Strategies for Tracing the Nonlinear Response Near Limit Points

8/9/2019 Strategies for Tracing the Nonlinear Response Near Limit Points

1/27

Strategies for

Tracing

the Nonlinear Response Near imit

Points

E RAMM

UOlversltat Stuttgart Germany

Abstract

For the prebucklmg range an extenslVe l l terature of effectIve

solutlOn techmq

ues

eXIsts for the numerIcal solutlOn of

st ructural

problems

but only a few algorIthms have been proposed to

t race

nonllnearresponsefrom the pre-l lmlt mto the post-l imIt range.

Among

these are the sImple method of suppressmg eqUlllbrlUm

IteratlOns, the mtroductlOn of

artIflClal

sprIngs, the dIsplace

ment control method

and

the constant-arc-Iength method

of

Rlks /Wempner.

t S the

purpose

of thIs paper

to

reVIew these

methods

and to dISCUSS the modIfIcatIons to a program that are

necessary for theIr ImplementatIon. Selected numerIcal

exam

ples

show that a modlfled Rlks /Wempner method

can

be espe

cIally recommended.

1. IntroductIon

Usually postcrIhcal

states

are

not tolerated

In the desIgn of

a

structure. However, the

predlctlOn of

response In

thIS

range

may stIll be of great value. A

tYPIcal example

S

the

Imperfechon

senSItIvIty of cer tam structures whlCh m general S dIrectly

related to the postcrIi lcal response.

In

partIcular this S true

for structures exhlbltmg a decreasIng post-lImIt characterIstic.

ThIS

may res ult a dynamIC

snap- through

or snap- back phenom

enon dependlllg on

whether

the load

or

the

dis

placement controls

the

system. However,

a

statIc analysIs t races the

wQole

post

crItIcal

range

allowmg

for

a

better

Judgement

of

the

overall

st ructural response.

t S

well

known that the us ually applIed N

ewton-Raphson

iteratlOn

methods are not

very

effICIent and often fall m the neighborhood

W. Wunderlich et al. (eds.), Nonlinear Finite Element Analysis in Structural Mechanics

Springer-Verlag Berlin Heidelberg 1981


2/27

64

of cri t ical points. The stiffness matrix

approaches smgularity

resul t

mg m

an

mcreasing number of iterations and smaller and smaller

load

steps. Fmally

the solution diverges.

In recent

years several

strategIes

have

been proposed

to

overcome

these

problems

and to

t race the

response beyond

the

critical point.

t

is the purpose of

this

paper to describe some of the most commonly

used

techmques. These are the

method

of suppressing the eqUlllbrlUm

IteratIOns in

the

neighborhood

of

the

critical point,

the method

of

art lficial

springs,

the displacement control techmque

and

the

con

stant -

arc

-

length method

of

Riks

[1] , [2J and Wempner [

3J.

In

partlcular an

attempt S

made to show the correlation of

the

latter

procedures. Special emphasis

S

given to some

modifIcations

of

the

Rlks /Wempner method

leadmg

to an efflclent i terative technique

throughout the

entire

range of loadmg and not only near the crIt ical

pomt. Other methods for solvmg the

same

type of

problem, e. g.

the

perturbation

method

or dynamIc relaxation, are not studied.

The dIscussIOn refers to lImIt

points

only.

BifurcatIOn problems

may

be mcluded either by introducing

a

small

perturbation m

geometry

or load

(Imperfect

approach) or

by

superImposing on the displacement

field

of the critlcal

load

a

part

of the eigenmode (perfect approach).

The

procedures are

descrIbed

in conJunction WIth

the Newton-Raphson

method

in

its

standard

or modifled

versions. A

combination

WIth

accelerated quasi Newton methods IS possible. Proportional

loading

S assumed but few

changes

are

necessary for

non

proportIOnal

load

ing.

2.

Start ing Pomt and Notation

The study

S

based on the incremental/I teratIve solutIOn

procedure

m a nonlinear

fmite

element

analysIs,

1. e.

the nonlmear

problem is

stepWIse

linearIzed

and the lInearizatIOn

r ro r

is corrected

by

addi

tIOnal eqUIlibrIum iteratIOn s, see for

instance [4J. A

left supers cript

mdlcates

the current confIguration

of the total dlsplacementsmu the


3/27

load vector

p ,

the mternal forces

mF

and the

out-of-

balance

forces

~ .

For proportional loadmg the loads

may

be expressed

by one load factor r:n. .

