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Engineering Structures 31 (2009) 2766–2778

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Multi-stage production cost optimization of semi-rigid steel frames usinggenetic algorithms

Nizar Bel Hadj Ali a,∗, Mohamed Sellami b, Anne-Françoise Cutting-Decelle c, Jean-Claude Mangin d

a Structural Engineering Institute, EPFL-ENAC-IS-IMAC, EPFL, 1015 Lausanne, Switzerlandb Department of Civil Engineering, Ecole Nationale d’Ingénieurs de Gabès, 6029 Gabès, Tunisiac Department of Organization and Production Engineering, Every University, Franced Laboratory of Design Optimization and Environmental Engineering, Polytech’Savoie, France

a r t i c l e i n f o

Article history:

Received 6 October 2008Received in revised form29 April 2009Accepted 2 July 2009Available online 17 July 2009

Keywords:

Optimum structural designSemi-rigid connectionsSteel frameGenetic algorithmMinimum cost

a b s t r a c t

The response of a steel structure is closely related to the behavior of its joints. This means that it isnecessary to take explicit account of joint properties in order to ensure a consistent approach to designoptimization of steel frames. Semi-rigid design has been introduced into steel construction standardssuch as Eurocode 3 and AISC. However, in the absence of appropriate guidelines, engineers encounterdifficulties when bringing in semi-rigid design to everyday engineering practice. Moreover, connectiondesign significantly affects the production cost of steel frame structures. Thus, a realistic optimizationof frame design should take into account the effective costs of different stages of production includingmanufacturing and erection activities. This paper presents a Genetic Algorithm based method for multi-stage cost optimization of steel structures. In the objective function, the total cost of different productionstagesis minimized. A newcost model is presentedthatitemizescostsof allstages of production (materialsupply, manufacturing, erection and foundation). Design examples are used to validate the proposedmethodology. Numerical validation shows that the multi-stagedesign optimization results in substantial

cost benefits between 10% and 25% compared to traditional design of steel frames. Furthermore, thedeveloped methodology is shown to be capable of measuring the possible impact of design choices inthe early design stage thus assisting designers to make better design decisions.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

During the conceptual design phase of a steel building project,designers are calledupon to establish a well defined structural sys-tem that serves as a basis for detailing analysis and manufactur-ing activities. As observed in real-world design practice, decisionstaken at the conceptual stage, especially those related to joint de-sign, have a major impact on the total production cost of the final

design. Due to time and resource constraints, designers have oftenlimited the range of alternative conceptual configurations consid-ered during the early design phase. In the absence of generic ap-proaches that support the designer’s activity, usage andexperiencehave generally prevailed.

In the past three decades, considerable research has beencarried out to assess the actual behavior of steel joints. Thesestudies have resulted in the development of various connectionmodels and connection databases [1–7]. In the same context, a

∗ Corresponding address: EPFL ENAC IS IMAC, GC G1 587, Station 18, CH-1015Lausanne, Switzerland. Tel.: +41 21 693 24 98; fax: +41 21 693 47 48.

E-mail address: [email protected] (N. Bel Hadj Ali).

large effort has also been devoted to incorporate the semi-rigidconnection behavior into computer-based frame analysis [8–16].Summarizing the advances made in connection research overthe last years, Nethercot [17] stated that the most surprisingis the change from a somewhat neglected area to one of fro-ntline importance. In the European standard for steel structures(Eurocode 3), the importance of structural joints is recognized anda specific standard for the design of steel joints is also created [18].

However, using the new standards is still time consuming forthe designer if no appropriate tools are available to assist in thedesign task [19]. In this context, numerous authors focused onthe merits of semi-rigid design and introduced practical designtools for its application in daily practice. Jaspart [4] presentedsimplified procedures for the design of structural joints allowingfor a reduction of the amount of calculation in comparison with thedirect application of Eurocode 3. In a similar approach, Weynandand Feldman [20] proposed design tools established for easydesign of structural joints. Design tools consist of design tablesand sheets concerning a set of commonly used beam-to-columnconnections. Practical design methods are also proposed by Chenin which joint properties are determined from a database [21].Steenhuis et al. [22] discussed the impact of joint classification on

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N. Bel Hadj Ali et al. / Engineering Structures 31 (2009) 2766–2778 2767

the economy of steel building frames. With the same objective,handbooks have also been prepared in the UK, France and TheNetherlands [23,24]. Bayo et al. [7], proposed an effective methodfor the global analysis of steel and composite frames based on arevised form of the component method employed for modelingsemi-rigid connections. In a recent overview, Bijlaard [25] pointedout that despite all the development in joint design, there is stilla need for reliable generic design approaches that enable thedesigner to use semi-rigid joints easily.

Structural optimization has been widely studied over the lastdecades and extensive work has been done in the case of opti-mal design of steel frames. However, engineers have few tools toapproach cost optimization in a systematic manner. The designof optimal structures requires methods for accurate cost estima-tion. Some studies have focused on cost modeling and estimationfor structural optimization. Tizani et al. [26] proposed an object-orientedcostmodelusedfortubulartrussdesign.Watsonetal. [27]developed concepts for steelwork costing in which different com-ponents of the cost were taken into account. A more detailed costfunction was presented by Jarmai and Farkas [28] with applica-tions to welded structures. Klansek and Kravanja [29] proposedsome basic self-manufacturing cost estimation used for compos-ite and steel structures. In the context of design optimization of semi-rigid steel frames, Xu et al. [30] considered a combined costof members and connections where the cost of each connectionis estimated according to its rotational stiffness. Using a similarapproach, Simões [31] accounted for connection cost to optimizesemi-rigid frames. Pavlovèiè et al. [32] considered manufactur-ing cost as well as material cost in a cost function implementedinto an optimization system for planar steel frames. Cabrero andBayo[19] developed a practicaldesignmethod for semi-rigidstruc-tures where joint design is optimized to fit optimal theoretical val-ues. Several papers cited here employ GA approaches due to theirbenefits. GA has been used in building engineering since the1990smostly in the area of structural optimization, such as minimumweight design [33–37] and minimum cost design [38–40], but also

