research article reliability-based design optimization for

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


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)


1198923(XY)=1205903minus[120590]

Normalstr

essa

tdangerous

point(3)

(1-1sectio

n)


1198924(XY)=120591 1minus[120591]

Maxim

umshearstre

ssatthem

iddleo

fmainweb

(2-2

section)


1198925(XY)=120591 2minus[120591]

Horizon

talshear

stressinthefl

ange

(2-2

section)


1198926(XY)=120590max4minus[1205901199034]

Fatig

uestr

engthatdangerou

spoint

(4)(1-1

section)


1198927(XY)=120590max5minus[1205901199035]

Fatig

uestr

engthatdangerou

spoint

(5)(1-1

section)


1198928(XY)=119891Vminus[119891

V]Ve

rticalstaticdeflectionof

them

idspan


1198929(XY)=119891ℎminus[119891ℎ]

Horizon

talstatic

displacemento

fthe

midspan


11989210(XY)=[119891119881]minus119891119881

Verticalnaturalvibratio

nfre

quency

ofthem

idspan


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


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


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)


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)


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


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


No

Yes

The stopping criterion is satisfied

No

Yes



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


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


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


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


[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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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


2

2

1

1


Rail

Main girder

End carriage

L






119866119895(119895 =


119876


119867and 119865





②③④

⑤

b

Y

①

Y

X X

e

ee

h

t

t

PH

be

t1

t2Fq

FH

sumP


PS

PS

PHFH

FHPH

L


PG PGPG sumP Fq




119895119901(XY) 119895119901 =




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



mm

which

facilitates

welding


alto

15119905so

asto

guaranteethe

localstabilityof

flangeo

verhanging

andther

ailinstallatio


fthe

RBDOprob

lem


Limitsta

tefunctio

nsDescriptio


1198921(XY)=1205901minus[120590]

Maxim

umstr

essc

ombinedatdangerou

spoint

(1)(1-1

section)


1198922(XY)=1205902minus[120590]

Normalstr

essa

tdangerous

point(2)

(1-1sectio

n)


1198923(XY)=1205903minus[120590]

Normalstr

essa

tdangerous

point(3)

(1-1sectio

n)


1198924(XY)=120591 1minus[120591]

Maxim

umshearstre

ssatthem

iddleo

fmainweb

(2-2

section)


1198925(XY)=120591 2minus[120591]

Horizon

talshear

stressinthefl

ange

(2-2

section)


1198926(XY)=120590max4minus[1205901199034]

Fatig

uestr

engthatdangerou

spoint

(4)(1-1

section)


1198927(XY)=120590max5minus[1205901199035]

Fatig

uestr

engthatdangerou

spoint

(5)(1-1

section)


1198928(XY)=119891Vminus[119891

V]Ve


them

idspan


1198929(XY)=119891ℎminus[119891ℎ]

Horizon

talstatic

displacemento

fthe

midspan


11989210(XY)=[119891119881]minus119891119881


nfre

quency

ofthem

idspan


11989211(XY)=[119891119867]minus119891119867

Horizon

taln

aturalvibrationfre

quency

ofthem

idspan


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



weld


perm

issible

stressesfor

points(4)a

nd(5)respectiv


verticalandho

rizon

talp


hich

aretaken

equalto119871800


elyand

119871isthe

span

oftheg

irder[119891119881]and[119891119867]arethe

verticalandho

rizon

talp


hich

aretaken

equalto2sim

4Hza

nd15sim2H

zrespectiv

ely119889

lower

119894and119889up

per

119894arethe

lower

andup

perb

ound

softhe

desig

nvaria

ble119889

119894

where


119909+11987211991011988210158401015840

119910

1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840

119909120590max5=1198721199091198821015840101584010158401015840



119872119909119872119910arethebend

ingmom



tetheinertia

lmod

uli1198821199091198821015840 11990911988210158401015840

119909119882101584010158401015840

1199091198821015840101584010158401015840

1199091198821199101198821015840 11991011988210158401015840

119910deno

tethestr

ength

mod


testhe

topflang

estatic

mom

ent11986001198601198890representn

etarea

atdifferent


eighto


eighto

fgird

eratthee


mod

ulus119892

isgravity

constant120593

isim


tingcoeffi

ciento

fthe

craneb


etelo

ngationof

thes

teelwire

ropeTh

eother

parametersa

rethes

amea

sbefore


0 10h

b

k

E

Sour

ces

Sour

ces

minus10 0 10 20

0 20 40 minus40 minus20 0 20



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 fH ()



119895119901(XY) 119895119901 =










119895] are





119895 119895 = 1 2 119899 where 119899 is the



119895le 119889

119895le 119889

upper119895




d119905119894= array

1[119898

1] array

2[119898

2] array

119899[119898

119899]119905

119894

= [1198891 1198892 119889

119899]119905

119894

(3)


119895(119898




array119895[119898



120578119895[119898

119895] =

119872119895minus 119898

119895+ 1

119872119895

(4)


1[119898

1] storing





119895[119898


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)


array119895[119898

119895] (119896 = 119895+1 and 119896 le 119899) 119879

119896[119898

119896] and 120578

119896(119898

119896) denote


119896[119898




120578119896(119898


described



1[119898

1] array

2[119898

2]

array119899[119898



1015840



[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)


[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


1198882 if array

119895[119898


1198883 otherwise

(7)


119865(d119905119894PX119905



119895] belongs to


1ge 119888

3ge 119888

2) depending



119865 (dPXY)

= 119886 sdot exp


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)


119894119889(dP) 119891








119895119901(XY) =


119877119895119901= prob (119892

119895119901 (XY) le 0)

= prob (119892119895119901 (S) le 0)


119891119904(1199041 1199042 119904

119899) 119889119904

11198891199042sdot sdot sdot 119889119904

119899

(10)


119891119904(1199041 1199042 119904




119904(1199041 1199042 119904



119899)




119895119901= 119892




119899) +

119899

sum119894=1

(119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(11)




119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

2

1

1


Rail

Main girder

End carriage

L






119866119895(119895 =


119876


119867and 119865





②③④

⑤

b

Y

①

Y

X X

e

ee

h

t

t

PH

be

t1

t2Fq

FH

sumP


PS

PS

PHFH

FHPH

L


PG PGPG sumP Fq




119895119901(XY) 119895119901 =




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



mm

which

facilitates

welding


alto

15119905so

asto

guaranteethe

localstabilityof

flangeo

verhanging

andther

ailinstallatio


fthe

RBDOprob

lem


Limitsta

tefunctio

nsDescriptio


1198921(XY)=1205901minus[120590]

Maxim

umstr

essc

ombinedatdangerou

spoint

(1)(1-1

section)


1198922(XY)=1205902minus[120590]

Normalstr

essa

tdangerous

point(2)

(1-1sectio

n)


1198923(XY)=1205903minus[120590]

Normalstr

essa

tdangerous

point(3)

(1-1sectio

n)


1198924(XY)=120591 1minus[120591]

Maxim

umshearstre

ssatthem

iddleo

fmainweb

(2-2

section)


1198925(XY)=120591 2minus[120591]

Horizon

talshear

stressinthefl

ange

(2-2

section)


1198926(XY)=120590max4minus[1205901199034]

Fatig

uestr

engthatdangerou

spoint

(4)(1-1

section)


1198927(XY)=120590max5minus[1205901199035]

Fatig

uestr

engthatdangerou

spoint

(5)(1-1

section)


1198928(XY)=119891Vminus[119891

V]Ve


them

idspan


1198929(XY)=119891ℎminus[119891ℎ]

Horizon

talstatic

displacemento

fthe

midspan


11989210(XY)=[119891119881]minus119891119881


nfre

quency

ofthem

idspan


11989211(XY)=[119891119867]minus119891119867

Horizon

taln

aturalvibrationfre

quency

ofthem

idspan


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



weld


perm

issible

stressesfor

points(4)a

nd(5)respectiv


verticalandho

rizon

talp


hich

aretaken

equalto119871800


elyand

119871isthe

span

oftheg

irder[119891119881]and[119891119867]arethe

verticalandho

rizon

talp


hich

aretaken

equalto2sim

4Hza

nd15sim2H

zrespectiv

ely119889

lower

119894and119889up

per

119894arethe

lower

andup

perb

ound

softhe

desig

nvaria

ble119889

119894

where


119909+11987211991011988210158401015840

119910

1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840

119909120590max5=1198721199091198821015840101584010158401015840



119872119909119872119910arethebend

ingmom



tetheinertia

lmod

uli1198821199091198821015840 11990911988210158401015840

119909119882101584010158401015840

1199091198821015840101584010158401015840

1199091198821199101198821015840 11991011988210158401015840

119910deno

tethestr

ength

mod


testhe

topflang

estatic

mom

ent11986001198601198890representn

etarea

atdifferent


eighto


eighto

fgird

eratthee


mod

ulus119892

isgravity

constant120593

isim


tingcoeffi

ciento

fthe

craneb


etelo

ngationof

thes

teelwire

ropeTh

eother

parametersa

rethes

amea

sbefore


0 10h

b

k

E

Sour

ces

Sour

ces

minus10 0 10 20

0 20 40 minus40 minus20 0 20



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 fH ()



119895119901(XY) 119895119901 =










119895] are





119895 119895 = 1 2 119899 where 119899 is the



119895le 119889

119895le 119889

upper119895




d119905119894= array

1[119898

1] array

2[119898

2] array

119899[119898

119899]119905

119894

= [1198891 1198892 119889

119899]119905

119894

(3)


119895(119898




array119895[119898



120578119895[119898

119895] =

119872119895minus 119898

119895+ 1

119872119895

(4)


1[119898

1] storing





119895[119898


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)


array119895[119898

119895] (119896 = 119895+1 and 119896 le 119899) 119879

119896[119898

119896] and 120578

119896(119898

119896) denote


119896[119898




120578119896(119898


described



1[119898

1] array

2[119898

2]

array119899[119898



1015840



[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)


[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


1198882 if array

119895[119898


1198883 otherwise

(7)


119865(d119905119894PX119905



119895] belongs to


1ge 119888

3ge 119888

2) depending



119865 (dPXY)

= 119886 sdot exp


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)


119894119889(dP) 119891








119895119901(XY) =


119877119895119901= prob (119892

119895119901 (XY) le 0)

= prob (119892119895119901 (S) le 0)


119891119904(1199041 1199042 119904

119899) 119889119904

11198891199042sdot sdot sdot 119889119904

119899

(10)


119891119904(1199041 1199042 119904




119904(1199041 1199042 119904



119899)




119895119901= 119892




119899) +

119899

sum119894=1

(119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(11)




119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











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



mm

which

facilitates

welding


alto

15119905so

asto

guaranteethe

localstabilityof

flangeo

verhanging

andther

ailinstallatio


fthe

RBDOprob

lem


Limitsta

tefunctio

nsDescriptio


1198921(XY)=1205901minus[120590]

Maxim

umstr

essc

ombinedatdangerou

spoint

(1)(1-1

section)


1198922(XY)=1205902minus[120590]

Normalstr

essa

tdangerous

point(2)

(1-1sectio

n)


1198923(XY)=1205903minus[120590]

Normalstr

essa

tdangerous

point(3)

(1-1sectio

n)


1198924(XY)=120591 1minus[120591]

Maxim

umshearstre

ssatthem

iddleo

fmainweb

(2-2

section)


1198925(XY)=120591 2minus[120591]

Horizon

talshear

stressinthefl

ange

(2-2

section)


1198926(XY)=120590max4minus[1205901199034]

Fatig

uestr

engthatdangerou

spoint

(4)(1-1

section)


1198927(XY)=120590max5minus[1205901199035]

Fatig

uestr

engthatdangerou

spoint

(5)(1-1

section)


1198928(XY)=119891Vminus[119891

V]Ve


them

idspan


1198929(XY)=119891ℎminus[119891ℎ]

Horizon

talstatic

displacemento

fthe

midspan


11989210(XY)=[119891119881]minus119891119881


nfre

quency

ofthem

idspan


11989211(XY)=[119891119867]minus119891119867

Horizon

taln

aturalvibrationfre

quency

ofthem

idspan


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



weld


perm

issible

stressesfor

points(4)a

nd(5)respectiv


verticalandho

rizon

talp


hich

aretaken

equalto119871800


elyand

119871isthe

span

oftheg

irder[119891119881]and[119891119867]arethe

verticalandho

rizon

talp


hich

aretaken

equalto2sim

4Hza

nd15sim2H

zrespectiv

ely119889

lower

119894and119889up

per

119894arethe

lower

andup

perb

ound

softhe

desig

nvaria

ble119889

119894

where


119909+11987211991011988210158401015840

119910

1205911=15119865119890ℎ119889(1199051+1199052)+11987911989912119860119889011990511205912=15119865119890119867119905(2119887+3119890+119887119890)+1198791198991211986011988901199051120590max4=119872119909119882101584010158401015840

