Research ArticleReliability-Based Design Optimization for Crane MetallicStructure Using ACO and AFOSM Based on China Standards
Xiaoning Fan and Xiaoheng Bi
Mechanical Engineering College Taiyuan University of Science and Technology Taiyuan Shanxi 030024 China
Correspondence should be addressed to Xiaoning Fan fannyfxn163com
Received 9 July 2014 Revised 4 January 2015 Accepted 19 January 2015
Academic Editor Paolo Lonetti
Copyright copy 2015 X Fan and X Bi This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The design optimization of crane metallic structures is of great significance in reducing their weight and cost Although it is knownthat uncertainties in the loads geometry dimensions and materials of crane metallic structures are inherent and inevitable andthat deterministic structural optimization can lead to an unreliable structure in practical applications little amount of research onthese factors has been reported This paper considers a sensitivity analysis of uncertain variables and constructs a reliability-baseddesign optimization model of an overhead traveling crane metallic structure An advanced first-order second-moment method isused to calculate the reliability indices of probabilistic constraints at each design point An effective ant colony optimization with amutation local search is developed to achieve the global optimal solution By applying our reliability-based design optimization toa realistic crane structure we demonstrate that compared with the practical design and the deterministic design optimization theproposed method could find the lighter structure weight while satisfying the deterministic and probabilistic stress deflection andstiffness constraints and is therefore both feasible and effective
1 Introduction
As tools for moving and transporting goods cranes are usedin various settings to aid the development of economy andundertake the heavy logistical handling tasks in factoriesrailways ports and so on Metallic structure is mainly madeout of rolledmerchant steel and plate steel byweldingmethodaccording to certain structural organization rules Cranemetallic structures (CMSs) also called crane bridge bear andtransfer the burden of crane and their own weights CMSsare mechanical skeleton and form the main componentsof cranes Their design qualities have direct impact on thetechnical and economic benefit as well as the safety of thewhole crane
Generally a CMS requires a large quantity of steel andconsequently weighs a considerable amount Its cost accountsfor over a third of the total cost of the crane Thus under thecondition of satisfying the relevant design codes improvingthe performances of the CMS saving material and reducingweight have important significance in terms of cost-savingsAs a consequence various scholars have researched on this
problem and current optimization methods mainly includefinite element method [1] neural networks [2] and Lagrangemultipliers [3] amongst others [4ndash6] However these meth-ods are all based on deterministic optimum designs anddo not consider the effect of randomness in the structureandor load Studies have shown that the loads effected onthe CMS and the materials and geometric dimensions of thestructure itself are uncertain Deterministic optimumdesignsare pushed to the limits of their constraints boundariesleaving no room for uncertainty Optimum designs obtainedwithout consideration of such uncertainties are thereforeunreliable [7 8] Recently there have been a few reports aboutreliability-based design of crane structure [9 10] Literature[9] researched on the reliability-based design of structure oftower crane based on the finite element analysis (FEA) andresponse surface method (RSM) Meng et al [10] analyzedthe reliability and sensitivity of crane metal structure bymeans of BP neural network based on finite element andfirst-order second-moment (FOSM) method Neverthelessdesign only considering reliability could not guarantee thebest work performance and the optimal design parameters
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 828930 12 pageshttpdxdoiorg1011552015828930
2 Mathematical Problems in Engineering
The aim of a design is to achieve adequate safety at minimumcost under the condition of meeting specified performancerequirements Hence the optimization based on reliabilityconcepts appears to be a more rational design philosophywhich is why reliability-based design optimization (RBDO)has been developed RBDO incorporates the optimization ofdesign parameters and reliability calculations for specifiedlimit states At present it is attracting increased attentionboth in theoretical research and practical applications [11ndash13] Despite advances in this area few RBDO approachesspecific to CMS have appeared in the technical literatureTherefore this paper develops an RBDO methodology foroptimizing CMS that both minimizes the weight and guar-antees structural reliability The main structural behaviorsare modeled by the crane design code (China Standard)[14] based on material mechanics structural mechanics andelasticity theory
CMSs are engaged in busy and heavy work It musthave sufficient strength stiffness and stability under complexoperating conditions Their design calculations involve thehyperstatic problem of space structures Therefore both thecalculating model and design calculation are very com-plicated Furthermore in practical production structuraldimensions are usually taken for integral multiples of mil-limeters and the specified thickness of the steel plates [15]Due to these manufacturing limitations the design variablescannot be considered as continuous but should be treatedas discrete in a large number of practical design situationswhich means that the CMS design optimization is a con-strained nonlinear optimization problem with discrete vari-ables known as NP-complete combinatorial optimizationTo solve such problems recent studies have focused on thedevelopment of heuristic optimization techniques such asgenetic algorithms (GAs) [16 17] particle swarm optimiza-tion (PSO) [18ndash20] ant colony optimization (ACO) [21 22]big bang-big crunch (BB-BC) [23 24] imperialist competi-tive algorithm (ICA) [25] and charged system search (CSS)[26] These algorithms can overcome most of the limitationsfound in traditional methods such as becoming trapped inlocal minima and impractical computational complexity [2728] In view of the simple operation easy implementationand the suitability of ACO for computational problemsinvolving discrete variables and combinatorial optimizationthe optimization process of BRDO described in this paper isperformed using ACOwith a mutation local search (ACOM)[29]
Structural reliability can be analyzed using analyticalmethods such as first- and second-order second moments(FOSM and SOSM) [30] and advanced first-order secondmoment (AFOSM) [31] or with simulation methods such asMonte Carlo sampling (MCS) method The FOSM methodis very simple and requires minimal computation effortbut sacrifices accuracy for nonlinear limit state functionsThe accuracy of the SOSM method is improved comparedwith that of the FOSM but its computation effort is greatlyincreased and this makes it not frequently used in practicesMCS method is accurate however it is computationallyintensive as it needs a large number of samples to evaluatesmall failure probabilities The AFOSM method a more
accurate analytical approach than the FOSM method isable to efficiently handle low-dimensional uncertainties andnonlinear limit state functions [32] and is applied in mostpractical cases It is used to calculate the reliability indices ofRBDO in this paper
The paper is organized as follows Section 2 outlines thegeneral formulation of discrete RBDO and then Section 3constructs the RBDO model of an overhead traveling CMS(OTCMS) Section 4 develops the ACOM algorithm usedfor the optimization process of the RBDO and Section 5describes the AFOSMmethod applied for reliability analysisTheRBDOprocedure is illustrated in Section 6 Some appliedexamples that demonstrate the potential of the proposedapproach for solving realistic problem are presented inSection 7 followed by concluding remarks in Section 8
2 Formulation of Discrete RBDO
In contrast to deterministic design optimization (DDO)RBDO assumes that quantities related to size materialsand applied loads of a structure have a random nature toconform to the actual one The parameters characterizingthese quantities are called random variables and these needto be taken into account in reliability analysis These randomvariables may be either random design variables or randomparameter variables In optimization process themean valuesof the random design variables are treated as optimizationvariables The formulation of discrete RBDO problem isgenerally written as follows
Find d
minimizing 119891 (dP)
subject to 119892119894119889 (dP) le 0 119894119889 = 1 119873
1
119877119895119901= prob (119892
119895119901 (XY) le 0) ge 119877119886119895119901
119895119901 = 1 2 1198732
(1)
where
dlower le d le dupper d isin 119877NDV P isin 119877NPV (2)
d = 120583(X) and P = 120583(Y) are the mean value vectorsof the random design vector X and random parametervector Y respectively 119891(dP) is the objective function(ie the structure weight or volume) 119892
119894119889(dP) le 0 and
119877119886119895119901
minus prob(119892119895119901(XY) le 0) le 0 are the deterministic
and probabilistic constraints prob(119892119895119901(XP) le 0) denotes
the probability of satisfying the 119895119901th performance function119892119895119901(XY) le 0 and this probability should be no less than
the desired design reliability 119877119886119895119901
1198731 119873
2are the number
of deterministic and probabilistic constraints respectivelyd can only take values from a given discrete set 119877NDVwhere NDV and NPV are the number of random design andparameter vectors respectively
Mathematical Problems in Engineering 3
2
2
1
1
End carriage Trolley
Rail
Main girder
End carriage
L
Figure 1 Metal structure for overhead travelling crane
3 RBDO Modeling of an OTCMS
Cranes are mechanically applied to moving loads withoutinterfering in activities on the ground As overhead trav-eling cranes are the most widely used a typical OTCMSis selected as the study object As shown in Figure 1 it iscomposed of two parallel main girders that span the widthof the bay between the runway girders The OTCMS moveslongitudinally and the two end carriages located on eithersides of the span house the wheel blocks Because the maingirders are the principal horizontal beams that support thetrolley and are supported by the end carriages they are theprimary load-carrying components and account for morethan 80 of the total weight of the OTCMS Therefore theRBDO of OTCMS mainly focuses on the design of thesemain girders The cranersquos solid-web girder is usually a boxsection fabricated from steel plate for themain and vicewebstop and bottom flanges as shown in Figure 2 Thereforegiven a span its RBDO is to obtain the minimum deadweight that is the minimum main girder cross-sectionalarea which simultaneously satisfies the required determin-istic and probabilistic constraints associated with strengthstiffness stability manufacturing process and dimensionallimits [14] In this paper a practical bias-rail box main girderis considered The mathematical model of its RBDO is givenin Table 1
A calculation diagram of section 1-1 of the main girder isshown in Figure 2 Figure 3 is a force diagram for this maingirder in the vertical and horizontal planes In Table 1 andFigures 2 and 3 119865
119902represents the uniform load 119875
119866119895(119895 =
1 2 ) denotes the concentrated load sum119875 stands for thetrolley wheel load applied by the sumof the lifting capacity119875
119876
and trolley weight 119875119866119909 119875
119867and 119865
119867represent the horizontal
concentrated anduniform inertial load respectively 119875119904is the
lateral forceIn the RBDO model the section dimensions are random
variables while their mean values are treated as designvariables (see Table 1) The independent uncertain variablesinclude section dimensions and some important parameters(Table 1) and the other parameters are considered to be
②③④
⑤
b
Y
①
Y
X X
e
ee
h
t
t
PH
be
t1
t2Fq
FH
sumP
Figure 2 Cross section 1-1 of main girder
PS
PS
PHFH
FHPH
L
(a) On the horizontal plane
PG PGPG sumP Fq
(b) On the vertical plane
Figure 3 Force diagram for main girder
deterministic It is assumed that all random variables arenormally distributed around their mean value Effects of eachrandom variable on the active constraints at the optimalpoint are illustrated in Figure 4 The inequality constraintsof the RBDO problem define in turn the failure by stressdeflection fatigue and stiffness These structural behaviorsare described by the assumed limit state functions givenin Table 1 [15] With respect to estimating the most criticalconfiguration of the trolley on themain girder there are threeconfigurations (i) The limit state functions 119892
119895119901(XY) 119895119901 =
1 2 3 correspond to the position of the maximum bendingmoment when a fully loaded trolley is lowering and braking
4 Mathematical Problems in Engineering
Table1Mathematicalmod
elof
theR
BDOforO
TCMS
Designvaria
blem
eanvalues
oftheo
ptim
izationprob
lem
d=[11988911198892119889311988941198895]119879=120583(X)=[120583(1199091)120583(1199092)120583(1199093)120583(1199094)120583(1199095)]119879
Designvaria
blem
eanvalue
1198891
1198892
1198893
1198894
1198895
Symbo
lℎ
119905119905 1
119905 2119887
Rand
omdesig
nvaria
bles
andparameterso
fthe
optim
izationprob
lem
S=[XY]119879=[11990911199092119909311990941199095119910111991021199103]119879=[11990411199042119904311990441199045119904611990471199048]119879
Varia
blea
ndparameter
1199091
1199092
1199093
1199094
1199095
1199101
1199102
1199103
Symbo
lℎ
119905119905 1
119905 2119887
119875119876
119875119866119909
119864
Objectiv
efun
ction
minim
izingthec
ross-sectio
nalareao
fthe
girder119891(dP)=ℎsdot(119905 1+119905 2)+119905sdot(2sdot119887+3sdot119890+119887 119890)
119890istakenequalto20
mm
which
facilitates
welding
and119887 119890issetequ
alto
15119905so
asto
guaranteethe
localstabilityof
flangeo
verhanging
andther
ailinstallatio
nInequalityconstraintso
fthe
RBDOprob
lem
Reliabilityconstraints
Limitsta
tefunctio
nsDescriptio
n1198771198861minusprob(1198921(XY)le0)le0
1198921(XY)=1205901minus[120590]
Maxim
umstr
essc
ombinedatdangerou
spoint
(1)(1-1
section)
1198771198862minusprob(1198922(XY)le0)le0
1198922(XY)=1205902minus[120590]
Normalstr
essa
tdangerous
point(2)
(1-1sectio
n)
1198771198863minusprob(1198923(XY)le0)le0
1198923(XY)=1205903minus[120590]
Normalstr
essa
tdangerous
point(3)
(1-1sectio
n)
1198771198864minusprob(1198924(XY)le0)le0
1198924(XY)=120591 1minus[120591]
Maxim
umshearstre
ssatthem
iddleo
fmainweb
(2-2
section)
1198771198865minusprob(1198925(XY)le0)le0
1198925(XY)=120591 2minus[120591]
Horizon
talshear
stressinthefl
ange
(2-2
section)
1198771198866minusprob(1198926(XY)le0)le0
1198926(XY)=120590max4minus[1205901199034]
Fatig
uestr
engthatdangerou
spoint
(4)(1-1
section)
1198771198867minusprob(1198927(XY)le0)le0
1198927(XY)=120590max5minus[1205901199035]
Fatig
uestr
engthatdangerou
spoint
(5)(1-1
section)
1198771198868minusprob(1198928(XY)le0)le0
1198928(XY)=119891Vminus[119891
V]Ve
rticalstaticdeflectionof
them
idspan
1198771198869minusprob(1198929(XY)le0)le0
1198929(XY)=119891ℎminus[119891ℎ]
Horizon
talstatic
displacemento
fthe
midspan
11987711988610minusprob(11989210(XY)le0)le0
11989210(XY)=[119891119881]minus119891119881
Verticalnaturalvibratio
nfre
quency
ofthem
idspan
11987711988611minusprob(11989211(XY)le0)le0
11989211(XY)=[119891119867]minus119891119867
Horizon
taln
aturalvibrationfre
quency
ofthem
idspan
Deterministicconstraints
Natureo
fcon
straint
Descriptio
n1198921(dP)=ℎ119887minus3le0
Stabilityconstraint
Height-to-width
ratio
oftheb
oxgirder
1198921198951(dP)=119889119894minus119889up
per
119894le0
Boun
dary
constraint
Upp
erbo
unds
ofdesig
nvaria
bles1198951
=2410119894=125
1198921198952(dP)=119889lower
119894minus119889119894le0
Boun
dary
constraint
Lower
boun
dsof
desig
nvaria
bles1198952
=3511119894=125
Materialp
ermissibleno
rmalstr
ess[120590]=120590119904133andperm
issibleshearstre
ss[120591]=[120590]radic3120590119904representsmaterialyield
stress[1205901199034]and[1205901199035]arethe
weld
edjointtensilefatigue
perm
issible
stressesfor
points(4)a
nd(5)respectiv
ely[119891V]and[119891ℎ]arethe
verticalandho
rizon
talp
ermissiblestaticstiffnessesw
hich
aretaken
equalto119871800
and1198712000respectiv
elyand
119871isthe
span
oftheg
irder[119891119881]and[119891119867]arethe
verticalandho
rizon
talp
ermissibledynamicstiffnessesw
hich
aretaken
equalto2sim
4Hza
nd15sim2H
zrespectiv
ely119889
lower
119894and119889up
per
119894arethe
lower
andup
perb
ound
softhe
desig
nvaria
ble119889
119894
where
1205901=radic[119872119909119882119909+119872119910119882119910]2+[120601sum119875(4ℎ119892+100)1199051]2minus[119872119909119882119909+119872119910119882119910][120601sum119875(4ℎ119892+100)1199051]+3[119865119901119878119910119868119909(1199051+1199052)+119879119899211986001199051]21205902=1198721199091198821015840 119909+1198721199101198821015840 1199101205903=11987211990911988210158401015840
119909+11987211991011988210158401015840
119910
1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840
119909120590max5=1198721199091198821015840101584010158401015840
119909119891V=sum119875119871348119864119868119909119891ℎ=(119875119867119871348119864119868119910)(1minus341199031)+(51198651198671198714384119864119868119910)(1minus451199031)
119891119881=(12120587)radic119892(1199100+1205820)(1+120573)1199100=119875119876119871396119864119868119909120573=((119865119902119871+119875119866119909)119875119876)(1199100(1199100+1205820))2119891119867=(12120587)radic192119864119868119910119898119890(41199031minus3)1198713119898119890=(119865119902119871+119875119876+119875119866119909)119892
119872119909119872119910arethebend
ingmom
ents119865119901119865119890119865119890119867aretheshearforcesand1198791198991198791198991arethetorquesa
tdifferentlocations119868119909119868119910deno
tetheinertia
lmod
uli1198821199091198821015840 11990911988210158401015840
119909119882101584010158401015840
1199091198821015840101584010158401015840
1199091198821199101198821015840 11991011988210158401015840
119910deno
tethestr
ength
mod
uliand119878119910deno
testhe
topflang
estatic
mom
ent11986001198601198890representn
etarea
atdifferent
sectionsℎ119892representstheh
eighto
frailandℎ119889representstheh
eighto
fgird
eratthee
nd-span119864iselastic
mod
ulus119892
isgravity
constant120593
isim
pactfactor1199031iscompu
tingcoeffi
ciento
fthe
craneb
ridgeand1205820isthen
etelo
ngationof
thes
teelwire
ropeTh
eother
parametersa
rethes
amea
sbefore
Mathematical Problems in Engineering 5
0 10h
b
k
E
Sour
ces
Sour
ces
minus10 0 10 20
0 20 40 minus40 minus20 0 20
minus40 minus20 0 20 minus40 minus20 0 20
Effect on fV ()minus20
minus40 minus20
minus10
PGx
PQ
k2
k1
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1 Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Effect on f ()
Effect on 1205902 () Effect on 1205901 ()
Effect on 1205903 ()
Effect on fH ()
Figure 4 Effects of the random variables on the active constraints
in the midspan and at the same time the crane bridge is start-ing or braking (ii) The limit state functions 119892
119895119901(XY) 119895119901 =
4 5 correspond to the position of the maximum shearstress when a fully loaded trolley is lowering and brakingat the end of the span and at the same time the cranebridge is starting or braking (iii) The limit state functions119892119895119901(XY) 119895119901 = 9 10 11 12 correspond to the position of
maximum deflection This is when a fully loaded trolley ispositioned in the midspan The fatigue limit state functions119892119895119901(XY) 119895119901 = 7 8 correspond to the normal working
condition of the crane The local stability (local buckling) ofthe main girder can be guaranteed by arranging transverseand longitudinal stiffeners to form the grids according tothe width-to-thickness ratio of the flange and the height-to-thickness ratio of the web Thus only the global stabilityof the main girder is considered here Therefore the RBDOmodel of the OTCMSmain girder is a five-dimensional opti-mization problem with 11 deterministic and 11 probabilisticconstraints
4 The ACO for the OTCMS
ACO simulates the behavior of real life ant colonies inwhich individual ants deposit pheromone along a path whilemoving from the nest to food sources and vice versaTherebythe pheromone trail enables individual to smell and selectthe optimal routs The paths with more pheromone aremore likely to be selected by other ants bringing on furtheramplification of the current pheromone trails and producinga positive feedback process This behavior forms the shortestpath from the nest to the food source and vice versa Thefirst ACO algorithm called ant system (AS) was appliedto solve the traveling salesman problem Because the searchparallelism of ACO is based on the components of a solutionlevel it is very efficient Thus since the introduction of ASthe ACO metaheuristic has been widely used in many fields[33 34] including for structural optimization and has shownpromising results for various applications Therefore ACO isselected as optimization algorithm in the present study
6 Mathematical Problems in Engineering
41 Representation and Initialization of Solution In our ACOalgorithm each solution is composed of different arrayelements that correspond to different design variables Oneantrsquos search path represents a solution to the optimizationproblem or a set of design schemes The path of the 119894th antin the 119899-dimensional search space at iteration 119905 could bedenoted by d119905
119894(119894 = 1 2 popsize popsize denotes the
population size) Continuous array elements array119895[119898
119895] are
used to store the mean values of the discrete design variable119889119895in the nondecreasing order (119898
119895denotes the array sequence
number for the 119895th design variable mean value 119889119895 This is
integer with 1 le 119898119895le 119872
119895 119895 = 1 2 119899 where 119899 is the
number of design variables and 119872119895is the number of discrete
values available for 119889119895 119889lower
119895le 119889
119895le 119889
upper119895
where 119889lower119895
and119889upper119895
are the lower and upper bounds of 119889119895) Thus consider
d119905119894= array
1[119898
1] array
2[119898
2] array
119899[119898
119899]119905
119894
= [1198891 1198892 119889
119899]119905
119894
(3)
The initial solution is randomly selectedWe set the initialpheromone level [119879
119895(119898
119895)]0 of all array elements in the space
to be zero The heuristic information 120578119895(119898
119895) of array element
array119895[119898
119895] is expressed by (4) so as to induce subsequent
solutions to select smaller variable values as far as possibleand accelerate optimization process
120578119895[119898
119895] =
119872119895minus 119898
119895+ 1
119872119895
(4)
42 Selection Probability and Construction of Solutions Asthe ants move from node to node to generate paths theywill ceaselessly select the next node from the unvisitedneighbor nodes This process forms the ant paths and thusin our algorithm constructs solutions In accordance withthe transition rule of ant colony system (ACS) [35] eachant begins with the first array element array
1[119898
1] storing
the first design variable mean value 1198891and selects array
elements in proper until array119899[119898
119899] In this way a solution
is constructed Hence for the 119894th ant on the array elementarray
119895[119898
119895] the selection probability of the next array element
array119896[119898
119896] is given by
119878119894(119895 119896) =
maxarray119896[119898119896]isin119869119896
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
if 119902 lt 1199020
119901119894(119895 119896) otherwise
(5a)
119901119894(119895 119896)
=
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
sumarray119896[119898119896]isin119869119896(119903)
[120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)]
if array119896[119898
119896] isin 119869
119896
0 otherwise
(5b)
where 119869119896is the set of feasible neighbor array elements of
array119895[119898
119895] (119896 = 119895+1 and 119896 le 119899) 119879
119896[119898
119896] and 120578
119896(119898
119896) denote
the pheromone intensity and heuristic information on arrayelements array
119896[119898
119896] respectively 120572 and 120585 give the relative
importance of trail 119879119896[119898
119896] and the heuristic information
120578119896(119898
119896) respectively Other parameters are as previously
described
43 Local Search Based on Mutation Operator It is wellknown that ACO easily converges to local optima under pos-itive feedback A local search can explore the neighborhoodand enhance the quality of the solution
GAs are a powerful tool for solving combinatorial opti-mization problems They solve optimization problems usingthe idea of Darwinian evolution Basic evolution opera-tions including crossover mutation and selection makeGAs appropriate for performing search In this paper themutation operation is introduced to the proposed algorithmto perform local searches We assume that the currentglobal optimal solution dbest = array
1[119898
1] array
2[119898
2]
array119899[119898
119899]best has not been improved for a certain number
of stagnation generations sgen One or more array elementsare chosen at random from dbest and these are changedin a certain manner Through this mutation operation weobtain the mutated solution d1015840best If d
1015840
best is better than dbestwe replace dbest with d1015840best Otherwise the global optimalsolution remains unchanged
44 Pheromone Updating The pheromone updating rules ofACO include global updating and local rules When ant 119894 hasfinished a path the pheromone trails on the array elementsthroughwhich the ant has passed are updated In this processthe pheromone on the visited array elements is consideredto have evaporated thus increasing the probability thatfollowing ants will traverse the other array elements Thisprocess is performed after each ant has found a path it is alocal pheromone update rule with the aim of obtaining moredispersed solutions The local pheromone update rule is
[119879119895(119898
119895)]119905+1
119894= (1 minus 120574) sdot [119879
119895(119898
119895)]119905
119894
119895 = 1 2 119899 1 le 119898119895le 119872
119895
(6a)
When all of the ants have completed their paths (whichis called a cycle) a global pheromone update is applied tothe array elements passed through by all ants This processis applied in an iterative mode [29] The rule is described asfollows
[119879119895(119898
119895)]119905+1
119894= (1 minus 120588) sdot [119879
119895(119898
119895)]119905
119894+ 120582 sdot 119865 (d119905
119894PX119905
119894Y) (6b)
where120582
=
1198881 if array
119895[119898
119895] isin the globally (iteratively) best tour
1198882 if array
119895[119898
119895] isin the iterationworst tour
1198883 otherwise
(7)
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
The aim of a design is to achieve adequate safety at minimumcost under the condition of meeting specified performancerequirements Hence the optimization based on reliabilityconcepts appears to be a more rational design philosophywhich is why reliability-based design optimization (RBDO)has been developed RBDO incorporates the optimization ofdesign parameters and reliability calculations for specifiedlimit states At present it is attracting increased attentionboth in theoretical research and practical applications [11ndash13] Despite advances in this area few RBDO approachesspecific to CMS have appeared in the technical literatureTherefore this paper develops an RBDO methodology foroptimizing CMS that both minimizes the weight and guar-antees structural reliability The main structural behaviorsare modeled by the crane design code (China Standard)[14] based on material mechanics structural mechanics andelasticity theory
CMSs are engaged in busy and heavy work It musthave sufficient strength stiffness and stability under complexoperating conditions Their design calculations involve thehyperstatic problem of space structures Therefore both thecalculating model and design calculation are very com-plicated Furthermore in practical production structuraldimensions are usually taken for integral multiples of mil-limeters and the specified thickness of the steel plates [15]Due to these manufacturing limitations the design variablescannot be considered as continuous but should be treatedas discrete in a large number of practical design situationswhich means that the CMS design optimization is a con-strained nonlinear optimization problem with discrete vari-ables known as NP-complete combinatorial optimizationTo solve such problems recent studies have focused on thedevelopment of heuristic optimization techniques such asgenetic algorithms (GAs) [16 17] particle swarm optimiza-tion (PSO) [18ndash20] ant colony optimization (ACO) [21 22]big bang-big crunch (BB-BC) [23 24] imperialist competi-tive algorithm (ICA) [25] and charged system search (CSS)[26] These algorithms can overcome most of the limitationsfound in traditional methods such as becoming trapped inlocal minima and impractical computational complexity [2728] In view of the simple operation easy implementationand the suitability of ACO for computational problemsinvolving discrete variables and combinatorial optimizationthe optimization process of BRDO described in this paper isperformed using ACOwith a mutation local search (ACOM)[29]
