research article reliability-based design optimization for crane...

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; [email protected]

Received 9 July 2014; Revised 4 January 2015; Accepted 19 January 2015

Academic Editor: Paolo Lonetti

Copyright © 2015 X. Fan and X. Bi. This is an open access article distributed under the Creative Commons Attribution License,which 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 factories,railways, 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-savings.As 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 [4–6]. However, these meth-ods are all based on deterministic optimum designs anddo not consider the effect of randomness in the structureand/or 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 boundaries,leaving 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. Nevertheless,design 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 pageshttp://dx.doi.org/10.1155/2015/828930

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 philosophy,which 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 attention,both in theoretical research and practical applications [11–13]. Despite advances in this area, few RBDO approachesspecific to CMS have appeared in the technical literature.Therefore, 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 situations,which means that the CMS design optimization is a con-strained nonlinear optimization problem with discrete vari-ables, known as NP-complete combinatorial optimization.To solve such problems, recent studies have focused on thedevelopment of heuristic optimization techniques, such asgenetic algorithms (GAs) [16, 17], particle swarm optimiza-tion (PSO) [18–20], 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 [27,28]. In view of the simple operation, easy implementation,and the suitability of ACO for computational problemsinvolving discrete variables and combinatorial optimization,the 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 functions.The 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 practices.MCS 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 analysis.TheRBDOprocedure 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, materials,and 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 𝑓 (d,P)

subject to 𝑔𝑖𝑑 (d,P) ≤ 0, 𝑖𝑑 = 1, . . . , 𝑁1

𝑅𝑗𝑝= prob (𝑔

𝑗𝑝 (X,Y) ≤ 0) ≥ 𝑅𝑎,𝑗𝑝,

𝑗𝑝 = 1, 2, . . . , 𝑁2,

(1)

where

dlower ≤ d ≤ dupper, d ∈ 𝑅NDV, P ∈ 𝑅NPV. (2)

d = 𝜇(X) and P = 𝜇(Y) are the mean value vectorsof the random design vector X and random parametervector Y, respectively; 𝑓(d,P) is the objective function(i.e., the structure weight or volume); 𝑔

𝑖𝑑(d,P) ≤ 0 and

𝑅𝑎,𝑗𝑝

− prob(𝑔𝑗𝑝(X,Y) ≤ 0) ≤ 0 are the deterministic

and probabilistic constraints; prob(𝑔𝑗𝑝(X,P) ≤ 0) denotes

the probability of satisfying the 𝑗𝑝th performance function𝑔𝑗𝑝(X,Y) ≤ 0 and this probability should be no less than

the desired design reliability 𝑅𝑎,𝑗𝑝

; 𝑁1, 𝑁

2are the number

of deterministic and probabilistic constraints, respectively.d can only take values from a given discrete set 𝑅NDV,where 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 crane’s solid-web girder is usually a boxsection fabricated from steel plate, for themain and vicewebs,top, and bottom flanges, as shown in Figure 2. Therefore,given 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 strength,stiffness, 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, 𝐹

𝑞represents the uniform load; 𝑃

𝐺𝑗(𝑗 =

1, 2, . . .) denotes the concentrated load; ∑𝑃 stands for thetrolley wheel load applied by the sumof the lifting capacity𝑃

𝑄

and trolley weight 𝑃𝐺𝑥; 𝑃

𝐻and 𝐹

𝐻represent the horizontal

concentrated anduniform inertial load, respectively; 𝑃𝑠is the

lateral force.In 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

∑P

Figure 2: Cross section 1-1 of main girder.

PS

PS

PHFH

FHPH

L

(a) On the horizontal plane

PG PGPG ∑P 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 stress,deflection, 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 𝑔

𝑗𝑝(X,Y), 𝑗𝑝 =

1, 2, 3 correspond to the position of the maximum bendingmoment, when a fully loaded trolley is lowering and braking


Table1:Mathematicalmod

elof

theR

BDOforO

TCMS.

