Mixed Variable Formulations for Truss Topology Optimization

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Tampereen teknillinen yliopisto. Julkaisu 1134

Tampere University of Technology. Publication 1134

Kristo Mela

Mixed Variable Formulations for Truss Topology

Optimization

Thesis for the degree of Doctor of Science in Technology to be presented with due

permission for public examination and criticism in Konetalo Building, Auditorium K1702,

at Tampere University of Technology, on the 20

th

_{of June 2013, at 12 noon.}

Tampereen teknillinen yliopisto - Tampere University of Technology

Tampere 2013

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ISBN 978-952-15-3076-0 (printed)

ISBN 978-952-15-3116-3 (PDF)

ISSN 1459-2045

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A study on formulating truss topologyoptimization problems using ontinuous and binary variables is presented. The ground stru ture approa h, where members and nodes are allowedto vanish from an initial dense truss, is adopted. Member ross-se tionsare hosenfrom adis reteset ofalternatives. Thebinaryvariablesare used todeterminetheexisten eof membersand nodesas wellas thesele tionofaprole for the trussmembers. Normal for es of the members and nodal displa ements are hosen as ontinuous variables. The equations of stru tural analysis are written as onstraintsof theoptimizationproblem. Further onstraintsensurethatthe trussis able to arry the loads and is kinemati ally stable. Memberstrength and bu kling onstraintsareformulateda ordingtothedesignrulesofEuro ode3.

Theaimoftheoptimization problems onsideredis tonde onomi altrussdesigns. The weight of the truss serves as the default riterion as it an be easily evaluated anditisrelatedtothetotal ost. However,itiswell-knownthatthea tualminimum ostdesign andierfromtheminimumweighttruss. Therefore,afeature-based ost fun tion is also devised for tubular plane trusses for ost optimization. For design situations, where the ost data is not available, a multi riterion optimization prob-lem where weight is minimized simultaneously with the number of truss members, nodesandprolesusedin thedesign,isformulatedandParetooptimalsolutionsare generated.

Theproposedformulationsleadtomixed-integerlinearoptimizationproblems. State-of-the-artsoftwareis employedto solvea setof ben hmarkproblems that verify the formulationsanddemonstratetheee tofdierent onstraints. Optimumtopologies forEuler bu kling andEuro ode3bu kling onstraintsare ompared. Topology op-timizationofarooftrussispresentedasa asestudy. The oni tofweightand ost isstudiedin onju tionwiththerooftruss.

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Thisstudywas ondu tedfrom thebeginningof 2009to thelastdaysof2012. From therathervaguetopi topologyoptimizationwithstress onstraints,thesubje twas rstnarroweddowntotrussesandlaterspe iedtotrusstopologyoptimizationwith dis rete ross-se tionsusingmixed variableformulations.

Nothinggrowsinthevoid,andI'mgrateful forhavingso manypeoplefrom dierent ba kgroundshelpingme alongthe way. Le turer Risto Silvennoinen(Departmentof Mathemati s,TUT)deservesthanks for ignitingthe sparkforoptimization. Emeri-tus Professor Juhani Koskitransformed that spark into ame by introdu ing me to therealmof stru turaloptimization andengineeringme hani s. Hisdeep knowledge onstru tural optimization and ourmyriad dis ussionshave preparedme to be ome anindependent resear her, and I feel privileged for theopportunity to ondu t this resear hunderhissupervision.

AfterProfessorKoskiretired,ProfessorArtoLehtovaara(DepartmentofEngineering Design, TUT) took over as my supervisor. Even though ourresear h elds did not oin ide,heprovidedsoundadvi eforpreparingthemanus riptthroughout. Our reg-ularmeetingsandhisdevotionhelpedmetokeeptheworkorganizedandins hedule.

In late 2010, the thesis got an unexpe ted thrust that gavethe resear h wonderful momentum and a new and interesting dire tion. I got a quainted with Professor MarkkuHeinisuo (DepartmentofStru turalEngineering,TUT),whointrodu ed me to the Euro odes and other aspe ts of pra ti ing stru tural engineer. In addition to providing supporting funding for my resear h, his open-mindedness, rm belief in thepotentialof stru turaloptimization, and hisvaluable pra ti e-orientedadvi e en ouragedme to venturedeeperinto theeld of stru turaldesign. His help wasso remarkablethateventuallyhebe amemyothersupervisor,whi hI amgratefulfor.

In autumn2011, I de ided it was ne essaryto seek outsidehelp in order to hasten things. I onta tedSeniorS ientistMathiasStolpe(Te hni alUniversityofDenmark (DTU),DTU WindEnergy), whose earlierwork onmixed variableformulationshad inspiredme. DuringmytwoshortstaysatDTUthat hehosted,Iwasabletoa hieve themain resultsofthethesis. Mathias'un onditionalhelp andguidan e,as ifI was oneof hisPhDStudents,wasastoundingandasignofatrues ientist. Iamgreatly indebtedtohis eortsin aidingmebysharinghisexperien esandthoughts.

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ofEngineeringMe hani s. I wouldliketothanktheleaderoftheprogram,Professor JukkaTuhkuri(DepartmentofAppliedMe hani s,AaltoUniversity),forthisunique opportunity to do basi resear h and to meet other graduate students from other Finnishuniversities.

I havebeenfortunate to havethe endlesssupport from myfriendsand family. I am forevergratefulto my parentsSinikkaand Kari fortheiren ouragement,and to my brothers, JaakkoandMikko,forlisteningtomyrantsovertheyears.

MywarmestthanksgotoSnow hangeCooperativeforshowingmeotherwaysofbeing in theworldandremindingmethat notallknowledgeisin thebooks.

Finally,IexpressmygratitudeformydearwifeJenni,thear hite tofmyheart,who never easedtobelieveandunderstand.

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Abstra t iii Prefa e v Nomen lature xi 1 Introdu tion 1 1.1 Ba kground . . . 1 1.2 LiteratureReview . . . 4

1.2.1 GeneralTheoreti alResults . . . 4

1.2.2 Dis reteCross-Se tionsand DesignCodes . . . 7

1.2.3 MixedVariableFormulations . . . 8

1.2.4 Dis ussion . . . 9

1.3 S opeandAimsoftheThesis . . . 10

1.4 MainContribution . . . 11

2 Design ofTubularTrusses 13 2.1 Introdu tion. . . 13

2.2 Stru turalAnalysis . . . 14

2.3 Classi ationofCross-Se tions . . . 15

2.4 Resistan eofCross-Se tions . . . 17

2.5 Bu klingResistan eofMembers . . . 17

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2.6.1 DesignofJoints . . . 18

2.6.2 ChordsasContinuousBeams . . . 19

2.7 Dis ussion . . . 19

3 MixedVariableFormulations for Dis reteSe tions 21 3.1 Introdu tion. . . 21 3.2 Variables . . . 23 3.2.1 ProleSele tion . . . 23 3.2.2 MemberFor es . . . 24 3.2.3 NodalVariables. . . 25 3.3 Constraints . . . 27 3.3.1 NodalEquilibrium . . . 27 3.3.2 StrengthConstraints . . . 30 3.3.3 StabilityConstraints . . . 31 3.3.4 MemberGrouping . . . 37 3.4 Criteria . . . 37 3.4.1 Weight. . . 38 3.4.2 NumberofMembers . . . 39 3.4.3 NumberofProles . . . 39 3.4.4 NumberofConne tions . . . 40 3.4.5 Cost . . . 40 3.5 AlternativeFormulations. . . 47 3.5.1 Formulation1. . . 48 3.5.2 Formulation2. . . 49 3.5.3 Formulation3. . . 50 3.6 Dis ussion . . . 51

