Global size optimization of statically determinate trusses considering displacement, member, and joint constraints

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CONSIDERING DISPLACEMENT, MEMBER, AND JOINT CONSTRAINTS

Roxane VanMellaert 1

,GeertLombaert, Mattias S hevenels

Abstra t

Realisti trussdesignoptimizationproblemsareoftengovernedbypra ti al onstraints. Be ause

ofthe omplexityofthese onstraints,usuallyonlymember onstraintsaretakenintoa ountduring

theoptimization,andjoint onstraintsarea ountedforinamanualpostpro essingstep. Thispaper

proposesamethodto a ountfor joint onstraintsinthe globaldis retesize optimizationofasteel

trussstru ture. The design of anN-type trussgirder is onsidered rst withoutand thenwiththe

joint onstraintsspe iedinEuro ode3. Inorderto guaranteeglobaloptimalityinboth ases,the

optimization problem is reformulated as a mixed-integer linearprogram. A stati ally determinate

analysis model is adopted soas to ensure that all joint onstraints an be reformulated as linear

fun tions. Ifthejoint onstraintsarenot onsideredintheoptimization,adesignisobtainedwhere

thejoints need additional strengthening. This an be done bymanually sele ting heavierse tions,

whi hoftenleads toa suboptimalresult, or bystrengthening thejoints(e.g.bymeans ofstiening

plates), whi h hasa serious impa t on the fabri ation ost. If thejoint onstraints are onsidered

inthe optimization,they are automati allysatised bythenal design. The weight of this design

is about 15% higher than inthe rst ase. This shows thatthejoint onstraints have a signi ant

impa tontheoptimaldesign. Ifthejoint onstraintsarea ountedforinasuboptimalway(e.g.by

manuallysele tingheavierse tions),theadditionalweightmaybeevenhigher. Takingintoa ount

joint onstraintsinthe optimizationleads toa ostredu tionat two levels: intermsof engineering

ost(nomanualpostpro essingstepisneeded)aswellasfabri ation ost(usingunne essarilyheavy

se tions aswell asjoint strengthening areavoided).

Keywords: Truss design, dis rete design optimization, joint onstraints, mixed-integer linear

pro-gram reformulation.

INTRODUCTION

Numeri aloptimizationmethodshaveagreatpotentialtosupportstru turalengineersinnding

theoptimaldesign,andsotokeep the onsumption ofnaturalresour es ofthebuildingindustry to

aminimum. However,pra ti ing stru tural engineersappeartoberelu tant to adopt optimization

as a daily design tool, even for relatively simple but tedious tasks su h as the sizing of a steel

truss girder. One of the reasonsis that real-world design problems are often governed by a large

number of onstraints and pra ti al issues. For a steel truss girder with welded joints, the usual

displa ement, memberfor e, andbu kling onstraintsasformulated inEuro ode3areimposed. In

addition,thefollowingpra ti al onstraintsmustbesatised: thememberse tions mustbe hosen

from agiven se tion atalog, and the joints mustobey ertain geometri al rules inorder to ensure

stru tural integrityand weldability,aswell asme hani al rules inorderto avoid hord web failure,

hordshearfailure,andbra efailure. Mostexistingdesignoptimizationalgorithms annottakeinto

a ount allthesepra ti al onstraints. Asa onsequen e,amanualpostpro essingstepisrequired,

where theoptimized design is modied to satisfy the onstraints whi h are not onsidered during

theoptimization. Thisoperationis umbersome,it ostspre iousengineeringtime,anditmaylead

to asuboptimal design or adesign thatno longer fulllsthestress anddispla ement onstraints.

1

Postprint submittedto Journal of Stru turalEngineering. Published version: R. Van Mellaert, G.Lombaert,

M.S hevenels. GlobalSizeOptimizationofStati allyDeterminateTrussesConsideringDispla ement,Member,and

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mentse tionsareiteratively updateduntilthestressinea helementequalsthemaximumallowable

value, whi h may lead to an optimal stru tural weight (Razani, 1965; Mueller etal., 2002). In its

original formulation, the FSD method is only useful for stress onstrained optimization problems.

AmodiedFullyUtilizedDesignmethod(FUD)isproposedbyPatnaiketal. (Patnaiketal.,1998)

to take into a ount both stress and displa ement onstraints. Only in ertain ases the result of

theFSDmethodisoptimal(PatnaikandHopkins,1998). However,forapra ti alsteeltrussdesign

problemthejointsremaintobedesignedmanually. Dependentonthese tions hosenforthebra es

and the hords, some ofthe jointswill need to be strengthened bymeans ofstiening platesor by

lo ally using a heavier se tion (Wardenier etal., 1992). Su h interventions have little inuen e on

theweight ofthe stru ture,but require additional welding aswell astesting ofthewelds,and this

hasa signi ant impa ton thefabri ation ost.

An additional drawba k of the FSD method is the fa t that it only an handle ontinuous

variables. For a pra ti al steel truss design problem the prole of the members has to be hosen

from a steel atalog, however. The optimization problem is therefore dis rete. Several algorithms

for dis rete optimization have been proposedinthe literature(Thanedar andVanderplaats, 1995).

The most popular algorithms that an handle dis rete variables are evolutionary algorithms, su h

as simulated annealing (Balling, 1991), geneti algorithms (Camp et al., 1998; Rajeev and

Krish-namoorthy,1992), ant olony optimization (Camp et al.,2005), rey algorithm (Gandomi et al.,

2011), arti ial bee olony algorithm (Sonmez, 2011), and parti le swarm optimization (Venter

and Sobiesz zanski-Sobieski, 2003). These methods explore thedesign spa e ina random fashion,

therebyusinginformation olle ted fromprevious analysestogradually movetowards abetter

per-forming design. Evolutionary algorithms owe their popularity to the fa t that they are easy to

understand and to implement. They an ope with dis rete parameters and are able to take into

a ount omplex onstraints. However,evolutionaryalgorithms onvergeslowly,involvealgorithmi

parametersthatrequire arefultuning, andglobal optimality annotbeguaranteedsin eno

on lu-sive onvergen e he ks an be made. Inorder to properlyassess the inuen e ofjoint onstraints

ontheoptimal designoftrussstru tures, itisimportant thatglobaloptimality anbeguaranteed.

Evolutionaryalgorithms arethereforenot suitable.

The method used in this paper is to reformulate theoptimization problem asa Mixed-Integer

Linear Program (MILP), whi h is solved with the bran h-and-bound method in order to a hieve

globaloptimality. ThisMILPisobtainedbymeansofbinaryde isionvariablesandtheSimultaneous

ANalysisandDesign(SAND)approa h: thestatevariables(thestru turalnodaldispla ementsand

thememberend for es)are onsidered asadditional design variables and thestate equations (the

equilibriumequations)areenfor ed by meansof additionalequality onstraints. This optimization

method has originally been proposed by Grossmann et al. (1992) for dis rete size optimization

problems and is extended by Rasmussen and Stolpe (2008) for truss topology design problems.

