Strategies for Tracing the Nonlinear Response Near Limit Points

Strategies for Tracing the Nonlinear Response Near Limit Points E.RAMM UOlversltat Stuttgart, Germany

Abstract For the prebucklmg range an extenslVe llterature of effectIve solutlOn techmq ues eXIsts for the numerIcal solutlOn of structural problems but only a few algorIthms have been proposed to trace nonllnearresponsefrom the pre-llmlt mto the post-limIt range. Among these are the sImple method of suppressmg eqUlllbrlUm IteratlOns, the mtroductlOn of artIflClal sprIngs, the dIsplacement control method and the "constant-arc-Iength method" of Rlks /Wempner. It IS 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 examples show that a modlfled Rlks /Wempner method can be especIally 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 IS the Imperfechon senSItIvIty of certam structures whlCh m general IS dIrectly related to the postcrIilcal response. In partIcular this IS true for structures exhlbltmg a decreasIng post-lImIt characterIstic. ThIS may res ult III a dynamIC snap- through or snap- back phenomenon dependlllg on whether the load or the dis placement controls the system. However, a statIc analysIs traces the wQole postcrItIcal range allowmg for a better Judgement of the overall structural response. It IS 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

64

of critical points. The stiffness matrix approaches smgularity resultmg 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 trace the response beyond the critical point. It 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 artlficial springs, the displacement control techmque and the "constant - arc - length method" of Riks [1], [2J and Wempner [ 3J. In partlcular an attempt IS made to show the correlation of the latter procedures. Special emphasis IS given to some modifIcations of the Rlks /Wempner method leadmg to an efflclent iterative technique throughout the entire range of loadmg and not only near the crItical 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 IS assumed but few changes are necessary for non proportIOnal loading.

2.

Starting Pomt and Notation

The study IS based on the incremental/IteratIve solutIOn procedure m a nonlinear fmite element analysIs,

1.

e. the nonlmear problem is

stepWIse linearIzed and the lInearizatIOn error is corrected by additIOnal eqUIlibrIum iteratIOns, see for instance [4J. A left supers cript mdlcates the current confIguration of the total dlsplacementsmu , the

65

load vector mp , the mternal forces mF forces ~

and the out-of- balance

. For proportional loadmg the loads may be expressed

r:n.".

by one load factor

(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

FIgure 1:

Ju

displacement

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 Jp

AU(J),

= mp

AP(J) and t:.A (J), respectively:

+ P(I)+AP(J) p(J)

Ju

=mU

and

JA =m A +

A(,)

+

.dA(J}

A(J)

+ U(I) +AU(J) U'{J)

(2)

66

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 caseanmtermediateposltlon J' forthe same load level A = A IS mtroduced before the final state J IS reached (flgure 1). Supposedly conflguratlon

1

has already been determmed and the in-

cremental eqUllIbrIUm equatlOns may be expressed by the lInearIzed stlffness expresslOn.

(3 a)

If the out-of-balance forces

The tangent stlffness matrIX

lR

lK

Ip _ IF are mserted

at posltlon

1

may mclude all possIble

nonlmear effects. It may be kept unchanged through severallteratlOn 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 LI U C IS the eigenmode of the crItlcal pomt. The smgularIty is usually checked by the determmant

det cK

= 0

(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 terms should be rnoni-

to red to detect negatIve eigenvalues. ThIS IS the pomt when the lImIt load IS passed and unloadmg should start.

67

3.

DescnptlOn of Some Iterative Techmques

3.1

Suppressing EquilIbrlUm IteratlOns

As mentlOned the eqUllIbrlUm 1terations usually break down near the hm1t pomt even 1f the load mcrement 1S small. The slmplest way of avo1dmg th1S d1fficulty 1S to suppress the iterations m the cr1tical zone. Th1S procedure 1S used w1th great success by Bergan [5J who mtroduced the "current stiffness parameter" to gU1de the algonthm (figure 2).

load

B

C ___ ---OC I

Increm. + deration

F1gure 2:

Suppressmg 1terations due to Bergan [5J

At a prescnbed value of the stiffness parameter the 1teratIon procedure 1S d1scontinued (pomt A). Then pure mcrementatlOn is used. If the EuclIdean norm of the d1S placement mcrements exceeds a certam prescr1bed lIm1t (pomt C') load and d1splacements are linearly scaled back (pomt C). Here negative d1agonal elements may be detected m Wh1Ch 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 equll1brlUm 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