for architectural design automation [41].Inthepastfewyears,GAshave been used by some number of researchers to find the op-timum design of steel structures with semi-rigid joints [42–46].Kameshki and Saka [42,43] proposed a genetic algorithm forweight optimization of planar steel frames with various semi-rigidconnections. Hayalioglu and Degertekin [47] presented an opti-mization technique based on genetic algorithm for steel frameswith semi-rigid connections and column bases. They consideredmembers and connection cost calculated according to the modeldefined by Xu et al. [30]. All these studies have shown that GAscan be a powerful tool for design optimization. However, design-ers cannot take advantage ofthe use of GAs unless they are coupledwith a realistic model of the design objectives.

Design practices in the field of steel structures have shown

that minimum weight designs may be up to 20% more expensivethan design solutions where fabrication costs have also been takeninto consideration [32]. This is due to the fact that, for a givensteel frame system, the total cost is mostly dependent upon thedegree of design complexity [37]. Analytical cost estimations usedin major cited works are mostly average values based on historicaldata for known operations [48]. Usually, average values used in thecost estimation model are functions of piece size and weight. Thus,such information cannot completely reflect the impact of designdecisions on the production cost of a steel structure. Furthermore,studies have shown that connection fabrication costs may bemore than 30% of the total fabrication cost of a steel frame [49,50]. These observations suggest that designing optimal structuresmust be based on accurate cost estimation taking appropriately

into consideration elementary cost components for all productionstages of a steel building project.

In this paper, we present a GA based optimization methodfor the structural design of steel frames with semi-rigid joints.Design optimization is based on a new approach where bothstructural members and joint detailing are chosen for minimalproduction cost. Rather than using historical data, a novel costmodel is presented where the production cost of a steel frame isexplicitly estimated by itemizing all steps of production. The costmodel is based on a study of the flow of work through a structuralsteel workshop. Data are collected for different manufacturingactivities and are used to estimate costs of producing structuralcomponents. Material supply, fabrication, erection and foundationstages are taken into account to ensure a realistic estimation of theproduction cost of a steel building project. The design algorithmfinds the solution with the least production cost and also ensuresthat stresses and displacements are within the limits defined byEurocode 3. Beams and columns are sized using standard steelsections, while connections are selected from a set of commonlyused beam-to-column joint configurations.

2. Multi-stage design optimization of steel structures

In steel frames, structural members are usually designed using

sections available in standard sizes. In addition, since a steel frameis made up of prefabricated components produced in a workshop,repetition of dimensions, shapes and details will streamline themanufacturing process and are major factors in economic design.Hence,designoptimizationof steel structures canbe formulated asa discrete optimization problem where both structural membersand joints are considered as discrete variables. In this approach,connections are also chosen from a set of commonly used ones.This may encourage both designers and fabricators to use a simpledesign of joints, leading to an increase in standardization andimproved economy.

2.1. Optimization problem

The design optimization problem of steel structures, whereminimizing multi-stage production cost is taken as the objectiveand the constraints are implemented according to Eurocode 3, canbe stated as follows:

Min C (I , X c , X s) (1)

Subject to:

N sd,i

N R,i+

M sd,i

M R,i≤ γ M i = 1, . . . , nc (2)

N sd,i

χi.N R,i+

ki.M sd,i

M R,i≤ 1 i = 1, . . . , nb (3)

vi ≤ vlim,i i = 1, . . . , nv (4)

ui ≤ ulim,i i = 1, . . . , nu. (5)The objective function (Eq. (1)) estimates the production cost of adesign solution for a steel frame project. I , X c and X s are the vectorsof design variables: cross sectional sizes of structural members,connections and column bases, respectively.

The design constraints are formulated according to Eurocode 3.Eqs. (2) and (3) define the local capacity and buckling checks forbeams and columns. N sd,i and M sd,i are the ultimate axial force andthe ultimate bending momentof member i. N R,i and M R,i aretheax-ial force capacity and the moment capacity of the member i aboutthe major axis. In Eqs. (2) and (3), nc and nb are the total numbersof capacity and buckling checks, respectively. In addition, γ M , χi

and ki are safety factors and coefficients defined by Eurocode 3.Eqs. (4) and (5) define the serviceability limit state requirements

for the structure. In Eq. (4), vertical deflections of beams are lim-ited. Eurocode 3 limits beam deflection under serviceability limit



2768 N. Bel Hadj Ali et al. / Engineering Structures 31 (2009) 2766–2778

state load cases to span/250. In Eq. (5), horizontal deflections of columnsarelimited.Underultimatelimitstateloadcases,thehori-zontal deflection of columns, in each storeyof height h,islimitedtoh/250. In addition, the global horizontal deflection of the structureis limited to H /420, where H is the total height of the structure.In these two equations, nv and nu are the numbers of vertical andhorizontal restricted deflections in the structure.

2.2. Multi-stage production cost model

Cost modeling in steel construction projects is difficult mainlybecause information about production costs is usually tightly re-lated to engineering experience and intuition. Generally, whenpricing a job, fabricators determine the cost of the supply of mate-rials, the number of hours involved in fabrication and the costs forsurface treatment and erection. These costs are usually expressedin rates per ton. Therefore, these rates prove to be unsatisfactory indetermining the cost of variations or refining designs. The reasonis that, for a given steel frame system, the total cost is highly de-pendent on particular design details. Cost estimation through ratesstill very general and may not help to assess the impact of specialproject details and the associated degree of design complexity.

An accurate costing is essential for the design and constructionof economical structures. We propose here a rational costingmethod where the production cost of a steel structure is appraisedusing an estimation of costs throughout all production stages of aproject. It is essential to point out that predicting the absolute costof a steel structure is still too ambitious. Theaim of this cost modelis to be able to evaluate the relative cost of feasible alternativedesign solutions in an optimization context.

The production cost of a steel frame as expressed in Eq. (6)consists of superstructure cost (C S) and foundation cost (C F ).

Production cost C = C S + C F . (6)

Eq. (7) defines the superstructure cost (C S) which can be split into:material cost (C Mat ), Fabrication cost (C Fab) andErection cost (C Ere).

C S = C Mat + C Fab + C Ere. (7)

2.2.1. Material cost (C Mat )

The material cost of a steel structure is relatively easy toquantify based on information gathered from suppliers of steelmaterials and components. It is calculated in accordance with theunit prices of the different steel parts of the structure (profiles, joints, supports, etc.). Unit prices are determined for differentsection types and for different steel grades. This price covers thecosts of all materials including hot rolled sections and plates, aswell as items such as bolts and anchor bolts.

2.2.2. Fabrication cost (C Fab)Fabrication cost estimation is a complex area because of its de-

pendency on factors such as operating practices, equipments, staff qualifications and workshop size. The accurate estimation of steelstructure fabrication cost is a challenging task and plays a key rolein the multi-stage production cost model presented here. The fab-rication cost is appraised by adding costs of elementary manufac-turing operations. Manufacturing costs are based on time neededto fabricate structural elements. The time is converted into a priceby applying a cost per hour of workshop labor.The fabrication timeis evaluated by adding the durations of elementary operations re-quired for each structural element (cutting-up, machining, weld-ing, bolting, etc.). The time considered in cost estimation takes intoaccount the differentphaseswithin an operationof manufacturing,

i.e. preparation, fabrication and handling. In addition, elementarycosts consist of labor and machine operating costs.

Basictimes of different manufacturing operations are evaluatedaccordingtoafabricationcostmodelofjointsdevelopedattheLab-oratory of the Design Optimization and Environment Engineering(LOCIE) of Polytech’Savoie (France). This cost model was first de-veloped by Hamchaoui [49] and incorporated into a computerizedmodule for joint design. Bel Hadj Ali updated the cost model forstructural optimization with semi-rigid joints [51,52].

The developed cost model is based on a study of the flow of work through a fabricator steel workshop. This study includestiming of individual manufacturing activities. Information aboutcost estimation was gathered in two stages: first, initial interviewsgathered background information about design and fabricationof traditional steel structures and methods used in estimatingfabrication costs. Subsequently, data for different manufacturingactivities were collected from a typical workshop. A data baseof manufacturing activities is then created to serve as a basisfor manufacturing time estimation. One example is presented inTables 1 and 2. It concerns a workstation producing gusset plates.The last column of Table 2 gives the average time required foreach operation carried out at this workstation. For example, theseaverage values can be used to estimate the manufacturing time fora plate with x holes and within a batch of n identical parts T (n, x)

using Eq. (8).

T (n, x) =t 1 + t 5

n+ t 3. x+ (t 2 + t 4) . (8)

Furthermore, an appropriate process plan is defined for each typeof connection and column base used in this study. The processplan is presented as a technical sheet prescribing all the individualmanufacturing operations required to fabricate each structuralelement. Fig.1 shows the process of fabrication timeestimation forajoint [49,51]. Forfurther detailsabout fabrication cost estimation,readers are referred to [51,52].

2.2.3. Erection cost (C Ere)

Erection cost includes the labor required to unload, lift, placeand connect the components of the structural steel frame. Thetotal erection cost is simply the cost of the field time requiredto assemble the structure. Erection cost is directly related to thenumber of lifts required and the size and the mass of structuralcomponent. Thus, it can be estimated according to the total massof the structure. In the developed cost model, we considered anerection cost per kg of steelwork. This can be justified since nowelds have to be done on-site forall connections andcolumn basesconsidered in this study. Consequently,erection cost is not directlyaffected by design choices related to connections and column-bases.

C Ere = kEre.M str . (9)

Erection cost estimation refers to the total mass of the structure(M str ) and is calculated using Eq. (9). kEre is a cost factor covering

man labor as well as machine power involved in the erectionprocess.

2.2.4. Foundation cost (CF)

The second part of the multi-stage production cost is thefoundation cost which can be an important factor in the overalleconomics of a steel project. Column base design directly affectsthe foundation design and cost. For poor soil conditions and rigidcolumn bases, expensive foundations are required because of theneed for thicker and larger base plates and the stiffening that isnecessary. Furthermore, large footings are required to resist thebase moments. Alternatively, pinned column bases with minimumfoundation cost can lead to a greater size for columns. Theseobservations suggest that foundation cost of a steel structure beappropriately accounted for in the cost model.

CF = kEx.V Ex + kF .V F . (10)




Table 1

Description of elementary manufacturing operations.

Workstation: Gussets machine (LP 703)

No. Phase Elementary job description Part Application level Parameters

1 Preparation Transfer and validation of program, changing tools, . . . Plates Batch (n plates) 1/n None2 Handling Part setting and clamping on the machine table Idem. Unit part 1 Weight3 Fabrication Execution of holes Idem. Hole ( x hole/plate) x Thickness4 Handling Cleaning and setting of part on a palette Idem Unit part 1 Weight

5 Match-marking Chalk marking of the last part idem Batch 1/n None

Table 2

Average time required for different elementary manufacturing operations.

