review article mathematical and metaheuristic applications

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 271031 33 pageshttpdxdoiorg1011552013271031

Review ArticleMathematical and Metaheuristic Applications in DesignOptimization of Steel Frame Structures An Extensive Review

Mehmet Polat Saka1 and Zong Woo Geem2

1 Department of Civil Engineering University of Bahrain PO Box 32038 Isa Town Bahrain2Department of Energy and Information Technology Gachon University Seongnam 461-701 Republic of Korea

Correspondence should be addressed to Zong Woo Geem zwgeemgmailcom

Received 28 September 2012 Accepted 4 December 2012

Academic Editor Sheng-yong Chen

Copyright copy 2013 M P Saka and Z W Geem This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The type of mathematical modeling selected for the optimum design problems of steel skeletal frames affects the size andmathematical complexity of the programming problem obtained Survey on the structural optimization literature reveals that thereare basically two types of design optimization formulation In the first type only cross sectional properties of frame membersare taken as design variables In such formulation when the values of design variables change during design cycles it becomesnecessary to analyze the structure and update the response of steel frame to the external loading Structural analysis in this typeis a complementary part of the design process In the second type joint coordinates are also treated as design variables in additionto the cross sectional properties of members Such formulation eliminates the necessity of carrying out structural analysis in everydesign cycle The values of the joint displacements are determined by the optimization techniques in addition to cross sectionalproperties The structural optimization literature contains structural design algorithms that make use of both type of formulationIn this study a review is carried out on mathematical and metaheuristic algorithms where the effect of the mathematical modelingon the efficiency of these algorithms is discussed

1 Introduction

Structural analysis and structural design are two inseparabletools of a structural designer Design process necessitatesfinding out the cross-sectional properties of the members ofa steel frame such that the frame with these members has therequired strength to withstand the external loadings and itsdeflected shape is within the limitations specified by designcodes Structural designer can determine the cross-sectionalproperties of steel frame members in one step if the frameis statically determinate if there are no restraints on jointdisplacements In such frames computation ofmember forcesdoes not require the prior information of cross-sectionalproperties and the design can be completed within onestep of structural analysis Despite the design of staticallyindeterminate frames structural analysis cannot be carriedout without knowing the values of cross-sectional propertieswhich makes the design process iterative Designer has tofirst assume certain values for the cross-sectional properties

or select certain steel profiles from the available list ofsteel sections for the frame members before the analysis ofthe frame can be carried out in order to find out internalforces and moments in the members With these assumedor preselected values of member cross-sectional propertiesthe structural response of the frame may not be within thelimitations imposed by the design codes or the responsemight be far away from the bounds which are an indicationof an overdesign In this case the designer has to changethe values of the originally adopted cross sectional propertiesin order to satisfy the conditions that the values of jointdisplacements and strength of its members are within thelimitations imposed by design codes It is not difficult toenvisage that designer has to carry out several trials before thedesired set of cross-sectional properties is obtained Althoughsome engineering experience and intuition can be used inassigning the cross-sectional properties of members it isneedless to mention that finding the best combination ofthese cross-sectional properties is quite time consuming and

2 Mathematical Problems in Engineering

a complex task It requires tremendous computational timeto try all the possible combinations such that the strengthand displacement constraints imposed by the design codesare satisfied and the amount of steel which is required tobuild the frame is the minimum However the emergence ofcomputational methods of mathematical programming andimprovements that took place in computers in early 1960rsquoshas provided another approach for handling the structuraldesign problems In this new approach the design problem isformulated as a decision-making problemwhere an objectivefunction is to be minimized or maximized while number ofconstraint functions is satisfiedThe formulation of structuraldesign problems as decision-making problems has yieldeda new branch in structural engineering called structuraloptimization [1ndash12]

The mathematical model of a decision-making problemhas the following form

Minimize 119882 = 119891 (119909119894) 119894 = 1 119899 (1a)

Subject to ℎ119895

(119909119894) = 0 119895 = 1 ne (1b)

119892119895

(119909119894) le 0 119895 = ne + 1 119898 (1c)

119909ℓ

119894le 119909

119894le 119909

119906

119894 (1d)

where 119909119894represents decision variable 119894 Decision variables

may take continuous or discrete values Continuous decisionvariable can take any real value within the range shown in(1d) The decision variables in structural optimization arecalled design variables which are the parameters that controlthe geometry or material properties of a steel frame Forexample the moment of inertias of beams and columns in aframe can be considered as continuous design variable if theseelements are to be manufactured locally for that particularframe However if they are to be selected from commerciallyavailable set of steel sections design variables can only takeisolated values which make them discrete variables

119891(119909119894) in (1a) represent the objective function which can

be used as a measure of effectiveness of the decision Instructural optimization problems generally minimumweightor cost of the structure is taken as objective functionThe equalities ℎ

119895(119909

119894) and inequalities 119892

119895(119909

119894) in (1b) and

(1c) respectively represent the limitations imposed on thebehavior of the structure by the design codes These may beserviceability limitations or ultimate strength requirementsif the design code considered is based on limit state designHowever if the design code is based on allowable stressdesign then the constraints given in (1b) and (1c) becomethe stiffness equalities displacement andor allowable stresslimitations

The review of the mathematical formulation of designoptimization of steel frames is carried out according tothe historical developments that took place in the designmethods Earlier steel design codes were based on allowablestress design concept while the later design codes use limitstate design perception

2 Formulation of Steel Frame DesignOptimization Problem

Survey on the structural optimization literature reveals thatthere are basically two types of design optimization formu-lation for the optimum design of elastic steel skeletal framedstructures although some variations of each type do exist Inthe first type of formulation which is also called conventionalformulation or coupled analysis and design (CAND) onlycross-sectional properties of steel frame members are takenas design variables Joint displacements are not treated asdesign variables In such formulation the stiffness equationsare excluded from the mathematical model Consequentlywhen the values of cross-sectional properties change duringdesign cycles it becomes necessary to analyze the structureand update the joint displacements and members forcesHence structural analysis is a complementary part of thedesign process In the second type of formulation jointdisplacements are also treated as design variables in addi-tion to the cross-sectional properties of frame membersIn such formulation the stiffness equations are included asdesign constraints in addition to other limitations in themathematicalmodelThis eliminates the necessity of carryingout structural analysis in every design cycle The values ofthe joint displacements are determined by the optimizationtechniques similar to those of cross-sectional properties It isapparent that in this type of formulation the total numberof design variables is much more than the first type offormulation This type of formulation is called simultaneousanalysis and design (SAND) It should be pointed out thatboth ways of formulation yield the same optimum resultHowever depending on the type of formulation adopted inthe design problem the size and mathematical complexity ofthe model acquired differ

21 Coupled Analysis and Design (CAND) In this type offormulation design phase is separated from the analysisphase This is achieved by only treating the cross-sectionalproperties such as areas or moment of inertias of membersas design variables The general outlook of the mathematicalmodel of this type of formulation is given in the following ifallowable stress design method is considered for the frame

Minimize 119882 = 120588

ng

sum

119894=1

ℓ119894119860119894 119894 = 1 ng (2a)

Subject to 120575119895

(119860119894) le 120575

119895119906 119895 = 1 rd (2b)

120590119896

(119860119894) le 120590all 119896 = 1 nm (2c)

119860119894119897

le 119860119894

le 119860119894119906

(2d)

where 120588 is the density of steel 119860119894represents cross-sectional

area for group and 119894 and ℓ119894is the total length of the

members in group 119894 in the steel frame 120575119895(119860

119894) is the restricted

119895th displacement 120575119895119906

is its upper bound and rd is thetotal number of such restricted displacements 120590

119896(119860

119894) is the

maximum stress in member 119896 and 120590all is the allowable stressfor steel nm is the total number of members in the structure

Mathematical Problems in Engineering 3

Table 1 119886119894and 119887

119894values for W sections

119894 119886119894

119887119894

1 13162 216502 05971 182203 10160 15710

In the previously mentioned model joint displacements andstresses that develop in a frame due to external loading areexpressed as function of cross-sectional areas of members Itis apparent that determination of the response of the frameunder the applied loads requires the values of not only areasof members groups but the other cross-sectional propertiessuch as moment of inertia and sectional modulus Thereforein order not to increase the number of design variables inthe programming problem it becomes necessary to relatethe other cross-sectional properties to sectional areas Thisis achieved by applying the least square approximation tosectional properties of practically available steel profilesThese relationships can be expressed in the following form

119868119909

= 11988611198601198871 119868

119910= 119886

21198601198872 119885 = 119886

31198601198873 (3)

where 119860 is the area 119868119909and 119868

119910are the moments of inertia

about 119909-119909 and 119910-119910 axis and 119885 is the sectional modulus ofthe cross-section 119886

1 119886

2 119886

3 1198871 1198872 1198873are constants The values

of these constants are obtained for the W sections of AISC[13] by making use of the least square approximation in [14]The values of these constants are given in Table 1 It should benoted that in their computation the unit of centimeter (1 cm= 10mm) was used

It is apparent from (2b) that restricted displacementsare required to be computed to find out whether theysatisfy the displacement limitations The joint displacementsin framed structures are related to external loads throughstiffness equations [119870]119863 = 119875 where [119870] is the overallstiffness matrix 119863 is the vector of joint displacementsand 119875 is the vector of joint loads The joint displacementscan be computed from the stiffness equations as 119863 =

[119870]minus1

119875 Although here the inverse of overall stiffnessmatrixis required to be calculated in order to obtain the jointdisplacements this is avoided due to its high computationalcost in the application of the optimization techniques used inthe solution of the problem Instead derivatives of the overallstiffnessmatrix with respect to design variables are calculateddepending on the optimization technique selected

22 Simultaneous Analysis and Design (SAND) In this typeof formulation joint displacements are also treated as designvariables in addition to cross-sectional properties Sincethe cross-sectional properties and joint displacements areimplicitly related through stiffness equations it becomesnecessary to include the stiffness equations as design con-straints in the mathematical model The general outlook of

the mathematical model of such formulation is given in thefollowing

Minimize 119882 = 120588

ng

sum

119894=1

ℓ119894119860119894 119894 = 1 ng (4a)

Subject to 119870 (119860119894) 119863 = 119875 (4b)

120575119895

le 120575119895119906

119895 = 1 rd (4c)

120590119896

(119860119894 120575

119895) le 120590all 119896 = 1 nm (4d)

119860119894ℓ

le 119860119894

le 119860119894119906

(4e)

where 120588 is the density of steel 119860119894represents the cross-

sectional area selected for group 119894 and ℓ119894is the total length

of the members in group 119894 in the frame ng is the totalnumber of groups in the steel frame119860

119894ℓand119860

119894119906are the lower

and upper bounds imposed on the cross-sectional area 119894Equality constraint (4b) represents stiffness equations 119863 isthe vector of joint displacements which are treated as designvariables 120575

119895is the restricted joint displacement and 120575

119895119906is its

bound rd is the total number of restricted displacements120590119896(119860

119894 120575

119895) represents stress constraint which is a function

of area and end joint displacements of member 119896 120590all isthe allowable stress for steel nm is the total number of themembers in the structure

This way of formulation of the design problem is anintegrated formulation which combines design and analysisphases in the same step that eliminates the need of separatestructural analysis in each design cycle Furthermore in thisformulation the displacement constraints becomes very sim-ple Because they are treated as design variables they turn outto be just upper bound constraints However the number ofdesign variables and constraints become very large comparedto the previous type of formulation which necessitates use ofpowerful optimization techniques that are efficient for large-scale problems It should be noticed that further variabletransformation is necessary for joint displacement variablesin this type of formulation if mathematical programmingtechniques are used for the solution of the design problemdue to the fact that while joint displacements can be positiveor negative mathematical programming techniques operatewith only positive variables This brings further mathemat-ical burden to a designer Furthermore the difference inmagnitude between the cross-sectional properties and jointdisplacements causes numerical difficulties in the solution ofthe design problem which makes it necessary to normalizethe design variables

Saka [15 16] carried out reviews of the mathematicalmodeling and solution algorithms presented in recent yearsfor the optimum design of skeletal structuresThe review firstintroduces the general formulation of the general structuraloptimization problem based on linear elastic behavior with-out referencing to a particular design code Later implemen-tation of the design requirements defined in design codes isalso covered The techniques developed for the solution ofoptimum design problems are reviewed in a historical order


Arora and Wang [17] also presented the review of math-ematical modeling for structural and mechanical systemoptimization In their review it is stated that the formulationsare classified into three broad categories The first type offormulation is known as the conventional formulation whereonly structural design variables are treated as optimizationvariables In the second type of formulation state variablessuch as node displacements andor element stresses aretreated as design variables in addition to the cross-sectionalproperties This type of formulation is called as simultaneousanalysis and design (SAND)The third type of formulation isknown as a displacement-based two-phase approach wheredisplacements are treated as unknowns in the outer loop andthe cross-sectional properties are treated as the unknownsin the inner loop The review also covers more general for-mulations that are applicable to economics optimal controlmultidisciplinary problems and other engineering fieldsAltogether 254 references are included in the review

Optimum design problem of steel frames turns out tobe a nonlinear programming problem if allowable stressdesign is adopted whichever formulation method explainedpreviously is used The numerical optimization techniquesavailable in the literature for obtaining the solution ofnonlinear programming problems can be broadly classifiedinto two groups such as deterministic and stochastic algo-rithms Deterministic optimization techniques make use ofderivatives of the objective function and constraints in thesearch of the optimum solution and involve no randomnessin the development of solution techniques The other groupsof techniques that are also called metaheuristic optimizationtechniques do not require the derivatives of the objectivefunction and the constraints They rely on random searchparadigms based on simulation of natural phenomenaThesemethods are nontraditional search and optimization meth-ods and they are very suitable and efficient in finding thesolution of combinatorial optimization problems

3 Deterministic SolutionTechniques for ElasticSteel Frame Design Optimization Problems

Historically mathematical programming techniques werefirst to be used to obtain the solution of optimum designproblem [1ndash12] The developments took place in computersand in the methods of mathematical programming and theyhave initiated a new era in designing engineering structuresThe need to design aircraft components to have minimumweight and later the requirements of space programs hasprovided the necessary funds and motivation to engineers todevelop effective design tools As a result of a large number ofresearch works numerous numerical techniques were devel-oped for the solution of nonlinear programming problems[1ndash12] However when these techniques were applied to thedesign of real-size practical steel frames numerical difficul-ties were encountered It is found out that mathematicalprogramming-based structural optimization methods were

only capable of designing steel frames with few dozen ofdesign variables

Optimality criteria approach was introduced in late 1960with the purpose of overcoming the previously mentioneddiscrepancies of mathematical programming techniques Itwas shown that the optimality criteria method was notdependent on the size of design problem and obtained thenear-optimum solution with few structural analyses It wasthis feature that made the optimality criteria techniquesattractive over the mathematical programming methodsOptimality criteria approaches were widely utilized in thedesign of large-size practical structures

One of the common features of the both methods is thatthey consider continuous design variables In reality variablesin most of the steel frame design problems are to be selectedfrom a set of discrete values that are available in practiceThis fact made it necessary to extend the mathematicalprogramming techniques to be capable of handling discreteprogramming problems This necessity has brought furtherdifficulties and the capability of the methods developed tosolve such programming problems was found to be limitedArora [18] recently presented comprehensive review of themethods available for discrete variable structural optimiza-tion In spite of the fact that the number of algorithmshas been developed for obtaining the solution of discretevariables steel frame design optimization problems they havenot found widespread application owing to their complexityand inefficiency in dealing with large size structures

31 Mathematical Programming Based Steel Frame DesignOptimization Techniques Mathematical programming tech-niques start the search for the optimum solution at a prese-lected initial point and compute the gradients of the objectivefunction and constraints at this point They take a step in thenegative direction of the gradient of the objective functionin the case of minimization problems to determine the nextpoint They continue this process of finding a new pointuntil there is no significant change in the values of designvariables within two consecutive iterations There are severalmathematical programming techniques available in the liter-ature In this paper only those that are used in developingstructural optimization algorithms for the optimum designof steel frames are consideredThese can be collected in threebroad groups Each group uses different concept In the firstgroup of algorithms approximation is carried out through theuse of linearization concept The second group converts theconstrained problem into unconstrained one and the thirdgroup handles the mathematical programming problem as itis without transforming it into another form

311 Sequential Linear Programs The first group of algo-rithms employs sequential linear programming methoddeveloped by Griffith and Stewart [19] This method makesuse of Taylor expansions to transfer the nonlinear pro-gramming problem into a linear programming one This isachieved by taking first two terms after applying Taylorrsquosexpansion to nonlinear functions in the programming prob-lem The design problems (2a) (2b) (2c) (2d) (4a) (4b)


(4c) (4d) and (4e) can be rewritten in a general form as inthe following

Minimize 119882 =

119898

sum

119896=1

119908119896

(119909119894) 119894 = 1 119899 (5a)


(119909119894) = 0 119895 = 1 ne (5b)

119892119895

(119909119894) le 0 119895 = ne + 1 nc (5c)

where 119909119894represents the design variable 119894 regardless of the type

of the mathematical formulation followed and 119899 is the totalnumber of design variables the problem ℎ

119895(119909

119894) represents

nonlinear equality constraints and ne is the total numberof such constraints 119892

119895(119909

119894) represents a nonlinear inequality

constraint 119895 nc is the total number of constraints in the designproblem The objective function and constraints functionsneed not be convex It is assumed that the region defined by(5a) (5b) and (5c) is bounded This nonlinear programmingproblem is locally approximated into a linear programmingproblem by taking two terms of Taylor expansion of everynonlinear function in (5a) (5b) and (5c) about the currentdesign point119909

119896 To assure that the approximation is adequatethe region of permissible solutions must be kept smallThis isaccomplished by applyingmove limits to the design variables

Assuming that all the constraints are nonlinear theprogramming problem (5a) (5b) and (5c) can be linearizedabout 119909

119896 which is the vector of design variables 119909119896

=

119909119896

1119909119896

2sdot sdot sdot 119909

119896

119899 as

Min 119882 = 119908 (119909119896) + nabla119908 (119909

119896) (119909

119896+1minus 119909

119896)

subject to ℎ119895

(119909119896) + nablaℎ

119895(119909

119896) (119909

119896+1minus 119909

119896) = 0

119895 = 1 ne

119892119895

(119909119896) + nabla119892

119895(119909

119896) (119909

119896+1minus 119909

119896) le 0

119895 = 119899 + 1 nc

(1 minus 119898) 119909119896

le 119909119896minus1

le (1 + 119898) 119909119896

(6)

where the value of every function is known at the design point119909119896 and nabla119908(119909

119896) nablaℎ

119895(119909

119896) and nabla119892

119895(119909

119896) are (1 times 119899) gradient

vectors of the functions at 119909119896 The last constraint in (6)

defines the region of permissible solutions that is constructedby using some preselected percentage of the current designpoint 119898 is known as move limit which is 0 le 119898 le 1The gradient vector nabla119908(119909

119896) consists of the first derivatives of

function 119908(119909119896)

nabla119908 (119909119896) = [

120597119908

120597119909119896

1

120597119908

120597119909119896

2

120597119908

120597119909119896

119899

] (7)

The linear programming problem (6) can be arranged in thefollowing form

Min 119882 = 119862119909119896+1

minus 1198620

subject to 119860119909119896+1

(=le

) 119861

(8)

where 119862119900is a constant 119862 is a vector of order (1 times 119899)

coefficient matrix 119860 is [(nc+ 2119899) times 119899] and the right-hand sidevector is of order of [(nc + 2119899) times 1] They have the followingform

119862119900

= 119908 (119909119896) minus nabla119908 (119909

119896) 119909

119896

C = nabla119908 (119909119896)

[119860] =[[[

[

nablaℎ119895

(119909119896)

nabla119892119895

(119909119896)

II

]]]

]

[119861] =

[[[[[

[

nablaℎ119895

(119909119896) 119909

119896minus ℎ

119895(119909

119896)

nabla119892119895

(119909119896) 119909

119896minus 119892

119895(119909

119896)

(1 + 119898) 119909119896

(1 + 119898) 119909119896

]]]]]

]

(9)

which are constructed at current design point 119909119896

The problem (8) is the linear approximation of the origi-nal nonlinear programming problem of (5a) (5b) and (5c)The simplex method can now be employed for the solutionof this problem to obtain 119909

119896+1 The previous design vector119909119896 is replaced by 119909

119896+1 and the linearization is carried outabout this new point This process is repeated with graduallydecreasing move limits until no significant improvementoccurs in the value of the objective function It was foundreasonable to start with 119898 = 1 and reduce to 01 each cycleuntil it reaches 01 [3 5 20] Reinschmidt et al [21] Pope[22] and Johnson and Brotton [23] are some of the studieswhich were based on this linearization technique In allthese works the cross-sectional properties of members werethe only design variables Saka [3 5] used the linearizationtechnique and treated joint displacements as design variablesin addition to cross-sectional areas Such formulation has alsobeen used by Arora and Haug [24]

Sequential linear-programming-based design algorithmsare attractive due to the availability of reliable linear program-ming packages to most of the computer users Initial designpoint required to be selected by user prior to employingthe algorithm can be feasible or infeasible However themethod has a number of disadvantages Firstly it requiresseveral optimization cycles to reach the optimum solutionwhich makes it computationally costly Secondly choice ofmove limits is important because it affects the total numberof iterations to find the optimum solution Unfortunatelythe selection of these limits is problem dependent Thirdlywhen the starting point is infeasible linearized problem maynot have a feasible solution which means that intermediatesolutions are practically useless Furthermore the conver-gence difficulties arise in the design of steel frames with largenumbers of design variables In such problems linearizationerrors become large and the linearized problemmay not havea feasible solution In both cases it becomes necessary to relaxthe move limits and restart of the application of the method[7 10]

Sequential quadratic programming also uses the conceptof linearization and it is one of the most effective mathe-matical programming techniques for nonlinearly constrainedoptimization problems [12 25 26] The method consists ofapproximating the original nonlinearly constrained problem


with a quadratic subproblem and solving the subproblemsuccessively until convergence has been achieved on theoriginal problem [27] It is shown in [28] that sequentialquadratic programming is quite efficient in obtaining thesolution of large-scale structural optimization problems

Wang and Arora [29] have presented two alternativeformulations based on the concept of simultaneous analysisand design (SAND) for optimumdesign of framed structuresNodal displacements andmember forces are treated as designvariables in addition to member cross-sectional propertiesThis leads to having stiffness equations as equality constraintsin the mathematical model The objective function andother constraints are expressed as explicit functions of theoptimization variables A sequential quadratic programmingmethod is used to obtain the solution of the design problemIt is concluded that the SAND formulation works quite wellfor optimization of framed structures

Wang and Arora [30] studied the sparsity features ofsimultaneous analysis and design (SAND) type of mathe-matical modeling for large-scale structures It is stated thatalthough SAND formulation has a large number of designvariables gradients of the functions and Hessian of theLagrangian are quite sparsely populated This property ofthe formulation is exploited with optimization algorithmswith sparse matrix capability such as sequential quadraticprogramming (SQP) for solving large-scale problems It isconcluded that in terms of the wall-clock time or the CPUtimeiteration the SAND formulations are more efficientthan the conventional formulation due to the fact that thenumber of call-to-analysis routine was much smaller

312 Penalty Function Methods The second group of algo-rithms uses sequential unconstrained minimization tech-niques of nonlinear programming methods to obtain thesolution of the design problem These methods replacethe constrained nonlinear programming problem by a newuncontained one so that one of the unconstrained mini-mization algorithms can be utilized for its solution Uncon-strainedminimizationmethods are quite general and efficientcompared to constrainedminimization algorithmsThus thebasic idea is to place a penalty on constraint violation sothat the solution is forced to satisfy the constraint functionsMost of the work in this area is based on the developmentintroduced by Carroll [31] and Fiacco and McCormack [32]These methods are widely used in structural design and havesome practical advantages There are two types of penaltyfunction methods exterior and interior penalty functions

Exterior penalty function method transforms the con-strained programming problem (5a) (5b) and (5c) into anuncontained one by the following function

Min 120601 (119909 119903) = 119882 (119909) + 119903

nesum

119895=1

ℎ2

119895(119909)

+1

119903

ncsum

119895 = ne+1min [0 119892

119895 (119909)]2

(10)

for 119903 = 1 01 001 0001 119903119894

rarr 0 119903119894

gt 0 and so forthEach penalty is directly associated with the corresponding

constraint violation The inequality terms are treated differ-ently from the equality terms because the penalty applies onlyfor constrained violation The positive multiplier 119903 controlsthe magnitude of the penalty termsThe problem is solved bydecreasing 119903 in successive steps In exterior penalty functionmethods all intermediate solutions lie in the feasible regionThe solution may be started from an infeasible point whichprovides flexibility for the designer eliminating the needfor finding the initial feasible design point However thealgorithm cannot find a feasible solution before reaching theoptimumAll intermediate solutions are infeasible and do notprovide a usable design

Interior penalty function method is employed when onlyinequality constraint is present in the problem It has thefollowing form

120601 (119909 119903) = 119882 (119909) + 119903

ncsum

119895 = 1

1

119892119895 (119909)

(11)

or in a logarithmic form

120601 (119909 119903) = 119882 (119909) minus 119903

ncsum

119895=1

log119890

[119892119895 (119909)]

119903 = 1 01 001 997888rarr 0 119903119894

gt 0

(12)

The interior penalty function method requires feasible initialdesign point to start with This provides an advantage overexterior penalty method such that all intermediate solutionsare feasible hence provide usable designs Furthermore theconstraints become critical only near the end of the solutionprocessThis provides advantage to the designers in selectingthe near optimal solution which is a less critical designinstead of optimal design

The structural optimization algorithms based on thepenalty function methods are numerically reliable for prob-lems of moderate complexity They are general and suitablefor various optimum structural design formulations How-ever they have the drawback of requiring large numberof structural analysis which is computationally expensiveNumerical difficulties may also arise in the case of ill-conditioned minimization problems In the field of optimumstructural design thesemethods have been used byKavlie andMoe [33 34] Griswold andMoe [35] and De Silva and Grant[36]

313 Gradient Method The third group of techniquesapproach to the solution of a nonlinear programming prob-lem in a direct manner Instead of transforming the probleminto another form they deal with the constrained one as itis These methods approach to the optimum solution stepby step At each step solution moves from current values ofdesign variables to the next values along a suitable directionThis direction is determined by making use of the gradientvectors of the objective and constraint functions This is whythey are called gradient methods The gradient-projectionmethod proposed by Rosen [37] for linear constraints isbased on a projecting search into the subspace defined bythe active constraints This method has been modified by


Haug and Arora [38] for nonlinear constraints and used todevelop a structural optimization algorithm The method offeasible direction proposed by Zoutindijk [39] is employed byVanderplaats [40 41] in the popular CONMIN programThefeasible direction method starts with some design point 119909

∘ atthe boundary of the feasible region and then searches for thebest direction and step size tomovewithin the feasible regionto a new point 119909

1 such that the objective function values areimproved

1199091

= 119909∘

+ 120572119878∘ (13)

where 120572 is a step size and 119878∘ is the direction vector In

determining the direction vector two conditions must besatisfiedThe first is that the directionmust be feasibleThis isachieved in problems where the constraints at a point curveinward if 119878

119879nabla119892j lt o If a constraint is linear or outward

curved it may require 119878119879

Δ119892119895

ge o The second condition isthat the direction must be usable and the value of objectivefunction is improved This is satisfied if 119878

119879nabla119891 lt o

Despite not being very widely used in structural opti-mization there are other nonlinear programming techniquesthat are employed in solving the steel frame design problemAmong these geometric programming [42] is applied toobtain optimum design of a number of different structuralproblems [43] On the other hand dynamic programming[44] was used in the optimum design of continuous beamsand multistory frames by Palmer [45 46] Both methods areapplicable to problems with specialized form This is whythese techniques have not been used extensively in the fieldof steel frame design

It can be concluded that mathematical programmingtechniques are general It is possible to include displacementstress stiffness and instability constraints in the optimumdesign formulation of the framed structures Inclusion ofstability constraints results in highly nonlinear programmingproblem Among the large number of available methods[47 48] the techniques used in the structural optimizationcan be collected in three groups The literature survey showsthat the optimum design algorithms developed are good onlyin designing structures with few members Unfortunatelythere is no single general nonlinear programming algorithmthat can effectively be used in solving all types of structuraloptimization problems Those that perform well in small- ormoderate-size structures run into numerical difficulties inlarge-size structures Except few computer implementationof these algorithms mostly is cumbersome They requireseveral structural analyses to be carried out in each designcycle to update the response of the structure This makesthem computationally expensive Overall it is reasonable tostate that among the existing mathematical programmingmethods the ones that make use of approximation tech-niques such as sequential quadratic programming result inan efficient structural optimization algorithms even for large-scale optimum design problems if the design variables arecontinuously not discrete

32 Optimality Criteria Methods Based on Steel Frame DesignOptimization Techniques Optimality criteria methods differ

from the mathematical programming methods in the waythey solve the optimumdesign problemWhilemathematicalprogramming techniques attempt to minimize the objectivefunction directly considering the constraint functions theoptimality criteria methods derive a criterion based onintuition such as fully stressed design or based on a math-ematical statement such as Kuhn-Tucker conditions Theythen establish an iterative procedure to achieve this criterionIt is expected that when this is accomplished the optimumsolution will be obtained Optimality criteria methods arenot as general as the mathematical programming techniquesHowever they are computationally more efficient Their per-formance does not depend on the size of the design problemand they are easier to implement for computers Number ofstructural analysis required to reach the final design is muchless than mathematical programming methods However incertain cases theymaynot converge to the optimumsolution

Optimality criteria methods were originated by Pragerand his coworkers [49] for continuous systems using uni-form energy distribution criteria Venkayya et al [50ndash52]Berke [53 54] and Khot et al [55ndash57] have later appliedthis idea to discrete systems Ragsdell has reviewed thesetechniques in [58] In these methods a prior criterion isderived using Kuhn-Tucker conditions This criterion thatis required to be satisfied in the optimum solution providesa basis in producing a recursive relationship for designvariables This relationship is used to resize the structuralelements during the optimum design cycles The designprocess is terminated when convergence is attained in thevalue of objective function The optimality criteria methodis applied to optimum design of pin-jointed structures byFeury and Geradin [59] Fleury [60] and Saka [61] It isapplied to optimum design of steel framed structures byTabak and Wright [62] Cheng and Truman [63] Khan[64] and Sadek [65] Khot et al [55] have carried out thecomparison of different optimality criteria methods In allthese algorithms displacement stress and minimum-sizeconstraints on the design variables were all considered Itis shown in these works that the design methods basedon the optimality criteria approach are straightforward andeasy to program Their behavior in obtaining the optimumsolution does not change depending on the number ofdesign variables Number of structural analysis required toreach to optimum design is relatively small compared tomathematical programming based techniques which makesoptimality criteria based structural optimization algorithmsefficient and attractive for practicing engineers

However none of the previously mentioned methodshave considered the code specifications in the formulationof the design problem Saka [66] has presented optimalitycriteria based minimum weight design algorithm for pinjointed steel structures where the design requirements werethe same as the ones specified in AISC [13] In this work analgorithm is developed for the optimumdesign of pin-jointedstructures under the number of multiple loading cases whileconsidering displacements stress buckling and minimum-size constraints In contrast to the previous work fixed-value allowable compressive stresses were not utilized for thecompression members instead they are left to the algorithm


to be decided The values of critical stresses are computed inevery design step by making use of the current values of thedesign variables In order to achieve this the critical stressesare first expressed in terms of design variables as nonlinearlinear equations by making use of the relationships given indesign codesThese equations are then solved by theNewton-Raphson method to obtain the value of the design variablethat satisfies the buckling constraint

321 Linear Elastic Steel Frames The optimum design prob-lem of a rigidly jointed plane frame formulated according toASD-AISC (Allowable Stress Design American Institute ofSteel Construction) [13] can be expressed as follows

Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894

Subject to 120575119895ℓ

le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982

119891bxnl119865bxnl

le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)

where 119860119896is the area of members belonging to group 119896

119898119896 is the total number of members in group 119896 120588119894and ℓ

119894

are the density of the material and the length of membersin group 119894 ng is the total number of member groups inthe structure 120575

119895ℓis the displacement of joint 119895 under the

load case ℓ and 120575119895ℓ119906

is its upper bound 119901 is the totalnumber of restricted displacements nℓc is the total numberof load cases that the frame is subjected to 119891anl and 119891bxnlare the computed axial stress and compressive bending stressin member 119899 under load case ℓ 119865anl is the allowable axialcompressive stress considering that the member is loadedby axial compression only 119865bxnl is the allowable compressivebending stress considering that the member 119894 is loaded inbending only 119898

1is a constant and 119898

2is an expression given

in [13] nm is the total number of members in the frame 119860119896ℓ

is the lower bound on the design variable 119860119896 This bound is

required to prevent member areas being reduced to zero inthe optimum solution

In the optimum design problem (14) only the cross-sectional areas of different groups in the frame are treatedas design variables It is apparent that determination of theresponse of the frame under the applied loads necessitates thevalues of the other cross-sectional properties such asmomentof inertia and sectional modulus Therefore it becomesnecessary to use the relationship (3) to relate the other cross-sectional properties to sectional areas in order not to increasethe number of design variables in the programming problem

The solution of the design problem (14) is obtained inan iterative manner The area variables are changed duringiterations with the aim of obtaining a better design It isapparent that the new values of design variables are decidedby the most severe design constraint in every design cycleThis necessitates obtaining an expression for updating these

variables depending upon whether the displacement stressor the buckling constraints are dominant in the designproblem If neither of these constraints is dominant then thearea variable is decided by the minimum-size limitation

Dominance of Displacement Constraints In the case wherethe displacement constraints are dominant in the designproblem the new values of the design variables are decidedby these constraints If the design problem (14) is rewritten byonly considering the displacement constraints the followingmathematical model is obtained

Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894

subject to 119892119894ℓ

(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)

Employing language multipliers this constrained problemcan be transformed into unconstrained form as

120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)

where 120582119895ℓis the Lagrangian parameter for the 119895th constraint

under the ℓth load case The necessary conditions for thelocal constrained optimum are obtained by differentiating(16) with respect to the design variable 119860

119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)

By using the virtual workmethod 120575119895119897 the 119895th displacement of

the frame under the load case ℓ can be expressed as

120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)

where 119883119895is the virtual displacement vector due to virtual

loading corresponding to the 119895th constraint This loadingcase is setup by applying a unit load in the direction of therestricted displacement 119895 119870 is the overall stiffness matrixof the frame and 119883

ℓis the displacement vector due to the

applied load case ℓ Equation (18) can be decomposed as sumof the contributions of each member in the frame as

120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)

where 119891119894119895119897is known as the flexibility coefficient

119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)

where 119870119894is the contribution of member 119894 to the overall

stiffness matrix Substituting (20) into the derivative of thedisplacement constraint leads to the following

120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)


which only involves with the members in group 119896 Hence (17)becomes

119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)

Rearranging (22) considering constant 120588 throughout theframe and multiplying both sides by 119860

119903

119896and then taking the

119903th root leads to

119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)

where 120584 is the iteration number This is known as theoptimality criterion It can be used to obtain the new valuesof design variables in each iteration provided that Lagrangeparameters are known There are several suggestions for thecomputation of Lagrange parameters They are summarizedin [55] Among them the one suggested byKhot [56] is simpleand easy to apply

120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)

where 120584 is the current and 120584+1 is the next iteration number 119888

is a preselected constant known as a step sizeThe value of 05provides steady convergence It is apparent that in order to use(24) it is necessary to select some initial values for Lagrangemultipliers

Dominance of Strength Constraints In the case where thestrength constraints are dominant in the design problem itbecomes necessary to derive an expression from which thenew values of area variables can be computed In this studythis expression is based on ASD-AISC [13] inequalities whichare repeated below for only considering in-plane bending

119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)

where 119909 with subscripts 119887 119898 and 119890 is the axis of bendingwhich a particular stress or design property applies 119865

119910is

the yield stress and 119891119886and 119891

119887are the computed axial stress

and bending stress at the point under consideration 119865119886is

the allowable axial compressive stress considering that themember is only axially loaded119865

119887is the allowable compressive

bending stress considering that the member is only underbending loading 119862

119898is a factor and is given as 085 for

compression members in frames subject to joint translation1198651015840

119890119909is Euler stress divided by a factor

1198651015840

119890119909=

121205872119864

231198782 (27)

where 119878 = 120578119897119887119903119887is the slenderness ratio of member 119897

119887is

the actual unbraced length in the plane of bending 119903119887is the

corresponding radius of gyration 120578 is the effective lengthfactor in plane of bending and 119864 is the modulus of elasticity

The inequalities (25) and (26) are required to be satisfiedfor those framemembers subjected to both axial compressionand bending the others that are subjected to axial tensionand bending require satisfying the inequality (26) In theseinequalities the allowable stresses 119865

119886 119865

119887 and 119865

119890are related

to the instability of the member They are function of variouscross-sectional properties that are to be expressed in terms ofarea variables

Axial Stability Constraints The allowable critical stress 119865119886for

a compression member 119894 under load case ℓ is given in ASD-AISC as the following

If 119878119894

ge 119862 elastic buckling is

119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894

le 119862 plastic buckling is

119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)

where 119878119894is the slenderness ratio of the member 119894 and 119862

is given as radic(21205872119864119865119910

) It is apparent from (28) and (29)that critical stress of a compression member is function ofits slenderness ratio which in turn is related to the radiusof gyration of the cross section of a member The cross-sectional areas change from one design step to another as wellas from one member to another during the design processThis results in having varying critical stress limits in thedesign optimization problemof the steel frameConsequentlyit becomes necessary to express the allowable critical stressof a compression member in terms of area variables sothat its value can be calculated whenever the cross-sectionalarea of a member changes This is achieved by employingthe approximate relationship given in (3) Computation ofallowable critical stress of a compression member differsdepending on its slenderness ratio as given in (28) and (29) Itis foundmore suitable here to identify whether a compressionmember buckles in elastic region or in plastic region throughthe value of the critical load119875crThis load is obtained by usingthe fact that for the value of 119878

119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)

where 119860cr is the critical area value for member 119894 Since acompression member can buckle about its 119909-119909 or 119910-119910 axisdepending onwhichever slenderness ratio is critical it is thennecessary to express both radii of gyration in terms of areavariables by using the relationship

119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871

minus 1) and 1198882

= 05(1198872

minus 1) It follows from119904119909119894

= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897

119910119894are the effective

length of member 119894 about 119909-119909 and 119910-119910 axis respectively Thecritical area value is then easily obtained from

119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)

where the subscript 119896 of 119897119896is either 119909 or 119910 and subscript 119896 of

119886119896and 119888

119896is either 1 or 2 whichever applies

After the computation of 119875cr from (37) the check for theelastic or plastic buckling can proceed as the following

If 119875119886

le 119875cr elastic buckling is

119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886

ge 119875cr plastic buckling is

119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)

where119875119886is the compression force inmember 119894 under the load

case ℓFinally the previously mentioned allowable stresses can

be related to area variables by substituting 119878119894

= 119897119896(119886

05

119896119860119888119896)

which yields to the following expressionsIf 119875

119886le 119875cr elastic buckling is

119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)

where the constants are

1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)

In the previously mentioned expressions

if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)

Consequently the allowable axial compressive stress 119865119886of

inequality (25) is replaced by either (35) or (36) whicheverapplies

Lateral Stability ConstraintsThe allowable compressive stress119865119887of (25) and (26) is given by the following formulae in ASD-

AISC

when radic70830119862

119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)

where the units of the yield stress 119865119910and the allowable

compressive bending stress 119865119887are kNcm2

In (40) and (41) the value of 119865119887cannot be more than 06

119865119910and 119903

119905are the radii of gyration of I section comprising the

flange plus one-third of the compressionweb area taken aboutminor axis 119897

119890is the lateral effective length of the beam The

coefficient 119862119887is given a

119862119887

= 175 + 15120573 + 031205732

le 23 (42)

in which 120573 is the ratio of the small moment 1198721to the larger

moment 1198722acting at the ends of the member 120573 has positive

sign if the member is in single curvature has negative signotherwise Since the end moments of the frame membersare available at every design step from the analysis of framewhich is carried out at every design step 119862

119887becomes a

constant which makes 119865119887only a function of 119903

119905 In I shaped

sections 119903119905may be approximated as 12119903

119910as given in [67] 119903

119905

can be expresses in terms of area variable by employing therelationship (3) as

119903119905

= 12119903119910

= 1211988605

21198601198882 (43)

Substitution of (43) into (44) yields

119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)

And substituting in (41) gives

119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)


Depending on the lateral slenderness ratio of the beam either(44) or (46) is substituted in (25) and (26)

Combined Stress ConstraintsTheEuler stress given in (27) canalso be expressed in terms of area variables by substituting119878119894

= 119897119909 (119886

05

11198601198881) which yields to the following expression

1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)

The axial and bending stresses in the member due to theapplied loads are

119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872

119909are the axial force and the maximum

bending moment in the memberSubstituting (48) and (50) into (25) with 119862

119898= 085 and

carrying out simplifications the combined stress constraintcan be reduced to the following nonlinear equation

12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881

minus 1 and the constants are

1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)

It is apparent from (35) (36) (44) and (46) that 119865119886and 119865

119887

are also nonlinear expressions of area variables Substitutionof these into (51) increases the nonlinearity of the equationeven further Similarly substitution of (50) into the combinedstress constraint of (26) reduces this equation into a nonlinearequation of area variable

11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)

The equations (51) and (53) are solved using Newton-Raphson method to obtain the value of area variables thatsatisfy the combined stress constraints The solution of theseequations does not introduce any difficulty although theyare highly nonlinear The derivatives with respect to areavariables that are required by Newton-Raphson method areevaluated analytically since119865

119886and119865

119887are already expressed in

terms of area variables Numerical experiments have shownthat it takes not more than 5 to 6 iterations to obtain the rootsof these nonlinear equations The larger value obtained fromthese equations is adopted as the member area for the next

design step for the case where the combined stress constraintsare dominant in the design problem

Optimum Design Algorithm The steps of the design proce-dure described previously are given in the following

(1) Select the initial values of area variables and Lagrangemultipliers

(2) Analyze the frame with these values of area variablesand obtain the joint displacements as well as memberend forces under the external loads and unit loadsapplied in the direction of the restricted displace-ments

(3) Compute the Lagrange multipliers from (24)(4) Take each area group in the frame respectively and

compute the new value of the area variable which isin the cycle from (23)

(5) Compute the lower bound on the same area variablefor the combined stress constraints from (51) and (53)Select the largest

(6) Compare the three area values obtained from dom-inant displacement constraints the combined stressconstraints and the minimum size restrictions Takethe largest among three as the new value of the areavariable for the next design cycle

(7) Carry out the steps 4 5 and 6 for all the area groupsin the frame

(8) Check the convergence on the weight of the frame Ifit is satisfied go to step 9 else go to step 2

(9) Checkwhether the displacement and combined stressconstraints are satisfied If not then go to step 2 elsestop

The initial design point required to initiate the designprocedure can be selected in any way desired It can befeasible or even be infeasible The area variables in the initialdesign point can be selected as all are equal to each other forsimplicity or can be decided using engineering judgmentThenumerical experimentation carried out has shown that thedesign algorithm works equally well with all these differentinitial points

SODA developed by Grieson and Cameron [68] was thefirst commercial structural optimization software for prac-tical buildings design This software considered the designrequirements from Canadian Code of Standard Practicefor Structural Steel Design (CANCSA-S16-01 Limit StatesDesign of Steel Structures) and obtained optimum steelsections for the members of a steel frame from an availableset of steel sections However the software was limited torelatively small skeleton framework

Optimality criteria based structural optimizationmethodis presented in [69] for the optimum designs of multi-story reinforced concrete structures with shear walls Thealgorithm considers displacement ultimate axial load andbending moment and minimum size constraints Memberdepths are treated as design variables The values of design


variables are evaluated from recursive relationships in everydesign cycle depending on the dominance of design con-straints The largest value of the design variable amongthose computed from the displacement limitations ultimateaxial and bending moment constraints and minimum sizerestrictions defines the new value of the design variable inthe next optimum design cycle This process is repeated untilconvergence is obtained on the objection function

It is shown in [70] that optimality criteria approach canalso be utilized in the optimum geometry design of rooftrusses In this work the slope of upper chord of a trussis treated as a design variable in addition in to area vari-ables Additional optimality criteria were developed for slopevariables under the displacement constraints the algorithmdetermines the optimum values of the cross-sectional areasin the roof truss as well as optimum height of the apex

In another application [71] the optimality criteria basedstructural optimization algorithm is developed for the opti-mum design of steel frames with tapered membersThe algo-rithm considers the width of an I section as constant togetherwith web and flange thickness while the depth is assumedto be varying linearly between joints The depth at each jointin the frame where the lateral restraints are assumed to beprovided is treated as a design variableTheobjective functiontaken as the weight of the frame is expressed in terms ofthe depth at each joint The displacement and combinedaxial and flexural strength constraints are considered in theformulation of the design problem in accordance with Loadand Resistance Factor Design (LRFD) of AISC code [72]The strength constraints that take into account the lateraltorsional buckling resistance of the members between theadjacent lateral restraints are expressed as a nonlinear func-tion of the depth variables The optimality criteria methodis then used to obtain a recursive relationship for the depthvariables under the displacement and strength constraintsThe algorithm basically consists of two steps In the first onethe frame is analyzed under the external and unit loadingsfor the current values of the design variables In the secondthis response is utilized together with the values of Lagrangemultipliers to compute the new values of the depth variablesThis process is continued until convergence obtained inthe objective function The optimum design of number ofpractical pitched roof frames is presented to demonstrate theapplication of the algorithm

Al-Mosawi and Saka [73] employed optimality criteriaapproach to develop an algorithm for the optimum designof single-core shear walls subjected to combined loading ofaxial force biaxial bending moment and torsional momentThe algorithm is based on the limit state theory and considersdisplacement limitations in addition to strain constraintsin concrete and yielding constraints in rebars It takes intoaccount the effect ofwarpingwhich can be considerable in thecomputation of normal stresses when the thin-walled sectionis subjected to a torsional moment The design algorithmmakes use of sectional properties of section which is quiteuseful in describing deformations and stresses when theplane cross-section no longer remains plane A numericalprocedure is developed to calculate the shear center ofreinforced concrete thin-walled section in addition to the

sectional and sectional properties Furthermore an iterativeprocedure is presented for finding the location of the neutralaxis in reinforced concrete thin-walled sections subjected toaxial force biaxial moments and torsional moment It isshown that optimality criteria method can effectively be usedin obtaining the optimum solution of highly nonlinear andcomplex design problem

Al-Mosawi and Saka [74] presented an algorithm thatobtains the optimum cross-sectional dimensions of cold-formed thin-walled steel beams subjected to axial forcebiaxial moments and torsional loadings The algorithmtreats the cross-sectional dimensions such as width depthand wall thickness as design variables and considers thedisplacement as well as stress limitations The presence oftorsional moments causes wrapping of thin-walled sectionsThe effect of warping in the calculations of normal stressesis included using Vlasov [75] theorems The design problemturns out to be highly nonlinear problem It is shown that theoptimality criteria method can effectively be used to obtainthe solution

Chan and Grierson [76] and Chan [77] presented apractical optimization technique based on the optimalitycriteria approach for the design of tall steel building frame-works where cross-sectional areas are selected from thestandard steel section tables This computer based methodconsiders the multiple interstory drift strength and sizingconstraints in accordancewith building code and fabricationsrequirements The optimality criteria approach and sectionpropertiesrsquo regression relationship are used to solve the designproblem To achieve a final optimum design using standardsteels section a pseudo-discrete optimality criteria techniqueis applied to assign standard steel sections to the members ofthe structure while maintaining the least change in structureweight A full-scale 50 story three-dimensional steel frameis designed to demonstrate the application of the automaticoptimal design method

Soegiarso and Adeli [78] presented an algorithm for theminimum weight design of steel moment resisting spaceframe structures with or without bracing based on LRFDspecifications of AISC The algorithm is based on the opti-mality criteria method The LRFD constraints for momentresisting frames are highly nonlinear and implicit functionsof design variablesThe structure is subjected to wind loadingaccording to the Uniform Building Code in addition to thedead and live loads The algorithm is applied to the optimumdesign of four large high-rise steel building structures rangingup in height from 20 to 81 stories

322 Nonlinear Elastic Steel Frames Optimality criteria ap-proach has been effectively employed in the optimum designof nonlinear skeleton structures Khot [79] was one of theearly researchers who presented an optimization algorithmbased on an optimality criteria approach to design a min-imum weight space truss with different constraint require-ments on system stability In order to reduce the imperfectionsensitivity of a structure the Eigen values associated with thesystem buckling modes are separated by a specified intervalThis requirement is included as a constraint in the design


problem This design obtained for the various constraintconditions was analyzed with and without specified geomet-ric imperfections using nonlinear finite element programthat accounts geometric nonlinearity The results obtainedfor various designs were compared for their imperfectionsensitivity Later Khot and Kamat [80] presented optimalitycriteria based algorithm for the minimum weight designof geometrically nonlinear pin-jointed space trusses Thenonlinear critical load is determined by finding the loadlevel at which the Hessian of the potential energy becomesnegative A recurrence relationship is based on the criteriathat at the optimum structure the nonlinear stain energydensity must be equal in all members This relationship isused to resize the space truss members The number ofexamples is considered to demonstrate the application of thealgorithm

Saka [81] also presented an optimum design algorithmthat takes into account the response of space trusses beyondthe elastic behavior The algorithm is developed by couplinga nonlinear analysis technique with an optimality criteriaapproach The nonlinear analysis method used determinesthe changes in the axial stiffness of space truss membersfrom the nonlinear load-deformation curves These curvesare obtained from the tensile stress-strain curve for thetension members and from the stress-strain curve undercompression that varies as the slenderness ratio of themember changes These nonlinear curves are approximatedby straight lines The intersection points of these lines arecalled as critical points As the load increases the slope oflinear segments changes This implies the variation in theaxial stiffness of the member Once the axial stiffness valuesof all members are specified up to the failure the nonlinearanalysis is easily carried out by allowing these changes inthe stiffness of members during the increase of the externalloads The optimality criteria approach is employed to obtaina recurrence relationship for area variables Considerationof postbuckling and postyielding behavior of truss membersmakes the stress and buckling constraints redundant Thenumber of space trusses is designed by the algorithm and itis shown that optimum designs are obtained after relativelyfewer members of iterations

Saka and Ulker [82] developed a structural optimizationalgorithm for geometrically nonlinear space trusses subjectedto displacement and stress and size constraints Tangentstiffness method is used to obtain the nonlinear responseof the space truss This response is used by the optimalitycriteria method to determine the values of area variables inthe next design cycle During the nonlinear analysis tensionmembers are loaded up to yield stress and compressionmembers are stressed until their critical limits The overallloss of elastic stability is checked throughout the steps of thealgorithm It is shown that the consideration of geometricnonlinearity in the optimum design of space trusses makesit possible to achieve further reduction in its overall weightIt is shown that inclusion of the geometric nonlinearitycaused 11 reduction in the overall weight in the optimumdesign of 120-bar laminar dome compared to minimumweight design considering linear behavior Furthermore by

considering the geometrical nonlinearity in the optimumdesign the algorithm takes into account the realistic behaviorof space trusses Geometric nonlinearity becomes importantin shallow-framed domes

Saka and Hayalioglu [83] present a structural optimiza-tion algorithm for geometrically nonlinear elastic-plasticframes The algorithm is developed by coupling the optimal-ity criteria approach with large deformation analysis methodfor elastic-plastic frames The optimality criteria method isused to obtain a recursive relationship for the design variablesconsidering the displacement constraints The nonlinearresponse of the frame used by the recursive relationship isbased on a Euclidian formulation which includes elastic-plastic effects Localmember force-deformation relationshipsare extended to cover the geometric nonlinearities Anincremental load approach with Newton-Raphson iterationsis adopted for the computational procedure These iterationsare terminated when the prescribed load factor reached It isshown in the optimum design of number of rigid frames con-sidered in the study that inclusion of geometric and materialnonlinearity in the optimum design does not only lead to amore realistic approach but also yields to lighter frames Thereduction in the overall weight of the frames varied from 20to 30 when compared to the linear-elastic optimum framedesigns Later the algorithm is extended to geometricallynonlinear elastic-plastic steel frames with tapered members[84] It is shown that consideration of nonlinear behaviorin the minimum weight design of pitched roof frame withtapered members yields almost 40 reduction in the weightcompared to the optimumdesignwith linear-elastic behaviorIt is stated in both works that the optimality criteria approachwas quite effective in handling the displacement constraintsin such complex design problems It was noticed that most ofthe computational time was consumed by the large deforma-tion analysis of elastic-plastic frame

Saka and Kameshki [85] employed optimality criteriaapproach to develop an algorithm for the optimum design ofunbraced rigid large frames that takes into account the non-linear response of P-120575 effect The algorithm considers swayconstraints and combined stress limitations in the designproblem A recursive relationship is developed for usingoptimality criteria approach for the sway limitations andthe combined strength constraints are reduced to nonlinearequations of the design variables The algorithm initiates thedesign process by carrying out nonlinear analyses of theframe in which in each analysis cycle the overall stabilityis checked When the nonlinear response of the frame isobtained without loss of stability the new values of designvariables are computed from the recursive relationshipsThis process of reanalyses and resizing is repeated until theconvergence is obtained in the objection function It wasnoticed that the nonlinear value of the top storey sway of 24-story 3-bay unbraced frame due to P-120575 effect was 10 morethan its linear value This clearly indicates the importance ofincluding the geometric nonlinearity in the optimum designof unbraced tall steel frames

This algorithm is later extended to the optimum designof three-dimensional rigid frames by Saka and Kameshki[86] The algorithm considers the displacement limitations


and restricts the combined stresses not to be more than yieldstress The stability functions for three-dimensional beam-columns are used to obtain the P-120575 effect in the frame Thesefunctions are derived by considering the effect of axial forceon flexural stiffness and effect of flexure on axial stiffnessThealgorithm employs the optimality criteria approach togetherwith nonlinear overall stiffness matrix to develop a recursiverelationship for design variables in the case of dominantdisplacement constraints The combined stress constraintsare reduced to nonlinear equations of design variables Thealgorithm initiates the optimum design at the selected loadfactor and carries out elastic instability analysis until theultimate load factor is reachedDuring these iterations checksof the overall stability of the space rigid frame are conductedIf the nonlinear response is obtained without loss of stabilitythe algorithm proceeds to the next design cycle It is shownin the design examples considered that P-120575 effect playsimportant role in the optimum design of framed domes andits consideration does not only provide more economy in theweight but also produces more realistic results

The research work reviewed previously clearly shows thatthe optimality criteria approach is quite effective in obtainingthe optimum design of skeleton structures The number ofdesign cycles required to reach to the optimum frame isrelatively low and it is independent of the size of frame Theoptimality criteria approach made it possible to obtain theoptimum design of large size realistic structures It is alsoshown that these methods are general and can be employedin the optimum design of linear elastic nonlinear elastic andelastic-plastic frames Furthermore it is shown that they caneven be used in the shape optimization of skeleton structuresThus it is apparent that in structural engineering problemswhere the design variables can have continuous values theoptimality criteria based structural optimization algorithmeffectively provides solutions However the optimum designof steel frames necessitates selection of steel profiles for itsmembers from available list of steel sections which containsdiscrete values not continuous Altering optimality criteriaalgorithms to cater this necessity is cumbersome and donot yield techniques that can be efficiently used in theoptimum design of large-size steel frames This discrepancyof mathematical programming and optimality criteria baseddesign optimization algorithms forced researchers to comeup with new ideas which caused the emergence of stochasticsearch techniques

4 Discrete Optimum Design Problem ofSteel Frames to LRFD-AISC

The design of steel frames requires the selection of steelsections for its columns and beams from a standard steelsection tables such that the frame satisfies the serviceabilityand strength requirements specified by LRFD-AISC (Loadand Resistance Factor Design-American Institute of SteelConstitution) [72] while the economy is observed in theoverall or material cost of the frame When formulated as a

programming problem it turns out to be a discrete optimumdesign problem which has the following mathematical form

Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)

where (55a) defines the weight of the frame 119898119903is the unit

weight of the steel section selected from the standard steelsections table that is to be adopted for group 119903 119905

119903is the total

number of members in group 119903 and ng is the total numberof groups in the frame 119897

119904is the length of the member 119904 which

belongs to the group 119903Inequality (55b) represents the interstorey drift of the

multistorey frame 120575119895and 120575

119895minus1are lateral deflections of two

adjacent storey levels and ℎ119895is the storey height ns is the

total number of storeys in the frame Equation (55c) definesthe displacement restrictions that may be required to includeother-than-drift constraints such as deflections in beamsnd is the total number of restricted displacements in theframe 120575

119895119906is the allowable lateral displacement LRFD-AISC

limits the horizontal deflection of columns due to unfactoredimposed load andwind loads to height of column400 in eachstorey of a buildingwithmore than one storey 120575

119894119906is the upper

bound on the deflection of beams which is given as span360if they carry plaster or other brittle finish

Inequalities (55d) and (55e) represent strength con-straints for doubly and singly symmetric steel memberssubjected to axial force and bending If the axial force inmember 119896 is tensile force the terms in these equationsare given as the following 119875

119906119896is the required axial tensile

strength 119875119899119896

is the nominal tensile strength 120601 becomes 120601119905

in the case of tension and called strength reduction factorwhich is given as 090 for yielding in the gross section and075 for fracture in the net section120601

119887is the strength reduction


factor for flexure given as 090 119872119906119909119896

and 119872119906119910119896

are therequired flexural strength 119872

119899119909119896and 119872

119899119910119896are the nominal

flexural strength about major and minor axis of member 119896respectively It should be pointed out that required flexuralbending moment should include second-order effects LRFDsuggests an approximate procedure for computation of sucheffects which is explained in C1 of LRFD In this case theaxial force in member 119896 is a compressive force and theterms in inequalities (55d) and (55e) are defined as thefollowing 119875

119906119896is the required compressive strength 119875

119899119896is

the nominal compressive strength and 120601 becomes 120601119888which

is the resistance factor for compression given as 085 Theremaining notations in inequalities (55d) and (55e) are thesame as the definition given previously

The nominal tensile strength of member 119896 for yielding inthe gross section is computed as 119875

119899119896= 119865

119910119860119892119896

where 119865119910is the

specified yield stress and 119860119892119896

is the gross area of member 119896The nominal compressive strength of member 119896 is computedas 119875

119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864

In these expressions 119864 is the modulus of elasticity and 119870

and 119897 are the effective length factor and the laterally unbracedlength of member 119896 respectively

Inequality (55f) represents the strength requirements forbeams in load and resistance factor design according toLRFD-F2 119872

119906119909119905and 119872

119899119909119905are the required and the nominal

moments about major axis in beam 119887 respectively 120601119887is the

resistance factor for flexure given as 090 119872119899119909119905

is equal to119872

119901 plastic moment strength of beam 119887 which is computed

as 119885119865119910where 119885 is the plastic modulus and 119865

119910is the specified

minimum yield stress for laterally supported beams withcompact sections The computation of 119872

119899119909119887for noncompact

and partially compact sections is given in Appendix F ofLRFD Inequality (55f) is required to be imposed for eachbeam in the frame to ensure that each beam has the adequatemoment capacity to resist the applied moment It is assumedthat slabs in the building provide sufficient lateral restraint forthe beams

Inequality (55g) is included in the design problem toensure that the flangewidth of the beam section at each beam-column connection of storey 119904 should be less than or equal tothe flange width of column section

Inequalities (55h) and (55i) are required to be included tomake sure that the depth and the mass per meter of columnsection at storey 119904 at each beam-column connection are lessthan or equal to width and mass of the column section at thelower storey 119904 minus 1 nu is the total number of these constraints

The solution of the optimum design problems describedpreviously requires the selection of appropriate steel sectionsfrom a standard list such that theweight of the frame becomesthe minimum while the constraints are satisfied This turnsthe design problem into a discrete programming problem Asmentioned earlier the solution techniques available amongthe mathematical programming methods for obtaining thesolution of such problems are somewhat cumbersome andnot very efficient for practical use Consequently structuraloptimization has not enjoyed the same popularity among thepracticing engineers as the one it has enjoyed among the

researchers On the other hand the emergence of new com-putational techniques called as stochastic search techniquesor metaheuristic optimization algorithms that are based onthe simulation of paradigms found in nature has changedthis situation altogether These techniques are inspired byanalogies with physics with biology or with ethology

5 Stochastic Solution Techniques forSteel Frame Design Optimization Problems

The stochastic search techniques make use of ideas inspiredfrom nature and they are equally efficient for obtainingthe solution of both continuous and discrete optimizationproblems The basic idea behind these techniques is tosimulate the natural phenomena such as survival of the fittestimmune system swarm intelligence and the cooling processofmoltenmetals through annealing to a numerical algorithm[87ndash107]These methods are nontraditional stochastic searchand optimization methods and they are very suitable andeffective in finding the solution of combinatorial optimiza-tion problems They do not require the gradient informationor the convexity of the objective function and constraints andthey use probabilistic transition rules not deterministic rulesFurthermore they donot even require an explicit relationshipbetween the objective function and the constraints Insteadthey are based stochastic search techniques that make themquite effective and versatile to counter the combinatorialexplosion of the possibilities An extensive and detailedreview of the stochastic search techniques employed indeveloping optimum design algorithms for steel frames isgiven in [16] This review covers the articles published inthe literature until 2007 In this paper firstly the summaryof stochastic search algorithms is given and the relevantpublications after 2007 are reviewed in detail

51 Evolutionary Algorithms Evolutionary algorithms arebased on the Darwinian theory of evolution and the survivalof the fittest They set up an artificial population that consistsof candidate solutions to optimum design problem whichare called individuals These individuals are obtained bycollecting the randomly selected values from the discrete setfor each design variable For example in a design problem ofa steel frame with three design variables the 11th 23119903119889 and41st steel sections in the standard section list that contains64 sections can randomly be selected for each design vari-able First these sequence numbers are expressed in binaryform as 001011 010111 and 101001 This is called encodingThere are different types of encodings such as binary real-number integer and literal permutation encodings Whenthese binary numbers are combined together an individualconsisting of zeros and ones such as 001011010111101001 isobtained This individual is inserted to the population as acandidate solutionThe reason 6 digits are used in the binaryrepresentation is to allow the possibility of covering total 64sections available in the standard section list The number ofindividuals can be generated sameway and collected togetherto set up an artificial population The binary representationof the individual is called its genome or chromosome Each


genome consists of a sequence of genes A value of a geneis called an allele or string It is apparent that all theseterms are taken from cellular biology It can be noticedthat a new individual can be obtained by just changing oneallele Evolutionary algorithms do exactly this They modifythe genomes in the population to obtain a new individualwhich are called offspring or a child Each individual isthen checked for its quality which indicates its fitness asa solution for the design problem under consideration Anew population which is called a new generation is thenobtained by keeping many of those offsprings with highfitness value and letting the ones with low fitness value todie off Populations are generated one after another untilone individual dominates either certain percentage of thepopulation or after a predetermined number of generationsThe individual with the highest fitness value is considered theoptimum solution in both approaches An extensive surveysof evolutionary computation and structural design are carriedout in [90 108ndash111]

There are varieties of evolutionary algorithms though ingeneral they all are population-based stochastic search algo-rithms They differ in the production of the new generationsHowever those that are used in steel frame optimization canbe collected under two main titles These are genetic algo-rithms developed by Holland [87] and evolution strategiesdeveloped by Goldberg and Santani [112] Gen and Chang[113] and Chambers [98]

511 Genetic Algorithms Genetic algorithm makes use ofthree basic operators to generate a new population from thecurrent one [16 90 114] The first one is known as selectionwhich involves selection of the individuals from the currentpopulation for mating depending on their fitness value Foreach individuals a fitness value is calculated which representsthe suitability of the individualrsquos potential to be selectedas a solution for the design More highly fit designs sendmore copies to the mating pool The fitness of individuals iscalculated from the fitness criteria In order to establish fitnesscriteria it is necessary to transform the constrained designproblem into an unconstrained oneThis is achieved by usinga penalty function that generates a penalty to the objectivefunction whenever the constraints are violated There arevarious forms of penalty functions used in conjunction withgenetic algorithms The details of these transformations aregiven in [98 113 115]

The second operator is called crossover in which thestrings of parents selected from the mating pool are brokeninto segments and some of these segments are exchangedwith corresponding segments of the other parentrsquos stringThecrossover operator swaps the genetic information betweenthe mating pair of individuals There are many differentstrategies to implement crossover operator such as fixedflexible and uniform crossovers [108 115]

After the application of crossover new individuals aregenerated with different strings These constitute the newpopulation Before repeating the reproduction and crossoveronce more to obtain another population the third operatorcalled mutation is used Mutation safeguards the process

from a complete premature loss of valuable genetic materialduring reproduction and crossover To apply mutation fewindividuals of the population are randomly selected and thestring of each individual at a random location if 0 is switchedto 1 or vice versaThemutation operation can be beneficial inreintroducing diversity in a population

Although initially genetic algorithms used the binaryalphabet to code the search space solutions there are alsoapplications where other types of coding are utilized Realcoding seemsparticularly naturalwhen tackling optimizationproblems of parameterswith variables in continuous domains[116] Leite and Topping [117] proposed modifications toone-point crossover by defining effective crossover site andusing multiple off-spring tournaments Modifications alsocovered to improve the efficiency of genetic operators toallow reduction in computation time and improve the resultsGenetic algorithm has been extensively used in discreteoptimization design of steel-framed structures [118ndash133]

Camp et al [124 125] developed a method for theoptimum design of rigid plane skeleton structures subjectedto multiple loading cases using genetic algorithm Thedesign constraints were implemented according to Allow-able Stress Design specifications of American Institutes ofSteels Construction (AISC-ASD) The steel sections wereselected from AISC database of available structural steelmembers Several strategies for reproduction and crossoverwere investigated Different penalty functions and fitnesspolicies were used to determine their suitability to ASDdesign formulation Saka and Kameshki [126] presentedgenetic algorithm based algorithm design of multistory steelframes with sideway subjected to multiple loading casesThe design constraints that include serviceability as well asthe combined strength constraints were implemented fromBritish Standards BS5950 Saka [127] used genetic algorithmin the minimum design of grillage systems subjected todeflection limitations and allowable stress constraints Wel-dali and Saka [128] applied the genetic algorithm to theoptimum geometry and spacing design of roof trusses sub-jected to multiple loading casesThe algorithm obtains a rooftruss that has the minimum weight by selecting appropriatesteel sections from the standard British Steel Sections tableswhile satisfying the design limitations specified by BS5950Saka et al [129] presented genetic algorithm based optimumdesign method that find the optimum spacing in additionto optimum sections for the grillage systems Deflectionlimitations and allowable stress constraints were consideredin the formulation of the design problem Pezeshk et al[130] also presented a genetic algorithm based optimizationprocedure for the design of plane frames including geometricnonlinearity The design constraints are accounted for AISC-LRFD specifications Hasancebi and Erbatur [131] carried outevaluation of crossover techniques in genetic algorithm Forthis purpose commonly used single-point 2-point multi-point variable-to-variables and uniform crossover are usedin the optimum design of steel pin jointed frames They haveconcluded that two-point crossover performed better thanother crossover types Erbatur et al [132] developed GAOSprogram which implements the constraints from the designcode and determines the optimum readymade hot-rolled


sections for the steel trusses and frames Kameshki and Saka[133] used genetic algorithm to compare the optimum designof multistory nonswaying steel frames with different types ofbracing Kameshki and Saka [134] presented an applicationof the genetic algorithm to optimum design of semirigid steelframes The design algorithm considers the service abilityand strength constraints as specified in BS5950 Andre etal [135] discussed the trade-off between accuracy reliabilityand computing time in the binary-encoded genetic algorithmused for global optimization over continuous variables Largeset of analytical test functions are considered to test theeffectiveness of the genetic algorithm Some improvementssuch as Gray coding and double crossover are suggested fora better performance Toropov and Mahfouz [136] presentedgenetic algorithm based structural optimization techniquefor the optimumdesign of steel frames Certainmodificationsare suggested which improved the performance of the geneticalgorithm The algorithm implements the design constraintsfrom BS5950 and considers the wind loading from BS6399The steel sections are selected from BS4 Hayalioglu [137]developed a genetic algorithm based optimum design algo-rithm for three-dimensional steel frames which implementsthe design constraints from LRFD-AISC The wind loadingis taken from Uniform Building Code (UBC) The algorithmis also extended to include the design constraints accordingto Allowable Stress Design (ASD) of AISC for comparisonJenkins [138] used decimal coding instead of binary codingin genetic algorithm The performance of the decimal codedgenetic algorithm is assessed in the optimum design of 640-bar space deck It is concluded that decimal coding avoidsthe inevitable large shifts in decoded values of variables whenmutation operates at the significant end of binary stringKameshki and Saka [139] extended thework of [134] to frameswith various semirigid beam-to-column connections suchas end plate without column stiffeners top and seat anglewith double web angle top and seat angle without doubleweb angle Yun and Kim [140] presented optimum designmethod for inelastic steel frames Kaveh and Shahrouzi [141]utilized a direct index coding within the genetic algorithmsuch a way that the optimization algorithm can simulta-neously integrate topology and size in a minimal lengthchromosome Balling et al [142] also presented a geneticalgorithm based optimum design approach for steel framesthat can simultaneously optimize size shape and topologyIt finds multiple optimums and near optimum topologiesin a single run Togan and Daloglu [143] suggested theadaptive approach in genetic algorithm which eliminatesthe necessity of specifying values for penalty parameter andprobabilities of crossover and mutation Kameshki and Saka[144] presented an algorithm for the optimum geometrydesign of geometrically nonlinear geodesic domes that usesgenetic algorithm and elastic instability analysis

Degertekin [145] compared the performance of thegenetic algorithm with simulated annealing in the optimumdesign of geometrically nonlinear steel space frames wherethe design formulation is carried out considering both LRFDand ASD specifications of AISC It is concluded that sim-ulated annealing yielded better designs Later in [146] theperformance of genetic algorithm is compared with that of

tabu search in finding the optimum design of steel spaceframes and found that tabu search resulted in lighter framesIssa andMohammad [147] attempted to modify a distributedgenetic algorithm to minimize the weight of steel framesThey have used twin analogy and a number of mutationschemes and reported improved performance for geneticalgorithm Safari et al [148] have proposed new crossoverand mutation operators to improve the performance of thegenetic algorithm for the optimum design of steel framesSignificant improvements in the optimum solutions of thedesign examples considered are reported

512 Evolution Strategies Evolution strategies were origi-nally developed for continuous structural optimization prob-lems [149] These algorithms are extended to cover discretedesign problems by Cai and Thierauf [150] The basic differ-ences between discrete and continuous evolution strategiesare in the mutation and recombination operators Evolutionstrategies algorithm randomly generates an initial populationconsisting of120583parent individuals It then uses recombinationmutation and selection operators to attain a new populationThe steps of the method are as follows [16 149]

(1) RecombinationThe population of 120583 parents is recom-bined to produce 120582 off springs in 119894th generation inorder to allow the exchange of design characteristicson both levels of design variables and strategy param-eters For every offspring vector a temporary parentvector 119904 = 119904

1 1199042 119904

119899 is first built by means of

recombination The following operators can be usedto obtain the recombined 119904

1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)

2global intermediate

(56)

where 119904119886and 119904

119887refer to the 119904 component of two

parent individuals which are randomly chosen fromthe parent population First case corresponds to norecombination case in which 119904

1015840 is directly formedby copying each element of first parent 119904

119886119894 In the

second 1199041015840 is chosen from one of the parents under

equal probability In the third the first parent iskept unchanged while a new second parent 119904

119887119895is

chosen randomly from the parent population Thefourth and fifth cases are similar to second and thirdcases respectively however arithmetic means of theelements are calculated

(2) Mutation This operator is not only applied to designvariables but also strategy parameters Carrying outthe mutation of strategy parameters first and thedesign variables later increases the effectiveness ofthe algorithm It is suggested in [149] that not all ofthe components of the parent individuals but only a


few of them should randomly be changed in everygeneration

(3) SelectionThere are two types of selection applicationIn the first the best 120583 individuals are selected from atemporary population of (120583 + 120582) individuals to formthe parents of the next generation In the second the120583 individuals produce 120582 off springs (120583 le 120582) andthe selection process defines a new population of 120583

individuals from the set of 120582 off springs only(4) Termination The discrete optimization procedure is

terminated either when the best value of the objectivefunction in the last 4 119899120583120582 or when the mean value ofthe objective values from all parent individuals in thelast 4 119899120583120582 generations has not been improved by lessthan a predetermined value 120576

There are different types of constraint handling in evo-lution strategies method An excellent review of these isgiven in [151] In nonadaptive evolution strategies constrainthandling is such that if the individual represents an infeasibledesign point it is discarded from the design space Henceonly the feasible individuals are kept in the populationFor this reason many potential designs that are close toacceptable design space are rejected that results in a lossof valuable information The adaptive evolution strategiessuggested in [152] use soft constraints during the initialstages of the search the constraints become severe as thesearch approaches the global optimum The detailed stepsof this algorithm are given in [149] where the procedureis explained in a simple example of three-bar steel trussstructure Later two steel space frames are designed withevolution strategies in which the effect of different selectionschemes and constraint handling is studied

Rajasekaran et al [153] applied the evolutionary strategiesmethod to the optimum design of large-size steel space struc-tures such as a double-layer grid with 700 degrees of freedomIt is reported that evolutionary strategies method workedeffectively to obtain the optimum solutions of these large-size structures Later Rajasekaran [154] used the samemethodto obtain the optimum design of laminated nonprismaticthin-walled composite spatial members that are subjected todeflection buckling and frequency constraints Evolutionarystrategies technique is applied to determine the optimal fibreorientation of composite laminates

Ebenau et al [155] combined evolutionary strategiestechnique with an adaptive penalty function and a specialselection scheme The algorithm developed is applied todetermine the optimumdesign of three-dimensional geomet-rically nonlinear steel structures where the design variableswere mixed discrete or topological

Hasancebi [156] has compared three different reformu-lations of evolution strategies for solving discrete optimumdesign of steel frames Extensive numerical experimentationis performed in the design examples to facilitate a statisticalanalysis of their convergence characteristics The resultsobtained are presented in the histograms demonstrating thedistribution of the best designs located by each approachFurther in [157] the same author investigated the applicationof evolution strategies to optimize the design of a truss bridge

The design problem involved identifying the optimum topol-ogy shape and member sizes of the bridge The designproblem consisted of mixed continuous and discrete designvariables It is shown that evolutionary strategies efficientlyfound the optimum topology configuration of a bridge in alarge and flexible design space

Hasancebi [158] has improved the computational perfor-mance of evolution strategies algorithm by suggesting self-adaptive scheme for continuous and discrete steel framedesign problems A numerical example taken from the litera-ture is studied in depth to verify the enhanced performance ofthe algorithm as well as to scrutinize the role and significanceof this self-adaptation scheme It is shown that adaptiveevolution strategies algorithms are reliable and powerful toolsand well-suited for optimum design of complex structuralsystems including large-scale structural optimization

52 Simulated Annealing Simulated annealing is an iterativesearch technique inspired by annealing process of metalsDuring the annealing process a metal first is heated up tohigh temperatures until it melts which imparts high energyto it In this stage all molecules of the molten metal canmove freely with respect to each other The metal is thencooled slowly in a controlled manner such that at each stagea temperature is kept of sufficient duration The atoms thenarrange themselves in a low-energy state and leads to acrystallized state which is a stable state that corresponds toan absolute minimum of energy On the other hand if thecooling is carried out quickly the metal forms polycrystallinestate which corresponds to a local minimum energy Themetal reaches thermal equilibrium at each temperature level119879 described by a probability of being in a state 119894 with energy119864119894given by the Boltzman distribution

119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)

where 119885(119879) is a normalization factor and 119896119861is Boltzmann

constant The Boltzmann distribution focuses on the stateswith the lowest energy as the temperature decreases Ananalogy between the annealing and the optimization canbe established by considering the energy of the metal asthe objective function while the different states during thecooling represent the different optimum solutions (designs)throughout the optimization process In the application of themethod first the constrained design problem is transferredinto an unconstrained problem by using penalty functionIf the value of the unconstrained objective function of therandomly selected design is less than the one in the currentdesign then the current design is replaced by the new designOtherwise the fate of the new design is decided according tothe Metropolis algorithm as explained in the following

119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)

where 119901119894119895is the acceptance probability of selected design

Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882

119894are the objective


function values of the selected and current designs Δ119882 isa normalization constant which is the running average ofΔ119882

119894119895 and119879

119896is the strategy temperatureΔ119882 is updatedwhen

Δ119882119894119895

gt 0 as explained in [88]The strategy temperature 119879

119896is gradually decreased while

the annealing process proceeds according to a cooling sched-ule For this the initial and the final values of the temperature119879119904and 119879

119891are to be known Once the starting acceptance

probability119875119904is decided the starting temperature is calculated

from 119879119904

= minus1 ln(119875119904) The strategy temperature is then

reduced using 119879119896+1

= 120572119879119896where 120572 is the cooling factor

and is less than one While 119879 approaches zero 119901119894119895also

approaches zero For a given final acceptance probability thefinal temperature is calculated from 119879

119891= minus1 ln(119875

119891) After

119873 cycles the final temperature value 119879119891can be expressed as

119879119891

= 119879119904120572119873minus1

Simulated annealing is originated by Kirkpatrick et al[88] and Cerny [159] independently at the same time whichis based on the previously mentioned phenomenon Thesteps of the optimum design algorithm for steel frames usingsimulated annealing method are given in the following Themethod is based on the work of Bennage and Dhingra [160]and the normalization constant in the Metropolis algorithm[161] is taken from the work of Balling [162] The details ofthese steps are given in [16 145]

(1) Select the values for 119875119904 119875

119891 and 119873 and calculate

the cooling schedule parameters 119879119904 119879

119891 and 120572 as

explained previously Initialize the cycle counter 119894119888 =

1(2) Generate an initial design randomly where each

design variable represents the sequence number ofthe steel section randomly selected from the list Theinitial design is assigned as current design Carry outthe analysis of the frame with steel section selectedin the current design and obtain its response underthe applied loads Calculate the value of objectivefunction 119882

(3) Determine the number of iterations required per cyclefrom the equation mentioned later

119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894

119891are the number of iterations per cycle

required at the initial and final temperatures 119879119904and

119879119891

(4) Select a variable randomly Apply a random perturba-tion to this variable and generate a candidate designin the neighbourhood of the current design ComputeΔ119882

119894119895

(5) Accept this candidate design as the current design ifΔ119882

119894119895le 0

(6) If Δ119882119894119895

gt 0 then update Δ119882 Calculate the accep-tance probability 119901

119894119895from (11) Generate a uniformly

distributed random number 119903 over interval [0 1] If119903 lt 119901

119894119895go to next step otherwise go to step 4

(7) Accept the candidate design as the current design Ifthe current design is feasible and is better than theprevious optimum design assign it temporarily as theoptimum design

(8) Update the temperature119879119896using119879

119896+1= 120572119879

119896 Increase

the cycle counter by one 119894119888 = 119894119888 + 1 If 119894119888 gt 119873

terminate the algorithm and take the last temporaryoptimum as the final optimum design Otherwisereturn to step 3

Balling [162] adapted simulated annealing strategy for thediscrete optimum design of three-dimensional steel framesThe total frame weight is minimized subject to design-code-specified constraints on stress buckling and deflectionThree loading cases are considered In the first loading casea uniform live load was placed on all floors and the roof Inthe second and third loading cases horizontal seismic loadsin the global 119883 and 119885 directions were placed at variousnodes respectively according to Uniform Building CodeMembers in the frame were selected from among the discretestandardized shapes Some approximation techniques wereused to reduce the computation time Later in [163] a filteris suggested through which each design candidate must passbefore it is accepted for computationally intensive structuralanalysis is set-up A scheme is devised whereby even lessfeasible candidates can pass through the filter in a proba-bilistic manner It is shown that the filter size can speed upconvergence to global optimum

Simulated annealing is applied to optimum design oflarge tetrahedral truss for minimum surface distortion in[164 165] In [166] the simulated annealing is applied tolarge truss structures where the standard cooling schedule ismodified such that the best solution found so far is used as astarting configuration each time the temperature is reduced

Topping et al [167] used simulated annealing in simulta-neous optimum design for topology and size The algorithmdeveloped is applied to truss examples from the literaturefor which the optimum solutions were known The effects ofcontrol parameters are discussed Later in [168] implemen-tation of parallel simulated annealing models is evaluatedby the same authors It is stated in this work that efficiencyof parallel simulated annealing is problem dependant andsome engineering problems solution spaces may be verycomplex and highly constrained Study provides guidelinesfor the selection of appropriate schemes in such problemsTzan and Pantelides [169] developed an annealing methodfor optimal structural design with sensitivity analysis andautomatic reduction of the search range

Hasancebi and Erbatur [170] presented simulated anneal-ing based structural optimization algorithm for large andcomplex steel frames Two general and complementaryapproaches with alternative methodologies to reformulatethe working mechanism of the Boltzmann parameter areintroduced to accelerate the convergence reliability of simu-lated annealing Later in [171] they have developed simulatedannealing based algorithm for the simultaneous optimumdesign of pin-jointed structures where the size shape andtopology are taken as design variables In the examples con-sidered while the weight of trusses is minimized the design


constraints such as nodal displacements member stress andstability are implemented from design code specifications

Degertekin [172] proposed a hybrid tabu-simulatedannealing algorithm for the optimum design of steel framesThe algorithm exploits both tabu search and simulatedannealing algorithms simultaneously to obtain near optimumsolutionThe design constraints are implemented from LRD-AISC specification It is reported that hybrid algorithmyielded lighter optimum structures than those algorithmsconsidered in the study

Hasancebi et al [173] have suggested novel approachesto improve the performance of simulated annealing searchtechnique in the optimum design of real-size steel framesto eliminate its drawbacks mentioned in the literature Itis reported that the suggested improvements eliminated thedrawbacks in the design examples considered in the studyIn [174] the same author has used simulated annealing basedoptimumdesignmethod to evaluate the topological forms forsingle span steel truss bridgeThe optimum design algorithmattains optimum steel profiles for the members and the trussas well as optimum coordinates of top chord joints so thatthe bridge has the minimum weight The design constraintsand limitations are imposed according to serviceability andstrength provisions of ASD-AISC (Allowable Stress DesignCode of American Institute of Steel Institution) specification

53 Particle Swarm Optimizer Particle swarm optimizer isbased on the social behavior of animals such as fish schoolinginsect swarming and birds flocking This behavior is con-cernedwith grouping by social forces that depend on both thememory of each individual as well as the knowledge gainedby the swarm [175 176] The procedure involves a numberof particles which represent the swarm and are initializedrandomly in the search space of an objective function Eachparticle in the swarm represents a candidate solution of theoptimumdesign problemTheparticles fly through the searchspace and their positions are updated using the currentposition a velocity vector and a time increment The stepsof the algorithm are outlined in the following as given in[177 178]

(1) Initialize swarm of particles with positions 119909119894

0and ini-

tial velocities 119907119894

0randomly distributed throughout the

design space These are obtained from the followingexpressions

119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)

where the term 119903 represents a random numberbetween 0 and 1 and 119909min and 119909max represent thedesign variables of upper and lower bounds respec-tively

(2) Evaluate the objective function values119891(119909119894

119896) using the

design space positions 119909119894

119896

(3) Update the optimum particle position 119901119894

119896at the

current iteration 119896 and the global optimum particleposition 119901

119892

119896

(4) Update the position of each particle from 119909119894

119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894

119896+1is the position of particle 119894

at iteration 119896 + 1 119907119894

119896+1is the corresponding velocity

vector and Δ119905 is the time step value(5) Update the velocity vector of each particle There are

several formulas for this depending on the particularparticle swarm optimizer under consideration Theone given in [176 177] has the following form

119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903

2are random numbers between 0 and

1 119901119894

119896is the best position found by particle 119894 so far

and 119901119892

119896is the best position in the swarm at time 119896

119908 is the inertia of the particle which controls theexploration properties of the algorithm 119888

1and 119888

2are

trust parameters that indicate how much confidencethe particle has in itself and in the swarm respectively

(6) Repeat steps 2ndash5 until stopping criteria are met

Fourie and Groenwold [178] applied particle swarmoptimizer algorithm to optimal design of structures withsizing and shape variables Standard size and shape designproblems selected from the literature are used to evaluate theperformance of the algorithm developed The performanceof particle swarm optimizer is compared with three-gradientbased methods and genetic algorithm It is reported thatparticle swarm optimizer performed better than geneticalgorithm

Perez and Behdinan [179] presented particle swarm basedoptimum design algorithm for pin jointed steel frames Effectof different setting parameters and further improvements arestudied Effectiveness of the approach is tested by consideringthree benchmark trusses from the literature It is reported thatthe proposed algorithm found better optimal solutions thanother optimum design techniques considered in these designproblems

In [180] particle swarm optimizer is improved by intro-ducing fly-back mechanism in order to maintain a fea-sible population Furthermore the proposed algorithm isextended to handle mixed variables using a simple schemeand used to determine the solution of five benchmarkproblems from the literature that are solved with differentoptimization techniques It is reported that the proposedalgorithm performed better than other techniques In [181]particle swarm optimizer is improved by considering apassive congregation which is an important biological forcepreserving swarm integrity Passive congregation is an attrac-tion of an individual to other groupmembers butwhere thereis no display of social behavior Numerical experimentationof the algorithm is carried out on 10 benchmark problemstaken from the literature and it is stated that this improve-ment enhances the search performance of the algorithmsignificantly


Li et al [182] presented a heuristic particle swarm opti-mizer for the optimum design of pin jointed steel framesThe algorithm is based on the particle swarm optimizerwith passive congregation and harmony search scheme Themethod is applied to optimum design of five-planar andspatial truss structures The results show that proposedimprovements accelerate the convergence rate and reache tooptimum design quicker than other methods considered inthe study

Dogan and Saka [183] presented particle swarm basedoptimum design algorithm for unbraced steel frames Thedesign constraints imposed are in accordance with LRFD-AISC codeThedesign algorithm selects optimumWsectionsfor beams and columns of unbraced steel frames such thatthe design constraints are satisfied and the minimum frameweight is obtained

54 Ant Colony Optimization Ant colony optimization tech-nique is inspired from the way that ant colonies find theshortest route between the food source and their nest Thebiologists studied extensively for a long timehowantsmanagecollectively to solve difficult problems in a natural way whichis too complex for a single individual Ants being completelyblind individuals can successfully discover as a colony theshortest path between their nest and the food source Theymanage this through their typical characteristic of employingvolatile substance called pheromones They perceive thesesubstances through very sensitive receivers located in theirantennas The ants deposit pheromones on the ground whenthey travel which is used as a trail by other ants in the colonyWhen there is a choice of selection for an ant between twopaths it selects the onewhere the concentration of pheromoneis more Since the shorter trail will be reinforced more thanthe long one after a while a greatmajority of ants in the colonywill travel on this route Ant colony optimization algorithmis developed by Colorni et al [184] and Dorigo [185 186]and used in the solution of travelling salesman The stepsof optimum design algorithm for steel frames based on antcolony optimization are as follows [187 188]

(1) Calculate an initial trail value as 1205910

= 1119908min where119908min is the weight of the frame from assigning thesmallest steel sections from the database to eachmember group of the frame

(2) Assign a member group to each ant in the colonyrandomly and then select a steel section from thedatabase so that the ant can start its tour Thisselection is carried out according to the followingdecision process

119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)

where 119895 is the steel section assigned to member group119894 and 119899119904

119894is the total number of steel sections in the

database 120573 is a constant The probability 119901ℓ

119894119895(119905) that

the ant ℓ assigns a steel section 119895 to member group 119894

at time 119905 is given as

119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)

(3) After the steel section is selected by the first ant asexplained in step 2 the intensity of trail on this path islowered using local update rule 120591

119894119895(119905) = 120585120591

119894119895(119905) where

120577 is an adjustable parameter between 0 and 1(4) The second ant selects a steel section from the

database for its member group 119894 and a local updaterule is applied This procedure is continued until allthe ants in the colony select a steel section for thestartingmember group in their position on the frameAfter completing the first iteration of the tour eachant selects another steel section to its next membergroup 119894 + 1 This procedure is repeated until eachant in the colony has selected a steel section fromthe database for each member group in the frameHencewhen the selection process is completed the antcolony has 119898 different designs for the frame where 119898

represents the total number of ants selected initially(5) Frame is analyzed for these designs and the violations

of constraints corresponding to each ant are calcu-lated and substituted into the penalty function of 119865 =

119882(1 + 119862)120572 where 119882 is the weight of the frame 119862

is the total constraint violation and 120572 is the penaltyfunction exponent All designs are in a cycle and areranked by their penalized weights and the elitist antthat selected the frame with the smallest penalizedweight in all cycles is determinedThe amount of trailΔ120591

119890

119894119895= 1119882

119890is added to each path chosen by this elitist

ant where 119882119890is the penalized frame weight selected

by the elitist ant(6) Carry out the global update of trails from 120591

119894119895(119905 + 1) =

120588120591119894119895

(119905) + (1 minus 120588)Δ120591119894119895where 120588 is a constant having value

between 0 and 1 that represents the evaporation rateand Δ120591

119894119895= sum

119898

119896=1Δ120591

119896

119894119895in which 119898 is the total number

of ants selected initially

Camp and Bichon [187] developed discrete optimumdesign procedure for space trusses based on ant colonyoptimization technique The total weight of the structureis minimized while stress and deflection limitations areconsideredThe design problem is transformed intomodifiedtravelling salesman problem where the network of travellingsalesman is taken as the structural topology and the length ofthe tour is theweight of the structureThenumber of trusses isdesigned using the algorithm developed and results obtainedare compared with genetic algorithm and gradient basedoptimization methods Later in [188] the work is extended torigid steel frames The serviceability and strength constraintsare implemented fromLRFD-AISC-2001 code A comparisonis presented between ant colony optimization frame designand designs obtained using genetic algorithm

Kaveh and Shojaee [189] also used ant colony opti-mization algorithm to develop an approach for the discrete


optimum design of steel frames The design constraintsconsidered consist of combined bending and compressioncombined bending and tension and deflection limitationswhich are specified according to ASD-AISC design codeThedetailed explanation of ant colony optimization operatorsis given Optimum designs of six plane steel structuresare obtained using the algorithm developed Some of theresults are compared to those that are achieved by geneticalgorithms which are taken from the literature Bland [190]also used ant colony optimization to obtain the minimumweight of transmission tower which has over 200 membersKaveh and Talatahari [191] presented an improved ant colonyoptimization algorithm for the design of steel frames Thealgorithm employs suboptimizationmechanism based on theprincipals of finite element method to reduce the searchdomain the size of trail matrix and the number of structuralanalyses in the global phase and the optimum design isobtained by considering W sections list in the neighborhoodof the result attained in the previous phase Wang et al[192] presented an algorithm for a partial topology andmember size optimization for wing configuration Ant colonyoptimization is used to determine the optimum topologyof the wing structure and gradient based optimizationmethod is used for component size optimization problemIt is stated that this combined procedure was effective insolving wing structure optimization problem In [193] antcolony algorithm is used to develop design optimizationtechnique for three-dimensional steel frames where the effectof elemental warping is taken into account

55 Harmony Search Method One other recent additionto metaheuristic algorithms is the harmony search methodoriginated by Geem et al [194ndash206] Harmony search algo-rithm is based on natural musical performance processesthat occur when a musician searches for a better state ofharmony The resemblance for an example between jazzimprovisation that seeks to find musically pleasing harmonyand the optimization is that the optimum design processseeks to find the optimum solution as determined by theobjective function The pitch of each musical instrumentdetermines the aesthetic quality just as the objective functionvalue is determined by the set of values assigned to eachdecision variable

Harmony search algorithm consists of five basic stepsThe detailed explanation of these steps can be found in [196]which are summarized in the following

(1) Initialize the harmony search parameters A possiblevalue range for each design variable of the optimumdesign problem is specified A pool is constructedby collecting these values together from which thealgorithm selects values for the design variables Fur-thermore the number of solution vectors in harmonymemory (HMS) that is the size of the harmonymemory matrix harmony considering rate (HMCR)pitch adjusting rate (PAR) and themaximumnumberof searches is also selected in this step

(2) Initialize the harmony memory matrix (HM) Eachrow of harmony memory matrix contains the values

of design variables which randomly selected feasiblesolutions from the design pool for that particulardesign variable Hence this matrix has 119899 columnswhere 119899 is the total number of design variables andHMS rows which is selected in the first step HMSis similar to the total number of individuals in thepopulation matrix of the genetic algorithm

(3) Improvise a new harmony memory matrix In gen-erating a new harmony matrix the new value of the119894th design variable can be chosen from any discretevalue within the range of 119894th column of the harmonymemory matrix with the probability of HMCR whichvaries between 0 and 1 In other words the new valueof 119909

119894can be one of the discrete values of the vector

1199091198941

1199091198942

119909119894hms

119879with the probability of HMCRThe same is applied to all other design variables Inthe random selection the new value of the 119894th designvariable can also be chosen randomly from the entirepool with the probability of 1-HMCR That is

119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms

119879 with probability HMCR119909119894

isin 1199091 119909

2 119909ns

119879with probability (1 minus HMCR)

(64)

where ns is the total number of values for the designvariables in the pool If the new value of the designvariable is selected among those of harmonymemorymatrix this value is then checked for whether itshould be pitch-adjusted This operation uses pitchadjustment parameter PAR that sets the rate of adjust-ment for the pitch chosen from the harmonymemorymatrix as follows

Is 119909new119894

to be pitch-adjusted

Yes with probability of PARNo with probability of (1 minus PAR)

(65)

Supposing that the new pitch adjustment decisionfor 119909

new119894

came out to be yes from the test and if thevalue selected for 119909

new119894

from the harmony memory isthe 119896th element in the general discrete set then theneighboring value 119896 + 1 or 119896 minus 1 is taken for new 119909

new119894

This operation prevents stagnation and improves theharmony memory for diversity with a greater changeof reaching the global optimum

(4) Update the harmony memory matrix After selectingthe new values for each design variable the objectivefunction value is calculated for the new harmonyvector If this value is better than the worst harmonyvector in the harmony matrix it is then included inthe matrix while the worst one is taken out of thematrix The harmony memory matrix is then sortedin descending order by the objective function value

(5) Repeat steps 3 and 4 until the termination criteria issatisfied


Lee andGeem [196] presented harmony search algorithmbased discrete optimum design technique for plane trussesEight different plane trusses are selected from the literatureand optimized using the approach developed to demonstratethe efficiency and robustness of the harmony search algo-rithm In all these examples the proposed algorithm hasfound the minimum weight that is lighter than those deter-mined by other techniques In [197] authors have extendedthe harmony search algorithm to deal with continuous engi-neering optimization problems Various engineering designexamples includingmathematical functionminimization andstructural engineering optimization problems are consideredto demonstrate the effectiveness of the algorithm It is con-cluded that the results obtained indicated that the proposedapproach is a powerful search and optimization techniquethat may yield better solutions to engineering problems thanthose obtained using current algorithms

Saka [207] presented harmony search algorithm basedoptimum geometry design technique for single-layergeodesic domes It treats the height of the crown as designvariable in addition to the cross-sectional designations ofmembers A procedure is developed that calculates the jointcoordinates automatically for a given height of the crownThe serviceability and strength requirements are consideredin the design problem as specified in BS5950-2000 Thiscode makes use of limit state design concept in whichstructures are designed by considering the limit statesbeyond which they would become unfit for their intendeduse The design examples considered have shown thatharmony search algorithm obtains the optimum height andsectional designations for members in relatively less numberof searches Later this technique is extended to cover theoptimum topology design of nonlinear lamella and networkdomes [208ndash210] where geometric nonlinearity is also takeninto account

Saka and Erdal [211] used harmony search algorithmto develop a discrete optimum design method for grillagesystems The displacement and strength specifications areimplemented from LRFD-AISC The algorithm selects theappropriate W sections from the database for transverse andlongitudinal beams of the grillage system The number ofdesign examples is considered to demonstrate the efficiencyof the proposed algorithm

Saka [212] presented harmony search algorithm baseddiscrete optimum design method for rigid steel frames Theobjective is taken as the minimum weight of the frameand the behavioral and performance limitations are imposedfrom BS5950 The list of Universal Beam and UniversalColumn sections of The British Code is considered for theframe members to be selected from The combined strengthconstraints that are considered for beam columns take intoaccount the effect of lateral torsional buckling The effectivelength computations for columns are automated and decidedby the algorithm itself depending upon the steel sectionsadopted for beams and columns within that design cycleIt is demonstrated that harmony search algorithm is quiteeffective and robust in finding the solution of minimumweight steel frame design Degertekin [213] also presentedharmony search method based discrete optimum design

method for steel frame where the design constraints areimplemented according to LRFD-AISC specifications Theeffectiveness and robustness of harmony search algorithm incomparison with genetic algorithm simulated annealing andcolony optimization based methods and were verified usingthree planar and two-space steel frames The comparisonsshowed that the harmony search algorithm yielded lighterdesigns for the presented examples Degertekin et al [214]used harmony search method to develop an optimum designalgorithm for geometrically nonlinear semirigid steel frameswhere the steel sections are selected from European wideflange steel sections (HE sections) It is stated that harmonysearch method efficiently obtained the optimum solution ofcomplex design problem requiring relatively less computa-tional time

Hasancebi et al [215 216] developed adaptive harmonysearch method for optimum design of steel frames wheretwo of the three main operators of classical harmony searchmethods namely harmony memory considering rate andpitch adjusting rate are adjusted automatically by the algo-rithm itself during the design iterations The initial valuesselected for these parameters directly affect the performanceof the algorithm This novel technique eliminates problem-dependent value selection of these parameters It is shownthat adaptive harmony search technique performs muchbetter than the standard harmony searchmethod in obtainingthe optimum designs of real-size steel frames

Erdal et al [217] formulated design problem of cellularbeams as an optimum design problem by treating the depththe diameter and the total number of holes as designvariables The design problem is formulated considering thelimitations specified inThe Steel Construction Institute Pub-lication Number 100 which is consisted BS5950 parts 1 and 3The solution of the design problem is determined by using theharmony search algorithm and particle swarm optimizers Inthe design examples considered it is shown that bothmethodsefficiently obtained the optimum Universal beam section tobe selected in the production of the cellular beam subjectedto general loading the optimum hole diameters and theoptimum number of holes in the cellular beam such that thedesign limitations are all satisfied

Saka et al [218] have evaluated the recent enhance-ments suggested by several authors in order to improvethe performance of the standard harmony search methodAmong these are the improved harmony search method andglobal harmony search method by Mahdavi et al [219 220]adaptive harmony search method [215] improved harmonysearch method by Santos Coelho and de Andrade Bernert[221] and dynamic harmony search method by Saka et al[218] The optimum design problem of steel space framesis formulated according to the provisions of LRFD-AISC(Load and Resistance Factor Design-American Institute ofSteel Corporation) Seven different structural optimizationalgorithms are developed each ofwhich is based on one of thepreviously mentioned versions of harmony search methodThree real-size space steel frames one of which has 1860members are designed using each of these algorithms Theoptimumdesigns obtained by these techniques are comparedand performance of each version is evaluated It is stated


that among all the adaptive and dynamic harmony searchmethods outperformed the others

56 Big Bang Big Crunch Algorithm Big Bang-Big Crunchalgorithm is developed by Erol and Eksin [222] which is apopulation based algorithm similar to the genetic algorithmand the particle swarm optimizer It consists of two phasesas its name implies First phase is the big bang in whichthe randomly generated candidate solutions are randomlydistributed in the search space In the second phase thesepoints are contracted to a single representative point thatis the weighted average of randomly distributed candidatesolutions First application of the algorithm in the optimumdesign of steel frames is carried out by Camp [223]The stepsof the method are listed later as it is given in [223]

(1) Decide an initial population size and generate initialpopulation randomly These candidate solutions arescattered in the design domain This is called the bigbang phase

(2) Apply contraction operation This operation takesthe current position of each candidate solution inthe population and its associated penalized fitnessfunction value and computes a center of mass Thecenter ofmass is theweighted average of the candidatesolution positions with respect to the inverse of thepenalized fitness function values

119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)

where 119909cm is the position of the center of mass 119909119894is

the position of candidate solution 119894 in 119899 dimensionalsearch space 119891

119894is the penalized fitness function value

of candidate solution I and np is the size of the initialpopulation

(3) Compute the position of the candidate solutionsin the next iteration using the following expressionconsidering that they are normally distributed aroundthe center of mass 119909cm

119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)

where 119909next119894

is the position of the new candidatesolution 119894 119903 is the random number from a standardnormal distribution 120572 is the parameter that limits thesize of the search space 119909max and 119909min are the upperand lower bounds on the design variables and 119904 isthe number of big bang iterations In the case wheresteel sections are to be selected from the availablesteel sections list then it becomes necessary to workwith integer numbers In this case 119909

next119894

of (67) isrounded to the nearest integer number as 119868

next119894

=

ROUND(119909next119894

) where 119868next119894

represents index numberthe steel profile from the tabular discrete list

(4) Repeat steps 2 and 3 until termination criteria aresatisfied

Camp [223] developed optimum design algorithm forspace trusses based on big bang-big crunch optimizer Thenumber of benchmark design examples having design vari-ables continuous as well discrete is taken from the literatureand designed with the developed algorithm The results arecompared with those algorithms of quadratic programminggeneral geometric programming genetic algorithm particleswarm optimizer and ant colony optimization It is reportedthat big bang-big crunch optimizer has relatively small num-ber of parameters to definewhich provides the algorithmwithbetter performance over the other techniques considered inthe study It is also concluded that the algorithm showed sig-nificant improvements in the consistency and computationalefficiency when compared to genetic algorithm particleswarm optimizer and ant colony technique

Kaveh and Talatahari [224] also presented big bang-big crunch-based optimum design algorithm for size opti-mization of space trusses In this study large size spacetrusses are designed by the algorithm developed as well asthose stochastic search techniques of genetic algorithm antcolony optimization particle swarm optimizer and standardharmony search method It is stated that big bang-big crunchalgorithm performs well in the optimum design of large-size space trusses contrary to other metaheuristic techniqueswhich presents convergence difficulty or get trapped at a localoptimum

Kaveh and Talatahari [225] developed optimum topologydesign algorithm based on hybrid big bang-big crunchoptimization method for schwedler and ribbed domes Thealgorithm determines the optimum configuration as well asoptimummember sizes of these domes It is reported that bigbang-big crunch optimizationmethod efficiently determinedthe optimum solution of large dome structures

Kaveh and Abbasgholiha [226] presented the optimumdesign algorithm for steel frames based on big bang-bigcrunch optimizer The design problem is formulated accord-ing to BS5950 ASD-AISCF and LRFD-AISC design codesand the optimumresults obtained by each code are comparedIt is stated that LRFD design codes yield to the lightest steelframe as expected among other codes

57 Hybrid and Other Stochastic Search Techniques Stochas-tic search techniques have two drawbacks although they arecapable of determining the optimum solution of discretestructural optimization problems First one is that there is noguarantee that the solution found at the end of predeterminednumber of iterations is the global optimum There is nomathematical proof available in these techniques due tothe fact that they use heuristics not mathematically derivedexpressionThe optimum solution obtained may very well bea local optimumor near optimum solution Second drawbackis that they need large amount of structural analysis to reachthe near optimum solution Some work is carried out toimprove the performance of the metaheuristic optimizationtechniques by hybridizing them Some of these algorithms arereviewed below

Kaveh andTalatahari [227 228] developed hybrid particleswarm and ant colony optimization algorithm for the discrete


optimum design of steel frames The algorithm uses particleswarm optimizer for global search and ant colony optimiza-tion for local search It is reported that the hybrid algorithmis quite effective in finding the optimum solutions

Kaveh and Talatahari [229 230] have also combined thesearch strategies of the harmony search method particleswarm optimizer and ant colony optimization to obtainefficient metaheuristic technique called HPSACO for theoptimum design of steel frames In this technique particleswarm optimization with passive congregation is used forglobal search and the ant colony optimization is employedfor local search The harmony search based mechanism isutilized to handle the variable constraints It is demonstratedin the design examples considered that proposed hybridtechnique finds lighter optimum designs than each standardparticle swarm optimizer and ant colony optimization aswell as harmony search method Further improvements aresuggested for the algorithm in [231]

Kaveh and Rad [232] presented another hybrid geneticalgorithm and particle swarm optimization technique for theoptimum design of steel frames They have introduced thematuring phenomenonwhich is mimicked by particle swarmoptimizer where individuals enhance themselves based onsocial interactions and their private cognition Crossoveris applied to this society Hence evolution of individualsis no longer restricted to the same generation The resultsobtained from the design examples have shown that thehybrid algorithm shows superiority in optimum design oflarge-size steel frames

Kaveh and Talatahari [233] presented a structural opti-mization method based on sociopolitically motivated strat-egy called imperialist competitive algorithm Imperialistcompetitive algorithm initiates the search by consideringmultiagents where each agent is considered to be a countrywhich is either a colony or an imperialist Countries formcolonies in the search space and they move towards theirrelated empires During this movement weak empires col-lapses and strong ones get stronger Such movements directthe algorithm to optimum point Presented algorithm is usedin the optimum design of skeletal steel frames

Kaveh and Talatahari [234] also introduced a novelheuristic search technique based on some principles ofphysics and mechanics called charged system search Themethod is a multiagent approach where each agent is acharged particle These particles affect each other based ontheir fitness values and distances among them The quantityof the resultant force is determined by using electrostatic lawsand the quality of movement is determined using Newtonianmechanics laws It is stated that charged system searchalgorithm is compared with other metaheuristic algorithmon benchmark examples and it is found that it outperformedthe others The algorithm is applied to optimum design ofskeletal structures in [235 236] to grillage systems in [236]to truss structures in [237] and to geodesic dome in [238] Itis stated in all these works that charged system search algo-rithm shows better performance than other metaheuristictechniques considered In [239] an improvement is suggestedfor the algorithm to enhance its performance even further

The algorithm is applied to optimum design of steel framesin [240]

Kaveh and Bakhspoori [241] used cuckoo search methodto develop optimum design algorithm for steel framesThe design problem is formulated according to Load andResistance Factor Design code of American Institute of SteelConstruction [72]The optimum designs obtained by cuckoosearch algorithm are compared with those attained by otheralgorithms on benchmark frames Cuckoo search algorithmis originated by Yang and Deb [242] which simulates repro-duction strategy of cuckoo birds Some species of cuckoobirds lay their eggs in the nests of other birds so that whenthe eggs are hatched their chicks are fed by the other birdsSometimes they even remove existing eggs of host nest inorder to give more probability of hatching of their own eggsSome species of cuckoo birds are even specialized to mimicthe pattern and color of the eggs of host birds so that host birdcould not recognize their eggs which gives more possibilityof hatching In spite of all these efforts to conceal their eggsfrom the attention of host birds there is a possibility that hostbird may discover that the eggs are not its own In such casesthe host bird either throws these alien eggs away or simplyabandons its nest and builds a new nest somewhere else Theengineering design optimization of cuckoo search algorithmis carried out in [243]

58 Evaluation of Stochastic Search Techniques It is apparentthat there are a lot of metaheuristic algorithms that can beused in the optimum design of steel frames The questionof which one of these algorithms outperforms the othersrequires an answer It should be pointed out that the per-formance of metaheuristic algorithms is dependent upon theselection of the initial values for their parameters which isquite problem dependentThe following works try to providean answer to the previously mentioned problem

Hasancebi et al [244 245] evaluated the performanceof the stochastic search algorithm used in structural opti-mization on the large-scale pin jointed and rigidly jointedsteel frames Among these techniques genetic algorithmssimulated annealing evolution strategies particle swarmoptimizer tabu search ant colony optimization andharmonysearch method are utilized to develop seven optimum designalgorithms for real-size pin and rigidly connected large-scalesteel frames The design problems are formulated accordingto ASD-AISC (Allowable Stress Design Code of AmericanInstitute of Steel Institution)The results reveal that simulatedannealing and evolution strategies are the most powerfultechniques and standard harmony search and simple geneticalgorithmmethods can be characterized by slow convergencerates and unreliable search performance in large-scale prob-lems

Kaveh and Talatahari [246] used particle swarm opti-mizer ant colony optimization harmony search method bigbang-big crunch hybrid particle swarm ant colony optimiza-tion and charged system search techniques in the optimumdesign of single-layer Schwedler and lamella domes Thedesign problem is formulated according to LRFD-AISCspecifications It is stated that among these algorithm hybrid


particle swarm ant colony optimization and charged systemsearch algorithms have illustrated better performance com-pared to other heuristic algorithms

6 Conclusions

The review carried out reveals the fact that the mathematicalmodeling of the optimum design problem of steel framescan be broadly formulated in different ways In the first waythe cross-sectional properties of frame members treated onlythe optimization variables In this case joint displacementsare not part of optimization model and their values arerequired to be obtained through structural analysis in everydesign cycle This type of formulation is called coupledanalysis and design (CAND) Naturally in this the type offormulation the total number of analysis is large which iscomputationally inefficient In the second type of formulationthe joint displacements are also treated as optimizationvariables in addition to cross-sectional propertiesThismakesit necessary to include the stiffness equations as constraintsin the mathematical model as equality constraints Such for-mulation is called simultaneous analysis and design (SAND)which eliminates the necessity of carrying out structuralanalysis in every design cycle Hence the total number ofstructural analysis required to reach the optimum solutionbecomes quite less compared to the first type of modelingHowever in the second type the total number of optimizationvariables is quite large and it becomes necessary to utilizepowerful optimization techniques that work efficiently insolving large-size optimization problems

The design code that is to be considered in the modelingof the optimum design problem also affects the complexityof the optimization problem obtained Formulating the opti-mum design of steel frames without referencing any designcode brings out relatively simple optimization problem iflinear elastic structural behavior is assumed Furthermore ifcontinuous design variables assumption is also made thenmathematical programming or optimality criteria algorithmsefficiently finds the optimum solution of the optimizationproblem in both ways of modeling Optimality criteriaalgorithms also provide optimum solutions without anydifficulty even if nonlinear elastic behavior is consideredin the optimum design of steel frames Among the math-ematical programming techniques it seems that sequentialquadratic programming method is the most powerful and itis reported in several works that it can attain the optimumsolution without any difficulty in large-size steel framedesign optimization If mathematical modeling of the framedesign optimization problem is to be formulated such thatthe allowable stress design code specifications such as thedisplacement and stress limitations are required to be satisfiedin the optimum design the optimality criteria approachesprovide efficient algorithms for that purpose However itshould be pointed out that in the optimum solution thedesign variables will have values attained from a continuousvariables assumptionOn the other handpracticing structuraldesigner needs cross-sectional properties selected from theavailable steel sections list This necessitates first finding thecontinuous optimum solution and then round these to the

nearest available values Such move may yield loss of whatis gained through optimization Altering the mathematicalprogramming or optimality criteria technique to work withdiscrete variables makes the algorithms complicated andthey present difficulties in obtaining the optimum solutionof real-size steel frames

Emergence of the stochastic search techniques providessteel frame designers with new capabilities These new tech-niques do not need the gradient calculations of objectivefunction and constraints They do not use mathematicalderivation in order to find a way to reach the optimumInstead they rely on heuristics These new optimizationtechniques use nature as a source of inspiration to developnumerical optimization procedures that are capable of solv-ing complex engineering design problems They are particu-larly efficient in obtaining the optimum solution where thedesign variables are to be selected from a tabulated list Assummarized in this paper there are several stochastic searchtechniques that are successfully used in the optimum designsteel frames where the steel sections for the frame membersare to be selected from the available steel sections list whilethe limit state design code specifications such as serviceabilityand strength are to be satisfied These techniques do provideoptimum solution that can be directly used by the practicingdesigners in their projects However they also have somedrawbacks The first one is that because they do not usemathematical derivations it is not possible to prove whetherthe optimum solution they attain is the global optimumor it is near optimum The second is that they work withrandom numbers and they have a number of parameterswhich need to be given values by the users prior to theirapplication Selection of these values affects the performanceof algorithms This requires a sensitivity works with differentvalues of these parameters in order to find the appropriatevalues for the problem under consideration Although sometechniques are available such as adaptive genetic algorithmand adaptive harmony search method where the values ofthese parameters are adjusted automatically by the algorithmitself this is not the case for other techniques The thirddrawback is that they need a large number of structural analy-sis which becomes computationally very expensive for large-size steel frames Among the existing techniques some ofthem excel and outperform others It is apparent that furtherresearch is required to reduce the total number of structuralanalysis required by stochastic search algorithm which iscomputationally expensive to a reasonable amount Howeverthe search for finding better stochastic search techniques iscontinuing It is difficult at this moment to conclude whichone of these techniques will become the standard one thatwill be used in the design tools of the finite element packagesthat are used in everyday practice However it is not difficultto conclude that metaheuristic techniques are going to bestandard design optimization tools in the near future

Acknowledgments

This work was supported by the Gachon University ResearchFund of 2012 (GCU-2012-R259) However there is no conflict


of interests which exists when professional judgment con-cerning the validity of research is influenced by a secondaryinterest such as financial gain

References

[1] L A Schmit ldquoStructural Design by Systematic Synthesisrdquo inProceedings of the ASCE 2nd Conference on Electronic Compu-tation pp 105ndash132 1960

[2] K I Majid Optimum Design of Structures Butterworths Lon-don UK 1974

[3] M P Saka Optimum design of structures [PhD thesis] Univer-sity of Aston Birmingham UK 1975

[4] L A A Schmit and H Miura ldquoApproximating concepts forefficient structural synthesisrdquo Tech Rep NASA CR-2552 1976

[5] M P Saka ldquoOptimum design of rigidly jointed framesrdquo Com-puters and Structures vol 11 no 5 pp 411ndash419 1980

[6] E Atrek R H Gallagher K M Ragsdell and O C ZienkewicsEdsNew Directions in Optimum Structural Design JohnWileyamp Sons Chichester UK 1984

[7] R T Haftka ldquoSimultaneous analysis and designrdquoAIAA Journalvol 23 no 7 pp 1099ndash1103 1985

[8] J S Arora Introduction toOptimumDesignMcGraw-Hill 1989[9] R T Haftka and Z Gurdal Elements of Structural Optimization

Kluwer Academic Publishers The Netherlands 1992[10] U Kirsch Structural Optimization Springer 1993[11] J S Arora ldquoGuide to structural optimizationrdquo ASCE Manuals

and Reports on Engineering Practice number 90 ASCE NewYork NY USA 1997

[12] A D Belegundi and T R ChandrupathOptimization Conceptsand Application in Engineering Prentice-Hall 1999

[13] American Institutes of Steel Construction Manual of SteelConstruction Allowable Stress Design American Institutes ofSteel Construction Chicago Ill USA 9th edition 1989

[14] M P Saka ldquoOptimum design of steel frames with stabilityconstraintsrdquo Computers and Structures vol 41 no 6 pp 1365ndash1377 1991

[15] M P Saka ldquoOptimum design of skeletal structures a reviewrdquo inProgress in Civil and Structural Engineering Computing H V BTopping Ed chapter 10 pp 237ndash284 Saxe-Coburg 2003

[16] M P Saka ldquoOptimum design of steel frames using stochasticsearch techniques based on natural phenoma a reviewrdquo inCivil Engineering Computations Tools and Techniques B H VTopping Ed chapter 6 pp 105ndash147 Saxe-Coburg 2007

[17] J S Arora and Q Wang ldquoReview of formulations for structuraland mechanical system optimizationrdquo Structural and Multidis-ciplinary Optimization vol 30 no 4 pp 251ndash272 2005

[18] J S Arora ldquoMethods for discrete variable structural optimiza-tionrdquo in Recent Advances in Optimum Structural Design S ABurns Ed pp 1ndash40 ASCE USA 2002

[19] R E Griffith and R A Stewart ldquoA non-linear programmingtechnique for the optimization of continuous processing sys-temsrdquoManagement Science vol 7 no 4 pp 379ndash392 1961

[20] M P Saka ldquoThe use of the approximate programming inthe optimum structural designrdquo in Proceedings of InternationalConference on Mathematics Black Sea Technical UniversityTrabzon Turkey September 1982

[21] K F Reinschmidt C A Cornell and J F Brotchie ldquoIterativedesign and structural optimizationrdquo Journal of Structural Divi-sion vol 92 pp 319ndash340 1966

[22] G Pope ldquoApplication of linear programming technique in thedesign of optimum structuresrdquo in Proceedings of the AGARSymposium Structural Optimization Istanbul Turkey October1969

[23] D Johnson and D M Brotton ldquoOptimum elastic design ofredundant trussesrdquo Journal of the Structural Division vol 95no 12 pp 2589ndash2610 1969

[24] J S Arora and E J Haug ldquoEfficient optimal design of structuresby generalized steepest descent programmingrdquo InternationalJournal for Numerical Methods in Engineering vol 10 no 4 pp747ndash766 1976

[25] P E Gill W Murray andM HWright Practical OptimizationAcademic Press 1981

[26] J Nocedal and J S Wright Numerical optimization SpringerNew York NY USA 1999

[27] S P Han ldquoA globally convergent method for nonlinear pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 22 no 3 pp 297ndash309 1977

[28] M-W Huang and J S Arora ldquoA self-scaling implicit SQPmethod for large scale structural optimizationrdquo InternationalJournal for Numerical Methods in Engineering vol 39 no 11 pp1933ndash1953 1996

[29] QWang and J S Arora ldquoAlternative formulations for structuraloptimization an evaluation using framesrdquo Journal of StructuralEngineering vol 132 no 12 Article ID 008612QST pp 1880ndash1889 2006

[30] Q Wang and J S Arora ldquoOptimization of large-scale trussstructures using sparse SAND formulationsrdquo International Jour-nal for NumericalMethods in Engineering vol 69 no 2 pp 390ndash407 2007

[31] C W Carroll ldquoThe crated response surface technique foroptimizing nonlinear restrained systemsrdquo Operations Researchvol 9 no 2 pp 169ndash184 1961

[32] A V Fiacco and G P McCormack Nonlinear ProgrammingSequential Unconstrained Minimization Techniques JohnWileyamp Sons New York NY USA 1968

[33] D Kavlie and J Moe ldquoAutomated design of framed structuresrdquoJournal of Structural Division vol 97 no 1 33 pages 62

[34] D Kavlie and J Moe ldquoApplication of non-linear programmingto optimum grillage design with non-convex sets of variablesrdquoInternational Journal for Numerical Methods in Engineering vol1 no 4 pp 351ndash378 1969

[35] K M Griswold and J Moe ldquoA method for non-linear mixedinteger programming and its application to design problemsrdquoJournal of Engineering for Industry vol 94 no 2 12 pages 1972

[36] B M E de Silva and G N C Grant ldquoComparison of somepenalty function based optimization procedures for synthesisof a planar trussrdquo International Journal for Numerical Methodsin Engineering vol 7 no 2 pp 155ndash173 1973

[37] J B Rosen ldquoThe gradient-projection method for nonlinearprogramming-Part IIrdquo Journal of the Society for Industrial andApplied Mathematics vol 9 pp 514ndash532 1961

[38] E J Haug and J S Arora Applied Optimal Design John WileyNew York NY USA 1979

[39] G ZoutindijkMethods of Feasible Directions Elsevier ScienceAmsterdam The Netherlands 1960

[40] GN Vanderplaats ldquoCONMIN a fortran program for con-strained functionminimizationrdquo Tech RepNASATNX-622821973

[41] G N Vanderplaats Numerical Optimization Techniques forEngineering Design McGraw-Hill New York NY USA 1984


[42] C Zener Engineering Design by Geometric Programming John-Wiley London UK 1971

[43] A J Morris ldquoStructural optimization by geometric program-mingrdquo in Foundation in Structural Optimization A UnifiedApproach A J Morris Ed pp 573ndash610 John Wiley amp SonsChichester UK 1982

[44] RE Bellman Dynamic Programming Princeton UniversityPress Princeton NJ USA 1957

[45] AC Palmer ldquoOptimum structural design by dynamic pro-grammingrdquo Journal of the Structural Division vol 94 pp 1887ndash1906 1968

[46] D J Sheppard and A C Palmer ldquoOptimal design of trans-mission towers by dynamic programmingrdquo Computers andStructures vol 2 no 4 pp 455ndash468 1972

[47] D M Himmelblau Applied Nonlinear Programming McGraw-Hill New York NY USA 1972

[48] E D Eason and R G Fenton ldquoComparison of numericaloptimization methods for engineering designrdquo Journal of Engi-neering for Industry Series B vol 96 no 1 pp 196ndash200 1974

[49] W Prager ldquoOptimality criteria in structural designrdquo Proceedingsof National Academy for Science vol 61 no 3 pp 794ndash796 1968

[50] V BVenkayyaN S Khot andV S Reddy ldquoEnergy distributionin optimum structural designrdquo Tech Rep AFFDL-TM-68-156Flight Dynamics laboratoryWright Patterson AFBOhio USA1968

[51] V B Venkayya N S Khot and L Berke ldquoApplication ofoptimality criteria approaches to automated design of largepractical structuresrdquo in Proceedings of the 2nd Symposium onStructural Optimization AGARD-CP-123 pp 3-1ndash3-19 1973

[52] V B Venkayya ldquoOptimality criteria a basis for multidisci-plinary design optimizationrdquo Computational Mechanics vol 5no 1 pp 1ndash21 1989

[53] L Berke ldquoAn efficient approach to the minimum weight designof deflection limited structuresrdquo Tech Rep AFFDL-TM-70-4-FDTR 1970

[54] L Berke and NS Khot ldquoStructural optimization using opti-mality criteriardquo in Computer Aided Structural Design C A MSoares Ed Springer 1987

[55] N S Khot L Berke and V B Venkayya ldquocomparison ofoptimality criteria algorithms for minimum weight design ofstructuresrdquo AIAA Journal vol 17 no 2 pp 182ndash190 1979

[56] N S Khot ldquoalgorithms based on optimality criteria to designminimum weight structuresrdquo Engineering Optimization vol 5no 2 pp 73ndash90 1981

[57] N S Khot ldquoOptimal design of a structure for system stabilityfor a specified eigenvalue distributionrdquo in New Directions inOptimum Structural Design E Atrek R H Gallagher K MRagsdell and O C Zienkiewicz Eds pp 75ndash87 John Wiley ampSons New York NY USA 1984

[58] KM Ragsdell ldquoUtility of nonlinear programming methods forengineering designrdquo in New Direction in Optimality Criteria inStructural Design E Atrek et al Ed Chapter 18 John Wiley ampSons 1984

[59] C Feury and M Geradin ldquoOptimality criteria and mathemati-cal programming in structural weight optimizationrdquo Journal ofComputers and Structures vol 8 no 1 pp 7ndash17 1978

[60] C Fleury ldquoAn efficient optimality criteria approach to the min-imum weight design of elastic structuresrdquo Journal of Computersand Structures vol 11 no 3 pp 163ndash173 1980

[61] M P Saka ldquoOptimum design of space trusses with bucklingconstraintsrdquo in Proceedings of 3rd International Conference

on Space Structures University of Surrey Guildford UKSeptember 1984

[62] E I Tabak and P M Wright ldquoOptimality criteria method forbuilding framesrdquo Journal of Structural Division vol 107 no 7pp 1327ndash1342 1981

[63] F Y Cheng and K Z Truman ldquoOptimization algorithm of 3-Dbuilding systems for static and seismic loadingrdquo inModeling andSimulation in Engineering W F Ames Ed pp 315ndash326 North-Holland Amsterdam The Netherlands 1983

[64] M R Khan ldquoOptimality criterion techniques applied to frameshaving general cross-sectional relationshipsrdquoAIAA Journal vol22 no 5 pp 669ndash676 1984

[65] E A Sadek ldquoOptimization of structures having general cross-sectional relationships using an optimality criterion methodrdquoComputers and Structures vol 43 no 5 pp 959ndash969 1992

[66] M P Saka ldquoOptimum design of pin-jointed steel structureswith practical applicationsrdquo Journal of Structural Engineeringvol 116 no 10 pp 2599ndash2620 1990

[67] C G Salmon J E Johnson and F A Malhas Steel StructuresDesign and Behavior Prentice Hall New York NY USA 5thedition 2009

[68] D E Grieson and G E Cameron SODA-Structural Optimiza-tion Design and Analysis Release 3 1 User manual WaterlooEngineering Software Waterloo Canada 1990

[69] M P Saka ldquoOptimum design of multistorey structures withshear wallsrdquo Computers and Structures vol 44 no 4 pp 925ndash936 1992

[70] M P Saka ldquoOptimum geometry design of roof trusses byoptimality criteria methodrdquo Computers and Structures vol 38no 1 pp 83ndash92 1991

[71] M P Saka ldquoOptimum design of steel frames with taperedmembersrdquo Computers and Structures vol 63 no 4 pp 797ndash8111997

[72] AISC Manual Committee ldquoLoad and resistance factor designrdquoinManual of Steel Construction AISC USA 1999

[73] S S Al-Mosawi andM P Saka ldquoOptimum design of single coreshear wallsrdquo Computers and Structures vol 71 no 2 pp 143ndash162 1999

[74] S Al-Mosawi and M P Saka ldquoOptimum shape design of cold-formed thin-walled steel sectionsrdquo Advances in EngineeringSoftware vol 31 no 11 pp 851ndash862 2000

[75] V Z Vlasov Thin-Walled Elastic Beams National ScienceFoundation Wash USA 1961

[76] C-M Chan and G E Grierson ldquoAn efficient resizing techniquefor the design of tall steel buildings subject to multiple driftconstraintsrdquo inThe Structural Design of Tall Buildings vol 2 no1 pp 17ndash32 John Wiley amp Sons West Sussex UK 1993

[77] C-M Chan ldquoHow to optimize tall steel building frameworksrdquoinGuideTo StructuralOptimization ASCEManuals andReportson Engineering Practice J S Arora Ed no 90 Chapter 9 pp165ndash196 ASCE New York NY USA 1997

[78] R Soegiarso andH Adeli ldquoOptimum load and resistance factordesign of steel space-frame structuresrdquo Journal of StructuralEngineering vol 123 no 2 pp 184ndash192 1997

[79] N S Khot ldquoNonlinear analysis of optimized structure withconstraints on system stabilityrdquo AIAA Journal vol 21 no 8 pp1181ndash1186 1983

[80] N S Khot and M P Kamat ldquoMinimum weight design of trussstructures with geometric nonlinear behaviorrdquo AIAA Journalvol 23 no 1 pp 139ndash144 1985


[81] M P Saka ldquoOptimum design of nonlinear space trussesrdquoComputers and Structures vol 30 no 3 pp 545ndash551 1988

[82] M P Saka and M Ulker ldquoOptimum design of geometricallynonlinear space trussesrdquo Computers and Structures vol 42 no3 pp 289ndash299 1992

[83] M P Saka and M S Hayalioglu ldquoOptimum design of geo-metrically nonlinear elastic-plastic steel framesrdquoComputers andStructures vol 38 no 3 pp 329ndash344 1991

[84] M S Hayalioglu and M P Saka ldquoOptimum design of geo-metrically nonlinear elastic-plastic steel frames with taperedmembersrdquoComputers and Structures vol 44 no 4 pp 915ndash9241992

[85] M P Saka and E S Kameshki ldquoOptimum design of unbracedrigid framesrdquo Computers and Structures vol 69 no 4 pp 433ndash442 1998

[86] M P Saka and E S Kameshki ldquoOptimum design of nonlinearelastic framed domesrdquo Advances in Engineering Software vol29 no 7-9 pp 519ndash528 1998

[87] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan press Ann Arbor Minn USA 1975

[88] S Kirkpatrick C D Gerlatt and M P Vecchi ldquoOptimizationby simulated annealingrdquo Science vol 220 pp 671ndash680 1983

[89] D E Goldberg and M P Santani ldquoEngineering optimizationvia generic algorithmrdquo in Proceedings of 9th Conference onElectronic Computation pp 471ndash482 ASCE New York NYUSA 1986

[90] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison Wesley 1989

[91] R Paton Computing With Biological Metaphors Chapman ampHall New York NY USA 1994

[92] RHorst andPM Pardolos EdsHandbook ofGlobalOptimiza-tion Kluwer Academic Publishers New York NY USA 1995

[93] R Horst and H Tuy Global Optimization DeterministicApproaches Springer 1995

[94] C Adami An Introduction to Artificial Life Springer-Telos1998

[95] C Matheck Design in Nature Learning from Trees SpringerBerlin Germany 1998

[96] M Mitchell An Introduction to Genetic Algorithms The MITPress Boston Mass USA 1998

[97] G W Flake The Computational Beauty of Nature MIT PressBoston Mass USA 2000

[98] L Chambers The Practical Handbook of Genetic AlgorithmsApplications Chapman amp Hall 2nd edition 2001

[99] J Kennedy R Eberhart and Y Shi Swarm Intelligence MorganKaufmann Boston Mass USA 2001

[100] G A Kochenberger and F Glover Handbook of Meta-Heuristics Kluwer Academic Publishers New York NY USA2003

[101] L N De Castro and F J Von Zuben Recent Developments inBiologically Inspired Computing Idea Group Publishing USA2005

[102] J Dreo A Petrowski P Siarry and E TaillardMeta-Heuristicsfor Hard Optimization Springer Berlin Germany 2006

[103] T F Gonzales Handbook of Approximation Algorithms andMetaheuristics Chapman amp HallsCRC 2007

[104] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2008

[105] X -S Yang Engineering Optimization An Introduction WithMetaheuristic Applications John Wiley New York NY USA2010

[106] S Luke ldquoEssentials of Metaheuristicsrdquo 2010 httpcsgmuedusimseanbookmetaheuristics

[107] P Lu S Chen and Y Zheng ldquoArtificial intelligence in civilengineeringrdquo Mathematical Problems in Engineering vol 2012Article ID 145974 22 pages 2012

[108] R Kicinger T Arciszewski and K De Jong ldquoEvolutionarycomputation and structural design a survey of the state-of-the-artrdquoComputers and Structures vol 83 no 23-24 pp 1943ndash19782005

[109] I Rechenberg ldquoCybernetic solution path of an experimentalproblemrdquo in Royal Aircraft Establishment no 1122 LibraryTranslation Farnborough UK 1965

[110] I Rechenberg Evolutionsstrategie optimierung technischer sys-teme nach prinzipen der biologischen evolution Frommann-Holzboog Stuttgart Germany 1973

[111] H -P Schwefel ldquoKybernetische evolution als strategie derexperimentellen forschung in der stromungstechnikrdquo in Diplo-marbeit Technishe Universitat Berlin Germany 1965


[113] M Gen and R Chang Genetic Algorithm and EngineeringDesign John Wiley amp Sons New York NY USA 2000


[115] S Rajev and C S Krishnamoorthy ldquoDiscrete optimizationof structures using genetic algorithmsrdquo Journal of StructuralEngineering vol 118 no 5 pp 1233ndash1250 1992

[116] F Herrera M Lozano and J L Verdegay ldquoTackling real-coded genetic algorithms operators and tools for behaviouralanalysisrdquo Artificial Intelligence Review vol 12 no 4 pp 265ndash319 1998

[117] J P B Leite and B H V Topping ldquoImproved genetic operatorsfor structural engineering optimizationrdquo Advances in Engineer-ing Software vol 29 no 7ndash9 pp 529ndash562 1998

[118] WM Jenkins ldquoTowards structural optimization via the geneticalgorithmrdquo Computers and Structures vol 40 no 5 pp 1321ndash1327 1991

[119] W M Jenkins ldquoStructural optimization via the genetic algo-rithmrdquoTheStructural Engineer vol 69 no 24 pp 418ndash422 1991

[120] P Hajela ldquoStochastic search in structural optimization geneticalgorithms and simulated annealingrdquo in Structural Optimiza-tion Status and Promise chapter 22 pp 611ndash635 AmericanInstitute of Aeronautics and Astronautics 1992

[121] H Adeli and N T Cheng ldquoAugmented lagrange geneticalgorithm for structural optimizationrdquo Journal of StructuralEngineering vol 7 no 3 pp 104ndash118 1994

[122] H Adeli and N T Cheng ldquoConcurrent genetic algorithmsfor optimization of large structuresrdquo Journal of AerospaceEngineering vol 7 no 3 pp 276ndash296 1994

[123] S D Rajan ldquoSizing shape and topology design optimization oftrusses using genetic algorithmrdquo Journal of Structural Engineer-ing vol 121 no 10 pp 1480ndash1487 1995

[124] C V Camp S Pezeshk and G Cao ldquoDesign of framedstructures using a genetic algorithmrdquo in Advances in StructuralOptimization D M Frangopol and F Y Cheng Eds pp 19ndash30ASCE NY USA 1997


[125] C Camp S Pezeshk and G Cao ldquoOptimized design of two-dimensional structures using a genetic algorithmrdquo Journal ofStructural Engineering vol 124 no 5 pp 551ndash559 1998

[126] M P Saka and E Kameshki ldquoOptimum design of multi-storeysway steel frames to BS5950 using a genetic algorithmrdquo inAdvances in Engineering Computational Technology B H VTopping Ed pp 135ndash141 Civil-Comp Press Edinburgh UK1998

[127] M P Saka ldquoOptimum design of grillage systems using geneticalgorithmsrdquo Computer-Aided Civil and Infrastructure Engineer-ing vol 13 no 4 pp 297ndash302 1998

[128] S H Weldali and M P Saka ldquoOptimum geometry and spacingdesign of roof trusses based onBS5990 using genetic algorithmrdquoDesign Optimization vol 1 no 2 pp 198ndash219 2000

[129] M P Saka A Daloglu and F Malhas ldquoOptimum spacingdesign of grillage systems using a genetic algorithmrdquo Advancesin Engineering Software vol 31 no 11 pp 863ndash873 2000

[130] S Pezeshk C V Camp and D Chen ldquoDesign of nonlinearframed structures using genetic optimizationrdquo Journal of Struc-tural Engineering vol 126 no 3 pp 387ndash388 2000

[131] O Hasancebi and F Erbatur ldquoEvaluation of crossover tech-niques in genetic algorithm based optimum structural designrdquoComputers and Structures vol 78 no 1 pp 435ndash448 2000

[132] F Erbatur O Hasancebi I Tutuncu and H Kılıc ldquoOptimaldesign of planar and space structures with genetic algorithmsrdquoComputers and Structures vol 75 pp 209ndash224 2000

[133] E S Kameshki and M P Saka ldquoGenetic algorithm basedoptimum bracing design of non-swaying tall plane framesrdquoJournal of Constructional Steel Research vol 57 no 10 pp 1081ndash1097 2001

[134] E S Kameshki and M P Saka ldquoOptimum design of nonlinearsteel frames with semi-rigid connections using a genetic algo-rithmrdquo Computers and Structures vol 79 no 17 pp 1593ndash16042001

[135] J Andre P Siarry and T Dognon ldquoAn improvement of thestandard genetic algorithm fighting premature convergence incontinuous optimizationrdquo Advances in Engineering Softwarevol 32 no 1 pp 49ndash60 2001

[136] V V Toropov and S Y Mahfouz ldquoDesign optimization ofstructural steelwork using a genetic algorithm FEM and asystem of design rulesrdquo Engineering Computations vol 18 no3-4 pp 437ndash459 2001

[137] M S Hayalioglu ldquoOptimum load and resistance factor designof steel space frames using genetic algorithmrdquo Structural andMultidisciplinary Optimization vol 21 no 4 pp 292ndash299 2001

[138] W M Jenkins ldquoA decimal-coded evolutionary algorithm forconstrained optimizationrdquo Computers and Structures vol 80no 5-6 pp 471ndash480 2002

[139] E S Kameshki and M P Saka ldquoGenetic algorithm based opti-mumdesign of nonlinear planar steel frames with various semi-rigid connectionsrdquo Journal of Constructional Steel Research vol59 no 1 pp 109ndash134 2003

[140] YM Yun and B H Kim ldquoOptimum design of plane steel framestructures using second-order inelastic analysis and a geneticalgorithmrdquo Journal of Structural Engineering vol 131 no 12 pp1820ndash1831 2005

[141] A Kaveh and M Shahrouzi ldquoSimultaneous topology and sizeoptimization of structures by genetic algorithm using minimallength chromosomerdquo Engineering Computations vol 23 no 6pp 644ndash674 2006

[142] R J Balling R R Briggs and K Gillman ldquoMultiple optimumsizeshapetopology designs for skeletal structures using agenetic algorithmrdquo Journal of Structural Engineering vol 132no 7 Article ID 015607QST pp 1158ndash1165 2006

[143] V Togan and A T Daloglu ldquoOptimization of 3D trusseswith adaptive approach in genetic algorithmsrdquo EngineeringStructures vol 28 pp 1019ndash1027 2006

[144] E S Kameshki and M P Saka ldquoOptimum geometry design ofnonlinear braced domes using genetic algorithmrdquo Computersand Structures vol 85 no 1-2 pp 71ndash79 2007

[145] S O Degertekin ldquoA comparison of simulated annealing andgenetic algorithm for optimum design of nonlinear steel spaceframesrdquo Structural and Multidisciplinary Optimization vol 34no 4 pp 347ndash359 2007

[146] S O Degertekin M P Saka and M S Hayalioglu ldquoOptimalload and resistance factor design of geometrically nonlinearsteel space frames via tabu search and genetic algorithmrdquoEngineering Structures vol 30 no 1 pp 197ndash205 2008

[147] H K Issa and F A Mohammad ldquoEffect of mutation schemeson convergence to optimum design of steel framesrdquo Journal ofConstructional Steel Research vol 66 no 7 pp 954ndash961 2010

[148] D Safari M R Maheri and A Maheri ldquoOptimum design ofsteel frames using a multiple-deme GA with improved repro-duction operatorsrdquo Journal of Constructional Steel Research vol67 no 8 pp 1232ndash1243 2011

[149] N D Lagaros M Papadrakakis and G Kokossalakis ldquoStruc-tural optimization using evolutionary algorithmsrdquo Computersand Structures vol 80 no 7-8 pp 571ndash589 2002

[150] J Cai and G Thierauf ldquoDiscrete structural optimization usingevolution strategiesrdquo in Neural Networks and CombinatorialOptimization in Civil and Structural Engineering B H V Top-ping and A I Khan Eds pp 95ndash100 Civil-Comp EdinburghUK 1993

[151] C A C Coello ldquoTheoretical and numerical constraint-handling techniques used with evolutionary algorithms asurvey of the state of the artrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 11-12 pp 1245ndash12872002

[152] T Back and M Schutz ldquoEvolutionary strategies for mixed-integer optimization of optical multilayer systemsrdquo in Proceed-ings of the 4th Annual Conference on Evolutionary ProgrammingR McDonnel R G Reynolds and D B Fogel Eds pp 33ndash51MIT Press Cambridge Mass USA

[153] S Rajasekaran V S Mohan and O Khamis ldquoThe optimisationof space structures using evolution strategies with functionalnetworksrdquo Engineering with Computers vol 20 no 1 pp 75ndash872004

[154] S Rajasekaran ldquoOptimal laminate sequence of non-prismaticthin-walled composite spatial members of generic sectionrdquoComposite Structures vol 70 no 2 pp 200ndash211 2005

[155] G Ebenau J Rottschafer and G Thierauf ldquoAn advancedevolutionary strategy with an adaptive penalty function formixed-discrete structural optimisationrdquo Advances in Engineer-ing Software vol 36 no 1 pp 29ndash38 2005

[156] O Hasancebi ldquoDiscrete approaches in evolution strategiesbased optimum design of steel framesrdquo Structural Engineeringand Mechanics vol 26 no 2 pp 191ndash210 2007

[157] O Hasancebi ldquoOptimization of truss bridges within a specifieddesign domain using evolution strategiesrdquo Engineering Opti-mization vol 39 no 6 pp 737ndash756 2007


[158] O Hasancebi ldquoAdaptive evolution strategies in structural opti-mization Enhancing their computational performance withapplications to large-scale structuresrdquo Computers and Struc-tures vol 86 no 1-2 pp 119ndash132 2008

[159] V Cerny ldquoThermodynamical approach to the traveling sales-man problem An efficient simulation algorithmrdquo Journal ofOptimization Theory and Applications vol 45 no 1 pp 41ndash511985

[160] W A Bennage and A K Dhingra ldquoSingle and multiobjectivestructural optimization in discrete-continuous variables usingsimulated annealingrdquo International Journal for NumericalMeth-ods in Engineering vol 38 no 16 pp 2753ndash2773 1995

[161] N Metropolis A W Rosenbluth M N Rosenbluth A HTeller and E Teller ldquoEquation of state calculations by fastcomputing machinesrdquo The Journal of Chemical Physics vol 21no 6 pp 1087ndash1092 1953

[162] R J Balling ldquoOptimal steel frame design by simulated anneal-ingrdquo Journal of Structural Engineering vol 117 no 6 pp 1780ndash1795 1991

[163] S A May and R J Balling ldquoA filtered simulated annealingstrategy for discrete optimization of 3D steel frameworksrdquoStructural Optimization vol 4 no 3-4 pp 142ndash148 1992

[164] R K Kincaid ldquoMinimizing distortion and internal forcesin truss structures by simulated annealingrdquo in Proceedingsof 31st AIAAASMEASCEAHSASC Structural Materials andDynamics Conference pp 327ndash333 Long Beach Calif USAApril 1990

[165] R K Kincaid ldquoMinimizing distortion and internal forces intruss structures by simulated annealingrdquo in Proceedings of32nd AIAAASMEASCEAHSASC Structural Materials andDynamics Conference pp 327ndash333 Baltimore Md USA April1990

[166] G S Chen R J Bruno and M Salama ldquoOptimal placementof activepassive members in truss structures using simulatedannealingrdquo AIAA Journal vol 29 no 8 pp 1327ndash1334 1991

[167] B H V Topping A I Khan and J P B Leite ldquoTopologicaldesign of truss structures using simulated annealingrdquo inNeuralNetworks and Combinatorial Optimization in Civil and Struc-tural Engineering B H V Topping and A I Khan Eds pp151ndash165 Civil-Comp Press UK 1993

[168] J P B Leite and B H V Topping ldquoParallel simulated annealingfor structural optimizationrdquo Computers and Structures vol 73no 1-5 pp 545ndash564 1999

[169] S R Tzan and C P Pantelides ldquoAnnealing strategy for optimalstructural designrdquo Journal of Structural Engineering vol 122 no7 pp 815ndash827 1996

[170] O Hasancebi and F Erbatur ldquoOn efficient use of simulatedannealing in complex structural optimization problemsrdquo ActaMechanica vol 157 no 1ndash4 pp 27ndash50 2002

[171] O Hasancebi and F Erbatur ldquoLayout optimisation of trussesusing simulated annealingrdquo Advances in Engineering Softwarevol 33 no 7ndash10 pp 681ndash696 2002

[172] S O Degertekin M S Hayalioglu and M Ulker ldquoA hybridtabu-simulated annealing heuristic algorithm for optimumdesign of steel framesrdquo Steel and Composite Structures vol 8no 6 pp 475ndash490 2008

[173] O Hasancebi S Carbas and M P Saka ldquoImproving the per-formance of simulated annealing in structural optimizationrdquoStructural and Multidisciplinary Optimization vol 41 no 2 pp189ndash203 2010

[174] O Hasancebi and E Dogan ldquoEvaluation of topological formsfor weight-effective optimum design of single-span steel trussbridgesrdquo Asian Journal of Civil Engineering vol 12 no 4 pp431ndash448 2011

[175] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[176] J Kennedy and R Eberhart Swarm Intellegence Morgan Kauf-man 2001

[177] G Venter and J Sobieszczanski-Sobieski ldquoMultidisciplinaryoptimization of a transport aircraft wing using particle swarmoptimizationrdquo Structural and Multidisciplinary Optimizationvol 26 no 1-2 pp 121ndash131 2004

[178] P C Fourie and A A Groenwold ldquoThe particle swarm opti-mization algorithm in size and shape optimizationrdquo Structuraland Multidisciplinary Optimization vol 23 no 4 pp 259ndash2672002

[179] R E Perez and K Behdinan ldquoParticle swarm approach forstructural design optimizationrdquo Computers and Structures vol85 no 19-20 pp 1579ndash1588 2007

[180] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[181] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[182] L J Li Z B Huang F Liu and Q H Wu ldquoA heuristic particleswarm optimizer for optimization of pin connected structuresrdquoComputers and Structures vol 85 no 7-8 pp 340ndash349 2007

[183] E Dogan and M P Saka ldquoOptimum design of unbraced steelframes to LRFD-AISC using particle swarm optimizationrdquoAdvances in Engineering Software vol 46 pp 27ndash34 2012

[184] A ColorniMDorigo andVManiezzo ldquoDistributed optimiza-tion by ant colonyrdquo in Proceedings of 1st European Conference onArtificial Life pp 134ndash142 USA 1991

[185] M Dorigo Optimization learning and natural algorithms[PhD thesis] Dipartimento Elettronica e InformazionePolitecnico di Milano Italy 1992

[186] M Dorigo and T Stutzle Ant Colony Optimization A BradfordBook MIT USA 2004

[187] C V Camp and B J Bichon ldquoDesign of space trusses using antcolony optimizationrdquo Journal of Structural Engineering vol 130no 5 pp 741ndash751 2004

[188] 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

[189] A Kaveh and S Shojaee ldquoOptimal design of skeletal structuresusing ant colony optimizationrdquo International Journal forNumer-ical Methods in Engineering vol 70 no 5 pp 563ndash581 2007

[190] J A Bland ldquoAutomatic optimal design of structures usingswarm intelligencerdquo in Proceedings of the Annual Conferenceof the Canadian Society for Civil Engineering pp 1945ndash1954Canada June 2008

[191] 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

[192] W Wang S Guo and W Yang ldquoSimultaneous partial topologyand size optimization of a wing structure using ant colony andgradient based methodsrdquo Engineering Optimization vol 43 no4 pp 433ndash446 2011


[193] I Aydogdu andM P Saka ldquoAnt colony optimization of irregularsteel frames including elemental warping effectrdquo Advances inEngineering Software vol 44 no 1 pp 150ndash169 2012

[194] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[195] ZW Geem J H Kim and G V Loganathan ldquoHarmony searchoptimization application to pipe network designrdquo InternationalJournal of Modelling and Simulation vol 22 no 2 pp 125ndash1332002

[196] K S Lee and Z W Geem ldquoA new structural optimizationmethod based on the harmony search algorithmrdquo Computersand Structures vol 82 no 9-10 pp 781ndash798 2004

[197] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36-38 pp 3902ndash3933 2005

[198] Z W Geem K S Lee and C L Tseng ldquoHarmony searchfor structural designrdquo in Proceedings of the Genetic and Evo-lutionary Computation Conference (GECCO rsquo05) pp 651ndash652Washington DC USA June 2005

[199] Z W Geem ldquoOptimal cost design of water distribution net-works using harmony searchrdquo Engineering Optimization vol38 no 3 pp 259ndash280 2006

[200] ZW Geem ldquoNovel derivative of harmony search algorithm fordiscrete design variablesrdquo Applied Mathematics and Computa-tion vol 199 no 1 pp 223ndash230 2008

[201] Z W Geem Ed Music-Inspired Harmony Search AlgorithmSpringer 2009

[202] Z W Geem ldquoParticle-swarm harmony search for water net-work designrdquo Engineering Optimization vol 41 no 4 pp 297ndash311 2009

[203] ZWGeem EdRecentAdvances inHarmony SearchAlgorithmSpringer 2010

[204] Z W Geem Ed Harmony Search Algorithms for StructuralDesign Optimization Springer 2010

[205] Z W Geem ldquoHarmony search algorithmrdquo 2011 httpwwwharmonysearchinfo

[206] Z W Geem and W E Roper ldquoVarious continuous harmonysearch algorithms for web-based hydrologic parameter opti-mizationrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 3 pp 231ndash226 2010

[207] M P Saka ldquoOptimumgeometry design of geodesic domes usingharmony search algorithmrdquoAdvances in Structural Engineeringvol 10 no 6 pp 595ndash606 2007

[208] S Carbas and M P Saka ldquoA harmony search algorithm foroptimum topology design of single layer lamella domesrdquo in Pro-ceedings of The 9th International Conference on ComputationalStructures Technology B H V Topping and M PapadrakakisEds no 50 Civil-Comp Press Scotland UK 2008

[209] S Carbas and M P Saka ldquoOptimum design of single layernetwork domes using harmony search methodrdquo Asian Journalof Civil Engineering vol 10 no 1 pp 97ndash112 2009

[210] S Carbas and M P Saka ldquoOptimum topology design ofvarious geometrically nonlinear latticed domes using improvedharmony searchmethodrdquo Structural andMultidisciplinaryOpti-mization vol 45 no 3 pp 377ndash399 2011

[211] M P Saka and F Erdal ldquoHarmony search based algorithmfor the optimum design of grillage systems to LRFD-AISCrdquoStructural and Multidisciplinary Optimization vol 38 no 1 pp25ndash41 2009

[212] M P Saka ldquoOptimum design of steel sway frames to BS5950using harmony search algorithmrdquo Journal of ConstructionalSteel Research vol 65 no 1 pp 36ndash43 2009

[213] S O Degertekin ldquoOptimumdesign of steel frames via harmonysearch algorithmrdquo Studies in Computational Intelligence vol239 pp 51ndash78 2009

[214] S O Degertekin M S Hayalioglu and H Gorgun ldquoOptimumdesign of geometrically non-linear steel frames with semi-rigid connections using a harmony search algorithmrdquo Steel andComposite Structures vol 9 no 6 pp 535ndash555 2009

[215] M P Saka and O Hasancebi ldquoAdaptive harmony searchalgorithm for design code optimization of steel structuresrdquo inHarmony Search Algorithms for Structural Design OptimizationZWGeem Ed chapter 3 SCI 239 pp 79ndash120 Springer BerlinGermany 2009

[216] O Hasancebi F Erdal and M P Saka ldquoAn adaptive harmonysearchmethod for structural optimizationrdquo Journal of StructuralEngineering vol 136 no 4 pp 419ndash431 2010

[217] F Erdal E Doan and M P Saka ldquoOptimum design of cellularbeams using harmony search and particle swarm optimizersrdquoJournal of Constructional Steel Research vol 67 no 2 pp 237ndash247 2011

[218] M P Saka I Aydogdu O Hasancebi and Z W GeemldquoHarmony search algorithms in structural engineeringrdquo inComputational Optimization and Applications in Engineeringand Industry X-S Yang and S Koziel Eds Chapter 6 SpringerBerlin Germany 2011

[219] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

[220] M G H Omran and M Mahdavi ldquoGlobal-best harmonysearchrdquo Applied Mathematics and Computation vol 198 no 2pp 643ndash656 2008

[221] L D Santos Coelho and D L de Andrade Bernert ldquoAnimproved harmony search algorithm for synchronization ofdiscrete-time chaotic systemsrdquoChaos Solitons and Fractals vol41 no 5 pp 2526ndash2532 2009

[222] O K Erol and I Eksin ldquoA new optimizationmethod Big Bang-Big CrunchrdquoAdvances in Engineering Software vol 37 no 2 pp106ndash111 2006

[223] C V Camp ldquoDesign of space trusses using big bang-big crunchoptimizationrdquo Journal of Structural Engineering vol 133 no 7pp 999ndash1008 2007

[224] 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

[225] A Kaveh and S Talatahari ldquoOptimal design of Schwedler andribbed domes via hybrid Big Bang-Big Crunch algorithmrdquoJournal of Constructional Steel Research vol 66 no 3 pp 412ndash419 2010

[226] A Kaveh and H Abbasgholiha ldquoOptimum design of steel swayframes using Big Bang-Big Crunch algorithmrdquo Asian Journal ofCivil Engineering vol 12 no 3 pp 293ndash317 2011

[227] A Kaveh and S Talatahari ldquoA discrete particle swarm antcolony optimization for design of steel framesrdquo Asian Journalof Civil Engineering vol 9 no 6 pp 563ndash575 2008

[228] A Kaveh and S Talatahari ldquoA particle swarm ant colony opti-mization for truss structures with discrete variablesrdquo Journal ofConstructional Steel Research vol 65 no 8-9 pp 1558ndash15682009


[229] A Kaveh and S Talatahari ldquoHybrid algorithm of harmonysearch particle swarm and ant colony for structural designoptimizationrdquo Studies in Computational Intelligence vol 239pp 159ndash198 2009

[230] 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

[231] A Kaveh S Talatahari and B Farahmand Azar ldquoAn improvedhpsaco for engineering optimum design problemsrdquo Asian Jour-nal of Civil Engineering vol 12 no 2 pp 133ndash141 2010

[232] A Kaveh and SM Rad ldquoHybrid genetic algorithm and particleswarm optimization for the force method-based simultaneousanalysis and designrdquo Iranian Journal of Science and TechnologyTransaction B vol 34 no 1 pp 15ndash34 2010

[233] A Kaveh and S Talatahari ldquoOptimum design of skeletalstructures using imperialist competitive algorithmrdquo Computersand Structures vol 88 no 21-22 pp 1220ndash1229 2010

[234] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 pp267ndash289 2010

[235] 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

[236] A Kaveh and S Talatahari ldquoCharged system search for opti-mum grillage system design using the LRFD-AISC coderdquoJournal of Constructional Steel Research vol 66 no 6 pp 767ndash771 2010

[237] A Kaveh and S Talatahari ldquoA charged system search with afly to boundary method for discrete optimum design of trussstructuresrdquo Asian Journal of Civil Engineering vol 11 no 3 pp277ndash293 2010

[238] A Kaveh and S Talatahari ldquoGeometry and topology optimiza-tion of geodesic domes using charged system searchrdquo Structuraland Multidisciplinary Optimization vol 43 no 2 pp 215ndash2292011

[239] A Kaveh andA Zolghadr ldquoShape and size optimization of trussstructures with frequency constraints using enhanced chargedsystem search algorithmrdquoAsian Journal of Civil Engineering vol12 no 4 pp 487ndash509 2011

[240] A Kaveh and S Talatahari ldquoCharged system search for optimaldesign of frame structuresrdquo Applied Soft Computing vol 12 pp382ndash393 2012

[241] A Kaveh and T Bakhspoori ldquoOptimum design of steel framesusing cuckoo search algorithm with levy flightsrdquoThe StructuralDesign of Tall and Special Buildings In press

[242] X -S Yang and S Deb ldquoEngineering Optimization by CuckooSearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[243] A H Gandomi X-S Yang and A H Alavi ldquoCuckoo searchalgorithm a metaheuristic approach to solve structural opti-mization problemsrdquo Engineering with Computers In press

[244] O Hasancebi S Carbas E Dogan F Erdal and M P SakaldquoPerformance evaluation of metaheuristic search techniquesin the optimum design of real size pin jointed structuresrdquoComputers and Structures vol 87 no 5-6 pp 284ndash302 2009

[245] O Hasancebi S Carbas E Dogan F Erdal and M P SakaldquoComparison of non-deterministic search techniques in theoptimum design of real size steel framesrdquo Computers andStructures vol 88 no 17-18 pp 1033ndash1048 2010

[246] A Kaveh and S Talatahari ldquoOptimal design of single layerdomes using meta-heuristic algorithms a comparative studyrdquoInternational Journal of Space Structures vol 25 no 4 pp 217ndash227 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of


a complex task It requires tremendous computational timeto try all the possible combinations such that the strengthand displacement constraints imposed by the design codesare satisfied and the amount of steel which is required tobuild the frame is the minimum However the emergence ofcomputational methods of mathematical programming andimprovements that took place in computers in early 1960rsquoshas provided another approach for handling the structuraldesign problems In this new approach the design problem isformulated as a decision-making problemwhere an objectivefunction is to be minimized or maximized while number ofconstraint functions is satisfiedThe formulation of structuraldesign problems as decision-making problems has yieldeda new branch in structural engineering called structuraloptimization [1ndash12]

The mathematical model of a decision-making problemhas the following form

Minimize 119882 = 119891 (119909119894) 119894 = 1 119899 (1a)


(119909119894) = 0 119895 = 1 ne (1b)

119892119895

(119909119894) le 0 119895 = ne + 1 119898 (1c)

119909ℓ

119894le 119909

119894le 119909

119906

119894 (1d)

where 119909119894represents decision variable 119894 Decision variables

may take continuous or discrete values Continuous decisionvariable can take any real value within the range shown in(1d) The decision variables in structural optimization arecalled design variables which are the parameters that controlthe geometry or material properties of a steel frame Forexample the moment of inertias of beams and columns in aframe can be considered as continuous design variable if theseelements are to be manufactured locally for that particularframe However if they are to be selected from commerciallyavailable set of steel sections design variables can only takeisolated values which make them discrete variables

119891(119909119894) in (1a) represent the objective function which can

be used as a measure of effectiveness of the decision Instructural optimization problems generally minimumweightor cost of the structure is taken as objective functionThe equalities ℎ

119895(119909

119894) and inequalities 119892

119895(119909

119894) in (1b) and

(1c) respectively represent the limitations imposed on thebehavior of the structure by the design codes These may beserviceability limitations or ultimate strength requirementsif the design code considered is based on limit state designHowever if the design code is based on allowable stressdesign then the constraints given in (1b) and (1c) becomethe stiffness equalities displacement andor allowable stresslimitations

The review of the mathematical formulation of designoptimization of steel frames is carried out according tothe historical developments that took place in the designmethods Earlier steel design codes were based on allowablestress design concept while the later design codes use limitstate design perception

2 Formulation of Steel Frame DesignOptimization Problem

Survey on the structural optimization literature reveals thatthere are basically two types of design optimization formu-lation for the optimum design of elastic steel skeletal framedstructures although some variations of each type do exist Inthe first type of formulation which is also called conventionalformulation or coupled analysis and design (CAND) onlycross-sectional properties of steel frame members are takenas design variables Joint displacements are not treated asdesign variables In such formulation the stiffness equationsare excluded from the mathematical model Consequentlywhen the values of cross-sectional properties change duringdesign cycles it becomes necessary to analyze the structureand update the joint displacements and members forcesHence structural analysis is a complementary part of thedesign process In the second type of formulation jointdisplacements are also treated as design variables in addi-tion to the cross-sectional properties of frame membersIn such formulation the stiffness equations are included asdesign constraints in addition to other limitations in themathematicalmodelThis eliminates the necessity of carryingout structural analysis in every design cycle The values ofthe joint displacements are determined by the optimizationtechniques similar to those of cross-sectional properties It isapparent that in this type of formulation the total numberof design variables is much more than the first type offormulation This type of formulation is called simultaneousanalysis and design (SAND) It should be pointed out thatboth ways of formulation yield the same optimum resultHowever depending on the type of formulation adopted inthe design problem the size and mathematical complexity ofthe model acquired differ