Structural reliability can be analyzed using analyticalmethods such as first- and second-order second moments(FOSM and SOSM) [30] and advanced first-order secondmoment (AFOSM) [31] or with simulation methods such asMonte Carlo sampling (MCS) method The FOSM methodis very simple and requires minimal computation effortbut sacrifices accuracy for nonlinear limit state functionsThe accuracy of the SOSM method is improved comparedwith that of the FOSM but its computation effort is greatlyincreased and this makes it not frequently used in practicesMCS method is accurate however it is computationallyintensive as it needs a large number of samples to evaluatesmall failure probabilities The AFOSM method a more
accurate analytical approach than the FOSM method isable to efficiently handle low-dimensional uncertainties andnonlinear limit state functions [32] and is applied in mostpractical cases It is used to calculate the reliability indices ofRBDO in this paper
The paper is organized as follows Section 2 outlines thegeneral formulation of discrete RBDO and then Section 3constructs the RBDO model of an overhead traveling CMS(OTCMS) Section 4 develops the ACOM algorithm usedfor the optimization process of the RBDO and Section 5describes the AFOSMmethod applied for reliability analysisTheRBDOprocedure is illustrated in Section 6 Some appliedexamples that demonstrate the potential of the proposedapproach for solving realistic problem are presented inSection 7 followed by concluding remarks in Section 8
2 Formulation of Discrete RBDO
In contrast to deterministic design optimization (DDO)RBDO assumes that quantities related to size materialsand applied loads of a structure have a random nature toconform to the actual one The parameters characterizingthese quantities are called random variables and these needto be taken into account in reliability analysis These randomvariables may be either random design variables or randomparameter variables In optimization process themean valuesof the random design variables are treated as optimizationvariables The formulation of discrete RBDO problem isgenerally written as follows
Find d
minimizing 119891 (dP)
subject to 119892119894119889 (dP) le 0 119894119889 = 1 119873
1
119877119895119901= prob (119892
119895119901 (XY) le 0) ge 119877119886119895119901
119895119901 = 1 2 1198732
(1)
where
dlower le d le dupper d isin 119877NDV P isin 119877NPV (2)
d = 120583(X) and P = 120583(Y) are the mean value vectorsof the random design vector X and random parametervector Y respectively 119891(dP) is the objective function(ie the structure weight or volume) 119892
119894119889(dP) le 0 and
119877119886119895119901
minus prob(119892119895119901(XY) le 0) le 0 are the deterministic
and probabilistic constraints prob(119892119895119901(XP) le 0) denotes
the probability of satisfying the 119895119901th performance function119892119895119901(XY) le 0 and this probability should be no less than
the desired design reliability 119877119886119895119901
1198731 119873
2are the number
of deterministic and probabilistic constraints respectivelyd can only take values from a given discrete set 119877NDVwhere NDV and NPV are the number of random design andparameter vectors respectively
Mathematical Problems in Engineering 3
2
2
1
1
End carriage Trolley
Rail
Main girder
End carriage
L
Figure 1 Metal structure for overhead travelling crane
3 RBDO Modeling of an OTCMS
Cranes are mechanically applied to moving loads withoutinterfering in activities on the ground As overhead trav-eling cranes are the most widely used a typical OTCMSis selected as the study object As shown in Figure 1 it iscomposed of two parallel main girders that span the widthof the bay between the runway girders The OTCMS moveslongitudinally and the two end carriages located on eithersides of the span house the wheel blocks Because the maingirders are the principal horizontal beams that support thetrolley and are supported by the end carriages they are theprimary load-carrying components and account for morethan 80 of the total weight of the OTCMS Therefore theRBDO of OTCMS mainly focuses on the design of thesemain girders The cranersquos solid-web girder is usually a boxsection fabricated from steel plate for themain and vicewebstop and bottom flanges as shown in Figure 2 Thereforegiven a span its RBDO is to obtain the minimum deadweight that is the minimum main girder cross-sectionalarea which simultaneously satisfies the required determin-istic and probabilistic constraints associated with strengthstiffness stability manufacturing process and dimensionallimits [14] In this paper a practical bias-rail box main girderis considered The mathematical model of its RBDO is givenin Table 1
A calculation diagram of section 1-1 of the main girder isshown in Figure 2 Figure 3 is a force diagram for this maingirder in the vertical and horizontal planes In Table 1 andFigures 2 and 3 119865
119902represents the uniform load 119875
119866119895(119895 =
1 2 ) denotes the concentrated load sum119875 stands for thetrolley wheel load applied by the sumof the lifting capacity119875
119876
and trolley weight 119875119866119909 119875
119867and 119865
119867represent the horizontal
concentrated anduniform inertial load respectively 119875119904is the
lateral forceIn the RBDO model the section dimensions are random
variables while their mean values are treated as designvariables (see Table 1) The independent uncertain variablesinclude section dimensions and some important parameters(Table 1) and the other parameters are considered to be
②③④
⑤
b
Y
①
Y
X X
e
ee
h
t
t
PH
be
t1
t2Fq
FH
sumP
Figure 2 Cross section 1-1 of main girder
PS
PS
PHFH
FHPH
L
(a) On the horizontal plane
PG PGPG sumP Fq
(b) On the vertical plane
Figure 3 Force diagram for main girder
deterministic It is assumed that all random variables arenormally distributed around their mean value Effects of eachrandom variable on the active constraints at the optimalpoint are illustrated in Figure 4 The inequality constraintsof the RBDO problem define in turn the failure by stressdeflection fatigue and stiffness These structural behaviorsare described by the assumed limit state functions givenin Table 1 [15] With respect to estimating the most criticalconfiguration of the trolley on themain girder there are threeconfigurations (i) The limit state functions 119892
119895119901(XY) 119895119901 =
1 2 3 correspond to the position of the maximum bendingmoment when a fully loaded trolley is lowering and braking
4 Mathematical Problems in Engineering
Table1Mathematicalmod
elof
theR
BDOforO
TCMS
Designvaria
blem
eanvalues
oftheo
ptim
izationprob
lem
d=[11988911198892119889311988941198895]119879=120583(X)=[120583(1199091)120583(1199092)120583(1199093)120583(1199094)120583(1199095)]119879
Designvaria
blem
eanvalue
1198891
1198892
1198893
1198894
1198895
Symbo
lℎ
119905119905 1
119905 2119887
Rand
omdesig
nvaria
bles
andparameterso
fthe
optim
izationprob
lem
S=[XY]119879=[11990911199092119909311990941199095119910111991021199103]119879=[11990411199042119904311990441199045119904611990471199048]119879
Varia
blea
ndparameter
1199091
1199092
1199093
1199094
1199095
1199101
1199102
1199103
Symbo
lℎ
119905119905 1
119905 2119887
119875119876
119875119866119909
119864
Objectiv
efun
ction
minim
izingthec
ross-sectio
nalareao
fthe
girder119891(dP)=ℎsdot(119905 1+119905 2)+119905sdot(2sdot119887+3sdot119890+119887 119890)
119890istakenequalto20
mm
which
facilitates
welding
and119887 119890issetequ
alto
15119905so
asto
guaranteethe
localstabilityof
flangeo
verhanging
andther
ailinstallatio
nInequalityconstraintso
fthe
RBDOprob
lem
Reliabilityconstraints
Limitsta
tefunctio
nsDescriptio
n1198771198861minusprob(1198921(XY)le0)le0
1198921(XY)=1205901minus[120590]
Maxim
umstr
essc
ombinedatdangerou
spoint
(1)(1-1
section)
1198771198862minusprob(1198922(XY)le0)le0
1198922(XY)=1205902minus[120590]
Normalstr
essa
tdangerous
point(2)
(1-1sectio
n)
1198771198863minusprob(1198923(XY)le0)le0
1198923(XY)=1205903minus[120590]
Normalstr
essa
tdangerous
point(3)
(1-1sectio
n)
1198771198864minusprob(1198924(XY)le0)le0
1198924(XY)=120591 1minus[120591]
Maxim
umshearstre
ssatthem
iddleo
fmainweb
(2-2
section)
1198771198865minusprob(1198925(XY)le0)le0
1198925(XY)=120591 2minus[120591]
Horizon
talshear
stressinthefl
ange
(2-2
section)
1198771198866minusprob(1198926(XY)le0)le0
1198926(XY)=120590max4minus[1205901199034]
Fatig
uestr
engthatdangerou
spoint
(4)(1-1
section)
1198771198867minusprob(1198927(XY)le0)le0
1198927(XY)=120590max5minus[1205901199035]
Fatig
uestr
engthatdangerou
spoint
(5)(1-1
section)
1198771198868minusprob(1198928(XY)le0)le0
1198928(XY)=119891Vminus[119891
V]Ve
rticalstaticdeflectionof
them
idspan
1198771198869minusprob(1198929(XY)le0)le0
1198929(XY)=119891ℎminus[119891ℎ]
Horizon
talstatic
displacemento
fthe
midspan
11987711988610minusprob(11989210(XY)le0)le0
11989210(XY)=[119891119881]minus119891119881
Verticalnaturalvibratio
nfre
quency
ofthem
idspan
11987711988611minusprob(11989211(XY)le0)le0
11989211(XY)=[119891119867]minus119891119867
Horizon
taln
aturalvibrationfre
quency
ofthem
idspan
Deterministicconstraints
Natureo
fcon
straint
Descriptio
n1198921(dP)=ℎ119887minus3le0
Stabilityconstraint
Height-to-width
ratio
oftheb
oxgirder
1198921198951(dP)=119889119894minus119889up
per
119894le0
Boun
dary
constraint
Upp
erbo
unds
ofdesig
nvaria
bles1198951
=2410119894=125
1198921198952(dP)=119889lower
119894minus119889119894le0
Boun
dary
constraint
Lower
boun
dsof
desig
nvaria
bles1198952
=3511119894=125
Materialp
ermissibleno
rmalstr
ess[120590]=120590119904133andperm
issibleshearstre
ss[120591]=[120590]radic3120590119904representsmaterialyield
stress[1205901199034]and[1205901199035]arethe
weld
edjointtensilefatigue
perm
issible
stressesfor
points(4)a
nd(5)respectiv
ely[119891V]and[119891ℎ]arethe
verticalandho
rizon
talp
ermissiblestaticstiffnessesw
hich
aretaken
equalto119871800
and1198712000respectiv
elyand
119871isthe
span
oftheg
irder[119891119881]and[119891119867]arethe
verticalandho
rizon
talp
ermissibledynamicstiffnessesw
hich
aretaken
equalto2sim
4Hza
nd15sim2H
zrespectiv
ely119889
lower
119894and119889up
per
119894arethe
lower
andup
perb
ound
softhe
desig
nvaria
ble119889
119894
where
1205901=radic[119872119909119882119909+119872119910119882119910]2+[120601sum119875(4ℎ119892+100)1199051]2minus[119872119909119882119909+119872119910119882119910][120601sum119875(4ℎ119892+100)1199051]+3[119865119901119878119910119868119909(1199051+1199052)+119879119899211986001199051]21205902=1198721199091198821015840 119909+1198721199101198821015840 1199101205903=11987211990911988210158401015840
119909+11987211991011988210158401015840
119910
1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840
119909120590max5=1198721199091198821015840101584010158401015840
119909119891V=sum119875119871348119864119868119909119891ℎ=(119875119867119871348119864119868119910)(1minus341199031)+(51198651198671198714384119864119868119910)(1minus451199031)
119891119881=(12120587)radic119892(1199100+1205820)(1+120573)1199100=119875119876119871396119864119868119909120573=((119865119902119871+119875119866119909)119875119876)(1199100(1199100+1205820))2119891119867=(12120587)radic192119864119868119910119898119890(41199031minus3)1198713119898119890=(119865119902119871+119875119876+119875119866119909)119892
119872119909119872119910arethebend
ingmom
ents119865119901119865119890119865119890119867aretheshearforcesand1198791198991198791198991arethetorquesa
tdifferentlocations119868119909119868119910deno
tetheinertia
lmod
uli1198821199091198821015840 11990911988210158401015840
119909119882101584010158401015840
1199091198821015840101584010158401015840
1199091198821199101198821015840 11991011988210158401015840
119910deno
tethestr
ength
mod
uliand119878119910deno
testhe
topflang
estatic
mom
ent11986001198601198890representn
etarea
atdifferent
sectionsℎ119892representstheh
eighto
frailandℎ119889representstheh
eighto
fgird
eratthee
nd-span119864iselastic
mod
ulus119892
isgravity
constant120593
isim
pactfactor1199031iscompu
tingcoeffi
ciento
fthe
craneb
ridgeand1205820isthen
etelo
ngationof
thes
teelwire
ropeTh
eother
parametersa
rethes
amea
sbefore
Mathematical Problems in Engineering 5
0 10h
b
k
E
Sour
ces
Sour
ces
minus10 0 10 20
0 20 40 minus40 minus20 0 20
minus40 minus20 0 20 minus40 minus20 0 20
Effect on fV ()minus20
minus40 minus20
minus10
PGx
PQ
k2
k1
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1 Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Effect on f ()
Effect on 1205902 () Effect on 1205901 ()
Effect on 1205903 ()
Effect on fH ()
Figure 4 Effects of the random variables on the active constraints
in the midspan and at the same time the crane bridge is start-ing or braking (ii) The limit state functions 119892
119895119901(XY) 119895119901 =
4 5 correspond to the position of the maximum shearstress when a fully loaded trolley is lowering and brakingat the end of the span and at the same time the cranebridge is starting or braking (iii) The limit state functions119892119895119901(XY) 119895119901 = 9 10 11 12 correspond to the position of
maximum deflection This is when a fully loaded trolley ispositioned in the midspan The fatigue limit state functions119892119895119901(XY) 119895119901 = 7 8 correspond to the normal working
condition of the crane The local stability (local buckling) ofthe main girder can be guaranteed by arranging transverseand longitudinal stiffeners to form the grids according tothe width-to-thickness ratio of the flange and the height-to-thickness ratio of the web Thus only the global stabilityof the main girder is considered here Therefore the RBDOmodel of the OTCMSmain girder is a five-dimensional opti-mization problem with 11 deterministic and 11 probabilisticconstraints
4 The ACO for the OTCMS
ACO simulates the behavior of real life ant colonies inwhich individual ants deposit pheromone along a path whilemoving from the nest to food sources and vice versaTherebythe pheromone trail enables individual to smell and selectthe optimal routs The paths with more pheromone aremore likely to be selected by other ants bringing on furtheramplification of the current pheromone trails and producinga positive feedback process This behavior forms the shortestpath from the nest to the food source and vice versa Thefirst ACO algorithm called ant system (AS) was appliedto solve the traveling salesman problem Because the searchparallelism of ACO is based on the components of a solutionlevel it is very efficient Thus since the introduction of ASthe ACO metaheuristic has been widely used in many fields[33 34] including for structural optimization and has shownpromising results for various applications Therefore ACO isselected as optimization algorithm in the present study
6 Mathematical Problems in Engineering
41 Representation and Initialization of Solution In our ACOalgorithm each solution is composed of different arrayelements that correspond to different design variables Oneantrsquos search path represents a solution to the optimizationproblem or a set of design schemes The path of the 119894th antin the 119899-dimensional search space at iteration 119905 could bedenoted by d119905
