂⋯ ⋯
1 1 1 11,1 1, 3 31,1 3 31,
2 2 2 2 2
1 11 31,1 31,
3
=2 2
= + +2 2
n n nn n
nn n
N N h x N N h x x s n s ns s s s s
h x h N n N n
s
∂ ∂∂ ∂ ∂+ + + + +∂ ∂ ∂ ∂ ∂
∂+
∂
⋯ ⋯
⋯
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
2 1 1 12,1 2, 3 32,1 3 32,
1 1 1 1 1
2 1 1 12,1 2, 3 32,1 3 32,
2 2 2 2 2
2 11 32,1
3
=2 2
=2 2
= +2
n n n
n n
n n nn n
N N h x N N h x x s n s n
s s s s s
N N h x N N h x x s n s n
s s s s s
x h N n
s
∂ ∂∂ ∂ ∂+ + + + +
∂ ∂ ∂ ∂ ∂
∂ ∂∂ ∂ ∂+ + + + +
∂ ∂ ∂ ∂ ∂
∂
∂
⋯ ⋯
⋯ ⋯
32,+2
nn n
h N n+⋯
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3 1 1 13,1 3, 3 33,1 3 33,
1 1 1 1 1
3 1 1 13,1 3, 3 33,1 3 33,
2 2 2 2 2
3 11 33,1
3
=2 2
=2 2
= +2
n n n
n n
n n nn n
x N N h N N h x s n s n
s s s s s
x N N h N N h x s n s n
s s s s s
x h N n
s
∂ ∂ ∂∂ ∂+ + + + +
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂∂ ∂+ + + + +
∂ ∂ ∂ ∂ ∂
∂
∂
⋯ ⋯
⋯ ⋯
33,+2
nn n
h N n+⋯
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
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1 1 11 1
1 2 31 21,1 1,
2 2 22,1 2,
1 2 3
3,1 3,
3 3 31 2
1 2 3
0
0
n
n
n n n
x x x N N
s s ss s x x
x x x J x x
s s s
x x N N x x xs s
s s s
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
∂ ∂
∂ ∂
= = + ∂ ∂ ∂ ∂
⋯
⋯ ⋮ ⋮ ⋮
⋯
1 1 1 1 1 N h N h hs s N
∂ ∂
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
31,1 31,
32,1 32,
33,1 33,
n
n
n
n n
n n
n n
⋯
⋯
⋯
1 2
3 3
1 2
2 2 2
2 2 2
n n n n nn
s s
N h N h hs s N
s s
∂ ∂
∂ ∂ ∂ ∂
⋮ ⋮ ⋮
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1 1 1
1 2 3
2 2 21 2
1 2 3
3 3 3
1 2 3
x x xs s s
x x x J J J
s s s
x x x
s s s
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= =
1 1 x ∂ ∂ 1 x ∂
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1 2
2 21
1 2
3 3
1 2
s s
x J
s s
x
s s
∂ ∂=
∂ ∂ ∂ ∂
∂ ∂
3
22
3
3
3
s
J s
x
s
∂
∂=
∂ ∂
∂
31 3
T T V V n
N N J X s L H
s s
∂ ∂= +
∂ ∂
32
T
n V J L H N =
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, 3 3 ,
1 2
ni k k k
ij i k i k
k j j j
x N N h J x s n
s s s=
∂ ∂ ∂= = + ∂ ∂ ∂
∑ for 1 i 3 1 j 2∧
3 3 ,13 2
ni k
i k i k k
x h
J N ns =
∂= =
∂ ∑ for 1 i 3
≤ ≤ ≤ ≤
≤ ≤
31 1 1 1 2 1 = xu u x u x u ∂∂ ∂ ∂ ∂ ∂ ∂
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1 1 1 2 1 3 1
31 1 1 1 2 1
2 1 2 2 2 3 2
31 1 1 1 2 1
3 1 3 2 3 3 3
= + +
= + +
s x s x s x s
xu u x u x u
s x s x s x s
u u x u x u
s x s x s x s
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
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32 2 1 2 2 2
1 1 1 2 1 3 1
32 2 1 2 2 2
2 1 2 2 2 3 2
32 2 1 2 2 2
3 1 3 2 3 3 3
= + +
= + +
= + +
u u x u x u
s x s x s x s
xu u x u x u
s x s x s x s
xu u x u x u
s x s x s x s
∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
3 3 3 3 31 2
1 1 1 2 1 3 1
3 3 3 3 31 2
2 1 2 2 2 3 2
3 3 3 3 31 2
3 1 3 2 3 3 3
= + +
= + +
= + +
u u u u x x x
s x s x s x s
u u u u x x x
s x s x s x s
u u u u x x x
s x s x s x s
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂
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1 1 1 1 1 1 1 1 1
1 2 3 1 2 3 1 2 3
2 2 2 2 2 2 2 2 2
1 2 3 1 2 3 1 2 3
3 3 3 3 3 3 3 3 3
1 2 3 1 2 3 1 2 3
u u u u u u x x x
s s s x x x s s s
u u u u u u x x x
s s s x x x s s s
u u u u u u x x x
s s s x x x s s s
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
= ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
u u J s x
∂ ∂=∂ ∂
1g s
J ε ε
−=
1 T
gt gt t sT J T ε ε − =
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1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 2 3
s
u M
u u u
s s s
u u u
s s s
u u u
s s s
D U
ε
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
,1 ,1 ,1
1 2 3
,2 ,2 ,2
1 2 3
,3 ,3 ,3
1 2 3
u u u
u u u
u
u u u
N N N s s s
N N N D
s s s
N N N s s s
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
0 0
0 0
0 0
M
U
U U
U
=
1 T
gt gt t
u M T D U J T ε − =
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1,1 1 1 1 1 1 13 11,1 3 12,1 3 13,1
11 1 1 1
3 11, 3 12, 3 13,
1 1 1 1
,1 1 1 13
2 2 2
N = 0 0
2 2 s 2
0 02 2 2
= 0 02
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
⋯
⋯
u
n n n n n n nn n n
u
N N N h N h hs P s P s P
s s s s
N N h N h N hs P s P s P
s s s s
N N N hs
s s s
1 1 1 111,1 3 12,1 3 13,1
2 2
2 2
∂ ∂
∂ ∂⋯
N h N hP s P s P
s s
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
3 11, 3 12, 3 13,
2 2 2 2
,1 1 1 11 11,1 1 12,1 1 1
3
0 0
2 2 2
= 0 0 02 2 2
∂ ∂ ∂ ∂
∂
∂
⋯ n n n n n n nn n n
u
s P s P s P
s s s s
N h h h N P N P N P
s3,1
11, 12, 13,
0 0 0 2 2 2
⋯
⋯
n n n
n n n n n n
h h h
N P N P N P
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31
,2 1 1 1 1 1 1 13 21,1 3 22,1 3 23,1
1 1 1 1 1
3 21, 3 22, 3 23,
1 1 1 1
,2 1 1 13 21,1
2 2 2
= 0 02 2 2
0 02 2 2
= 0 02
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
⋯
⋯
u
n n n n n n nn n n
u
N N N h N h N hs P s P s P
s s s s s
N N h N h N hs P s P s P
s s s s
N N N hs P
s s s
1 1 1 13 22,1 3 23,1
2 2
2 2
∂ ∂
∂ ∂
∂ ∂ ∂ ∂
⋯ N h N h
s P s Ps s
N N h N h N h
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
3 21, 3 22, 3 23,
2 2 2 2
,2 1 1 11 21,1 1 22,1 1 23,1
3
2 2 2
= 0 0 02 2 2
∂ ∂ ∂ ∂
∂
∂
⋯
⋯
n n n
u
s s s
s s s s N h h h
N P N P N Ps
21, 22, 23, 0 0 0 2 2 2
⋯
n n n
n n n n n n
h h h
N P N P N P
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,3 1 1 1 1 1 1 13 31,1 3 32,1 3 33,1
1 1 1 1 1
3 31, 3 32, 3 33,
1 1 1 1
,3 1 1 13 31,
2 2 2
= 0 02 2 2
0 02 2 2
= 0 02
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
⋯
⋯
u
n n n n n n nn n n
u
N N N h N h N hs P s P s P
s s s s s
N N h N h N hs P s P s P
s s s s
N N N hs P
s s s
1 1 1 11 3 32,1 3 33,1
2 2
2 2
∂ ∂
∂ ∂⋯
N h N hs P s P
s s
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
3 31, 3 32, 3 33,
2 2 2 2
,3 1 1 11 31,1 1 32,1 1 33,1
3
0 0
2 2 2
= 0 0 02 2 2
∂ ∂ ∂ ∂
∂
∂
⋯ n n n n n n nn n n
u
s P s P s P
s s s s
N h h h N P N P N P
s
31, 32, 33, 0 0 0 2 2 2