Workstation: Gussets machine (LP 703)

No. Phase Estimation category Timing Average value

1 Preparation None t 1 200 s2 Handling Manuel handling t 2 40 s

Mechanical handling : crane bracket 10 s3 Fabrication Drilling t 3 0.5–1 mm/s/hole

Punching 4 s/hole4 Handling Manuel handling t 4 10 s

Mechanical handling : crane bracket 55 s5 Match-marking None t 5 20 s

Fig. 1. Fabrication time estimation procedure.

For each structural design solution, foundations are predesignedbased on base efforts and moments. The foundation cost is thenestimated using Eq. (10). The foundation cost includes excavationand footing manufacturing costs. The excavation cost is calculatedaccording to excavation volume (V Ex) and a cost factor (kEx).

Manufacturing cost of footings is calculated based on the wholefooting volume (V F ) and considering a cost factor (kF ). The costfactor (kF ) evaluates the cost per m3 for footings and integrates:shuttering, concrete manufacturing, reinforcement and pouring.Cost factors are quantified according to the unit prices for materialand manufacturing activities.

2.3. Optimization variables

Three design variables are considered: the cross-sectional sizeof structural members, the type of beam to column connectionsand the type of column bases. Structural members are selectedfrom available steel profiles; we used IPE and HEB profiles. Beamsare selected from a set of 18 IPE profiles and columns are selected

from a set of 19 HEB profiles. It must be pointed out here thatstructural members are deliberately limited to standard profiles

considered in the SPRINT model used in this work to characterizethe semi-rigid connections. The SPRINT model was created inthe frame of a European project aiming to establish simplifiedprocedures for structural joint design [24].

For beam to column connections, five configurations are

considered. Fig. 2 gives details of these connection types. Thenumber of different connection types is deliberately limited to themost commonly used connections in Europe.

For structural analysis, joint 1 will be idealized as fully rigidbecause of the use of stiffeners between flanges. Joint 5 will beidealized as nominally pinned according to its weak rotationalstiffness. On the other hand, joints 2, 3 and 4 are known toexhibit semi-rigid behavior. The response of these joints in termsof flexural stiffness and resistance is determined according to theAnnexe J of Eurocode 3 [18]. Eurocode 3 proposes a simple bilinearidealization of the nonlinear moment–rotation characteristics of the joint. The half initial secant stiffness method allows for a linearelastic idealization of the moment–rotation curve up to the joint’smoment capacity (M j,Rd). The stiffness is assumed to be equal

to half of the joint’s initial stiffness (S j,ini) with the addition of a horizontal plastic branch as shown in Fig. 3. To perform the




Stiffened extended end plate joint (1) Extended end plate joint (2)

Flush end plate joint (3) Flange cleated joint (4)

Web cleated joint (5)

Fig. 2. Type of joints considered.

Fig. 3. Elastic–plastic joint idealization (EC3, Annex J).

structural analysis for each design solution we used the SPRINTdata base for semi-rigid joints created in the frame of the SPRINTEuropean project [24]. SPRINT design tables provide the userswith the neededmechanicalcharacteristics (initial stiffness, designmoment, and shear resistance) of standard combinations of jointsof different profiles.

For column bases, two configurations are considered. Simplecolumn base idealized as pinned support and stiffened columnbase considered as rigidly supported (Fig. 4). In general, columnbases are designed with unstiffened base plates, but stiffened baseplates may be used when the connection is required to transferhigh bending moments. The column base is usually supported byconcrete footing or a slab.

3. Analysis of semi-rigid steel frames

Over the past three decades, great efforts have been madeto incorporate semi-rigid connections into computer-based frame

Simple column base Stiffened column base

Fig. 4. Type of column bases considered.

analysis. The general approach to incorporate connection behaviorin frame analysis is based on modeling connections as rotationalsprings attached to member ends. Monforton and Wu [53]introduced a fixity factor to define the rotational stiffness of aconnection relative to the attached member. They derived thestiffness matrix of a beam member with semi-rigid connections.The implementation of this approach requires small modificationsin existing analysis programs which made it widely used insemi-rigid frame analysis. Moreover, semi-rigid steel frames areknown to exhibit more flexibility than conventional rigid frames.This suggests that second order effects should be considered

in design practice. Xu [54] derived the geometrical stiffnessmatrix for a semi-rigid member by accounting for second order




Fig. 5. Coding of design solution.

effects and obtained the overall stiffness matrix in which bothconnection flexibility and second order effects are considered. Inthe present work, the stiffness matrices derived in [53,54] are usedfor beam members with semi-rigid connections. The structuralstiffness equations including effects of geometric nonlinearity andconnection stiffness are nonlinear and thus require an iterativesolution procedure.

As explained in previous section, structural analysis is per-formed considering a linear elastic idealization of the momentrotation characteristic of beam-to-column joints. The constantstiffness is defined as half of the initial stiffness (0.5 S j,ini) as rec-ommended by Eurocode 3. A preliminary joint design is performed

to ensure that the design moment M j,Sd experienced by the joint isless than the joint moment capacity M j,Rd.

In the implemented analysis procedure for semi-rigid frames,applied loads are divided into a number of small load incrementsandthe structuralequilibriumequation is writtenin the incremen-tal form as shown in Eq. (11).

[K ] .D = F . (11)

In Eq. (11), [K ] is the structure stiffness matrix, F is theincremental load vector, and D is the incremental displacementvector. For each loading step the displacements are calculatedusing a linear analysis. The stiffness matrix is then updated basedon the geometry and internal forces existing at the beginning of any loading step. The solutions for all load increments are added

up to obtain the total nonlinear response of the structure.

4. GA based optimization methodology

Genetic algorithms are global search methods that belong tothe class of stochastic search algorithms. They were originallyproposed by John Holland, but the success of the method owesmuch to the work of Goldberg [55]. Genetic Algorithms are robustalgorithms capable of traversing large and complex search spaceto provide optimal solutions. They have been used efficientlysince the early 1990s to solve structural optimization problems.As formulated in this study, the design optimization problem ischaracterized by a finite number of discrete variables. With threetypes of design variables, the number of design configurations

generated for a structural system increases exponentially. Theconstraints (Eqs. (2)–(5)) are implicit functions of the designvariables. We have chosen to use GA to solve the problem. Inthe following sections, we illustrate the main features of the GAoptimization procedure.

4.1. Coding of design variables

To apply genetic algorithm efficiently, it is required totransform the design space representation into a genetic spacerepresented by chromosome strings. Each design variable iscoded in to a substring and these substrings are concatenated toform a chromosome string representing a design solution. Thispaper considers three types of design variables: member sizing,

connections andcolumn bases variables. Hence, each chromosomeconsists of three parts corresponding to the three types of design

variables considered. The first part of a chromosome correspondsto structural members, the second to beam-to-column jointsand the last one to column bases. Integer coding is chosen fordesign variables based on the fact that with increasing number of variables, binary strings becomes longer and convergence speedis slowed down [56]. Each design variable is represented by anunsigned integer and the length of each chromosome correspondsto the numberof discrete variables(Fig.5). Forstructural members,standard profiles are coded by integer numbers corresponding tothe type of the profile and its number (100 to 117 for IPE profilesand 200 to 218 for HEB profiles). Beam-to-column connections arecoded with integers (1 to 5) corresponding to the five joint types

considered in this study. Column bases can only be rigid or pinned,a (0/1) code is used in this case.

4.2. Constraint handling

Genetic algorithms are directly applicable to only unconstr-ained optimization problems. Various constraint-handling meth-ods using GA have been proposed, including methods based onpenalty function and repair algorithms [57–59]. The concept of penalty function is used here to handle design constraints. It con-sists of penalizing individualsviolating constraints,and thus givinga lower probability of survival to these individuals. In this man-ner, the search for optimum solutions is directed towards the fea-sible regions of the search space. The penalty function approach

is implemented by adding an additional term to the objectivefunction. This additional term corresponds to the cost of violatingconstraints.

In this study, constraints are normalized so that infeasiblesolutions are assigned a penalty cost that is in proportion to thedegree of constraint violation. The normalized form of the designconstraints are expressed as follows:

g 1,i =

N sd,i

N R,i+

M sd,i

M R,i

− γ M i = 1, . . . , nc (12)

g 2,i =

N sd,i

χi.N R,i+

ki.M sd,i

M R,i

− 1 i = 1, . . . , nb (13)

g 3,i = vi/vlim,i

− 1 i = 1, . . . , nv (14)

g 4,i =

ui/ulim,i− 1 i = 1, . . . , nu. (15)

A new objective function is then defined (Eq. (16)) where theviolated constraints are penalized.

F (I , X c , X s) = C (I , X c , X s) + K .

4i=1

m j=1

P i, j (I , X c , X s). (16)

I , X c and X s are the vectors of design variables. C is the originalcost function (Eq. (6)). K is a constant selected depending onthe problem and intended to amplify the penalty term. P i, j is thepenalty function. m is the total number of constraints. For eachconstraint written in normalized form, the penalty function iscomputed using Eq. (17).

P i, j(I , X c , X s) =

0 if g i, j ≤ 0 g i, j if g i, j > 0. (17)




Fig. 6. Evaluation procedure for design configurations.

4.3. GA methodology

The Genetic Algorithm first generates a fixed size populationof individuals. Each individual is randomly generated and corre-sponds to a possible design solution for the analyzed frame. Eachindividual is then evaluated with respect to the objective functionand is assigned a fitness value. The evaluation process consists of analyzing the design solution, checking design constraints to cal-culate eventual penalization and computing its production costaccording to the multi-stage cost model explained in Section 2.2.Fig. 6 shows the general procedure for evaluation of design con-figurations. In the evaluation procedure, every structural solutionmust be analyzed to determine stresses anddisplacements. To per-form the structural analysis required foreach design configuration,we implemented a specific program devoted to structural analysisof plane frames with semi-rigid joints. The program is based on fi-

niteelement analysis where both connection flexibility andsecondorder effects are considered [52].The Genetic Algorithm iteratively builds new generations of

population through selection, crossover and mutation until con-vergence is achieved.

The fitness of each design configuration is calculated based onmodified objective function (Eq. (16)), which combine the totalproduction cost of the structure with the penalty function. Takingintoaccount that the design optimization problem is formulated asa minimization problem, the fitness of each individual is calculatedusing Eq. (18).

(Fitness)i = (F max − F i) / (F max − F min) . (18)

In Eq. (18), F max and F min refer to the modified objective function

values of the worst and the best individuals in a given population,respectively, and F i is the modified objective function value of the ith individual. Using this equation, fitness values are scaledbetween 0 and 1.

The reproduction operator chooses individuals from a parentpopulation according to their fitness. A new population is gen-erated using a two-point crossover scheme applied with a speci-fied probability (P c ). Crossover operator exchanges correspondinggenes from paired parents which creates offspring and exploresnew regions of the design space. A mutation operator is then ap-plied to the created offspring with a specified probability (P m).Gene values on a mutated locus are replaced according to the de-sign variable considered. The current population is then replacedby the new population and the procedure is repeated until the dif-

ference between average fitness values of consecutive generationsfalls below a specified tolerance.

Table 3

Applied loads.

Load Load value (kN/m)

qsw-0 7.80qsw-1 6.50qil-0 11.20qil-1 3.20qw 3.80

5. Design examples

Design examples are presented to demonstrate the applicationof the multi-stage cost optimization approach. The performance of the proposed methodology is first discussed for a three bay, twostoreyframestructurecitedbyCabreroandBayo[ 19].Next,athreebay,three storey frame structure is studied. Results are also shownfordifferentstoreyheights andbay widths.The third example usedis a three bay, ten storey steel frame optimized for unbraced andbraced configurations.

5.1. Three bay, two storey frame example

In this section, the performance of the proposed optimizationtechnique is demonstrated using a three bay, two storey frameexample studied by Cabrero and Bayo [19]. They have developed apractical design method for semi-rigid steel structures. The frameconfiguration is shown in Fig. 7 and load details are shown inTable 3.

The used steel grade is S275, with a modulus of elasticity of 210 000 MPa and yield stress of 275 MPa. European sections areused (HEB for columns and IPE for beams). The authors consideredload factors of 1.3 for self-weight loads (qsw), and 1.5 for imposed(qil) and wind (qw) loads. Four variables are defined for members.The columns are supposed to be continuous, and the beams aregrouped so as to share the same section. Four groups of beam-to-column connections (C 1 , C 2 , C 3 and C 4) are also considered.

The proposed algorithm is used to obtain the optimal solutionfor the structure. Several tests were carried out to fix the GAparameters. For the example proposed here, optimization resultswere satisfactory for a population size of 50 individuals evaluatedover 100 generations. Crossover and mutation probabilities wereset as 0.8 and 0.05 respectively. The penalty amplificationcoefficient K is set to 3000. The best solution generated over asequence of 10 runs using different random seeds is consideredas the optimal design solution. Optimal designs obtained withdifferent rigid and semi-rigid joints are presented in Table 4.Resulting profiles are presented for frame optimization with:extended end plate joints, flange cleated joints and finally withstiffened extended end plate joints. Table 4 also shows designsolutions given by Cabrero and Bayo [19] for the case when semi-

rigid connections are considered and also in thecase of pinned andrigid joints. The semi-rigid joint considered by these authors is anunstiffened extended end-plate joint. Cost estimations presentedby Cabrero and Bayo [19] include only steel and connectioncosts and are based on prices obtained from major Spanish steelfabricators. It must be pointed out that these costs cannot becompared to those obtained using multi-stage production costestimation. Costs are presented here only to demonstrate semi-rigid structure competitiveness.

From the cost details of the different stages of productionpresented in Table 5, we can see that, considering the frame withextended end plate joints as the reference solution, the differencein the cost is 20.8% in the frame with flange cleated joints and upto 22.6% in the frame having stiffened extended end plate joints.

The multi-stage minimum costs obtained using different typesof joints are compared in Fig. 8. It is noticed that optimum frames




Fig. 7. Configuration of the three bay, two storey frame.

Table 4

Resulting profiles and costs for different types of connections.

Members Our approach Cabrero and Bayo [19]

Extended end plate Flange cleated Stiffened extended end plate Semi-rigid Pinned Rigid1, 2, 3 IPE 180 IPE 200 IPE 200 IPE 200 IPE 240 IPE 2004, 5, 6 IPE 240 IPE 270 IPE 240 IPE 270 IPE 330 IPE 2707, 10 HEB 140 HEB 140 HEB 140 HEB 140 HEB 120 HEB 1408, 9 HEB 140 HEB 160 HEB 140 HEB 160 HEB 140 HEB 160Cost (e) 6878.6 8312.1 8434.2 4124.6 4545.4 5440.9

Table 5

Cost details for different design solutions.

Connection type Cost calculation (e)

Material Shop labor Erection labor Foundation Total

Extended end plate 3141.5 1510.8 1668.7 557.6 6878.6Flange cleated 3604.6 2102.0 2050.0 555.7 8312.1Stiffened extended end plate 3241.2 2855.0 1750.0 578.1 8434.2

Fig. 8. Costs for production stages with different type of joints.

with semi-rigid connections, such as the extended end plate,result in total cost saving. Cost savings are mainly obtained at themanufacturing stage which is not surprising since manufacturingcosts aresignificantlyinfluenced by joint complexity. In traditional joint design, designers systematically used transverse columnstiffeners in combination with thick end-plates when defining thegeometry of rigid joints. These elements are labor-intensive detail

materials leading to an increase in manufacturing cost. On theother hand, the use of flange cleated joints which have relatively

weak stiffnesscomparedto other joints increasesthe material cost,and consequently erection cost, since greater sections are neededfor beams and columns.

Itisshownthatthedegreeofconstructioncomplexityrelatedto joint design can be taken into account explicitly through the multi-stage production cost presented here. Additionally, when framesystems are to be conceptually estimated, this approach can be

very helpful to assist designers in judging the impact of designchoices on the total production cost.




Fig. 9. Three bay, three storey optimized frame.

Table 6

Applied loads.

Load case Loads (kN/m)

Ultimate limit state g 1 = 35.1 g 2 = 24.9 g 3 = 3.3Serviceability limit state q1 = 25.0 q2 = 18.3 q3 = 2.2

5.2. Three bay, three storey frame example

For the second numerical example we chose a three-storey,

three-bay frame (Fig. 9). The used steel grade is S235, with amodulus of elasticity of 210 000 MPa and yield stress of 235 MPa.European sections are used (HEB for columns and IPE for beams).Loads and load cases are defined according to Eurocode 1 andare shown in Table 6. Optimization variables are grouped into sixgroupsof profiles, four groupsof beam-to-column connections (C 1 ,C 2, C 3 and C 4) and two groups of column bases (S1 and S2).

For this example, optimization results were quite satisfactoryfor a population size of 50 individuals running for 100 generations.Crossover and mutation probabilities were fixed as 0.8 and 0.05respectively. The penalty amplification coefficient K is set to10000.

The design leading to a minimum production cost is presentedin Table 7. This table also shows the optimum solution for the case

when semi-rigid connections are not considered. Table 8 showsdetails of production cost for the two design solutions.

Cost comparison between design solutions shown in Table 8demonstrates the importance of considering semi-rigid joints ina global design approach. Compared to traditional frame design

Table 8

Cost comparison between design solutions.

Design solution

Considering semi-rigid joints Without semi-rigid joints

Total cost (EUR) 12 722.7 16 534.9Connection cost (%) 15.7% 29.9%Foundation cost (%) 9.7% 8%Total weight (kN) 45.0 49.6Connections weight (%) 3.8% 6.4%

approach in which joints are regarded as either pinned or rigid,considering semi-rigid joint behavior results in a total cost savingof 23% in this case. Cost saving is mainlyobtained in manufacturingand erection costs. Using stiffened joints results in over-designedconnections and uneconomical configurations, as the fabricationis highly dependent on the degree of stiffening. Furthermore, thestiffening usually prevents easy erection on-site which tends toincrease costs.

On the other hand, design optimization with semi-rigid jointsgives a substantial decrease in structural weight. In fact, atraditional design solution is 10% heavier than a semi-rigid designsolution. Consequently, benefits arising out of using less materialcan be realized considering the rising prices characterizing steelmaterials over the last five years. For example, according to the

French Federation of Steel (FFS) steel profile prices increased by195% between 2003 and 2008 in France and this is also real in allEuropean countries. Besides, a semi-rigid design solution is moreeconomical considering foundation costs. In fact, the use of semi-rigid joints results in a better distribution of internal forces in the

Table 7

Resulting design solutions.

Design variables Design solution

Considering semi-rigid joints Without semi-rigid joints

Members 1, 2 HEB 180, HEB 180 HEB 180, HEB 1803, 4 IPE 240, IPE 270 IPE 270, IPE 3005, 6 IPE 220, IPE 220 IPE 240, IPE 240

Joints C 1 , C 2 and C 3 Extended end plate Stiffened extended end plateSupports S1 Stiffened end plate (Rigid) Stiffened end plate (Rigid)

S2 Simple end plate (pinned) Simple end plate (pinned)




Table 9

Optimum designs for unbraced three bay, ten storey steel frame.

Member group Beam-to-column connection type

Stiffened extended end plate Extended end plate Flush end plate Flange cleated

1 HEB 450 HEB 400 HEB 400 HEB 5002 HEB 450 HEB 500 HEB 500 HEB 6003 HEB 300 HEB 300 HEB 300 HEB 3004 HEB 360 HEB 360 HEB 360 HEB 360

5 HEB 280 HEB 260 HEB 260 HEB 2606 HEB 300 HEB 300 HEB 320 HEB 3007 HEB 240 HEB 240 HEB 240 HEB 2408 HEB 240 HEB 260 HEB 260 HEB 2809 HEB 200 HEB 200 HEB 200 HEB 200

10 HEB 200 HEB 220 HEB 260 HEB 20011 IPE 450 IPE 450 IPE 500 IPE 45012 IPE 450 IPE 550 IPE 600 IPE 50013 IPE 450 IPE 450 IPE 550 IPE 60014 IPE 450 IPE 500 IPE 600 IPE 60015 IPE 450 IPE 400 IPE 450 IPE 60016 IPE 450 IPE 450 IPE 500 IPE 60017 IPE 450 IPE 500 IPE 500 IPE 55018 IPE 400 IPE 500 IPE 550 IPE 60019 IPE 500 IPE 400 IPE 400 IPE 45020 IPE 400 IPE 330 IPE 330 IPE 400Total cost (EUR) 107 314.3 95 193.1 100 602.4 109 566.4

Connection cost (%) 22.6 11.1 10.0 11.5Total weight (kN) 324.2 321.35 350.22 387.6

Fig. 10. Cost saving for H = 4 m.

structure. Less stiffening for column bases and smaller footings arethen required.

Studies are also performed for different combinations of spanand height for the three-storey, three-bay frame. We considerthree values for storey height (H = 4, 5 and 6 m) and four valuesfor bay span (L = 5, 6, 7 and 8 m; L1 = L and L2 = L + 1). Allthese different configurations are optimized and results are shownin Figs. 10–12. The production costs in these figures are estimatedusing the multi-stage production cost model and thus include allcost components (material, shop labor, erection and foundationcosts).

In all studied cases, multi-stage design optimization with semi-rigid joints shows substantial cost benefits running between 10%and 25%. Cost savings are for the major part resulting from areduction of fabrication and erection costs. Additionally, there isalso some reduction due to a decrease in structural weight.

5.3. Three bay, ten storey frame example

The third numerical example is a three bay, ten storey steelframe designed by Xu and Grierson [8]. Kameshki and Saka [43]as well as Foley and Schinler [46] performed weight optimizationon the same example using Genetic Algorithms and evolutionarycomputation. The frame configuration, dimensions and loading are

shown in Fig. 13. The used steel grade is S235, with a modulus of elasticity of 210 000 MPa and yield stress of 235 MPa.


