Research ArticleEvolutionary Modeling to Evaluate the Shear Behavior ofCircular Reinforced Concrete Columns

Alessandra Fiore Giuseppe Carlo Marano Daniele Laucelli and Pietro Monaco

Department of Science of Civil Engineering and Architecture Technical University of Bari (Politecnico di Bari) Via Orabona 470125 Bari Italy

Correspondence should be addressed to Alessandra Fiore afiorepolibait

Received 27 February 2014 Revised 4 August 2014 Accepted 18 August 2014 Published 7 September 2014

Academic Editor Hossein Moayedi

Copyright copy 2014 Alessandra Fiore et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Despite their frequent occurrence in practice only limited studies on the shear behavior of reinforced concrete (RC) circularmembers are available in the literature Such studies are based on poor assumptions about the physical model often resultingin being too conservative as well as technical codes that essentially propose empirical conversion rules On this topic in thispaper an evolutionary approach named EPR is used to create a structured polynomial model for predicting the shear strength ofcircular sections The adopted technique is an evolutionary data mining methodology that generates a transparent and structuredrepresentation of the behavior of a system directly from experimental data In this study experimental data of 61 RC circularcolumns as reported in the technical literature are used to develop the EPR models As final result physically consistent shearstrength models for circular columns are obtained to be used in different design situations The proposed formulations arecompared withmodels available from building codes and literature expressions showing that EPR technique is capable of capturingand predicting the shear behavior of RC circular elements with very high accuracy A parametric study is also carried out to evaluatethe physical consistency of the proposed models

1 Introduction

It is well known that columns are the most vulnerable ele-ments in reinforced concrete (RC) structuresThese elementsare principally designed to bear axial loads but as a resultof lateral loads for example wind pressure or earthquakeground motion they could deal with relevant shear loadsand thus should inevitably be designed to avoid possibleshear failures [1ndash6] Composite and steel columns [7] areparticularly suitable at this aim

Circular columns are very popular for bridge pier designdue to simplicity of construction and because their strengthcharacteristics under wind and seismic loads are similar inany direction Circular elements are also used extensivelyas columns in buildings or as piles for foundations or assecant piling to form diaphragm walls Despite their frequentoccurrence in practice only limited researches on the shearbehavior of RC circular members have been carried out

Most of the researchers and codes state that the shearstrength 119881Rd of a beam is the sum of the contributions

of concrete 119881119888and shear reinforcement 119881

119904 if present The

first takes into account the shear stress transferred by thecompressed zone of the beam the dowel action the aggregateinterlock and the arch effect and depends on many factorssuch as the tensile longitudinal steel ratio concrete gradeaggregate size or shear span among othersThe evaluation ofthis term is debatable and originates on empirical methodsTraditional codes therefore propose empirical formulasderived from various tests developed in rectangular sectionsand then extrapolating results for arbitrary sections suchas circular ones Also the evaluation of the contribution ofthe hoops obtained from the truss analogy for rectangularsections was later extended to arbitrary sections but nospecific address was developed for circular ones

A very limited number of shear models for circularsection members exist in the literature Ghee et al [8]proposed a model for columns subjected to cyclic loadingconsisting of a purely empirical concrete and a classical shearreinforcement contribution where shear crack inclination

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2014 Article ID 684256 14 pageshttpdxdoiorg1011552014684256

2 Advances in Civil Engineering

was obtained by using a lower bound plasticity model Theconcrete contribution for low flexural ductility was definedas the ldquoinitial concrete strengthrdquo and calculated as strength atmaximum lateral load the transverse reinforcement capacitywas calculated from 45-degree truss mechanisms assumingthat transverse reinforcement exposed by a presumed 45-degree diagonal crack is at yield At higher flexural ductility adegradation of the concrete term was assumed and the ldquofinalconcrete strengthrdquo was proposed Different adjustments tothis basic model have been added over the years by variousauthors Priestley et al [9] modified the model proposed byGhee et al [8] and separated the shear capacity term of theaxial load arch mechanism 119881

119901 from the concrete term 119881

119888

Other modifications to Ghee et al original model [8] weredeveloped by Wong et al [10] and by Kowalsky and Priestley[11] In this work the authors revised the model proposedby Priestley et al [9] and the effect of the aspect ratio andlongitudinal reinforcement ratio was incorporated

In addition the truss mechanism component developedby Ghee et al [8] has been slightly modified assuming thatin themember compression zone the cracks are by definitionclosed

A different approach was due to Collins et al [12] bymeans of the application of the Modified Compression FieldTheory (MCFT) [13] to deal with shear strength predictionof circular members Moreover this type of approach ismore complex than those usually based on strut and tiemodels which are still used in many applications till theirfirst implementation in early 1900 thanks to their simplicityThe practical application of the MCFT requires its imple-mentation as a calculation algorithm within a computerroutine thus strongly increasing the complexity level also fora standard code formulation

Bentz et al [14] described a simpler version of theMCFT from which the shear design rules described inAASHTO LRFD [15] were derived and which also containsrules for shear design of circular members Turmo et al [16]proposed an analytical model for evaluating the contributionof the transverse reinforcement in concrete members ofsolid and hollow circular cross section consisting in anaccurate formula for evaluating the shear transferred byspiral reinforcement in solid members The calculation ofhollow core circular columns with both vertical and spiralreinforcement was also deduced

Recently Merta [17] proposed a model based on the trussanalogy by adding a concrete contribution term to the capac-ity of the shear reinforcement An additional deviatoric shearresisting mechanism of hoops present exclusively in mem-berswith curved transverse reinforcement was identified andexpressed analytically It is explained by the fact that a curvedreinforcing bar under tension induces compression in radialdirection as well The component of this compressive forcein the direction of external shear could thus be consideredas an additional shear enhancing mechanism of the hoopsIts magnitude is expressed through the friction force thatis present between the concrete and steel after the sectionis cracked and the bond is partially destroyed Under theseassumptions the concrete shear capacity was derived by aparametric study

11 Experimental Approaches to Shear Capacity of CircularMembers Modeling As for theoretical researches also exper-imental studies on the shear capacity of circular membersare quite limited Assuming that the extrapolation of tests onrectangular beamsmay not be accurate for evaluating circularmembers shear strength some researchers performed differ-ent test campaigns in such structural elements Moreoverall the above-mentioned models are derived starting froma mechanical simplified model whose parameters comefrom the extrapolation of experimental tests for evaluatingcircular member shear strength In their paper Clarke andBirjandi [18] reported also test results from different previousexperimental studies in particular

(i) Capon and Cossio [19] who tested 21 membersmainly of 250mm in diameter having only longitu-dinal steel but four that had stirrups eleven of thespecimens are reported as failing in shear two withstirrups and nine without

(ii) Khalifa and Collins [20] who tested five columns allof 445mm in diameter these were subject to an axialload of 1000 kN equivalent to a stress of 64MPa andall specimens failed in shear

(iii) Nagato [21] whose tests involved 16 columns all witha diameter of 300mm with and without transversereinforcement and with and without axial stress allspecimens are reported as failing in shear

Later on Ghee et al [8] tested 25 circular columnsunder axial load and cyclic lateral inelastic displacements24 of them failing in shear The columns had a diameterof 400mm Variables in the test included axial load levellongitudinal reinforcement ratio transverse reinforcementratio and aspect ratio They addressed for the first timethe efficiency of circular hoops and deduced the formulafor calculating 119881

119904presuming a 45-degree diagonal tension

crack and vertical reinforcement The work of Ghee et al[8] was completed by Priestley et al [9 22] which takesinto account the potential development of steeper angles ofcracking other than 45-degree one Various experimentalresearches on circular RC members loaded monotonicallyin shear were aimed at verifying the possibility to extend tocircular sections the use of design equations developed forrectangular sections [18ndash20]