21 Coupled Analysis and Design (CAND) In this type offormulation design phase is separated from the analysisphase This is achieved by only treating the cross-sectionalproperties such as areas or moment of inertias of membersas design variables The general outlook of the mathematicalmodel of this type of formulation is given in the following ifallowable stress design method is considered for the frame

Minimize 119882 = 120588

ng

sum

119894=1

ℓ119894119860119894 119894 = 1 ng (2a)

Subject to 120575119895

(119860119894) le 120575

119895119906 119895 = 1 rd (2b)

120590119896

(119860119894) le 120590all 119896 = 1 nm (2c)

119860119894119897

le 119860119894

le 119860119894119906

(2d)

where 120588 is the density of steel 119860119894represents cross-sectional

area for group and 119894 and ℓ119894is the total length of the

members in group 119894 in the steel frame 120575119895(119860

119894) is the restricted

119895th displacement 120575119895119906

is its upper bound and rd is thetotal number of such restricted displacements 120590

119896(119860

119894) is the

maximum stress in member 119896 and 120590all is the allowable stressfor steel nm is the total number of members in the structure


Table 1 119886119894and 119887


119894 119886119894

119887119894

1 13162 216502 05971 182203 10160 15710


119868119909

= 11988611198601198871 119868

119910= 119886

21198601198872 119885 = 119886

31198601198873 (3)




1 119886

2 119886




[119870]minus1




Minimize 119882 = 120588

ng

sum

119894=1

ℓ119894119860119894 119894 = 1 ng (4a)

Subject to 119870 (119860119894) 119863 = 119875 (4b)

120575119895

le 120575119895119906

119895 = 1 rd (4c)

120590119896

(119860119894 120575

119895) le 120590all 119896 = 1 nm (4d)

119860119894ℓ

le 119860119894

le 119860119894119906

(4e)




119894ℓand119860




119895119906is its


119894 120575

















Minimize 119882 =

119898

sum

119896=1

119908119896

(119909119894) 119894 = 1 119899 (5a)


(119909119894) = 0 119895 = 1 ne (5b)

119892119895

(119909119894) le 0 119895 = ne + 1 nc (5c)



119895(119909

119894) represents


119895(119909






=

119909119896

1119909119896


119896

119899 as

Min 119882 = 119908 (119909119896) + nabla119908 (119909

119896) (119909

119896+1minus 119909

119896)


(119909119896) + nablaℎ

119895(119909

119896) (119909

119896+1minus 119909

119896) = 0

119895 = 1 ne

119892119895

(119909119896) + nabla119892

119895(119909

119896) (119909

119896+1minus 119909

119896) le 0

119895 = 119899 + 1 nc

(1 minus 119898) 119909119896

le 119909119896minus1

le (1 + 119898) 119909119896

(6)


119896) nablaℎ

119895(119909

119896) and nabla119892

119895(119909





function 119908(119909119896)

nabla119908 (119909119896) = [

120597119908

120597119909119896

1

120597119908

120597119909119896

2

120597119908

120597119909119896

119899

] (7)


Min 119882 = 119862119909119896+1

minus 1198620

subject to 119860119909119896+1

(=le

) 119861

(8)



119862119900

= 119908 (119909119896) minus nabla119908 (119909

119896) 119909

119896

C = nabla119908 (119909119896)

[119860] =[[[

[

nablaℎ119895

(119909119896)

nabla119892119895

(119909119896)

II

]]]

]

[119861] =

[[[[[

[

nablaℎ119895

(119909119896) 119909

119896minus ℎ

119895(119909

119896)

nabla119892119895

(119909119896) 119909

119896minus 119892

119895(119909

119896)

(1 + 119898) 119909119896

(1 + 119898) 119909119896

]]]]]

]

(9)













Min 120601 (119909 119903) = 119882 (119909) + 119903

nesum

119895=1

ℎ2

119895(119909)

+1

119903

ncsum

119895 = ne+1min [0 119892

119895 (119909)]2

(10)

for 119903 = 1 01 001 0001 119903119894

rarr 0 119903119894




120601 (119909 119903) = 119882 (119909) + 119903

ncsum

119895 = 1

1

119892119895 (119909)

(11)


120601 (119909 119903) = 119882 (119909) minus 119903

ncsum

119895=1

log119890

[119892119895 (119909)]

119903 = 1 01 001 997888rarr 0 119903119894

gt 0

(12)








1199091

= 119909∘

+ 120572119878∘ (13)





Δ119892119895


119879nabla119891 lt o










Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982


le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)



119894



load case ℓ and 120575119895ℓ119906











Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)


120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)



119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)



120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)





120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)


119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)



120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)



119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 119886119894and 119887


119894 119886119894

119887119894

1 13162 216502 05971 182203 10160 15710


119868119909

= 11988611198601198871 119868

119910= 119886

21198601198872 119885 = 119886

31198601198873 (3)




1 119886

2 119886




[119870]minus1




Minimize 119882 = 120588

ng

sum

119894=1

ℓ119894119860119894 119894 = 1 ng (4a)

Subject to 119870 (119860119894) 119863 = 119875 (4b)

120575119895

le 120575119895119906

119895 = 1 rd (4c)

120590119896

(119860119894 120575

119895) le 120590all 119896 = 1 nm (4d)

119860119894ℓ

le 119860119894

le 119860119894119906

(4e)




119894ℓand119860




119895119906is its


119894 120575

















Minimize 119882 =

119898

sum

119896=1

119908119896

(119909119894) 119894 = 1 119899 (5a)


(119909119894) = 0 119895 = 1 ne (5b)

119892119895

(119909119894) le 0 119895 = ne + 1 nc (5c)



119895(119909

119894) represents


119895(119909






=

119909119896

1119909119896


119896

119899 as

Min 119882 = 119908 (119909119896) + nabla119908 (119909

119896) (119909

119896+1minus 119909

119896)


(119909119896) + nablaℎ

119895(119909

119896) (119909

119896+1minus 119909

119896) = 0

119895 = 1 ne

119892119895

(119909119896) + nabla119892

119895(119909

119896) (119909

119896+1minus 119909

119896) le 0

119895 = 119899 + 1 nc

(1 minus 119898) 119909119896

le 119909119896minus1

le (1 + 119898) 119909119896

(6)


119896) nablaℎ

119895(119909

119896) and nabla119892

119895(119909





function 119908(119909119896)

nabla119908 (119909119896) = [

120597119908

120597119909119896

1

120597119908

120597119909119896

2

120597119908

120597119909119896

119899

] (7)


Min 119882 = 119862119909119896+1

minus 1198620

subject to 119860119909119896+1

(=le

) 119861

(8)



119862119900

= 119908 (119909119896) minus nabla119908 (119909

119896) 119909

119896

C = nabla119908 (119909119896)

[119860] =[[[

[

nablaℎ119895

(119909119896)

nabla119892119895

(119909119896)

II

]]]

]

[119861] =

[[[[[

[

nablaℎ119895

(119909119896) 119909

119896minus ℎ

119895(119909

119896)

nabla119892119895

(119909119896) 119909

119896minus 119892

119895(119909

119896)

(1 + 119898) 119909119896

(1 + 119898) 119909119896

]]]]]

]

(9)













Min 120601 (119909 119903) = 119882 (119909) + 119903

nesum

119895=1

ℎ2

119895(119909)

+1

119903

ncsum

119895 = ne+1min [0 119892

119895 (119909)]2

(10)

for 119903 = 1 01 001 0001 119903119894

rarr 0 119903119894




120601 (119909 119903) = 119882 (119909) + 119903

ncsum

119895 = 1

1

119892119895 (119909)

(11)


120601 (119909 119903) = 119882 (119909) minus 119903

ncsum

119895=1

log119890

[119892119895 (119909)]

119903 = 1 01 001 997888rarr 0 119903119894

gt 0

(12)








1199091

= 119909∘

+ 120572119878∘ (13)





Δ119892119895


119879nabla119891 lt o










Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982


le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)



119894



load case ℓ and 120575119895ℓ119906











Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)


120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)



119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)



120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)





120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)


119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)



120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)



119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Minimize 119882 =

119898

sum

119896=1

119908119896

(119909119894) 119894 = 1 119899 (5a)


(119909119894) = 0 119895 = 1 ne (5b)

119892119895

(119909119894) le 0 119895 = ne + 1 nc (5c)



119895(119909

119894) represents


119895(119909






=

119909119896

1119909119896


119896

119899 as

Min 119882 = 119908 (119909119896) + nabla119908 (119909

119896) (119909

119896+1minus 119909

119896)


(119909119896) + nablaℎ

119895(119909

119896) (119909

119896+1minus 119909

119896) = 0

119895 = 1 ne

119892119895

(119909119896) + nabla119892

119895(119909

119896) (119909

119896+1minus 119909

119896) le 0

119895 = 119899 + 1 nc

(1 minus 119898) 119909119896

le 119909119896minus1

le (1 + 119898) 119909119896

(6)


119896) nablaℎ

119895(119909

119896) and nabla119892

119895(119909





function 119908(119909119896)

nabla119908 (119909119896) = [

120597119908

120597119909119896

1

120597119908

120597119909119896

2

120597119908

120597119909119896

119899

] (7)


Min 119882 = 119862119909119896+1

minus 1198620

subject to 119860119909119896+1

(=le

) 119861

(8)



119862119900

= 119908 (119909119896) minus nabla119908 (119909

119896) 119909

119896

C = nabla119908 (119909119896)

[119860] =[[[

[

nablaℎ119895

(119909119896)

nabla119892119895

(119909119896)

II

]]]

]

[119861] =

[[[[[

[

nablaℎ119895

(119909119896) 119909

119896minus ℎ

119895(119909

119896)

nabla119892119895

(119909119896) 119909

119896minus 119892

119895(119909

119896)

(1 + 119898) 119909119896

(1 + 119898) 119909119896

]]]]]

]

(9)













Min 120601 (119909 119903) = 119882 (119909) + 119903

nesum

119895=1

ℎ2

119895(119909)

+1

119903

ncsum

119895 = ne+1min [0 119892

119895 (119909)]2

(10)

for 119903 = 1 01 001 0001 119903119894

rarr 0 119903119894




120601 (119909 119903) = 119882 (119909) + 119903

ncsum

119895 = 1

1

119892119895 (119909)

(11)


120601 (119909 119903) = 119882 (119909) minus 119903

ncsum

119895=1

log119890

[119892119895 (119909)]

119903 = 1 01 001 997888rarr 0 119903119894

gt 0

(12)








1199091

= 119909∘

+ 120572119878∘ (13)





Δ119892119895


119879nabla119891 lt o










Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982


le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)



119894



load case ℓ and 120575119895ℓ119906











Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)


120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)



119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)



120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)





120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)


119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)



120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)



119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Min 120601 (119909 119903) = 119882 (119909) + 119903

nesum

119895=1

ℎ2

119895(119909)

+1

119903

ncsum

119895 = ne+1min [0 119892

119895 (119909)]2

(10)

for 119903 = 1 01 001 0001 119903119894

rarr 0 119903119894




120601 (119909 119903) = 119882 (119909) + 119903

ncsum

119895 = 1

1

119892119895 (119909)

(11)


120601 (119909 119903) = 119882 (119909) minus 119903

ncsum

119895=1

log119890

[119892119895 (119909)]

119903 = 1 01 001 997888rarr 0 119903119894

gt 0

(12)








1199091

= 119909∘

+ 120572119878∘ (13)





Δ119892119895


119879nabla119891 lt o










Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982


le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)



119894



load case ℓ and 120575119895ℓ119906











Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)


120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)



119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)



120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)





120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)


119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)



120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)



119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199091

= 119909∘

+ 120572119878∘ (13)





Δ119892119895


119879nabla119891 lt o










Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982


le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)



119894



load case ℓ and 120575119895ℓ119906











Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)


120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)



119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)



120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)





120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)


119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)



120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)



119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Min 119882 =

ng

sum

119896=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


le 120575119895ℓ119906

119895 = 1 119901

ℓ = 1 nℓc

119891anl119865anl

+1198981

1198982


le 1 119899 = 1 nm

119860119896

ge 119860119896ℓ

119896 = 1 ng

(14)



119894



load case ℓ and 120575119895ℓ119906











Min 119882 =

ng

sum

119870=1

119860119896

119898119896

sum

119894=1

120588119894ℓ119894


(119860119896) = 120575

119895ℓminus 120575

119895ℓ119906le 0 119895 = 1 119901

ℓ = 1 nℓc

(15)


120601 (119860119896 120582

119895ℓ) =

ng

sum

119896=1

119860119896

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

119892119895ℓ

(119860119896) (16)



119896as

120597120601 (119860119896 120582

119895ℓ)

120597119860119896

=

119898119896

sum

119895=1

120588119894ℓ119894

+

ncℓsum

ℓ=1

119875

sum

119869=1

120582119895ℓ

120597119892119895ℓ

(119860119896)

120597119860119896

= 0 (17)



120575119895ℓ

= 119883119879

ℓ119870119883

119895 (18)





120575119895119897

=

nmsum

119894=1

119883119879

119894119897119870119894119883119894119895

=

nmsum

119894=1

119891119894119895119897

119860119894

(19)


119891119894119895119897

= 119883119879

119894119897119870119894119883119894119895

119860119894 (20)



120597119892119895ℓ

(119860119896)

120597119860119896

= minus1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

(21)



119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898119896

sum

119894=1

120588119894ℓ119894

minus

nℓcsum

ℓ=1

119901

sum

119895=1

120582119895ℓ

1

1198602

119896

119898119896

sum

119895=1

119891119894119895ℓ

= 0 (22)


119903



119860120584+1

119896= 119860

120584

119896[

[

1

120588 sum119898119896

119894=1119897119894

nℓcsum

ℓ=1

119875

sum

119895=1

120582119895119897

1198602

119896

119898119896

sum

119869=1

119891119894119895ℓ

]

]

1119903

119896 = 1 ng

(23)


120582120584+1

119895ℓ= 120582

119907

119895ℓ(

120575119895ℓ

120575119895ℓ119906

)

1119888

119895 = 1 119901 (24)




119891119886

119865119886

+119862119898119909

119891119887119909

(1 minus 1198911198861198651015840

119890119909) 119865

119887119909

le 1 (25)

119891119886

06119865119910

+119891119887119909

119865119887119909

le 1 (26)


119910is










1198651015840

119890119909=

121205872119864

231198782 (27)


119887is




119886 119865

119887 and 119865

119890are related




If 119878119894


119865119886

=12120587

2119864

231198782

119894

(28)

If 119878119894


119865119886

=

119865119910

(1 minus 1198782

1198942119862

2)

119899

119899 =5

3+

3

8

119878119894

119862minus

1198783

119894

81198623

(29)


is given as radic(21205872119864119865119910


119894= 119862 (28) and (29) give the same

value That is

119878119894

= 119862 119875cr = 119865119886119860cr 119875cr =

121205872119864119860cr

23119862 (30)


119903119909

= 11988605

11198601198881 119903

119910= 119886

05

21198601198882 (31)


where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1198881

= 05(1198871


= 05(1198872


= 119897119909119894

119903119909119894and 119878

119910119894= 119897

119910119894119903119910119894where 119897

119909119894and 119897



119878119894

=119897119896

(11988605

119896119860119888119896

cr) 119860cr = [

119897119896

(11988605

119896119862)

]

119888119896

(32)


119886119896and 119888



If 119875119886


119865119886

=12120587

2119864

231198782

119894

(33)

If 119875119886


119865119886

=

119865119910

(241198623

minus 121198621198782

119894)

401198623 + 91198781198941198622 minus 3119878

3

119894

(34)




= 119897119896(119886

05

119896119860119888119896)



119865119886

= 11990561198602119888119896 (35)

If 119875119886


119865119886

=11989011198603119888119896 minus 119890

2119860119888119896

11989031198603119888119896 + 119890

41198602119888119896 minus 119890

5

(36)


1199056

=12120587

2119864119886

119896

(231198972

119896)

1198901

= 24119865119910

119862311988615

119896 119890

2= 12119865

1199101198972

119896119862119886

05

119896

(37)

1198903

= 40119862311988615

119896 119890

4= 9119862

2119897119896119886119896 119890

5= 3119897

3

119896 (38)


if 119878119909119894

le 119878119910119894

then 119897119896

= 119897119910119894

119886119896

= 1198862 119888

119896= 119888

2

if 119878119909119894

gt 119878119910119894

then 119897119896

= 119897119909119894

119886119896

= 1198861 119888

119896= 119888

1

(39)




AISC


119887

119865119910

le119897119890

119903119905

le radic354200119862

119887

119865119910

then 119865119887

= [

[

2

3minus

119865119910

(119897119890119903119905)2

1055 times 106119862119887

]

]

119865119910

(40)

when119897119890

119903119905

gt radic354200119862

119887

119865119910

then 119865119887

=117 times 10

5119862119887

(119897119890119903119905)2

(41)




119865119910and 119903





119862119887

= 175 + 15120573 + 031205732

le 23 (42)




119887becomes a


119905 In I shaped


119910as given in [67] 119903

119905


119903119905

= 12119903119910

= 1211988605

21198601198882 (43)


119865119887

= (1199051

minus 1199052119860minus21198882) (44)

where

1199051

=

2119865119910

3 119905

2=

1198652

1199101198972

119890

(1519 times 1061198862119862119887)

(45)


119865119887

= 119905411986021198882 (46)

where

1199054

= 16848 times 1051198862119862119887119897minus2

119890 (47)




= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












= 119897119909 (119886

05


1198651015840

119890119909= 119905

511986021198881 (48)

where

1199055

=12120587

2119864119886

1

(231198972119909)

(49)


119891119886

=119875119886

119860 119891

119887=

119872119909

(11988631198601198873)

(50)

where 119875119886and 119872



119898= 085 and


12057211198651198871198601198873 minus 120572

21198651198871198601198883 + 120572

3119865119886119860 minus 120572

41198651198871198651198861198601198873+1

+ 12057251198651198871198651198861198601198883+1

= 0

(51)

where 1198883

= 1198873

minus 21198881


1205721

= 11988631199055119875119886 120572

2= 119886

31198752

119886 120572

3= 085119872

1199091199055

1205724

= 11988631199055 120572

5= 119886

3119875119886

(52)


119887


11990611198651198871198601198873 + 119906

2119860 minus 119906

31198651198871198601198873+1

= 0 (53)


1199061

= 1198863119875119886 119906

2= 06119865

119910119872

119909 119906

3= 06119865

1199101198863 (54)


119886and119865









































Minimize =

ng

sum

119903=1

119898119903

119905119903

sum

119904=1

ℓ119904 (55a)

Subject to(120575

119895minus 120575

119895minus1)

ℎ119895

le 120575119895119906

119895 = 1 ns (55b)

120575119894

le 120575119894119906

119894 = 1 nd (55c)

119875119906119896

120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601119875119899119896

ge 02

(55d)119875119906119896

2120601 119875119899119896

+8

9(

119872119906119909119896

120601119887119872

119899119909119896

+

119872119906119910119896

120601119887119872

119899119910119896

) le 1

for119875119906119896

120601 119875119899119896

lt 02

(55e)

119872119906119909119905

le 120601119887119872

119899119909119905 119905 = 1 119899119887 (55f)

119861119904119888

le 119861119904119887

119904 = 1 nu (55g)

119863119904

le 119863119904minus1

(55h)

119898119904

le 119898119904minus1

(55i)



119903is the total














strength 119875119899119896






and 119872119906119910119896


119899119909119896and 119872

119899119910119896are the nominal



119899119896is




119899119896= 119865

119910119860119892119896




119899119896= 119860

119892119896119865cr where 119865cr = (0658

1205822

119888 )119865119910for 120582

119888le 15 and

119865cr = (08771205822

119888)119865

119910for 120582

119888gt 15 and 120582

119888= (119870119897119903120587)radic119865

119910119864




119906119909119905and 119872




is equal to119872





























1 1199042 119904



1015840

1199041015840

119894=

119904119886119894

no recombination

119904119886119894

or 119904119887119894

discrete

119904119886119894

or 119904119887119895119894

global discrete

119904119886119894

+(119904119887119894

minus 119904119886119894

)

2intermediate

119904119886119894

+

(119904119887119895119894

minus 119904119886119894

)


(56)

where 119904119886and 119904




119886119894 In the



119887119895is















119875119903

=1

119885 (119879)exp(

minus119864119894

119896119861

119879) (57)



119901119894119895

(119879119896) =

1 if Δ119882119894119895

le 0

exp(

minusΔ119882119894119895

Δ119882 119879119896

) if Δ119882119894119895

gt 0(58)


Δ119882119894119895

= 119882119895

minus 119882119894in which 119882

119895and 119882




119894119895 and119879


Δ119882119894119895






from 119879119904






119891= minus1 ln(119875

119891) After


119879119891

= 119879119904120572119873minus1





119891 and 120572 as





119894 = 119894119891

+ (119894119891

minus 119894119904) (

119879 minus 119879119891

119879119891

minus 119879119904

) (59)

where 119894119904and 119894



119879119891


119894119895


119894119895le 0

(6) If Δ119882119894119895







119896+1= 120572119879

119896 Increase













0and ini-




119909119894

0= 119909min + 119903 (119909max minus 119909min)

119907119894

0= [

(119909min + 119903 (119909max minus 119909min))

Δ119905]

(60)



119896) using the


119896


119896at the


119892

119896


119896+1=

119909119894

119896+ 119907

119894

119896+1Δ119905 where 119909

119894


at iteration 119896 + 1 119907119894




119907119894

119896+1= 119908119907

119894

119896+ 119888

11199031

(119901119894

119896minus 119909

119894

119896)

Δ119905+ 119888

21199032

(119901119892

119896minus 119909

119894

119896)

Δ119905 (61)

where 1199031and 119903


1 119901119894


and 119901119892



1and 119888

2are













119886119894119895 (119905) =

[120591119894119895 (119905)] [119907

119894119895]120573

sum119899119904119894

119896=1[120591

119894119896 (119905)][119907119894119896

]120573

(62)




119894119895(119905) that



119901ℓ

119894119895(119905) =

119886119894119895 (119905)

sum119899119904119894

119896=1119886119894ℓ (119905)

(63)


119894119895(119905) = 120585120591

119894119895(119905) where







119890

119894119895= 1119882




119894119895(119905 + 1) =

120588120591119894119895



119894119895= sum

119898

119896=1Δ120591

119896














1199091198941

1199091198942

119909119894hms


119909new119894

= 119909119894

isin 1199091198941

1199091198942

119909119894hms


isin 1199091 119909

2 119909ns


(64)


Is 119909new119894



(65)


new119894


new119894


new119894


















119909cm = [

np

sum

119894=1

119909119894

119891119894

] divide [

119899

sum

119894=1

1

119891119894

] (66)






119909next119894

= 119909cm +119903120572 (119909max minus 119909min)

119904 (67)



next119894


next119894

=


) where 119868next119894






















6 Conclusions





Acknowledgments




References




































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article mathematical and metaheuristic applications

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