(1)

where P

IS

a

vector

of

reference

loads. WIthin one

increment from

confIguratIOn

m

to

m

+ 1, the posItIons

1

and

J = 1

+

1, before

and

after an arbItrary Iteratlon cycle, are dIstmgUlshed (figure

1).

load

1

1=1,2/3

U Ju displacement

FIgure

1: NotatIOn

The

total

Increments

between

positions m

and

1 are denoted by u(i),

P

1)

and

A

1)

whereas the

changes

in increments from

1

to J are

denoted by AU(J), AP(J) and t: A J), respectively:

p =

m p

+

P(I)+AP(J)

p J)

J

u

=

mU + U I) +AU(J)

U {J)

and

J

A

=m

A

+

A ,)

+

.dA J}

A J)

(2)

65


4/27

In VIew of the fact that Iteration takes place m

the

displacement and

load s pace the load level may change from

one

Iterate

to

the

other.

J 1

In

thIS caseanmtermediateposl t lon

J

forthe

same

load

level

A

=

A

IS

mtroduced before

the

final

state

J IS

reached

(flgure

1).

Supposedly

conflguratlon has already been determmed and

the in

cremental

eqUllIbrIUm

equatlOns may be

expressed by

the

lInearIzed

stlffness expresslOn.

3

a)

f

the out-of-balance forces

lR

Ip _

I

are mserted

The

tangent stlffness matrIX l at posltlon 1 may mclude

all

possIble

nonlmear

effects. t may be

kept unchanged

through severall teratlOn

cycles

followmg the modIfied

Newton-Raphson

technique. Eq.

3)

IS

the basIc

relation

used as the startmg pomt for the dIfferent Iterative

techmques described below.

The statIc

stabilIty CrIterlOn indIcates

a lImIt

or

blfurcatlon

pomt by

4)

where L U

IS

the eigenmode

of the

crIt lcal

pomt.

The

smgularIty

is

usually checked by the determmant

det c

=

5 )

The determmant can

easIly

be

calculated

as

the

product of

all

diagonal

terms m the

trIangularIzed

matrIX durmg GaussIan

elimmation. Note

that

a posltlve

determmant

IS not a sufficIent

CrIterlOn

for stable

equIlIbrIum.

Rather

the sIgns

of

the

dIagonal

t e rms

should be rnoni-

to

red to detect negatIve eigenvalues. ThIS IS the pomt

when

the lImIt

load IS

passed

and unloadmg should start .


5/27

3.

DescnptlOn of

Some

Iterative Techmques

3.1 Suppressing EquilIbrlUm IteratlOns

As mentlOned the eqUllIbrlUm 1terations usually break down

near

the

hm1t

pomt even

f

the

load mcrement S small.

The

slmplest way of

avo1dmg th1S d1fficulty S to suppress

the

i terations m the cr1tical

zone.

Th1S

procedure S

used w1th

great success by Bergan

[5J who

mtroduced

the current

stiffness

parameter

to

gU1de the algonthm

(figure 2 .

F1gure

2:

load

Increm.

derat ion

B C

OC

I

Suppressmg 1terations due to Bergan [5J

At

a prescnbed value of the stiffness parameter the 1teratIon

proce

dure S d1scontinued (pomt

A .

Then pure mcrementatlOn

is

used. I f

the EuclIdean norm of

the

d S

placement

mcrements exceeds a certam

prescr1bed

lIm1t

(pomt

C ) load

and

d1splacements are

linearly

scaled

back

(pomt

C).

Here

negative

d1agonal

elements

may

be

de

tected m

Wh Ch case negative

load

mcrements

are

applied

(point D .

The 1teratlOn procedure

is

resumed when the stiffness parameter

agamreaches

its prescr1bed

value

(pomt E).

The

lIm1t point

is located

by a zero

value

of the stiffness parameter. The techmque requ1res

very small load mcrements to aVOld

dnftmg

away from the equll1b

rlUm path.

3.2

Artlficial-

Sprmg

-

Method

ThiS method

was

developed for frames by Wright and Gaylord [ 6J

and has been apphed

to

arch

systems by Shanf1 and

Popov

[7J and

67


6/27

68

to shell

structures

by

the author [ 8J.

The

techmque

is

based on the

observatlOn that a snap-through problem may

be

transformed mto one

w1th a

pos1hve defmite character1stIc

f

linear artificIal sprmgs

are

added to

the

system

(f1gure

3 .

load

I

If

p

F1gure

3:

ArtlfiClal

s

prmg method

The method IS descr1bed

m

detaIl

m

append1x

I

t IS

an

essentlal

requ1rement that a separatlOn

of the real

problem must

be possIble

after the analys1s of the stiffened system

S

obtamed, i e. for each

stage only one load-reduchon factor

S defmed.

Furthermore the

symmetry

of

the augmented stiffness

matrix

should

be preserved.