Designvaria

blem

eanvalues

oftheo

ptim

izationprob

lem

d=[𝑑1,𝑑2,𝑑3,𝑑4,𝑑5]𝑇=𝜇(X)=[𝜇(𝑥1),𝜇(𝑥2),𝜇(𝑥3),𝜇(𝑥4),𝜇(𝑥5)]𝑇

Designvaria

blem

eanvalue

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5

Symbo

lℎ

𝑡𝑡 1

𝑡 2𝑏

Rand

omdesig

nvaria

bles

andparameterso

fthe

optim

izationprob

lem

S=[X,Y]𝑇=[𝑥1,𝑥2,𝑥3,𝑥4,𝑥5,𝑦1,𝑦2,𝑦3]𝑇=[𝑠1,𝑠2,𝑠3,𝑠4,𝑠5,𝑠6,𝑠7,𝑠8]𝑇

Varia

blea

ndparameter

𝑥1

𝑥2

𝑥3

𝑥4

𝑥5

𝑦1

𝑦2

𝑦3

Symbo

lℎ

𝑡𝑡 1

𝑡 2𝑏

𝑃𝑄

𝑃𝐺𝑥

𝐸

Objectiv

efun

ction:

minim

izingthec

ross-sectio

nalareao

fthe

girder𝑓(d,P)=ℎ⋅(𝑡 1+𝑡 2)+𝑡⋅(2⋅𝑏+3⋅𝑒+𝑏 𝑒)

𝑒istakenequalto20

mm

which

facilitates

welding

and𝑏 𝑒issetequ

alto

15𝑡so

asto

guaranteethe

localstabilityof

flangeo

verhanging

andther

ailinstallatio

n.Inequalityconstraintso

fthe

RBDOprob

lem

Reliabilityconstraints

Limitsta

tefunctio

nsDescriptio

n𝑅𝑎,1−prob(𝑔1(X,Y)≤0)≤0

𝑔1(X,Y)=𝜎1−[𝜎]

Maxim

umstr

essc

ombinedatdangerou

spoint

(1)(1-1

section).

𝑅𝑎,2−prob(𝑔2(X,Y)≤0)≤0

𝑔2(X,Y)=𝜎2−[𝜎]

Normalstr

essa

tdangerous

point(2)

(1-1sectio

n).


𝑔3(X,Y)=𝜎3−[𝜎]

Normalstr

essa

tdangerous

point(3)

(1-1sectio

n).


𝑔4(X,Y)=𝜏 1−[𝜏]

Maxim

umshearstre

ssatthem

iddleo

fmainweb

(2-2

section).


𝑔5(X,Y)=𝜏 2−[𝜏]

Horizon

talshear

stressinthefl

ange

(2-2

section).


𝑔6(X,Y)=𝜎max4−[𝜎𝑟4]

Fatig

uestr

engthatdangerou

spoint

(4)(1-1

section)


𝑔7(X,Y)=𝜎max5−[𝜎𝑟5]

Fatig

uestr

engthatdangerou

spoint

(5)(1-1

section)


𝑔8(X,Y)=𝑓V−[𝑓

V]Ve

rticalstaticdeflectionof

them

idspan


𝑔9(X,Y)=𝑓ℎ−[𝑓ℎ]