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 1f linear artificIal sprmgs are added to the system (f1gure 3).

load IG If

Ip

F1gure 3:

ArtlfiClal s prmg method

The method IS descr1bed m detaIl m append1x I. It IS an essentlal requ1rement that a separatlOn of the real problem must be possIble after the analys1s of the stiffened system 1S obtamed, i. e. for each stage only one load-reduchon factor 1S 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 art1fic1al sprmgs the total number of 1terations can nevertheless be reduced compared to the analys1s without springs. Numerical experIence shows that the method is successful only in real snap-through

69

problems where the sprlllgs can keep the destabllizlllg structure alIve. The method cannot be recommended for structures wIth local buckllllg or when a tendency to bIfurcation IS present. 3. 3

DIsplacement - Control

The most often used method to avoId the slllgularIty at the crItIcal POlllt IS the mterchange of dependent and mdependent varIables. Here a smgle dIsplacement component selected as a controllIng parameter IS prescrIbed and the correspondmg load level IS taken as unknown. The procedure was llltroduced fIrst by ArgyrIs [9J but

III

the meantIme

has been modIfIed by several authors. For SImplICIty let us assume that the stIffness expreSSIOn, eq. (3), IS reordered so that the prescrIbed component one m the dIsplacement vector

f::,

u~J) = U2

IS the last

AU(J). Then equatIOn (3) may be de-

composed llltO two parts

(6 )

Interchanglllg the varIables

(7)

It IS ObVIOUS that the loss of the symmetrIcal and banded structure of the shffness matrIX IS a severe handIcap. Later It was recogmzed that the solutIon of eq. (7) could be formed m two parts. The fIrst lme of eq. (7) +

IS

IR _ IK 1

A

12' U2

(8)

lInear m the unknown mcrement at the load parameter 6. A (J)

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,

AU W -

'K 11 . AU(J)n, - 'R ,

(10 a)

-

'K12 · u"2

Q---------~

2'

2'

O------~~

IU,

Figure 4:

(lOb)

IU

Displacement - Control Method

i

The displacement mcrement ;:1"1 J ), eq. (9), is mtroduced mto the second part of eq. (7). This allows the determination of the load parameter t::. A(J):

(11 )

2

71

Thus lllstead of solvmg an unsymmetrIcal equation the modifled shffness express lOn, eq. (8), lS analysed for two rIght hand sldes provlded that

1

K 11

lS not slllgular. Smce the dlsplacement 112 lS held

flxeddurlllgthelterahonthe underhned terms in equations (10 b) and (11) are oml tted

III

all further itera tlOn cycles.

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 gl ves a physlcallllterpretatlOn of the two step method. Sablr and Lock [12J exphcltly mtroduced the out-of-balance terms mto the formulatIon. The method was also descrlbed

III

detall 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 NewtonRaphson 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

lK 11)

the authors point out that it lS not very likely

to obtam exactly the singular pomt. Hence the original matrIx 1 K

may

shll be used and equatlOns (10) are replaced by (12 a)

(12 b) where the underhned term

III

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 first iterahon cycle (m

-7

J = 1)

72

allows the determmation of the mcremental load parameter

(14)

Supposedly the structure 1S m an eqUllibriUm state at the begmmng

u~J)I1. Then constramt t:, u~l) = u2 .

ofastepsotheout-of-balanceforcesvamshand so does t:,

X. (1) 1S slmply a scalmg factor prov1dmg the

t:,

Batoz and Dhatt [15J even drop th1S first cycle. They update the displacement field only by 1tS component

t:,

u~l)

and start to iterate.

For all further cycles u (J) does not change 2 X. (J) 1S

1.

e.

t:,

u (J) 1S zero and 2

t:,

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 components are adJusted and the new equilibriUm pos1tion 1S found (fig. 4). The d1splacement control method 1S usually used only m the ne1ghborhood 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. It 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)

73

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 mcrements must vanIsh: (17 a)

or In matrIX notahon (17 b)

J

=2,3

'A new to ngent 'normal plane'

tan gent

FIgure 5:

Constant - Arc - Length Method

'u

74

The constramt equations orIgmally were added to the incremental stIffness expressIOn destroymg symmetry and the banded structure of the matrIx. It 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. It IS thIS Idea followed m thIS study. i~) Agam the unknown vector D. Q{J) IS 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

IR

P (D. A = 1)

or the out-of-balance forces

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 contams the mfluence of alldlsplacement components man 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. It 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 lteration. In many degree-of-freedom systems the value D. A(I) m eq. (19) does not play an lmPOrtant role and may be suppressed.

i~)

Durmg the preparatIOn of thls study the author became aware of the valuable paper by Crlsfleld [ 17J devoted to the same subJect.