These

requirements

lead to sprmgs

at

all

loaded

degrees of

freedom,

WhICh

are

coupled,

and

depend on

one

smgle

reference shffness.

Th1S

parameter has

to

be

found

by

trial.

The coupling

of all

arhflclal

stlffnesses may destroy the banded nature of the stiffness matrix.

In

[8J

the elements outside the band

were

omItted

from

the

stiffness

matrix

but

were retamed

on the right

hand

sIde to find the proper

mternal forces. Augmentmg the sprmg stiffnesses on the band

by

a

factor

of

three

to

five accelerates

the convergence.

Because the nonlinearity of the system

is d1mmished

by the art1fi

c1al sprmgs the total number of 1terations can

nevertheless be re

duced compared to the analys1s without springs. Num erical experI

ence

shows

that

the method

is

successful

only

in

real snap-through


7/27


8/27

70

Therefore

its

solution may be decomposed mto (figure 4)

9)

correspondmgtothe two parts of

the

right

hand

side of eq. 8). That

is,

both solutiOns

are obtamed

simultaneously using two different

load vectors

K

P

, - ,

(10 a)

'K

.

AU(J)n- 'R

u 2

, - , -

12

(lOb)

2

'

2

Q ~ O ~ ~

IU

Figure 4:

Displacement - Control

Method

IU

2

The

displacement

mcrement ;:1 1 i

J

, eq. 9), is mtroduced mto the

second

part of

eq.

7).

This allows

the determination of

the

load

parameter t:: A

J):

(11 )


9/27

Thus

lllstead

of

solvmg

an unsymmetrIcal equation the modifled shff

ness express lOn,

eq. 8),

lS

analysed for two rIght hand sldes pro-

1

vlded

that K

11 lS not slllgular. Smce the dlsplacement

2

lS held

f lxeddurl l lgthel terahonthe

underhned

terms

in

equations

10 b)

and

11) re

oml

ted all further i tera lOn cycles.

7

ThlS

modlfled

dlsplacement control

method

was descrIbed

flrst

by

Plan and Tong

[10J

wlthout

menhomng

the out-of-balance terms.

Zlenkiewlcz

[ l1J refers to the standard programmlllg techmque

and

glves

a physlcalll l terpretatlOn

of the two

step method.

Sablr and

Lock

[12J

exphclt ly mtroduced the

out-of-balance

terms mto the formula

tIon.

The method

was also descrlbed detal l

by

Stncklm et

al. [13J.

A slmllar procedure

has been applied by

Nemat-Nasser

and Shatoff

[14J

who

used a dlreCt subshtutlOn method

instead

of

the

Newton-

Raphson techmque.

A valuable slmphflcatlOn was uhhzed by Batoz and Dhatt [15J. Slllce

the

techmque above

descnbed

reqUlres a

modiflcatlOn

of the

shffness

matrIX 1

K

-7

l

11) the

authors point

out that

it

lS

not

very

likely

to

obtam exactly

the singular

pomt.

Hence the

original

matrIx 1K

may

sh l l be used and equatlOns 10) are replaced by

12 a)

12 b

where the underhned t e rm eq. 10 b) lS not

required

to be formed.

Agalll both solutlOns are

added:

13

a

The vector

lllcludes

also the

prescrIbed component

13 b)

ThlS constramt equatlOn used m the f irs t i terahon cycle m -7

J

= 1)


10/27

72

allows

the determmation of the mcremental

load parameter

14)

Supposedly

the

structure 1S m

an eqUllibriUm state at the

begmmng

ofastepsotheout-of-balanceforcesvamshand

so does t: u ~ J I 1 . Then

t: X 1)

1S

slmply a scalmg factor prov1dmg the constramt t: u ~ l =

u

2

.

Batoz

and Dhatt

[15J

even

drop

th1S first cycle.

They

update

the

dis

placement field only by

1tS

component t: u ~ l and

start to

iterate.

For

all

further

cycles

u J)

does

not change

1.

e.

t:

u

J)

1S

zero

and

2 2

t:

X

J)

1S

J=2 3

15)

Applymg

the mod1fied Newton-Raphson techmque eq. 12 a) needs to

be

solved

only

when

the

stiffness matrix 1S

updated.

Then

no

additional

computer

time

1S

required and

the

only add1tional vector stored 1S

~ U

1

) I.

The

1teration is contmued until all

other dis

placement com

ponents are

adJusted and the new equilibriUm

pos1tion

1S

found

fig. 4).

The d1splacement

control

method 1S

usually

used only

m the

ne1ghbor

hood

of

the critical pomt although

it

may

be applied throughout

the

entire

load range. Obv10usly the

method

falls whenever the

structure

snaps back from one load level to a lower one

see

example 5.2).