119894(119894 = 1 2 popsize popsize denotes the
population size) Continuous array elements array119895[119898
119895] are
used to store the mean values of the discrete design variable119889119895in the nondecreasing order (119898
119895denotes the array sequence
number for the 119895th design variable mean value 119889119895 This is
integer with 1 le 119898119895le 119872
119895 119895 = 1 2 119899 where 119899 is the
number of design variables and 119872119895is the number of discrete
values available for 119889119895 119889lower
119895le 119889
119895le 119889
upper119895
where 119889lower119895
and119889upper119895
are the lower and upper bounds of 119889119895) Thus consider
d119905119894= array
1[119898
1] array
2[119898
2] array
119899[119898
119899]119905
119894
= [1198891 1198892 119889
119899]119905
119894
(3)
The initial solution is randomly selectedWe set the initialpheromone level [119879
119895(119898
119895)]0 of all array elements in the space
to be zero The heuristic information 120578119895(119898
119895) of array element
array119895[119898
119895] is expressed by (4) so as to induce subsequent
solutions to select smaller variable values as far as possibleand accelerate optimization process
120578119895[119898
119895] =
119872119895minus 119898
119895+ 1
119872119895
(4)
42 Selection Probability and Construction of Solutions Asthe ants move from node to node to generate paths theywill ceaselessly select the next node from the unvisitedneighbor nodes This process forms the ant paths and thusin our algorithm constructs solutions In accordance withthe transition rule of ant colony system (ACS) [35] eachant begins with the first array element array
1[119898
1] storing
the first design variable mean value 1198891and selects array
elements in proper until array119899[119898
119899] In this way a solution
is constructed Hence for the 119894th ant on the array elementarray
119895[119898
119895] the selection probability of the next array element
array119896[119898
119896] is given by
119878119894(119895 119896) =
maxarray119896[119898119896]isin119869119896
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
if 119902 lt 1199020
119901119894(119895 119896) otherwise
(5a)
119901119894(119895 119896)
=
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
sumarray119896[119898119896]isin119869119896(119903)
[120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)]
if array119896[119898
119896] isin 119869
119896
0 otherwise
(5b)
where 119869119896is the set of feasible neighbor array elements of
array119895[119898
119895] (119896 = 119895+1 and 119896 le 119899) 119879
119896[119898
119896] and 120578
119896(119898
119896) denote
the pheromone intensity and heuristic information on arrayelements array
119896[119898
119896] respectively 120572 and 120585 give the relative
importance of trail 119879119896[119898
119896] and the heuristic information
120578119896(119898
119896) respectively Other parameters are as previously
described
43 Local Search Based on Mutation Operator It is wellknown that ACO easily converges to local optima under pos-itive feedback A local search can explore the neighborhoodand enhance the quality of the solution
GAs are a powerful tool for solving combinatorial opti-mization problems They solve optimization problems usingthe idea of Darwinian evolution Basic evolution opera-tions including crossover mutation and selection makeGAs appropriate for performing search In this paper themutation operation is introduced to the proposed algorithmto perform local searches We assume that the currentglobal optimal solution dbest = array
1[119898
1] array
2[119898
2]
array119899[119898
119899]best has not been improved for a certain number
of stagnation generations sgen One or more array elementsare chosen at random from dbest and these are changedin a certain manner Through this mutation operation weobtain the mutated solution d1015840best If d
1015840
best is better than dbestwe replace dbest with d1015840best Otherwise the global optimalsolution remains unchanged
44 Pheromone Updating The pheromone updating rules ofACO include global updating and local rules When ant 119894 hasfinished a path the pheromone trails on the array elementsthroughwhich the ant has passed are updated In this processthe pheromone on the visited array elements is consideredto have evaporated thus increasing the probability thatfollowing ants will traverse the other array elements Thisprocess is performed after each ant has found a path it is alocal pheromone update rule with the aim of obtaining moredispersed solutions The local pheromone update rule is
[119879119895(119898
119895)]119905+1
119894= (1 minus 120574) sdot [119879
119895(119898
119895)]119905
119894
119895 = 1 2 119899 1 le 119898119895le 119872
119895
(6a)
When all of the ants have completed their paths (whichis called a cycle) a global pheromone update is applied tothe array elements passed through by all ants This processis applied in an iterative mode [29] The rule is described asfollows
[119879119895(119898
119895)]119905+1
119894= (1 minus 120588) sdot [119879
119895(119898
119895)]119905
119894+ 120582 sdot 119865 (d119905
119894PX119905
119894Y) (6b)
where120582
=
1198881 if array
119895[119898
119895] isin the globally (iteratively) best tour
1198882 if array
119895[119898
119895] isin the iterationworst tour
1198883 otherwise
(7)
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
2
2
1
1
End carriage Trolley
Rail
Main girder
End carriage
L
Figure 1 Metal structure for overhead travelling crane
3 RBDO Modeling of an OTCMS
Cranes are mechanically applied to moving loads withoutinterfering in activities on the ground As overhead trav-eling cranes are the most widely used a typical OTCMSis selected as the study object As shown in Figure 1 it iscomposed of two parallel main girders that span the widthof the bay between the runway girders The OTCMS moveslongitudinally and the two end carriages located on eithersides of the span house the wheel blocks Because the maingirders are the principal horizontal beams that support thetrolley and are supported by the end carriages they are theprimary load-carrying components and account for morethan 80 of the total weight of the OTCMS Therefore theRBDO of OTCMS mainly focuses on the design of thesemain girders The cranersquos solid-web girder is usually a boxsection fabricated from steel plate for themain and vicewebstop and bottom flanges as shown in Figure 2 Thereforegiven a span its RBDO is to obtain the minimum deadweight that is the minimum main girder cross-sectionalarea which simultaneously satisfies the required determin-istic and probabilistic constraints associated with strengthstiffness stability manufacturing process and dimensionallimits [14] In this paper a practical bias-rail box main girderis considered The mathematical model of its RBDO is givenin Table 1
A calculation diagram of section 1-1 of the main girder isshown in Figure 2 Figure 3 is a force diagram for this maingirder in the vertical and horizontal planes In Table 1 andFigures 2 and 3 119865
119902represents the uniform load 119875
119866119895(119895 =
1 2 ) denotes the concentrated load sum119875 stands for thetrolley wheel load applied by the sumof the lifting capacity119875
119876
and trolley weight 119875119866119909 119875
119867and 119865
119867represent the horizontal
concentrated anduniform inertial load respectively 119875119904is the
lateral forceIn the RBDO model the section dimensions are random
variables while their mean values are treated as designvariables (see Table 1) The independent uncertain variablesinclude section dimensions and some important parameters(Table 1) and the other parameters are considered to be
②③④
⑤
b
Y
①
Y
X X
e
ee
h
t
t
PH
be
t1
t2Fq
FH
sumP
Figure 2 Cross section 1-1 of main girder
PS
PS
PHFH
FHPH
L
(a) On the horizontal plane
PG PGPG sumP Fq
(b) On the vertical plane
Figure 3 Force diagram for main girder
deterministic It is assumed that all random variables arenormally distributed around their mean value Effects of eachrandom variable on the active constraints at the optimalpoint are illustrated in Figure 4 The inequality constraintsof the RBDO problem define in turn the failure by stressdeflection fatigue and stiffness These structural behaviorsare described by the assumed limit state functions givenin Table 1 [15] With respect to estimating the most criticalconfiguration of the trolley on themain girder there are threeconfigurations (i) The limit state functions 119892
119895119901(XY) 119895119901 =
1 2 3 correspond to the position of the maximum bendingmoment when a fully loaded trolley is lowering and braking
4 Mathematical Problems in Engineering
Table1Mathematicalmod
elof
theR
BDOforO
TCMS
Designvaria
blem
eanvalues
oftheo
ptim
izationprob
lem
d=[11988911198892119889311988941198895]119879=120583(X)=[120583(1199091)120583(1199092)120583(1199093)120583(1199094)120583(1199095)]119879
Designvaria
blem
eanvalue
1198891
1198892
1198893
1198894
1198895
Symbo
lℎ
119905119905 1
119905 2119887
Rand
omdesig
nvaria
bles
andparameterso
fthe
optim
izationprob
lem
S=[XY]119879=[11990911199092119909311990941199095119910111991021199103]119879=[11990411199042119904311990441199045119904611990471199048]119879
Varia
blea
ndparameter
1199091
1199092
1199093
1199094
1199095
1199101
1199102
1199103
Symbo
lℎ
119905119905 1
119905 2119887
119875119876
119875119866119909
119864
Objectiv
efun
ction
minim
izingthec
ross-sectio
nalareao
fthe
girder119891(dP)=ℎsdot(119905 1+119905 2)+119905sdot(2sdot119887+3sdot119890+119887 119890)
119890istakenequalto20
mm
which
facilitates
welding
and119887 119890issetequ
alto
15119905so
asto
guaranteethe
localstabilityof
flangeo
verhanging
andther
ailinstallatio
nInequalityconstraintso
fthe
RBDOprob
lem
Reliabilityconstraints
Limitsta
tefunctio
nsDescriptio
n1198771198861minusprob(1198921(XY)le0)le0
1198921(XY)=1205901minus[120590]
Maxim
umstr
essc
ombinedatdangerou
spoint
(1)(1-1
section)
1198771198862minusprob(1198922(XY)le0)le0
1198922(XY)=1205902minus[120590]
Normalstr
essa
tdangerous
point(2)
(1-1sectio
n)
1198771198863minusprob(1198923(XY)le0)le0
1198923(XY)=1205903minus[120590]
Normalstr
essa
tdangerous
point(3)
(1-1sectio
n)
1198771198864minusprob(1198924(XY)le0)le0
1198924(XY)=120591 1minus[120591]
Maxim
umshearstre
ssatthem
iddleo
fmainweb
(2-2
section)
1198771198865minusprob(1198925(XY)le0)le0
1198925(XY)=120591 2minus[120591]
Horizon
talshear
stressinthefl
ange
(2-2
section)
1198771198866minusprob(1198926(XY)le0)le0
1198926(XY)=120590max4minus[1205901199034]
Fatig
uestr
engthatdangerou
spoint
(4)(1-1
section)
1198771198867minusprob(1198927(XY)le0)le0
1198927(XY)=120590max5minus[1205901199035]
Fatig
uestr
engthatdangerou
spoint
(5)(1-1
section)
1198771198868minusprob(1198928(XY)le0)le0
1198928(XY)=119891Vminus[119891
V]Ve
rticalstaticdeflectionof
them
idspan
1198771198869minusprob(1198929(XY)le0)le0
1198929(XY)=119891ℎminus[119891ℎ]
Horizon
talstatic
displacemento
fthe
midspan
11987711988610minusprob(11989210(XY)le0)le0
11989210(XY)=[119891119881]minus119891119881
Verticalnaturalvibratio
nfre
quency
ofthem
idspan
11987711988611minusprob(11989211(XY)le0)le0
11989211(XY)=[119891119867]minus119891119867
Horizon
taln
aturalvibrationfre
quency
ofthem
idspan
Deterministicconstraints
Natureo
fcon
straint
Descriptio
n1198921(dP)=ℎ119887minus3le0
Stabilityconstraint
Height-to-width
ratio
oftheb
oxgirder
1198921198951(dP)=119889119894minus119889up
per
119894le0
Boun
dary
constraint
Upp
erbo
unds
ofdesig
nvaria
bles1198951
=2410119894=125
1198921198952(dP)=119889lower
119894minus119889119894le0
Boun
dary
constraint
Lower
boun
dsof
desig
nvaria
bles1198952
=3511119894=125
Materialp
ermissibleno
rmalstr
ess[120590]=120590119904133andperm
issibleshearstre
ss[120591]=[120590]radic3120590119904representsmaterialyield
stress[1205901199034]and[1205901199035]arethe
weld
edjointtensilefatigue
perm
issible
stressesfor
points(4)a
nd(5)respectiv
ely[119891V]and[119891ℎ]arethe
verticalandho
rizon
talp
ermissiblestaticstiffnessesw
hich
aretaken
equalto119871800
and1198712000respectiv
elyand
119871isthe
span
oftheg
irder[119891119881]and[119891119867]arethe
verticalandho
rizon
talp
ermissibledynamicstiffnessesw
hich
aretaken
equalto2sim
4Hza
nd15sim2H
zrespectiv
ely119889
lower
119894and119889up
per
119894arethe
lower
andup
perb
ound
softhe
desig
nvaria
ble119889
119894
where
1205901=radic[119872119909119882119909+119872119910119882119910]2+[120601sum119875(4ℎ119892+100)1199051]2minus[119872119909119882119909+119872119910119882119910][120601sum119875(4ℎ119892+100)1199051]+3[119865119901119878119910119868119909(1199051+1199052)+119879119899211986001199051]21205902=1198721199091198821015840 119909+1198721199101198821015840 1199101205903=11987211990911988210158401015840
119909+11987211991011988210158401015840
119910
1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840
119909120590max5=1198721199091198821015840101584010158401015840
119909119891V=sum119875119871348119864119868119909119891ℎ=(119875119867119871348119864119868119910)(1minus341199031)+(51198651198671198714384119864119868119910)(1minus451199031)
119891119881=(12120587)radic119892(1199100+1205820)(1+120573)1199100=119875119876119871396119864119868119909120573=((119865119902119871+119875119866119909)119875119876)(1199100(1199100+1205820))2119891119867=(12120587)radic192119864119868119910119898119890(41199031minus3)1198713119898119890=(119865119902119871+119875119876+119875119866119909)119892
119872119909119872119910arethebend
ingmom
ents119865119901119865119890119865119890119867aretheshearforcesand1198791198991198791198991arethetorquesa
tdifferentlocations119868119909119868119910deno
tetheinertia
lmod
uli1198821199091198821015840 11990911988210158401015840
119909119882101584010158401015840
1199091198821015840101584010158401015840
1199091198821199101198821015840 11991011988210158401015840
119910deno
tethestr
ength
mod
uliand119878119910deno
testhe
topflang
estatic
mom
ent11986001198601198890representn
etarea
atdifferent
sectionsℎ119892representstheh
eighto
frailandℎ119889representstheh
eighto
fgird
eratthee
nd-span119864iselastic
mod
ulus119892
isgravity
constant120593
isim
pactfactor1199031iscompu
tingcoeffi
ciento
fthe
craneb
ridgeand1205820isthen
etelo
ngationof
thes
teelwire
ropeTh
eother
parametersa
rethes
amea
sbefore
Mathematical Problems in Engineering 5
0 10h
b
k
E
Sour
ces
Sour
ces
minus10 0 10 20
0 20 40 minus40 minus20 0 20
minus40 minus20 0 20 minus40 minus20 0 20
Effect on fV ()minus20
minus40 minus20
minus10
PGx
PQ
k2
k1
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1 Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Effect on f ()
Effect on 1205902 () Effect on 1205901 ()
Effect on 1205903 ()
Effect on fH ()
Figure 4 Effects of the random variables on the active constraints
in the midspan and at the same time the crane bridge is start-ing or braking (ii) The limit state functions 119892
119895119901(XY) 119895119901 =