75

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 LI U (1):

(20)

Instead of Iteratlllg

III

the " plane " normal to the tangent ~(1) 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 llleverYlteratlOncycle (fIgure 6 b). That IS, (1)

placed by the total lllcrement U

III

eq. (19)

~u(l)

IS re-

. It was found that except for very

large load steps the dIfferences resultlllg from these formulations are mInor.

IU

bJ

oj

ModlflCatlOn of constant - arc - length method

FIgure 6:

NumerIcal experIence has shown that thIS IteratIve techmque IS very effIcIent in the entIre load range partlCularly when automatic load lllcrementatlOn based on eq. (16) IS used. The only addItIonal storage reqUlred

1S

the vector ilU(1). The extra computer tlme

~s

neghglb1e.

76

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. • It was found that a factor 1

n.1 In 1 res ults in oscillatlOns

m the number of iterations requIred from step to step so that

JIl]"n' 1 1

IS recommended. If 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 evaluation 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), b)

AdJust the step slze to the desired number of iteratlOns

e. g. c)

3.

ilU(l).

Jfl{rl . 1 1

n.

l'

Check the trlangulanzed matrlx for unloadmg.

a)* Update the stiffness matnx lK b)

and, sImultaneously, determine the out-of- balance forces1R

c)* Solve for d)

P

to determine ,dU (J)I.

and, slmultaneously, solve for the out-of- balance forces

lR

to determme iI U (J) II. Note:

* mdicates a step which is omItted m the modIfIed NewtonRaphson procedure.

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 mcrements

~U (J).

(If needed use acceleratlon 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 matnx and start a new step by returmng to 2.

5.

Numencal 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 stress element IS used [4J. The plate and shell structures are IdealIzed by degenerated ISOparametnc 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 already been analysed

III

[8J applying the artlficlal sprlllg method

(c 11 = 28 lb/m), see also [7J. Ten 8 node Isoparametnc plane stress 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;;;I

1

wIth a desired number of IteratIOns

~ = 5. In addItion, the load lllcrement was reduced to 50 1

% whenever

It alternated and the absolute value decreased. Now only 9 steps are

78

suffICIent. The number of iteratlOns required are indicated flgure. The diagram also shows the starting point

III

III

the

each step after

the flrst Newton-Raphson Iterate. Compared to the artifIcial sprlllg technique consIderable savlllgs are achIeved.

--03}0

14------------~----------------------

~

•••• ~ 000

basIc load step

p=1.

12 a.

10.

06 R=100

In

02=008

0.4

02

I

h=2b= 2

I

E=10 7psl ,

In

J

fJ =025

R

001~----~----~--~----~----~----~

0.00 FIgure 7:

5.2

0.02

004

w/R

0.06

Shallow cIrcular arch

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

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.

= 0.4 kN was chosen. Agalll the load steps

As the basIc load step, P

were adJusted wIth ~ and the acceleratIOn scheme descrIbed for 1

1

the arch was applIed. The entlre load deflectIOn dIagram IS obtallled III

one solutIOn wIth 15 steps and 3 to 9 Iteratlons per step as llldlcated

III

the fIgure. If the acceleratIOn techmque was not used the number

of IteratIOns Increased consIderably especIally at the mlmmum load.

0.6

•• Sablr an d Lock [201

fI'----n.....

P

I

"-

1

[KNl

I

0.4

'

4

'\ ",W,

,,

'\

5

\

" ,

5

•

•

0.;:

E=3103 KN/mm2,J..l=03

2L

I

L

-02-

-0.4

7 \.

hi n ged

R=10 L= 2540 mm h=635mm 0

10

FIgure 8:

Shallow cyhndrIcal shell

I I

I

\A \~

20

WC,W,

ThIS part of the load-deflechon curve IS numerically

[m m 1

d~fflcult

30

because

of the abrupt changes of the response at, for lllstance, the POlllt i at

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 reduced 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

Pcr

06

04

I~

b

----'l .~1 .

