Problem Des ription

ConsiderthedesigndomainofthemaingirderofahallbuildingshowinFig.7.1. The boundaryofthedomainisxed, andthegoalisto determinetheoptimumlayoutof thebra ingmembers,pla edbetweenthe hords. Thespanofthetruss

L = 24000

mm, and theheightfrom thelower hordto thesupportsis

h = 2000

mm. Thein lination oftheupper hordis1:20,whi hmeansa

2.86

◦

anglewiththe

x

-axis. Theangleofline segmentABwiththe

y

-axisis

α = 30

◦

. Theline load

q = 25.1

kN/m. Thisin ludes thesnowloadtypi alforsouthernFinland,andtheweightoftheroong. Thesetwo loads are ombineda ordingto theEuro ode. ThesteelgradeS355is used, thatis

f

y

= 355

MPa,

E = 210

GPa,and

ρ = 7850

kg

/

m 3

.

Thegroundstru tureis reatedasfollows. Thedesigner hoosesthenumberofnodes on thehalfof thelower hord(linesegmentBC in Fig.7.1). Thesenodes arepla ed equidistantly. The samenumber of nodes is pla ed on the upper hord, with

x

o- ordinates orrespondingto thenodesof thelower hord. Then,amemberis reated between a pair of nodes, if the angle between the member and both hords is at least

30

◦

. This restri tionis to guaranteefavourable welding onditionsas statedin (EN 1993182005).

Chainsare reatedatthe hords. Amaximum hainmemberlength,

Lmax= 6000

mm is pres ribedinorderto redu ethenumberofmembers. Thepurposeisto eliminate unrealisti ally long hordmembersbyengineeringjudgment.

Symmetry of theground stru ture with respe tto theline dened by points Cand D is enfor ed to keepthe number of variables as small as possible. Note, however, that thetrue optimum stru ture mightnot be symmetri . Thesymmetry ondition isin ludedintheproblembyrelatingtheexisten eandprolevariablesofamember

withitssymmetri ounterpart. Similarpro edureis appliedforthenodalvariables. In order to allow the node lo ated at point C to vanish, members overlapping this nodeare reatedonthelower hord.

Six ground stru turesare onsidered. Inthe oarsest groundstru ture, 5nodes are pla edonthehalf ofthelower hord. This number isthen in reasedup to10. The nodal oordinatesandelement onne tivitytables anbefoundinAppendixB.3. For ea h ground stru ture, the numberof prole alternatives is variedfrom 15 to 30 in stepsof5. Inthelast ase,40prolesareavailable. Theproledata anbefoundin AppendixA .Theprolesofthedierent asesareas follows(seeTableA.1): i)16to 30(15alternatives);ii)11to30(20alternatives);iii)11to35(25alternatives);iv)6 to35(30alternatives);v)all40proles.

When the ground stru ture is xed and only the number of prole alternatives is in reased, the minimum weight solution of the previous problem an be used as a startingpointforthenextproblemwithmoreproles.

For ea h ground stru ture, the minimum weight and minimum ost problems are

solved. Memberbu klingisa ordingtoEuro ode3,andastabilizingloading ondi- tionisin ludedto guaranteekinemati stability. Formulation3isemployed, that is, memberexisten e variables, nodal variables and prolevariables are allin luded in theproblem.

The ost fun tion presented in Chapter 3 is employed and modied. For bra es, the ost fun tion omponents are easy to determine, sin e ea h bra ing member is onne tedto both hordsin onstantanglesdenedbythegroundstru ture.

The hordsaremanufa turedasuniformlongmemberstowhi hthebra esarewelded. Ifthemembersofthestru turalmodel(pin-jointedtruss)wereusedtodeterminethe ostof the hords, unne essarilymanysawingandwelding would bein ludedin the ost obje tive fun tion. Therefore, the ost omponents of the hords have to be treatedseparately.

Theupper hordismadeoftwoparts thatareweldedtogetherusingabuttweld. A similar weld is employed at the supports to take into a ount the ost of atta hing the truss to the olumns of the building. Thus, altogether four ends are sawn and three endsare welded. Asthe hordmembersare grouped to havethesameprole, thegroupingvariables

wj

anbeemployedto omputethesawingand welding osts ofthe entire upper hord. Denote by

CW j

and

CSj

thewelding andsawing osts of prole

j

of theupper hord,respe tively. The ostoftheupper hordis then

C_SWup

(x) =

nS

X

j=1

(2CSj+ 3CW j)wj

(7.1)

Note that

CSj

in ludes thesawing of both ends ofone half of the hord. The om- ponents

CSj

and

CW j

anbedeterminedasdes ribedinChapter3. Forthewelding, onlythedimensionsandtheangles oftheproleareneeded. However,thelengthof thememberisneededforthesawing ost. Thelengthofone half oftheupper hord is

Lup=

24000

mm

This valueisusedfor omputingthe

CSj

.

Forthelower hord,onlythesawing omponentsmustbedetermined. Allthewelding related to the lower hordis in luded in the ost omponentsof thebra es. During optimization, the lengthofthe lower hord an vary. This possibilityis negle tedin the ost omputation,andthelengthofthelower hordisdeterminedfromtheground stru ture:

Llow

= 24000

mm

− 2 · (2000

mm

) · tan (30

◦_{) = 21691.6}

mm (7.3)

Thelower hordsawing ostisthen

CSlow(x) =

nS

X

j=1

CSjwj

(7.4)

The material, blasting, and painting osts ofthe hords an bein luded in the ost fun tion forthe hordmemberswithrespe ttothevariables

yij

.

The values for parametersof the ost fun tion that depend on thesize of the truss are as follows. For bothpainting and assembly by welding ost enters, the areaof the working spa eis

(24 + 2) × (2.6 + 2)

m

2

= 119.6

m 2

. UsingEq. (2) fromHaapio (2012),thevalues

cReA= cReP

= 0.0487 e/

minand

cSeA= cSeP

= 0.0712 e/

minare obtained.

In document Mixed Variable Formulations for Truss Topology Optimization (Page 113-115)