4 Solving MixedVariable Problems 53 4.1 Introdu tion. . . 53

4.2 Bran h-and-Cut. . . 54

4.3 SolutionSoftware . . . 57

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5 Ben hmark Problems 59

5.1 Introdu tion. . . 59

5.2 CantileverTruss . . . 60

5.2.1 The2-by-2GroundStru ture . . . 62

5.2.2 The4-by-4GroundStru ture . . . 66

5.3 L-ShapedTruss . . . 70 5.3.1 Aluminium Members. . . 70 5.3.2 Steel Members . . . 74 5.4 Dis ussion . . . 78 6 Multi riterion Formulations 81 6.1 Introdu tion. . . 81 6.2 ProblemStatements . . . 83 6.3 Coni tofCriteria . . . 84

6.4 GeneratingParetoOptimalSolutions. . . 86

6.5 Multi riterionOptimization ofL-ShapedTruss . . . 87

6.6 Dis ussion . . . 92

7 Case Study: Design ofRoof Truss 95 7.1 Introdu tion. . . 95

7.2 ProblemDes ription . . . 96

7.3 Results. . . 98

7.4 Dis ussion . . . 102

8 Summaryand Con lusions 109 A Sele tion of Proles 113 B Ground Stru tures 115 B.1 CantileverTruss . . . 115 B.2 L-ShapedTruss . . . 117 B.3 RoofTruss . . . 118 B.3.1 GroundStru ture1 . . . 118 B.3.2 GroundStru ture2 . . . 119 B.3.3 GroundStru ture3 . . . 120 B.3.4 GroundStru ture4 . . . 121 B.3.5 GroundStru ture5 . . . 122 B.3.6 GroundStru ture6 . . . 124

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Abbreviations

DM De isionmaker (page86)

LP Linearprogramming (page4)

MILP Mixed-integerlinearprogramming (page8)

MINLP Mixed-integernonlinearprogramming (page8)

NAND Nestedanalysis anddesign (page22)

NLP Nonlinearprogramming (page4)

RHS Re tangularhollowse tion (page14)

SAND Simultaneous analysisanddesign (page5)

Weldingenergy ost[

e

h

Horizontal(solid)partofthesawn ross-se tion[mm

2

℄ (page42)

A

t

Cross-se tionalareaof asawnprole[mm

2

℄ (page43)

c

M,i

Unit ostofatubular prole[

e

/kg℄ (page41)

F

sp

Thi kness-dependentsawbladedurabilityparameter (page43)

F

s

Sawbladedurabilityparameter (page43)

p

SB

Pri eofthesawblade[

e

℄ (page43)

Q

Sawinge ien y[mm

2

/min℄ (page42)

S

Verti alfeedingspeedofsawblade[mm/min℄ (page42)

S

m

Materialfa torforsawing (page42)

S

t

Durabilityofthesawblade[mm

2

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Sets

A

Setofavailable ross-se tionalareas (page23)

J

c

Interiornodesof hain

c

(page33)

E

V

c

Indi esofnodesof hain

c

(page33)

E

o

c

(s)

Membersof hain

c

overlappingthenode

v

s

(page35)

N

c

by onse utivenodes

v

s

and

v

s+1

(page33)

C

Setof hains (page33)

G

N

Groundstru turenodes (page23)

P

Available proles (page23)

N

S

Setofsupportednodes (page26)

Me hani s

α

Imperfe tion fa tor (page18)

ˆ

B

Extendedstati smatrixofatruss (page28)

χ

Redu tionfa torforbu kling (page17)

ε

k

i

Axialstrainofmember

i

inloading ondition

k

(page15)

f

y

Yieldstrength (page16)

B

Stati smatrixofatruss (page15)

p

γ

M0

Partialsafetyfa torforresistan eof ross-se tions (page17)

γ

M1

Partialsafetyfa torforresistan eofmembersto bu kling (page17)

¯

λ

Non-dimensionalslenderness (page18)

σ

k

i

˜

N

Memberfor esin theauxiliaryloading ondition. (page36)

p

k

L

n

Bu klinglengthofatrussmember (page18)

N

k

i

W

Weightofthetruss (page38)

W

i

Weightofmember

i

[kg℄ (page41)

Numbers

N

B

Numberof binaryvariables (page72)

N

C

Numberof ontinuousvariables (page72)

n

d

Numberof nodaldegreesoffreedom (page15)

n

E

Numberof groundstru turemembers (page15)

N

Outersurfa eareaofprole

j

perunitlength[mm 2

/mm℄ (page46)

c

Sidelengthofthehollowpartofare tangularse tion (page16)

H

Sidelengthofare tangularse tion (page16)

ˆ

I

Anavailablemomentofinertia (page23)

t

Wallthi knessofahollowse tion (page16)

A

i

Cross-se tionalareaofatrussmember (page15)

I

Momentofinertia (page18)

I

Variables

α

j

Binaryauxiliaryvariableforprole ounting (page39)

ˆ

u

Variableforsele tingaproleforagroupofmembers (page37)

z

ℓ

Nodalvariable(binary) (page25)

y

ij

Binaryvariablesforprolesele tion (page23)

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Introdu tion

Thequestion begstheanswer anyouforgivemesomehow?

TomWaits

1.1 Ba kground

Stru turaldesignisataskthatrequiresexpertiseinstru turalme hani s,engineering ingenuity,and reative ollaborationwithotherdis iplinesinvolvedin thedesign pro- ess. Intrussdesign,thegoalistondane onomi alstru turewhi hisableto arry thegiven loadsand whi h an bemanufa turedby available te hnologies. Thisgoal is often pursuedby trial and error, where the designer gradually modies an initial stru ture. Whilethisdesignmethodology anbee ientinsimpleandtothedesigner familiar ases,it an beverytime- onsumingformore omplexsituations,espe ially ifanentirelynew on eptualdesignisdesired. Forsu hinstan es,amoresystemati approa hshouldbe onsidered.

Stru turaloptimizationisaresear heldwhi hprovidesane ienttoolfor reating a synthesis of design, fabri ation and e onomy of stru tures. By formulating the designtask at hand as an optimization problem, e onomi alsolutions an be found systemati allybynumeri aloptimization algorithms. Thisapproa hhasat leastthe followingbenetsoverthetraditionaldesignmethodology:

•

Any design aspe t or quantity that an be expressed mathemati ally an be taken into a ount. Thus, omplex stru tural systems, where dependen ies of dierentquantitiesaredi ultforthedesigner toquantify, anbe onsidered.

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•

Thesolutionis notentirelydependent onthe experien eof thedesigner. This meansthatnewe onomi alsolutionsthatarebeyondtheintuitionofthedesigner an be found. Obviously, thedesigner is stillneeded forameaningfulproblem formulationandinterpretationof theresults.

•

Thetimeneededto nde onomi aldesignsisredu ed.

Trussesareespe ially suitableforoptimization,sin etheiranalysis issimple(if they are onsideredas pin-jointedstru tureswith loads atthenodes),andtheypossessa great deal of mathemati al stru ture that an be exploited in the solution pro ess. Trusses are also frequently used in pra ti al appli ations, for example, in ivil and aerospa eengineering.

Trussoptimizationproblemsaregenerallydividedintothree ategories. Before intro-du ing them, aremark on the terminology is in pla e. Inthe literature, the terms layout, onguration,topology,geometry,andshapeofthetrussare ommonlyused. Inthisthesis,thesetermsaredenedas follows.

The layout of the truss means the number of nodes and members, the lo ation of the nodes, and the onne tivity of the members. The topology of the truss onsists of the member onne tivity, whi h in ludes the number of nodes as well. However, topologydoesnot ontaininformationaboutnodalposition. Thegeometryofthetruss means thenode lo ationsfor xedtopology. The terms onguration and shape are equivalentto layout and geometry,respe tively,but theyarenotusedinthisthesis.

Thesimplesttrussoptimizationproblemis sizingoptimization (Fig.1.1a), wherethe optimum ross-se tions of truss members are to be determined for a xed layout. This problem an be extended in twodire tions, both leadingto substantiallymore ompli ated formulations.