Some knowledge

of

the fa1lure mode is requ1red for a proper choice

of the controlling dIsplacement. t m1ght even

be

necessary to change

the prescribed parameter. Therefore an ObViOUS mod1fication IS to

relate

the procedure to

a measure mcludmg

all

dIsplacements

rather

than

to one smgle

component.

Th1S

1S

dIscussed m the next section.

3.4 ModifIed Constant -

Arc

- Length - Method of R1ks Wempner

Th1S 1terative techmque has been mdependently

mtroduced

by R1ks

[ IJ . [2J and

Wempner

[3J. Both

authors

limit

the

load step t:

X

1)


11/27

by

the

constramt

equatIOn

(16)

That

IS,

the

generahzed

arc

length of the tangent at m

IS

fIxed to a

prescrIbed

value

ds. Then the IteratIOn

path

follows a

plane

normal

to the tangent (fIgure 5), so the

scalar

product of the tangent 1 1) and

the vector

t,-+u J)

contammg the

unknown

load and

dIsplacement mcre-

ments must vanIsh:

(17

a

or In matrIX notahon

(17

b)

J =

2,3

A

new to ngent

normal

plane

tan

gent

u

FIgure 5:

Constant

- Arc -

Length

Method

73


12/27

7

The constramt equations

orIgmally

were added to

the incremental

stIffness

expressIOn destroymg symmetry and the banded structure of

the

matrIx.

t was realIzed

by Wessels [16J based on geometrIcal

consIderatIOns

that these

dIffICultIes

could be

removed

by

a

two

step

techmque

SImIlar

to that descrIbed

m

the previous sectIon. t

S thIS

Idea followed m thIS study.

Agam the

unknown vector

D.

Q{J) S formed in two parts

18 a

or

m

matrIX

notatIOn

equIvalent to eq.

13

a).

18 b

Also here AU{J I and AU{J II are obtamed

by

equatIons 12) usmg

eIther

the

reference load vector P

D.

A =1) or the out-of-balance forces

R

as rIght

hand SIdes.

Then

eq. 18) lS

mserted mto the

constramt

eq. (17) and solved

for the

unknown load mcrement

D. A

J)

(19)

GeometrIcally thls lS the

mtersectIOn

J of

the

new

tangent

t{J) wlth the

normal plane

(flgure

5). Eq. 19) lS

equivalent

to

eq.

15) but

con

tams the mfluence of alldlsplacement

components

m n mtegral sense.

The

load

mcrement

D A

1) m

the

denommator,

which

ObVIOusly

has

another dlmensIOn,

expresses the

different scalIng of the load aXlS

Wlth

respect

to the

dlsplacement

space.

t

may be seen for the one

degree-of-freedom system mflgure 6 a that a low value

D.

A

(I)

tends

to

a dlsplacement control

and

a large

value to

a load control of

the l tera

tion.

In many degree-of-freedom systems the value

D.

A(I) m eq. 19)

does not

play

an lmPOrtant role and may be

suppressed.

Durmg

the preparatIOn of

thls

study the author became aware

of the valuable paper by Crlsfleld [ 17J devoted to the same

subJect.


13/27

Agam

the

modIfIed Newton-Raphson techmque slmphfles the

method

because eq. 12 a) IS solved only once

at

the beglllmng

of

the step and

may even

be

replaced by

the

flrst solutlOn L U

(1):

20)

~ 1 )

Instead of

Iteratlllg

the

plane

normal

to

the tangent t It

mIght

be

useful

to deflne a sphere

wIth

a center

at

m and a

radius

ds [ 17J

(see

appendIx

II). Alternatlvely the

normal plane

may

be updated

l l leverYlteratlOncycle

(fIgure

6

b).

That

IS,

eq.

(19)

~ u l )

IS

r

(1)

placed by

the

total

lllcrement

U . t was found

that

except

for very

large load steps

the dIfferences

resultll lg from these

formulations

are

mInor.

o

b

IU

FIgure 6:

ModlflCatlOn of constant - arc - length

method

NumerIcal experIence has shown

that

thIS IteratIve techmque IS very

effIcIent in the entIre

load

range partlCularly when automatic load lll

crementatlOn based on eq. (16) IS used. The only addItIonal storage

reqUlred 1S the

vector ilU 1). The

extra computer t lme neghglb1e.

7


14/27

7

In

addItion

to the

constant-arc-length

the step SIze may be scaled

by relatmgthe

number

of iterations, n

used

in

the

previous step

to

a

1

desIred

value,

II t was

found

that

a

factor n

n

res ults in oscillatlOns

1 1 1

m

the

number

of

iterations requIred from step

to

step so that

JIl] n'

1 1

IS recommended. f materIal nonlmeantIes

are

involved smaller load

steps should be defmed

to

avoId drIfting.