Horizon

talstatic

displacemento

fthe

midspan

𝑅𝑎,10−prob(𝑔10(X,Y)≤0)≤0

𝑔10(X,Y)=[𝑓𝑉]−𝑓𝑉

Verticalnaturalvibratio

nfre

quency

ofthem

idspan

𝑅𝑎,11−prob(𝑔11(X,Y)≤0)≤0

𝑔11(X,Y)=[𝑓𝐻]−𝑓𝐻

Horizon

taln

aturalvibrationfre

quency

ofthem

idspan

Deterministicconstraints

Natureo

fcon

straint

Descriptio

n𝑔1(d,P)=ℎ/𝑏−3≤0

Stabilityconstraint

Height-to-width

ratio

oftheb

oxgirder

𝑔𝑗1(d,P)=𝑑𝑖−𝑑up

per

𝑖≤0

Boun

dary

constraint

Upp

erbo

unds

ofdesig

nvaria

bles𝑗1

=2,4,...,10,𝑖=1,2,...,5

𝑔𝑗2(d,P)=𝑑lower

𝑖−𝑑𝑖≤0

Boun

dary

constraint

Lower

boun

dsof

desig

nvaria

bles𝑗2

=3,5,...,11,𝑖=1,2,...,5

Materialp

ermissibleno

rmalstr

ess[𝜎]=𝜎𝑠/1.33andperm

issibleshearstre

ss[𝜏]=[𝜎]/√3;𝜎𝑠representsmaterialyield

stress;[𝜎𝑟4]and[𝜎𝑟5]arethe

weld

edjointtensilefatigue

perm

issible

stressesfor

points(4)a

nd(5),respectiv

ely;[𝑓V]and[𝑓ℎ]arethe

verticalandho

rizon

talp

ermissiblestaticstiffnesses,w

hich

aretaken

equalto𝐿/800

and𝐿/2000,respectiv

ely,and

𝐿isthe

span

oftheg

irder;[𝑓𝑉]and[𝑓𝐻]arethe

verticalandho

rizon

talp

ermissibledynamicstiffnesses,w

hich

aretaken

equalto2∼

4Hza

nd1.5∼2H

z,respectiv

ely.𝑑

lower

𝑖and𝑑up

per

𝑖arethe

lower

andup

perb

ound

softhe

desig

nvaria

ble𝑑

𝑖.

where

𝜎1=√[𝑀𝑥/𝑊𝑥+𝑀𝑦/𝑊𝑦]2+[𝜙∑𝑃/(4ℎ𝑔+100)𝑡1]2−[𝑀𝑥/𝑊𝑥+𝑀𝑦/𝑊𝑦][𝜙∑𝑃/(4ℎ𝑔+100)𝑡1]+3[𝐹𝑝𝑆𝑦/𝐼𝑥(𝑡1+𝑡2)+𝑇𝑛/2𝐴0𝑡1]2,𝜎2=𝑀𝑥/𝑊 𝑥+𝑀𝑦/𝑊 𝑦,𝜎3=𝑀𝑥/𝑊

𝑥+𝑀𝑦/𝑊

𝑦,

𝜏1=1.5𝐹𝑒/ℎ𝑑(𝑡1+𝑡2)+𝑇𝑛1/2𝐴𝑑0𝑡1,𝜏2=1.5𝐹𝑒𝐻/𝑡(2𝑏+3𝑒+𝑏𝑒)+𝑇𝑛1/2𝐴𝑑0𝑡1,𝜎max4=𝑀𝑥/𝑊

𝑥,𝜎max5=𝑀𝑥/𝑊

𝑥,𝑓V=∑𝑃𝐿3/48𝐸𝐼𝑥,𝑓ℎ=(𝑃𝐻𝐿3/48𝐸𝐼𝑦)(1−3/4𝑟1)+(5𝐹𝐻𝐿4/384𝐸𝐼𝑦)(1−4/5𝑟1)

𝑓𝑉=(1/2𝜋)√𝑔/(𝑦0+𝜆0)(1+𝛽),𝑦0=𝑃𝑄𝐿3/96𝐸𝐼𝑥,𝛽=((𝐹𝑞𝐿+𝑃𝐺𝑥)/𝑃𝑄)(𝑦0/(𝑦0+𝜆0))2,𝑓𝐻=(1/2𝜋)√192𝐸𝐼𝑦/𝑚𝑒(4𝑟1−3)𝐿3,𝑚𝑒=(𝐹𝑞𝐿+𝑃𝑄+𝑃𝐺𝑥)/𝑔

𝑀𝑥,𝑀𝑦arethebend

ingmom

ents,𝐹𝑝,𝐹𝑒,𝐹𝑒𝐻aretheshearforces,and𝑇𝑛,𝑇𝑛1arethetorquesa

tdifferentlocations.𝐼𝑥,𝐼𝑦deno

tetheinertia

lmod

uli,𝑊𝑥,𝑊 𝑥,𝑊

𝑥,𝑊

𝑥,𝑊

𝑥,𝑊𝑦,𝑊 𝑦,𝑊

𝑦deno

tethestr

ength

mod

uli,and𝑆𝑦deno

testhe

topflang

estatic

mom

ent.𝐴0,𝐴𝑑0representn

etarea

atdifferent

sections,ℎ𝑔representstheh

eighto

frail,andℎ𝑑representstheh

eighto

fgird

eratthee

nd-span.𝐸iselastic

mod

ulus,𝑔

isgravity

constant,𝜑

isim

pactfactor,𝑟1iscompu

tingcoeffi

ciento

fthe

craneb

ridge,and𝜆0isthen

etelo

ngationof

thes

teelwire

rope.Th

eother

parametersa

rethes

amea

sbefore.


