The Finite Element Method for the Analyyysis of Linear Systemsarchiv.ibk.ethz.ch/emeritus/fa/education/ss_FE/FE_I_FS09/Lecture_10.pdf · xyz xy zrtesian coordinates of -th nodek:

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


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



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=∑


,x uat Gauss integration point at Gauss integration point2 2n k n

k∑


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



G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


• 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


vector at node )k


G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


• 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


0 0 02 1k k k

n= ×V V V


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= =∑ ∑



G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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

= =∑ ∑



G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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



G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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

− −

− − −

∂ ∂ ∂= +

∂ ∂ ∂∂ ∂⎛ ⎞+ +⎜ ⎟


⎣ ⎦ 1 1 111 12 13

k k kx kG t J J J h

r s⎛ ⎞= + +⎜ ⎟∂ ∂⎝ ⎠


G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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


yy y y⎣ ⎦


G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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

−⎜ ⎢ ⎥⎜ ⎢ ⎥⎜ −⎢ ⎥⎜ ⎢ ⎥⎣ ⎦⎝ ⎠

⎟⎟⎟⎟


2⎢ ⎥⎣ ⎦⎝ ⎠


G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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


3 cos( , ),y

x tl = e e 3 3 cos( , ), cos( , )y t z tm n= =e e e e


G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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



G l Sh ll El t

s

t

kw0 k

nV

Element midsurface


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


and the plane stress elements we do not integrate in the t-direction.


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


x 10


G l Sh ll El t

y

z

1.20.8 2

10

t 45o23

0 2nV0 3

nV


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


1 1 1 1

0⎢ ⎥⎢ ⎥⎣ ⎦


G l Sh ll El t

y

z

1.20.8 2

10

t 45o23

0 2nV0 3

nV


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


,2 4 8 8

1 1 0.45 0.9141/ 2n n⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦



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


[ ]r s t ydr ds dt

g g g



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) .




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

îjn

ij DIij k ij k

khε

=

=∑ B u

Better general performance is achieved from the MITC9 and MITC16 elements by Bucalem and Bathe.




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

îj

ij DIij k ij k

khε

=

=∑ B u

1 1s

interpolation functionsk

3 3

r

35

r

35




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

îj DIij k ij k

khε

=

=∑ B u

1 1s

interpolation functionsk

3 3

r

13r13



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



B d C ditiBoundary Conditions

Let us consider an example:

,z w

,y v,x u L

h Rigid support

L

1hL

rigid



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


soft – only whard - also θ

Displacement in all directions


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∂


Displacement in all directions

w and the derivatives i.e. dx


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 !


Documents

The Finite Element Method for the Analyyysis of Linear Systemsarchiv.ibk.ethz.ch/emeritus/fa/education/ss_FE/FE_I_FS09/Lecture_10.pdf · xyz xy zrtesian coordinates of -th nodek: