Engineering Structures 33 (2011) 2320–2329

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Engineering Structures

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Design of tall bridge piers by ant colony optimizationFrancisco J. Martínez 1, Fernando González-Vidosa ∗, Antonio Hospitaler 1, Julián Alcalá 1

Universidad Politécnica de Valencia, Instituto Ciencia Tecnología del Hormigón, Dept. Ingeniería Construcción, 46022 Valencia, Spain

a r t i c l e i n f o

Article history:Received 21 November 2010Received in revised form11 March 2011Accepted 1 April 2011Available online 7 May 2011

Keywords:Tall bridge piersStructural designEconomic optimizationAnt colony optimizationConcrete structures

a b s t r a c t

This paper describes a methodology for the analysis and design of Reinforced Concrete (RC) tall bridgepiers with hollow rectangular sections, which are typically used in deep valley bridge viaducts. Piers areusually considered tall when the shaft has a height of 50 m or more. Three different types of rectangularhollow tall piers have been studied for road piers of 90.00 m in height: RTRA90, RLON90 and RLT90.RTRA90 has the two side walls inclined, RLON90 has the two frontal walls inclined and RLT90 has all fourwalls inclined. The procedure used in the present study to solve the combinatorial problem is a variant ofthe ant colony optimization. RTRA90 leads to the most economical pier, both in column and foundationcost, since it is the most efficient set up for horizontal loads. Regarding the cost of the vertical columnonly, i.e. excluding the foundation, the cost of RTRA90 and RLON90 are similar, but the cost of the columnRLT90 is higher due to its larger unit cost of interior formwork.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Bridge piers are crucial for the design of prestressed concreteviaducts, especially when the piers are tall, since they can makeup more than 50% of the total cost of the viaduct. Fig. 1 showsa frontal and a side elevation of a hollow rectangular tall bridgepier. Tall bridge piers include a bottom foundation, which canbe either a surface footing as in Fig. 1 or include deep piles; themain hollow shaft with inclined walls; and the top block endpart that sustains the reactions due to the pair of pot bearingsof the bridge deck. Piers are usually considered tall when theheight of the shaft reaches 50 m or more. Shafts shorter than50.00 m are generally considered as high piers. High piers donot generally require inclined walls because a constant cross-section is sufficient and facilitates the construction procedure.The construction sequence is normally done in shaft stages ofapproximately 5.00 m in height. The main parameters affectingtheir design are the pier height, the vertical and horizontal loadstransferred by the bridge deck and the permissible ground stress.The behavior of a tall pier resembles that of a loaded cantilever.Rectangular hollow cross-sections aremost frequently used for tallpiers (see Fig. 2). These sections efficiently distribute the weightof an area and resist axial loading and the bending momentsdue to the eccentric traffic loading, together with the bending

∗ Corresponding author. Tel.: +34 963879563; fax: +34 963877569.E-mail addresses: fmm@postesa.com (F.J. Martínez), fgonzale@upv.es,

fgonzale@cst.upv.es (F. González-Vidosa), ahospitaler@upv.es (A. Hospitaler),jualgon@upv.es (J. Alcalá).1 Tel.: +34 963879563; fax: +34 963877569.

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.04.005

due to the horizontal loads at the top of the pier and along thecolumn. Additionally, the high radius of gyration of rectangularhollow cross-sections improves the strength against instability dueto second order effects. Piers are generally calculated to sustainthe actions prescribed by the loading code considered in theanalysis [1] and must comply with the limit states prescribed bythe concrete code under consideration [2].

An engineering model for the optimum design of high pierswith constant cross-sectionswas developed in a previous study [3],which described the optimization model in terms of cost function,design variables, parameters and structural constraints. Theoptimization methodology was based on two types of algorithms:population algorithms (ant colony and genetic) and neighborhood-based algorithms (simulated annealing and threshold accepting).The ant colony algorithm appeared to be more robust and waschosen for the present study of optimum design of tall piers.While the initial publication concentrated on the development ofan automatic design model for high piers, the present publicationupgrades and generalizes the model so as to cater to the analysisand design of tall piers of any height and variable cross-sections. Itis worth noting that high piers can be designed by a combinationof simplified buckling methods, spreadsheet calculations andelemental software for cross-section computations which areavailable to most postgraduate specialists. On the other hand, tallpiers require specialized software which is rarely available.

Most traditional procedures for structural concrete adopt initialdesigns based on cross-section dimensions, steel reinforcementand material grades arising from sanctioned common practice.The selection of initial solutions in the traditional approach isfollowed by the analysis of the structure and checking the passivereinforcement. Should the dimensions, reinforcement or material

F.J. Martínez et al. / Engineering Structures 33 (2011) 2320–2329 2321

Fig. 1. Typical frontal and side elevations of a tall bridge pier.

