Design of reinforced concrete bridge frames by heuristic ...mech.fsv.cvut.cz/~leps/teaching/mom/data/TA_paper1.pdfDesign of reinforced concrete bridge frames by heuristic optimization

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Advances in Engineering Software xxx (2007) xxx–xxx

Design of reinforced concrete bridge frames by heuristic optimization

Cristian Perea 1, Julian Alcala 2, Victor Yepes 3,Fernando Gonzalez-Vidosa *, Antonio Hospitaler 4

School of Civil Engineering, Construction Engineering Department, Technical University of Valencia, Campus de Vera, 46022 Valencia, Spain

Received 20 December 2006; accepted 19 July 2007

Abstract

This paper deals with the economic optimization of reinforced concrete box frames used in road construction. It shows the efficiencyof four heuristic algorithms applied to a problem of 50 design variables. Heuristic methods used are the random walk and the descentlocal search. The metaheuristic methods are the threshold accepting and the simulated annealing. The four methods have been applied tothe same frame of 13 m of horizontal span. The comparison of the four heuristic algorithms leads to the conclusion that the proposedthreshold accepting is more efficient, since it improves cost results of the random walk and descent local search by 7.5% and 1.4%, respec-tively, while improving deviation of random results of the simulated annealing. Finally, the inclusion of the deflections and fatigue limitstates appears to be crucial, since their ignorance leads to 3.9% more economic but unsafe results.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Structural design; Economic optimization; Heuristics; Concrete structures

1. Introduction

Since its appearance in the mid-1950s, artificial intelli-gence has dealt with a variety of fields [1,2], such as auto-matic programming, the solution of constrainedproblems, operational research, planning, neuronal net-works, etc. The design of structures is a problem of selec-tion of design variables subject to structural constraintsfor which artificial intelligence is very much suited. In spiteof the potential capabilities of artificial intelligence, presentdesign of economic concrete structures is much conditionedby the experience of structural engineers. Most proceduresare based on the adoption of cross-section dimensions and

0965-9978/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.advengsoft.2007.07.007

* Corresponding author. Tel.: +34 963879563; fax: +34 963877569.E-mail addresses: [email protected] (C. Perea), jualgon@

cst.upv.es (J. Alcala), [email protected] (V. Yepes), [email protected](F. Gonzalez-Vidosa), [email protected] (A. Hospitaler).

1 Tel.: +34 963879563; fax: +34 963877569.2 Tel.: +34 963879563; fax: +34 963877569.3 Tel.: +34 963879563; fax: +34 963877569.4 Tel.: +34 963877566; fax: +34 963877569.

Please cite this article in press as: Perea C et al., Design of reinforced(2007), doi:10.1016/j.advengsoft.2007.07.007

material grades based on sanctioned common practice.Once the structure is defined, it follows the analysis ofstress resultants and the computation of passive and activereinforcement that satisfy the limit states prescribed byconcrete codes. Should the dimensions or material gradesbe insufficient, the structure is redefined on a trial and errorbasis. Such process leads to safe designs, but the economyof the concrete structures is, therefore, very much linked tothe experience of the structural designer.

The methods of structural optimization may be classi-fied into two broad groups: exact methods and heuristicmethods. The exact methods are the traditional approach.They are usually based on the calculation of optimal solu-tions following iterative techniques of linear programmingof the expressions of the objective function and the struc-tural constraints [3,4]. The objective function is the expres-sion to be optimized (e.g. the cost of a concrete structure,the weight of the elements of a steel structure, etc.) andthe structural constraints are the limit states to complywith. These methods are computationally very efficientwhen the number of variables is limited, since they requirea small number of iterations. However, they have to solve

concrete bridge frames by heuristic optimization, Adv Eng Softw

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every iteration a problem of linear conditioned optimiza-tion, which is much computer time consuming for a largenumber of variables.

The second main group are the heuristic methods, whoserecent development is linked to the evolution of artificialintelligence procedures. This group includes a broad num-ber of artificial intelligence search algorithms such asgenetic algorithms, simulated annealing, threshold accept-ing, tabu search, ant colonies, etc. [5–9]. These methodshave been successful in areas different to structural engi-neering. Civil engineering applications can be found forinstance in hydraulics, project planning and vehicle routeorganization [10]. These methods consist of simple algo-rithms, but they require a great computational effort, sincethey include a large number of iterations in which theobjective function is evaluated and the structural con-straints are checked.

In a thorough 1994s review on structural optimizationby Cohn and Dinovitzer [11], they underline that there isa great gap between theoretical work and the practicalapplication in aeronautical and civil engineering. They alsostate that there is a dominance on the study of algorithmsversus the solution of structural problems of interest. Theyalso notice that most of the published work deals with steelstructures, whereas structural concrete is only addressed in4% of the cases. They end up pointing out the great poten-tial of genetic algorithms although they were then only intheir origins. Among the first works of heuristic optimiza-tion applied to structures, the contributions of Jenkins andof Rajeev and Krishnamoorthy in 1991–1992 are to bementioned [12–14]. Both authors applied genetic algo-rithms to the optimization of the weight of steel structures.In addition, it is also worth mentioning the early work ofGrierson and Pak on genetic algorithms applied to steelskeletal structures [15]. As regards RC structures, earlyapplications in 1997 include the work of Coello et al.[16], who applied genetic algorithms to the economic opti-mization of RC beams; together with the work of Leite andTopping who applied genetic algorithms to prestressedconcrete beams [17]. Another early work includes the1998s work of Kousmousis and Arsenis on genetic algo-rithms applied to concrete members [18]; together withthe work of Rafiq and Southcombe on genetic algorithmsapplied to RC columns [19]. Recently, there has been anumber of RC applications, such as the work of Hrstkaet al. [20] and Leps and Sejnoha [21], who also optimizeseveral cases of RC beams; and the contributions of Leeand Ahn [22] and Camp et al. [23], who both optimizeRC building frames by genetic algorithms. And morerecently, the work of Rafiq et al. on the design of biaxialcolumns [24]. Also recently, our research group has appliedsimulated annealing and threshold acceptance to the opti-mization of walls, portal and box road frames and buildingframes [25–29].

The road box frames object of this work are those usu-ally built of RC in road construction. They are used withspans between 3.00 and 20.00 m for solving the intersection


of transverse hydraulic or traffic courses with the mainupper road. The parts of the box frame are the following(see Fig. 1): the top slab that sustains the earth cover andthe traffic loads, the lateral walls that transfer the reactionsof the top slab and contain the lateral earth fill, and thebottom slab that acts as the bearing foundation of thestructure. An alternate design consists of portal RC framesthat have footings instead of a slab as a foundation. Boxframes are preferred when there is a low bearing strengthterrain or when there is a risk of scour due to flooding.The depth of the top and bottom slab is typically designedbetween 1/10 and 1/15 of the horizontal free span; and thedepth of the walls is typically designed between 1/12 of thevertical free span and the depth of the slabs. The main dataor parameters that affect their design are the horizontal freespan (Lh), the vertical free span (Lv) and the earth cover(He). The box frames are calculated to sustain the trafficand earth loads prescribed by the codes and have to satisfyall the limit states required as a RC structure.

The objective of this work is to investigate on the heuris-tic optimization of this type of road box frame RC struc-ture. (The optimization of alternate portal road RCframes has been the object of parallel work and can befound elsewhere [30].) The method followed has consistedin the development of an evaluation computer modulewhere cross-section dimensions, materials and steel rein-forcement have been taken as discrete variables. This mod-ule computes the cost of a solution and checks all therelevant limit states. Heuristic algorithms are then usedto search the solution space. The four heuristic methodsprogrammed have been the random walk, the local descentsearch, the simulated annealing and the threshold accep-tance method. The random walk is a basic heuristic methodthat basically finds feasible solutions by random choice ofthe design variables. The descent local search is a heuristicmethod that starts with a feasible solution and then gradu-ally alters the values of the variables so far as the new solu-tions improve the objective function. It is well-known thatit gets trapped in local optima. As regards the simulatedannealing and the threshold acceptance, they are similarto the descent local search but they also accept solutionsthat worsen the objective function in the course of thesearch. The four heuristic methods are described thor-oughly in Section 3.

2. Optimization problem definition

2.1. Problem definition

The problem of structural concrete optimization that isput forward consists of an economic optimization. It dealswith the minimization of the objective function F of expres-sion (2.1), satisfying also the constraints of expression (2.2).

F ðx1; x2; . . . xnÞ ¼X

i¼1;r

pi � miðx1; x2; . . . ; xnÞ ð2:1Þ

gjðx1; x2; . . . ; xnÞ 6 0 ð2:2Þ


Fig. 1. Typical reinforced concrete road box frame.

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Note that the objective function in expression (2.1) is aneconomic function expressed as the sum of unit prices mul-tiplied by the measurements of construction units (con-crete, steel, formwork, etc.). And that the constraints inexpression (2.2) are all the service and ultimate limit statesthat the structure has to satisfy, as well as the geometricaland constructability constraints of the problem.

2.2. Variables

A total number of 50 variables define a solution of RCroad box frame (see Fig. 2 for the reinforcement variables).

Fig. 2. Reinforcement variables of the RC


These variables define the geometry, the type of concretegrades and the passive reinforcement of the frame. The restof data necessary to compute a particular box frame arewhat are defined as parameters in Section 2.3. Logically,parameters are a priori data and they are kept constantin the optimization search; though they will be the objec-tive of future parametric studies.

The first three variables are geometrical and they corre-spond to the depth of the lateral walls and the depth of theupper and lower slabs. The second three variables relate tothe wall and slabs concrete grades. And the remaining 44variables relate to the definition of a passive reinforcement

road box frame optimization problem.


Table 1Parameters of the RC road box frame of 13 m of horizontal free span

Geometric parameters

Horizontal free span: 13.00 mVertical free span: 6.17 mEarth cover: 1.50 m

Loading related parameters

Earth density: 20.00 kN/m3

Coefficient of active earth pressure: 0.33Coefficient of resting earth pressure: 0.50

Ground bearing characteristics

Ballast coefficient: 10 MN/m3

Economic related parameters

Unit costs: see Table 2

Legislative related parameters

Code regulations: EHE/EC2/IAP-98Security coefficients: EHE/EC2/IAP-98

Exposure related parameters

Internal ambient conditions: IIb (EHE)External ambient conditions: IIa (EHE)


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set up. Steel grade corresponds to EHE Spanish B500S inall cases (yield stress of 500 MPa). Flexural bars includevariable length corner reinforcement bars and positivebending reinforcement in the top and bottom slabs. Shearreinforcement includes two zones of variable definition inslabs (corner and mid-span) and three variable zones inwalls (bottom, middle and top). It is important to note thatbar distances are arranged and checked against constructa-bility, which means that any solution considered has to becompatible in terms of bars arrangement. In this sense,three basic bar distance arrangements are allowed, i.e.150, 200 and 250 mm and the rest of the bar distancesallowed have to be a multiple or subdivision of the chosenbasic bar distance. In total, eight bar distances are allowed,i.e. 50, 75, 100, 125, 150, 200, 250 and 300 mm. The pro-gram checks that any solution considered is constructablein the sense that there is certain order in the arrangementsof all bars including compatibility of bar distances betweenwalls and slabs. For example a slab reinforcement spaced75 mm is incompatible with a wall reinforcement spaced100 mm, but it is compatible with a spacing of 75, 150and 300 mm. Corner reinforcement compatibility is alsoimplemented and, hence, combinations of variables leadalways to practical designs, i.e. designs that would be wellaccepted by contractors and steel subcontractors. Never-theless, details that are not fully practical can be reviewedand modified by the contractor.

It is worth noting that variables are discrete and notcontinuous. Thus, for example, the depth of the top slabcan vary between a minimum of 400 mm and a maximumof 1650 mm in steps of 50 mm. As regards concrete grades,they can vary between 25 and 50 MPa cylinder compressivestrength in steps of 5 MPa. Finally, steel bars can varybetween 6 and 40 mm following a standard sequence.

The set of combinations of values of the 50 variablesmay be defined as the solution space. Such space is in prac-tice unlimited due to what is known as combinatorialexplosion; the number of combinations in our case is ofthe order of 1040. Each vector of 50 variables defines a solu-tion that has an economic cost following expression (2.1).Solutions that satisfy the constraints of the limit states ofexpression (2.2) will be called feasible solutions. Those thatdo not verify all constraints will be called unfeasiblesolutions.

2.3. Parameters

The parameters of the RC box frame are all the magni-tudes taken as data and not being part of the optimizationsearch. They consist of geometrical values, properties ofthe ground and of the earth fill, partial coefficients of safetyand durability data. As mentioned above, the main geomet-rical parameters are the horizontal free span (Lh), the verti-cal free height (Lv) and the fill height over the box frame(He). The main parameter of the ground is the stiffness mod-ulus of the foundation. The data of the fill are its density andthe coefficients of active and resting earth pressure. As


regards durability, two cases are considered depending onthe concrete face covered by the fill or exposed to trafficand weather conditions. Table 1 summarises the parametersof the RC box frame of Lh = 13 m used throughout thiswork for the calibration and comparison of the four heuris-tic algorithms used below in Sections 4 and 5.

2.4. Cost function

The objective function considered is the cost functiondefined in expression (2.1), where pi are the unit pricesand mi are the measurements of the 10 units in which theconstruction of the RC box frame is split. The cost functionincludes the value of materials (concrete and steel) and allthe entries required to evaluate the whole cost of the frameper linear meter. It includes, for example, the excavation ofthe foundation and the lateral fill of the walls. The basicprices considered are given in Table 2. These prices havebeen obtained from local Valencian contractors of roadconstruction in October 2003.

Given the 50 variables of the present problem, the mea-surement and cost evaluation of a particular solution isstraightforward. The main computational work goes tothe evaluation of the constraints of the limit states of thefollowing epigraph 2.5. It is important to note that manyworks transform constrained problems in unconstrainedby using penalty functions. Penalty costs are small for lightlacks of compliance and strong for larger ones. This workrestricts to feasible solutions only and, therefore, penaltyfunctions are not used.

2.5. Structural constraints

The structural constraints of expression (2.2) are all thelimit states that the structure and its foundation should


Table 2Basic prices of the cost function of the RC road box frame

Unit Cost (€)

Kg of steel (B-500S) 0.583m2 of lower slab formwork 18.030m2 of wall formwork 18.631m2 of upper slab formwork 30.652m3 of scaffolding 6.010m3 of lower slab concrete (labour) 5.409m3 of wall concrete (labour) 9.015m3 of upper slab concrete (labour) 7.212m3 of concrete pump rent 6.010m3 of concrete HA-25 45.244m3 of concrete HA-30 49.379m3 of concrete HA-35 53.899m3 of concrete HA-40 58.995m3 of concrete HA-45 63.803m3 of concrete HA-50 68.612m3 of earth removal 3.005m3 of earth fill-in 4.808


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comply with. A previous step to the verification of limitstates consists of the calculation of stress envelopes dueto the fill and traffic loads. The latter are in accordanceto Spanish Code IAP [31] that prescribes a uniform distrib-uted load of 4 kN/m2 and a heavy vehicle of 600 kN. Fati-gue loads are taken into account by the action of a vehicleof 390 kN and a dynamic coefficient of 1.2. The structuralmodel used is shown in Fig. 3 and it includes 40 bars andnodal points. Note that the bottom slab is supported onsprings whose stiffness is proportional to the ballast coeffi-cient of the ground. Calculations are per linear meter andthey are performed by means of an internal matrix methodprogram. As regards actions, permanent actions includeself-weight, the weight of the earth cover and the lateralactive pressure. Variable actions include the traffic loads,a possible increment of lateral pressure up to the restingstate in both of the lateral walls and a superficial embank-ment uniform load of 10 kN/m2. The structural model istwo-dimensional and, hence, it has limitations since longi-tudinal bending moments cannot be computed.

Fig. 3. Structural model for the ana


Once the stress resultants are known, it follows checkingall the service and ultimate limit states prescribed by con-crete codes [32–34]. In this respect, Spanish EHE is fol-lowed except for deflections and fatigue that are checkedin accordance to Eurocode 2. Note that once the 50 vari-ables that define a box solution are chosen, then geometry,materials and passive reinforcement are fully defined. It isimportant to note that no attempt is made to computethe passive reinforcement according to usual design rules.Such common design procedures follow a conventionalorder of obtaining reinforcement bars from flexural-shearultimate limit states and, then, check service limit statesand fatigue and redefine if necessary. This order is effective,but it ignores other possibilities that heuristic search algo-rithms do not oversee. In this sense, for example, it is pos-sible to suppress shear reinforcement by increasing flexuralreinforcement, which may result in more economic designs,as it has already been shown for earth retaining walls [25].

The calculation of the ULS of flexure checks whether theacting resultants Nd–Md are inside the ultimate interactiondiagram Nu–Mu. As regards shear, the acting shear is com-pared with the two ultimate values. Both flexural and shearminimum amount of reinforcement, as well as geometricalminimum, are also dealt with. The SLS of cracking includescompliance of the crack width limitation for the existingdurability conditions. The ULS of fatigue is verified shouldthe concrete and steel stress increments be smaller thanEurocode 2 limits. Given the structural model adopted,every end of the 40 bars of the mesh is checked against allsectional limit states. As for deflections of the top slab, a lim-itation of 1/250 for the quasi-permanent loading conditionsis imposed following the general recommendation of Euro-code 2. As mentioned earlier, longitudinal resultants are notcomputed and, hence, longitudinal reinforcement is taken asnot smaller than a practical one-fifth of the correspondingtransverse reinforcement and not smaller than minimumgeometrical limits. It is important to note that the inclusionof the SLS of deflections and the rarely checked ULS of fati-

lysis of the RC road box frame.


Working Solution P0

Current Solution P1

SA-TA Acceptance

Criterion

Structural Performance

Yes

Yes

No

No

Upd

ate

Wor

king

Sol

utio

n

Fig. 4. Flow-chart of the simulated annealing and threshold acceptingsearch algorithms.


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gue is crucial in the optimization search, as will be shownbelow in Section 4.4.

3. Applied heuristic search methods

The four heuristic search methods used in this work arethe random walk, the descent local search, the simulatedannealing and the threshold accepting method. The firsttwo methods are called heuristics, which means methods ofsearch that obtain good solutions; but they do not generallyconverge to the global optimum. The two second methodsare called metaheuristics, since they are capable of solvingdifficult problems of combinatorial optimization and theyhave proved applicable to problems of different nature.

The first method used is the random walk, which consistsof generating solutions by random choice of the variables ofthe optimization problem. The cost of each random solu-tion is evaluated as well as the structural constraints tocheck whether the solution is feasible or not. The processis repeated a certain number of iterations and the lower costsolution is kept as the best result. The algorithm has nointelligence and it does not reach any optimal value, butit is worth to explore the solution space and to determinethe percentage of feasible solutions from the total numberof generated solutions. Moreover, this random generationof solutions is useful as the starting point of other heuris-tics. First results indicated that the percentage of feasiblesolutions was smaller than 1% and it was then decided torestrict the search only to structures that satisfied minimumsteel reinforcement. Additionally, repairs of minor unfeasi-ble solutions were performed, so as to improve the meth-od’s performance. Results for 500,000,000 iterations willbe presented below in Section 4.1.

The second method used is the descent local search. Thismethod requires a feasible solution as an initial solution.This solution is gradually altered by applying moves tothe values of the variables. A move is a small random var-iation up or down to the values of some of the variablesthat define the current solution. Given a current solution,a move is applied and, hence, a new solution is obtained.This new solution is evaluated and it is adopted as thenew current solution when it improves the cost and it isalso feasible. Otherwise, it is rejected and a new attemptto improve is performed. This method improves generallyrandom walk results, but it is known that it gets trappedin local optima. Eight types of random moves have beentried: m03, m05, m09, m17, M03, M05, M09 and M17;where the number relates to the number of variables ran-domly changed in every iteration; and the lower/upper casem/M differentiates whether the counter of variables is arandom up to a number or a total number (e.g. m05 statesfor a random up to five variables modified in each iteration,whereas M05 indicates that a total of five variables aremodified in each iteration). It is worth noting that, sinceresults are random, every test is repeated 25 times to getmean and minimum values of the search. The main aimof programming this method is to choose the most efficient


moves to incorporate them in the following two metaheu-ristics (simulated annealing and threshold acceptance).

The third method is the simulated annealing (SA hence-forth), that was originally proposed by Kirkpatrick et al.for the design of electronic circuits [35]. The SA algorithmis based on the analogy of crystal formation from massesmelted at high temperature and let cool slowly. At hightemperatures, configurations of greater energy than previ-ous ones may randomly form, but, as the mass cools, theprobability of higher energy configurations formingdecreases. The process is governed by Boltzmann expres-sion exp(�DE/T), where DE is the increment of energy ofthe new configuration and T is the temperature. Theflow-chart of the algorithm is shown in Fig. 4. It starts witha feasible solution randomly generated and a high initialtemperature. The initial working solution, Po, is changedby a move like in the descent local search explained above.The new current solution, P1, is evaluated in terms of cost.Greater cost solutions are accepted when a 0–1 randomnumber is smaller than the expression exp(�DE/T), whereDE is the cost increment and T is the current temperature.The current solution is then checked against structuralrestrictions and if it is feasible, it is adopted as the newworking solution. The initial temperature is decreased geo-metrically (T = kT) by means of a coefficient of cooling k.A number of iterations called Markov chains is allowed ateach step of temperature. The algorithm stops when thetemperature is a small percentage of the initial temperature(typically 1%). The SA method is capable of surpassinglocal optima at high-medium temperatures and graduallyconverges as temperature reduces to zero. The methodrequires calibration of the initial temperature, of the lengthof the Markov chains and of the cooling coefficient;together with calibration of the stop criterion as a percent-age of the initial temperature.



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The last method used is the metaheuristic called thresh-old accepting method (TA henceforth), which was origi-nally proposed by Dueck and Scheuer [36]. Fig. 4 alsoshows the flow-chart of this algorithm, which is similar tothe SA chart. It starts with an initial randomly generatedworking solution, Po, and an initial high threshold valuefor accepting solutions. As for SA, TA changes the solutionby a move. The new current solution, P1, is accepted if ithas a lower cost or when the cost increment is smaller thanthe current threshold value. Again, the current solution ischecked against structural constraints and if it is feasible,it is adopted as the new working solution Po. The initialthreshold is reduced geometrically and gradually by meansof a coefficient similar to the cooling coefficient of SA. Anumber of iterations at each threshold step is performedin the same way as for Markov chains of SA. TA methodis also capable of surpassing local optima and graduallyconverges as the threshold value reduces to zero. Themethod requires calibration of the initial threshold, of thelength of the cycle chains and of the threshold reducingcoefficient; and again, together with calibration of the stopcriterion as a percentage of the initial threshold.

4. Results of heuristic search methods

4.1. Random walk

As mentioned above, first results of this kind of searchindicated that less than 1% of the generated solutions werefeasible (solutions that satisfy the structural constraints). It

0

2000

4000

6000

8000

10000

12000

0 5000 10000 15000time (sec

cost

(eur

os)

Fig. 5. Cost results of a standard rando


was then decided to restrict the search to structures thatsatisfied minimum steel reinforcement. Additionally, minorunfeasible solutions were repaired to improve the perfor-mance of the method. Following this approach, Fig. 5shows the results for a 500,000,000 iterations search witha running time of about 28,800 s. (Times are for compiledCompaq Visual Fortran Professional 6.6.0 in a Pentium IVof 2.4 GHz.) The best result found has a cost of 4726.12€(see Table 1 for data of the parameters of theLh = 13 m). The main characteristics of this solution arean upper and lower slab thickness of 750 mm of C35 and800 mm of C25 and a wall thickness of 500 mm of C35;the amount of positive span reinforcement in the top slabis 1.12% and of 0.40% in the upper negative corners.

4.2. Descent local search

This method requires an initial feasible solution whichhas been obtained by random generation. The averageresults of the four types of random moves, M03, M05,M09 and M17 are shown in Fig. 6 for different numberof iterations without cost improvement. As describedabove, the number in Mnn indicates the number of vari-ables that are changed randomly in every move. Resultswere obtained for the four types of moves and for a stopcriterion based on 10, 50, 100, 500, 1000, 5000 and10,000 iterations without improvement. It was concludedthat move M09 and 500 iterations without improvementas stop criterion were the most efficient combination forthis algorithm. Similarly, the four random moves m03,

20000 25000 30000)

RW

RW_min

m walk for 500,000,000 iterations.


Fig. 6. Medium cost results of descent local search for moves M03, M05, M09 and M17.


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m05, m09 and m17 cited above were also investigated butproved less effective [37]. The minimum result was a boxframe with a cost of 4433.19€, which improves by 6.2%the results of the random walk. (The main characteristicsof this solution are an upper and lower slab thickness of950 and 800 mm of C25 and a wall thickness of 400 mmof C30; the amount of positive span reinforcement in thetop slab is of 0.71% and of 0.15% in the upper negative cor-ners.) As regards computer running times, they are reducedto only 2.45 s, compared to the 28,800 s of the 500,000,000random walk.

4.3. Simulated annealing

The simulated annealing algorithm was also applied tothe same road box frame of Lh = 13 m. The methodrequires an initial feasible solution and selection of the ini-tial temperature, length of the Markov chains, a coolingcoefficient and a stop criterion. As for the descent localsearch method, randomly generated initial solutions weretried so as to study the influence of the initial solution onthe results. The initial temperature was adjusted followingthe method proposed by Medina [38], which consists inchoosing an initial value and checking whether the percent-age of acceptances of higher energy solutions is between alower and upper limit. If the percentage is greater than theupper limit, the initial temperature is halved; and if it smal-ler than the lower limit, the initial temperature is doubled.The limits considered have been 10–30% and 20–40%. Thecalibration of the Markov chains and the cooling coeffi-cient was done trying chains of 500, 1000, 2000 and 3000iterations and cooling coefficients of 0.80, 0.85, 0.90 and0.95. As for the stop criterion, three percentages of the ini-tial temperature were tried (nominally 1%, 0.5% and 0.1%).


A total of 96 heuristics were tried. It was concluded toadopt Markov chains of 2000 iterations, a cooling coeffi-cient of 0.90, the 10–30% limits for Medina’s method forthe initial temperature and 1% of the initial temperatureas a stop criterion. All 96 heuristics were done with moveM09, as recommended in Section 4.2 for the local descentstudy. Nevertheless, moves M05 and M17 were also triedfor the best combination of parameters and proved lesseffective than results for move M09. Table 3 shows themean, minimum and cost results for 25 runs with randominitial solutions. It can be seen that results have a deviationwith respect to the best of 1.41%, and, therefore, it may beconcluded that the method is almost insensitive to theselection of the initial solution. Fig. 7 shows a typical curveof cost reduction and temperature decrease following SA.Best results are a box frame of 4373.23€, which improveby 1.4% the cost obtained by the descent local search(4433.19€). (The main characteristics of this solution arean upper and lower slab thickness of 950 and 800 mm ofC25 and a wall thickness of 400 mm of C30; the amountof positive span reinforcement in the top slab is of 0.67%and of 0.16% in the upper negative corners.) Computerrunning times amount an average of 80 s, compared tothe 2.45 s of the local descent search and to the 28,800 sof the 500,000,000 random walk. It is worth noting thatthe 80 s · 25 runs amounts a total of 33.33 min for thewhole statistical analysis of a structure, which is a runningtime well acceptable for practical design purposes.