Ingeometryoptimization(Fig.1.1b),theoptimumlo ationsofsele tedjointsaretobe determined in additionto member ross-se tions. Thetopologyofthetruss remains xed during optimization. Introdu ing nodal oordinates as designvariablesgreatly in reasesthenonlinearityoftheoptimizationproblem,makingthenumeri alsolution substantiallymoredi ult.

The other extension of sizing optimization is topology optimization (Fig. 1.1 ). As suggestedbythename,thegoalistodeterminetheoptimumtopologyforgivenloads, supports, and material properties. Even though the topology of the truss does not in lude informationabout thelo ationof thenodes,the optimum topologydepends strongly on where the nodes are pla ed. Thus, determining the optimum topology automati ally in ludes nding theoptimumnode positions as well as member ross-se tions. Consequently,byoptimizingthetopology,theoptimumlayout isalsofound. Inthisthesis,onlyproblemsoftopologyoptimizationare onsidered.

Topologyoptimization anbeunderstoodasanextensionofsizingoptimization,ifthe so- alledgroundstru tureapproa h (Dorn,Gomory&Greenberg1964)isadopted. In the groundstru ture approa h, aninitial truss, alled thegroundstru ture, with an ex essivenumberof membersand jointsis employed. Duringoptimization the joint lo ations are xed, but members(andjoints) areallowedto vanish. This pro edure an then beviewed as sizingoptimization with zero ross-se tionallowed. However, removingmembersfromthegroundstru tureleadstoserioustheoreti alandnumeri al

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F

(a)Sizingoptimization.

h

F

α

F

(b)Geometryoptimization.

F

( )Topologyoptimization.

Figure 1.1: Trussoptimizationproblemtypes.

di ulties that maketopology optimization themost di ultoptimization problem intrussoptimization.

Topology optimization an also be performed on more general stru tures that are modelledusing ontinuumme hani s. Thegoalis,roughlyspeaking,todeterminethe optimalnumber,lo ationandshapeoftheholesandouterboundaryofthestru ture. Thisapproa hdiersfromtrusstopologyoptimizationsubstantially,eventhoughthere aresomesimilarissuesaswell. Topologyoptimizationof ontinuumstru turesisnot onsideredinthisthesis. Foranoverviewofthetopi ,see(Es henauer&Olho2001) and(Bendsøe&Sigmund2003).

Theworkof thedesignerisregulatedbylawandaseriesof design odes. For exam-ple,theEuro odesprovidemandatoryrulesfordesigningstru turesintheEuropean Union. Therefore,inordertomaketheresultsofoptimizationappli able,the require-mentsof therespe tivedesign odesshould be in ludedin theproblem formulation. Any requirementthat isex luded from optimization must be he ked separately for thesolution. Ifsomerequirementsareviolated,thesolutionofoptimizationmustbe modiedappropriately.

Ontheotherhand,ifoptimizationisappliedtonda on eptualdesign,oradraftof thedesignis neededqui kly, simplied problemformulationsthat areeasierto solve anbeemployed. Insu hinstan es,itisa knowledgedthattheresultofoptimization hastobemodied, butoptimization provides agoodstartingpointformorerened design. This approa h isoftentheonlypossibilitytoapply optimizationin pra ti e, as taking into a ount allthe ne essaryrequirementswould leadto an optimization problem that is intra table by urrent solution methods. By pertinent resear h on stru tural optimization, the gap between the stru tural designer and theresear her

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anbegraduallynarrowed. Thisisespe iallythe asefortrusstopologyoptimization, whi hholdsagreatpotentialin produ ingnew on eptualdesigns.

1.2 Literature Review

Extensiveresear hontrusstopologyoptimizationhasbeen arriedoutsin ethelatter partofthe20th entury,espe iallysin ethe1980s. Startingfromthesimplest prob-lem formulations, the theoreti al pitfalls lurking within the subje t as well as some propertiesofoptimumtopologieshavebeenun overed. Furtherresear hhasextended the problem formulations, and methods for ir umventing thetheoreti al di ulties havebeenproposed.

Elsewhere, stru tural optimization under design odes has been studied. This re-sear hrarely onsiders trusstopologyoptimization, butprovidesvaluableinsightfor extendingthe onventionalformulationsoftopologyoptimizationtoin ludethedesign odes.

In order to pro eed with the resear h, the results and a hievements of the resear h ommunityneedtobeunderstood.

1.2.1 General Theoreti al Results

Theearliestproblemthathasbeen onsideredintheliteratureistheminimizationof weight of the truss with onstraintson member strength. The ross-se tionalareas ofthegroundstru turemembersandtheirnormalfor esaretakenasthe ontinuous design variables. If thekinemati ompatibility onditionsare negle ted, theweight minimizationproblem an onsequentlybewrittenasalinearprogramming(LP) prob-lem (Dornet al.1964). For athoroughreviewofthis formulationanditsproperties, see Kirs h(1989)andRozvany,Bendsøe&Kirs h(1995).

Under a single loading ondition, the minimum weight topology is stati ally deter-minate (Sved 1954, Barta 1957, Dorn et al. 1964, Fleron 1964). As a stati ally de-terminate stru ture satises the ompatibility onditions automati ally, the a tual minimum weight design for stress onstraints an be found by the LP formulation. However, under multiple loading onditions, the minimum weight truss is typi ally stati ally indeterminate. Inthis ase,theLP formulationisunable to ndthe opti-mumsolution,providingonlyalowerbound fortheminimumweight.

Ifstati allydeterminateoptimumtopology annotbeguaranteed,thekinemati om-patibility onditions must be in luded in the problem formulation. For ontinuous ross-se tions, this leads to a nonlinear programming (NLP) formulation (Cheng & Guo 1997, Guo, Cheng & Yamazaki 2001, Stolpe 2004), whi h imposes both theo-reti al and numeri alissues. The theoreti al di ulties are solely due to vanishing members.

Therstobservationthatneedsattentionistheo urren eofsingularandlo aloptima (Kirs h 1990, Kirs h 1993). The feasibleset usually ontainsdegenerate parts with dimension smallerthanthe dimensionofthe feasibleset. These parts orrespondto designs,where membershavebeeneliminatedfromthegroundstru ture. Theglobal

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optimummay lie in su h a degenerate part. Unfortunately, gradient-basedsolution methodsthat are ommonlyused,areunableto ndsolutionsin theseparts. This is duetothefa tthatthe onstraintquali ationoftheKarush-Kuhn-Tu ker onditions doesnotholdinthedegenerateregions. Lo aloptimaappearasa onsequen eofthe non onvexityoftheproblem.

The essential problem with vanishing members is that the stru tural model is not alteredduring theoptimization. Inthe niteelementsetting, that is frequently em-ployed,thismeansthatthestru turein ludesmemberswithzero ross-se tionalarea, whi h anleadtoasingularstinessmatrix. Bruns(2006)identiestwo ases,where thesingularitymayappear: theremovalofmembersmayleadtoanisolatednodeor anisolatedelement. Bothofthese asesleadtosingularstinessmatrix,whi h auses thenumeri aloptimizationpro edure tohalt.

Thesingularityofthestinessmatrix anbe ir umventedbyimposingasmallpositive lowerboundonmemberareas. Atthesolution,membershavingthislowerboundarea areremovedfromthedesign,ee tivelyroundingthememberareastozero. However, this approa h may fail to nd the global optimum, as explained by Cheng & Guo (1997).

Anotherapproa hfortreatingthesingularityofthestinessmatrixistoformulatethe optimization problema ordingto the prin ipleofsimultaneous analysis anddesign (SAND),wherethenodaldispla ementsaretakenasvariablesintheoptimizationand thestinessequationis onsideredasasetofequality onstraints(Sankaranarayanan, Haftka&Kapania1994). Su haformulationrequiresspe ialsolutionmethods. Con-sequently,thetheoreti alissueswith vanishingmembersaretransferred to omputa-tional matters whi h an be approa hed bydevelopingnumeri aloptimization algo-rithms.

