degenerated isoparametric finite elements in

Published as: I. Kreja & Z. Cywiński Degenerated Isoparametric Finite Elements in Nonlinear Analysis of 2D-Problems Computers & Structures, Vol. 41, No. 5, 1991, 1029 - 1040.

DEGENERATED ISOPARAMETRIC FINITE ELEMENTS IN NONLINEAR ANALYSIS OF 2D-PROBLEMS *

IRENEUSZ KREJA

and

ZBIGNIEW CYWINSKI

Faculty o'f Civil Engineering, Technical University o'f Gdansk ul_

Majakowskiego 11/12. 80-952 Gdansk. Poland

Abstract -

Geometrically nonlinear

elastic

astic

oblems of' structur-al mechanics, based on

o'f of'

the

application

ated i soparametri c 'fi ni te el ements. is bei ng

The i nvesti gati ons concentrate equilibrium equations

using

on the

Large rotation increments and a loading

are

taken

into

the

del" i vati on

Total

account.

rule and the von Mises

dependent

developing.

in

Prandtl-Reuss

astic yield criterion

Riks-Wempner constant arc length ion.

are

consequence.

are pr'esented and compar'ed with

with

numerical

analytical

matrix.

re'ference

The

Newton

incremental sample

For 'flow

adopted.

method.

Sever al

pressure

associated

iterations. are employed - to solve the linear equations ln

nonl i near

'formulation.

additional components o'f the corresponding sti'f'fness the elastic-plastic behaviour the

per'for-med.

of'

an

displacement

analysis

non­

problems solutions.

The pr'oo'f on a good accur'acy of' the presented method is given,

*

This study represents. besides [lJ. another' compact

the PhD

Engng

dissertation

[2J

o'f

the

'first

portion

author.

par'tly

updated. seconded in 1989 at the Technical University o'f Gdatlsk.

t

o'f

INTRODUCTION

Since fini~e

of

work of Ahmad et al.

~he

elemen~

in~egra~ion

of

~his

In

isoparame~ric

~o

has been

~his

repor~

of

essen~ials

emphasis

~he

pu~

on

~he:

deriva~ion

of

~he

of

effec~s

au~hor's

nonlinear

Lagrangian CTL)

To~al

~o

large

i

of qualified sample problems and

problems where

~o

wi~h

~his

~he

plane

assump~ion ~he

arches),

or

applied:

plas~ic

~he

Kreja [2].

Brief

plane

presen~ed

accoun~

~he

and

of

a

~o

examine

arc

equa~ions,

on some

and of

~he

~he

~o

of

rule, me~hod

en~er

those

2

be

direc~ion

of

axial frames,

~he

hold

par~iculars ~he

in

numerical

~he

von

of

Mises

bodies,

~he

solving

~he

algori~hms

original work of

problems

Specifics of

associa~ed

by Kreja and Cywinski in [1].

can

(beams,

elas~ic-plas~ic

leng~h

~hose

cylinders).

observed

flow

and

i. e.

s~a~es

st~resses

~he

associa~ed

concerning

commen~s

comparison

~heir

occurring in one plane

(slender

reader is r-ecommended

~echnique

~he

perpendicular

from

announced by Kreja also in (5] and [6]. in~egra~ion

using

cons~ruc~ion

~he

resul~ing

cons~an~

nonlinear

developed.

along

s~rains

In order

anal

~he

revolu~ion),

cri~erion

Riks-Wempner-Ramm

par~icular

2D-problems,

~o

displacemen~s

~ha~

Prand~l-Reuss

yield

incremen~al

~he

rela~ions

plane

correspondingly.

limi~ed

behaviour of

(shells of

symme~ry

~he

paper is

~he

wi~h

cer~ain

solu~ions.

described by

sufficien~ly

only,

reference

~his

na~ure

pressure loading, and

presen~a~ion

The scope of

briefly

ncremen~s

dependen~

exis~ing

reduced

S~egmi.iller.

in~o

displacemen~

wi~h

~he

equa~ions

~aking

ro~a~ion

concep~

s~a~e-of-~he-ar~

[2].

equilibrium and

~he

The origin and

presen~

~hesis

formula~ion

incorpor-a~ed

by

differen~

a chance

~ake

firs~

s~ruc~ures,

suppor~ed

in

based

displacemen~

review [4) of Ramm and

~he

au~hors

~he

popular.