Whenever

a negative

element

m

the trlangularlzed matrIx IS encountered unloadmg IS ImtIated. The

convergence

may

be eIther monotomc or alternating and may m some

cases be slow. Then

relaxation

factors

may

accelerate the IteratlOn

process.

For mstance,

m

the

alternatmg case a

cut-back

of the next

load

change

to 50

resulted m a conSIderable

Improvement.

4.

Summary of the

DIsplacement Control and

ModIfied

Rlks IWempner

Method

The

algonthms

for the

displacement control

method and the modified

Rlks IWempner method dIffer only

m

the equation used for the evalua

tion

of

D

A( )

The algorIthm IS summarized as follows:

1. Select

a basIc load mcrement

as the

reference load

P ,

thus

defmmg the length ds

m

the first step

(eq.

16).

2. In any

step:

a)

Solve

the equihbrlUm

equations

for

P

and linearly

scale

the

load

and dIsplacements

to produce

the length

ds.

ThIS

determmes

D.A(I), ilU(l).

b

AdJust the step slze to the desired

number

of

iteratlOns

n

l

e.

g.

Jfl{r l .

1 1

c) Check

the

trlangulanzed matrlx for unloadmg.

3.

a)*

Update the

stiffness matnx

lK

b) and, sImultaneously, determine

the

out-of- balance forces1R

c)* Solve

for

P to determine

dU

(J)I.

d)

and,

slmultaneously, solve

for

the

out-of- balance

forces

l

to determme i

U (J) II.

Note: * mdicates a step which is omItted m the modIfIed Newton

Raphson

procedure.


15/27

77

4. Use

constralllt eq. 15)

or 19) to determllle the

load

lllcrement

/:: A J) and eq. 13 a) ' eq. 18 b) to determllle dIsplacement

m

crements

~ U J). If

needed

use accelerat lon factors. )

5.

Update the load

level

and the

dIS placement

fIeld.

6. Repeat steps

3 - 5

untll the desIred accuracy

IS

achIeved.

7.

Reformulate the stlffness m tnx and st r t

a

new

step by re-

turmng to

2.

5.

Numenca l

Examples

The

examples

have been analysed on CDC

6600/Cyber 174 computers

USlllg

the nonlmear fmlte

element code NISA

[18J. The geometncal

nonlmeanty IS

based on

the total LagrangIan formulatlon.

For

the

arch example,

an

8 node

Isoparametnc plane s t ress element IS used

[4J . The plate and

shell

structures are IdealIzed

by

degenerated ISO

p r metnc elements developed

m

[8J , [19J. The modIfIed Rlks

/

Wempner method,

m

comblllatlOn wIth

the modlfled Newton-Raphson

techmque, has been applIed exclusIvely. The ratlo of the change of

the

mcremental

dIsplacements to

the

total dIsplacement mcrements ,

uSIng EuclIdean norms, IS

used

for the convergence

cnterIOn.

5. 1

Shallow Arch

The shallow

cIrcular arch

under umform pressure

flgure

7)

has

al

ready been analysed [8J applying

the artlficlal

sprlllg

method

c

11

= 28

lb/m), see

also

[7J .

Ten 8

node Isoparametnc plane s t ress

elements

were

used

for one half of the arch.

The

analysIs wIth a basIc

loadolp= 0.3 and usmg the

constant-arc-length

constraint shows

the

tYPIcal

step

SIze reductIOn m

the

neIghborhood of

the

lImIt pOlllt.

ThIrty steps wIth 1 to 2 IteratIOns

per

step were

needed.

The analySIS

has been

repeated for

a

basIc

load

step

of

p =

1.

O.

The

step

SIze

has

been

adJusted

by the factor rn wIth a desired number of IteratIOns

I 1

~ = 5. In addItion,

the load

lllcrement

was

reduced to 50 whenever

1

It

alternated

nd

the absolute value

decreased. Now

only

9 steps are


16/27

78

suffICIent.

The

number of

iteratlOns required are indicated

the

flgure. The diagram also shows the star t ing

point

each step after

the f l rst Newton-Raphson Iterate.

Compared to the artifIcial

sprlllg

technique

consIderable

savlllgs

are

achIeved.

a.

10

1 4 - - - - - - - - - - - - ~ - - - - - - - - - - - - - - - - - - - - - -

03}

12

06

0 4

02

~ - 0

basIc load step

p=1.