Lee et al [23] tested 4 full scale circular column spec-imens which show shear related behavior under cycliclateral load with constant axial force The test variables areaspect ratio transverse steel configuration (with or withoutcrossties) and longitudinal steel ratio The columns showedflexural failure or flexure-shear failure depending on the testvariables The test results were compared with shear strengthequations adopted by the bridge seismic design specificationsor guidelines and proposed by other researchers The studyshowed that the accuracy of each method for shear strengthprediction also depends on the aspect ratio and the relativesteel amount of longitudinal reinforcement and transversereinforcement

More recently Jensen and Hoang [24] presented theresults of a test series on heavily shear reinforced circular

Advances in Civil Engineering 3

concrete members The specimens had shear reinforcementpercentages up to more than three times the maximumpercentage found in existing tests The test results indicatethat it is possible to obtain shear strengths which exceedthe upper limit usually imposed on rectangular membersThe test results are compared with a recently developedplasticity-based shear model for circular members Satisfac-tory agreement was found Comparisons were also madewith calculations using the AASHTO LRFD [15] design codeIt was found that the AASHTO LRFD design code givesreasonable results for members with small amounts of shearreinforcement while it underestimates the shear strength forheavily shear reinforced members

This paper proposes an innovative approach mainlybased on experimental data in order to develop an explicitcompact expression to assess the shear strength of RC circularcolumns Such approach is based on the evolutionary poly-nomial regression (EPR) [25] which is a hybrid data-driventechnique that combines the effectiveness of evolutionarysearch with the advantage of classical numerical regressionfor developing data-driven models (ie input-output rela-tionships) featured by explicit mathematical expression TheEPR technique has been successfully applied to modelinga wide range of complex engineering problems includingconstitutive modeling of soils stability of slopes settlementof foundations liquefaction of soils due to earthquake and anumber of other applications in civil engineering [26ndash28]

The EPR implements a multiobjective genetic algorithms(MOGA) optimization paradigm in order to search formodel expressions as trade-off between fitness to data andcomplexity of mathematical formulations [29] The mainadvantage of the MOGA EPR approach is the possibility toselect models from a set of optimal solutions according to thespecific application and to the physical insight of the analyst

2 Proposed Models for Shear Strength andContemporary Code Provisions of CircularReinforced Concrete Columns

Shear strength of reinforced concrete has received consid-erable attention in research during the past century Severalmodels for column shear strength have been proposed andused for design of new buildings and assessment of existingbuildings

Shear reinforcements are used to ensure that the elementfails in flexure the addition shear reinforcement affects themechanism by which shear is carried by a beam in sev-eral ways shear reinforcement carries tensile actions acrosscracks confines the compression-zone concrete increasing itsshear capacity encloses the flexural reinforcement and canprevent dowel-splitting of the concrete Moreover for a givenapplied load equilibrium of a cracked section with stirruprequires a shorter crack length but a larger crack width thatone without stirrup and the shape of the crack will also differShear transfer mechanism in beamwith stirrups has not beenexamined in somuch detail as that in beams without stirrups

Truss analogies are most commonly used in the designThe assumed internal equilibrium state comprises tensile

shear reinforcement and inclined compressive struts of con-crete The original Morsch truss analogy [30] uses a 45∘ strutangle and predicts failure when the shear reinforcementsare at yield The modified truss analogy [31] established anoptimal lower bound for the shear capacity by varying thecompressive strut angle to give reinforcement yield and webconcrete failure simultaneously

Even if shear transfer mechanisms are qualitatively wellknown there is no agreement on the quantification of theshear strength of concrete members As the value of 119881

119904can

be easily calculated with rational models such as the trussanalogy research focuses on the elaboration of methods foran accurate evaluation of 119881

119888 The concrete shear capacity

is the capacity of the section without shear reinforcementdue to the complex stress redistribution after cracking theshear transfer mechanisms of RC element have not beenclearly understood yet Its evaluation is very controversial andalways relies somehow on empirical methods consideringits dependence from various factors such as the tensile lon-gitudinal steel ratio concrete grade aggregate size or shearspan among others Present shear researches concern useof self-training algorithms to predict 119881

119888or the development

and application of equilibrium methods and equilibriummdashcompatibility methods such as the Modified CompressionField Theory [13] The same interest has not been dedicatedto improve the evaluation of the shear reinforcement contri-bution

In this framework the shear strength of circular concretecolumns has not received as much attention as rectangularones and it is interesting to note that there are not guidelinesfor shear capacity in some international codes likewisemanyothers prescribe that the shear capacity of a circular sectionequals the capacity of an equivalent rectangular section [32]In effect this kind of approach has not been fully validatedby test evidence and it obviously is questionable because incircular sections hoops contribute differently to the shearstrength compared with rectangular stirrups

In this section a review of the shear strength provisionsof various contemporary design code and proposed modelsis developed for circular columns Most design codes includecontributions from concrete119881

119888and transverse reinforcement

119881119904to analyze the shear strength of circular columns The

two components are then summed to estimate the total shearstrength in

119881 = 119881119888+ 119881119904 (1)

One should point out that generally codified methods ofdesign cannot be considered as predictive equations sincethey are intended to provide only a conservative and safelower bound to strength Therefore most shear design codesuse empirical or semiempirical models for predicting shearstrengths The American code [15] and the New Zealand one[33] explicitly refer to circular concrete members but withsimplified methods These recommend the calculation of theshear carried by the truss mechanism by representing thesection as an equivalent rectangle having a width equal to thediameter119863 and an effective depth equal to 08119863

Based on a large number of tests on circular cantilevercolumns under uniaxial load and multidirectional cyclic

4 Advances in Civil Engineering

displacements Ghee et al [8] proposed a model for the shearstrength of circular sections under cyclic load by adopting anadditive approach based on the contribution from concreteand transverse reinforcement The concrete contribution forlow flexural ductility (120583 lt 2) was defined as the ldquoinitialconcrete strengthrdquo and calculated as a strength at maximumlateral load as follows

119881119888= 037120572(1 +

3119875

1198911015840119888119860119892

)radic1198911015840119888119860119890

(in N) (2)

where 119875 is the axial load 119860119892is the cross-sectional area

and 1198911015840

119888is the concrete compressive strength The effective

area 119860119890is proposed as 08 times the cross-sectional area 119860

119892

which approximately corresponds to the area of the confinedconcrete core

For low aspect ratios (119886119863 lt 2) the shear capacityenhancement factor 120572 was proposed as

120572 =2

119886119863ge 10 (3)

where 119886 is the shear span and 119863 is the sectionrsquos diameterThe transverse reinforcement capacity was calculated from45-degree truss mechanisms as follows

119881119904=120587

2

119860 sh119891yh1198631015840

119904 (4)

where 119860 sh is the area of the shear reinforcement 119904 is thehooprsquos spacing 119891yh is their yield strength and 119863

1015840 is the corediameter measured at the center line of the transverse hoopor spiral It was assumed that all transverse reinforcementsexposed by a presumed 45∘ diagonal crack are at yield Anintegral averaging was imposed assuming that the spacingof the shear reinforcement is sufficiently small comparedto the diameter It was noted that for low 119863

1015840

119904 ratios theequation can be up to 10 nonconservative At higherflexural ductilities (120583 gt 2) the degradation of the concreteproduces a reduced concrete contribution and then a ldquofinalconcrete strengthrdquo is proposed One should point out thatthis formulation such as other ones implicitly assumes thatall hoops exposed by a crack are at yield and also consider aldquosmearedrdquo distribution of transverse reinforcement whereasin reality transverse reinforcement consists of a discretenumber of hoops that traverse a diagonal crack plane asshown in Figure 1 The later assumption is valid just if thepitch is sufficiently small compared to the diameter1198631015840

Clarke and Birjandi [18] proposed to use for circular sec-tions the same shear design approach as given by the Britishcodes of practice BS 5400 [34] for rectangular sections

However a modification was suggested that is that thesectionrsquos effective depth should be considered as the distancefrom the extreme compression fiber to the centroid of thetension reinforcement

Following this the effective shear area represents the areacorresponding to the effective depth

The design approach for rectangular section in Britishcodes of practice [34] consists in adding together the concrete

c

d

cov

0 1

D

x

y

Vd

Aswfyw

i ntmiddot middot middotmiddot middot middot

120579

(D minus c minus cov)cot120579

Figure 1 Geometrical and mechanical characteristics affecting theshear strength of a circular column

and transverse reinforcement contributions to the shearcapacity The concrete term is defined as

119881119888= [027120572(

100119860 sl119887119908119889

)

13

(500

119889)

14

(119891cu)13

] 119887119908119889 (5)

where 119860 sl is the area of longitudinal steel 119887119908 is the sectionrsquoswidth 119889 is the effective depth and 119891cu is the concrete cubestrength For loads applied at a distance119886V closer than 2119889 fromthe support the concrete shear capacity is increased by

120572 =2119889

119886Vge 10 (6)

The shear force carried by transverse reinforcement is calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (7)

where 119860 sh is the cross-sectional area of the linkrsquos both legs atthe sectionrsquos neutral axis In amember with axial compressionload 119875 the shear capacity should be multiplied by (1 +

005119875119860119892)

Dancygier [35] modified Gheersquos model by adopting adiscrete computation of tension-force components developedby reinforcing hoops in the direction of the shear force givenby the following expression

119881119904= 2119860 sh119891yh (

119899

sum

119894=1

cos120572119894) (8)

where 119894 is the index of the hoop that is crossed by the inclinedcrack 120572 is the angle between its force (119860 sh119891yh) and thedirection of the shear force given by

sin120572119894= 1 minus

2119886119894

1198631015840 119886119894= [1199040+ (119894 minus 1) 119904] cot 120579 (9)

and 119899 is the number of the hoops that are crossed by the crack

119899 = roundup(1198631015840

119904tan 120579 minus

1199040

119904) (10)

where 119904 is the pitch 120579 is crackrsquos inclination angle and 1199040is

the distance from the end of the crack to the first hoop that iscrossed by it

Advances in Civil Engineering 5

Kowalsky and Priestley [11] discussed Gheersquos modelconsidering that in the derivation of the truss mechanismequation it was assumed that a diagonal crack is ableto mobilize transverse reinforcement along a crack lengthextending the full width of the confined core of the concreteIn the compression zone of the column however all cracksare by definition closed Therefore if the crack is closedshear cannot be transferred across it by tension strain inthe transverse reinforcement Therefore it is evident that areduced column width minus119863 minus 119888 minus cov minus (Figure 1) is appropri-ate for calculating the number of hoops mobilized by thecracks between the compression struts and then a revisedtruss component equation should be adopted The authorsobtained an approximated equation given by

119881119904=120587

2119860119904119908119891119910119908

119863 minus 119888 minus cov119904

cot 120579 (11)

where 119860119904119908

is the area of the hooprsquos one leg 119891119910119908

is the hooprsquosyield strength 119888 is the compression zonersquos depth and cov isthe concrete cover

With reference to the concrete mechanism the followingequation was discussed in the cited work

119881119888= 120572120573120574radic1198911015840

119888(08119860

119892) (12)

where the parameter 120572 accounts for the column aspectratio the factor 120573 is a modified factor that accounts forthe longitudinal steel ratio and 120574 represents the reductionin strength of the concrete shear resisting mechanism withincreasing ductility

Also Kim and Mander [36] reported a deficiency of themodel originally proposed by Ghee et al [8] and verifiedthat the applied integral averaging by the reinforcement termrestricts its use only to members with diameter at leastfour times the spacing of the shear reinforcement For allother ratios the formula could be even more than 50nonconservative For this reason they proposed a correctivecoefficient to convert the amount of shear implied by acontinuous truss model to the discrete model

The current ACI 318-M-08 [37] considers a portion ofthe design shear force to be carried by the concrete shearresisting mechanism 119881

119888 with the remainder carried by

trussmechanism119881119904 involving transverse reinforcementThe

code proposes the following equation to calculate 119881119888for

members subjected to combined shear moment and axialcompression

119881119888= 017(1 +

119875

14119860119892

)radic1198911015840119888119887119908119889 (13)

The transverse reinforcement contribution is also calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (14)

where the meaning of various terms has been explainedbefore

Merta [17] developed a simple shear capacitymodel of RCcircular sections It was based on the truss analogy by addingan empirical concrete contribution term to the capacity of theshear reinforcement obtaining finally the following relation

119881 = [37120588119897+ 018 + 008(

119875

119860119892

)03] 119896radic119891101584011988807119860119892

+ 119860119904119908119891119910119908

(18119899119905+ 120582119899119889minus 1)

(15)

where 119899119905represents the number of hoops across the crack

and is evaluated considering only the hoops outside thecompression zone

119899119905=119863 minus 119888 minus cov

119904cot 120579 (16)

while 120582 is assumed to be equal to 053 and 119899119889is given by

119899119889= INT [

1198632 minus cov119904

cot 120579] (17)

being 119888 cong 03119863 The influence of the main variables on theshear capacity such as the longitudinal reinforcement ratio120588

119897

the axial load level 119875119860119892 and the shear span-to-depth ratio

119886119863 has been determined empirically based on a total of44 pieces of data of circular cross-section specimens withoutshear reinforcement under monotonic load The term 07119860

119892

represents the sectionrsquos effective shear area while 119896 = 1 if119886119863 gt 25 and 125 if 119886119863 le 25

3 A Data-Driven Approach for Estimationof Shear Strength of Reinforced ConcreteCircular Columns The EvolutionaryPolynomial Regression Paradigm

Numerical Regression is the most powerful and commonlyapplied form of regression that provides a solution to theproblem of finding the best model to fit the observed data(eg fitting a line through a set of points) However thefunctional form (linear exponential logarithmic etc) hasto be selected before fitting commences On the other handgenetic pogramming uses simple but very powerful artificialintelligence tactics for computer learning inspired by naturalevolution to find the appropriate mathematical model to fita set of points The computer produces and evolves a wholepopulation of functional expressions based on how closelyeach of them fits the data The evolutionary polynomialregression (EPR) is a recently developed hybrid regressionmethod by Giustolisi and Savic [25] that integrates the bestfeatures of numerical regression with genetic programming[39 40] An example of the generalmodel structures that EPRcan manage is reported in

Y = 1198860+

119898

sum

119895=1

119886119895sdot (X1)ES(1198951)

sdot sdot (X119896)ES(119895119896)

sdot 119891 ((X1)ES(119895119896+1)

sdot sdot (X119896)ES(1198952119896)

)

(18)

6 Advances in Civil Engineering

where 119898 is the number of additive terms 119886119895are numerical

parameters to be estimated X119894are candidate explanatory

variables ES(119895 119911) (with 119911 = 1 2119896) is the exponentof the 119911th input within the 119895th term in (18) and 119891 is auser-selected function among a set of possible alternatives(including no function selection)The exponents ES(119895 119911) areselected from a user-defined set of candidate values Notethat it is recommended to include 0 among such exponentsin order to allow EPR to exclude some of the candidateexplanatory variables inX

