2-Problems Of Structural Optimization For Post-Buckling Behaviour.pdf

Researchpaper

StructMultidiscOptim25,423–435(2003)

DOI10.1007/s00158-003-0330-7

Problems of structural optimization for post-buckling behaviour

B.Bochenek

Abstract A proposal of a new approach to the op-

timal design of structures under stability constraints is

presented.Itisshownthatthestandardproblemofmax-

imizationoftheinstabilityloadmaybemodiﬁedsoasto

obtainaspeciﬁedpost-criticalbehaviourofthedesigned

structure.Themodiﬁedoptimalstructurerepresentssta-

blepost-bucklingbehavioureitherlocally,thatis,inthe

vicinity of the critical point, or for a speciﬁed range

of generalized displacements. First, some rigid–elastic

ﬁnite-degree-of-freedommodelsareoptimized,givingan

insight into the modiﬁed design problems. Then a clas-

siﬁcationofthenewoptimizationproblemsispresented.

Various forms of instability are taken into account and

abroadselectionofobjectiveaswellasconstraintfunc-

tions is proposed. Based on the presented classiﬁcation

and following the proposed optimization concept, de-

tailed formulations of nonlinear problems of design for

post-bucklingbehaviouraregiven.

imperfectionsare,ingeneral,notincludedinsuchastan-

dard formulation and therefore important information

aboutthebehaviourofadesignedelementafterbuckling

is not provided. Very often the standard optimal struc-

ture represents unstable post-buckling behaviour and is

verysensitivetoimperfections.Thisisadrawbackofthe

design and it indicates that the combination of geomet-

rically nonlinear analysis with the design procedure is

necessary, especially from a practical point-of-view. Be-

causeofitscomplexity,thisareaofresearchhasnotbeen

broadly investigated so far. Only recently have papers

beenpublisheddealingwiththeoptimizationofgeomet-

ricallynonlinearstructuresexposedtoalossofstability

(Godoy1996;Mr´ozandPiekarski1996,1998;Perryand

Gurdal 1996; Pietrzak 1996; Cardoso etal. 1997; Sousa

etal. 1999; Sorokin and Terentiev 2001). It has been

shownthatifgeometricalnonlinearityisallowedforand

nonlinearinstabilityanalysisisperformed,moreaccurate

informationaboutthebehaviouroftheoptimizedstruc-

turecanbeprovided.Itispossibletoevaluatethequality

of the design and, if necessary, to reject solutions that

are not applicable. Furthermore, it is possible to imple-

ment nonlinear constraints into the formulation of the

optimization problem and hence to modify the design.

Post-bucklingconstraintsofaspecialformthatdepends

on the type of instability are added to the mathemat-

ical programming problem, which allows the nonlinear

equilibriumpathoftheoptimizedstructuretobealtered

andastablepost-bucklingpathtobecreated.Thiscon-

ceptwasproposedbyBochenek(1993),andthenapplied

tosolvingmanynonlinearoptimizationproblems(Boch-

enek 1996, 1997a,b, 1999a,b; Bochenek and Kru˙zelecki

2001;BochenekandBielski2001).

Key words instability, post-buckling behaviour, opti-

maldesign

Introduction

Themaximizationoftheinstabilityloadforaprescribed

volume of a designed element is a standard problem of

optimizationunderstabilityconstraints.Theanalysisof

nonlinear post-buckling behaviour and the inﬂuence of

Received: 8January 2002

Published online:30October 2003

Springer-Verlag2003

B.Bochenek

InstituteofMechanicsandMachineDesign,CracowUniversity

ofTechnology,JanaPawlaII37,31-864Krakow,Poland

e-mail:Bogdan.Bochenek@pk.edu.pl

A concept of modiﬁed optimization

Theaimofthissectionistopresenttheideaofanewap-

proachtooptimizationagainstinstability.Severalsimple

rigid–elasticﬁnite-degree-of-freedommodelsthatconsist

424

of rigid rods connected by elastic joints and equipped

withextensionalandrotationalspringsarechosenforthis

purpose. For each ofthem, an instability analysisbased

onenergyconsiderationsisperformed,leadingtoanan-

alyticalexpressionfortheloadasafunctionofthegener-

alizeddisplacementthatdescribesthenonlinearequilib-

riumpath.

The ﬁrst model shown in Fig.1 consists of two rigid

rods connected by an elastic joint. A rotational spring

of stiﬀness C and an extensional spring K ,bothwith

linearly elastic characteristics,are added to the system.