0 10h

b

k

E

Sour

ces

Sour

ces

−10 0 10 20

0 20 40 −40 −20 0 20

−40 −20 0 20 −40 −20 0 20

Effect on fV (%)−20

−40 −20

−10

PGx

PQ

k2

k1

h

b

k

E

PGx

PQ

k2

k1

Sour

ces

h

b

k

E

PGx

PQ

k2

k1 Sou

rces

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 𝜎2 (%) Effect on 𝜎1 (%)

Effect on 𝜎3 (%)

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 𝑔

𝑗𝑝(X,Y), 𝑗𝑝 =

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 functions𝑔𝑗𝑝(X,Y), 𝑗𝑝 = 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 functions𝑔𝑗𝑝(X,Y), 𝑗𝑝 = 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 versa.Thereby,the 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 AS,the 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.


4.1. Representation and Initialization of Solution. In our ACOalgorithm, each solution is composed of different arrayelements that correspond to different design variables. Oneant’s search path represents a solution to the optimizationproblem or a set of design schemes. The path of the 𝑖th antin the 𝑛-dimensional search space at iteration 𝑡 could bedenoted by d𝑡

𝑖(𝑖 = 1, 2, . . . , popsize; popsize denotes the

population size). Continuous array elements array𝑗[𝑚

𝑗] are

used to store the mean values of the discrete design variable𝑑𝑗in the nondecreasing order (𝑚

𝑗denotes the array sequence

number for the 𝑗th design variable mean value 𝑑𝑗. This is

integer with 1 ≤ 𝑚𝑗≤ 𝑀

𝑗. 𝑗 = 1, 2, . . . , 𝑛, where 𝑛 is the

number of design variables and 𝑀𝑗is the number of discrete

values available for 𝑑𝑗. 𝑑lower

𝑗≤ 𝑑

𝑗≤ 𝑑

upper𝑗

, where 𝑑lower𝑗

and𝑑upper𝑗

are the lower and upper bounds of 𝑑𝑗). Thus, consider

d𝑡𝑖= {array

1[𝑚

1] , array

2[𝑚

2] , . . . , array

𝑛[𝑚

𝑛]}𝑡

𝑖

= [𝑑1, 𝑑2, . . . , 𝑑

𝑛]𝑡

𝑖.

(3)

The initial solution is randomly selected.We set the initialpheromone level [𝑇

𝑗(𝑚

𝑗)]0 of all array elements in the space

to be zero. The heuristic information 𝜂𝑗(𝑚

𝑗) of array element

array𝑗[𝑚

𝑗] is expressed by (4), so as to induce subsequent

solutions to select smaller variable values as far as possibleand accelerate optimization process:

𝜂𝑗[𝑚

𝑗] =

𝑀𝑗− 𝑚

𝑗+ 1

𝑀𝑗

. (4)

4.2. 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, thus,in our algorithm, constructs solutions. In accordance withthe transition rule of ant colony system (ACS) [35], eachant begins with the first array element array

1[𝑚

1] storing

the first design variable mean value 𝑑1and selects array

elements in proper until array𝑛[𝑚

𝑛]. In this way, a solution

is constructed. Hence, for the 𝑖th ant on the array elementarray

𝑗[𝑚

𝑗], the selection probability of the next array element

array𝑘[𝑚

𝑘] is given by

𝑆𝑖(𝑗, 𝑘) =

{{{{{{{{{

maxarray𝑘[𝑚𝑘]∈𝐽𝑘

{𝛼 ⋅ 𝑇𝑘(𝑚

𝑘) + 𝜉 ⋅ 𝜂

𝑘(𝑚

𝑘)} ,

if 𝑞 < 𝑞0,

𝑝𝑖(𝑗, 𝑘) , otherwise,

(5a)