119894from those present in the returned

models thus selecting themost influencing input variables onthe target prediction

Starting from a general structure such as that in (18)the final model expression is obtained through a populationbased strategy that mimics the evolution of the individuals innature it is a combination of a genetic algorithm (GA) [41]paradigm for finding the best exponents to be assigned toeach candidate input and a least squares (LS) regression [42]for the identification of the constant values 119886

119895[25]

Recently the EPR has been upgraded to MOGA EPRthrough the implementation of a search pattern of thebest models based on a multiobjective genetic algorithm(MOGA) [43] optimization scheme being able to find severalmodel expressions maximizing fitness to data and parsimonyof mathematical formulations simultaneously [29] MOGAEPR explores the space of 119898-terms formulas using at leastthree objectives among the following conflicting ones (i)maximization of model accuracy (ii) minimization of thenumber of model coefficients and (iii) minimization ofthe number of actually used model inputs (ie whoseexponent is not null in the resulting model structure)Based on the Pareto dominance criterion [44] MOGAEPR finally obtains a set of optimal solutions (ie thePareto front) which can be considered as trade-offs betweenstructural complexity and accuracy MOGA EPR displaysthe returned formulas within the Pareto front as rankedaccording to their structural complexity this way the com-parisons among formulas according to different criteria (egselected inputs structure etc) are facilitated as well as theselection of models suited for different modeling purposes[29]

In particular this possibility has been exploited in thispaper in order to find the best model for the estimationof shear strength of reinforced concrete circular columnsThus the first step was to define the MOGA EPR searchoptions (candidate model attributes candidate exponentsfor attributes maximum number of parameters objectivefunctions etc) then the set of optimal models with respectto the objective functions produced by MOGA EPR hasbeen analyzed by considering (i) the model structure withrespect to physical insight of the phenomenon at stake(ii) the similarities andor recurrent groups of variableswithin the mathematical structures of the Pareto models(iii) the generalization performance of models in terms ofstatistical indicators andmathematical parsimony Moreoverthe returnedmodels have been also evaluated looking at theirconsistency with existing codestheories about the physicalphenomenon as described above All these issues can lead toa quite robust model selection Figure 2 is illustrative of the

decision support framework allowed by MOGA EPR as hereimplemented

The present analyses have been performed using theEPR MOGA-XL v1 which is a Microsoft Office Excel add-infunction freely available at wwwhydroinformaticsit whichemploys a MOGA optimizer named optimized multiobjec-tive genetic algorithm (OPTIMOGA) [45]

4 Results and Comparison with Literature andTechnical Code Models

The proposed model has been verified by comparing it toexisting ones and to experimental tests The experimentaldata herein considered have been obtained from Merta [17]Capon and Cossio [19] Khalifa and Collins [20] Clarke andBirjandi [18] and Kim [38] and include the results of severaltests performed by different authorsThe test specimens referto circular columns characterized by six different diametersand having both shear and longitudinal reinforcementsLongitudinal reinforcement consistsmainly of high yield barsplaced evenly at eight locations round the pitch circle Shearreinforcement is generally in the form of 6mm or 8mmmild steel links The compressive strength of concrete rangesfrom 132MPa to 493MPa the stirrup percentage rangesfrom 01 to 045 the yielding stress of shear reinforcementranges from 250MPa to 1728MPa and the longitudinal steelpercentage ranges from 22 to 56

A total of 61 circular columns have been selected withdifferent geometric and mechanical parameters such asdimension of the section diameter (119863) area of concrete(119860119892) total cross-section of links at the neutral axis of the

section (119860 sh) spacing of links along themember (119904) concretecompressive strength (119891

119888) yield strength of closed links (119891yh)

total area of longitudinal reinforcement (119860 sl) yield strengthof longitudinal reinforcement (119891yl) concrete cover (cov)effective depth (119889) and axial compression load (119875)The rangeof datasets is listed in Table 1

The main objective of this study is to investigate theapplication of EPR MOGA technique to get possible newformulations of shear strength of circular RC columns able toreproduce the experimental data better than the existing codeexpressions In fact the correlation coefficient 1198772 betweenexperimental data and previsions of ACI 318M-08 [37] andBS 8100 [33] are 9015 and 7692 respectively thus thereis some room for possible improvements

Since most of experimental data refer to pure-shear teststhe dependence of shear strength from the axial force 119875 isnot investigated It can be accounted for by adding a termrepresentative of the axial force contribution as the oneproposed in Merta or in technical code models

41 EPR MOGA Search Settings Bearing in mind whatabove mentioned about MOGA EPR search paradigm somecompact expressions will be sought for in terms of bothnumber of parameters and explanatory variables involvedThis in turn would facilitate their physical interpretationand technical soundness for practical purposes The base

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Advances in Civil Engineering

was obtained by using a lower bound plasticity model Theconcrete contribution for low flexural ductility was definedas the ldquoinitial concrete strengthrdquo and calculated as strength atmaximum lateral load the transverse reinforcement capacitywas calculated from 45-degree truss mechanisms assumingthat transverse reinforcement exposed by a presumed 45-degree diagonal crack is at yield At higher flexural ductility adegradation of the concrete term was assumed and the ldquofinalconcrete strengthrdquo was proposed Different adjustments tothis basic model have been added over the years by variousauthors Priestley et al [9] modified the model proposed byGhee et al [8] and separated the shear capacity term of theaxial load arch mechanism 119881

119901 from the concrete term 119881

119888

Other modifications to Ghee et al original model [8] weredeveloped by Wong et al [10] and by Kowalsky and Priestley[11] In this work the authors revised the model proposedby Priestley et al [9] and the effect of the aspect ratio andlongitudinal reinforcement ratio was incorporated

In addition the truss mechanism component developedby Ghee et al [8] has been slightly modified assuming thatin themember compression zone the cracks are by definitionclosed

A different approach was due to Collins et al [12] bymeans of the application of the Modified Compression FieldTheory (MCFT) [13] to deal with shear strength predictionof circular members Moreover this type of approach ismore complex than those usually based on strut and tiemodels which are still used in many applications till theirfirst implementation in early 1900 thanks to their simplicityThe practical application of the MCFT requires its imple-mentation as a calculation algorithm within a computerroutine thus strongly increasing the complexity level also fora standard code formulation

Bentz et al [14] described a simpler version of theMCFT from which the shear design rules described inAASHTO LRFD [15] were derived and which also containsrules for shear design of circular members Turmo et al [16]proposed an analytical model for evaluating the contributionof the transverse reinforcement in concrete members ofsolid and hollow circular cross section consisting in anaccurate formula for evaluating the shear transferred byspiral reinforcement in solid members The calculation ofhollow core circular columns with both vertical and spiralreinforcement was also deduced

Recently Merta [17] proposed a model based on the trussanalogy by adding a concrete contribution term to the capac-ity of the shear reinforcement An additional deviatoric shearresisting mechanism of hoops present exclusively in mem-berswith curved transverse reinforcement was identified andexpressed analytically It is explained by the fact that a curvedreinforcing bar under tension induces compression in radialdirection as well The component of this compressive forcein the direction of external shear could thus be consideredas an additional shear enhancing mechanism of the hoopsIts magnitude is expressed through the friction force thatis present between the concrete and steel after the sectionis cracked and the bond is partially destroyed Under theseassumptions the concrete shear capacity was derived by aparametric study

