92 Barbero, 3-d Finite Element

8/18/2019 92 Barbero, 3-d Finite Element

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3/9

A -D finite element for laminated composites

STANDARD

PLATE

NEW 3-D ELEMENT

26S

- - - -

_ :I :.

- - -

I

I

I

I

i

I

•

I

I

____

A .... _

u.c

a c

8 Bridge deck-and-beammonithic connecIcn

b Folded plate

•

N=2

30lC

I

I

N=4

. cifftcUty

(c) Tapered ~ nber beam

t--

n L

-

t

~

- i I

__

_ c _ :

1 ~ 3 0 ~ L C I

d PIy

drop.off and

la p joi1t

Fig. 1. Special

applications for

the new 3-D

element compared to the approximations introduced

by

standard sheD elements.

equivalent to FSDT and is the least expensive

and

least accurate of alltbe possible models of

I laminated composite that can be obtained with

this

element.

f

one element per layer

is

used, the

model is equivalent to LeST and in the most accurate

and expensive although less expensive than a model

constructed with conventional 3-D continuum

elements. Any number of elements through the thick

ness

can be used, each representing clusters

of

layers.

This

approach leads to significant savings while

rttaining accuracy an intelligent choice of clusters

made based on the stacking sequence and the

type of results desired e.g., maximum interlaminar

stresses .

Unlike plate elements based on LCST formulation,

the position

of

the middle surface

is

irrelevant. There

fore, variable number of layers and thicknesses can be

modeled e.g., ply drop oIrs, lap joints, etc . The

modeling process

is

very similar to the modeling

using conventional 3-D continuum m n t s ~ The

connectivity array is used to tie together aU the

elements through the thickness at a particular

location and to enforce the incompressibility of the

transverse nomals. The l w 3DLC element with this


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6/9

268

E.

J. BADERO

3. ASSE.\tBLY PROCEDURE AND CONDENSATION OF

REDt.:NDANT OOF

Fig. S.

Layer-wise

constant representation or

the

shear

strain through the thickness

of

an isotropic plate.

22

25

-

1

12

26.

17

19

1

14

6

5

2

4. NUMERICAL INTEGRATION AND

COMPl ATION OF STRESSES

Constitutive equations are used

to

obtain

all

six

components of stress at

the reduced

Gauss

points.

The distribution

of

in-phase stresses J J and

S1

is

linear through the

thickness. The

distribution

of

inter-laminar stresses J and J r: is layer-wise con

stant. Quadratic interJaminar stresses that satisfy the

shear boundary conditions

at

the top and bottom

surfaces of the she)) can be obtained by postprocess

ing

20, 21 t 29]. AU components of stress obtained

at

the

integration points are extrapolated to the

nodes

using the procedure described by Cook [30].

This

is

done to facilitate

the

graphic output using a

commercial pre- and postprocessing package.

Selective

reduced integration is used on the shear

related terms. The 3DLC element reduces to FSDT

• 0 u v nodes

• w nodes

Fig. 6. Element nodal numbering. Nodes 1-18 have

two

DOF

u,

v and nodes 19-27

one

M ).

During the assembly

process,

further condensation

is required since

the

w-displacements of all the

elements that form a laminate are identical at a

particular position on the surface of the shell due to

the incompressibility condition. For example, tbe

nodes on the bottom surface in Fig. 4

were

chosen

to

be

the

global w-nodes

for

aU

the

elements through

the

thickness. Condensation

can

be efficiently accom

plished by the usual

assembly

algorithms

by

assignina

the

same

global node number to

all

of the

w-nodes

located on a particular normal to the shell.

This

technique

has

the following ad

vantages:

a) It eliminates the need

for

complex bookkeeping

to identify individual sets of elements stacked

to form a laminate.

b)

It allows the front-\\idth or band-width

optim

ization algorithm to take into account

the

thickness direction as

well as

the surface

direc

tion in the search

fo r th e

optimum

element or

node assembly order.

c)

It eliminates the need

for

elements with large

number of DOF that otherwise result

if the

assembly

through

the

thickness

is

perfonned

II

priori

[18,

21, 23]. This is

particularly

useful for

the implementation in commercial FEA codes.

14)

30LC

30LC

[K J{cS} = tf l

or in contracted form

The integration of the element stiffness matrix is

performed

as

for

a standard 18-node element

with

three DOF per node

u,

w but with the appropriate

shape functions H; and

M;

described previously.

Since

is

constant through the thickness. the contribution

of .

to

the shear stiffness terms must

be

integrated

over nine nodes only top or bottom). For

th e same

reason, the transverse strain

vanishes.

Since

the

transverse deflection

w is

constant through the thick

ness of the element, the DOF of

two

nodes aligned

through the thickness can

be

reduced to a single

DOF. This is done at the element level

by

rearranging

the element stiffness

matrix

so that the DOF corre

sponding to the displacements u and v on the surface

of

the shell for

aU

nodes e.g.,

8

nodes) are

con-

sidered first. The remaining OOF corresponding

to

M- -dispJacements

are assigned to a

new

set of nodes

e.g. nine nodes) called w-nodes in this work. The

M

-nodes are independent of the original nodes

to

facilitate the assembly procedure. However.

they

can

be any

set

of

nodes located at one

of

the interfaces

of the laminate. The location

of

the w-nodes through

the thickness of the laminate does not

affect th e

results. At the element level, the resulting element

has

8

nodes with two DOF

u

and

t

per node

plus

nine

additional nodes

with

one DOF

w

per node Fig. 6).

f ..

