4.MASAT 2012-2013 MindlinShell

8/11/2019 4.MASAT 2012-2013 MindlinShell

1/24

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

[email protected]

www.civil.uminho.pt


2/24

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


3/24

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



++++

++++

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


4/24

4

T

mx

v

y

u

y

v

x

u

+=

,,

T

xyxy

fxyyx

=

,,

Joaquim Barros



T

yxcx

w

y

w

+=

,

{ }fmmf z +=


5/24

5

+

x u3 3,

x2+

x u1 1,

x u2 2,

x u3 3,

Joaquim Barros



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

= +


6/24

6

x1

x u3 3,

x u1 1,

x u2 2,

x u3 3,

x xu

x1 1

3

2

=

Joaquim Barros



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

+

+

+=


7/24

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



=

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


8/24

8

{ }Tyixiiiii wvuU ,,,,=

{ }T

iimi vuU ,=

T

Joaquim Barros



fciyixii

cifi

UwUU === ,,


9/24


10/24

10

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



( )( ) 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


11/24

11

( )( ) ( )

( ) ( ) ( )

+=

e

c

e

f

e

fm

e

mf

e

me

bKKK

KKK

( )

( )= eV mb

mf

T

m

e

m dVBDBK

Joaquim Barros



( ) ( )= eV fbmfTmemf dVBDBzK

( ) ( )

( )

==e

V m

b

mf

T

f

Te

mf

e

fm dVBDBzKK

( )

( )= eV fb

mf

T

f

e

f dVBDBzK2


12/24

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



( )=

=

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,


13/24

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



( (= = ki

k

s

k

m

k

b

kmf zzzD1

2

,

( ) ( )[ ] 3/ 133

,==

nc

k

ik

sk

bkmf

b

f zzDD


14/24

14

( ) [ ]( ) 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



( ) 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

, = =

=


15/24

15

Element

Integration scheme Shape function Linear Quadratic Cubic

Formulation Lagrange Serendipity Lagrange

Joaquim Barros



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


16/24

16

7 Internal forces

( )

( )

=

eV

Te

bdVBf

int,

( )dAdBB

mfh TTe

=2/

Joaquim Barros



c

A he

2/int

( ) ( )( ) ++= e eeA AT

cA

T

f

T

m dAQBdAMBdANB


17/24

17

{ }

( )

=

=

==

nc

k

i

k

s

kkmf

h

h mf

T

xyyx

zz

dzNNNN

1

,

2/

2/,,

==hT 2/

Joaquim Barros



( )=

=

nc

k

i

k

s

k

m

kkmf

h mxyzyzx

zzz1

,

2/

{ }

c

h

h c

T

xzyz

h

dzQQQ

=

== 2/

2/,


18/24

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



is

nr

i

is hN ,1

, =

=

isis

nr

i

is zhM ,,1

,

=

=


19/24

19

z z

y y

y

xyyxx

y

yz

yz

xz

Joaquim Barros



xx

x yx distr ibuio real

distribuio aproximadaxz

(a) (b)

xy


20/24

20

z z

y y

x

NxNy

Nyx

Nyx NyNxy

Nxy

M

Mxy

Myx

MyxMzy

Mzy

Mzx

Mzx

Joaquim Barros



x xy

(c) (d)z

y

x

Qyz

Qxz

Qyz

Qxz

(e)


21/24

21

Joaquim Barros




22/24

22

Joaquim Barros




23/24

23

Joaquim Barros




24/24

24

Joaquim Barros



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4.MASAT 2012-2013 MindlinShell