T

0.2

a

..i LLJLLL-4-LLl...L.LJ- ~

P=6h

~b/2~

b =4a =1680 mm E =210 kN/mm2

I

b

2

h= 6mm

JJ=03

OO~----~----~------~----~----~----~

o

FIgure 9:

10

20

We

[mml

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

81

maXImum amplItude of 0.294 mm. The YIeld lImIt

(J

y

of the elastIc-

Ideally plastlc steel IS 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 IS nondlmenslOnalIzed WIth the lInear elastIc buckling load P

cr

of the plate

wIth umform load on the entIre boundary:

Pcr = k

b

n 2 ·E·h 3 12 (1-fJ 2 ) 0 2

(21 )

p=

The basIc load step chosen was

0.25. In fIgure 9 the normalIzed

load IS plotted versus the center lateral dIsplacement. The plate falls under combllled geometncal and matenal faIlure. The mltlal YIeld POlllt at a deflectlOn of about 6 mm IS ImmedIately followed by the lImIt POlllt at about 8. 3 mm. ThIrty steps with 1 or 2 lteratlons per step were used. The elasto-plastlc analysIs was supplemented by a purely elastlc solutlOn also shown

III

the flgure. Here the typlcallll-

creaslllg postbucklIng response of plates IS recogmzed.

5.4

CylIndncal Shell under Wllld Load

The buckllllg analysIs of the closed cylmdncal shell under wmd load (flgure 10) studIed III [21] has been extended to the postbucklIng range.

~

p

T

10

1350

L

1

-10

R= L/2 = 220 mm E=6 87 10 4 N/mm2 FIgure 10.

I

I

h =0.105 mm }J =0.3

Geometry and load functIon of a cyhndrlcal shell

18

1t

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 III fIgure 10 IS 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

= 0918

E(1/

.h.- f1[ - 0 RVh

657

is

=

p

(22)

One quarter of the sheills 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 first 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 direction, located

III

III

the circumferentIal

the compression zone, is indicated. The post-

bucklIng mimmum of the load-deflection diagram (figure 12) IS 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 times the wall thickness. The

83

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. It should be noted that the example IS 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.

15.-------------------------------------~

P ~I

perfect shell

10

/

"..--0--__ "

~~

~ I mperfect shell

05

I I I I I

/

/

---0------

_.....,.,

~~~--~o_~

/

oo+-----------~----------~--------~----~

o

5

FIgure 12:

6.

10

w/h

15

Load - deflectIon - dIagram of a wllld loaded shell

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 small load steps. if

The method of artlflclal s prlllgs IS based on numerIcal experience and trIal solutIOns. For local fallure It may not be suc-

cessful.

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 equatIOn does not need to be solved simultaneously with the equihbnum 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.

85

References [IJ

Rlks, E.: The ApphcatlOn of Newton's Method to the ProblemofElastlcStabllity. J. Appl. Mech. 39 (1972) 1060-1066.

[2J

Rlks, E. : An IncrementalApproach to the SolutlOn of SnappmgandBucklmg Problems. Int. J. Sohds Struct. 15 (1979) 529-551.

[3J

Wempner, G. A. : Dlscrete Approxlmations Related to Nonlinear 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", H¢Vlk, Norway 1979, pubhshed by A. S. Computas.

[6J

Wnght, E. W. , Gaylord, E. H. : Analysls of Unbraced MultiStorySteelRlgldFrames. 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. : Geometnschmchtlmeare Elastostabk undflmte Elemente. Habllitationsschnft, Umversltat Stuttgart, 1975.

[9J

Argyns, J. H. : Contmua and Dlsconbnua. Proc. 1st Conf. "Matnx Meth. Struct. Mech. ", Wrlght-Patterson A. F. B., OhlO 1965, 11-189.

[ 10J

Plan, T. H. H. , Tong, P. : Variabonal FormulatlOn of Finite DlsplacementAnalysis. IUTAMSymp. on "Hlgh Speed Computmg of Elastic Structures", LH~ge 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 Geometncally 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 IncrementatlOnmNonhnear StructuralAnalysls by the Self-CorrectmgMethods. 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.

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[15J

Batoz, J. -L., Dhatt, G.: Incremental Displacement Algorithms for Nonlmear Problems. Int. J. Num. Meth. Engng. 14 (1979) 1262-1267.

[16J

Wessels, M.: Das statlsche und dynamische Durchschlagsproblem der imperfekten flachen Kugels chale bei elastlscher rotahonssymmetrischer Verformung. DissertatlOn, TU Hannover, 1977, Mitteil. Nr. 23 des Instltuts fUr Stahk.

[ 17J

Crisfield, M. A. : A F