Mela and Koski (2013) in luded all member onstraints spe ied by the Euro ode in the truss

topology designproblem. Inthispaper,the fo usisrestri ted tosize optimization,but allrelevant

onstraints pre ribed by the Euro ode (European Committee for Standardization, 2005a,b) are

takeninto a ount, in ludingboth the member onstraintsand thejoint onstraints.

In orderto ensure thatall joint resistan e onstraints an be reformulatedaslinear onstraints

intermsofthedesign variables,thes ope ofthispaperislimitedto stati allydeterminateanalysis

models. Sin ethememberfor esofstati allydeterminatemodelsdonotdependonthese tionsand

remain onstantintheoptimization,theydonothavetobe onsideredasadditionaldesignvariables.

The joint onstraints - some of whi h would be quadrati if the member for es are onsidered as

design variables - anthenbereformulated asmixed-integer linear onstraints. Stati determina y

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a ura y assuming pinned onne tions. External stati determina y imposes a stronger limit on

theappli ability of the method asit only holds for simplysupported trusses. It does not hold for

ontinuoustrusses or trusses thatarepartof aportal frame.

The example problem onsidered in this paper is the dis rete size optimization of an N-type

truss girder with welded joints under stati nodal loading as shown in gure 1. The top hord

members aresteelH-se tions(HEA),thebottom hordmembersaresteel hannel-se tions (UPN),

andthebra esaresteelRe tangularHollowSe tions(RHS).ThisN-typetrussgirderiswidelyused

inpra ti e.

The paperisorganizedasfollows. Inse tion2,asummaryof allgoverningdesign requirements

fortrussstru turesisgiven: thedispla ement,memberfor eandbu kling onstraintsasformulated

in part 1-1 of Euro ode 3 (European Committee for Standardization, 2005a) as well as the joint

resistan e onstraintsasformulatedinpart1-8ofEuro ode 3(European Committee for

Standard-ization, 2005b). In se tion 3, the mixed-integer linear formulation for stati ally determinatetruss

stru tures is introdu ed, and the example truss is optimized onsidering only displa ement and

member onstraints. In se tion 4, the joint onstraints are also taken into a ount. In se tion 5,

theresultsaredis ussed. Theoptimaldesignwithout joint onstraintsandtheoptimaldesignwith

joint onstraints are ompared.

DESIGNOF TRUSS STRUCTURES WITH WELDED JOINTS

This se tion gives an overview of the design pro edure for the example truss a ording to the

European building odes.

First, a stru tural analysis is performed to obtain the member for es and the nodal

displa e-ments. Se ond, the imposed onstraints are he ked. In theservi eabilitylimit state,the

displa e-ment onstraintsareveried. Intheultimatelimitstate,therearetwotypesof apa ity onstraints:

member onstraintsand joint onstraints. Themember onstraintsare spe iedinpart 1-1of

Eu-ro ode 3 and an be subdivided inmember resistan e onstraints (to avoidyielding) and member

stability onstraints(toavoidbu kling). The joint onstraintsarespe ied inpart1-8ofEuro ode

3and an be subdividedinjoint geometry onstraintsandjoint resistan e onstraints. Inaddition,

theglobalstabilityofthestru turehastobe he ked,butinthispaperitisassumedthattheglobal

stabilityis guaranteedbythese ondary stru tureor bymeans ofextrastieners.

Stru tural models

Two dierent models an be used for the stru tural analysis of trusses. In the rst model, all

membersarepin onne tedasshowningure2a. Asa onsequen e,onlynormalfor eso ur. This

model provides a good approximation for trusses withslender members and where the enterlines

ofjoinedmembersinterse tea hotherinasinglepoint(Wardenieretal.,2008). Unfortunately,the

latterisoftennot truefor reasonsofweldability and utting. Asa onsequen e,theinterse tionof

the enterlines ofthe bra esislo atedata ertaindistan efromthe enterlineofthe hords. These

nodal e entri ities ause bending moments inthe hords, whi h are alled e entri ity moments.

Thee entri ity moments have to be taken into a ount inthe hord member design. Asproposed

in the CIDECT design guide (Wardenier et al., 2008), they are a ounted for in an approximate

wayby distributingthem equallyoverboth hord members oneitherside ofthe joint.

Inthese ondmodel, the hordsare ontinuousandthebra es arepin onne tedwithinnitely

sti members at a distan e of the e entri ity to the hords as shown ingure 2b. This model is

internallystati allyindeterminate.

In order to ensure that all joint resistan e onstraints an be taken into a ount in the MILP

formulation,onlystati allydeterminatemodelsare onsidered. Thereforetherstmodelisadopted

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A ording to Euro ode 0, the verti al displa ement

u

z

of roof stru tures should be limited as follows:

|u

z

| 6

l

200

(1)

where

l

isthespanof thestru ture. Member resistan e onstraints

The apa itiesofthe members are he ked a ordingto part1-1of Euro ode 3. Theresistan e

of the ross-se tionsof thebra es is he ked asfollows:

|N| γ

M0

Af

y

6

1

(2)

where

N

isthedesign valueof thenormalfor e,

A

isthe ross-se tionarea,

f

y

istheyieldstrength

of the material,and

γ

M0

is thepartialfa tor for resistan eof ross-se tions,whi his equal to 1.

The resistan eof the ross-se tionsof the hordsis he ked asfollows:

|N| γ

M0

Af

y

+

|M| γ

M0

W

pl

f

y

6

1

(3)

where

M

isthe design valueof the bending moment and

W

pl

istheplasti se tion modulus.

Stability onstraints

Thebu klingresistan eofmembersin ompressionand/orbendingshouldbeveried.

Depend-ing on theo urringfor es - ompression, bending, or both - and these tion type, dierent types

ofbu klingshouldbe he ked. Forea hbu klingmode,adierent redu tionfa tor

χ

forthedesign resistan eis al ulated. Allredu tionfa torsaredeterminedfromtherelevantbu kling urvewhi h

issele ted inagreement withthetype of ross-se tion. Thebu kling urvesdependon these tion,

thebu klinglength,and the yieldstrength ofthe material.

The exuralbu klingresistan eis he ked asfollows:

−

χ

y

Af

y

γ

M

1

6 N

(4)

−

χ

z

Af

y

γ

M

1

6 N

(5) where

γ

TF

Af

y

γ

M

1

6 N

(7) where

χ

T and

χ

TF

(5)

should also be he ked. For the HEA top hord members in bending and axial ompression, the

bu klingresistan eis he ked asfollows (assuming lass1 or lass 2 ross-se tions):

|N| γ

M

1

χ

y

f

y

A

+ k

yy

|M| γ

M

1

χ

LT

f

y

W

pl

,y

6

1

(8)

|N| γ

M

1

χ

z

f

y

A

+ k

zy

|M| γ

M

1

χ

LT

f

y

W

pl

,y

6

1

(9) where

χ

LT

isthe redu tionfa tor due to lateral-torsional bu klingand

k

yy

and

k

zy

are intera tion fa torswhi h an bedetermined a ordingto oneof themethods des ribedinannexA orannexB

of Euro ode 3. Inthis paper,themethoddes ribedinannex Bisused.

Joint geometry onstraints

Ea h joint has to satisfygeometri onstraints that eitherfollow from pra ti al onsiderations

su h as weldability or areimposed to ensure that the design remains within the range of validity

ofthe joint apa ity onstraintsdis ussed inthenext subse tion. Whenthereisagap betweenthe

bra es that are onne ted to the hord, the joint is alled a gap joint. When the bra es overlap

ea h other, the joint is alledan overlap joint. To limit the e entri ity moments, the onne tions

ofthe RHSbra esandHEA top hordsare hosentobegapjointsandthe onne tionsoftheRHS

bra es and UPN bottom hordsare hosen to be overlap joints. Thetwo typesof jointsapplied in

the N-truss example as well as the notations used for all member dimensions are shown in gure

3. The notation used in the Euro ode is adopted: properties of hord members aredenoted by a

subs ript

0

and properties of bra es are denoted by the subs ripts i and j. For overlap joints, it isimportant to distinguish between theoverlapping and theoverlapped bra e; here thesubs ripti

refersto the overlapping bra e and the subs ript j to theoverlapped bra e. For the gap jointsthe

gap

g

has a positive value. In order to ensure weldability, the gap of thejoint should be at least aslarge as thesum of the thi knesses

t

i and

t

j

of the two bra es. For the overlap jointsthe gap

g

hasanegative value. Theoverlap

λ

ov

is hosento be100%for pra ti al onsiderations: theend of

theoverlapping bra eonlyhasto be utinoneangle. Asa onsequen e theoverlap

g

should be at leastaslarge asthe width

(squarese tion) (15)

h

j

b

j

= 1

(squarese tion) (16)

2.5

mm

6 t

i

6

25

mm (17)

2.5

mm

6 t

j

6

25

mm (18)

g > t

i

+ t

j (19)

h

i

b

i

6

2

(25)

0.5 6

g = −b

i (31)

The onstraint given by equation (31) imposes an overlap of 100% for pra ti al onsiderations,

whi h isstri terthan the onstraint spe iedinthe Euro ode.

In addition, lass 2 ross-se tions mustbeusedfor the hords and lass1 ross-se tionsfor the

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For onne tions between hollow se tions and open se tions the following types of failure an

o ur: hord fa e failure, hord web failure, hord shear failure, pun hing shear failure, bra e

failure, and lo albu klingfailure. A ording to part 1-8of Euro ode 3, only ertain failuremodes

mustbe onsideredforea h typeofjoint. For HEA-RHSgapjointsthefollowing riteria shouldbe

he ked: hordwebfailure,bra efailure,and hordshearfailure. ForUPN-RHSoverlapjointswith

an overlap of minimum 80% onlybra e failureof theoverlapping bra emember mustbe he ked.

In order to avoid hord web failure in the HEA-RHS joints (gure 4), the normal for e

N

i

in

bra eiis limitedasfollows:

|N

i

| 6

f

y

t

w

b

w

sin θ

i

1 γ

M

5

(32) where

f

y

is the yield strength of the hord,

γ

M

5

is the partial safety fa tor for the resistan e of

jointsinhollowse tion latti e girder,whi h isequal to

1

,and

b

w

is theee tive width fortheweb

of the hord, whi his obtained as:

b

w

= min

h

i

sin θ

i

+ 5 (t

f

+ r

0

) , 2t

i

+ 10 (t

f

+ r

0

)

(33)

Thenormal for e

N

j

inbra ej islimitedinthesame way.

In orderto avoid bra efailureintheHEA-RHSjoints (gure5),thenormal for e

N

i inbra e i islimitedasfollows:

|N

i

| 6 2f

yb

t

i

p

e

1 γ

M

5

(34) where

f

yb

is the yield strength of the bra e, and

p

e

is the ee tive length of the onta t area of

thebra ememberonto the fa eof the hord,whi h is al ulated as:

p

e

= min

t

w

+ 2r

0

+ 7t

f

j

inbra ej islimitedinthesame way.

The last joint resistan e he kfor theHEA-RHS joints isrelated to hord shearfailure (gure

6). Thisfailure modeis avoided by limitingthe normal for e

N

i inbra e iasfollows:

|N

i

| 6

f

y

A

v

0

√

3 sin θ

i

1 γ

M

5

(36)

where the shear area

A

v

0

= A

0

− (2 − α) b

0

t

f

+ (t

w

+ 2r) t

f , and

α = 1 + 4g

2

_/3t

2

f

−

1

2

for RHS

se tions. The normalfor e

N

j

inbra e jis limited inthe same way. In addition, the normal for e

of the hord

N

0

islimited asfollows:

|N

0

| 6



(A

0

− A

v

0

) f

y

+ A

v

0

f

y

0

s

1 −

V

pl

,

Rd

2





1 γ

M

5

(37)

where thedesignshear for e

V = max (|N

i

| sin θ

i

, |N

j

| sin θ

j

)

andtheplasti design shearresistan e

V

pl,Rd

= f

y

0

A

v

0

/γ

M

5

√

3

.

ForUPN-RHSoverlapjointswithanoverlapofatleast80%,onlybra efailurehastobe he ked

(gure 7). This failure mode is avoided by limiting the normal for e in the overlapping bra e as

1 γ

M

5

(38)

where theee tivewidth

b

e,ov

ofthe onne tionbetween theoverlapping bra eandtheoverlapped

bra eis al ulated as:

b

e,ov

= min

10t

2

°, and the

angleof thediagonal bra esis

θ

j

= 45

°. Thevalueofthe verti al loadintheultimate limitstate is

F = 100

kNandisusedto he kthememberandjoint onstraints, thevalueoftheverti al loadin theservi eabilitylimitstateis

F = 74.07

kNandisusedto he kthedispla ements. Themaximum allowabledispla ement

u

max

is

l/200 = 0.1

m . Allse tions aresteelse tions withYoung's modulus

E = 210

GPa ,and density

ρ = 7850

kg

/

m

3

.

The top hord se tions are hosenfroma atalog withtwenty-four HEA-se tionsgiven intable

1 (Ar elorMittal, 2015). The bottom hord se tionsare hosen froma atalog witheighteen

UPN-se tionsgivenintable2(Ar elorMittal, 2015). These tionsofthebra es are hosenfroma atalog

witheighty-two old formedsquareRHS-se tions given intable3 (vanEldik, 2006). All top hord

members musthave the samese tion, andall bottom hords membersmusthavethesame se tion.

For the hords steel grade S355 is hosen and for the bra es steel grade S275 is hosen. For the

analysisof thetruss,themodelshowningure2a isadopted.

Obje tive fun tion and ompatibility onstraints

Inordertosolvethedis retesizeoptimizationproblemtoglobaloptimality,itisreformulatedas

a Mixed-Integer Linear Program (MILP). This approa h wasoriginally proposed to solve dis rete

size optimization problems (Grossmann et al., 1992) and was extended later for truss topology

optimization problems (Stolpe and Svanberg, 2003; Stolpe, 2007; Rasmussen and Stolpe, 2008;

Melaand Koski, 2013;Mela, 2013).

InordertoobtainanMILP,theSimultaneousANalysisandDesignapproa h(SAND)isadopted.

Thedesign variables intheoptimization problem are omplemented witha setof ontinuous state

variables,in ludingthenodaldispla ementsandnormalfor es, whiletheequilibriumequationsare

in orporatedasequality onstraints(Haftka,1985;AroraandWang,2005),sonoexpli itstru tural

analysisis made.

The analysis model adopted in this paper is stati ally determinate. The member for es are

independent of the hosen se tions and an be al ulated a priori. As a onsequen e, thenormal

for esarenotadoptedasdesignvariablesandtheequilibrium onstraintsaredropped. Theoriginal

MILP proposed fordis rete size optimization isthus simplied.

ij

= 0

. Althoughthenumberofoptimization variables and onstraintsbe omeslargeinthisapproa h,the

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solvers basedonthe bran h-and-bound method.

In this subse tion, the obje tive fun tion and the ompatibility onstraints are formulated. In

thefollowing subse tions, the Euro ode provisions regardingthe maximumdispla ements and the

strengthofthemembersareaddedtotheproblemasinequality onstraintsinorderto ompletethe

MILP.Inorder to formulate the MILP for the optimization of theN-type truss, themembers and

atalogs of proles are subdivided indierent sets: the set of all members

M

, all hord members

M

C

,top hord members

M

TC

, bottom hord members

M

BC

and bra es

M

B

, and the atalog of

proles

C

i

for member

i

. Theoptimization problem(without Euro ode onstraints)is given bythe following equations: min

y∈B

n

b

,u∈R

n

dof

ρ

X

i∈M

X

j∈C

i

L

i

A

ij

y

ij

(39) s.t.

1 =

X

j∈C

i

y

ij

∀ i ∈ M

(40)

X

j∈C

i

A

_i

1

TC

j

y

i

1

TC

j

=

X

j∈C

i

A

ij

y

ij

∀ i ∈ M

TC (41)

X

j∈C

i

A

_i

1

BC

j

y

i

1

BC

j

=

X

j∈C

i

A

ij

y

ij

∀ i ∈ M

BC (42)

E

L

i

A

ij

b

T

i

u

− N

i

6 (1 − y

ij

)N

ij

∀ i ∈ M, ∀ j ∈ C

i

(43)

E

L

i

A

ij

b

T

i

u

− N

i

>

(1 − y

ij

)N

ij

∀ i ∈ M, ∀ j ∈ C

i

. The onstraint in equation(40)ensuresthata singlese tion

j

is hosenfromthe atalog

C

i

formember

ij

and

N

ij

arearti ial upperandlowerboundsintrodu ed to ensure feasibility of theoptimization problem when se tion

j

is not sele ted for element

i

and

y

ij

= 0

. Inthisexample,the values oftheupperand lower boundsof the ompatibility onstraints inequations(43)and(44)are al ulatedbasedontheminimumandmaximumalloweddispla ement

(Rasmussenand Stolpe,2008).

The total numberof members inthe stru ture isdenoted by

n

m

,thetotal number ofavailable

se tionsfor ea hmember

i

isdenoted by

n

s

,i

,thetotalnumber ofdegrees offreedomis denotedby

n

dof

,andthe totalnumberofjointsisdenotedby

n

g

. Asa onsequen e,thetotal numberofbinary

de ision variables is

n

b

=

P

n

m

i=1

n

s

,i

. Displa ement onstraints

The displa ement onstraints aregiven byequation (45). Here

u

= −0.1

mand

u

= 0.1

mare theminimumandmaximumallowed displa ement, respe tively.

(10)

u 6 u 6 u

(45) Member onstraints

In this subse tion, the member resistan e and stability onstraints des ribed in the previous

se tion are in orporated in the MILP.The joint onstraints are not onsidered yet; it is therefore

assumed that the members an be onne ted in su h a way that e entri ities are avoided. No

e entri itymoments arethereforetaken into a ount intheoptimization.

The stress onstraints given by equation (2) and the in-plane and out-of-plane exural

bu k-ling,torsional bu kling andtorsional-exuralbu kling onstraintsgiven byequations (4)to(7)are

reformulated as:

N

i

y

ij

6 f

y

A

ij

y

ij

∀ i ∈ M

C

, ∀ j ∈ C

i

(46)

N

i

y

ij

>

− min (1, χ

y

, χ

z

, χ

T

, χ

TF

) f

y

A

ij

y

ij

>

− min (1, χ

y

, χ

z

) f

yb

A

ij

y

ij

∀ i ∈ M

B

, ∀ j ∈ C

i

(49) where

f

y and

f

yb

are the yield strength of the hords and bra es, respe tively. Following the

re ommendations intheCIDECTguide(Wardenieretal.,1992),thebu klinglengthsareassumed

as follows: for the hords

L

r,y

= L

r,z

= 0.9L

i

, and for the bra es

L

r,y

= L

r,z

= 0.75L

i

. The

onstraints inequations (46) to (49) arelinear equations in terms of thebinary de ision variables

y

ij

. Results

The MILP given by equations (39) to (49) onsists of 1133 design variables, in luding 1112

binarydesign variables and 21 ontinuousdesign variables, 4519 onstraints, and 11698 non-zeros.

TheMILP issolved bymeans ofGurobi5.6(Gurobi Optimization In .,2013), whi huses the

ut-and-bran hmethod,ona omputerwitha 4threads IntelCore(TM)2 QuadCPUQ9550pro essor

and4.0GBRAM.Theresultsaregivenintable4. ThetotalweightoftheN-trussis1826.3kg. The

nodaldispla ements aregiven in table 5. It is observed that the displa ement onstraints arenot

riti al. Gurobiis apableofsolvingthisproblemintheprepro essingstage. Withoutperforminga

presolveorgenerating uts,theproblemissolvedalsoattherootnode. The omputationtimeisless

than 0.2 se onds inboth ases. This isprobably due to thefa tthat thedispla ement onstraints

do not be ome a tive. Without displa ement onstraints, the problem be omes trivial due to the

stati determina yofthetruss. Therefore,itispossibletoverifythesolution: theminimumse tion

areasthatarerequireda ordingtothememberstrengthandstability onstraints anbe al ulated

manually. This leadsto the same results.

It is now veried to what extent the optimized design satises the onstraints that are not

expli itly onsideredintheMILP.Inordertoensureweldabilityofthejoints,minimale entri ities

are introdu ed as spe ied in table 6. The gaps of the top joints are hosen to be equal to the

sum of the thi kness of the two bra es of the joint. The gaps of the bottom joints are hosen so

that theoverlap is 100%in orderto avoid di ult utting of theoverlapping bra e. Table 7 gives

the utilization ratios for all the onstraints that must be he ked a ording to part 1-1 and part

1-8 of Euro ode 3 as des ribed in previous se tion. These utilization ratios are al ulated as the

ratio between the a tual valueand themaximumallowed value. Dueto theintrodu tion of nodal

e entri ities after the optimization, the stability ratios are not guaranteed to be smaller than 1.

However, as an be observed in table 7, the ee t of the e entri ities is limited and all stability

(11)

the joints have to be strengthened inorder to satisfyall onstraints. This an be done by either

sele ting dierent proles, or bylo allystrengthening thejoints e.g. bymeans of stiening plates.

The rst approa h would lead to a suboptimal result, as it is very di ult to determine whi h

members should be made heavier and whi h se tions should be sele ted. The se ond approa h

would only have a limited impa t on the weight of the truss, but the fabri ation osts would be

mu hhigher.

OPTIMAL DESIGN WITH JOINTCONSTRAINTS

Problem des ription

In thisse tion, the displa ement onstraints, themember onstraints, and thejoint onstraints

are onsideredintheoptimization. Duetotheintrodu tionofthejointgeometry onstraints,trusses

withzeroe entri itiesbe omeinfeasible-non-zero e entri ities arethereforeallowed. Inorder to

ensurethattheimpa tofthee entri itiesontheweightofthetrussremainsminimal,thegapsizes

(whi h ontrol the e entri ities) are onsidered as additional design variables in the optimization

problem. Asa onsequen e, thedesign variables now onsistof thenodaldispla ements

u

∈ R

n

dof ,

thebinary de isionvariables

y

of joints.

Obje tive fun tion and ompatibility onstraints

Inthissubse tion,theobje tivefun tionandthe ompatibility onstraintsareformulated. Inthe

followingsubse tions, theEuro odeprovisionsregardingthemaximumdispla ements, thestrength

of the members, and the strength of thejointsare added to theproblem asinequality onstraints

inorderto omplete theMILP.In orderto formulate theMILPfor theoptimization oftheN-type

truss, themembers and atalogsof prolesare subdividedindierent sets: theset of all members

M

J

,the set ofall joints witha hord on the left-handside is denoted by

J

CL

, theset of all joints witha hord on the right-hand side is denoted by

J

CR

, the set of all

top jointsis denotedby

J

TJ

,and thesetof all bottom jointsis denotedby

J

BJ

. The optimization

problem(without Euro ode onstraints) isgiven bythefollowing equations:

min

y∈B

n

b

,u∈R

n

dof

,g∈R

n

g

ρ

X

i∈M

X

j∈C

i

L

i

A

ij

y

ij

(50) s.t.

1 =

X

j∈C

i

y

ij

∀ i ∈ M

(51)

X

j∈C

i

A

_i

1

TC

j

y

i

1

TC

j

=

X

j∈C

i

A

ij

y

ij

∀ i ∈ M

TC (52)

X

j∈C

i

A

_i

1

BC

j

y

i

1

BC

j

=

X

j∈C

i

A

ij

y

ij

∀ i ∈ M

BC (53)

E

L

i

A

ij

b

T

i

u

− N

i

6 (1 − y

ij

)N

ij

∀ i ∈ M, ∀ j ∈ C

i

(54)

E

L

i

A

ij

b

T

i

u

− N

i

>

(1 − y

ij

)N

ij

∀ i ∈ M, ∀ j ∈ C

i

(55) Theseequationsareidenti altoequations(39)to(44)forthe asewithoutjoint onstraints,ex ept

(12)

The displa ement onstraints are identi al to the displa ement onstraints given by equation

(45)inprevious se tion:

u 6 u 6 u

(56)

Member onstraints

Sin e thee entri itymomentsdepend onthegap sizes, themixed-integer linearreformulation

of the ombined stress and stability onstraintsdepends not only on thebinary de ision variables

y

ij

,but also on the gap variables

g

m

. The resistan e of the ross-se tions of the hords is he ked a ordingtoequation(3). Thedesignvaluesofthenormal for es areobtained beforeoptimization.

Thedesignvaluesofthe bendingmoments,however, depend onthee entri ities,whi hdependon

thegap size

g

m

. The e entri ity

e

m

injoint

m

is al ulated as(Wardenieretal.,1992):

e

m

=

g

m

+

b

i

BL

m

2 sin θ

_i

BL

g

+

θ

m

is the angle between the two bra es of the joint, whi h is al ulated as

θ

m

= π − θ

i

BL

m

− θ

i

BR

m

. The e entri ity moment

M

e

,m

injoint

m

is al ulated as(Wardenieretal.,1992):

M

e

,m

=

∆N

m

e

m

2

(58) where

∆N

m

=

N

i

CL

m

− N

of the hords at the right-hand sideof the joint is he ked a ordingto equation(3) asfollows:

N

i

CR

m

f

y

A

i

CR

m

+

|M

e

,m

|

f

y

W

pl

,y,i

CR

m

6

1

(59) where

f

y

is the yield strength of the hords,

W

pl

,y,i

CR

+

∆N

m

|e

m

|

2f

y

W

pl

,y,i

CR

m

6

1

(60)

Repla ingthee entri ity

|e

W

pl

,y,i

CR

m

g

m

+

b

_i

BL

m

2 sin θ

_i

BL

m

+

b

i

BR

m

2 sin θ

_i

BR

m

tan θ

m

−

h

_i

CR

m

2

6

1

(61)

(13)

belongingto dierent design entities areseparated:

∆N

m

2 tan θ

m

g

m

|

{z

}

gapsize

+

∆N

m

b

i

BL

m

4 tan θ sin θ

_i

BL

m

|

{z

}

leftbra e

+

∆N

m

b

i

BR

m

4 tan θ sin θ

_i

BR

m

|

{z

}

rightbra e

+





−∆N

m

h

_i

CR

m

4 − f

y

A

_i

CR

m





|

{z

}

right hord

6

0

(62)

−

∆N

m

2 tan θ

m

g

m

|

{z

}

gapsize

−

∆N

m

b

i

BL

m

4 tan θ sin θ

i

BL

m

|

{z

}

leftbra e

−

∆N

m

b

i

BR

m

4 tan θ sin θ

i

BR

m

|

{z

}

rightbra e

+





∆N

m

h

i

CR

m

4 − f

y

A

i

CR

m





|

{z

}

right hord

6

0

(63)

In reality, multiple proles areavailable for ea h member. For the hord at the right hand side of

joint

m

,equations (62) and(63)are thenrewrittenas:

∆N

m

2 tan θ

g

m

+

X

j∈C

_i

BL

m

∆N

m

b

i

BL

m

j

4 tan θ sin θ

i

BL

m

y

i

BL

m

j

+

X

j∈C

_i

BR

m

∆N

m

b

i

BR

m

j

4 tan θ sin θ

i

BR

m

y

i

BR

m

j

+

X

j∈C

_i

CR

m

−∆N

m

h

i

CR

m

j

4 − f

y

W

pl

,y,i

CR

m

j

+

N

i

CR

m

W

pl

i

CR

m

j

4 − f

y

W

pl

,y,i

CR

m

j

+

N

y

i

CR

m

j

6

0 ∀ m ∈ J

CR (65)

For the hordsattheleftsideofjoint

m

,equations (64)and(65)arereformulatedinthesameway, nowusingthe index

i

CR

m

insteadof

i

CL

m

and

J

CR insteadof

J

CL .

Asmentionedinthepreviousse tionandexpressedbyequations(8)and(9),thelateraltorsional

bu klingresistan eformembersbentintheplaneofhighestexuralrigidityshouldbe he kedalong

thex-axisand alongthe z-axis. Inthis ase only thetop hords mustbe he ked. The onstraints

(14)

∆N

m

2 tan θ

g

m

+

X

j∈C

_i

BL

m

∆N

m

b

i

BL

m

j

4 tan θ sin θ

i

BL

m

y

i

BL

m

j

+

X

j∈C

_i

BR

m

∆N

m

b

i

BR

m

j

4 tan θ sin θ

i

BR

m

y

i

BR

m

j

+

X

j∈C

_i

CR

m







−∆N

m

h

i

CR

m

4 −

1 −

N

i

CR

m

χ

_y,i

CR

m j

f

y

A

i

CR

m j

f

y







y

i

CR

m

j

6

0 ∀ m ∈ J

TJ

∩ J

CR (66)

−

∆N

m

2 tan θ

g

m

−

X

j∈C

_i

BL

m

∆N

m

b

i

BL

m

j

4 tan θ sin θ

i

BL

m

y

i

BL

m

j

−

X

j∈C

_i

BR

m

∆N

m

b

i

BR

m

j

4 tan θ sin θ

i

BR

m

y

i

BR

m

j

+

X

j∈C

_i

CR

m







∆N

m

h

i

CR

m

j

4 −

1 −

N

i

CR

m

χ

_y,i

CR

m j

f

y

A

i

CR

m j

f

y







y

i

CR

m

j

6

0 ∀ m ∈ J

TJ

∩ J

CR (67) where

χ

y,i

CR

m

j

and

χ

LT

,i

CR

m

j

aretheredu tionfa torsfortheresistan eofthe hordattheright-hand

side of the joint due to in-plane exural bu kling and lateral torsional bu kling, respe tively, and

k

_yy,i

CR

m

j

is an intera tion fa tor. The bu kling onstraints for the hords at the left side of the

joint are reformulated in the same way, now using the index

i

CR

m

instead of

i

CL

m

and

J

CR instead of

J

CL

. The bu kling onstraints along the z-axis are reformulated in thesame way, by repla ing

theredu tion fa tordue to in-plane exuralbu kling

χ

y,i

CR

m

m

j

. All

intera tion fa torsare al ulated a ordingto the method des ribed inannexBof Euro ode3.

Forthebra es,thestress onstraintsasgivenbyequation(2),andthein-planeandout-of-plane

exuralbu kling,torsionalbu klingandtorsional-exuralbu kling onstraintsasgivenbyequations

(4)to (7)remain the sameasintheprevious se tion:

N

i

y

ij

6 f

yb

A

ij

y

ij

∀ i ∈ M

B

, ∀ j ∈ C

i

(68)

N

i

y

ij

>

− min (1, χ

y

, χ

z

) f

yb

A

ij

y

ij

∀ i ∈ M

B

, ∀ j ∈ C

i

(69)

Joint onstraints

Thejointgeometry onstraintsgivenbyequations(10)to(18)and(20)to(28)andthoserelated

to the ross-se tion lasses areimposedby onlyin luding allowablese tions intheprole atalog.

The geometry onstraints for the top hords given byequation(19) arereformulated as:

X

j∈C

i

BL

m

t

_i

BL

m

j

y

i

BL

m

j

+

X

j∈C

i

BR

m

t

_i

BR

m

j

y

i

BR

m

j

− g

m

6

0 ∀ m ∈ J

TJ (70) where

i

i

BC

j

y

i

BC

j

−

X

j∈C

i

∈ M

B

C

, i

B

∈ M

B (71)

g

m

+

X

j∈C

i

(15)

where

i

BR

m

is theindex of theoverlappingbra e injoint

j

y

i

BL

m

j

−

X

j∈C

i

The design resistan e for hord web failure of the top joints given by equation (32) ontains

theee tive widthfor the web of the hord

b

w

,whi h hasto be al ulated astheminimumof two

expressions. The hord webfailure onstraintsforthetopjointsarethereforereformulatedinto two

linear onstraints. For thebra es at theright hand sideof thetop joints, therst onstraint given

by equation(32)is reformulated as:

X

j∈C

i

CL

m

N

i

BR

y

_i

CL

m

j

−

X

j∈C

i

BR

m

h

_i

BR

m

j

sin θ

i

BR

m

y

_i

BR

m

j

6

0 ∀ m ∈ J

TJ

∩ J

CL (74)

These ond onstraint given by equation(32)is reformulatedas:

X

j∈C

i

CL

m

N

i

BR

y

_i

CL

m

j

−

X

j∈C

i

repla ing

i

BR

m

by

i

BL

m

inequations (74)and (75).

The bra e failure onstraints for the bra es at the right-hand side of the top joints given by

equation(34) arereformulated into twolinear onstraints:

X

j∈C

_i

BR

m

N

i

BR

m

2f

yb

t

i

BR

m

j

y

i

BR

m

j

−

X

j∈C

CL

m

j

f

yb

isthe yieldstrengthof the bra es. For the bra es at theleft handside of thetopjoints,

thebra efailure onstraintsarereformulatedinthesame waybyrepla ing

i

BR

m

by

i

BL

m

inequations (76)and (77).

The resistan e to hord shear failure for the top joints given by equations (36) and (37) does

not have to be he ked for the enter top joint. The hord shear failure onstraints for the other

(16)

N

i

BR

m

6 X

j∈C

i

CL

m

f

y

A

v

,i

CL

m

j

√

3 sin

θ

i

BR

m

1 γ

M

5 !

y

i

CL

m

j

∀ m ∈ J

TJ (78)

N

i

BL

m

6 X

j∈C

i

CL

m

f

y

A

v

,i

CL

m

j

√

3 sin

θ

i

BL

m

1 γ

M

5 !

y

i

CL

m

j

∀ m ∈ J

TJ (79)

N

i

CL

m

6 X

j∈C

_i

CL

m













A

i

CL

m

j

− A

v

,i

CL

m

j

f

y

+

A

v

,i

CL

m

j

f

y

v

u

t1

−





γ

M

5 √

3 max

N

i

BL

m

sin

θ

i

BL

m

, N

i

BR

m

sin

θ

i

BR

m

f

y

A

v

,i

CL g

j





2 





1 γ

M

5 





y

i

CL

m

j

∀ m ∈ J

TJ

∩ J

CL (80)

For the bottom joints, only bra e failure of the overlapping bra e has to be he ked as given

byequation (38). The design resistan efor bra efailureofthebottom joints ontains theee tive

width

b

e,ov

of the onne tion between the overlapping bra e and the overlapped bra e, whi h has

to be al ulated as theminimum of two expressions. The bra efailure onstraints for thebottom

jointsarethereforereformulatedinto twolinear onstraints. Therst onstraint isreformulatedas:

X

j∈C

_i

BR

m

N

i

BR

m

m

j

−

X

j∈C

_i

BL

m

10f

yb

t

i

BL

m

j

b

i

BL

m j

t

i

BL

m j

y

_i

BL

m

j

6

0 ∀ m ∈ J

BJ (81)

These ond onstraint is reformulated as:

N

i

BR

m

−

X

j∈C

_i

BR

m

The MILPof theN-trussgiven byequations(50)to (56),and(64)to (82) onsidering

displa e-ment,memberandjoint onstraints onsistof1143designvariables,in luding1112binaryvariables,

21 ontinuous nodaldispla ement variablesand10 ontinuousgapvariables. TheMILP onsistsof

4751 onstraints, in luding232additional memberandjoint onstraints,and 36344non-zeros. The

MILP is again solved by means of Gurobi 5.6 using the same omputer as for the previous ase.

The optimal solutionis againfound inthe prepro essing stage inlessthan 0.2 se onds. Whenthe

MILP is solved without performing a presolve and without generating uts, the problem an no

longer besolvedat theroot node. In this ase, 1110nodesareexplored, andthe omputationtime

is1.3 se onds. Theresults aregiven intable8. The total weight of theN-trussis 2091.0kg.

When the optimization problem is solved taking into a ount the displa ement onstraints,

member onstraints, andjoint onstraints, thedesign variablesalsoin lude thegapsinorderto be

abletoimplementthejointgeometry onstraintsandtotakeintoa ountthee entri itymoments.

The optimalgap values aregiven intable 9. The displa ements aregiven intable 10. Alsoin this

ase thedispla ement onstraints arenot riti al. Theutilization ratio for ea h onstraint isgiven

intable 11. In this ase all the utilizationratios - in ludingthose relatedto thejoint onstraints

-aresmallerthan 1.

The weight of the optimized stru ture is about 15% higher than in the ase where joint

on-straints arenot onsidered. This shows that takinginto a ount joint onstraintsduring the

(17)

Forapra ti altrussdesignoptimizationproblem,threetypesof onstraintshavetobetakeninto

a ount: displa ement onstraints, member onstraints, and joint onstraints. Most optimization

methodsdonottakeintoa ountthejoint onstraints. Asa onsequen e,thejoint onstraintsmust

be he ked after optimization and the jointsmay need to bestrengthened. This paperveriesthe

importan e of joint onstraints inthe optimal design of trusses and presentsa method to a ount

for joint onstraints duringtheoptimization. Theoptimization problemis solved byreformulating

it asa mixed-integer linearprogram (MILP). The ve tor of design variables onsists of (1)binary

de ision variables whi h sele t a se tion from a atalog for ea h member, (2) nodal displa ement

variables, and (3) joint gaps when joint onstraints are taken into a ount. The MILP is solved

withthe ut-and-bran h method whi h isimplementedintheGurobi optimizer. Theadvantage of

this approa h isthatthe problem an be solved to globaloptimality.

This papermakesa omparison between the results obtained after optimization of a stati ally

determinate steel N-truss girder with and without joint onstraints. The results show thattaking

into a ount joint onstraints during theoptimization leads to a result with a weight that is 15%

higher than in the ase where these onstraints are not onsidered. When only displa ement and

member onstraintsaretakenintoa ountduringtheoptimization,oneshouldbevery arefulwhen

he king thejoint onstraintsand adapting theoptimized designa posteriori inorderto satisfyall

onstraints, sin e the joint onstraints have a large impa t on the total weight of the stru ture.

Taking into a ount joint onstraints during the optimization therefore leads to a ost redu tion

at two levels: in terms of engineering ost (no manual postpro essing step is needed) as well as

fabri ation ost(joint strengthening isavoided).