⋯
⋯
n n n
n n n n n n
h h h
N P N P N P
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33
,
, 1 3 1, 3 2, 3 3,
1 1 1 1 1
= = 0 02 2 2
u i k k k k k k k u i i k i k i k
N N N h N h N h D s P s P s P
s s s s s
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ⋯ ⋯
,
, 2 3 1, 3 2, 3 3,= = 0 02 2 2
u i k k k k k k k u i i k i k i k
N N N h N h N h D s P s P s P
s s s s s
∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂⋯ ⋯
for 1 i 3 j =1∧≤ ≤
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
,, 3 1, 2, 3,
3
= = 0 0 02 2 2
u i k k k u i k i k k i k k i k N h h h D N P N P N P
s
∂
∂
⋯ ⋯
for 1 i 3 j =2∧≤ ≤
for 1 i 3 j =3∧≤ ≤
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34
1
2
12
23
13
t
t
mb
t t
s
B
U B U
B
ε
ε
ε γ
γ γ
= = − − =
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
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35
8 – Stress components
{ }1 2 12 23 13, , , ,T t t t t t t σ σ σ τ τ τ =
{ }1 2 12, ,T
t t t t mbσ σ σ τ =
{ 23 13,T t t t
sσ τ τ =
t I
3
Stresses in the tangential coordinate system
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
{ },T
t t t
mb sσ σ σ =
π
t I
1
I
t I
2σ t I 1
τ t t
I
1 2
τ t t
I
1 2
Surface of the integration
point in analysis
Generic integration point
Middle surface
τ τ t t
I
t t
I
2 3 3 2=
τ τ t t
I
t t
I
1 3 3 1=
τ t t
I
2 3τ t t
I
1 3
σ t
I
2
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36
T gt gt g t T T σ σ =
1 12 13
12 2 23
13 230
t t t
t t t t
t t
σ τ τ
σ τ σ τ
τ τ
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
1 12 13
12 2 23
13 23 3
g g g
g g g g
g g g
σ τ τ
σ τ σ τ
τ τ σ
=
{ }1 2 3 12 23 13, , , , ,T g g g g g g gσ σ σ σ τ τ τ =
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37
9 – Constitutive equations for linear elastic behaviourt t
Dσ ε =
0
0
t t
mbmb mbt t
ss s
D
D
σ ε
σ ε
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
2
1 0
1 01
10 0
2
mb E D
υ
υ υ
υ
= − −
1 00 1
s D F G =
0
0u d
t t t t mb mb T T mb
t t ss s
D
D
ε ε ε σ
σ ε
∆ ∆ − −
=
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38
10 – Stiffness matrix for linear elastic materials
int
T t t
V
W dV δ δ ε σ = ∫
int
T T t t
V
W U B D B dV U δ δ = ∫
T t t
K B D B dV =
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
V
,, , ,
, , ,,
E f E ff E fp E f
E pf E pp E p E E p
QU K K
U K K Q R
= +
, ,, ,,
,
E ff E fp E f E p E f
E f
K U Q K U
Q
= −
=
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39
11 – Load vector
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho – Department of Civil Engineering
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40
12 – Resultant stresses
{ } { }1 2 1 2 1 2 1 2
2
3
2
2
, , , ,
h
T T t m t t t t t t t t
h
h
N N N dt σ σ σ σ −
= = ∫ { } { }2 1 1 2 1 2 1 2
2
3 3
2
2
, , , ,
h
T T t b t t t t t t t t
h
h
M M M t dt σ σ σ σ −
= = ∫
Membrane forces:Bending moments:
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques University of Minho – Department of Civil Engineering
3
2
t
mb
h
dt σ
−
=
3 3
2
t
mb
h
t dt σ
−
=
{ } { }2 3 1 3 2 3 1 3
2
3
2
2
3
2
, ,
h
T T t
s t t t t t t t t
h
h
T t
s
h
V V dt
dt
σ τ τ
σ
−
−
= =
=
∫
∫
Shear forces:
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41
{ { }1 2 1 2 2 1 1 2 2 3 1 3, , , , , , , , ,
T T t t t t
m b s t t t t t t t t t t t t N N N M M M V V σ σ σ σ = =
V t t
I
2 3
M t t
I
1 2
M t
I
2
N t t
I
1 2
N t
I
2t
I
3
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques University of Minho – Department of Civil Engineering
I
Middle surface of
the shell element
Generic integration point
M t t I
1 2 M t
I 1
V t t
I
1 3 N t t
I
1 2
N t
I
1
t
I
2
t I
1
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{ }
( )
( )
1 2 1 2
2
3
2
1
31
1
3
1
, ,
2
1
2
u d
u d
h
T T t t
m mbt t t t
h
t t t
mb mb T T
t t t
mb mb T T
N N N dt
h
D ds
D h ds
σ σ
ε ε ε
ε ε ε
−
+
∆ ∆−
+
∆ ∆
−
= =
= − −
= − −
∫
∫
∫
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques University of Minho – Department of Civil Engineering
{ }
( )
1 2 1 2
2
3
2
1
3
1
1
3
1
, ,
2
22
u d
u
h
T T t t
m mbt t t t
h
t t t mb mb T T
t t
mb mb T
N N N dt
h D B U ds
h D B U ds
σ σ
ε ε
ε
−
+
∆ ∆
−
+
∆
−
= =
= − −
= −
∫
∫
∫
1
3
1
0d
t
T dsε
+
∆
−
=∫
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{ } ( )( )
( )( )
1 2 1 2
3
1 1
1
3
1 1
1
, , , ,2
, ,2
u
u
nGPT t t t
mb mb mbm T t t t t k k
k
nGPt t
mb mb T k k
k
h N N N D B s s P U W h D
h D B s s P U W N
σ ε ∆
=
∆
=
= = −
= −
∑
∑
u u
u
t t
mbT T
t
T
N h D
h
ε
σ
∆ ∆
∆
=
=
Joaquim BarrosModeling and Advanced Structural Analysis Techniques University of Minho – Department of Civil Engineering
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44
{ }
( )( )
2 1 1 2
2
3 3
2
1
3 31
12
3 3
1
, ,
2 2
4
u d
u d
h
T T t t
b mbt t t t
h
t t t
mb mb T T
t t t
mb mb T T
M M M t dt
h h D s ds
h D s ds
σ σ
ε ε ε
ε ε ε
−
+
∆ ∆
−
+
∆ ∆
−
= =
= − −
= − −
∫
∫∫
Joaquim BarrosModeling and Advanced Structural Analysis Techniques University of Minho – Department of Civil Engineering
( )
( )( )
12
3 3
1
2 3
1 1 3
1
2
1
4
2, ,
4 3
,4
u d
d
t t t
mb mb T T
nGPt t
mb mb T k k
k
t mb mb
h D B U s ds
h D B s s P U s W
h D B s
ε ε
ε
+
∆ ∆
−
∆
=
= − −
= −
=
∫
∑
( )( )
( )( )
23
1 3
1
2 3
1 1 3
1
,6
, ,4
d
d
nGP
t mb T k k
k
nGPt t
mb mb T k k
k
hs P U s W D
h D B s s P U s W M
ε ∆
=
∆
=
−
= −
∑
∑
1
3 31 0u
t
T s dsε
+
∆−
=
∫1 1
2
3 3 3 3
1 1
2
3d d d
t t t
T T T s ds s dsε ε ε + +
∆ ∆ ∆
− −
= =∫ ∫
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{ }2 3 1 3
2
3
2
1
3
1
1
3
1
,
2
2
h
T T t t
s st t t t
h
t
s s
t
s s
V V dt
h
D ds
h D B U ds
σ σ
ε
−
+
−
+
−
= =
=
=
∫
∫
∫
Joaquim BarrosModeling and Advanced Structural Analysis Techniques University of Minho – Department of Civil Engineering
( )( )3
1 1
1
, ,2
nGP
t s s k k
k
h D B s s P U W
=
= ∑
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