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REISSNER-MINDLIN SHELL FE FORMULATION
U n i v e r s i t y o f M i n h o
Department of Civil EngineeringGuimares, Portugal
www.civil.uminho.pt
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2
2 - Displacement field
z
w
uyzk
( ) ( ) ( )0,,0,,,, =+== zyxzzyxuzyxu y
( ) ( ) ( )0,,0,,,, === zyxzzyxvzyxv x
( ) ( )0,,,, == zyxwzyxw
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
zki
zkm
zk
zk
x
x
yy
u
v
k
ksuperfcie mdia da camada genrica
superfcie mdia do elemento
v zk x( )
x
ycamada n
camada k
camada 1
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3
3 - Strain field
++++
+
+
+
+
+
+
=
= wwv
x
vu
x
u
x
vu
y
w
y
v
y
u
y
v
x
w
x
v
x
u
x
u
xy
y
x
222
222
2
1
2
1
+
= xy
x
y
y
x
z
y
vxu
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
++++
++++
z
w
x
w
z
v
x
v
z
u
x
u
x
w
z
u
zw
yw
zv
yv
zu
yu
yw
zvxz
+
+
y
x
x
w
y
w
y
xy
=
+
=
c
mf
c
fm z
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4
T
mx
v
y
u
y
v
x
u
+=
,,
T
xyxy
fxyyx
=
,,
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
T
yxcx
w
y
w
+=
,
{ }fmmf z +=
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5
+
x u3 3,
x2+
x u1 1,
x u2 2,
x u3 3,
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
u
x
3
1
x2
u
x
3
1
u3
x u2 2,
x u1 1,
Normal superfcie mdia
aps a deformao
Deformao admitida
Deformao real
u
x x x3
12 2
+ ++ =
x xu
x2 23
1
+
+
= +
x x
u
x2 23
1
= +
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6
x1
x u3 3,
x u1 1,
x u2 2,
x u3 3,
x xu
x1 1
3
2
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
x1+
u
x
3
2
u
x
3
2
x u1 1,
x u2 2,
u3
Normal superfcie mdia aps a
deformao
Deformao admitida
Deformao real
u
x x x
3
21 1
+ + =
x xu
x1 13
2
+
= +
x x
u
x1 13
2
+
+
+=
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7
4 B matrices
( )en
i
i UB=
=
1
( ) ( ) ( )een
i ci
mfie
i
n
i ci
fimi
c
mfUBU
B
BU
B
BzB=
=
+=
== 11
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
=
x
N
y
Ny
Nx
N
B
ii
i
i
mi
0
0
=
y
N
x
Ny
Nx
N
B
ii
i
i
fi
0
00
00
=
i
i
i
i
ci
Nx
N
Ny
N
B
0
0
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8
{ }Tyixiiiii wvuU ,,,,=
{ }T
iimi vuU ,=
T
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
fciyixii
cifi
UwUU === ,,
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6 Stiffness matrix
( )= eVT
b dV int,
( )
) .2int,
dVUBDBUUBDzBU
UBDzBUUBDzBUUBDBU
cc
b
c
T
c
T
cff
b
mf
T
f
T
f
V mm
b
mf
T
f
T
fff
b
mf
T
m
T
mmm
b
mf
T
m
T
mb e
+
+++=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
( )( ) dV
BDB
BDBzBDBz
BDBzBDB
K eV
c
b
c
T
c
f
b
mf
T
f
m
b
mf
T
f
f
b
mf
T
mm
b
mf
T
m
e
b
+
=
2
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11
( )( ) ( )
( ) ( ) ( )
+=
e
c
e
f
e
fm
e
mf
e
me
bKKK
KKK
( )
( )= eV mb
mf
T
m
e
m dVBDBK
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
( ) ( )= eV fbmfTmemf dVBDBzK
( ) ( )
( )
==e
V m
b
mf
T
f
Te
mf
e
fm dVBDBzKK
( )
( )= eV fb
mf
T
f
e
f dVBDBzK2
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12
( )( )= eA m
b
m
T
m
e
m dABDBK
=2/
2/
h
h
b
mf
b
m dzDD
( )
( )= eA fb
mf
T
m
e
mf dABDBK
( )( )= eA m
b
fm
T
f
e
mf dABDBK
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
( )=
=
nc
k
i
k
s
k
b
kmf zzD1
,
==2/
2/
h
h
b
mf
b
fm
b
mf dzzDDD
( )==
nc
k
i
k
s
k
m
k
b
kmf zzzD
1,
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13
( )( )= eA f
b
f
T
f
e
f dABDBK
=2/
2/
2 hh
b
mf
b
f dzzDD
( )( )= eA c
b
c
T
c
e
c dABDBK
=2/
2/
h
h
b
c
b
c dzDD
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
( (= = ki
k
s
k
m
k
b
kmf zzzD1
2
,
( ) ( )[ ] 3/ 133
,==
nc
k
ik
sk
bkmf
b
f zzDD
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( ) [ ]( ) qpN
p
N
q
m
b
m
T
m
e
m WWJBDBKm m
qpdet
1 1
, = =
=
( )
[ ]( ) qpN
p
N
q
fbmf
Tm
emf WWJBDBK
m m
qp
det1 1
, = =
=
N Nm m
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
( ) qpp q
mfm
T
f
e
fm WWJBDBK qp det1 1
,
= ==
( ) [ ]( ) qp
N
p
N
q
f
b
f
T
f
e
f WWJBDBKm m
qp
det
1 1
, = =
=
( ) [ ]( ) qpN
p
N
q
c
b
c
T
c
e
c WWJBDBKc c
qpdet
1 1
, = =
=
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15
Element
Integration scheme Shape function Linear Quadratic Cubic
Formulation Lagrange Serendipity Lagrange
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Shear 2 X 2 3 X 3 3 X 3
Reduced Membrane and bending 2 X 2 2 X 2 2 X 2
Shear 1 X 1 2 X 2 2 X 2Selective Membrane and bending 2 X 2 3 X 3 3 X 3
Shear 1 X 1 2 X 2 2 X 2
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16
7 Internal forces
( )
( )
=
eV
Te
bdVBf
int,
( )dAdBB
mfh TTe
=2/
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
c
A he
2/int
( ) ( )( ) ++= e eeA AT
cA
T
f
T
m dAQBdAMBdANB
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{ }
( )
=
=
==
nc
k
i
k
s
kkmf
h
h mf
T
xyyx
zz
dzNNNN
1
,
2/
2/,,
==hT 2/
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
( )=
=
nc
k
i
k
s
k
m
kkmf
h mxyzyzx
zzz1
,
2/
{ }
c
h
h c
T
xzyz
h
dzQQQ
=
== 2/
2/,
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18
( ) ( ) ( ) ( )32
33
1,
22
,
i
k
s
knc
kf
b
kmf
i
k
s
k
m
b
kmf
zz
D
zz
DM
+
=
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
is
nr
i
is hN ,1
, =
=
isis
nr
i
is zhM ,,1
,
=
=
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z z
y y
y
xyyxx
y
yz
yz
xz
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
xx
x yx distr ibuio real
distribuio aproximadaxz
(a) (b)
xy
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20
z z
y y
x
NxNy
Nyx
Nyx NyNxy
Nxy
M
Mxy
Myx
MyxMzy
Mzy
Mzx
Mzx
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
x xy
(c) (d)z
y
x
Qyz
Qxz
Qyz
Qxz
(e)
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Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
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Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
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Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
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Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
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