𝑝𝑖(𝑗, 𝑘)

=

{{{{{{{{{{{

𝛼 ⋅ 𝑇𝑘(𝑚

𝑘) + 𝜉 ⋅ 𝜂

𝑘(𝑚

𝑘)

∑array𝑘[𝑚𝑘]∈𝐽𝑘(𝑟)

[𝛼 ⋅ 𝑇𝑘(𝑚

𝑘) + 𝜉 ⋅ 𝜂

𝑘(𝑚

𝑘)],

if array𝑘[𝑚

𝑘] ∈ 𝐽

𝑘,

0 otherwise,

(5b)

where 𝐽𝑘is the set of feasible neighbor array elements of

array𝑗[𝑚

𝑗] (𝑘 = 𝑗+1 and 𝑘 ≤ 𝑛). 𝑇

𝑘[𝑚

𝑘] and 𝜂

𝑘(𝑚

𝑘) denote

the pheromone intensity and heuristic information on arrayelements array

𝑘[𝑚

𝑘], respectively. 𝛼 and 𝜉 give the relative

importance of trail 𝑇𝑘[𝑚

𝑘] and the heuristic information

𝜂𝑘(𝑚

𝑘), respectively. Other parameters are as previously

described.

4.3. 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 = {array1[𝑚1], array2[𝑚2], . . . ,array

𝑛[𝑚

𝑛]}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 dbest. If d

best is better than dbest,we replace dbest with dbest. Otherwise, the global optimalsolution remains unchanged.

4.4. Pheromone Updating. The pheromone updating rules ofACO include global updating and local rules. When ant 𝑖 hasfinished a path, the pheromone trails on the array elementsthroughwhich the ant has passed are updated. In this process,the 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

[𝑇𝑗(𝑚

𝑗)]𝑡+1

𝑖= (1 − 𝛾) ⋅ [𝑇

𝑗(𝑚

𝑗)]𝑡

𝑖,

𝑗 = 1, 2, . . . , 𝑛, 1 ≤ 𝑚𝑗≤ 𝑀

𝑗.

(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:

[𝑇𝑗(𝑚

𝑗)]𝑡+1

𝑖= (1 − 𝜌) ⋅ [𝑇

𝑗(𝑚

𝑗)]𝑡

𝑖+ 𝜆 ⋅ 𝐹 (d𝑡

𝑖,P,X𝑡

𝑖,Y) ,(6b)

where𝜆

=

{{{{{{{{{

𝑐1, if array

𝑗[𝑚

𝑗] ∈ the globally (iteratively) best tour,

𝑐2, if array

𝑗[𝑚

𝑗] ∈ the iterationworst tour,

𝑐3, otherwise.

(7)


𝐹(d𝑡𝑖,P,X𝑡

𝑖,Y) represents the fitness function value of ant 𝑖 on

the 𝑡th iteration (see the next section). array𝑗[𝑚

𝑗] belongs to

the array elements of the path generated by ant 𝑖 on the 𝑡thiteration. 𝜆 is a phase constant (𝑐

1≥ 𝑐

3≥ 𝑐

2) depending

on the quality of the solutions to reinforce the pheromone ofthe best path and evaporate that of the worst. 𝛾 ∈ [0, 1] and𝜌 ∈ [0, 1] are the local and global pheromone evaporationrates, respectively. Other parameters are the same as before.

4.5. Evaluation of Solution. The aim of OTCMS RBDOis to develop a design that minimizes the total structureweight while satisfying all deterministic and probabilisticconstraints.The 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:

𝐹 (d,P,X,Y)

= 𝑎 ⋅ exp{{{{{

−𝑐 ⋅ 𝑓 (d,P) − 𝑏

⋅ [

[

𝑁1

∑𝑖𝑑=1

𝑓𝑖𝑑 (d,P) +

𝑁2

∑𝑗𝑝=1

𝑓𝑗𝑝 (X,Y)]

]

2

}}}}}

,

(8)

where𝑓𝑖𝑑 (d,P) = max [0, 𝑔𝑖𝑑 (d,P)] ,

𝑓𝑗𝑝 (X,Y) = max [0, 𝑅𝑎,𝑗𝑝 − prob (𝑔𝑗𝑝 (X,Y) ≤ 0)] .