The model is loaded by a conservative force P that re-

tainsitsdirectionafterbuckling.Iftheangle ϕ ischosen

asthegeneralizeddisplacementthatcontrolsthenonlin-

ear post-critical deformation, the total potential energy

ofthesystemcanbewrittenas

Π = 1

2 Cϕ 2 + 1

2 K ( L sin ϕ ) 2

−

P ( L + D

−

L cos ϕ

−

D cos ψ ) .

(1)

Fig. 2 Post-criticalpathsforselectedvaluesofγ

Fromthestationarityconditiond Π/ d ϕ =0followsanex-

pressionfor theload p vs displacement ϕ that describes

thenonlinearequilibriumpath:

problemcanbe formulated.Foragivenvalueof C , ﬁnd

γ soastomaximizethecriticalloadandsimultaneously

assure stable post-buckling behaviour of the optimized

structure:

maximize p cr ( γ )=(1+ γ ) κ

−

sin 2 ϕ ϕ + 1

2 γ sin2 ϕ

p ( ϕ )=

(2)

sin 2 ϕ sin ϕ + 1

−

2 sin2 ϕ

inwhichageometricalrelationanddimensionlessquanti-

tiesareintroducedas

subjectto ∂ 2 p

∂ϕ 2 (0; γ )

≥

0 .

(5)

sin ψ = 1

sin ϕ, p = PL

C , γ = KL 2

C ,

= D

Solvingtheproblemformulatedabove,inwhichthepost-

bucklingconstraintissetlocallyfor ϕ =0,oneobtains

(3)

γ opt = κ

3 +4

−

The critical (bifurcation) load can be found directly

from(2)as

3 +1) ,

(6)

p cr =(1+ γ ) κ

(4)

=2 . 0,gives γ = γ opt =7 / 9and p cr =32 / 27 .

Thepost-bucklingpathfortheoptimalsolutionisrepre-

sentedinFig.2byathicksolidline.

Ascanbeseenintheaboveexample,byimplementing

anappropriatelocalpost-bucklingconstraintintothefor-

mulationoftheoptimizationproblem,thedesiredmodi-

ﬁcationofthesymmetricpost-criticalpathwasachieved.

Itisworthstressingthatsuchalocalconstraintmaynot

besucientinallcases.Tomakemattersevenmorecom-

plicated, the behaviour of the structure at the critical

pointdoesnothavetobesymmetric.Whatcanbedone

ifthishappens?We shalldiscussthisissuebyanalyzing

another model, shown in Fig.3. Once again the model

consistsoftwobars,butthistimeonlytheoneoflength L

isrigid.Thelengthofthesecondonecanvaryanditsex-

tensibilityismodelledbyaspring K .Thetotalpotential

energyforthemodelisgivenby

If post-critical paths found for various values of the

stiﬀness γ areanalyzed,onecanseefromFig.2thatthe

post-bucklingbehaviourofthesystemcanbeeithersta-

bleorunstabledependingonthevalueof γ .Thismeans

thatbyselectingappropriatevaluesof γ ,thecreationof

aspeciﬁedbehaviourofthestructureispossible.Hence,

for γ as the design variable, the following optimization

Fig. 1 Rigid–elasticone-degree-of-freedomsystem,symmet-

ricbifurcation

2 Cϕ 2 + 1

2 K ( L k −

L 0 ) 2

−

PL (1

−

cos ϕ ) , (7)

L .

which,for

Π = 1

425

Fig. 3 Rigid–elasticone-degree-of-freedommodel,asymmet-

ricbifurcation

whichleadsto

p ( ϕ )= ϕ

sin ϕ + γ [1

−κ

tan α +

cot ϕ ]

−

(8)

cos α

(

−κ

tan α )+

cot ϕ

Fig. 4 Post-criticalpathsforselectedvaluesofγ

tan α +cos ϕ

−

1) 2 +(

+sin ϕ ) 2

and

somecases,lossofstabilitycanbetheresultofanincrease

intemperature.Asimplemodelthatisexposedtother-

malbucklingisshowninFig.5.Theextensibilityofabar

axisisrepresentedbyaspringofstiﬀness K andacoe-

cientofthermalexpansion α .Thetotalpotentialenergy

ofdeformation(withoutthermalenergy)canbewritten

Π = 1

p cr =1+ γ cos 2 α.