11 Experimental Approaches to Shear Capacity of CircularMembers Modeling As for theoretical researches also exper-imental studies on the shear capacity of circular membersare quite limited Assuming that the extrapolation of tests onrectangular beamsmay not be accurate for evaluating circularmembers shear strength some researchers performed differ-ent test campaigns in such structural elements Moreoverall the above-mentioned models are derived starting froma mechanical simplified model whose parameters comefrom the extrapolation of experimental tests for evaluatingcircular member shear strength In their paper Clarke andBirjandi [18] reported also test results from different previousexperimental studies in particular

(i) Capon and Cossio [19] who tested 21 membersmainly of 250mm in diameter having only longitu-dinal steel but four that had stirrups eleven of thespecimens are reported as failing in shear two withstirrups and nine without

(ii) Khalifa and Collins [20] who tested five columns allof 445mm in diameter these were subject to an axialload of 1000 kN equivalent to a stress of 64MPa andall specimens failed in shear

(iii) Nagato [21] whose tests involved 16 columns all witha diameter of 300mm with and without transversereinforcement and with and without axial stress allspecimens are reported as failing in shear

Later on Ghee et al [8] tested 25 circular columnsunder axial load and cyclic lateral inelastic displacements24 of them failing in shear The columns had a diameterof 400mm Variables in the test included axial load levellongitudinal reinforcement ratio transverse reinforcementratio and aspect ratio They addressed for the first timethe efficiency of circular hoops and deduced the formulafor calculating 119881

119904presuming a 45-degree diagonal tension

crack and vertical reinforcement The work of Ghee et al[8] was completed by Priestley et al [9 22] which takesinto account the potential development of steeper angles ofcracking other than 45-degree one Various experimentalresearches on circular RC members loaded monotonicallyin shear were aimed at verifying the possibility to extend tocircular sections the use of design equations developed forrectangular sections [18ndash20]

Lee et al [23] tested 4 full scale circular column spec-imens which show shear related behavior under cycliclateral load with constant axial force The test variables areaspect ratio transverse steel configuration (with or withoutcrossties) and longitudinal steel ratio The columns showedflexural failure or flexure-shear failure depending on the testvariables The test results were compared with shear strengthequations adopted by the bridge seismic design specificationsor guidelines and proposed by other researchers The studyshowed that the accuracy of each method for shear strengthprediction also depends on the aspect ratio and the relativesteel amount of longitudinal reinforcement and transversereinforcement

More recently Jensen and Hoang [24] presented theresults of a test series on heavily shear reinforced circular

Advances in Civil Engineering 3

concrete members The specimens had shear reinforcementpercentages up to more than three times the maximumpercentage found in existing tests The test results indicatethat it is possible to obtain shear strengths which exceedthe upper limit usually imposed on rectangular membersThe test results are compared with a recently developedplasticity-based shear model for circular members Satisfac-tory agreement was found Comparisons were also madewith calculations using the AASHTO LRFD [15] design codeIt was found that the AASHTO LRFD design code givesreasonable results for members with small amounts of shearreinforcement while it underestimates the shear strength forheavily shear reinforced members

This paper proposes an innovative approach mainlybased on experimental data in order to develop an explicitcompact expression to assess the shear strength of RC circularcolumns Such approach is based on the evolutionary poly-nomial regression (EPR) [25] which is a hybrid data-driventechnique that combines the effectiveness of evolutionarysearch with the advantage of classical numerical regressionfor developing data-driven models (ie input-output rela-tionships) featured by explicit mathematical expression TheEPR technique has been successfully applied to modelinga wide range of complex engineering problems includingconstitutive modeling of soils stability of slopes settlementof foundations liquefaction of soils due to earthquake and anumber of other applications in civil engineering [26ndash28]

The EPR implements a multiobjective genetic algorithms(MOGA) optimization paradigm in order to search formodel expressions as trade-off between fitness to data andcomplexity of mathematical formulations [29] The mainadvantage of the MOGA EPR approach is the possibility toselect models from a set of optimal solutions according to thespecific application and to the physical insight of the analyst

2 Proposed Models for Shear Strength andContemporary Code Provisions of CircularReinforced Concrete Columns

Shear strength of reinforced concrete has received consid-erable attention in research during the past century Severalmodels for column shear strength have been proposed andused for design of new buildings and assessment of existingbuildings

Shear reinforcements are used to ensure that the elementfails in flexure the addition shear reinforcement affects themechanism by which shear is carried by a beam in sev-eral ways shear reinforcement carries tensile actions acrosscracks confines the compression-zone concrete increasing itsshear capacity encloses the flexural reinforcement and canprevent dowel-splitting of the concrete Moreover for a givenapplied load equilibrium of a cracked section with stirruprequires a shorter crack length but a larger crack width thatone without stirrup and the shape of the crack will also differShear transfer mechanism in beamwith stirrups has not beenexamined in somuch detail as that in beams without stirrups

Truss analogies are most commonly used in the designThe assumed internal equilibrium state comprises tensile

shear reinforcement and inclined compressive struts of con-crete The original Morsch truss analogy [30] uses a 45∘ strutangle and predicts failure when the shear reinforcementsare at yield The modified truss analogy [31] established anoptimal lower bound for the shear capacity by varying thecompressive strut angle to give reinforcement yield and webconcrete failure simultaneously

Even if shear transfer mechanisms are qualitatively wellknown there is no agreement on the quantification of theshear strength of concrete members As the value of 119881

119904can

be easily calculated with rational models such as the trussanalogy research focuses on the elaboration of methods foran accurate evaluation of 119881

119888 The concrete shear capacity

is the capacity of the section without shear reinforcementdue to the complex stress redistribution after cracking theshear transfer mechanisms of RC element have not beenclearly understood yet Its evaluation is very controversial andalways relies somehow on empirical methods consideringits dependence from various factors such as the tensile lon-gitudinal steel ratio concrete grade aggregate size or shearspan among others Present shear researches concern useof self-training algorithms to predict 119881

119888or the development

and application of equilibrium methods and equilibriummdashcompatibility methods such as the Modified CompressionField Theory [13] The same interest has not been dedicatedto improve the evaluation of the shear reinforcement contri-bution

In this framework the shear strength of circular concretecolumns has not received as much attention as rectangularones and it is interesting to note that there are not guidelinesfor shear capacity in some international codes likewisemanyothers prescribe that the shear capacity of a circular sectionequals the capacity of an equivalent rectangular section [32]In effect this kind of approach has not been fully validatedby test evidence and it obviously is questionable because incircular sections hoops contribute differently to the shearstrength compared with rectangular stirrups

In this section a review of the shear strength provisionsof various contemporary design code and proposed modelsis developed for circular columns Most design codes includecontributions from concrete119881

119888and transverse reinforcement

119881119904to analyze the shear strength of circular columns The

two components are then summed to estimate the total shearstrength in

119881 = 119881119888+ 119881119904 (1)

One should point out that generally codified methods ofdesign cannot be considered as predictive equations sincethey are intended to provide only a conservative and safelower bound to strength Therefore most shear design codesuse empirical or semiempirical models for predicting shearstrengths The American code [15] and the New Zealand one[33] explicitly refer to circular concrete members but withsimplified methods These recommend the calculation of theshear carried by the truss mechanism by representing thesection as an equivalent rectangle having a width equal to thediameter119863 and an effective depth equal to 08119863

Based on a large number of tests on circular cantilevercolumns under uniaxial load and multidirectional cyclic

4 Advances in Civil Engineering

displacements Ghee et al [8] proposed a model for the shearstrength of circular sections under cyclic load by adopting anadditive approach based on the contribution from concreteand transverse reinforcement The concrete contribution forlow flexural ductility (120583 lt 2) was defined as the ldquoinitialconcrete strengthrdquo and calculated as a strength at maximumlateral load as follows

119881119888= 037120572(1 +

3119875

1198911015840119888119860119892

)radic1198911015840119888119860119890

(in N) (2)

where 119875 is the axial load 119860119892is the cross-sectional area

and 1198911015840

119888is the concrete compressive strength The effective

area 119860119890is proposed as 08 times the cross-sectional area 119860

119892

which approximately corresponds to the area of the confinedconcrete core

For low aspect ratios (119886119863 lt 2) the shear capacityenhancement factor 120572 was proposed as

120572 =2

119886119863ge 10 (3)

where 119886 is the shear span and 119863 is the sectionrsquos diameterThe transverse reinforcement capacity was calculated from45-degree truss mechanisms as follows

119881119904=120587

2

119860 sh119891yh1198631015840

119904 (4)

where 119860 sh is the area of the shear reinforcement 119904 is thehooprsquos spacing 119891yh is their yield strength and 119863

1015840 is the corediameter measured at the center line of the transverse hoopor spiral It was assumed that all transverse reinforcementsexposed by a presumed 45∘ diagonal crack are at yield Anintegral averaging was imposed assuming that the spacingof the shear reinforcement is sufficiently small comparedto the diameter It was noted that for low 119863

1015840

119904 ratios theequation can be up to 10 nonconservative At higherflexural ductilities (120583 gt 2) the degradation of the concreteproduces a reduced concrete contribution and then a ldquofinalconcrete strengthrdquo is proposed One should point out thatthis formulation such as other ones implicitly assumes thatall hoops exposed by a crack are at yield and also consider aldquosmearedrdquo distribution of transverse reinforcement whereasin reality transverse reinforcement consists of a discretenumber of hoops that traverse a diagonal crack plane asshown in Figure 1 The later assumption is valid just if thepitch is sufficiently small compared to the diameter1198631015840

Clarke and Birjandi [18] proposed to use for circular sec-tions the same shear design approach as given by the Britishcodes of practice BS 5400 [34] for rectangular sections

However a modification was suggested that is that thesectionrsquos effective depth should be considered as the distancefrom the extreme compression fiber to the centroid of thetension reinforcement

Following this the effective shear area represents the areacorresponding to the effective depth

The design approach for rectangular section in Britishcodes of practice [34] consists in adding together the concrete

c

d

cov

0 1

D

x

y

Vd

Aswfyw

i ntmiddot middot middotmiddot middot middot

120579

(D minus c minus cov)cot120579

Figure 1 Geometrical and mechanical characteristics affecting theshear strength of a circular column

and transverse reinforcement contributions to the shearcapacity The concrete term is defined as

119881119888= [027120572(

100119860 sl119887119908119889

)

13

(500

119889)

14

(119891cu)13

] 119887119908119889 (5)

where 119860 sl is the area of longitudinal steel 119887119908 is the sectionrsquoswidth 119889 is the effective depth and 119891cu is the concrete cubestrength For loads applied at a distance119886V closer than 2119889 fromthe support the concrete shear capacity is increased by

120572 =2119889

119886Vge 10 (6)

The shear force carried by transverse reinforcement is calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (7)

where 119860 sh is the cross-sectional area of the linkrsquos both legs atthe sectionrsquos neutral axis In amember with axial compressionload 119875 the shear capacity should be multiplied by (1 +

005119875119860119892)

Dancygier [35] modified Gheersquos model by adopting adiscrete computation of tension-force components developedby reinforcing hoops in the direction of the shear force givenby the following expression

119881119904= 2119860 sh119891yh (

119899

sum

119894=1

cos120572119894) (8)

where 119894 is the index of the hoop that is crossed by the inclinedcrack 120572 is the angle between its force (119860 sh119891yh) and thedirection of the shear force given by

sin120572119894= 1 minus

2119886119894

1198631015840 119886119894= [1199040+ (119894 minus 1) 119904] cot 120579 (9)

and 119899 is the number of the hoops that are crossed by the crack

119899 = roundup(1198631015840

119904tan 120579 minus

1199040

119904) (10)

where 119904 is the pitch 120579 is crackrsquos inclination angle and 1199040is

the distance from the end of the crack to the first hoop that iscrossed by it

Advances in Civil Engineering 5

Kowalsky and Priestley [11] discussed Gheersquos modelconsidering that in the derivation of the truss mechanismequation it was assumed that a diagonal crack is ableto mobilize transverse reinforcement along a crack lengthextending the full width of the confined core of the concreteIn the compression zone of the column however all cracksare by definition closed Therefore if the crack is closedshear cannot be transferred across it by tension strain inthe transverse reinforcement Therefore it is evident that areduced column width minus119863 minus 119888 minus cov minus (Figure 1) is appropri-ate for calculating the number of hoops mobilized by thecracks between the compression struts and then a revisedtruss component equation should be adopted The authorsobtained an approximated equation given by

119881119904=120587

2119860119904119908119891119910119908

119863 minus 119888 minus cov119904

cot 120579 (11)

where 119860119904119908

is the area of the hooprsquos one leg 119891119910119908

is the hooprsquosyield strength 119888 is the compression zonersquos depth and cov isthe concrete cover

With reference to the concrete mechanism the followingequation was discussed in the cited work

119881119888= 120572120573120574radic1198911015840

119888(08119860

119892) (12)

where the parameter 120572 accounts for the column aspectratio the factor 120573 is a modified factor that accounts forthe longitudinal steel ratio and 120574 represents the reductionin strength of the concrete shear resisting mechanism withincreasing ductility

Also Kim and Mander [36] reported a deficiency of themodel originally proposed by Ghee et al [8] and verifiedthat the applied integral averaging by the reinforcement termrestricts its use only to members with diameter at leastfour times the spacing of the shear reinforcement For allother ratios the formula could be even more than 50nonconservative For this reason they proposed a correctivecoefficient to convert the amount of shear implied by acontinuous truss model to the discrete model

The current ACI 318-M-08 [37] considers a portion ofthe design shear force to be carried by the concrete shearresisting mechanism 119881

119888 with the remainder carried by

trussmechanism119881119904 involving transverse reinforcementThe

code proposes the following equation to calculate 119881119888for

members subjected to combined shear moment and axialcompression

119881119888= 017(1 +

119875

14119860119892

)radic1198911015840119888119887119908119889 (13)

The transverse reinforcement contribution is also calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (14)

where the meaning of various terms has been explainedbefore

Merta [17] developed a simple shear capacitymodel of RCcircular sections It was based on the truss analogy by addingan empirical concrete contribution term to the capacity of theshear reinforcement obtaining finally the following relation

119881 = [37120588119897+ 018 + 008(

119875

119860119892

)03] 119896radic119891101584011988807119860119892

+ 119860119904119908119891119910119908

(18119899119905+ 120582119899119889minus 1)

(15)

where 119899119905represents the number of hoops across the crack

and is evaluated considering only the hoops outside thecompression zone

119899119905=119863 minus 119888 minus cov

119904cot 120579 (16)

while 120582 is assumed to be equal to 053 and 119899119889is given by

119899119889= INT [

1198632 minus cov119904

cot 120579] (17)

being 119888 cong 03119863 The influence of the main variables on theshear capacity such as the longitudinal reinforcement ratio120588

119897

the axial load level 119875119860119892 and the shear span-to-depth ratio

119886119863 has been determined empirically based on a total of44 pieces of data of circular cross-section specimens withoutshear reinforcement under monotonic load The term 07119860

119892

represents the sectionrsquos effective shear area while 119896 = 1 if119886119863 gt 25 and 125 if 119886119863 le 25