Bt D [B do{lS}

=

f

N T{q}

do

+ f< O

NJT{ P} dA 13)

Fig.

S . As

the number of layers increases the error

reduces and no shear correction factor is needed.

If·more than one material layer are encountered

in

a single element, the numerical integration of the

element stiffness matrix is performed layer

by layer.

Analytical integration

of

the constitutive equations,

like in classical lamination theory, is not

used.

There

Core, material nonlinearities can be easily incorpor

ated in the analysis.

The element stiffness matrix is obtained by using

the kinematic equations 7) and the constitutive

equations 12) into the equilibrium equations

JO)


7/9

A 3-D finite element for

laminated

composites

269

1

,bd only one element is

used

through the thickness.

fberefore.

the behavior of the nine-node Lagrangian

fSDT

element with selective reduced integration,

ret of shear locking, is also present in the proposed

ddJ1ent

when

severa) 3DLC elements are stacked

brough

the thickness. Furthermore, Barbero and

Itcddy

[3 } successfully used selective reduced inte

oon

on

their LeST element that has the same

kinematics

as a stack of 3DLC elements. As shown

bY

A

verit

32],

the

nine-node Lagrangian element

with

seJeCtive reduced integration does not exhibit locking

as the plate or in this paper, layer) becomes very thin.

/1bis is

a

remarkable advantage

of

3DLC elements

v r

conventional 3-D continuum elements with full

,integration.

be 3DLCelement

gives

a very good representation

oraD

stress components except

1 :

without

the

aspect

ratio limitations

of

conventional 3· continuum el

ements. When transverse stress

1 : is

needed, either

conventional 3-D continuum elements can

be

used

they are

fully

compatible with 3DLC elements) or

further

postprocessing of the 3DLC results can be

done

by

using

the third equilibrium

equation

o ~ i :

~

+ O ~ c c

+P 0

q -\ ?t1;

t ~

5.

~ A u D A n o N

F h e ~ n e w l D L C

element was incorporated into

I standard finite element program. A few

simple

esamples

are presented to validate the program. First,

I cantilever beam is modeled

with

two elements along

t beamand one element through the thickness. The

resulting

model is similar to FSDT and the numerical

results obtained with 3DLC and FSDT are identical.

The

rotations in FSDT can be related to the in-plane

displacements

Fig.

6)

of

3DLC as follows:

where I is the thickness

of

the element and

i

=1, ,9. The effect

of

the E/G ratio is shown in

Fig. 7 where can be seen that a large value of G

..

1.4

Fig.

7.

Tip

deflection of

a cantilever isotropic beam under

tip

load.

simulates the classical

beam

theory

Can.

The beam

results are obtained

by

setting the Poisson ratio equal

to UfO

1De

geometry of the beam

is

such that

s = II =20 and . 1 = 10 where I is tbe length, h is

the thickness, and w

is

the width of tbe beam. For

a two-element mesh, the aspect ratio

of

the elements

is 14 and no locking is observed. The material

properties are E = 30 x lr f psi and v =0.25. The

deflection

at

the loaded end is nondimensionalized

with

the

eBT

solution.

Next, a quarter of an orthotropic rectangular plate

of side aand thickness

It

is modeled with a 2

x

2mesh

on

the

plane

of the

plate and two elements through

the thickness,

with

a total of eight elements Fig. 4).

The aspect ratio of an element

is

r =0.707 a/h that

for the case of thickness ratio

alh

=

80

gives r =

57

and no shear locking

is

observed.

The

plate

is

simply

supported and subjected to a concentrated load at

its

center Fig.

4).

The eight-element mesh bas a total

of

175 DOF before imposition of tbe boundary con

ditions. 1 DOF correspond to the

II

and v displace

ments at 25 locations x

y

and 3 locations through the

thickness.

The

remaining

25

DOF correspond to

the

transverse

deflection WI

at the nine locations x

y

on

the

surface

of

the

plate. The transverse deflection w

is constant through the thickness of the plate

at

eacb

particular location

y

as represented by a single

w-DOF at each node on the bottom surface Fig. 4).

The reSults are nondimensionalizedWith the first-term

Rayleigh-Ritz approximation [2]. For homogeneous

material not laminated) tbe 3DLC solution coincides

with

the FSDT results.

1De effect of the

degree

of orthotropicity

EllE

2

and

thickness

ratio

S

=

Q

lit

is

shown in Fig.

8

for constant

shear

moduli

G

I2

= and

G

n

=

£ /5.

Shear defor

mations are correctly accounted for as shown

in

Fig. 9

for

various

values

of

the thickness ratio

=

alh

and

a

1

2

=

30. The present formulation predicts larger

transverse

def1ections

than CPT due to the effect

of

shear defonnation that is neglected in CPT. The

difference is more pronounced for small values of the

thickness

ratio.

4.0... ...

3.5

3.0

u

~ 2 . 0

0.5

+............... .... ......................................

0

E,/E

I

Fil. 8. Effect of the

orthotropidty ratio

and thickness ratio

on the center deftection or aD orthotropic.

s i m p l e · s u p p o r t ~

square

plate

under center load.

HAdM


8/9

2

70

E

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W.

4

;


9/9

92 Barbero, 3-d Finite Element

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Transcript of 92 Barbero, 3-d Finite Element