119909120590max5=1198721199091198821015840101584010158401015840



119872119909119872119910arethebend

ingmom



tetheinertia

lmod

uli1198821199091198821015840 11990911988210158401015840

119909119882101584010158401015840

1199091198821015840101584010158401015840

1199091198821199101198821015840 11991011988210158401015840

119910deno

tethestr

ength

mod


testhe

topflang

estatic

mom

ent11986001198601198890representn

etarea

atdifferent


eighto


eighto

fgird

eratthee


mod

ulus119892

isgravity

constant120593

isim


tingcoeffi

ciento

fthe

craneb


etelo

ngationof

thes

teelwire

ropeTh

eother

parametersa

rethes

amea

sbefore


0 10h

b

k

E

Sour

ces

Sour

ces

minus10 0 10 20

0 20 40 minus40 minus20 0 20



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 fH ()



119895119901(XY) 119895119901 =










119895] are





119895 119895 = 1 2 119899 where 119899 is the



119895le 119889

119895le 119889

upper119895




d119905119894= array

1[119898

1] array

2[119898

2] array

119899[119898

119899]119905

119894

= [1198891 1198892 119889

119899]119905

119894

(3)


119895(119898




array119895[119898



120578119895[119898

119895] =

119872119895minus 119898

119895+ 1

119872119895

(4)


1[119898

1] storing





119895[119898


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)


array119895[119898

119895] (119896 = 119895+1 and 119896 le 119899) 119879

119896[119898

119896] and 120578

119896(119898

119896) denote


119896[119898




120578119896(119898


described



1[119898

1] array

2[119898

2]

array119899[119898



1015840



[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)


[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


1198882 if array

119895[119898


1198883 otherwise

(7)


119865(d119905119894PX119905



119895] belongs to


1ge 119888

3ge 119888

2) depending



119865 (dPXY)

= 119886 sdot exp


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)


119894119889(dP) 119891








119895119901(XY) =


119877119895119901= prob (119892

119895119901 (XY) le 0)

= prob (119892119895119901 (S) le 0)


119891119904(1199041 1199042 119904

119899) 119889119904

11198891199042sdot sdot sdot 119889119904

119899

(10)


119891119904(1199041 1199042 119904




119904(1199041 1199042 119904



119899)




119895119901= 119892




119899) +

119899

sum119894=1

(119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(11)




119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10h

b

k

E

Sour

ces

Sour

ces

minus10 0 10 20

0 20 40 minus40 minus20 0 20



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 fH ()



119895119901(XY) 119895119901 =










119895] are





119895 119895 = 1 2 119899 where 119899 is the



119895le 119889

119895le 119889

upper119895




d119905119894= array

1[119898

1] array

2[119898

2] array

119899[119898

119899]119905

119894

= [1198891 1198892 119889

119899]119905

119894

(3)


119895(119898




array119895[119898



120578119895[119898

119895] =

119872119895minus 119898

119895+ 1

119872119895

(4)


1[119898

1] storing





119895[119898


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)


array119895[119898

119895] (119896 = 119895+1 and 119896 le 119899) 119879

119896[119898

119896] and 120578

119896(119898

119896) denote


119896[119898




120578119896(119898


described



1[119898

1] array

2[119898

2]

array119899[119898



1015840



[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)


[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


1198882 if array

119895[119898


1198883 otherwise

(7)


119865(d119905119894PX119905



119895] belongs to


1ge 119888

3ge 119888

2) depending



119865 (dPXY)

= 119886 sdot exp


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)


119894119889(dP) 119891








119895119901(XY) =


119877119895119901= prob (119892

119895119901 (XY) le 0)

= prob (119892119895119901 (S) le 0)


119891119904(1199041 1199042 119904

119899) 119889119904

11198891199042sdot sdot sdot 119889119904

119899

(10)