R=100 In

I h=2b=

2 In J

= 8

I

E=10

7

psl

f =025

R

0 0 1 ~ ~ ~ ~ ~ ~ ~

0 00

0.02

004

w/R

0 06

FIgure

7:

Shallow cIrcular arch

5.2 Shallow

CylIndrIcal

Shell

The shallow cylIndrical

shell

under

one

concentrated

load

figure 8

IS

hinged

at

the longitudlllal

edges

and free

at

the curved boundaries.

The

structure

exhibIts snap-through as well as snap-back phenomena

WIth

horizontal and vertIcal

tangents. The

shell

has been analysed by


17/27

79

Sablr and Lock [20J

who

used a comblllahon of the dIsplacement and

load control techmques. In the present study one quarter of the shell

has

been

IdealIzed by four 16 node blcublC degenerated shell elements.

As

the basIc

load

step,

P

=

0.4

kN

was

chosen.

Agalll

the

load

steps

were adJusted wIth and

the

acceleratIOn

scheme

descrIbed for

1 1

the

arch was applIed. The

entlre

load

deflectIOn dIagram IS

obtallled

I I I one solutIOn wIth

15

steps and

3

to

9

Iteratlons per

step

as llldlcated

I I I

the

fIgure. I f

the

acceleratIOn techmque

was not

used the

number

of

IteratIOns

Increased consIderably especIally at

the

mlmmum load.

0.6

fI ----n.....

P

-

[KNl 1

0.4

4

\

,W,

,

,

\

\

Sablr

an

d Lock

[201

5

, 5

0 ;:

E=3103 KN/mm

2

J..l=03

7

2L

02-

hi n ged

\

R=10 L= 2540 mm

h=635mm

-0.4

0

10

20

WC,W

,

FIgure 8:

Shallow

cyhndrIcal

shell

I

I

I

I

\A

[m

m

1

30

ThIS part of the

load-deflechon

curve IS numerically d ~ f f l c u l t

because

of the abrupt changes of the

response

at,

for

lllstance,

the POlllt

i at


18/27

80

the

free

edge. The structure has also been analysed

using 36

bIlinear

4

node degenerated elements in combmation with

an umform

1 x 1

re

duced mtegratIOnscheme. ApproxImately the same results have been

obtamed

but

at

about

20

of

the

CP-tlme.

5.3 Elastlc Plastlc Bucklmg of

a

Plate

P

p

cr

06

04

I ~

b

---- l

0 2

LLJ LLL

. . .L.LJ

~

~ b 2 ~

b =4a =1680

mm

I

E

=21

kN/mm2

T

a

..i

b

P=6h

2

h= 6mm

JJ= 3

O O ~ ~ ~ ~ ~ ~ ~

o

1

2

We

[mml

FIgure

9:

Buckhng of

a

long plate

The

sImply supported plate shown

m

fIgure

9

has

an

aspect

ratIO of

a = 1/4 and IS loaded only on

ItS

mIddle

part.

The

plate

has an

mltlal

geometrIcal

ImperfectIOn defmed by

a

double sm-functIOn

wIth

a

30


19/27

maXImum amplItude of 0.294 mm. The YIeld

lImIt

J of the

elastIc-

y

Ideally

plastlc

steel S

240 N

/mm2. EIghteen

blcublC degenerated

elements unevenly

spaced

were used for one quarter of the

plate. The

thIckness was

dIvIded

llltO

seven

layers.

The

total

load

P

S

non

dlmenslOnalIzed

WIth

the

lInear

elastIc

buckling load P of

the

plate

cr

wIth

umform

load

on the entIre boundary:

cr = k b

n

2

Eh

3

12 1-fJ

2

0

2

21 )

81

The

basIc

load step chosen

was p

= 0.25. In fIgure 9 the

normalIzed

load IS plotted versus the center la teral

dIsplacement.

The plate

falls

under combllled

geometncal and

matena l faIlure.

The

mltlal YIeld

POlllt at a deflectlOn of about 6

mm

S ImmedIately

followed

by the

lImIt POlllt at about 8. 3 mm. ThIrty steps with 1 or 2

l terat lons

per

step

were

used. The elasto-plastlc analysIs was supplemented by a

purely elastlc solutlOn

also

shown the

flgure.

Here the typlcalll l

creasl l lg postbucklIng

response

of

plates

IS

recogmzed.

5 4 CylIndncal Shell under Wllld Load

The

buckllllg

analysIs of the closed

cylmdncal

shel l under

wmd load

flgure 10) studIed [21] has been

extended

to the

postbucklIng

range.

p

10

T

L

1

10

R=

L 2 = 220 mm I h =0.105

mm

E=6 87

10

4

N mm2 I }J =0.3

5

FIgure

10.