4.4. Threshold accepting

Finally, the threshold acceptance algorithm was againapplied to the road box frame of Lh = 13 m. TA calibra-tion is similar to SA. It requires an initial feasible solution


Table 3SA–TA cost results and times for 25 runs with mean and minimum values

RUN TA cost (€) TA time (s) SA cost (€) SA time (s)

R01 4390.55 94.95 4413.92 111.49R02 4417.66 93.08 4445.82 58.78R03 4373.23 112.99 4395.94 67.78R04 4426.53 124.98 4461.05 112.30R05 4390.40 123.44 4394.56 59.02R06 4387.71 122.38 4500.93 108.61R07 4428.91 121.38 4458.59 105.66R08 4415.50 126.33 4395.15 63.77R09 4390.15 122.77 4480.87 108.23R10 4388.50 122.84 4405.54 62.84R11 4421.68 122.97 4483.25 89.31R12 4408.71 117.44 4396.74 64.27R13 4420.57 85.34 4397.39 60.91R14 4447.74 88.73 4466.15 106.59R15 4422.83 118.58 4499.71 101.98R16 4386.66 91.41 4481.26 113.24R17 4443.10 125.30 4392.99 66.34R18 4428.60 88.78 4394.38 63.72R19 4419.33 88.05 4445.45 63.56R20 4388.90 86.36 4373.23 60.27R21 4400.27 127.89 4398.31 67.03R22 4390.15 124.23 4416.97 103.41R23 4412.42 120.78 4447.65 57.81R24 4432.29 90.47 4479.80 69.22R25 4387.71 87.24 4442.15 58.67

Minimum 4373.23 85.34 4373.23 80.19Mean 4408.80 109.14 4434.70 57.81Best dev. 0.81 1.41


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and selection of an initial threshold value, length of thecycle chains and a reducing coefficient of the threshold.Random initial solutions were used and a similar conclu-sion about the insensitivity of the initial solution wasachieved. The initial threshold was also calibrated by Med-ina’s criterion. The limits tried have been 30–50% and 50–70%. Three moves of the descent local search study were

4000

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5500

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6500

0 10 20 30 40

Fig. 7. Typical cost variation and temperat


tried (M05, M09 and M17). Similarly, the calibration ofthe length of the cycle chains and the reducing coefficientwas done trying chains of 500, 1000, 2000 and 3000 itera-tions and threshold reduction coefficients of 0.80, 0.85,0.90 and 0.95. And the stop criterion was tried for 1%,0.5% and 0.1% of the initial threshold value. A total of288 heuristics were tried. It was concluded to adopt moveM05, chains of 1000 iterations, a reduction coefficient of0.95, an initial temperature for limits 30–50% and a stopcriterion for 0.1% of the initial threshold value. Table 3shows the mean, minimum and cost results for 25 runs withrandom initial solutions. It can be seen that results have adeviation with respect to the best of 0.81%, and, again, itmay be concluded that the method is almost insensitiveto the selection of the initial solution. Fig. 8 shows a typicalcurve of cost reduction and threshold decrease followingTA. Best result is a box frame of 4373.23€ and it coincideswith SA best result. It is worth noting that results in Table3 indicate that there is myriad of near optimal solutionswhich are obtained by slight variations in the thicknessesof the elements, the material strength and the multitudeof similar reinforcement configurations. Hence, it appearsthat the problem at hand is multi-modal and, this is whyit is so important to use metaheuristics to escape from localoptima. As regards the basic concrete configuration, thealgorithm is very robust since about 52% of the solutionsshare the same configuration depicted in Fig. 9 for thethicknesses of the slabs and walls and concrete qualities.Computer running times amount about 109 s, comparedto the 80 s of the SA.

Additional runs were performed to check the impor-tance of the deflections and the fatigue limit states, whoseimportance has been reported in a previous work [25,26].If both limit states are ignored, the cost reduces to4201.18€ (3.9% reduction compared to 4373.23€); it staysin 4373.23€ when only fatigue is ignored and it reduces

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ure reduction for simulated annealing.


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thre

shol

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Fig. 8. Typical cost variation and threshold reduction for threshold accepting.

Fig. 9. Most economic design of RC road box frame with 13 m of horizontal free span.


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to 4258.65€ (2.6% reduction) when only deflections areignored. It is clearly concluded that the inclusion of bothlimit states is essential, since neglecting them leads to slen-derer and unsafe structures.

5. Analysis of results

Table 4 summarises the results of best cost and com-puter running times of the four heuristic methods. (Asmentioned earlier, computer times are for compiled Com-paq Visual Fortran Professional 6.6.0 in a Pentium IV of


2.4 GHz.) It can be seen that both simulated annealingand threshold lead to the same result in terms of cost,4373.23€, respectively, though TA improve SA deviationwith respect to the best from 1.41% to 0.81%. On the otherhand, the proposed SA improves running times from 109 to80 s. Both SA-TA metaheuristics improve substantiallyresults of random walk and descent local search heuristicsin terms of cost. Cost reduction goes from 4726.12€ for theRW to 4433.19 for the DLS and to 4373.29€ for the SA–TA procedures (7.5% and 1.4% reduction for the SA–TA). As regards running times, it appears that both


Table 4Summary of minimum and mean cost and computer time of fouroptimization methods

Heuristic algorithm Mean time(s)

Mean cost(€)

Minimum cost(€)

Random walk(500,000,000)

28800 – 4726.12

Descent local search 2.45 4878.61 4433.19Simulated annealing 80.19 4437.71 4373.23Threshold accepting 109.15 4408.80 4373.23


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DLS–SA–TA are acceptable for practical purposes, whilerunning time of 28,800 s of the RW appears to be not prac-tical. From the above results, it is concluded that the pro-posed TA is the overall most efficient procedure. (It isworth noting that TA was calibrated using move M05,30–50% of acceptances for the initial temperature, chainsof cycles of 1000 iterations, a threshold reduction coeffi-cient of 0.95 and 0.1% of the initial threshold as a stopcriterion.)