(9)

𝐹(d,P,X,Y) is unconstrained objective function (the fitnessfunction); 𝑓(d,P) is the original constraint objective func-tion (see (1)); 𝑎, 𝑐, and 𝑏 are positive problem-specificconstants; and 𝑓

𝑖𝑑(d,P), 𝑓

𝑗𝑝(X,Y) are penalty functions cor-

responding to the 𝑖𝑑th deterministic and 𝑗𝑝th probabilistic,respectively. 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.

4.6. 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 functions𝑔𝑗𝑝(X,Y), which can represent the stress, displacement, stiff-

ness, and so on, is defined in terms of random vectorsS (define S ∈ (X,Y)). The limit states that separate thedesign space into “failure” and “safe” regions are 𝑔

𝑗𝑝(X,Y) =

0. Accordingly, the probability of structural reliability withrespect to the 𝑗𝑝th limit state function in the specified modeis

𝑅𝑗𝑝= prob (𝑔

𝑗𝑝 (X,Y) ≤ 0)

= prob (𝑔𝑗𝑝 (S) ≤ 0)

= ∬ ⋅ ⋅ ⋅ ∫𝐷

𝑓𝑠(𝑠1, 𝑠2. . . , 𝑠

𝑛) 𝑑𝑠

1𝑑𝑠2⋅ ⋅ ⋅ 𝑑𝑠

𝑛,

(10)

where 𝐷 denotes the safe domain (𝑔𝑗𝑝(S) ≤ 0).

𝑓𝑠(𝑠1, 𝑠2, . . . , 𝑠

𝑛) is the joint probability density function

(PDF) of the random vector S, and 𝑅𝑗𝑝

can be calculated byintegrating the PDF 𝑓

𝑠(𝑠1, 𝑠2, . . . , 𝑠

𝑛) over 𝐷. Nevertheless,

this integral is not a straightforward task, as 𝑓𝑠(𝑠1, 𝑠2, . . . , 𝑠

𝑛)

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 MPPS∗, and the reliability index 𝛽 is defined as the distance ofthe design point and the origin, 𝛽 = ‖S∗‖, which could becalculated by determining theMPP in randomvariable space.

Firstly, obtain a linear approximation of the performancefunction 𝑍

𝑗𝑝= 𝑔

𝑗𝑝(S) by using the first-order Taylor’s series

expansion about the MPP S∗:

𝑍 ≅ 𝑔 (𝑠∗1, 𝑠∗2, . . . , 𝑠∗

𝑛) +

𝑛

∑𝑖=1

(𝑠𝑖− 𝑠∗

𝑖)𝜕𝑔

𝜕𝑠𝑖

𝑠∗𝑖

. (11)


Table 2: Main technical characteristics and mechanical properties of the metallic structure.

Lifting capacity Lifting height Lifting speed WeightTrolley Cab Traveling mechanism

𝑃𝑄= 32000 kg 𝐻 = 16m V

𝑞= 13m/min 𝑃

𝐺𝑥= 11000 kg 𝑃

𝐺1= 2000 kg 𝑃

𝐺2= 800 kg

Span length Trolley velocity Crane bridge velocity Yield stress Poisson’s ratio Elasticity modulus𝐿 = 25.5m V

𝑥= 45m/min V

𝑑= 90m/min 𝜎

𝑠= 235MPa ] = 0.3 𝐸 = 2.11 × 1011 Pa

Since the MPP S∗ is on the limit state surface, the limitstate function equals zero 𝑔(𝑠∗

1, 𝑠∗2, . . . , 𝑠∗

𝑛) = 0. Here the

subscript 𝑗𝑝 has been dropped for the sake of simplicity ofthe subsequent notation.