3 A Data-Driven Approach for Estimationof Shear Strength of Reinforced ConcreteCircular Columns The EvolutionaryPolynomial Regression Paradigm

Numerical Regression is the most powerful and commonlyapplied form of regression that provides a solution to theproblem of finding the best model to fit the observed data(eg fitting a line through a set of points) However thefunctional form (linear exponential logarithmic etc) hasto be selected before fitting commences On the other handgenetic pogramming uses simple but very powerful artificialintelligence tactics for computer learning inspired by naturalevolution to find the appropriate mathematical model to fita set of points The computer produces and evolves a wholepopulation of functional expressions based on how closelyeach of them fits the data The evolutionary polynomialregression (EPR) is a recently developed hybrid regressionmethod by Giustolisi and Savic [25] that integrates the bestfeatures of numerical regression with genetic programming[39 40] An example of the generalmodel structures that EPRcan manage is reported in

Y = 1198860+

119898

sum

119895=1

119886119895sdot (X1)ES(1198951)

sdot sdot (X119896)ES(119895119896)

sdot 119891 ((X1)ES(119895119896+1)

sdot sdot (X119896)ES(1198952119896)

)

(18)

6 Advances in Civil Engineering

where 119898 is the number of additive terms 119886119895are numerical

parameters to be estimated X119894are candidate explanatory

variables ES(119895 119911) (with 119911 = 1 2119896) is the exponentof the 119911th input within the 119895th term in (18) and 119891 is auser-selected function among a set of possible alternatives(including no function selection)The exponents ES(119895 119911) areselected from a user-defined set of candidate values Notethat it is recommended to include 0 among such exponentsin order to allow EPR to exclude some of the candidateexplanatory variables inX

119894from those present in the returned

models thus selecting themost influencing input variables onthe target prediction

Starting from a general structure such as that in (18)the final model expression is obtained through a populationbased strategy that mimics the evolution of the individuals innature it is a combination of a genetic algorithm (GA) [41]paradigm for finding the best exponents to be assigned toeach candidate input and a least squares (LS) regression [42]for the identification of the constant values 119886

119895[25]

Recently the EPR has been upgraded to MOGA EPRthrough the implementation of a search pattern of thebest models based on a multiobjective genetic algorithm(MOGA) [43] optimization scheme being able to find severalmodel expressions maximizing fitness to data and parsimonyof mathematical formulations simultaneously [29] MOGAEPR explores the space of 119898-terms formulas using at leastthree objectives among the following conflicting ones (i)maximization of model accuracy (ii) minimization of thenumber of model coefficients and (iii) minimization ofthe number of actually used model inputs (ie whoseexponent is not null in the resulting model structure)Based on the Pareto dominance criterion [44] MOGAEPR finally obtains a set of optimal solutions (ie thePareto front) which can be considered as trade-offs betweenstructural complexity and accuracy MOGA EPR displaysthe returned formulas within the Pareto front as rankedaccording to their structural complexity this way the com-parisons among formulas according to different criteria (egselected inputs structure etc) are facilitated as well as theselection of models suited for different modeling purposes[29]

In particular this possibility has been exploited in thispaper in order to find the best model for the estimationof shear strength of reinforced concrete circular columnsThus the first step was to define the MOGA EPR searchoptions (candidate model attributes candidate exponentsfor attributes maximum number of parameters objectivefunctions etc) then the set of optimal models with respectto the objective functions produced by MOGA EPR hasbeen analyzed by considering (i) the model structure withrespect to physical insight of the phenomenon at stake(ii) the similarities andor recurrent groups of variableswithin the mathematical structures of the Pareto models(iii) the generalization performance of models in terms ofstatistical indicators andmathematical parsimony Moreoverthe returnedmodels have been also evaluated looking at theirconsistency with existing codestheories about the physicalphenomenon as described above All these issues can lead toa quite robust model selection Figure 2 is illustrative of the

decision support framework allowed by MOGA EPR as hereimplemented

The present analyses have been performed using theEPR MOGA-XL v1 which is a Microsoft Office Excel add-infunction freely available at wwwhydroinformaticsit whichemploys a MOGA optimizer named optimized multiobjec-tive genetic algorithm (OPTIMOGA) [45]

4 Results and Comparison with Literature andTechnical Code Models

The proposed model has been verified by comparing it toexisting ones and to experimental tests The experimentaldata herein considered have been obtained from Merta [17]Capon and Cossio [19] Khalifa and Collins [20] Clarke andBirjandi [18] and Kim [38] and include the results of severaltests performed by different authorsThe test specimens referto circular columns characterized by six different diametersand having both shear and longitudinal reinforcementsLongitudinal reinforcement consistsmainly of high yield barsplaced evenly at eight locations round the pitch circle Shearreinforcement is generally in the form of 6mm or 8mmmild steel links The compressive strength of concrete rangesfrom 132MPa to 493MPa the stirrup percentage rangesfrom 01 to 045 the yielding stress of shear reinforcementranges from 250MPa to 1728MPa and the longitudinal steelpercentage ranges from 22 to 56

A total of 61 circular columns have been selected withdifferent geometric and mechanical parameters such asdimension of the section diameter (119863) area of concrete(119860119892) total cross-section of links at the neutral axis of the

section (119860 sh) spacing of links along themember (119904) concretecompressive strength (119891

119888) yield strength of closed links (119891yh)

total area of longitudinal reinforcement (119860 sl) yield strengthof longitudinal reinforcement (119891yl) concrete cover (cov)effective depth (119889) and axial compression load (119875)The rangeof datasets is listed in Table 1

The main objective of this study is to investigate theapplication of EPR MOGA technique to get possible newformulations of shear strength of circular RC columns able toreproduce the experimental data better than the existing codeexpressions In fact the correlation coefficient 1198772 betweenexperimental data and previsions of ACI 318M-08 [37] andBS 8100 [33] are 9015 and 7692 respectively thus thereis some room for possible improvements

Since most of experimental data refer to pure-shear teststhe dependence of shear strength from the axial force 119875 isnot investigated It can be accounted for by adding a termrepresentative of the axial force contribution as the oneproposed in Merta or in technical code models

41 EPR MOGA Search Settings Bearing in mind whatabove mentioned about MOGA EPR search paradigm somecompact expressions will be sought for in terms of bothnumber of parameters and explanatory variables involvedThis in turn would facilitate their physical interpretationand technical soundness for practical purposes The base

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 3

concrete members The specimens had shear reinforcementpercentages up to more than three times the maximumpercentage found in existing tests The test results indicatethat it is possible to obtain shear strengths which exceedthe upper limit usually imposed on rectangular membersThe test results are compared with a recently developedplasticity-based shear model for circular members Satisfac-tory agreement was found Comparisons were also madewith calculations using the AASHTO LRFD [15] design codeIt was found that the AASHTO LRFD design code givesreasonable results for members with small amounts of shearreinforcement while it underestimates the shear strength forheavily shear reinforced members

This paper proposes an innovative approach mainlybased on experimental data in order to develop an explicitcompact expression to assess the shear strength of RC circularcolumns Such approach is based on the evolutionary poly-nomial regression (EPR) [25] which is a hybrid data-driventechnique that combines the effectiveness of evolutionarysearch with the advantage of classical numerical regressionfor developing data-driven models (ie input-output rela-tionships) featured by explicit mathematical expression TheEPR technique has been successfully applied to modelinga wide range of complex engineering problems includingconstitutive modeling of soils stability of slopes settlementof foundations liquefaction of soils due to earthquake and anumber of other applications in civil engineering [26ndash28]