(9)

The deﬁnitions of the dimensionless quantities are the

same as in the previous example, and once again, the

maximalcriticalloadsubjecttostablepost-bucklingbe-

haviour is sought. However, the formulation of the op-

timization problem is diﬀerent. Since the behaviour of

the structure at the critical point is asymmetric, post-

buckling constraints that are independent of each other

for positive and negative values of the generalized dis-

placement must be imposed. In addition, setting con-

straintsthatensuresymmetricbehaviourinthevicinity

ofthecriticalpointisnecessary.Forgivenvalues C and

α ,valuesof γ and

2 Cϕ 2 + 1

2 K [ αL 0 T

−

L + L 0 ] 2 .

(12)

Anonlinearequilibriumpath t ( ϕ ) that followsfrom the

stationarityconditionisgivenby

t ( ϕ )= 1

cos ϕ

cos ϕ .

(13)

aresoughtforwhichthecriticalload

ismaximalwithrespecttotheconstraintsforcingtheop-

timizedstructuretobehaveinastablewayinaspeciﬁed

intervaloftheangulardisplacement ϕ :

Thequantitiesin(13)aredeﬁnedas

t = αT, γ = KL 0

C .

(14)

maximize p cr ( γ,

)=1+ γ cos 2 α,

The optimal value of γ is sought so as to maximize the

criticaltemperature t (strain causedby temperaturein-

subjectto ∂p

∂ϕ (0; γ,

κ−

sin α cos α =0 , (10)

∂p

∂ϕ ( ϕ ; γ,

)sign ϕ

≥

0or ϕ

∈

[ ϕ 1 ,ϕ 2 ] .

(11)

In Fig.4, selected solutions (for α =60 ◦ ) that fulﬁll

equality constraint (10) are shown. The optimal solu-

tion

κ opt = √ 3 / 4 ,γ opt =0 . 74 ,p cr =1 . 185 , found for

90 ◦ ,ϕ 2 =90 ◦ , isrepresentedbyathicksolidline.

In the examples discussed above, instability was

caused by applied externalforces.Itisknownthatin

−

Fig. 5 Rigid–elastic one-degree-of-freedom model, thermal

buckling

sin ϕ cos 2 ϕ + 1

−

ϕ 1 =

426

crement T ) subject to stable post-critical behaviour of

thesystem:

inginpost-criticalregime.Althoughtheproblemisnon-

conservative, the static criterion of stability is sucient

aslongastheanalysisislimitedtonegativevaluesof η .

Fromtheequationofequilibriumonecanobtain

maximize t cr = 1

γ ,

∂ 2 t

∂ϕ 2 (0; γ )

Cϕ + KL 2 sin ϕ cos ϕ = PL cos ηϕ sin ϕ

−

subjectto

≥

0 .

(15)

PL sin ηϕ cos ϕ,

(16)

Solving(15)oneobtains γ opt =5 / 3 ,t cr =3 / 5,andthe

post-buckling path for the optimal solution is given by

athicksolidlineinFig.6.

The results obtained so far show that by changing

quantities that describe the stiﬀness of the structure or

its geometry, the post-buckling behaviour can be modi-

ﬁed and the desired stable behaviourafter buckling can

beobtained.Itisknownthatthedesignvariablesinthe

modiﬁeddesignproblemscanalsobechosenfromquan-

titiesdescribingadditionalsupportoradditionalloading.

Thenextexampleshowsthatevenaparametercontrol-

ling the behaviour of the loading after buckling can be

a design variable. The analyzed structure is shown in

Fig.7. The quantity η describes the direction of load-

which leads to (the former deﬁnitions of dimensionless

quantitieshold)

p ( ϕ )= ϕ + γ sin ϕ cos ϕ

sin(1

−

η ) ϕ

(17)

Foragiven γ thefollowingmodiﬁeddesignproblem,

maximize p cr ( η )= 1+ γ

−

η ,

subjectto ∂ 2 p

∂ϕ 2 (0; η )

≥

0 ,

(18)

leadstotheoptimalvalueofthedesignvariable η ,

2 γ

η opt =1

−

1+ γ .

(19)

√ 2 ,p cr = √ 2).