Even ifthestru ture that remainsafter vanishingmembersareremovedwouldhave anon-singularstiness matrix, thevanishing members ause severeproblems to the optimization. Arguablythe mostseriousissue appears with onstraintsthat depend onthestressofthemembers.

It was observedby Sved &Ginos (1968)that the omputationalvalue of the stress of avanishing membermay be non-zero whi h may leadto erroneous results, when gradient-basedoptimizationmethodsareapplied. Cheng&Jiang(1992)investigated thisissuefurtherandshowedthatthestressisadis ontinuousfun tionofthemember areaat zero. They alsoshowed that forsmall valuesof the ross-se tionalarea,the stress in a member an be too high, but as the member is removed, the resulting stru ture an be feasible. This observation provides another obsta le for gradient-basedsolutionmethods.

Toavoidtheproblemsasso iatedwithstress onstraintsandsingularoptima,several approa heshavebeenproposed. Cheng&Jiang(1992)suggestmultiplyingthestress onstraintfun tionbyaso- alledqualityfun tion,whi hhasthefollowingproperties: it is ontinuous, zero at zero ross-se tion and positive for positive ross-se tional areas. This modi ation removes the dis ontinuity issue. However, the feasible set stillin ludesdegenerateparts.

In order to allow numeri al optimization methods to nd solutions in these parts, Cheng& Guo(1997) introdu ed aso- alled

ǫ

-relaxation te hnique, where the stress

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onstraint

g(x) ≤ 0

A htziger(1999a)dis ussesbu kling onstraintsfor ir ularandsquarese tions,and without ompatibility onditions. Througha arefuldenitionof hains ( onse utive memberslayingonaline),a onstraintfortopologi al bu kling isintrodu ed. Inthis onstraint,thenode an ellation istakenintoa ountbydening thea tivebu kling length. However,as A htzigerpointsout,this problemformulationmaynothavean optimalsolution. Thisisduetothedis ontinuityofthea tivebu klinglength,whi h leadstonon- losedfeasibleset. Thisphenomenon anbe ir umventedbyaddingthe so- alledslenderness ondition (apositivelowerboundonmemberareasformembers present in the topology). The numeri al treatment is alleviated by noti ing that all members in a hain have the same axial for e. Then, the ompli ated bu kling onstraint needs to be introdu ed for only one member of that hain, while for the others, asimpleside- onstraintforthememberareaissu ient.

In the se ond part of the paper, A htziger (1999b) proposes a numeri al solution

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bu kling onstraints are transformed into regular onstraints by an approximation parameter, and the resulting problem is solved by asequential linearprogramming method. Thenumeri alexampleshowthat in ludingtopologi al stabilityleadsto a dierenttopologythanthesimplebu kling onstraints.

To ta kle the problem of unstable topologies that often appear with bu kling on-straints,Guo,Cheng&Olho(2005)applythefa tthatanunstabletrusshasazero riti alloadfa tor. Byaddingapositivelowerbound forthe riti alloadfa tor, un-stablesolutionsareavoided. Determiningthe riti alloadfa torrequiresthesolution ofthegeneralizedeigenvalueproblemoflinearstabilitytheory.

1.2.2 Dis rete Cross-Se tions and Design Codes

As a rst step towards a tual design situations, the assumption that the member areasare ontinuousisalteredsu hthatonlyanitesetofpredeneddis retevalues for ross-se tionalareasis available. This hange in problem formulation leadsto a dis rete optimization problem. The main impli ation is that the solution methods of ontinuousoptimizationbe omeinappli able, anddierentapproa hesareneeded. Surveysofmethodsfordis retestru turaloptimizationareprovidedbyArora,Huang &Hsieh (1994), Thanedar & Vanderplaats (1995) and Arora (2002). Textbooks on dis reteand mixed variableoptimization in lude thosebyFloudas (1995)(nonlinear problems)andNemhauser&Wolsey(1999)(linearproblems).

Tofurthertakeintoa ounttheneedsofthestru turaldesigner,design ode require-ments an be in luded in the optimization problem as onstraints. Galante (1996), Dominguez,Stiharu&Sedaghati(2006),andBalling,Briggs&Gillman(2006)in lude member bu kling and ross-se tions a ording to the AISC design ode. Erbatur, Hasançebi, Tütün ü &Klç (2000) optimizetrusses a ordingto the Turkish ode, Pedersen &Nielsen (2003)formulate sizing andgeometry optimization a ordingto theDanish ode,andShea&Smith(2006)in orporatetheSwiss odefortransmission toweroptimization. Inmost ases,the stru turesareanalyzed aspin-jointedtrusses withloadsatthenodes,su hthatbending,shear,andtorsionofthemembersarenot in luded.

In Europe, the Euro odes have be ome the unied design odes in the European

Union. Farkas&Jármai (1997,Chapter11)in lude onstraintsonmemberbu kling andjointstrengthforoptimizationoftubulartrussesa ordingto Euro ode. Similar formulations, where joint strength and e entri ity are taken into a ount, an be foundin(Farkas&Jármai2003, Farkas &Jármai2008). Jalkanen(2007)formulates tubulartruss optimization problems,where bending and torsionof themembersare in luded,aswellasjointstrength. ilih,Premrov&Kravanja(2005)optimizetimber trussesa ordingtoEuro ode5.

Inmost ases,wheredesign odesarein orporatedintheoptimizationproblem,only sizingand geometryoptimizationis onsidered. Ararestudy ontrusstopology opti-mizationunderEuro ode onstraintsispresentedbyilih&Kravanja(2008),butthe detailsoftheformulationarenotgiven.

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1.2.3 Mixed Variable Formulations

It was noted earlier, that the theoreti al issues related to the vanishing members an be ir umventedby employing theSAND formulation. However,as reported by Sankaranarayanan et al. (1994), onvergen e to the solution, where member areas a tually be ome zero is slow. Togain better ontrol overthe topology of the truss during optimization, binary variables an be introdu edto indi ate the existen eof membersandnodes. Inthe aseofdis retememberproles,binaryvariables analso beusedforprolesele tion.

Ghattas & Grossmann (1991) and Grossmann, Voudouris & Ghattas (1992) seem to be the rst to propose this approa h for truss topology optimization. As in the SANDformulation,theequationsofstru turalanalysisarein ludedas onstraintsin the optimization problem. However, in ontrast to Sankaranarayananet al. (1994), the equations of me hani s are written for ea h element separately. If the member areas are ontinuous, theproblem be omes nonlinear. Interestingly, a mixed-integer linear optimization (MILP) problem formulation is obtained in the ase of dis rete member areas. Member proles are sele ted by binary variables, whereas member for es,stressesandelongationsaswellasnodaldispla ementsare hosenas ontinuous statevariables.

A general framework for mixed variable formulations in stru tural optimization is presentedbyKravanja,Kravanja&Bedenik(1998). However,noexpli itformulations fortrusses arepresented.

Bollapragada,Ghattas&Hooker(2001)proposeaformulationfordis retesizingand topologyoptimization,wherethesele tionofthememberproleisstatedasalogi al disjun tion. Then, a logi -based bran h and bound algorithm is proposed, where a quasi-relaxation problemthat is formulatedas anLP problem issolved sequentially. Lower bounds are obtained by the solutions of the quasi-relaxation problem, and upperboundsareobtained,ifthebinaryvariablestakeintegervaluesatthesolution. Bran hingisperformedwithrespe tto memberareavariables,andalsotheso- alled logi uts an beemployed. Thebenetof thelogi -basedapproa his thefa t that the quasi-relaxation problem is substantially smaller than the MILP formulationof thetrussoptimizationproblem.

As mentionedabove,inthe aseof ontinuousmemberareas,themixedvariable for-mulationleadstoamixed-integernonlinearprogramming(MINLP)problem,whi his non onvex(Stolpe2004,Ohsaki&Katoh2005). Thenonlinearitiesappearasbilinear termsinequality onstraints. IfEulerbu klingisin luded,univariate on avetermsin inequality onstraintsappearaswell. Anonlinearbran h-and-bound algorithm anbe appliedtondtheglobaloptimumoftrusstopologyoptimizationproblems. However, these problemsareverydi ulttosolvetoglobaloptimum.