4 5 correspond to the position of the maximum shearstress when a fully loaded trolley is lowering and brakingat the end of the span and at the same time the cranebridge is starting or braking (iii) The limit state functions119892119895119901(XY) 119895119901 = 9 10 11 12 correspond to the position of
maximum deflection This is when a fully loaded trolley ispositioned in the midspan The fatigue limit state functions119892119895119901(XY) 119895119901 = 7 8 correspond to the normal working
condition of the crane The local stability (local buckling) ofthe main girder can be guaranteed by arranging transverseand longitudinal stiffeners to form the grids according tothe width-to-thickness ratio of the flange and the height-to-thickness ratio of the web Thus only the global stabilityof the main girder is considered here Therefore the RBDOmodel of the OTCMSmain girder is a five-dimensional opti-mization problem with 11 deterministic and 11 probabilisticconstraints
4 The ACO for the OTCMS
ACO simulates the behavior of real life ant colonies inwhich individual ants deposit pheromone along a path whilemoving from the nest to food sources and vice versaTherebythe pheromone trail enables individual to smell and selectthe optimal routs The paths with more pheromone aremore likely to be selected by other ants bringing on furtheramplification of the current pheromone trails and producinga positive feedback process This behavior forms the shortestpath from the nest to the food source and vice versa Thefirst ACO algorithm called ant system (AS) was appliedto solve the traveling salesman problem Because the searchparallelism of ACO is based on the components of a solutionlevel it is very efficient Thus since the introduction of ASthe ACO metaheuristic has been widely used in many fields[33 34] including for structural optimization and has shownpromising results for various applications Therefore ACO isselected as optimization algorithm in the present study
6 Mathematical Problems in Engineering
41 Representation and Initialization of Solution In our ACOalgorithm each solution is composed of different arrayelements that correspond to different design variables Oneantrsquos search path represents a solution to the optimizationproblem or a set of design schemes The path of the 119894th antin the 119899-dimensional search space at iteration 119905 could bedenoted by d119905
119894(119894 = 1 2 popsize popsize denotes the
population size) Continuous array elements array119895[119898
119895] are
used to store the mean values of the discrete design variable119889119895in the nondecreasing order (119898
119895denotes the array sequence
number for the 119895th design variable mean value 119889119895 This is
integer with 1 le 119898119895le 119872
119895 119895 = 1 2 119899 where 119899 is the
number of design variables and 119872119895is the number of discrete
values available for 119889119895 119889lower
119895le 119889
119895le 119889
upper119895
where 119889lower119895
and119889upper119895
are the lower and upper bounds of 119889119895) Thus consider
d119905119894= array
1[119898
1] array
2[119898
2] array
119899[119898
119899]119905
119894
= [1198891 1198892 119889
119899]119905
119894
(3)
The initial solution is randomly selectedWe set the initialpheromone level [119879
119895(119898
119895)]0 of all array elements in the space
to be zero The heuristic information 120578119895(119898
119895) of array element
array119895[119898
119895] is expressed by (4) so as to induce subsequent
solutions to select smaller variable values as far as possibleand accelerate optimization process
120578119895[119898
119895] =
119872119895minus 119898
119895+ 1
119872119895
(4)
42 Selection Probability and Construction of Solutions Asthe ants move from node to node to generate paths theywill ceaselessly select the next node from the unvisitedneighbor nodes This process forms the ant paths and thusin our algorithm constructs solutions In accordance withthe transition rule of ant colony system (ACS) [35] eachant begins with the first array element array
1[119898
1] storing
the first design variable mean value 1198891and selects array
elements in proper until array119899[119898
119899] In this way a solution
is constructed Hence for the 119894th ant on the array elementarray
119895[119898
119895] the selection probability of the next array element
array119896[119898
119896] is given by
119878119894(119895 119896) =
maxarray119896[119898119896]isin119869119896
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
if 119902 lt 1199020
119901119894(119895 119896) otherwise
(5a)
119901119894(119895 119896)
=
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
sumarray119896[119898119896]isin119869119896(119903)
[120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)]
if array119896[119898
119896] isin 119869
119896
0 otherwise
(5b)
where 119869119896is the set of feasible neighbor array elements of
array119895[119898
119895] (119896 = 119895+1 and 119896 le 119899) 119879
119896[119898
119896] and 120578
119896(119898
119896) denote
the pheromone intensity and heuristic information on arrayelements array
119896[119898
119896] respectively 120572 and 120585 give the relative
importance of trail 119879119896[119898
119896] and the heuristic information
120578119896(119898
119896) respectively Other parameters are as previously
described
43 Local Search Based on Mutation Operator It is wellknown that ACO easily converges to local optima under pos-itive feedback A local search can explore the neighborhoodand enhance the quality of the solution
GAs are a powerful tool for solving combinatorial opti-mization problems They solve optimization problems usingthe idea of Darwinian evolution Basic evolution opera-tions including crossover mutation and selection makeGAs appropriate for performing search In this paper themutation operation is introduced to the proposed algorithmto perform local searches We assume that the currentglobal optimal solution dbest = array
1[119898
1] array
2[119898
2]
array119899[119898
119899]best has not been improved for a certain number
of stagnation generations sgen One or more array elementsare chosen at random from dbest and these are changedin a certain manner Through this mutation operation weobtain the mutated solution d1015840best If d
1015840
best is better than dbestwe replace dbest with d1015840best Otherwise the global optimalsolution remains unchanged
44 Pheromone Updating The pheromone updating rules ofACO include global updating and local rules When ant 119894 hasfinished a path the pheromone trails on the array elementsthroughwhich the ant has passed are updated In this processthe pheromone on the visited array elements is consideredto have evaporated thus increasing the probability thatfollowing ants will traverse the other array elements Thisprocess is performed after each ant has found a path it is alocal pheromone update rule with the aim of obtaining moredispersed solutions The local pheromone update rule is
[119879119895(119898
119895)]119905+1
119894= (1 minus 120574) sdot [119879
119895(119898
119895)]119905
119894
119895 = 1 2 119899 1 le 119898119895le 119872
119895
(6a)
When all of the ants have completed their paths (whichis called a cycle) a global pheromone update is applied tothe array elements passed through by all ants This processis applied in an iterative mode [29] The rule is described asfollows
[119879119895(119898
119895)]119905+1
119894= (1 minus 120588) sdot [119879
119895(119898
119895)]119905
119894+ 120582 sdot 119865 (d119905
119894PX119905
119894Y) (6b)
where120582
=
1198881 if array
119895[119898
119895] isin the globally (iteratively) best tour
1198882 if array
119895[119898
119895] isin the iterationworst tour
1198883 otherwise
(7)
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Table1Mathematicalmod
elof
theR
BDOforO
TCMS
Designvaria
blem
eanvalues
oftheo
ptim
izationprob
lem
d=[11988911198892119889311988941198895]119879=120583(X)=[120583(1199091)120583(1199092)120583(1199093)120583(1199094)120583(1199095)]119879
Designvaria
blem
eanvalue
1198891
1198892
1198893
1198894
1198895
Symbo
lℎ
119905119905 1
119905 2119887
Rand
omdesig
nvaria
bles
andparameterso
fthe
optim
izationprob
lem
S=[XY]119879=[11990911199092119909311990941199095119910111991021199103]119879=[11990411199042119904311990441199045119904611990471199048]119879
Varia
blea
ndparameter
1199091
1199092
1199093
1199094
1199095
1199101
1199102
1199103
Symbo
lℎ
119905119905 1
119905 2119887
119875119876
119875119866119909
119864
Objectiv
efun
ction
minim
izingthec
ross-sectio
nalareao
fthe
girder119891(dP)=ℎsdot(119905 1+119905 2)+119905sdot(2sdot119887+3sdot119890+119887 119890)
119890istakenequalto20
mm
which
facilitates
welding
and119887 119890issetequ
alto
15119905so
asto
guaranteethe
localstabilityof
flangeo
verhanging
andther
ailinstallatio
nInequalityconstraintso
fthe
RBDOprob
lem
Reliabilityconstraints
Limitsta
tefunctio
nsDescriptio
n1198771198861minusprob(1198921(XY)le0)le0
1198921(XY)=1205901minus[120590]
Maxim
umstr
essc
ombinedatdangerou
spoint
(1)(1-1
section)
1198771198862minusprob(1198922(XY)le0)le0
1198922(XY)=1205902minus[120590]
Normalstr
essa
tdangerous
point(2)
(1-1sectio
n)
1198771198863minusprob(1198923(XY)le0)le0
1198923(XY)=1205903minus[120590]
Normalstr
essa
tdangerous
point(3)
(1-1sectio
n)
1198771198864minusprob(1198924(XY)le0)le0
1198924(XY)=120591 1minus[120591]
Maxim
umshearstre
ssatthem
iddleo
fmainweb
(2-2
section)
1198771198865minusprob(1198925(XY)le0)le0
1198925(XY)=120591 2minus[120591]
Horizon
talshear
stressinthefl
ange
(2-2
section)
1198771198866minusprob(1198926(XY)le0)le0
1198926(XY)=120590max4minus[1205901199034]
Fatig
uestr
engthatdangerou
spoint
(4)(1-1
section)
1198771198867minusprob(1198927(XY)le0)le0
1198927(XY)=120590max5minus[1205901199035]
Fatig
uestr
engthatdangerou
spoint
(5)(1-1
section)
1198771198868minusprob(1198928(XY)le0)le0
1198928(XY)=119891Vminus[119891
V]Ve
rticalstaticdeflectionof
them
idspan
1198771198869minusprob(1198929(XY)le0)le0
1198929(XY)=119891ℎminus[119891ℎ]
Horizon
talstatic
displacemento
fthe
midspan
11987711988610minusprob(11989210(XY)le0)le0
11989210(XY)=[119891119881]minus119891119881
Verticalnaturalvibratio
nfre
quency
ofthem
idspan
11987711988611minusprob(11989211(XY)le0)le0
11989211(XY)=[119891119867]minus119891119867
Horizon
taln
aturalvibrationfre
quency
ofthem
idspan
Deterministicconstraints
Natureo
fcon
straint
Descriptio
n1198921(dP)=ℎ119887minus3le0
Stabilityconstraint
Height-to-width
ratio
oftheb
oxgirder
1198921198951(dP)=119889119894minus119889up
per
119894le0
Boun
dary
constraint
Upp
erbo
unds
ofdesig
nvaria
bles1198951
=2410119894=125
1198921198952(dP)=119889lower
119894minus119889119894le0
Boun
dary
constraint
Lower
boun
dsof
desig
nvaria
bles1198952
=3511119894=125
Materialp
ermissibleno
rmalstr
ess[120590]=120590119904133andperm
issibleshearstre
ss[120591]=[120590]radic3120590119904representsmaterialyield
stress[1205901199034]and[1205901199035]arethe
weld
edjointtensilefatigue
perm
issible
stressesfor
points(4)a
nd(5)respectiv
ely[119891V]and[119891ℎ]arethe
verticalandho
rizon
talp
ermissiblestaticstiffnessesw
hich
aretaken
equalto119871800
and1198712000respectiv
elyand
119871isthe
span
oftheg
irder[119891119881]and[119891119867]arethe
verticalandho
rizon
talp
ermissibledynamicstiffnessesw
hich
aretaken
equalto2sim
4Hza
nd15sim2H
zrespectiv
ely119889
lower
119894and119889up
per
119894arethe
lower
andup
perb
ound
softhe
desig
nvaria
ble119889
119894
where
1205901=radic[119872119909119882119909+119872119910119882119910]2+[120601sum119875(4ℎ119892+100)1199051]2minus[119872119909119882119909+119872119910119882119910][120601sum119875(4ℎ119892+100)1199051]+3[119865119901119878119910119868119909(1199051+1199052)+119879119899211986001199051]21205902=1198721199091198821015840 119909+1198721199101198821015840 1199101205903=11987211990911988210158401015840
119909+11987211991011988210158401015840
119910
1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840
119909120590max5=1198721199091198821015840101584010158401015840
119909119891V=sum119875119871348119864119868119909119891ℎ=(119875119867119871348119864119868119910)(1minus341199031)+(51198651198671198714384119864119868119910)(1minus451199031)
119891119881=(12120587)radic119892(1199100+1205820)(1+120573)1199100=119875119876119871396119864119868119909120573=((119865119902119871+119875119866119909)119875119876)(1199100(1199100+1205820))2119891119867=(12120587)radic192119864119868119910119898119890(41199031minus3)1198713119898119890=(119865119902119871+119875119876+119875119866119909)119892
119872119909119872119910arethebend
ingmom
ents119865119901119865119890119865119890119867aretheshearforcesand1198791198991198791198991arethetorquesa
tdifferentlocations119868119909119868119910deno
tetheinertia
lmod
uli1198821199091198821015840 11990911988210158401015840
119909119882101584010158401015840
1199091198821015840101584010158401015840
1199091198821199101198821015840 11991011988210158401015840
119910deno
tethestr
ength
mod
uliand119878119910deno
testhe
topflang
estatic
mom
ent11986001198601198890representn
etarea
atdifferent
sectionsℎ119892representstheh
eighto
frailandℎ119889representstheh
eighto
fgird
eratthee
nd-span119864iselastic
mod
ulus119892
isgravity
constant120593
isim
pactfactor1199031iscompu
tingcoeffi
ciento
fthe
craneb
ridgeand1205820isthen
etelo
ngationof
thes
teelwire
ropeTh
eother
parametersa
rethes
amea
sbefore
Mathematical Problems in Engineering 5
0 10h
b
k
E
Sour
ces
Sour
ces
minus10 0 10 20
0 20 40 minus40 minus20 0 20
minus40 minus20 0 20 minus40 minus20 0 20
Effect on fV ()minus20
minus40 minus20
minus10
PGx
PQ
k2
k1
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1 Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Effect on f ()
Effect on 1205902 () Effect on 1205901 ()
Effect on 1205903 ()
Effect on fH ()
Figure 4 Effects of the random variables on the active constraints
in the midspan and at the same time the crane bridge is start-ing or braking (ii) The limit state functions 119892
119895119901(XY) 119895119901 =
4 5 correspond to the position of the maximum shearstress when a fully loaded trolley is lowering and brakingat the end of the span and at the same time the cranebridge is starting or braking (iii) The limit state functions119892119895119901(XY) 119895119901 = 9 10 11 12 correspond to the position of
maximum deflection This is when a fully loaded trolley ispositioned in the midspan The fatigue limit state functions119892119895119901(XY) 119895119901 = 7 8 correspond to the normal working
condition of the crane The local stability (local buckling) ofthe main girder can be guaranteed by arranging transverseand longitudinal stiffeners to form the grids according tothe width-to-thickness ratio of the flange and the height-to-thickness ratio of the web Thus only the global stabilityof the main girder is considered here Therefore the RBDOmodel of the OTCMSmain girder is a five-dimensional opti-mization problem with 11 deterministic and 11 probabilisticconstraints