Geometry

and load

functIon of a

cyhndrlcal shell

18

1t


20/27

82

The extremely thlll structure

with

a radIus to thickness

ratIO of over

2000 IS sImply supported at

both

ends. The

variation

of the wind load

defmed fIgure 10

S

taken

as

constant over the length

of

the cylinder.

The

maxImal

load

p

at the stagnatIon

point

is normalized

to

the

lInear

buckling load

of

the shell

under

uniform

pressure

= 918

E 1/

h f [

0

6 7

Vh

p

s =

(22)

One quarter

of

the sheil ls

Idealized by

2 x 18 bicubiC 16 node

elements.

Two

elements

of

unequal length

are

used

in the

aXIal

dIrectIOn, whIle

the 18 elements in the circumferential direction are concentrated near

the

stagnation

zone.

The f i rst

load

lllcrement defllled the

basic

step

SIze as p = 0.25. Both the perfect

and

an imperfect shell have

been

analysed. FIgure 11 shows the dIsplacement pattern of one

quarter

of

FIgure

11:

DIS placement pattern

the shell near

the lImIt pomt.

A faIlure

mode wIth one

half a wave

in

the

aXial direction

and

a

few bucklIng

waves the

circumferentIal

direction,

located

the

compression

zone,

is indicated. The post

bucklIng

mimmum

of

the

load-deflection

diagram

figure

12)

S

about

60

of

the limIt

pOlllt.

The

ImperfectIOn assumed for the

second

analysIs corresponds to the failure mode

of

the perfect structure. The

maXImum

Imperfection amplitude is 2. 5 t imes the wa ll thickness. The


21/27

load

deflectlOn

path flgure 12) llldlcates a reductlOn of the lImIt load

to

68

%of that for the perfect shell . The postbuckllllg mlllima

nearly

cOlllclde. t should be

noted

that the example S numerically very

sensitIve

because

of

the

extreme slenderness

ratIo

and

the local

nature

of

the failure mechamsm. In both cases over 60 steps were

necessary.

83

1 5 . ~

P

~ I

1

perfect

shell

..--0--__ _..... .

~ ~ ~ ~ ~ - - ~ o _ ~

/ 0

I

mperfect

shell

5

/

o o - - - - - - - - - - - ~ - - - - - - - - - - ~ - - - - - - - - ~ - - - - ~

o

5 1

w h

15

FIgure

12:

Load

-

deflectIon

-

dIagram

of

a

wllld

loaded

shell

6. Conclus

lOns

ThIS study on IteratIve techmques for passlllg lImIt pOlllts

allows

the

followlllg concluslOns:

SuppresslOn

of

eqUllIbrIum

IteratlOns near

the

lImIt pomt

may

be

a useful procedure but reqmres

very

smal l

load steps.

if

The method

of

artlf lclal s prlllgs S based on numerIcal experi-

ence

nd

t rIal

solutIOns.

For

local fallure It

may

not be suc-

cessful.


22/27

84

The displacement control method requires a proper selectIOn

of

the controlling parameter. It

fails m

snap-back

situatIOns.

The constant

arc

length

method

of

Riks /Wempner

seems to

be the most versatlle

techmq

ue

being advantageous m

the entire

load range.

Due

to modificatIOns of

the origmal method the

constramt

equa

tIOn does not need to be solved simultaneously with the equihb-

n u m equatIOns.

Automatic

adJustment of

the load step and acceleration schemes

may further Improve the performance. Only

mmor

changes m

coding are

necessary.

Applymg the

modifled

Newton-Raphson

technIque requIres the storage of one additIonal vector. The

extra computer

time is

neghgible.

Acknowledgement

The author would like to

thank

Professor

D. W. Murray

Umversity

of

Edmonton

currently at the

University

of

Stuttgart for valuable

discussIOns.


23/27

85

References

[IJ Rlks, E.:

The

ApphcatlOn of Newton's Method to the

Prob

lemofElastlcStabllity.

J. Appl. Mech.

39

(1972) 1060-1066.

[2J Rlks, E.

:

An

IncrementalApproach

to the

SolutlOn

of

Snapp

mgandBucklmg Problems. Int. J . Sohds Struct.

15 (1979)

529-551.

[3J

Wempner, G. A. : Dlscrete Approxlmations Related

to Non

l inear

Theones of Sohds. Int. J.

Solids

Struct. 7 (1971)

1581-1599.

[4J Bathe,

K. - J . Ramm, E .

Wllson, E. L.: Flmte

Element

FormulatlOns for Large DeformatlOn

Dynamlc

Analysis.

Int.

J.

Num. Meth. Engng. 9 (1975)

353-386.

[5J Bergan, P. G. : SolutlOn

Algonthms

for Nonlmear

Structural

Problems.

Int. Conf. on Engng. Appl. of the F.