Re ently,theMILPformulationresultingfordis retememberareashasbeenstudied and extended (Faustino, Júdi e, Ribeiro & Neves 2006, Rasmussen & Stolpe 2008, Kanno &Guo2010). Eventhoughwell-establishedsoftwareexistsforthese formula-tions,andtherearelessdi ultiesinndingtheglobaloptimumthaninthe ontinu-ous(nonlinear) ase,onlyrelativelysmallproblemshavebeensolvedintheliterature. This indi atesaneedforfurtherdevelopmentofthisapproa h.

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The binary variables that indi ate member and node existen e an be employed to express topologi al properties of the truss in the form of linear onstraints. Ohsaki & Katoh (2005) formulate onstraints for the minimum and maximum number of membersthat an be onne ted to an existing node. A dierent set of onstraints disallow members from interse ting ea h other. Rasmussen &Stolpe(2008) present inequality onstraintsthat expli itlyensurethatthestru tureisstati allyfeasible.

Aseriousdrawba kofthe onventionalformulationsoftopologyoptimizationisthat theoptimumstru turemaybekinemati allyunstable(Dornetal.1964,Kirs h1989). This meansthat the stru tureis in equilibrium with respe t to thegivenloads,but it is unstable with respe t to variations of the loads. This problem is presentboth in the LP and in the NLP formulations sin e the optimum solution of the two for-mulationsareequal,if theoptimumtopologyisstati allydeterminate. Thisissue is ommonlyhandled byintrodu ingsmallauxiliaryloadsatpredenednodes(Ben-Tal &Nemirovski1997, Rasmussen & Stolpe2008). The problem with thisapproa h is that it is very di ult to know a priori, whi h nodes will be present in the opti-maltopology. As allloadednodeswill bein ludedin theoptimumstru ture, adding auxiliaryloads to nodesthat arenotneeded in theoptimaltopologywilldistort the solution.

Theproblemofkinemati instability anbetreatedinthemixedvariableframeworkin anelegantmanner. Faustinoetal.(2006)writethene essary onditionforkinemati stability of the truss (Grubler's riterion) as a linear onstraint with respe t to the binary variables. To further enfor e kinemati stability, a new loading ondition is added to the problem, where ea h degree of freedom is given a small load that is applied, if the orresponding node is present. The truss is then required to be in equilibriumwith respe ttothese loads. Asimilarapproa hisproposed byKanno& Guo(2010).

1.2.4 Dis ussion

Themain observationfrom theliteraturereview is that trusstopology optimization problems with ontinuous member areaspossessesproperties that makestheir solu-tionverydi ult. Ifthenestedapproa his adopted,singularity phenomenarelated tovanishingmembersandproblemswithbu kling onstraintsposeseriousdi ulties, as wellas theissue of kinemati stability. The mixed variable formulations ir um-ventthese problems, but fa e theissues ofglobal optimization of MINLP problems. Presently, in the mixed variable framework, only relatively small problems an be solvedtoglobaloptimality.

Ifthememberprolesare hosenfrom apredened dis reteset, thenestedapproa h hastodealwiththesamedi ultiesasinthe ontinuous ase. Furthermore, gradient-basedsolutionmethodsbe omeunavailable,andalsodeterministi methodsofdis rete optimizationaredi ulttoapply,sin eanalyti alexpressionsofthe onstraint fun -tionsarenotavailable.Heuristi methods,su hasgeneti algorithms, anbeapplied tosolvedis rete topologyoptimizationproblems, but theyrequireahuge numberof stru turalanalyses,andarelimitedinthesizeofproblemsthey ansolve. Inthe dis- retesetting,themixedvariableformulationoftheSANDapproa hleadsto aMILP

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problem thatdoesnotsuerfromthesingularityissuesor othertheoreti alproblems oftopologyoptimization.

Even thoughtruss topologyoptimization hasbeenstudied forde ades, there isstill anobviousdemandforfurtherresear h. Forthegeneraldevelopment,theissueswith bu kling length and kinemati stabilityla k a satisfyingsolution. In order to make topology optimization more dire tly appli able for stru tural designers,the require-ments of the design odes should be introdu ed in the problem formulation where possible. Thepresentwork is astudy of trusstopologyoptimization with regardto these questions.

1.3 S ope and Aims of the Thesis

The main fo us of the thesis is the mixed variable approa h for truss topology op-timization with dis rete member ross-se tions. This approa h was hosen due to followingreasons:

•

the formulation is not sus eptible to singularities and other theoreti al issues ausedbyvanishingmembers;

•

ifthe solutionalgorithm ndsanoptimum solution,it isguaranteed to be the globaloptimum;

•

topologi al onstraints anbeexpressede ientlybybinaryvariables.

The rstgoalisto unifythemixed variable formulationspresentedin theliterature. Then, the formulationsare extended su h that memberbu klingand kinemati sta-bility of the truss are treated properly. Furthermore, requirements of Euro ode 3 are in orporatedintheproblem. Inthisthesis,memberstrength,stability, and sti-ness are onsidered,andthestru ture isanalyzed asapin-jointedtruss. Thus, joint strength and theee ts of bending and torsion are not in luded in the study. This restri tion learlyimpliesthatthesolutionprovidedbyoptimizationisnotne essarily appli abletothestru turaldesignerassu h, butmightneedfurther modi ationsto satisfytherequirementsthatwerenotin ludedintheproblemformulations.

Intruss optimization,typi allytheweightof thetrussisminimized, anditservesas thedefaultobje tivefun tioninthisthesisaswell. However,oftenthemostinteresting obje tivefun tionisthe ostofthestru ture. Inthisstudy,the ostfun tionofHaapio (2012) ismodiedfor tubular planetrusses and takenas an obje tivefun tion. For situations, where a tual ost data is not available, other quantities that ae t the ost are hosen as obje tive fun tions. These in lude the number of members, the number of joints, and the number of proles appearing in the truss. As none of these riteria arelikelyto yield asolutionthat is loseto the minimum ostdesign, multi riterion formulationsareproposed,wherethethree riteriaandtheweightare minimized simultaneously. Consequently, a set of Pareto optimal solutions, whi h representmathemati allybetterdesignsthantheotherfeasiblesolutions,isgenerated.

Afurtherinterestingissueistherelationshipbetweentheminimum ostandminimum weightstru tures. Thismatterisinvestigatedbyamulti riterionproblemformulation,

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where ostandmassofthetrussareminimized simultaneously. Bythisformulation, quantitativeinformationaboutthe oni tof ostandmassisobtainedinthe ontext oftopologyoptimization.

Thevariousformulationspresentedinthisstudyservetoanswerthefollowingresear h questions:

Q1. Whatpossibilitiesdoesthemixedvariableapproa hoerforsolvingthepresent openissuesoftrusstopologyoptimizationandfortakingintoa ounttheneeds ofthestru turaldesigner?

Q2. Is the mixed variable approa h a suitable tool for ost optimization? If the details ofa ostfun tionare notavailable,what otherpossibilitiesarethereto nde onomi alsolutions?

These ond majorgoal ofthe thesis is to investigatethe apability of modern algo-rithms to solve the optimization problems of the proposed formulation in pra ti al designsituations. Inthis study, ommer ialstate-of-the-artsoftwareisemployedfor solvingthemixedvariableproblems. Thispartofthethesiswillgiveguidelinesabout the asesthatare presentlysolvable. Theresear hquestionrelated tothenumeri al solutionpro essis:

Q3. What is thequalityof the solutionsthat an beobtained in giventime, when themixedvariableformulationsareappliedtopra ti aldesignsituations?

Thequality of the solution means, roughlyspeaking, how far the obtainedsolution is from the global optimum or its onservative approximation. Su h information is readilyavailable withthemethod appliedinthis thesis.