Fig. 2. Typical rectangular hollow section.

grades be insufficient, the structure is redefined on a trial-and-error basis. This process is not automatic and leads to safe designs,but the cost of the reinforced concrete (RC) pier is, consequently,highly dependent upon the experience of the structural designer.Modern artificial intelligence procedures define the structurebased on the design variables, automatically calculate and validate

the structure and then redefine it by means of an optimizationalgorithm that controls the flow of a large number of iterations inthe search for the optimum structure. This optimum structure hasto satisfy the limit states prescribed by concrete codes. Heuristicoptimization methods are a clear alternative to experience basedmethods. However, it is worth mentioning that experience is

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crucial for the development of computer design models sincedesign involves more than a mere application of codes of practice.This means that experience will move beyond preliminary designdecisions to the judgment required to develop computer designmodels like the one used for the present study. The recentdevelopment of heuristic methods, such as genetic algorithms,simulated annealing, threshold accepting, tabu search and antcolonies, among others [4–8], is linked to the evolution of personalcomputers. The first studies on heuristic structural optimizationwere applied to steel structures by Jenkins [9] and Rajeev andKrishnamoorthy [10]. These studies applied genetic algorithmsto the optimization of the weight of steel skeletal structures.Regarding RC structures, early applications include a pioneeringoptimization of RC beams by Coello et al. [11], and the applicationof genetic algorithms to prestressed concrete beams by Leite andTopping [12]. Another early work includes a study on geneticalgorithms applied to concrete members by Kousmousis andArsenis [13]; as well as a study on genetic algorithms appliedto RC columns by Rafiq and Southcombe [14]. Recently, therehave been a number of RC applications on RC beams and RCbuilding frames [15–18]. More recently, our research group hasapplied several metaheuristic algorithms to the optimization ofwalls, bridge frames, building frames, high bridge piers, vaults andpedestrian bridges [3,19–24].

The objective of the present publication is to guide designersto a methodology for the optimum modeling of tall piers. Thismethodology would reduce the time needed for preliminarydesign, analysis and overall design of tall bridge piers. Themethod followed consisted in formulating the optimizationproblem, modeling the structure on the basis of design variables,choosing the ant colony optimization algorithm and description ofnumerical examples.

2. Problem definition

2.1. Optimization problem definition

The problem of structural concrete design established inthe present study consists of an economic optimization of thestructural design of tall bridge piers. The structure has tominimizethe objective function F in expression (1), while also satisfying theconstraints of expression (2).

F(x1, x2, . . . , xn) =

−i=1,r

pi ∗ mi(x1, x2, . . . , xn) (1)

gj(x1, x2, . . . , xn) ≤ 0 (2)

xi ∈ (di1, di2, . . . , diqi). (3)

Note that x1, x2, . . . , xn are the design variables for the analysisdescribed in Section 2.2. Each design variable can take the discretevalues in a list in expression (3). The remaining data necessary tocalculate a pier are the parameters of the problem described inSection 2.3. The objective function in expression (1) and Section 2.4is an economic function expressed as the total unit prices, pi,multiplied by the construction unit measurements, mi (concrete,steel, formwork, etc.). Furthermore, the constraints in expression(2) and Section 2.5 are all the service and ultimate limit statesthat the structure must satisfy, as well as the geometrical andconstructability constraints of the problem.

2.2. Design variables

The design variables define all the magnitudes that change inthe optimization process. Design variables define the geometry,steel reinforcement and concrete type in the different parts ofthe pier. Any other data necessary to calculate a specific pier aredefined as parameters of the analysis. Logically, parameters are

not part of the optimization procedure, although their influencecan be studied in future design space studies. All variablesand parameters are discrete since the final solution has to beconstructable. The data considered in this paper are for pier P-1of the main viaduct over the river Palancia on the motorway A-23Sagunto–Somport [25]. This pier is the most heavily loaded pier ofthe viaduct with span lengths of 60–90–60 m. The pier supportsa 60 m span on the left side and a 90 m span on the right side.The deck width is 11.80 m, split in two lanes of 3.50 m in thecenter and 2.40 m for the shoulder and protection barrier on eachside. The height of the pier considered is 90.00 m. Note that a90 m pier is equivalent in height to a 28-story building. The pieris built in 19 stages. Stage 1 has 2.00 m of height, stage 2 to 18have 5.00 m of height, and the top stage has 3.00 m of height (seeFig. 1). In this study for tall piers, is included the possibility thatthe walls of the columns are not vertical. Three different designs ofrectangular hollow tall piers have been considered. The first typeof tall pier (RTRA90 henceforth) has the two side walls inclinedthat are parallel with the direction of the traffic and the other twofrontal walls are vertical, i.e. the side walls are inclined and haveconstant width, while the frontal walls are vertical and have avariable width. The second type of tall pier (RLON90 henceforth)has the two frontal walls inclined that are perpendicular to thedirection of the traffic and the other two side walls are vertical,i.e. the frontal walls are inclined and have a constant width andthe two side walls are vertical and have a variable width. Andthe third type of tall pier (RLT90 henceforth) has all four wallsinclined. Additionally, any inclination of the walls is constant fromthe bottom to the top of the piers, parallel walls have the sameinclination and the internal walls have the same inclination as theexternal walls, so all cross-sections of the pier maintain doublesymmetry. Solutions for these rectangular hollow piers add a totalof 279 variables for piers RTRA90 and RLON90 and 280 for pierRLT90.

The 279 design variables for the RTRA90 and RLON90 piersinclude 263 variables to define the column and 16 to define thefoundation. The RLT90 pier has one more variable since it has twovariables to define the slopes of the walls, while the RTRA90 andRLON90 piers have only one slope. The external dimensions ofthe cross-sections are fixed once the top section and the slopesare fixed. The frontal width of the top section is fixed in all casesto 5.24 m to adjust to the bottom transverse width of the bridgedeck. On the other hand, the side width of the top section isvariable in all cases. This variable can vary from 1.00 to 5.00 min steps of 0.05 m and it has to be larger than the dimensionof the pot bearing plus 0.20 m. Wall inclinations can vary from1/100 to 1/20 in steps of 1/2.50. Additionally, the option thatthere is no inclination is also included. The next 36 variables ofthe column are geometrical and correspond to the frontal andside thicknesses of the 18 hollow column stages into which thepier is split. The thicknesses of each stage must be equal toor smaller than those of the stage underneath. Thicknesses canvary between 0.25 and 0.75 m in steps of 0.025 m. The next19 column variables are the concrete qualities of the 19 columnstages, which must decrease with height. These qualities can varybetween the HA-25 and the HA-50 considered by the structuralcode EHE [2], the number indicating the characteristic compressivecylinder strength at 28 days. The remaining 206 column variablescorrespond to the steel reinforcement of the pier. The longitudinalvertical reinforcement of the column is defined by the spacing andthe diameter of the bars, which is different for the frontal andside walls and for the outer and inner faces. This means thereare 8 variables per stage and a total of 152 variables in the 19construction stages. The spacing can vary from 0.10 to 0.30 min steps of 0.02 m, and the diameters considered are 12, 16, 20,25 and 32 mm. The number and diameter of the bars must be

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equal to or smaller than those of the stage below. The shearhorizontal reinforcement accounts for 3 variables per hollow stage:the vertical spacing and the bar diameters in the frontal and sidewalls. The spacing varies from 0.10 to 0.30 m in steps of 0.025 m.This shear reinforcement involves a total of 54 variables (3 by18 hollow column stages). These 54 variables, together with the152 defining longitudinal reinforcement, total 206 variables forthe reinforcement. Finally, the reinforcement of the top stage ofthe pier is calculated and added to the measurement of passivereinforcement. It is important that all variables be discrete andnon-continuous. The tables of reinforcement include bar diametersand spacing, so all the ULS and SLS can be checked in detail.

There are 16 variables that define footing values. The first 5are geometrical and define the total depth of the footing, theplan dimensions of the footing and the plan dimensions of theplinth. The depth of the plinth is equal to half the total depthof the footing. The depth of the footing varies between 1.00 and4.00 m in steps of 0.10 m, and the plan dimensions of the footingmeasure between 8.00 and 25.00 m in steps of 0.25 m. The plandimensions of the plinth range from 4.00 to 25.00 m in steps of0.25 m. Another variable defines the type of concrete and the 10remaining variables define the reinforcement of the footing and theplinth.

The set of design value combinations for RTRA90 and RLON90(279 variables) and for the RLT90 (280 variables)may be defined asthe solution space. Such space is, in practice, unlimited due towhatis known as combinatorial explosion; the number of combinationsin this case is of the order of 1055 for RTRA90 and RLON90 and1056 for RLT90. Each vector of 279 or 280 variables, accordingto the case, defines a solution whose economic cost is given byexpression (1). Solutions that satisfy the constraints of the limitstates in expression (2) will be called feasible solutions. Those thatdonot satisfy all constraintswill be deemedas unfeasible solutions.

2.3. Parameters

The parameters of the analysis are all the magnitudes takenas fixed data. They are required to calculate the pier, but they donot vary during the optimization analysis. The parameters can begrouped as geometrical, actions on the pier, ground properties,partial factors of safety and durability exposure conditions. Aspreviously mentioned, the main geometrical parameter is theheight of the pier (90.00 m). Another geometrical parameter is thedimension of the frontal side of the top cross-section that is 5.24mand is given by the soffit of the bridge deck. The actions consideredtogether with the main parameters studied are summarized inTable 1. Other parameters are the number of bearings, their spacingand type of bearings. It has been considered that both are potbearings, one multidirectional and the other one unidirectional(i.e. it only allows the longitudinal movement). Concerning to theconstruction process, the parameters are the height of the columnstages (5.00 m) and the height of the top and bottom stages of thecolumn.

2.4. Cost function

The problem of optimization established in the present studyconsists of an economic optimization. The objective function inexpression (1) is the cost function, which is the sum of unit pricesmultiplied by the measurements of construction units, where piare the unit prices and mi are the measurements of the units intowhich the construction of the RC pier is split. The cost functionincludes the price of materials (concrete and steel) and all theentries required to evaluate the full cost of the pier, including,among others, the excavation of the foundation and its lateral fill.The basic prices considered are given in Table 2. These prices were

Table 1Basic parameters of geometry and actions of the pier.

Parameter Values

Transverse dimension of the top of the pier 5.24 mHeight of pier 90.00 mHeight of top end block 3.00 mHeight of formwork stage 5.00 mNumber of bearings 2Spacing of bearings 3.60 mEarth fill density 20.00 kN/m3

Permissible ground stress 500.00 kN/m2

Reactions permanent load SLS 12240, 12240 kNReactions maximum load SLS 15445, 14241 kNReactions maximum torque SLS 15690, 11442 kNReactions minimum loading SLS 11724, 11708 kNPot bearing unidirectional PU-1700 1040 × 1145 mmPot bearing multidirectional PL-1700 1040 × 1077 mmLongitudinal friction transmitted by the potbearings

1224 kN

Wind horizontal force 1627 kN

Table 2Basic prices of the cost function of the reported piers.

Unit Unit cost (e)

kg of steel (B-500S) 0.73m2 of foundation formwork 18.00m2 of exterior pier wall formwork 48.19m2 of interior pier wall formwork (2 vertical walls) 49.50m2 of interior pier wall formwork (4 inclined walls) 65.00m3 of footing concrete (labor) 6.20m3 of wall concrete (labor) 6.50m3 of concrete pump rent 6.01m3 HA-25 concrete 45.24m3 HA-30 concrete 49.38m3 HA-35 concrete 53.90m3 HA-40 concrete 59.00m3 HA-45 concrete 63.80m3 HA-50 concrete 68.61m3 of earth removal 3.01m3 of earth fill-in 4.81

obtained from national contractors of road construction in October2007. It is important to note that the price of the internal formworkis cheaper for piers RTRAN90 andRLON90 (2 verticalwalls) than forpier RLT90 (4 inclined walls).

2.5. Problem constraints

The problem constraints in expression (2) are structural con-straints. They are all the limit stateswithwhich the column and thefoundation must comply in accordance to EHE prescriptions [2].Additionally the problem includes the implicit constraints, whichare all constraints embedded in themodel definition regarding thegeometry, materials and the constructability of solutions. Theseimplicit constraints include, among others, the choice of a hollowrectangular section, the set of bar diameters, the set-up of rein-forcement, maximum and minimum thicknesses of the walls andgeometrical dimensions of the footing, and so on.

The structural constrains for bridge piers have been fullydiscussed in a previous publication [3], so they are only brieflyoutlined here. The columnmust comply with the ULS for buckling,shear and fatigue, and the SLS for cracking. The ULS for bucklingrequires the greatest amount of computing time. It was checkedwith the stiffness method as reported by Manterola [26]. First, aneccentricity is adopted in theweak direction from the constructionimperfection, forwhich the top value is that of Section 4.3.5.4 in theEurocode 2 [27], and a sine shape is assumed for the imperfection.From the factored actions and the construction imperfection, thedeformed shape is then calculated using the stiffness method,taking into consideration the stiffness of the different pier sections

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Fig. 3. Bottom cross-section for RTRA90 pier.

Fig. 4. Footing for RTRA90 pier.

calculated from the corresponding moment–curvature diagrams.This deformed shape gives the second-order bending momentson the pier which, added to the first-order bending moments,equals the total bending moments. It is then necessary tocheck the biaxial bending of all the sections which results ina new calculation of deformations. Should the biaxial bendingmoments exceed the resistance values, the solution is consideredas unfeasible. Deformations are calculated successively, and thecolumn is accepted as stable when the increment in deflectionsdecreases and converges. The process is repeated until thelongitudinal and transverse deflections differ by less than 5% fromthe value of the previous iteration. The procedure checks thatcompression and biaxial bending moments are acceptable in all

iterations. The integration of cracked sections is performed withthe Gauss–Legendre quadrature proposed by Bonet et al. [28]. Asregards the stress–strain relationships and the ULS domains fordeformation, the procedure uses those proposed in the EHE [2]corrected by 1 + ϕ∗, where ϕ∗ is the coefficient of reduced creepthat takes into account the percentages of axial and bendingmoments due to permanent loads as compared to the total values.In addition, the procedure checks all the constraints for minimumamounts of reinforcement due to flexural, shear and geometry asprescribed by EHE [2]. The footing is checked from the groundstresses calculated in the SLS. A trapezoidal block is used unlessthere is lifting, in which case a triangular distribution is used. Peakvalues can increase by 25% compared to the permissible ground

F.J. Martínez et al. / Engineering Structures 33 (2011) 2320–2329 2325

stress and the smallest value has to be positive. Reinforcementis checked in accordance with the EHE prescriptions, includingverification of flexure, shear, cracking and fatigue.

3. Ant colony model for tall bridge piers

The procedure used in the present work is a variant of theant colony optimization method [29,30]. The algorithm is basedon the behavior of ant colonies in their search to find sources offood. A single ant cannot do much on its own, but a group of antsbehaves as an intelligent system.When they leave the nest, the firsttrajectory of individual ants is primarily random. However, antsthat find food mark the path with a trace of pheromone. Hence,the trajectory of a second group of ants searching for food willdepend both on the trace of pheromone left by the first ants aswell as on a random component. Moreover, successive stages ofants strengthen the trace of already-explored paths or discovernew and shorter paths, where the trace pheromone is quicklyimproved since more ants follow the path in less time leavingadditional pheromone. Another factor is evaporation,which causeslonger paths to lose the trace of pheromone over time in contrastto shorter paths where the pheromone is replaced faster. In anycase, the random component of the search is never lost so that thediversity of the search is guaranteed. The details of the authors’ antcolony variants has been described in a previous publication [3],where all the details can be found. The ACO01 variant has beenchosen for this publication. The transition probability of thesevariants is given by expression (4):

P(t, k, i, j) = α(t) ·T (t, i, j)T (t, i)

+ β(t) · R (4)

where T (t, i, j) is the total trace at the end of stage t for variablei and position j, T (t, i) is the addition of all T (t, i, j), α and βdetermine whether the choice prefers the trace or the randomselection and R is a random number between 0 and 1. The resultsin the following section include results with initial values for αand β of 0.9–0.1 for the column and 0.8–0.2 for the footing, sothe influence of the trace left is more important than the randomchoice in the optimization process. In any case, α and β are madeto converge to 1 and 0 (α + β = 1) in order to converge to fulluse of the trace search with no exploration (random) search. Theconvergence ofα andβ to 1 and 0 is linearlymadewith the numberof stages, i.e.α = αo+(1−αo).t/tmax, where t is the number of thestage, and tmax is the total number of stages. Once the probabilityof each position j is known, the procedure generates ants bymeansof the roulette, taking into account the high or low probability ofchoosing a position.

4. Numerical results

The algorithm was programmed in Fortran Compaq VisualProfessional Edition 6.6.0. Computer runs were performed in aconventional PC computer with processor Core 2 Duo of 1.86 GHz.The optimization by ant colonies was applied to the same column(90.00 m in height), considering the three types of piers describedin Section 2.2, whose parameters are defined in Table 1. Theapplication of the algorithm described in Section 3 requires thedefinition of the initial values for α and β in expression (4), thenumber of ants in each stage, H , and the number of stages. Resultswere obtained for initial values for α and β of 0.9–0.1 for thecolumn and 0.8–0.2 for the footing. The values for α − β are madeto converge to 1 and 0 as the analysis progresses while α + β = 1.The number of ants considered in each stage is 50 and the numberof stages 100, so the product of the number of ants multipliedby the number of stages was kept constant at 5000 in the three

Fig. 5. Bottom cross-section for RLON90 pier.

Table 3Frontal width of the piers (transverse inclination equal to 1/20.00 for RTRA90 and1/22.50 for RLT90).

Height RTRA90 RLON90 RLT90

90.00 5.240 5.240 5.24087.00 5.390 5.240 5.37382.00 5.640 5.240 5.59677.00 5.890 5.240 5.81872.00 6.140 5.240 6.04067.00 6.390 5.240 6.26262.00 6.640 5.240 6.48457.00 6.890 5.240 6.70752.00 7.140 5.240 6.92947.00 7.390 5.240 7.15142.00 7.640 5.240 7.37337.00 7.890 5.240 7.59632.00 8.140 5.240 7.81827.00 8.390 5.240 8.04022.00 8.640 5.240 8.26217.00 8.890 5.240 8.48412.00 9.140 5.240 8.7077.00 9.390 5.240 8.9292.00 9.640 5.240 9.1510.00 9.740 5.240 9.240

cases studied. Due to the random nature of the results, multipleruns were performed for statistical purposes. The number of runswas fixed using a Student’s t-distribution and required that anapproximate 95% confidence interval of the population mean beestimated with an error less than 0.5% of the minor cost of theresults in the population in the first stage. The estimated erroris given by t2.5N−1

s√N, where t2.5N−1 is the Student’s t-distribution

coefficient, s is the standard deviation and N is the number ofruns.

Tables 3–8 show the different characteristics of the optimumpiers and Tables 9 and 10 the material measurements and the costdistribution, respectively. Figs. 3 and 4 show the bottom cross-section and the footing for RTRA90 pier. Similarly, Figs. 5 and 6show the same information for RLON90 and Figs. 7 and 8 for RLT90.

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Fig. 6. Footing for RLON90 pier.

Table 4Side width of the piers (longitudinal inclination equal to 1/20.00 for RLON90 and1/37.50 for RLT90).

Height RTRA90 RLON90 RLT90

90.00 5.000 3.750 3.50087.00 5.000 3.900 3.58082.00 5.000 4.150 3.71377.00 5.000 4.400 3.84772.00 5.000 4.650 3.98067.00 5.000 4.900 4.11362.00 5.000 5.150 4.24757.00 5.000 5.400 4.38052.00 5.000 5.650 4.51347.00 5.000 5.900 4.64742.00 5.000 6.150 4.78037.00 5.000 6.400 4.91332.00 5.000 6.650 5.04727.00 5.000 6.900 5.18022.00 5.000 7.150 5.31317.00 5.000 7.400 5.44712.00 5.000 7.650 5.5807.00 5.000 7.900 5.7132.00 5.000 8.150 5.8470.00 5.000 8.250 5.900

Table 5Thicknesses of the hollow column stages.

Stage RTRA90 RLON90 RLT90Frontal Side Frontal Side Frontal Side

19 0.250 0.250 0.250 0.250 0.250 0.25018 0.250 0.250 0.250 0.250 0.250 0.25017 0.250 0.250 0.275 0.250 0.250 0.25016 0.350 0.250 0.275 0.250 0.250 0.25015 0.350 0.250 0.275 0.250 0.250 0.25014 0.350 0.250 0.275 0.250 0.250 0.25013 0.350 0.250 0.275 0.250 0.250 0.25012 0.350 0.250 0.350 0.250 0.275 0.25011 0.350 0.250 0.350 0.250 0.275 0.25010 0.450 0.250 0.350 0.250 0.275 0.2509 0.475 0.250 0.375 0.250 0.275 0.2508 0.475 0.250 0.375 0.250 0.275 0.2507 0.475 0.250 0.375 0.250 0.275 0.2506 0.475 0.250 0.375 0.250 0.325 0.2505 0.500 0.250 0.375 0.250 0.325 0.2504 0.500 0.250 0.425 0.250 0.325 0.2503 0.500 0.250 0.525 0.275 0.375 0.2502 0.500 0.325 0.525 0.275 0.375 0.3501 0.575 0.350 0.750 0.275 0.425 0.650

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Fig. 7. Bottom cross-section for RLT90 pier.

Fig. 8. Footing for RLT90 pier.

Table 3 shows the frontal width of the column sections. Note thatthis dimension is the same in the top section and is given by thesoffit of the bridge deck (5.24 m). Table 4 shows the side width ofthe column sections. Table 5 shows the thicknesses of each stagein the frontal and side walls of the column. Table 6 shows theconcrete qualities of the column stages, which must decrease withheight. These qualities can vary between the HA-25 and the HA-50 considered by the structural code EHE, the number indicatingthe characteristic compressive cylinder strength at 28 days. Higher

resistances are required in the bottom sections because thesesections have to resist more compression. Note that the crackingSLS limits themaximum compression in concrete to 0.60fck, wherefck is the compressive characteristic strength. In addition, bottomsectionsmust also verify the fatigueULS,which ismore demandingin such locations.

Tables 7 and 8 show the vertical steel reinforcement in thefrontal and side walls, respectively. Regarding the horizontal shearreinforcement, it results in being equal to zero, except in the

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Table 6Concrete qualities of the column stages.

Stage RTRA90 RLON90 RLT90

19 25 25 2518 25 25 2517 25 25 2516 25 25 2515 35 30 3014 35 30 3513 35 30 3512 35 30 4011 35 30 4010 35 30 409 35 30 408 35 30 407 35 30 406 35 35 455 35 40 454 35 40 453 35 40 452 35 50 501 35 50 50

Table 7Vertical reinforcement in frontal walls.