𝑍’s mean value 𝜇𝑍and standard deviation 𝜎

𝑍could be

expressed as follows:

𝜇𝑍≅

𝑛

∑𝑖=1

(𝜇𝑠𝑖− 𝑠∗

𝑖)𝜕𝑔

𝜕𝑠𝑖

𝑠∗𝑖

, (12a)

𝜎𝑍= √

𝑛

∑𝑖=1

(𝜕𝑔

𝜕𝑠𝑖

)2𝑠∗𝑖

𝜎2𝑠𝑖. (12b)

The reliability index 𝛽 is shown as

𝛽 =𝜇𝑍

𝜎𝑍

=∑𝑛

𝑖=1(𝜇𝑠𝑖− 𝑠∗

𝑖) (𝜕𝑔/𝜕𝑠

𝑖)𝑠∗𝑖

∑𝑛

𝑖=1𝛼𝑖𝜎𝑠𝑖(𝜕𝑔/𝜕𝑠

𝑖)𝑠∗𝑖

, (13)

where

𝛼𝑖=

𝜎𝑠𝑖(𝜕𝑔/𝜕𝑠

𝑖)𝑠∗𝑖

[∑𝑛

𝑖=1(𝜕𝑔/𝜕𝑠

𝑖)2𝑠∗𝑖

𝜎2𝑠𝑖]1/2. (14)

Then

𝜇𝑠𝑖− 𝑠∗

𝑖− 𝛽𝛼

𝑖𝜎𝑠𝑖= 0. (15)

Finally, combining (13), (15), and the limit state function𝑔(S) = 0, the MPP 𝑠∗

𝑖and reliability index 𝛽 could be

calculated by an iterative procedure. Then, the reliability𝑅 could be approximated by 𝑅 = Φ(𝛽) where Φ(⋅) isthe standard normal cumulative distribution function. Here,random variables 𝑠

𝑖(𝑖 = 1, 2, . . . , 𝑛) 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 on2.26GHz Intel Dual Core processor and 1GB main memory.

Table 3: Statistical properties of the random variables 𝑠𝑖.

Random variable Distribution Mean value (𝜇) COV (𝜎/𝜇)Lifting capacity (𝑃

𝑄) Normal 32000 kg 0.10

Trolley weight (𝑃𝐺𝑥) Normal 11000 kg 0.05

Elasticity modulus (𝐸) Normal 2.11 × 1011 Pa 0.05Design variables (𝑑

𝑖) Normal 𝑑

𝑖mm 0.05

Table 4: Parameter values used in adopted ACO.

Parameters Values for the exampleFitness function parameters 𝑎 = 100, 𝑐 = 10−6, 𝑏 = 10Number of ants Popsize= 40Maximum number of iterations mgen= 500Fixed nonevolution generation sgen= 5 or 10Pheromone updating rule parameter 𝜌 = 0.2, 𝛾 = 0.4Selection probability parameters 𝛼 = 1, 𝛽 = 1Phase constants 𝑐1 = 2, 𝑐2 = 0, 𝑐3 = 1Transition rule parameter 𝑞

0= 0.9

7.1. 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 𝑀

1= 65

and 𝑀5

= 100. The step size intervals were set to be0.5mm for web and flange thicknesses of less than 30mmand to be 1mm for thicknesses more than 30mm [15]. Thus,the number of discrete mean values available for the designvariables (thicknesses of main web, vice web, and flange) was𝑀2= 𝑀

3= 𝑀

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 0.979 was set for 𝑔

1(X,Y)

to 𝑔7(X,Y), 0.968 for 𝑔

8(X,Y) and 𝑔

9(X,Y), and 0.759 for

𝑔10(X,Y) and 𝑔

11(X,Y). It should be noted that these target

reliabilities serve only as examples to illustrate the proposedRBDO approach and are not recommended design values.By 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

Calculate the fitness function

No

Yes

The stopping criterion is satisfied?