1.4 Main Contribution

Thestudy ontainsseveralresults. Inthefollowing,themostsigni ant ontributions arebrieydis ussed.

Uni ation of mixed variable formulations Ea h of the few papers dealing

withmixedvariableformulationsfortrusstopologyoptimizationemphasizesdierent aspe ts. Inthisthesis,theideaspresentedintheliteraturearebroughttogetherand uniedunderthesameproblemformulation.

Extensionsofbasi formulations Typi ally,theweightofthetrussisminimized subje ttostrength(stress) onstraints. Inthepresentstudy,thisbasi formulationis extended toin lude memberbu kling onstraints. BothEuler bu klingand bu kling a ording to Euro ode 3 are onsidered. The issue of the jump in bu kling length phenomenon is resolved by introdu ing hains to the groundstru ture. Chains are alsousedtoensurekinemati stabilityofthesolutionandtotreatlineloadingproperly.

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Multi riterion formulations E onomy of a design is generally not determined onlybyitsweight. Otherfa torsae tingthee onomyofthetrussareidentiedand written as riteria that are optimized simultaneously with the weight. The Pareto optimalsolutionsoftheresultingmulti riterionproblem anbefurther evaluated for ostandmanufa turingpurposes.

Computational studies The proposed formulations are veried by ben hmark

problems, most of them devised for the purposes of the thesis. The problem data and theresultsaregivenindetailsothey an beusedfor testingtheperforman eof future algorithms. Theappli abilityoftheformulationstopra ti aldesignsituations is exploredbya asestudy onsideringtheoptimizationofarooftruss.

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Design of Tubular Trusses

A ommonmistake thatpeoplemake whentrying todesign something ompletely foolproofisto underestimate theingenuityof omplete fools.

DouglasAdams

2.1 Introdu tion

In stru tural design a great number of de isions must be made, ranging from the hoi e of stru ture type and materials to the ne details of the onne tions. It is ommonlya epted,thatthede isionsmadeat the on eptualdesignphasehavethe strongestinuen eonthee onomyandperforman eofthestru ture. Intrussdesign, thismeansthatthelayoutofthetrussisthekeyelementfora hievingane onomi al stru ture. In this thesis, thes ope of designis limitedto determining theoptimum layout(topology)ofthetruss. Finerdetails,su hasjointdesign,arenot onsidered.

Adesignpro edurethat is ommonlydes ribedintheliterature(Trahair&Bradford 1988,Haapio 2012) an bestated fortrusses as follows. Based ontheloads and the spanorother geometri dataof thetruss,thedesigner hoosestheinitiallayoutand member se tions. Then, stru tural analysis is performed and the design rules are he ked. If some rules are violated or if the designer thinks that the solution an be improved, the member se tionsare modied. This iterativepro ess of stru tural analysisanddesignmodi ationisrepeateduntilthesolutionsatisesthedesignrules

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and ise onomi alenough. Finally,thedetailsofthetrussaredesignedsu hthat the truss an bemanufa tured.

Theabovedesignpro ess anbepartlyautomatedbythemeansofstru tural optimiza-tion. Thelevelofautomationdependsonwhi hofthethreeproblemtypespresented in Chapter1is hosen. Sizingoptimizationautomatestheloopofstru turalanalysis and veri ationofthedesign rules. Geometryandtopologyoptimization relievethe designer from xing the layoutof the truss, thus providing more exibility. Of the three problem types, topologyoptimization givesthe mostfreedom to the designer, sin eonlythegroundstru tureneedstobedeterminedinthebeginning.

For hoosing themember ross-se tionalternatives,manykindsof prolesare avail-able. Every ommer iallyavailableprolehasitsbenetsanddrawba ksin termsof me hani albehaviourandappli ations. Asane onomi aldesignismostoftensought, hoosing afavourable ross-se tiontypeis ru ial. Inthis thesis,theproles hosen forthetrussmembersaresquarehollowse tions(SHS)orre tangularhollow se tions (RHS) made ofsteel. However,the formulationspresentedare alsoappli able (with modi ations)forother ross-se tionshapesandmaterials. Inthis hapter,the anal-ysis and design of tubular trusses is onsidered as related to the thesis. Both the stru tural analysis model and the design ode requirementsare essentialfor formu-lating the optimization problems, as they provide the onstraintswhi h ensure the appli abilityofthesolution.

In truss design, sele tingthe appropriateshapeand size ofthe memberproles has agreat impa tonthee onomyof thestru ture. Tubular ross-se tionshavebe ome very popular due to their ex ellent me hani al properties whi h enable e onomi al designs. Presently, tubular trusses an be found in onstru tions su h as roofs of publi andindustrial buildingsandarenas,bridges,transmissiontowers,and ranes.

Compared to theirweight,hollowse tionshavehigh torsionaland bending stiness, and theyarewell-suitedfor ompression members. Hollowse tions arealsoless sus- eptibleto lateralandtorsionalbu kling. Also,theoutersurfa eof a losedshapeis relativelysmallwhi h,togetherwiththela kofsharp orners,redu esthe ostofre and orrosionprote tion.

A moredetailed dis ussion about the benets and appli ations of stru tural hollow se tions anbefoundin(Wardenier,Pa ker,Zhao&vanderVegte2010)and(Jalkanen 2007).

IntheEuropeanUnion,theEuro odesprovideaseries ofrulesthatasteeltrusshas to satisfyin any onstru tion(EN 199311 2005). Therules on ern thestrength, stability, durability, servi eability and re safety. In this work, the onstraints of theoptimizationproblem on erningmemberstrengthandbu klingarederivedfrom Euro ode3(EN1993112005). Inthefollowing,trussdesigna ordingtoEuro ode 3asrelatedtothepresentstudyisdis ussed.

2.2 Stru tural Analysis

A ordingtotheEuro ode,eitherelasti orplasti designmethodology anbeapplied in trussdesign(EN 1993112005, Clause 5). Inthis thesis,the stru turalanalysis

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is performed a ordingto the theory of linearelasti ity. Thetruss is modelled as a pin-jointed stru ture whi h is loaded at the nodes. Thus, the only stress resultant appearingin themembersisthenormalfor e.

Thebasisofstru turalanalysisisthedispla ementmethod,whi hisalsoimplemented inmostniteelementprograms. However,in thisthesis,thestinessequationofthe displa ementmethodisdisaggregatedinordertoavoidthesingularityofthestiness matrix, whi h is a ommon phenomenon in topology optimization as explained in Chapter1.

Intrussanalysis,thethree onditionsthatmustbesatisedare: equilibriumoffor es atthenodes, ompatibility onditions,andfor e-displa ementrelations.

Supposethetrussissubje tedto

n

L

loading onditions. Thenodalequilibrium equa-tions an bewrittenas

BN

k

= p

k

∀ k = 1, 2, . . . , n

L

(2.1)

where

B

∈ R

n

d

Theaxial strainofmember

i

in loading ondition

k

is

ε

k

i

=

1 L

i

b

T

i

u

k

(2.2)

where

L

i

isthelengthofthemember,

b

i

∈ R

n

d

isthe

i

th

olumnofthestati smatrix,

and

u

k

_{∈ R}

n

d

istheve torofglobalnodaldispla ementsinthe

k

th

loading ondition. ApplyingHooke'slawgivesthememberstress

σ

k

i

:

σ

k

i

= E

i

ε

k

i

=

E

i

L

i

b

T

i

u

k

(2.3)

where

E

i

N

k

i

=

E

i

A

i

L

i

b

T

i

u

k

(2.4)

where

A

i

isthe ross-se tionalareaofthemember. Eqs.(2.1)and(2.4) onstitutethe equationsofstru turalanalysisthatthetrussmustsatisfy. Alternatively,Eq.(2.3) an be usedalong withthe equation

N

k

i

= σ

k

i

A

i

insteadofEq. (2.4). Eq. (2.4)in ludes the ompatibility onditionsandfor e-displa ementrelations.