Stage RTRA90 RLON90 RLT90External Internal External Internal External Internal

19 31Ø12 21Ø12 49Ø12 18Ø12 50Ø12 18Ø1218 31Ø12 22Ø12 49Ø12 18Ø12 52Ø12 18Ø1217 31Ø12 23Ø12 49Ø12 18Ø12 54Ø12 18Ø1216 31Ø16 24Ø12 49Ø12 18Ø12 56Ø25 21Ø1215 31Ø16 25Ø12 49Ø12 18Ø12 58Ø25 21Ø1214 31Ø16 26Ø12 49Ø12 18Ø12 61Ø25 21Ø1213 31Ø16 27Ø12 49Ø12 18Ø12 63Ø32 21Ø1212 31Ø16 28Ø12 49Ø12 18Ø12 65Ø32 24Ø1211 31Ø16 29Ø12 49Ø12 18Ø12 67Ø32 24Ø1210 31Ø16 30Ø12 49Ø12 18Ø12 69Ø32 24Ø129 31Ø16 31Ø12 49Ø12 18Ø12 71Ø32 24Ø128 65Ø16 32Ø12 49Ø12 18Ø12 73Ø32 28Ø127 67Ø16 33Ø12 49Ø16 18Ø12 75Ø32 28Ø126 69Ø16 34Ø12 49Ø16 18Ø12 77Ø32 28Ø125 71Ø16 35Ø12 49Ø20 18Ø12 79Ø32 28Ø124 73Ø16 37Ø12 49Ø25 18Ø16 81Ø32 28Ø123 75Ø32 38Ø16 49Ø25 18Ø16 84Ø32 28Ø122 77Ø32 38Ø25 49Ø32 18Ø32 86Ø32 28Ø251 77Ø32 38Ø32 49Ø32 18Ø32 86Ø32 53Ø25

Table 8Vertical reinforcement in side walls.

Stage RTRA90 RLON90 RLT90External Internal External Internal External Internal

19 49Ø12 18Ø12 38Ø12 33Ø12 15Ø12 17Ø1218 49Ø12 18Ø12 41Ø12 36Ø12 15Ø12 17Ø1217 49Ø12 18Ø12 43Ø12 38Ø12 19Ø12 17Ø1216 49Ø12 18Ø12 46Ø12 40Ø16 19Ø12 17Ø1215 49Ø20 18Ø12 48Ø12 42Ø25 19Ø12 17Ø1214 49Ø20 18Ø12 51Ø12 45Ø32 19Ø12 17Ø1213 49Ø20 18Ø12 53Ø12 47Ø32 19Ø12 17Ø1212 49Ø32 18Ø12 56Ø20 48Ø32 19Ø12 17Ø1211 49Ø32 18Ø12 58Ø20 50Ø32 19Ø12 17Ø1210 49Ø32 18Ø12 61Ø32 53Ø32 41Ø12 17Ø129 49Ø32 18Ø12 63Ø32 55Ø32 42Ø12 17Ø128 49Ø32 18Ø12 66Ø32 57Ø32 43Ø16 17Ø127 49Ø32 18Ø12 68Ø32 60Ø32 44Ø20 17Ø126 49Ø32 18Ø12 71Ø32 62Ø32 45Ø20 17Ø125 49Ø32 18Ø16 73Ø32 64Ø32 46Ø20 35Ø124 49Ø32 18Ø16 76Ø32 65Ø32 47Ø20 36Ø123 49Ø32 18Ø32 78Ø32 65Ø32 49Ø25 36Ø122 49Ø32 18Ø32 81Ø32 65Ø32 50Ø25 36Ø201 49Ø32 18Ø32 81Ø32 65Ø32 50Ø32 36Ø20

bottom stage of RLT90, where it is �12 spaced 200 mm in thefrontal walls (see Fig. 7). The longitudinal reinforcement of thecolumn is defined by the spacing and the diameter of the bars,

so that the number of bars depends on the width of the section.Figs. 4, 6 and 8 show the footings for the different piers. Regardingthe geometry of the footings, all piers have a similar depth ofabout 4.00 m. RTRA90 and RLT90 have similar plan dimensions,but plan dimensions for RLON90 are larger. This latter footing hasits longitudinal dimension larger than the transverse dimension, asit also happens for the bottom cross-section of the column. Largedimensions of the pier footings are due to the smallest value ofground stress, which has to be positive since lifting of the footingsis not allowed. Concrete quality is the lowest (HA-25) for RTRA90and RLT90. The RLON90 footing has a greater HA-30 concretequality since the bigger footing plan dimension requires it. Thereinforcement in the footing is defined by the number of bars inthe longitudinal and transverse directions, the number of layersand the bar diameters.