Multi-stage production cost optimization is performed consid-ering only members as design variables while beam-to-columnconnections are specified to be of same type. Column bases aresupposed to be rigid. Optimization variables are thus limited to10 groups of beam members and 10 groups of column members.Beam members at each storey level are to have the same Euro-pean IPE section while exterior and interior column members areto have the same European HEB section over two stories (Fig. 13).Multi-stage production cost optimization is performed for bracedand unbraced configurations of the frame. For this example, op-timization results were quite satisfactory for a population size of

100 individuals running for 200 generations. This can be explainedby the significant number of design variables (20). Crossover andmutation probabilities were fixed as 0.8 and 0.05 respectively. Thepenalty amplification coefficient K is set to 15 000. The best solu-tion generated over a sequence of 10 runs using different randomseeds is considered as the optimal design solution.

Optimum designs obtained for frames with rigid and semi-rigidconnections are presented in Tables 9 and 10, for unbraced andbraced frame configurations, respectively. Details on productioncost and total weight for design solutions are also given in Tables 9and 10.

For an unbraced frame configuration, cost comparison betweendesign solutions obtained using four different beam-to-columnconnections shows that the minimum production cost is obtained

with extended end plate joints. If it is compared to the designsolution with stiffened extended end plate joints, the design





solution with extended end plate joints results in a production costsaving of about 11.3%. It is interesting to note here that the weightsof the two design solutions are very similar which suggests thatthe cost saving is mainly obtained in the manufacturing stage dueto the important contribution of the connection cost in the total

production cost of the structure. Optimization results show alsothat the production cost of the structure may be increased by morethan 2.0% if flange cleated joints are considered. These results arenot really surprising since relatively flexible connections, such asflange cleated connections, tend to increase second order effectsin the steel frame leading to bigger member sizes and thus greaterproduction costs.

Table 10 illustrates the optimum designs obtained for braced

frame configurations with rigid and semi-rigid connections. For abraced configuration of the frame, the minimum production cost isobtained with flush end plate connections. Compared to the designsolution in which joints are regarded as rigid, considering semi-rigid joints results in total productioncost benefits between8% and19.2%. It is also noted that, forthis bracedconfiguration, using flushend plate connections produces the lightest design solution.

Optimization results obtained for the three bay, ten storeyframe show clearly that considering semi-rigid connections resultsin greater production cost benefits for braced configurations. Forunbraced configurations, overall structural flexibility induced by joints with low rigidity can lead to heavier design solutions with agreater production cost.

Fig. 13. Three bay, ten storey optimized frame.




Table 10

Optimum designs for braced three bay, ten storey steel frame.

Member group Beam-to-column connection type

Stiffened extended end plate Extended end plate Flush end plate Flange cleated

1 HEB 300 HEB 300 HEB 280 HEB 2602 HEB 320 HEB 320 HEB 340 HEB 3203 HEB 280 HEB 280 HEB 260 HEB 2604 HEB 300 HEB 300 HEB 300 HEB 300

5 HEB 260 HEB 260 HEB 240 HEB 2606 HEB 260 HEB 280 HEB 240 HEB 2407 HEB 260 HEB 240 HEB 220 HEB 2208 HEB 220 HEB 220 HEB 220 HEB 2209 HEB 240 HEB 200 HEB 200 HEB 200

10 HEB 200 HEB 200 HEB 220 HEB 20011 IPE 330 IPE 330 IPE 330 IPE 40012 IPE 330 IPE 330 IPE 330 IPE 36013 IPE 330 IPE 330 IPE 330 IPE 36014 IPE 330 IPE 330 IPE 330 IPE 40015 IPE 330 IPE 330 IPE 330 IPE 40016 IPE 330 IPE 330 IPE 330 IPE 40017 IPE 330 IPE 330 IPE 330 IPE 40018 IPE 330 IPE 360 IPE 330 IPE 45019 IPE 330 IPE 330 IPE 330 IPE 45020 IPE 300 IPE 300 IPE 300 IPE 300Total cost (EUR) 82 034.0 69 207.7 66 294.2 75 572.3

Connection cost (%) 24.2 12.1 11.1 14.2Total weight (kN) 227.2 218.8 210.5 246.5

6. Conclusion

The multi-stage production cost optimization presented inthis paper provides useful support to steel structure designers infinding a final structural solution with minimum cost. The papershows that a structural engineer can obtain a possible lay-out of design choices early in the design stage. Concurrent generationand evaluation of many design alternative solutions in earlydesign is an important contribution of this study. Furthermore, thedeveloped cost model is capable of measuring the impact of designchoices on: material, manufacturing, erection and foundation costs

of a structural engineering project. The cost model is based ona systematic estimating approach and can be easily updated.By means of the multi-stage optimization method, semi-rigidbehavior of joints can be incorporated easily into structural designpractice allowing designers to take advantage of using simpledesign of joints. As stated by standards such as Eurocode 3, jointbehavior has a significant effect on the response of structuralframes. This is confirmed by optimization results where both thechoice of structural members and the detailing of joints are donewith respect to theeconomy. Results show also that the cost of the joints may represent more than 20% of the total cost of optimizedsteel frame structures. Therefore, using multi-stage productioncost optimization in early design can significantly reduce theeffective costs of steel structures.

Acknowledgements

The authors would like to thank Prof. Ian F.C. Smith, Head of theApplied Computing and Mechanics Laboratory at EPFL, Switzer-land, and Dr. Prakash Kripakaran for their valuable comments andsuggestions.

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