4 The ACO for the OTCMS
ACO simulates the behavior of real life ant colonies inwhich individual ants deposit pheromone along a path whilemoving from the nest to food sources and vice versaTherebythe pheromone trail enables individual to smell and selectthe optimal routs The paths with more pheromone aremore likely to be selected by other ants bringing on furtheramplification of the current pheromone trails and producinga positive feedback process This behavior forms the shortestpath from the nest to the food source and vice versa Thefirst ACO algorithm called ant system (AS) was appliedto solve the traveling salesman problem Because the searchparallelism of ACO is based on the components of a solutionlevel it is very efficient Thus since the introduction of ASthe ACO metaheuristic has been widely used in many fields[33 34] including for structural optimization and has shownpromising results for various applications Therefore ACO isselected as optimization algorithm in the present study
6 Mathematical Problems in Engineering
41 Representation and Initialization of Solution In our ACOalgorithm each solution is composed of different arrayelements that correspond to different design variables Oneantrsquos search path represents a solution to the optimizationproblem or a set of design schemes The path of the 119894th antin the 119899-dimensional search space at iteration 119905 could bedenoted by d119905
119894(119894 = 1 2 popsize popsize denotes the
population size) Continuous array elements array119895[119898
119895] are
used to store the mean values of the discrete design variable119889119895in the nondecreasing order (119898
119895denotes the array sequence
number for the 119895th design variable mean value 119889119895 This is
integer with 1 le 119898119895le 119872
119895 119895 = 1 2 119899 where 119899 is the
number of design variables and 119872119895is the number of discrete
values available for 119889119895 119889lower
119895le 119889
119895le 119889
upper119895
where 119889lower119895
and119889upper119895
are the lower and upper bounds of 119889119895) Thus consider
d119905119894= array
1[119898
1] array
2[119898
2] array
119899[119898
119899]119905
119894
= [1198891 1198892 119889
119899]119905
119894
(3)
The initial solution is randomly selectedWe set the initialpheromone level [119879
119895(119898
119895)]0 of all array elements in the space
to be zero The heuristic information 120578119895(119898
119895) of array element
array119895[119898
119895] is expressed by (4) so as to induce subsequent
solutions to select smaller variable values as far as possibleand accelerate optimization process
120578119895[119898
119895] =
119872119895minus 119898
119895+ 1
119872119895
(4)
42 Selection Probability and Construction of Solutions Asthe ants move from node to node to generate paths theywill ceaselessly select the next node from the unvisitedneighbor nodes This process forms the ant paths and thusin our algorithm constructs solutions In accordance withthe transition rule of ant colony system (ACS) [35] eachant begins with the first array element array
1[119898
1] storing
the first design variable mean value 1198891and selects array
elements in proper until array119899[119898
119899] In this way a solution
is constructed Hence for the 119894th ant on the array elementarray
119895[119898
119895] the selection probability of the next array element
array119896[119898
119896] is given by
119878119894(119895 119896) =
maxarray119896[119898119896]isin119869119896
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
if 119902 lt 1199020
119901119894(119895 119896) otherwise
(5a)
119901119894(119895 119896)
=
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
sumarray119896[119898119896]isin119869119896(119903)
[120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)]
if array119896[119898
119896] isin 119869
119896
0 otherwise
(5b)
where 119869119896is the set of feasible neighbor array elements of
array119895[119898
119895] (119896 = 119895+1 and 119896 le 119899) 119879
119896[119898
119896] and 120578
119896(119898
119896) denote
the pheromone intensity and heuristic information on arrayelements array
119896[119898
119896] respectively 120572 and 120585 give the relative
importance of trail 119879119896[119898
119896] and the heuristic information
120578119896(119898
119896) respectively Other parameters are as previously
described
43 Local Search Based on Mutation Operator It is wellknown that ACO easily converges to local optima under pos-itive feedback A local search can explore the neighborhoodand enhance the quality of the solution
GAs are a powerful tool for solving combinatorial opti-mization problems They solve optimization problems usingthe idea of Darwinian evolution Basic evolution opera-tions including crossover mutation and selection makeGAs appropriate for performing search In this paper themutation operation is introduced to the proposed algorithmto perform local searches We assume that the currentglobal optimal solution dbest = array
1[119898
1] array
2[119898
2]
array119899[119898
119899]best has not been improved for a certain number
of stagnation generations sgen One or more array elementsare chosen at random from dbest and these are changedin a certain manner Through this mutation operation weobtain the mutated solution d1015840best If d
1015840
best is better than dbestwe replace dbest with d1015840best Otherwise the global optimalsolution remains unchanged
44 Pheromone Updating The pheromone updating rules ofACO include global updating and local rules When ant 119894 hasfinished a path the pheromone trails on the array elementsthroughwhich the ant has passed are updated In this processthe pheromone on the visited array elements is consideredto have evaporated thus increasing the probability thatfollowing ants will traverse the other array elements Thisprocess is performed after each ant has found a path it is alocal pheromone update rule with the aim of obtaining moredispersed solutions The local pheromone update rule is
[119879119895(119898
119895)]119905+1
119894= (1 minus 120574) sdot [119879
119895(119898
119895)]119905
119894
119895 = 1 2 119899 1 le 119898119895le 119872
119895
(6a)
When all of the ants have completed their paths (whichis called a cycle) a global pheromone update is applied tothe array elements passed through by all ants This processis applied in an iterative mode [29] The rule is described asfollows
[119879119895(119898
119895)]119905+1
119894= (1 minus 120588) sdot [119879
119895(119898
119895)]119905
119894+ 120582 sdot 119865 (d119905
119894PX119905
119894Y) (6b)
where120582
=
1198881 if array
119895[119898
119895] isin the globally (iteratively) best tour
1198882 if array
119895[119898
119895] isin the iterationworst tour
1198883 otherwise
(7)
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
0 10h
b
k
E
Sour
ces
Sour
ces
minus10 0 10 20
0 20 40 minus40 minus20 0 20
minus40 minus20 0 20 minus40 minus20 0 20
Effect on fV ()minus20
minus40 minus20
minus10
PGx
PQ
k2
k1
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1 Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Sour
ces
h
b
k
E
PGx
PQ
k2
k1
Effect on f ()
Effect on 1205902 () Effect on 1205901 ()
Effect on 1205903 ()
Effect on fH ()
Figure 4 Effects of the random variables on the active constraints
in the midspan and at the same time the crane bridge is start-ing or braking (ii) The limit state functions 119892
119895119901(XY) 119895119901 =
4 5 correspond to the position of the maximum shearstress when a fully loaded trolley is lowering and brakingat the end of the span and at the same time the cranebridge is starting or braking (iii) The limit state functions119892119895119901(XY) 119895119901 = 9 10 11 12 correspond to the position of
maximum deflection This is when a fully loaded trolley ispositioned in the midspan The fatigue limit state functions119892119895119901(XY) 119895119901 = 7 8 correspond to the normal working
condition of the crane The local stability (local buckling) ofthe main girder can be guaranteed by arranging transverseand longitudinal stiffeners to form the grids according tothe width-to-thickness ratio of the flange and the height-to-thickness ratio of the web Thus only the global stabilityof the main girder is considered here Therefore the RBDOmodel of the OTCMSmain girder is a five-dimensional opti-mization problem with 11 deterministic and 11 probabilisticconstraints
4 The ACO for the OTCMS
ACO simulates the behavior of real life ant colonies inwhich individual ants deposit pheromone along a path whilemoving from the nest to food sources and vice versaTherebythe pheromone trail enables individual to smell and selectthe optimal routs The paths with more pheromone aremore likely to be selected by other ants bringing on furtheramplification of the current pheromone trails and producinga positive feedback process This behavior forms the shortestpath from the nest to the food source and vice versa Thefirst ACO algorithm called ant system (AS) was appliedto solve the traveling salesman problem Because the searchparallelism of ACO is based on the components of a solutionlevel it is very efficient Thus since the introduction of ASthe ACO metaheuristic has been widely used in many fields[33 34] including for structural optimization and has shownpromising results for various applications Therefore ACO isselected as optimization algorithm in the present study
6 Mathematical Problems in Engineering
41 Representation and Initialization of Solution In our ACOalgorithm each solution is composed of different arrayelements that correspond to different design variables Oneantrsquos search path represents a solution to the optimizationproblem or a set of design schemes The path of the 119894th antin the 119899-dimensional search space at iteration 119905 could bedenoted by d119905
119894(119894 = 1 2 popsize popsize denotes the
population size) Continuous array elements array119895[119898
119895] are
used to store the mean values of the discrete design variable119889119895in the nondecreasing order (119898
119895denotes the array sequence
number for the 119895th design variable mean value 119889119895 This is
integer with 1 le 119898119895le 119872
119895 119895 = 1 2 119899 where 119899 is the
number of design variables and 119872119895is the number of discrete
values available for 119889119895 119889lower
119895le 119889
119895le 119889
upper119895
where 119889lower119895
and119889upper119895
are the lower and upper bounds of 119889119895) Thus consider
d119905119894= array
1[119898
1] array
2[119898
2] array
119899[119898
119899]119905
119894
= [1198891 1198892 119889
119899]119905
119894
(3)
The initial solution is randomly selectedWe set the initialpheromone level [119879
119895(119898
119895)]0 of all array elements in the space
to be zero The heuristic information 120578119895(119898
119895) of array element
array119895[119898
119895] is expressed by (4) so as to induce subsequent
solutions to select smaller variable values as far as possibleand accelerate optimization process
120578119895[119898
119895] =
119872119895minus 119898
119895+ 1
119872119895
(4)
42 Selection Probability and Construction of Solutions Asthe ants move from node to node to generate paths theywill ceaselessly select the next node from the unvisitedneighbor nodes This process forms the ant paths and thusin our algorithm constructs solutions In accordance withthe transition rule of ant colony system (ACS) [35] eachant begins with the first array element array
1[119898
1] storing
the first design variable mean value 1198891and selects array
elements in proper until array119899[119898
119899] In this way a solution
is constructed Hence for the 119894th ant on the array elementarray
119895[119898
119895] the selection probability of the next array element
array119896[119898
119896] is given by
119878119894(119895 119896) =
maxarray119896[119898119896]isin119869119896
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
if 119902 lt 1199020
119901119894(119895 119896) otherwise
(5a)
119901119894(119895 119896)
=
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
sumarray119896[119898119896]isin119869119896(119903)
[120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)]
if array119896[119898
119896] isin 119869
119896
0 otherwise
(5b)
where 119869119896is the set of feasible neighbor array elements of
array119895[119898
119895] (119896 = 119895+1 and 119896 le 119899) 119879
119896[119898
119896] and 120578
119896(119898
119896) denote
the pheromone intensity and heuristic information on arrayelements array
119896[119898
119896] respectively 120572 and 120585 give the relative
importance of trail 119879119896[119898
119896] and the heuristic information
120578119896(119898
119896) respectively Other parameters are as previously
described
43 Local Search Based on Mutation Operator It is wellknown that ACO easily converges to local optima under pos-itive feedback A local search can explore the neighborhoodand enhance the quality of the solution
GAs are a powerful tool for solving combinatorial opti-mization problems They solve optimization problems usingthe idea of Darwinian evolution Basic evolution opera-tions including crossover mutation and selection makeGAs appropriate for performing search In this paper themutation operation is introduced to the proposed algorithmto perform local searches We assume that the currentglobal optimal solution dbest = array
1[119898
1] array
2[119898
2]
array119899[119898
119899]best has not been improved for a certain number
of stagnation generations sgen One or more array elementsare chosen at random from dbest and these are changedin a certain manner Through this mutation operation weobtain the mutated solution d1015840best If d
1015840
best is better than dbestwe replace dbest with d1015840best Otherwise the global optimalsolution remains unchanged
44 Pheromone Updating The pheromone updating rules ofACO include global updating and local rules When ant 119894 hasfinished a path the pheromone trails on the array elementsthroughwhich the ant has passed are updated In this processthe pheromone on the visited array elements is consideredto have evaporated thus increasing the probability thatfollowing ants will traverse the other array elements Thisprocess is performed after each ant has found a path it is alocal pheromone update rule with the aim of obtaining moredispersed solutions The local pheromone update rule is
[119879119895(119898
119895)]119905+1
119894= (1 minus 120574) sdot [119879
119895(119898
119895)]119905
119894
119895 = 1 2 119899 1 le 119898119895le 119872
119895
(6a)
When all of the ants have completed their paths (whichis called a cycle) a global pheromone update is applied tothe array elements passed through by all ants This processis applied in an iterative mode [29] The rule is described asfollows
[119879119895(119898
119895)]119905+1
119894= (1 minus 120588) sdot [119879
119895(119898