E. Method ,

HVlk

Norway 1979,

pubhshed

by

A. S. Computas.

[6J Wnght, E. W.

Gaylord,

E. H. : Analysls of Unbraced

Multi

StorySteelRlgldFrames. Proc.

ASCE, J . Struct. D1V 94

(1968)

1143-1163.

[7J Shanfl , P .

Popov, E. P . : Nonhnear Buckhng Analysls of

Sandwich

Arches.

Proc.

ASCE, J .

Engng. D1V

97 (1971)

1397-1412.

[8J Ramm, E. :

Geometnschm chtlmeare Elastostabk

undflmte

Elemente.

Hablli tationsschnft,

Umversltat Stuttgart, 1975.

[9J

Argyns, J. H. :

Contmua and

Dlsconbnua.

Proc.

1st Conf.

Matnx Meth.

Struct.

Mech.

, Wrlght-Pat terson

A.

F. B.,

OhlO 1965, 11-189.

[ 10J Plan,

T.

H. H.

Tong, P.

:

Variabonal

FormulatlOn of

Finite

DlsplacementAnalysis. IUTAMSymp. on Hlgh

Speed

Com

putmg of Elastic Structures , L H ~ g e 1970, 43-63.

[ l1J

Zlenkiewlcz,

O.C.: Incremental Dlsplacement mNon-Linear

Analysls. Int. J .

Num. Meth. Engng.

3 (1971)

587-588.

[12J Lock, A. C., Sablr, A. B. : Algonthm for Large

DeflectlOn

Geometncal ly Nonhnear Plane

and

Curved Structures.

In Mathematlcs of Flmte Elements and Apphcahons (ed.

J. R.

Whlteman), AcademlC Press ,

N. Y.

1973,483-494.

[13J Halsler , W., Stnckhn, J . Key, J . : Dlsplacement Incre

mentatlOnmNonhnear StructuralAnalysls by the Self-Cor

rectmgMethods.

Int. J. Num. Meth. Engng.

11

(1977) 3-10.

[14J Nemat-Nasser , S., Shatoff,

H. D. :

Numencal

Analysls

of

Pre - and PostcnhcalResponse of

Elastic Continua

at Fmite

Strams.

Compo Struct. 3 (1973) 983-999.


24/27

86

[15J Batoz, J . -L.

Dhatt, G.:

Incremental Displacement Al

gorithms for Nonlmear

Problems.

Int. J.

Num.

Meth.

Engng.

14 1979)

1262-1267.

[16J Wessels, M.:

Das

statlsche und dynamische Durchschlags

problem

der

imperfekten

flachen

Kugels

chale

bei

elastlscher

rotahonssymmetrischer Verformung. DissertatlOn, TU Han

nover,

1977, Mitteil. Nr. 23

des

Instltuts

fUr

Stahk.

[ 17J Crisfield, M. A. : A F


25/27

87

AppendIX

I:

The

Arhflcla l Sprlllg Method

Accordlllg

to

fIgure 3 the vector of the

total

external loads 1

G

of the

modIfIed sys tem

is

decomposed lllto the re l

load

vector p and the

part

resIs ted

by

the s prlllgs 1 f

+

A

1)

To retalll

the desIred r ho

of

specIfled

loads It S

required that all

components of the re l load can be

obtallled

by one common load-re

duchon-factor ly

A 2

That IS, all

components

of conflguratlOn 1 have the

same

ratlO

1=1 2 3

n

(A 3)

It follows

that

sprlllgs have to be attached to all

loaded

degrees-of

freedom and all

sprlllg

shffnesses are coupled. The sprlllg stiffness

matrix

C IS defllled by

f

=

C . U

A

4

Energy

pnnclples reqUlre

C to

be

a symmetr ical

matrIx

(c

kt

= c

tk

.

EquatIon A 3)

allows

the elements c

k t

of the

m tnx

to be determined

If

one

reference

shffness

c

11

is prescnbed

A 5)

or If the reference load

vector P

IS llltroduced

l l

T

C = - 2 - P P

P

A 6)

The

I terahon

equatIon, eq. 3

a), S modifIed

to

( K

C) L1U J)

= P f

-

F - c

u

A 7

J


26/27

88

The

ngh t hand sIde expresses the out-of-balance forces. After i tera

tlOn J ... m + 1) the rea l loads are determined by eq. A 2):

A

8

The load-reductIon

factor

IS obtallled by

eq.

A 3):

(A

9)

It

was

found that an effectIve

value

of c

l l

IS one

WhIch

leads to

o

Strategies for Tracing the Nonlinear Response Near Limit Points

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Transcript of Strategies for Tracing the Nonlinear Response Near Limit Points