~ruly

pic~ured

only

men~ion

elemen~s,

became

~echnique,

concep~

records,

developed for shell

me~hod

degenera~ed

wi~hin

[3),

have ~he

been

reduced

locking phenomena were

INCREMENTAL EQUATIONS OF MOTION

Total Lagrangian f'ormulation

Consider the body motion referred to the global cartesian assuming the convenience,

existence

of

static

effects

only.

the time variable t is being used

system.

For

sake

that

of

describes

the situation of the body and the action of its loading (see Bathe [7J and Kleiber

[8J).

The equilibrium positions

certain time instants O.

of

the

body

are approached.

where

the time increment.

The solution is being attained by

help

recurrency process;

for this purpose.

there are

known

~t.

solutions

interval 0 and t,

2~t.

for

inclusively.

body for the instant

time

left

by repeating

0.1.

and

2.

of

within

that

the body's

Particular configurations

superscripts

instants

If specifying the position

the total range of the time variation can be obtained.

is

~t

a

the assumption is taken that

all

then -

t+~t,

at

Thus,

are the

the

of

the

process

for

complete

path

characterized coordinates

by

of

an

arbitrary point P within the initial configuration (instant 0) are Ox • Ox • Ox ; similar coordinates for the current (instant 1

2

the incremented (instant and

2

x.

t)

3

2

x.

2

x.

123

Assuming that

configurations are

t+~t)

1

X. :I.

:I.

X. 2

and :I.

X. 3

respectively. the

approached

current

represents the state of equilibrium.

the

configuration virtual

wor'k

at

t+~t

principle

can be formulated in the following manner:

where

2 15W

i.

is the internal.

(1)

eo

L

and

2 15W

-

eo

the external

virtual

work;

the former can be expressed by equation 2 T.

LJ

with

2

T .. '-.J

being the

cartesian

15 e 2

components 3

(2)

i.j

of

the

Cauchy

stress

and

2

t.he

e"1.)

component.s

of

i nf- i n1 t.esi mal

t.he

st.r-ai n

t.ensor.

Since t.he t+At configurat.ion is unknown. t.he equat.ion (1) can

not.

be solved st.raight. orl; wit.hin t.he Tot.al Lagr-arlgian formulat.ion t.he

approximat.e solut.ion can be found. t.o t.he init.ial configurat.ion.

by referring all t.he

As

t.he

st.ress

configurat.ion. relat.ed t.o that. for O. Piola-Kirchhoff.

2 .....

o

the

variables

quant.it.y

2nd

stress

for

t+At

tensor

can be applied; similar' consider'ations

.:::>. I.J

st,rains result in the Green-Lagrange strain tensor.

As

of for­

shown

by

Bat.he in [7J. t.he following relation holds: 6 e"

i.j

2

I.J

2 dV

2

2.

g. F.Frey,

S.Cesco~~o.

Lagrangian

skonczonych

eLern..entow

PWN.

Some new

descrip~ion

ELern..ent

n i eLi n i

w

in

Method

0?J)€'

j

NonL inear

Warszawa-Poznan 1985. aspec~s

o:f

~he

in nonlinear anal

incremen~al

s,

To~al

Con:f.

In~.

on

"Finite El.ern..ents in NonLinear SOLid and Structural. Mechanics",

Geilo. Norway,

Vol.l,

10. K.Schweizerho:f and loads in nonlinear 18,

323-343 (1977).

E.Ramm, :fini~e

Displacemen~

elemen~

dependen~

analyses,

pressure

Corn..put.

Struc t.

1099-1114 (1984).

11. G.A.Cohen, on a shell.

Conserva~iveness

AIAA Journal. 4,

12. M.J.Sewell. On Mech.

Anal..

5truct.

31,

ac~ing

1886 (1966).

Con:figura~ion-dependen~

loading,

Arch.

Rational.

23. 327-351 (1967).

13. K.-M.Hsiao s~ruc~ures

o:f a normal pressure :field

and by

Y.-R.Chen,

degenera~ed

Nonlinear

isoparame~ric

427-438 (1989). 21

analysis shell

o:f

elemen~.

shell Corn..put.

14. J.T.Holden, On

~he

Solids & Struct.

15.

J.H.Argyris analysis or na~ural

Appl.

rini~e

elas~ic

Int.

beams,

J.

sys~ems

-

I.

under

rini~e

loading

nonconserva~ive

Quasis~a~ic

elemen~

problems.

Com.p-u.t.

Large

derlec~ions

l1eth.

Engrte 26, 75-123 (1981).

l1ech.

F.S.Manuel,

s~abili~y

~hin

Nonlinear

S.Symeonidis.

rormula~ion

16. S.L.Lee,

or

1051-1055 (1972).

8,

and

derlec~ions

or

and

E.C.Rossow,

rrames.

elas~ic

l1ech.

Div.

and

ASCE

94.

521-533 (1968). 17. ~ T.Ishizaki

and

displacemen~

and

K.

elas~ic-plas~ic

Com.put.

s~ruc~ur·es,

-J.Ba~he,

Struc t.

18. T.J.R.Hughes and W.K.Liu, shells. l1ech.

Par~

Ene.

II.

27,

12,

On

dynamic

167-181 (1981).

elemen~

analysis

or

large shell

309-318 (1980).

Nonlinear

Two-dimensional

rini~e

rini~e

shells.

elemen~

Com.p.

analysis or l1eth.

Appl.

FIGURE LEGENDS

Fig.

1. 3-node

elemen~

in

configura~ions

sys~em

of

~he

car~esian

coordina~es

Fig.

2.

Example 1 -

Displacemen~s

- five 4-node Fig.

3.

Example 1 -

~en

-

~he

can~ilever

~ip

. Model

I

of

~he

can~ilever

~ip

, Model II

elemen~s

Displacemen~s

3-node

of

elemen~s

Fig.

4. Example 1

Fig.

5.

Example 2 -

Displacemen~s

Fig.

6.

Example 2 -

Def-orma~ion

s~a~es

of

the

can~ilever

under

Deforma~ion

s~a~es

of

~he

can~ilever

under

Deforma~ion

s~a~es

of

of

~he

~he

can~ilever

can~ilever

~ip

"dead" load Fig.

7.

Example 2 -

follower load Fig.

8. Example 3 - Displacement..s at

Fig.

9. Example 3 - Frame

Fig.10. Example 4 _. Mid Fig.11. Example 5

deforma~ion

deflec~ion

Comple~e

load

~he

of

spherical

applica~ion

poin~

s~a~es

~he

cylindrical shell

shell

and

i~s

numer'ical

model Fig.12.

Example '3 -

Fig.13.

Example 5 - Influence of

Deflec~ions

of

~he

sphere pole

imperfec~ions

for spherical shell

Fig. 1. Three-node element configurations in system of the Cartesian coordinates

Fig. 2. Example 1 -Displacements of the cantilever tip;. Model I -five 4-node elements

Fig. 3. Example 1 Displacements of ~he cantilever tip;, Model II -ten 3-node elements

Fig. 4. Example 1 -Deformations states of the cantilever

Fig. 5. Example 2 -Displacements of the cantilever tip

Fig. 6. Example 2 -Deformation states of the cantilever under "dead" load

Fig. 7. Example 2 -Deformation states of the cantilever under follower load

Fig. 8. Example 3 -Displacement..s at the load application point

Fig. 9. Example 3 -Frame deformation states

Fig.10. Example 4 _. Mid deflection of the cylindrical shell

Fig.11. Example 5 Complete spherical shell and its numerical model

Fig.12. Example '3 -Deflections of the sphere pole

Fig.13. Example 5 -Influence of imperfections for spherical shell



 

 

 

   

 

   

     



 

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