119895)]119905
119894+ 120582 sdot 119865 (d119905
119894PX119905
119894Y) (6b)
where120582
=
1198881 if array
119895[119898
119895] isin the globally (iteratively) best tour
1198882 if array
119895[119898
119895] isin the iterationworst tour
1198883 otherwise
(7)
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
41 Representation and Initialization of Solution In our ACOalgorithm each solution is composed of different arrayelements that correspond to different design variables Oneantrsquos search path represents a solution to the optimizationproblem or a set of design schemes The path of the 119894th antin the 119899-dimensional search space at iteration 119905 could bedenoted by d119905
119894(119894 = 1 2 popsize popsize denotes the
population size) Continuous array elements array119895[119898
119895] are
used to store the mean values of the discrete design variable119889119895in the nondecreasing order (119898
119895denotes the array sequence
number for the 119895th design variable mean value 119889119895 This is
integer with 1 le 119898119895le 119872
119895 119895 = 1 2 119899 where 119899 is the
number of design variables and 119872119895is the number of discrete
values available for 119889119895 119889lower
119895le 119889
119895le 119889
upper119895
where 119889lower119895
and119889upper119895
are the lower and upper bounds of 119889119895) Thus consider
d119905119894= array
1[119898
1] array
2[119898
2] array
119899[119898
119899]119905
119894
= [1198891 1198892 119889
119899]119905
119894
(3)
The initial solution is randomly selectedWe set the initialpheromone level [119879
119895(119898
119895)]0 of all array elements in the space
to be zero The heuristic information 120578119895(119898
119895) of array element
array119895[119898
119895] is expressed by (4) so as to induce subsequent
solutions to select smaller variable values as far as possibleand accelerate optimization process
120578119895[119898
119895] =
119872119895minus 119898
119895+ 1
119872119895
(4)
42 Selection Probability and Construction of Solutions Asthe ants move from node to node to generate paths theywill ceaselessly select the next node from the unvisitedneighbor nodes This process forms the ant paths and thusin our algorithm constructs solutions In accordance withthe transition rule of ant colony system (ACS) [35] eachant begins with the first array element array
1[119898
1] storing
the first design variable mean value 1198891and selects array
elements in proper until array119899[119898
119899] In this way a solution
is constructed Hence for the 119894th ant on the array elementarray
119895[119898
119895] the selection probability of the next array element
array119896[119898
119896] is given by
119878119894(119895 119896) =
maxarray119896[119898119896]isin119869119896
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
if 119902 lt 1199020
119901119894(119895 119896) otherwise
(5a)
119901119894(119895 119896)
=
120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)
sumarray119896[119898119896]isin119869119896(119903)
[120572 sdot 119879119896(119898
119896) + 120585 sdot 120578
119896(119898
119896)]
if array119896[119898
119896] isin 119869
119896
0 otherwise
(5b)
where 119869119896is the set of feasible neighbor array elements of
array119895[119898
119895] (119896 = 119895+1 and 119896 le 119899) 119879
119896[119898
119896] and 120578
119896(119898
119896) denote
the pheromone intensity and heuristic information on arrayelements array
119896[119898
119896] respectively 120572 and 120585 give the relative
importance of trail 119879119896[119898
119896] and the heuristic information
120578119896(119898
119896) respectively Other parameters are as previously
described
43 Local Search Based on Mutation Operator It is wellknown that ACO easily converges to local optima under pos-itive feedback A local search can explore the neighborhoodand enhance the quality of the solution
GAs are a powerful tool for solving combinatorial opti-mization problems They solve optimization problems usingthe idea of Darwinian evolution Basic evolution opera-tions including crossover mutation and selection makeGAs appropriate for performing search In this paper themutation operation is introduced to the proposed algorithmto perform local searches We assume that the currentglobal optimal solution dbest = array
1[119898
1] array
2[119898
2]
array119899[119898
119899]best has not been improved for a certain number
of stagnation generations sgen One or more array elementsare chosen at random from dbest and these are changedin a certain manner Through this mutation operation weobtain the mutated solution d1015840best If d
1015840
best is better than dbestwe replace dbest with d1015840best Otherwise the global optimalsolution remains unchanged
44 Pheromone Updating The pheromone updating rules ofACO include global updating and local rules When ant 119894 hasfinished a path the pheromone trails on the array elementsthroughwhich the ant has passed are updated In this processthe pheromone on the visited array elements is consideredto have evaporated thus increasing the probability thatfollowing ants will traverse the other array elements Thisprocess is performed after each ant has found a path it is alocal pheromone update rule with the aim of obtaining moredispersed solutions The local pheromone update rule is
[119879119895(119898
119895)]119905+1
119894= (1 minus 120574) sdot [119879
119895(119898
119895)]119905
119894
119895 = 1 2 119899 1 le 119898119895le 119872
119895
(6a)
When all of the ants have completed their paths (whichis called a cycle) a global pheromone update is applied tothe array elements passed through by all ants This processis applied in an iterative mode [29] The rule is described asfollows
[119879119895(119898
119895)]119905+1
119894= (1 minus 120588) sdot [119879
119895(119898
119895)]119905
119894+ 120582 sdot 119865 (d119905
119894PX119905
119894Y) (6b)
where120582
=
1198881 if array
119895[119898
119895] isin the globally (iteratively) best tour
1198882 if array
119895[119898
119895] isin the iterationworst tour
1198883 otherwise
(7)
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
119865(d119905119894PX119905
119894Y) represents the fitness function value of ant 119894 on
the 119905th iteration (see the next section) array119895[119898
119895] belongs to
the array elements of the path generated by ant 119894 on the 119905thiteration 120582 is a phase constant (119888
1ge 119888
3ge 119888
2) depending
on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst 120574 isin [0 1] and120588 isin [0 1] are the local and global pheromone evaporationrates respectively Other parameters are the same as before
45 Evaluation of Solution The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraintsThe ACO algorithmwas originally developed forunconstrained optimization problems and hence it is neces-sary to somehow incorporate constraints into the ACO algo-rithm Constraint-handling techniques have been exploredby a number of researchers [36] and commonly employedmethods are penalty functions separation of objectives andconstraints and hybrid methods Penalty functions are easyto implement and in particular are suitable for discreteRBDO Hence the penalty function method is selected forconstraint handling The following fitness function is used totransform a constrained RBDO problem to an unconstrainedone
119865 (dPXY)
= 119886 sdot exp
minus119888 sdot 119891 (dP) minus 119887
sdot [
[
1198731
sum119894119889=1
119891119894119889 (dP) +
1198732
sum119895119901=1
119891119895119901 (XY)]
]
2
(8)
where119891119894119889 (dP) = max [0 119892
119894119889 (dP)]
119891119895119901 (XY) = max [0 119877
119886119895119901minus prob (119892
119895119901 (XY) le 0)] (9)
119865(dPXY) is unconstrained objective function (the fitnessfunction) 119891(dP) is the original constraint objective func-tion (see (1)) 119886 119888 and 119887 are positive problem-specificconstants and 119891
119894119889(dP) 119891
119895119901(XY) are penalty functions cor-
responding to the 119894119889th deterministic and 119895119901th probabilisticrespectively When satisfied these penalty functions returnto a value of zero otherwise the values would be amplifiedaccording to the square term in (8) Other parameters are thesame as before
46 Termination Criterion Each run is allowed to continuefor a maximum of 100 generations However a run may beterminated before this when no improvement in the bestobjective value is noticed
5 Structural Reliability Analysis
The goal of RBDO is to find the optimal values for a designvector to achieve the target reliability level In accordance
with the RBDO model of OTCMS randomness in thestructure is expressed as the random design vector X andthe random parameter vector Y The limit state functions119892119895119901(XY) which can represent the stress displacement stiff-
ness and so on is defined in terms of random vectorsS (define S isin (XY)) The limit states that separate thedesign space into ldquofailurerdquo and ldquosaferdquo regions are 119892
119895119901(XY) =
0 Accordingly the probability of structural reliability withrespect to the 119895119901th limit state function in the specified modeis
119877119895119901= prob (119892
119895119901 (XY) le 0)
= prob (119892119895119901 (S) le 0)
= ∬ sdot sdot sdot int119863
119891119904(1199041 1199042 119904
119899) 119889119904
11198891199042sdot sdot sdot 119889119904
119899
(10)
where 119863 denotes the safe domain (119892119895119901(S) le 0)
119891119904(1199041 1199042 119904
119899) is the joint probability density function
(PDF) of the random vector S and 119877119895119901
can be calculated byintegrating the PDF 119891
119904(1199041 1199042 119904
119899) over 119863 Nevertheless
this integral is not a straightforward task as 119891119904(1199041 1199042 119904
119899)
is not always available To avoid this calculation momentmethods and simulation techniques can be applied toestimate the probabilistic constraints FOSM method isbroadly used for RBDO applications owing to its effective-ness efficiency and simplicity and it was recommended bythe Joint Committee of Structural Safety It solves structuralreliability using mean value and standard derivation Atfirst the performance function is expanded using theTaylor series at some point then truncating the series tolinear terms the first-order approximate mean value andstandard deviation may be obtained and the reliability indexcould be solved Therefore it is called FOSM Accordingto the difference of the selected linearization point FOSMis divided into mean value first-order second moment(MFOSM) (the linearization point is mean value point)and advanced first-order second moment (AFOSM) alsocalled Hasofer-Lind and Rackwitz-Fiessler method (thelinearization point is the most probable failure point (MPP))The advantages of AFOSM are that it is invariant with respectto different failure surface formulations in spaces having thesame dimension and more accurate compared with FOSM[37 38] Consequently the MPP-based AFOSM is used toquantify probabilistic characterization in this research
AFOSM uses the closest point on the limit state surfaceto the origin in the standard normal space as a measure ofthe reliability The point is called as the design point or MPPSlowast and the reliability index 120573 is defined as the distance ofthe design point and the origin 120573 = Slowast which could becalculated by determining theMPP in randomvariable space
Firstly obtain a linear approximation of the performancefunction 119885
119895119901= 119892
119895119901(S) by using the first-order Taylorrsquos series
expansion about the MPP Slowast
119885 cong 119892 (119904lowast1 119904lowast2 119904lowast
119899) +
119899
sum119894=1
(119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(11)
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 2 Main technical characteristics and mechanical properties of the metallic structure
Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism
119875119876= 32000 kg 119867 = 16m V
119902= 13mmin 119875
119866119909= 11000 kg 119875
1198661= 2000 kg 119875
1198662= 800 kg
Span length Trolley velocity Crane bridge velocity Yield stress Poissonrsquos ratio Elasticity modulus119871 = 255m V
119909= 45mmin V
119889= 90mmin 120590
119904= 235MPa ] = 03 119864 = 211 times 1011 Pa
Since the MPP Slowast is on the limit state surface the limitstate function equals zero 119892(119904lowast
1 119904lowast2 119904lowast
119899) = 0 Here the
subscript 119895119901 has been dropped for the sake of simplicity ofthe subsequent notation
119885rsquos mean value 120583119885and standard deviation 120590
119885could be
expressed as follows
120583119885cong
119899
sum119894=1
(120583119904119894minus 119904lowast
119894)120597119892
120597119904119894
10038161003816100381610038161003816100381610038161003816119904lowast119894
(12a)
120590119885= radic
119899
sum119894=1
(120597119892
120597119904119894
)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894
1205902119904119894 (12b)
The reliability index 120573 is shown as
120573 =120583119885
120590119885
=sum119899
119894=1(120583119904119894minus 119904lowast
119894) (120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
sum119899
119894=1120572119894120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
(13)
where
120572119894=
120590119904119894(120597119892120597119904
119894)1003816100381610038161003816119904lowast119894
[sum119899
119894=1(120597119892120597119904
119894)210038161003816100381610038161003816119904lowast119894
1205902119904119894]12 (14)
Then
120583119904119894minus 119904lowast
119894minus 120573120572
119894120590119904119894= 0 (15)
Finally combining (13) (15) and the limit state function119892(S) = 0 the MPP 119904lowast
119894and reliability index 120573 could be
calculated by an iterative procedure Then the reliability119877 could be approximated by 119877 = Φ(120573) where Φ(sdot) isthe standard normal cumulative distribution function Hererandom variables 119904
119894(119894 = 1 2 119899) are assumed to be
normal distribution and are independent to each other andthe same assumption will be used throughout this paper
6 RBDO Procedure
Using the methods described in the previous sections theRBDOnumerical procedure illustrated in Figure 5 was devel-oped
7 Example of Reliability-BasedDesign Optimization
The proposed approach was coded in C++ and executed on226GHz Intel Dual Core processor and 1GB main memory
Table 3 Statistical properties of the random variables 119904119894
Random variable Distribution Mean value (120583) COV (120590120583)Lifting capacity (119875
119876) Normal 32000 kg 010
Trolley weight (119875119866119909) Normal 11000 kg 005
Elasticity modulus (119864) Normal 211 times 1011 Pa 005
Design variables (119889119894) Normal 119889
119894mm 005
Table 4 Parameter values used in adopted ACO