Summarizing the discussion of this section, one can

state that modiﬁcation of the standard optimization

problem is possible and the proposed approach allows

the speciﬁed behaviour of the optimized structure after

buckling to be obtained. The modiﬁed optimal struc-

tureexhibitsstablepost-criticalbehavioureitherlocally,

−

Fig. 6 Post-criticalpathsforselectedvaluesofγ

Fig. 7 Rigid–elastic one-degree-of-freedom model, buckling

undersubtangentialforce

Fig. 8 Post-criticalpathsforselectedvaluesofη

Selectedpost-bucklingpathsfor γ =1areshowninFig.8,

inwhichthethicksolidlinerepresentstheoptimalsolu-

tion( η opt =1

427

thatis,inthevicinityofthecriticalpointorinspeciﬁed

range of a generalized displacement. Moreover various

cases of loadings or design variables show that the im-

plementation of nonlinear post-buckling analysis in the

formulationofoptimizationproblemsopensmanypossi-

bilitiesfornewdesignproblems.Theproposednewcon-

cept ofoptimizationunderstability constraintsiscalled

themodiﬁedoptimization.

structuresexposedtoelasticinstabilitycanbeclassiﬁed

according to the form of instability. Selected objective

functions for standard and modiﬁed problems of struc-

tural optimization against instability are presented in

Figs.10and11.Thefollowingnotationwasappliedtode-

scribeparticularoptimizationtasks:

•

Upper-caseletters–typeofinstabilityloading:B-Bi-

furcation,M-Multimodalbifurcation,S-Snap-through

loading,L-Lowercriticalload,U-Uppercriticalload-

ing (leading to exhaustion of carrying capacity),

F-Flutter load, O-denotes the absence of a relevant

formulation;

General classiﬁcation

Inthemodiﬁeddesignproblems,themostimportantde-

cision to be made is the choice of post-buckling con-

straints.Onecanimposetheseconstraintseitherlocally

(i.e. in the vicinity of the critical point) orfor the spec-

iﬁed rangeofageneralizeddisplacement.The latterap-

proachiscalled“extendedlocal”here.Ifconstraintsare

set for any speciﬁc value of a generalized displacement,

it is called a “global” approach. The concept of post-

bucklingconstraintsispresentedinFig.9.

The design variables in the modiﬁed design prob-

lemcanbechosenfromquantitiesthatdescribethestiﬀ-

ness of a structure, the shape of its cross-sectionor the

shapeofitsaxis,additionalactiveorpassive(additional

support)loads,andeventhebehaviouroftheloadafter

buckling.

Theobjectiveinthemodiﬁeddesignproblemisusu-

allythesameasinthestandardproblemofoptimization

againstinstability,i.e.,bifurcationorsnap-throughload.

Sincenonlinearanalysisisallowedfor,theobjectivecan

alsobechosenasthemaximalloadonthenonlinearpost-

buckling path or the minimal load if the maximal load

isabsent.Whendesignvariablesdonotaﬀectthebuck-

lingloadbutcanchangethepost-criticalbehaviour,the

objectivecanbechosenasaspeciﬁedfunction.

Selecting the objective now and implementing the

post-buckling constraints, many new modiﬁed design

tasks may be proposed. These modiﬁed problems for

•

Lower-case letters – type of formulation: s-standard

formulation,m-modiﬁedformulation;

•

Superscripts – e-elasticity (modiﬁed problems can

be formulated for inelastic instability and then p-

plasticity,c-creepareused), (1)-singlecriterionopti-

mization,(2)-multi-criteriaoptimization;

•

Subscripts – 2-second order bifurcation, o-objective

function diﬀerent from critical load, d-displacement

forsnap-throughloadastheobjective;

•

Lower-caseletters in parentheses – type of approach

for post-buckling constraints:(l)-local approach,(f)-

extended local approach (for ﬁnite interval), (g)-

globalapproach.

Detailed formulations

Based on the presented classiﬁcation and following the

proposedoptimization concept,detailed formulationsof

selected nonlinear problems of design for post-buckling

behaviour are given. The particular tasks are deﬁned

withinthegroupsofproblemsspeciﬁedinSect.3.Math-

ematicalformulaeforthosetasksarepresented,aswellas

a graphical illustration of each subproblem. The ﬁgures

show the results of application of the modiﬁed formula-

tioncomparedwiththeresultsofthestandardoptimiza-

tion.

4.1

Structuraloptimizationagainst instabilityleading

to maximizationof single buckling load

Maximization of the bifurcation load subject to a con-

stanttotalvolumefortheoptimizedstructureisastan-

dardproblemofoptimizationunderstabilityconstraints:

maximize p cr ( a i ) ,

subjectto V ( a i )= V 0 .

(20)

Fig. 9 Local, extended local, and global post-buckling con-

straints

In(20), a i standsforthedesignvariablesand V isthevol-

umeofthestructure.Thestandardproblemisnowmodi-

ﬁed by implementing suitable post-buckling constraints

eitherinlocalorinextendedlocalform.Bothsymmetric

andasymmetricbifurcationaretakenintoaccount.

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