119891119904(1199041 1199042 119904




119904(1199041 1199042 119904



119899)




119895119901= 119892




119899) +

119899

sum119894=1

(119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(11)




119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119895] are





119895 119895 = 1 2 119899 where 119899 is the



119895le 119889

119895le 119889

upper119895




d119905119894= array

1[119898

1] array

2[119898

2] array

119899[119898

119899]119905

119894

= [1198891 1198892 119889

119899]119905

119894

(3)


119895(119898




array119895[119898



120578119895[119898

119895] =

119872119895minus 119898

119895+ 1

119872119895

(4)


1[119898

1] storing





119895[119898


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)


array119895[119898

119895] (119896 = 119895+1 and 119896 le 119899) 119879

119896[119898

119896] and 120578

119896(119898

119896) denote


119896[119898




120578119896(119898


described



1[119898

1] array

2[119898

2]

array119899[119898



1015840



[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)


[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


1198882 if array

119895[119898


1198883 otherwise

(7)


119865(d119905119894PX119905



119895] belongs to


1ge 119888

3ge 119888

2) depending



119865 (dPXY)

= 119886 sdot exp


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)


119894119889(dP) 119891








119895119901(XY) =


119877119895119901= prob (119892

119895119901 (XY) le 0)

= prob (119892119895119901 (S) le 0)


119891119904(1199041 1199042 119904

119899) 119889119904

11198891199042sdot sdot sdot 119889119904

119899

(10)


119891119904(1199041 1199042 119904




119904(1199041 1199042 119904



119899)




119895119901= 119892




119899) +

119899

sum119894=1

(119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(11)




119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119865(d119905119894PX119905



119895] belongs to


1ge 119888

3ge 119888

2) depending



119865 (dPXY)

= 119886 sdot exp


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)


119894119889(dP) 119891








119895119901(XY) =


119877119895119901= prob (119892

119895119901 (XY) le 0)

= prob (119892119895119901 (S) le 0)


119891119904(1199041 1199042 119904

119899) 119889119904

11198891199042sdot sdot sdot 119889119904

119899

(10)


119891119904(1199041 1199042 119904




119904(1199041 1199042 119904



119899)




119895119901= 119892




119899) +

119899

sum119894=1

(119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(11)




119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119875119876= 32000 kg 119867 = 16m V

119902= 13mmin 119875

119866119909= 11000 kg 119875

1198661= 2000 kg 119875

1198662= 800 kg


119909= 45mmin V

119889= 90mmin 120590

119904= 235MPa ] = 03 119864 = 211 times 1011 Pa



119899) = 0 Here the



119885could be


120583119885cong

119899

sum119894=1

(120583119904119894minus 119904lowast

119894)120597119892

120597119904119894

10038161003816100381610038161003816100381610038161003816119904lowast119894

(12a)

120590119885= radic

119899

sum119894=1

(120597119892

120597119904119894

)2100381610038161003816100381610038161003816100381610038161003816119904lowast119894

1205902119904119894 (12b)


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)




119894(119894 = 1 2 119899) are assumed to be


6 RBDO Procedure






119876) Normal 32000 kg 010




119894mm 005




0= 09



1= 65

and 1198725


3= 119872



1(XY)

to 1198927(XY) 0968 for 119892

8(XY) and 119892

9(XY) and 0759 for

11989210(XY) and 119892







using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













using AFOSM

Output the best

record the best



Update the local


No

Yes


No

Yes




NoYes

pheromone

pheromone

Update the global

individuals dti



solution dtbest





t gt sgen

t = 0

t = t + 1

t = t + 1


0 set random





[Tj(mj)]ti

[Tj(mj)]ti




1198892

1198893

1198894

1198895









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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












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


120590max 5 (MPa) le[1205901199031198945] 1016 0998 129959 0775 104977 0998


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


119881119891119881

119892119895119901(sdot) 119895119901 = 3 8 10


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


times104

Obj

ectiv

e fun

ctio

n va

lue (

mm

2)


(a)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Generation

Relia

bilit

y in

dex



(b)


119895119901(sdot) 119895119901 = 3 8 10)




3= 267 120573

8= 232 and 120573

10= 104)




8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











8 Conclusions




Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article reliability-based design optimization for

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