Parameters Values for the exampleFitness function parameters 119886 = 100 119888 = 10minus6 119887 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 120588 = 02 120574 = 04Selection probability parameters 120572 = 1 120573 = 1
Phase constants 1198881 = 2 1198882 = 0 1198883 = 1Transition rule parameter 119902
0= 09
71 Design Parameters The proposed RBDO method wasapplied to the design of a real-world OTCMS with a workingclass of A6 Its main technical characteristics and mechanicalproperties are presented in Table 2 The upper and lowerbounds of the design vectors were taken to be dupper =
1820 40 40 40 995 and dlower = 1500 6 6 6 500(units inmm) In practical design themain girder height andwidth are usually designed as integer multiples of 5mm sothe number of discrete mean values available for the designvariables (main girder height and width) was 119872
1= 65
and 1198725
= 100 The step size intervals were set to be05mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15] Thusthe number of discrete mean values available for the designvariables (thicknesses of main web vice web and flange) was1198722= 119872
3= 119872
4= 60 The statistical properties of the
random variables are summarized in Table 3 For the desiredreliability probabilities we refer to JCSS [39] and the currentcrane design code [14] a value of 0979 was set for 119892
1(XY)
to 1198927(XY) 0968 for 119892
8(XY) and 119892
9(XY) and 0759 for
11989210(XY) and 119892
11(XY) It should be noted that these target
reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design valuesBy means of a large number of trials and experience theparameter values of the constructed ACO algorithm were setin Table 4
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Calculate the fitness function
Check the deterministic
Check the probabilistic constraints
using AFOSM
Output the best
record the best
parameter vector P and heuristic
Produce next generation
Update the local
Calculate the fitness function
No
Yes
The stopping criterion is satisfied
No
Yes
Check the deterministic
Check the probabilistic constraints
Perform local search for the best solution
NoYes
pheromone
pheromone
Update the global
individuals dti
i lt popsizei = i + 1
constraints gid(dti P) le 0
solution dtbest
solution d0best the best solution dtbest
prob(gjp(Xti Y) le 0) ge Rajp using AFOSM
values F(dti PXti Y) record
dtbest using mutation
t gt sgen
t = 0
t = t + 1
t = t + 1
Initialize design vectors d0i andpheromone [Tj(mj)]
0 set random
information 120578j(mj) let t = 0
constraints gid(d0i P) le 0
prob(gjp(X0i Y) le 0) ge Rajp
values F(d0i PX0i Y)
[Tj(mj)]ti
[Tj(mj)]ti
Figure 5 Numerical procedure of RBDO
Table 5 Optimization results
Solution Optimization variable (mm) Objective and fitness function Generation Time (s)1198891
1198892
1198893
1198894
1198895
119891(dP) (mm2) 119865(dPXY)PD 1600 10 8 6 760 39700 961078 mdash mdashDDO 1765 7 6 6 610 30875 969597 58 01794RBDO 1815 75 65 6 785 35756 964875 23 423PD stands for practical design
72 RBDO Results Some final points concerning the prac-tical design and the deterministic and reliability-based opti-mization process are given in Table 5 Table 6 shows theperformance and reliability with respect to each optimumdesign Variations in the number of iterations required forthe reliability indices of active constraints and cross-sectionalarea to converge are illustrated in Figure 6
Tables 5 and 6 show that the deterministic design took 58generations to find the optimum area of 30875mm2 which isabout 8825mm2 less than that of the practical design Thusthe deterministic optimization found a better solution withinjust 01794 s The critical constraint of the optimum design isthe vertical natural vibration frequency 119891
119881at the midspan
point The corresponding constraint value is 200067Hz
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table 6 The performance and reliability with respect to optimum designs
Performance Constraints requirement Practical design The DDO The RBDOPerformance 119877
119886119895119901Per 119877
119895119901Per 119877
119895119901Per 119877
119895119901
1205901(MPa)
le176936 0999 122786 0996 989442 10
1205902(MPa) 1173 0999 148342 0888 11936 0997
1205903(MPa) 1345 0979 169924 0604 136865 0996
1205911(MPa)
le101 5115 0999 615463 0999 586088 09991205912(MPa) ge0979 912 0999 185513 0999 154756 0998
120590max 4 (MPa) le[1205901199031198944] 1081 0999 137608 0999 110975 0999
[1205901199031198944] (MPa) mdash 2163 mdash 214401 mdash 21545 mdash
120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998
[1205901199031198945] (MPa) mdash 1417 mdash 138996 mdash 14049 mdash
119891V (mm) le31875ge0968 2368 0968 288096 0728 221105 0989
119891ℎ(mm) le12750 286 0999 442906 0999 278834 09999
119891119881(Hz) ge20
ge0759 208 0759 200067 0494 211043 0851119891119867(Hz) ge15 2757 10 20638 10 26094 10
Feasible Infeasible FeasibleActive constraint 119891
119881119891119881
119892119895119901(sdot) 119895119901 = 3 8 10
Per stands for performance
944946948
95952954956958
96962964966968
97
Fitn
ess f
unct
ion
valu
e
Generation
2 4 6 8 10 12 14 16 18 20 22 24 26 28 300 5 15 25
3
35
4
45
5
55
6
Fitness function value
times104
Obj
ectiv
e fun
ctio
n va
lue (
mm
2)
Objective function value
(a)
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
Generation
Relia
bilit
y in
dex
Vertical static deflection 1205738Vertical natural vibration frequency 12057310
1205733Normal stress at point ③
(b)
Figure 6 Convergence histories for the RBDO of the OTCMS (a) for objective and fitness function and (b) for reliability indices of activeconstraints (119892
119895119901(sdot) 119895119901 = 3 8 10)
which is higher than the required value 2Hz and satisfies theperformance requirement Probabilistic analyses were alsoconducted for this deterministic optimum and the practicaldesign the results are shown inTable 6The active constraintsof the deterministic optimum have reliabilities of 06040728 and 0494 which are all below the desired reliabilitiesof 0979 0968 and 0759 respectively The results indicatethat the DDO can significantly reduce the structural area butits ability tomeet the design requirements for reliability underuncertainties is quite low To obtain a more reliable design
by considering uncertainties during the optimization processRBDO is needed
As shown in Tables 5 and 6 RBDO required 23 optimiza-tion iterationsThe reliabilities of the active constraints at theoptimum point are 0996 0989 and 0851 (corresponding toreliability indices 120573
3= 267 120573
8= 232 and 120573
10= 104)
which are all above the desired reliabilities of 0979 0968 and0759 ComparedwithDDO the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of159) and the CPU time is one order of magnitude more
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
than that of DDO However the reliability of the RBDOresults exhibits a significant increase and meets the desiredlevels Therefore considering the inherent uncertainties inmaterial dimensions and loads only the final RBDO designis both feasible and safe
8 Conclusions
This paper presented an RBDO methodology that combinesACOM and AFOSM This was applied to the design ofa real-world OTCMS under uncertainties in loads cross-sectional dimensions and materials for the first time Thedesign procedure directly couples structural performancecalculation numerical design optimization and structuralreliability analysis while considering different modes of fail-ure in the OTCMS From the results obtained the followingconclusions could be drawn The deterministic optimizationmethod can improve design quality and efficiency neverthe-less it is more likely to lead to unreliable solutions once weconsider uncertainty On the contrary RBDO can achieve amore compromised design that balances economic and safetyThe inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design methodIt is worth noting that in such high-risk equipment anincrease in the reliability that leads to a cost decrementis financially much more beneficial rather than increasingthe weight which results in the cost increments on a longviewThe constructed approach is applicable and efficient forOTCMSs RBDO and might also be useful for other metallicstructures with more design and random variables as well asmultiple objectives This will be studied in a future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant no 51275329
References
[1] C B Pinca G O Tirian A V Socalici and E D ArdeleanldquoDimensional optimization for the strength structure of a trav-eling cranerdquo WSEAS Transactions on Applied and TheoreticalMechanics vol 4 no 4 pp 147ndash156 2009
[2] ND Lagaros andM Papadrakakis ldquoApplied soft computing foroptimum design of structuresrdquo Structural andMultidisciplinaryOptimization vol 45 no 6 pp 787ndash799 2012
[3] RMijailovic andG Kastratovic ldquoCross-section optimization oftower crane lattice boomrdquoMeccanica vol 44 no 5 pp 599ndash6112009
[4] K Jarmai ldquoDecision support system on IBM PC for design ofeconomic steel structures applied to crane girdersrdquoThin-WalledStructures vol 10 no 2 pp 143ndash159 1990
[5] C Seeszligelberg ldquoAbout the cross-section optimization of weldedprofiles of run-way beams for top mounted overhead cranesrdquoStahlbau vol 72 no 9 pp 636ndash645 2003
[6] C Sun Y Tan J C Zeng J S Pan and Y F Tao ldquoThestructure optimization of main beam for bridge crane based onan improved PSOrdquo Journal of Computers vol 6 no 8 pp 1585ndash1590 2011
[7] G Sun G Li S Zhou H Li S Hou and Q Li ldquoCrash-worthiness design of vehicle by using multiobjective robustoptimizationrdquo Structural and Multidisciplinary Optimizationvol 44 no 1 pp 99ndash110 2011
[8] N D Lagaros V Plevris andM Papadrakakis ldquoNeurocomput-ing strategies for solving reliability-robust design optimizationproblemsrdquoEngineering Computations vol 27 no 7 pp 819ndash8402010
[9] L Yu Y Cao Q Chong and X Wu ldquoReliability-based designfor the structure of tower crane under aleatory and epistemicuncertaintiesrdquo in Proceedings of the International Conference onQuality Reliability Risk Maintenance and Safety Engineering(ICQR2MSE rsquo11) pp 938ndash943 June 2011
[10] W Meng Z Yang X Qi and J Cai ldquoReliability analysis-based numerical calculation of metal structure of bridge cranerdquoMathematical Problems in Engineering vol 2013 Article ID260976 5 pages 2013
[11] R Jafar and E Masoud ldquoLatin hypercube sampling appliedto reliability-based multidisciplinary design optimization of alaunch vehiclerdquoAerospace Science and Technology vol 28 no 1pp 297ndash304 2013
[12] J Fang Y Gao G Sun and Q Li ldquoMultiobjective reliability-based optimization for design of a vehicle doorrdquo Finite Elementsin Analysis and Design vol 67 pp 13ndash21 2013
[13] H M Gomes A M Awruch and P A M Lopes ldquoReliabilitybased optimization of laminated composite structures usinggenetic algorithms and Artificial Neural Networksrdquo StructuralSafety vol 33 no 3 pp 186ndash195 2011
[14] LWan G N Xu andDM Gu ldquoDesign rules for cranesrdquo TechRep GBT3811-2008 China Standards Press Beijing China2008
[15] Y F Tang X H Wang and Z M Pu Dimension Shape Weightand Tolerances for Hot-Rolled Steel Plates and Sheet (GBT 709-2006) China Standards Press Beijing China 2007
[16] A Kaveh andVKalatjari ldquoGenetic algorithm for discrete-sizingoptimal design of trusses using the force methodrdquo InternationalJournal for Numerical Methods in Engineering vol 55 no 1 pp55ndash72 2002
[17] A Kaveh and A Abdietehrani ldquoDesign of frames using geneticalgorithm force method and graph theoryrdquo International Jour-nal for Numerical Methods in Engineering vol 61 no 14 pp2555ndash2565 2004
[18] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011
[19] C K Dimou and V K Koumousis ldquoReliability-based optimaldesign of truss structures using particle swarm optimizationrdquoJournal of Computing in Civil Engineering vol 23 no 2 pp 100ndash109 2009
[20] A Xiao B Wang C Sun S Zhang and Z Yang ldquoFitnessestimation based particle swarm optimization algorithm forlayout design of truss structuresrdquo Mathematical Problems inEngineering vol 2014 Article ID 671872 11 pages 2014
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
[21] C V Camp B J Bichon and S P Stovall ldquoDesign of steelframes using ant colony optimizationrdquo Journal of StructuralEngineering vol 131 no 3 pp 369ndash379 2005
[22] A Kaveh and S Talatahari ldquoAn improved ant colony opti-mization for the design of planar steel framesrdquo EngineeringStructures vol 32 no 3 pp 864ndash873 2010
[23] O Hasancebi and S Kazemzadeh Azad ldquoDiscrete size opti-mization of steel trusses using a refined big bangmdashbig crunchalgorithmrdquo Engineering Optimization vol 46 no 1 pp 61ndash832014
[24] A Kaveh and S Talatahari ldquoSize optimization of space trussesusing Big Bang-Big Crunch algorithmrdquo Computers and Struc-tures vol 87 no 17-18 pp 1129ndash1140 2009
[25] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010
[26] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010
[27] J A Momoh R Adapa and M E El-Hawary ldquoA review ofselected optimal power flow literature to 1993 I Nonlinearand quadratic programming approachesrdquo IEEE Transactions onPower Systems vol 14 no 1 pp 96ndash104 1999
[28] J A Monoh M E EI-Hawary and R Adapa ldquoA reviewof selected optimal power flow literature to 1993 Part IINewton linear programming and Interior Point MethodsrdquoIEEE Transactions on Power Systems vol 14 no 1 pp 105ndash1111999
[29] X Fan Y Lin andZ Ji ldquoShip pipe routing design using theACOwith iterative pheromone updatingrdquo Journal of Ship Productionvol 23 no 1 pp 36ndash45 2007
[30] O Ditlevsen and H O Madsen Structural Reliability MethodsWiley Chichester UK 1996
[31] Y T Wu H R Millwater and T A Cruse ldquoAdvanced prob-abilistic structural analysis method for implicit performancefunctionsrdquo AIAA Journal vol 28 no 9 pp 1663ndash1669 1990
[32] Y-P Wang Z-Z Lu and Z-F Yue ldquoReliability analysis forimplicit limit state equationrdquoAppliedMathematics andMechan-ics (English Edition) vol 26 no 9 pp 1158ndash1164 2005
[33] H Hemmatian A Fereidoon A Sadollah and A Bahreinine-jad ldquoOptimization of laminate stacking sequence for mini-mizing weight and cost using elitist ant system optimizationrdquoAdvances in Engineering Software vol 57 no 3 pp 8ndash18 2013
[34] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009
[35] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997
[36] P W Jansen and R E Perez ldquoConstrained structural designoptimization via a parallel augmented Lagrangian particleswarm optimization approachrdquo Computers and Structures vol89 no 13-14 pp 1352ndash1366 2011
[37] A M Hasofer and N C Lind ldquoExact and invariant second-moment code formatrdquo Journal Engineering Mechanics Division-ASCE vol 100 no 1 pp 111ndash121 1974
[38] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers and Structures vol9 no 5 pp 489ndash494 1978
[39] JCSS Probabilistic Model Code ldquoThe Joint Committee onStructural Safetyrdquo httpwwwjcssethzch2
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of