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Tables

Table 1: HEA prole atalog

Index Name Index Name Index Name

1 HEA100 9 HEA 260 17 HEA 500

2 HEA120 10 HEA 280 18 HEA 550

3 HEA140 11 HEA 300 19 HEA 600

4 HEA160 12 HEA 320 20 HEA 650

5 HEA180 13 HEA 340 21 HEA 700

6 HEA200 14 HEA 360 22 HEA 800

7 HEA220 15 HEA 400 23 HEA 900

8 HEA240 16 HEA 450 24 HEA 1000

Table 2: UPN prole atalog

Index Name Index Name Index Name

1 UPN 50 7 UPN 160 13 UPN 280

2 UPN 65 8 UPN 180 14 UPN 300

3 UPN 80 9 UPN 200 15 UPN 320

4 UPN 100 10 UPN 220 16 UPN 350

5 UPN 120 11 UPN 240 17 UPN 380

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Index Name Index Name Index Name 1 RHS 20x20x2 29 RHS 80x80x3 57 RHS 140x140x8 2 RHS 25x25x2 30 RHS 80x80x4 58 RHS 140x140x10 3 RHS 25x25x3 31 RHS 80x80x5 59 RHS 150x150x4 4 RHS 30x30x2 32 RHS 80x80x6 60 RHS 150x150x5 5 RHS 30x30x3 33 RHS 80x80x8 61 RHS 150x150x6 6 RHS 30x30x4 34 RHS 90x90x3 62 RHS 150x150x8 7 RHS 35x35x2 35 RHS 90x90x4 63 RHS 150x150x10 8 RHS 35x35x3 36 RHS 90x90x5 64 RHS 160x160x5 9 RHS 40x40x2 37 RHS 90x90x6 65 RHS 160x160x6 10 RHS 40x40x3 38 RHS 100x100x3 66 RHS 160x160x8 11 RHS 40x40x4 39 RHS 100x100x4 67 RHS 160x160x10 12 RHS 45x45x2 40 RHS 100x100x5 68 RHS 180x180x6 13 RHS 45x45x3 41 RHS 100x100x6 69 RHS 180x180x8 14 RHS 45x45x4 42 RHS 100x100x8 70 RHS 180x180x10 15 RHS 50x50x2 43 RHS100x100x10 71 RHS 180x180x12.5 16 RHS 50x50x3 44 RHS 110x110x4 72 RHS 200x200x5 17 RHS 50x50x4 45 RHS 110x110x5 73 RHS 200x200x6 18 RHS 50x50x5 46 RHS 120x120x3 74 RHS 200x200x8 19 RHS 60x60x2 47 RHS 120x120x4 75 RHS 200x200x10 20 RHS 60x60x3 48 RHS 120x120x5 76 RHS 200x200x12.5 21 RHS 60x60x4 49 RHS 120x120x6 77 RHS 220x220x6 22 RHS 60x60x5 50 RHS 120x120x8 78 RHS 220x220x8 23 RHS 60x60x6 51 RHS120x120x10 79 RHS 220x220x10 24 RHS 70x70x2 52 RHS 125x125x5 80 RHS 250x250x6 25 RHS 70x70x3 53 RHS 125x125x6 81 RHS 250x250x8 26 RHS 70x70x4 54 RHS 140x140x4 82 RHS 250x250x10 27 RHS 70x70x5 55 RHS 140x140x5 28 RHS 70x70x6 56 RHS 140x140x6

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Member Prole A [mm

2

℄

Top hords HEA 180 4530

Bottom hords UPN 220 3740

Bra e 11 RHS110x110x5 2036 Bra e 12 RHS125x125x5 2336 Bra e 13 RHS120x120x4 1815 Bra e 14 RHS120x120x4 1815 Bra e 15 RHS100x100x4 1495 Bra e 16 RHS90x90x4 1335 Bra e 17 RHS100x100x3 1141 Bra e 18 RHS70x70x3 781 Bra e 19 RHS70x70x3 781 Bra e 20 RHS40x40x2 294 Bra e 21 RHS70x70x2 534 Weight: 1826.3 kg

Table 5: Nodaldispla ementsof theN-trussoptimized onsidering only displa ement and member

onstraints.

Nodenumber 1 2 3 4 5 6 7 8 9 10 11 12

Horizontaldispla ement[mm ℄ 7.40 6.70 5.45 3.82 1.95 0 -6.60 -6.60 -5.75 -4.24 -2.26 0

Verti aldispla ement[mm ℄ -1.73 -21.32 -39.27 -54.25 -65.52 -72.18 0 -19.58 -37.62 -52.71 -64.16 -70.86

Table 6: Gapsand e entri ities introdu ed to ensure weldability of thejoints ofthe N-truss

opti-mized onsidering only displa ement and member onstraints.

Joint number Gaps [mm ℄ E entri ity[mm℄

1 10.0 67.9 2 8.0 67.4 3 8.0 36.1 4 6.0 20.0 5 5.0 -17.2 8 -120.0 7.0 9 -100.0 13.5 10 -100.0 -7.8 11 -70.0 -6.9 12 -70.0 -28.1

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member onstraints.

Member onstraints Joint onstraints

Memberresistan eratio Stabilityratio Jointresistan eratioHEA-RHS Jointresistan eratioUPN-RHS

Mem b er n um b er Chord w eb failure Bra e failure Chord shear failure Bra e failure 1 0.54 0.60 0.39 2 0.60 0.67 0.53 3 0.69 0.81 0.67 4 0.76 0.91 0.75 5 0.78 0.94 6 0.07 7 0.41 8 0.71 9 0.83 10 0.93 11 0.89 1.00 1.01 1.49 0.96 12 0.99 0.83 1.90 0.86 13 0.90 0.99 0.87 1.68 0.82 1.01 14 0.99 0.65 1.85 0.64 15 0.85 0.97 0.74 1.31 0.64 1.00 16 0.96 0.47 1.32 0.46 17 0.80 0.91 0.53 1.24 0.43 0.87 18 0.99 0.32 1.06 0.26 19 0.70 0.91 0.37 0.75 0.25 0.80 20 0.87 0.13 0.85 0.08 21 0.68 0.88 0.24 0.75 0.77

Table8: Se tionsoftheN-trussoptimized onsideringdispla ement,member,andjoint onstraints.

Member Prole A [mm

2

℄

Top hords HEA 200 5380

Bottom hords UPN 220 3740

Bra e 11 RHS 100x100x8 2724 Bra e 12 RHS 100x100x10 3257 Bra e 13 RHS 100x100x8 2724 Bra e 14 RHS80x80x8 2084 Bra e 15 RHS90x90x5 1636 Bra e 16 RHS80x80x5 1436 Bra e 17 RHS80x80x4 1175 Bra e 18 RHS70x70x3 781 Bra e 19 RHS70x70x3 781 Bra e 20 RHS60x60x3 661 Bra e 21 RHS60x60x3 661 Weight: 2091.0 kg

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mized onsidering displa ement, member, andjoint onstraints.

Joint number Gap [mm ℄ E entri ity [mm℄

1 18.0 43.7 2 16.0 27.6 3 10.0 16.6 4 7.0 1.5 5 6.0 -11.6 8 -100.0 -0.7 9 -90.0 -9.8 10 -80.0 -4.8 11 -70.0 -6.9 12 -60.0 -9.0

Table 10: Nodal displa ements of the N-truss optimized onsidering displa ement, member, and

joint onstraints.

Nodenumber 1 2 3 4 5 6 7 8 9 10 11 12

Horizontaldispla ement[mm ℄ 6.23 5.64 4.59 3.21 1.64 0 -6.60 -6.60 -5.75 -4.24 -2.26 0

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joint onstraints.

Member onstraints Joint onstraints

Memberresistan eratio Stabilityratio Jointresistan eratioHEA-RHS Jointresistan eratioUPN-RHS

Mem b er n um b er Chord w eb failure Bra e failure Chord shear failure Bra e failure 1 0.36 0.39 0.30 2 0.45 0.51 0.44 3 0.56 0.65 0.56 4 0.63 0.73 0.63 5 0.66 0.76 6 0.01 7 0.35 8 0.68 9 0.82 10 0.93 11 0.67 0.78 0.90 0.86 0.91 12 0.71 0.69 0.87 0.82 13 0.60 0.70 0.81 0.77 0.80 0.56 14 0.86 0.60 0.85 0.62 15 0.78 0.92 0.66 0.96 0.55 0.75 16 0.90 0.43 0.97 0.39 17 0.77 0.96 0.49 0.86 0.36 0.79 18 0.99 0.27 0.97 0.22 19 0.70 0.91 0.31 0.68 0.21 0.80 20 0.39 0.10 0.38 0.07 21 0.55 0.79 0.22 0.53 0.61