The results of the most economic box frame are shownin Fig. 9. It is important to note the slenderness of the geo-metrical variables and the fairly high passive reinforce-ment. The depth of the upper slab is 950 mm of C25(25 MPa of characteristic strength), which represents aslender span/depth ratio of 13.7. And the depth of the wallis 400 mm in C30, which represents a vertical span/depth

Fig. 10. Economic design of 900 m artificial box tunnel –


ratio of 15.4. The positive span reinforcement of the topslab is 0.67%, which is 41.20% the balanced ratio. The neg-ative upper corner reinforcement of the top slab is 0.16%,which is 9.82% of the balanced ratio. As for the wall, thenegative upper corner reinforcement is 0.73%, which repre-sents 37.40% of the balanced ratio. As regards the shearreinforcement, the corner of the top slab includes12.56 cm2/m, which represents 1.5 times the minimumshear reinforcement; and the walls do not include any shearreinforcement at all. The overall ratio of reinforcement inthe top slab is 83.5 kg/m3 and 80.7 kg/m3 for the wholebridge frame. It may, hence, be concluded that results ofthe optimization search tend to slender and fairly high rein-forced structural box frames.

As regards deflections and fatigue limit states, theirinclusion has shown to be crucial. Neglecting both limitstates leads to a 3.9% more economical solution, but obvi-ously unsafe. It is worth noting that deflections mainlyaffect the design of the top slab, which, therefore, requiresadditional tension and compression steel (the latter reduceslong-term deflections). As for the wall, a C30 concrete isneeded due to fatigue considerations. It should be takeninto account that the upper corner of the wall has impor-tant compressions due to vertical loads and it is also sub-jected to important corner negative flexure. It isimportant to note that fatigue checks are usually consid-ered in railways designs but, on the other hand, they are

underground line Palma to University of Illes Balears.



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commonly neglected in road structures design and, as it hasbeen shown, that this may lead to unsafe designs; speciallywhen optimization search procedures are used.

As an example of the practical application of theabove research, Fig. 10 shows the cross-section of anadditional road box frame. It corresponds to the designof a 900 m artificial box tunnel for the underground linePalma – UIB (University of Illes Balears). The mainparameters are a horizontal free span of 8.90 m, a verticalfree span of 6.00 m and an earth cover of 4.00 m withroad traffic loading on top. Several cross-sections for dif-ferent earth covers were designed to specifications fromthe work site that restricted to a single concrete gradeof 25 MPa characteristic strength, no stirrups in the lat-eral walls and a general spacing of reinforcement of200 mm. Additionally, the same thickness for the topand bottom slab was adopted. The cost of this solutionis 3941€/m and savings from tentative design office solu-tions are estimated at about 5%. The underground tunnelis currently under construction and it will be completedby the end of 2006.

6. Conclusions

From the above work, the following conclusions may bederived:

– The random walk procedure is not efficient, since lessthan 1% of the generated solutions are feasible. Therestriction to solutions that comply with minimumamount of passive reinforcement, together with therepair of minor unfeasible solutions, leads to a goodalgorithm for the generation of feasible solutions.

– From the moves studied in the descent local search,move M09 (random variation of 9 of the 50 variables)has shown the most efficient of the four moves M03,M05, M09 and M17.

– The proposed threshold accepting algorithm has shownthe overall most efficient procedure of the four includedin the present study. TA improves cost results of ran-dom walk and descent local search by 7.5% and 1.4%,respectively and it also improves deviation of resultswith respect to the best of SA from 1.4% to 0.81%, whileachieving same results in terms of cost.

– Structural results of the heuristic optimization searchhave shown to be slender and fairly high reinforced.The span/depth ratio of the top slab is 14, which is closeto the upper limit of common design values of 10–15.The amount of 84 kg/m3 in the top slab is also fairlyhigh when compared to practical designs.

– The inclusion of the deflections and fatigue limit stateshas shown crucial, since neglecting them leads to moreeconomic results (3.9% reduction), but unsafe designs.

– The example of the tunnel of the underground linePalma-UIB shows the practical applicability of themodel developed herein.


Acknowledgements

The five authors would like to acknowledge the fundingreceived from the Spanish Ministry of Public Works underresearch project 2004/80016/A04.

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