2.3 Classi ation of Cross-Se tions

Member ross-se tionsaredividedintofour lassesa ordingtotheroleoflo al bu k-linginlimitingtheresistan eandrotation apa ityofthese tion(EN1993112005, Clause5.5). Inthisthesis, lasses1,2and3are onsidered. Cross-se tionsin lass1 andeveloptheirplasti momentresistan ewiththerotation apa ityas requiredby plasti analysis before lo al bu kling, i.e. bu klingof the ross-se tionwalls,o urs.

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t

H

c

R

Figure 2.1: Squarehollowse tion dimensions. Thedimension

R

istheouter ra-diusof the ornerrounding. The length

c

is theside length ofthe hollowpart,fromwhi htheinnerradiihavebeensubtra ted.

Class 2 se tions an also develop their plasti moment resistan e, but the rotation apa ityat the plasti hinge is limited bylo al bu kling. For lass3 ross-se tions, theyieldstrength anberea hed,butlo albu klingo ursbeforetheplasti moment resistan eisattained.

The lass of a ross-se tion is dened by its geometry and the type of loading the se tionissubje tedto. Forre tangularse tions,thelimitinginequalityis

c

t

≤ Cǫ

(2.5)

where

t

is the wall thi kness of the se tion,

c

is the length of the hollow part of the se tionwhere theinner radiiof theroundings havebeensubtra tedas shownin Fig.2.1,

C

isa onstantdependingontheloadingtypeand

ǫ =

s

235 f

y (2.6) where

f

y

is theyield strengthofthematerialofthese tion. Thevalueof

C

depends on whether the partof the rossse tion is subje tto bending, ompression or their ombination. For example, re tangular ompression members belong to lass 3, if Eq. (2.5)issatisedwith

C = 43

.

Thedimension

c

appearinginEq.(2.5) anbe omputedby

c = H − 2R

(2.7)

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2.4 Resistan e of Cross-Se tions

Members under axial for e

N

Ed

( ompression) (2.9)

wherethedesignresistan e is

N

t,Rd

= N

= 1.0

givenin (EN1993112005, lause6.1(1)) isusedasstatedin theFinnishNationalAnnexoftheEuro ode(EC3NA2005).

If the ommonsign onventionis employed, where the normal for e in ompression is negative, Eqs. (2.8) and (2.9) an be ombined to a form, whi h is suitable for optimization:

− A

f

y

γ

M0

≤ N

Ed

≤ A

f

y

γ

M0 (2.11)

Eq. (2.11) is ommonly used in truss topologyoptimization to express the strength (stress) onstraints.

2.5 Bu kling Resistan e of Members

Stabilityisthepredominantphenomenonthatdi tatesthesizingof ompression mem-bers. Typi allyin theliterature ontruss optimization, Eulerbu kling is onsidered. However,as shownbyFarkas &Jármai (1997,Chapter9), designa ordingto Euler bu klingmightleadtounsafedesigns,sin etheee tsofinitial rookednessand resid-ualstresses arenot takeninto a ount. Therefore, the designrules of theEuro ode shouldbe onsidered.

A ordingtoEuro ode3,membersin ompressionmustsatisfythe ondition(EN1993 112005,Clause6.3.1)

and

χ

istheredu tionfa tor. Itsvalueisdeterminedfrom thenon-dimensional slen-derness,

λ

¯

,by

χ =

1 Φ +

p

Φ

2 _{− ¯}

_λ

2 ,

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where

Φ = 0.5

1 + α(¯

λ − 0.2) + ¯

λ

2

(2.15) and

¯

λ =

Intheabove,

I

isthemomentofinertiaofthe ross-se tionwithrespe ttothemajor prin ipal axis,and

L

n

isthebu klinglengthofthemember.

Theimperfe tionfa tor,

α

,inEq.(2.15)dependsonthe ross-se tion. For oldformed re tangularse tions,

α = 0.49

(EN 1993112005,Tables6.1and6.2).

Employingthesamesign onventionasabove,thebu kling onstraint anbeexpressed as

N

Ed

≥ −

χAf

y

γ

M1

(2.18)

A ordingtotheFinnishNationalAnnex(EC3NA2005),thevalue

γ

M1

= 1.0

stated in (EN1993112005, lause6.1(1))isused.

2.6 Other Design Aspe ts

Therequirementsformemberstrengthandbu kling onstitutetheprimary onstraints of the optimization problems presented later in the thesis. The Euro ode ontains further rulestoguaranteethesafetyofthestru ture. Theserulesarenotin ludedin theoptimizationproblems ofthisthesis. Inthefollowing,ashortdis ussiononsome ofthedesignaspe tsthathavebeennegle ted,ispresented.

2.6.1 Design of Joints

Ingeneral,jointbehaviourshouldbe onsideredearlyinthedesignpro ess(Wardenier etal.2010). Intubulartrusses,mostlyweldedjointsareused. In(EN1993182005), design rules for joint strength are given. Ea h joint type has its own design rules, whi h orrespond to the dierent failure modes of the joint. The design rules are given in terms of the normal for es of the bra ing members, and the weld sizes are then determinedasmultiplesofmemberwallthi kness.

Inordertoapplythejointdesignrules,thedetailsofthegeometryofthejointmustbe known. Ifthe enter linesofthe onne tingbra esdonotinterse tat the enterline of the hord, ane entri ity appears at thenode, whi h leadsto abending moment tothe hord. Thise entri itymustbetakenintoa ountforthe ompression hord, and alsoin thejointdesign,ifthee entri ityisgreaterthangivenbounds.

The greatest hallenge for in orporating the joint strength rules in topology opti-mization is the fa t that the design rules depend on the type of the joint, whi h is

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determinedby thenumberof membersjoining at anode. Intopologyoptimization, thenumberofmemberspresentatanodeisnotxedbutmayvary. Thus,thedesign ruletoapplyalsovariesduringoptimization. Ifadensegroundstru tureisemployed, keepingtra kofthemembersatanodeandenfor ingthe orre tdesignrulebe omes verytedious.

Consequently,thejointstrengthisnot onsideredin thisthesis. Apossibleapproa h forin ludingjointstrengthintheoptimizationistopro eedin twophases. First,the optimumtopologyisdetermined withoutjointstrength onstraints. Then,thejoints of thedesign obtained from topologyoptimization are he ked. If anyjoint fails to satisfythedesignrules,asizingoptimization isperformedforthexedtopologyand withjointstrength onstraints.

2.6.2 Chords as Continuous Beams

Inthestru turalmodelofthetrussthatisemployedin thisthesis,onlyaxialfor eof membersispresentandtheee ts ofbendingandtorsion arenegle ted. Thismodel isjustiedinmany ases. However,therearealsosituations,where bending and tor-sion need to be onsidered. For example, if transversal loads are applied elsewhere thanatthenodesorifthee entri ityofthenodesistoolarge,theresultingbending momentsneedtobetakenintoa ount. Also,inrooftrusses,the hordsare manufa -turedas longmembers,extendingoverthenodesatwhi hthe bra esare onne ted. Consequently, transversalloads ause bending moment in the hords. Insu h ases, itisre ommendedto onsiderthe hordsas ontinuousbeamstowhi hthebra esare onne tedbypinjoints(EN1993182005,Clause5.1.5)and(Wardenieretal.2010, pp.68). This requires aframe analysis of thestru ture, whi h is onsiderablymore involvedthantheanalysisoftrusses. As thes opeofthethesisisspe i allylimited totrusses,treatingthestru turesasframesisleftforfuturestudies.

2.7 Dis ussion

Intheliteratureontrusstopologyoptimization,thedesign odesares ar elyin luded intheproblemformulations. Onereasonforthis ouldbethatmostrulesotherthan the strength onstraints are mathemati ally di ult to handle in the optimization. Forexample,theexpressionsfor omputingtheredu tionfa torformemberbu kling, Eqs.(2.14)(2.17),areratherinvolved,Eq.(2.14)beingnon-dierentiableat

¯

λ = 0.2

. The joint design rules are also very di ult mathemati ally: expressions of some fa torsdependonwhetherthememberisintensionorin ompression,andthedesign rules depend on the type of the joint whi h may vary at the node during topology optimization.