Table 9 shows the material measurements of the differentparts of the tall piers. RTRA90 and RLT90 footings have similarmeasurements regarding the plan dimensions (earth removal,earth fill-in, formwork and m3 of concrete). However, RLT90 has79870 kg of steel, while RTRA90 a lower quantity of 66268 kg ofsteel. The amount of steel kg/m3 of concrete is similar for RTRA90and RLON90 and is about 65 kg/m3, while in RLT90 it is about77 kg/m3. The measurements in the top block are similar forRLON90 and RLT90. These measurements are larger for RTRA90due to the larger dimensions of the top block. Regarding to thehollow section, RLON90 has the largest vertical reinforcement,but on the other hand it has the smallest m2 of wall formworkand concrete volume. The RTRA90 column has the smallestreinforcement, but the largest m2 of wall formwork and concretevolume. Consequently, RLON90 has the greatest amounts of kg ofreinforcement by column lineal meter and of kg of reinforcementby m3 of concrete. Table 10 shows the cost distribution of thematerialmeasurement in piers, aswell as the total cost. Total cost isthe sum of unit prices (Table 2) multiplied by themeasurements ofconstruction units (Table 9). The total costs of the RTRA90, RLON90and RLT90 piers are 468858, 535071 and 499901, respectively.RTRA90 pier has the column with the lowest cost, the maindifference with RLT90 column is the greater internal formworkcost in RLT90. Although measurements of formwork are similarin RTRA90 and RLT90 the unit price of the interior formwork isgreater in RLT90 than in RLON90 and RTRA90 because the fourwalls of the column in RLT90 are not vertical, so it is necessary toemploy a crane and other auxiliary means for taking the interiorformwork down to the ground and vary their the dimensions forthe next stage of the column. This maneuver is only necessary inRLT90 and increases the unit cost of the interior formwork. Thecolumn cost of RLON90 is similar to that of RTRA90, despite havingmore reinforcement. The RTRA90 footing is the least cost footingand this cost is similar in cost to the RLT90, the main differencebetween them being the cost in the reinforcement. The RLON90footing has the greatest cost, because of its larger measurements.

5. Concluding remarks

Three different types of rectangular hollow piers have beenstudied for tall road piers of 90 m in height: RTRA90, RLON90and RLT90, defined in Section 2.2. RTRA90 has the two side wallsinclined, RLON90 has the two frontal walls inclined and RLT90 hasthe four walls inclined. The optimization procedure used in thepresent study is a variant of the ant colony optimization describedin Section 3 and requires the definition of the initial values forα and β in expression (4). Numerical results indicate that themain cost difference between the shafts is due to the differencesin the unit price of the internal formwork, which is larger for 4inclined walls (RLT90) than for two inclined walls only (RTRAN90and RLON90). The difference of this cost is due to the fact when

F.J. Martínez et al. / Engineering Structures 33 (2011) 2320–2329 2329

Table 9Material measurements.

RTRA90 RLON90 RLT90

m3 of earth removal 1 664.30 2409.75 1682.10m3 of fill-in 645.94 962.55 676.76

Footing kg of steel (B-500S) 66268.25 101138.16 79870.32m2 of formwork 339.51 476.56 349.15m3 of concrete 1 048.73 1491.85 1037.74kg steel/m3 concrete 63.19 67.79 76.97kg of steel (B-500S) 6 268.66 5533.37 5419.19

Top Block m2 of formwork 88.88 74.87 72.35Pier m3 of concrete 79.72 60.13 56.36

kg steel/m pier 69.65 61.48 60.21kg steel/m3 concrete 78.63 92.02 96.15kg of steel (B-500S) 108942.62 160205.15 122626.70m2 of formwork 4141.71 3725.09 4001.54

Hollow pier m3 of concrete 747.34 568.16 570.53kg steel/m pier 1 210.47 1780.06 1362.52kg steel/m3 concrete 145.77 281.97 214.93

Table 10Cost distribution (euros).

RTRA90 RLON90 RLT90

Earth removal 5 009.54 7253.35 5063.12Earth fill-in 3 106.99 4629.87 3255.23

Footing Steel 48375.82 73830.86 58305.33Formwork 6111.13 8578.08 6284.61Concrete 60249.62 91883.04 59617.95Total 122853.10 186175.20 132526.24Steel 4 576.12 4039.36 3956.01

Top Block Formwork 6231.22 5085.08 4877.10Pier Concrete 4 604.12 3472.45 3254.60

Total 15 411.46 12596.89 12087.71Steel 79528.11 116949.76 89517.49

Hollow pier Formwork 202149.35 181811.22 224843.04Concrete 48916.67 37538.75 40927.49Total 330594.13 336299.73 355288.02

Pier Total 468858.69 535071.82 499901.97Cost/m pier 5 209.54 5945.24 5554.47

the four walls are not vertical, it is necessary to employ a crane andother auxiliarymeans for taking the interior formwork down to theground and vary its dimensions for the next stage of the column.This maneuver increases the unit cost of the internal formwork.On the other hand, the cost of columns RTRA90 and RLON90 issimilar, and hence it is not relevant for the column cost the choiceof which two parallel walls are not vertical. Nevertheless, RLON90is the pier with the largest cost due the cost of its footing. RTRA90is the most economic pier, its column being the cheapest and itsfooting being the cheapest as well. This so since the wind force isthe biggest horizontal force and the wind direction coincides withthe direction of variable wall widths for RTRA90.

Acknowledgments

This study was funded by the Spanish Ministry of Education(research project BIA2006-01444). The authors are grateful for thethorough revision of the manuscript by Ms. Lia Wallon.

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