No

Yes

Check the deterministic

Check the probabilistic constraints

Perform local search for the best solution

NoYes

pheromone

pheromone

Update the global

individuals dti

i < popsize?i = i + 1

constraints gid(dti ,P) ≤ 0

solution dtbest

solution d0best the best solution dtbest

prob(gjp(Xti ,Y) ≤ 0) ≥ Ra,jp using AFOSM

values F(dti ,P,Xti ,Y); record

dtbest using mutation

t > sgen?

t = 0

t = t + 1

t = t + 1

Initialize design vectors d0i andpheromone [Tj(mj)]

0; set random

information 𝜂j(mj); let t = 0

constraints gid(d0i ,P) ≤ 0

prob(gjp(X0i ,Y) ≤ 0) ≥ Ra,jp

values F(d0i ,P,X0i ,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)𝑑1

𝑑2

𝑑3

𝑑4

𝑑5

𝑓(d,P) (mm2) 𝐹(d,P,X,Y)PD 1600 10 8 6 760 39700 96.1078 — —DDO 1765 7 6 6 610 30875 96.9597 58 0.1794RBDO 1815 7.5 6.5 6 785 35756 96.4875 23 4.23PD stands for practical design.

7.2. 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. Thus,the deterministic optimization found a better solution withinjust 0.1794 s. The critical constraint of the optimum design isthe vertical natural vibration frequency 𝑓

𝑉at the midspan

point. The corresponding constraint value is 2.00067Hz,


Table 6: The performance and reliability with respect to optimum designs.

Performance Constraints requirement Practical design The DDO The RBDOPerformance 𝑅

𝑎,𝑗𝑝Per. 𝑅

𝑗𝑝Per. 𝑅

𝑗𝑝Per. 𝑅

𝑗𝑝

𝜎1(MPa)

≤17693.6 0.999 122.786 0.996 98.9442 1.0

𝜎2(MPa) 117.3 0.999 148.342 0.888 119.36 0.997

𝜎3(MPa) 134.5 0.979 169.924 0.604 136.865 0.996

𝜏1(MPa)

≤101 51.15 0.999 61.5463 0.999 58.6088 0.999𝜏2(MPa) ≥0.979 9.12 0.999 18.5513 0.999 15.4756 0.998

𝜎max 4 (MPa) ≤[𝜎𝑟𝑖4] 108.1 0.999 137.608 0.999 110.975 0.999[𝜎𝑟𝑖4] (MPa) — 216.3 — 214.401 — 215.45 —

𝜎max 5 (MPa) ≤[𝜎𝑟𝑖5] 101.6 0.998 129.959 0.775 104.977 0.998[𝜎𝑟𝑖5] (MPa) — 141.7 — 138.996 — 140.49 —

𝑓V (mm) ≤31.875 ≥0.968 23.68 0.968 28.8096 0.728 22.1105 0.989𝑓ℎ(mm) ≤12.750 2.86 0.999 4.42906 0.999 2.78834 0.9999

𝑓𝑉(Hz) ≥2.0

≥0.759 2.08 0.759 2.00067 0.494 2.11043 0.851𝑓𝐻(Hz) ≥1.5 2.757 1.0 2.0638 1.0 2.6094 1.0

Feasible Infeasible FeasibleActive constraint 𝑓

𝑉𝑓𝑉

𝑔𝑗𝑝(⋅), 𝑗𝑝 = 3, 8, 10

Per. stands for performance.

94.494.694.8

9595.295.495.695.8

9696.296.496.696.8

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

3.5

4

4.5

5

5.5

6

Fitness function value

×104

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 𝛽8Vertical natural vibration frequency 𝛽10

𝛽3Normal 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 (𝑔

𝑗𝑝(⋅), 𝑗𝑝 = 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 6.The active constraintsof the deterministic optimum have reliabilities of 0.604,0.728, and 0.494, which are all below the desired reliabilitiesof 0.979, 0.968, and 0.759, 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 process,RBDO is needed.

As shown in Tables 5 and 6, RBDO required 23 optimiza-tion iterations.The reliabilities of the active constraints at theoptimum point are 0.996, 0.989, and 0.851 (corresponding toreliability indices 𝛽

3= 2.67, 𝛽

8= 2.32, and 𝛽

10= 1.04),

which are all above the desired reliabilities of 0.979, 0.968, and0.759. ComparedwithDDO, the final area given by theRBDOprocess increases from 30875 to 35756mm2 (an increase of15.9%), 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 safety.The inherent nature of uncertain factors in the design ofCMSs means that RBDO is a more realistic design method.It 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 longview.The 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.

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