Themixed variable formulations studied in this thesis allowto in orporate some of thedesignrulesoftheEuro odein theoptimizationproblem. Forexample,member bu kling redu es to a linear onstraint in the employed formulations. This thesis provides a starting point for further studies for in luding the dierent rules of the Euro odesandpossiblyotherdesign odesintrusstopologyoptimization.

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Mixed Variable Formulations for Dis rete Se tions

Mathemati ians arelikeFren hmen: whatever yousaytothemthey translate intotheirownlanguage, andforthwithitissomethingentirely dierent.

Goethe

3.1 Introdu tion

Inthis hapter, trusstopologyoptimization problems areformulated. Themain as-sumptionisthatthemember ross-se tionsare hosenfromapredenedsetofdis rete alternatives. This orrespondsto ommondesignsituations,wherethedesignermust hoose the proles from ommer ially available atalogue provided by the manufa -turer.

Theformulationof optimization problems onsists of three main omponents: vari-ables, obje tive fun tion, and onstraints. Thevariables are the quantities that an be altered in order to improve the obje tive fun tion whi h is either minimized or maximized. Thevariables anbe ategorizedtodesignvariables, whi h onstitutethe a tualdesign,andstatevariables, whi hdeterminethestateof thedesign. Common hoi esfordesignvariablesarethe ross-se tionalareasofthetrussmembers,whereas thememberfor esandnodaldispla ementsareoften hosenasstatevariables.

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An optimizationproblemiswritten instandardform as

min

x

f (x)

subje tto

g

i

(x) ≤ 0

i = 1, 2, . . . , q

h

j

(x) = 0

j = 1, 2, . . . , p

(3.1)

where

x

is the ve tor of designvariables,

f

is the obje tivefun tion, and

g

i

and

h

j

are the inequalityand equality onstraint fun tions,respe tively. Problem Eq. (3.1) an bewrittenmore ompa tlyas

min

x∈Ω

f (x)

(3.2)

where

Ω = { x | g

i

(x) ≤ 0, h

j

(x) ≤ 0, ∀i = 1, 2, . . . , q, j = 1, 2, . . . , p }

(3.3) is thefeasibleset. Thepointsof thefeasiblesetare alled feasible designs.

There are several approa hesfor formulatingstru tural optimization problems. The prevailingphilosophiesare thenestedanalysis and design (NAND), and the simulta-neousanalysis anddesign (SAND)approa h(Arora&Wang2005).

InNANDformulations,theoptimizationvariablesaresolelythedesignvariables,and all the responses,su h as displa ements, stresses, and internal for es are treated as impli it fun tions of the design variables. Ea h time these impli it fun tions need to be evaluated, a stru tural analysis is performed for given design variable values. Furthermore,bythemeansofsensitivity analysis,thegradientsoftheresponses an be omputed.

Iftheoptimizationproblemisformulateda ordingtotheSANDapproa h,thestate variablesarein ludedasoptimizationvariables,andtheequationsofstru tural anal-ysis are treatedas equality onstraints. Consequently, nostru tural analysis is per-formed during optimization, but the onstraintsguarantee that theoptimum design satisestheequationsofme hani s.

The two approa hes have been ompared by Arora & Wang (2005). The main ad-vantagesoftheNANDapproa harethatfewervariablesand onstraintsarein luded in the optimizationproblem and thatthe equilibriumequations aresatisedat ea h iteration. On the other hand, solving the equilibrium equations at ea h iteration and performingsensitivityanalysis an beverytime- onsuming. Furthermore,as the responses areknown onlyimpli itly as fun tions of the designvariables, the mathe-mati al propertiesofthesefun tions annotbefullyutilizedinoptimization.

TheSAND approa hdoesnotsuerfrom thedisadvantages oftheNANDapproa h. As theequationsof stru turalanalysis arenowin luded asequality onstraints, nei-ther a separate stru tural analysis nor sensitivity analysis needs to be performed. Furthermore, in some instan es, ertain onstraints be ome linear in the variables. As theanalyti alexpressions ofstru tural analysis areavailable,their mathemati al properties anbeexploitedinthesolutionpro ess. Ontheotherhand,thenumberof optimization variables and onstraintsbe omes largein theSAND approa h. Thus, large-s ale optimization algorithms must be used for most problems in the SAND formulation. Another disadvantageof the SAND approa h is that the intermediate

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solutionsmightnotsatisfytheequilibriumequations. Consequently, onlyatthe end oftheoptimization, ausabledesignisguaranteed.

Forfurther dis ussiononthedierentformulationapproa hesinstru tural optimiza-tion,see(Arora&Wang2005).

An instan eof theSAND approa h fortrusstopologyoptimization ispresentedand studiedinthisthesis. Byintrodu ingbinaryvariablesformemberandnodeexisten e andby onsidering thenormalfor esandnodal displa ementsas statevariables,the topologyoptimization problem an be stated as amixed-integerlinear optimization problem. A ordingtotheSANDphilosophy,theequationsofstru turalanalysisare treatedas equality onstraints. The stiness equation is disaggregated as presented in Chapter 2. Consequently, the singularity issues and dis ontinuities indu ed by vanishingmembersareavoided.

3.2 Variables

}

.

3.2.1 Prole Sele tion

Supposethememberprolesaretobe hosenfromasetof

n

S

alternatives,whi hhave been orderedin in reasing ross-se tionalarea. Denote thesets ofavailablemember areasandmomentofinertias,respe tively,by

A

= { ˆ

A

1 , ˆ

A

2 , . . . , ˆ

A

n

S

}

(3.4)

I

= { ˆ

I

1 , ˆ

I

2 , . . . , ˆ

I

n

S

}

(3.5)

Here,

A

ˆ

j

< ˆ

A

j+1

forall

j

,but

I

ˆ

j

> ˆ

I

j+1

ispossible.

A proleis hosenforea h memberby binaryvariables,

y

ij

, dened by (Rasmussen &Stolpe2008)

y

ij

=

(

1

X

j=0

y

ij

= 1

(3.8)

The variables

y

i0

an beinterpreted either as sla k variables forthe inequality on-straintEq.(3.7),or variablesforsele tingaprolewithzero ross-se tionalarea.

The ross-se tionalpropertiesofmember

i

areexpressedby

A

i

=

n

S

X

j=1

ˆ

A

j

y

ij

(3.9)

I

i

=

n

S

X

j=1

ˆ

I

j

y

ij

(3.10)

i

=

n

S

X

j=1

ˆ

X

j

y

ij

(3.11)

Aseparatebinaryvariable,

y

i

, anbeassignedforea hmemberto ontroltheexisten e of the member. Thisvariable takesthe value1 ifmember

i

is in luded in the truss and 0otherwise,anditisrelatedto theprolesele tionvariablesby

y

i

=

n

S

X

j=1

y

ij

(3.12)

This onstraint repla es Eq. (3.7). Note that if the variables

y

i

are in luded, the variables

y

i0

be omeredundantandshouldbeex ludedfromtheproblemformulation. Furtherimpli ationsofthevariables

y

i

tothesolutionpro essaredis ussedlater. Thebinaryvariables anbe ompiledinave torformby

y

= {y

11 y

12 . . . y

1,n

S

. . . y

n

E

,1

y

n

E

,2

. . . y

n

E

,n

S

}

(3.13)

Y

= {y

1 y

2 . . . y

n

E

}

(3.14)

y

0 = {y

10 y

20 . . . y

n

E

,0

}

(3.15)

3.2.2 Member For es

The memberfor esare takenasstatevariables intheoptimization. Here,the devel-opmentsofRasmussen&Stolpe(2008)arefollowed. Forthepurposesoftheproposed problem formulation, it is bene ial to dene a separate memberfor e variable for ea havailableproleandea hloading ondition