The EPR implements a multiobjective genetic algorithms(MOGA) optimization paradigm in order to search formodel expressions as trade-off between fitness to data andcomplexity of mathematical formulations [29] The mainadvantage of the MOGA EPR approach is the possibility toselect models from a set of optimal solutions according to thespecific application and to the physical insight of the analyst

2 Proposed Models for Shear Strength andContemporary Code Provisions of CircularReinforced Concrete Columns

Shear strength of reinforced concrete has received consid-erable attention in research during the past century Severalmodels for column shear strength have been proposed andused for design of new buildings and assessment of existingbuildings

Shear reinforcements are used to ensure that the elementfails in flexure the addition shear reinforcement affects themechanism by which shear is carried by a beam in sev-eral ways shear reinforcement carries tensile actions acrosscracks confines the compression-zone concrete increasing itsshear capacity encloses the flexural reinforcement and canprevent dowel-splitting of the concrete Moreover for a givenapplied load equilibrium of a cracked section with stirruprequires a shorter crack length but a larger crack width thatone without stirrup and the shape of the crack will also differShear transfer mechanism in beamwith stirrups has not beenexamined in somuch detail as that in beams without stirrups

Truss analogies are most commonly used in the designThe assumed internal equilibrium state comprises tensile

shear reinforcement and inclined compressive struts of con-crete The original Morsch truss analogy [30] uses a 45∘ strutangle and predicts failure when the shear reinforcementsare at yield The modified truss analogy [31] established anoptimal lower bound for the shear capacity by varying thecompressive strut angle to give reinforcement yield and webconcrete failure simultaneously

Even if shear transfer mechanisms are qualitatively wellknown there is no agreement on the quantification of theshear strength of concrete members As the value of 119881

119904can

be easily calculated with rational models such as the trussanalogy research focuses on the elaboration of methods foran accurate evaluation of 119881

119888 The concrete shear capacity

is the capacity of the section without shear reinforcementdue to the complex stress redistribution after cracking theshear transfer mechanisms of RC element have not beenclearly understood yet Its evaluation is very controversial andalways relies somehow on empirical methods consideringits dependence from various factors such as the tensile lon-gitudinal steel ratio concrete grade aggregate size or shearspan among others Present shear researches concern useof self-training algorithms to predict 119881

119888or the development

and application of equilibrium methods and equilibriummdashcompatibility methods such as the Modified CompressionField Theory [13] The same interest has not been dedicatedto improve the evaluation of the shear reinforcement contri-bution

In this framework the shear strength of circular concretecolumns has not received as much attention as rectangularones and it is interesting to note that there are not guidelinesfor shear capacity in some international codes likewisemanyothers prescribe that the shear capacity of a circular sectionequals the capacity of an equivalent rectangular section [32]In effect this kind of approach has not been fully validatedby test evidence and it obviously is questionable because incircular sections hoops contribute differently to the shearstrength compared with rectangular stirrups

In this section a review of the shear strength provisionsof various contemporary design code and proposed modelsis developed for circular columns Most design codes includecontributions from concrete119881

119888and transverse reinforcement

119881119904to analyze the shear strength of circular columns The

two components are then summed to estimate the total shearstrength in

119881 = 119881119888+ 119881119904 (1)

One should point out that generally codified methods ofdesign cannot be considered as predictive equations sincethey are intended to provide only a conservative and safelower bound to strength Therefore most shear design codesuse empirical or semiempirical models for predicting shearstrengths The American code [15] and the New Zealand one[33] explicitly refer to circular concrete members but withsimplified methods These recommend the calculation of theshear carried by the truss mechanism by representing thesection as an equivalent rectangle having a width equal to thediameter119863 and an effective depth equal to 08119863

Based on a large number of tests on circular cantilevercolumns under uniaxial load and multidirectional cyclic

4 Advances in Civil Engineering

displacements Ghee et al [8] proposed a model for the shearstrength of circular sections under cyclic load by adopting anadditive approach based on the contribution from concreteand transverse reinforcement The concrete contribution forlow flexural ductility (120583 lt 2) was defined as the ldquoinitialconcrete strengthrdquo and calculated as a strength at maximumlateral load as follows

119881119888= 037120572(1 +

3119875

1198911015840119888119860119892

)radic1198911015840119888119860119890

(in N) (2)

where 119875 is the axial load 119860119892is the cross-sectional area

and 1198911015840

119888is the concrete compressive strength The effective

area 119860119890is proposed as 08 times the cross-sectional area 119860

119892

which approximately corresponds to the area of the confinedconcrete core

For low aspect ratios (119886119863 lt 2) the shear capacityenhancement factor 120572 was proposed as

120572 =2

119886119863ge 10 (3)

where 119886 is the shear span and 119863 is the sectionrsquos diameterThe transverse reinforcement capacity was calculated from45-degree truss mechanisms as follows

119881119904=120587

2

119860 sh119891yh1198631015840

119904 (4)

where 119860 sh is the area of the shear reinforcement 119904 is thehooprsquos spacing 119891yh is their yield strength and 119863

1015840 is the corediameter measured at the center line of the transverse hoopor spiral It was assumed that all transverse reinforcementsexposed by a presumed 45∘ diagonal crack are at yield Anintegral averaging was imposed assuming that the spacingof the shear reinforcement is sufficiently small comparedto the diameter It was noted that for low 119863

1015840

119904 ratios theequation can be up to 10 nonconservative At higherflexural ductilities (120583 gt 2) the degradation of the concreteproduces a reduced concrete contribution and then a ldquofinalconcrete strengthrdquo is proposed One should point out thatthis formulation such as other ones implicitly assumes thatall hoops exposed by a crack are at yield and also consider aldquosmearedrdquo distribution of transverse reinforcement whereasin reality transverse reinforcement consists of a discretenumber of hoops that traverse a diagonal crack plane asshown in Figure 1 The later assumption is valid just if thepitch is sufficiently small compared to the diameter1198631015840

Clarke and Birjandi [18] proposed to use for circular sec-tions the same shear design approach as given by the Britishcodes of practice BS 5400 [34] for rectangular sections

However a modification was suggested that is that thesectionrsquos effective depth should be considered as the distancefrom the extreme compression fiber to the centroid of thetension reinforcement

Following this the effective shear area represents the areacorresponding to the effective depth

The design approach for rectangular section in Britishcodes of practice [34] consists in adding together the concrete

c

d

cov

0 1

D

x

y

Vd

Aswfyw

i ntmiddot middot middotmiddot middot middot

120579

(D minus c minus cov)cot120579

Figure 1 Geometrical and mechanical characteristics affecting theshear strength of a circular column

and transverse reinforcement contributions to the shearcapacity The concrete term is defined as

119881119888= [027120572(

100119860 sl119887119908119889

)

13

(500

119889)

14

(119891cu)13

] 119887119908119889 (5)

where 119860 sl is the area of longitudinal steel 119887119908 is the sectionrsquoswidth 119889 is the effective depth and 119891cu is the concrete cubestrength For loads applied at a distance119886V closer than 2119889 fromthe support the concrete shear capacity is increased by

120572 =2119889

119886Vge 10 (6)

The shear force carried by transverse reinforcement is calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (7)

where 119860 sh is the cross-sectional area of the linkrsquos both legs atthe sectionrsquos neutral axis In amember with axial compressionload 119875 the shear capacity should be multiplied by (1 +

005119875119860119892)

Dancygier [35] modified Gheersquos model by adopting adiscrete computation of tension-force components developedby reinforcing hoops in the direction of the shear force givenby the following expression

119881119904= 2119860 sh119891yh (

119899

sum

119894=1

cos120572119894) (8)

where 119894 is the index of the hoop that is crossed by the inclinedcrack 120572 is the angle between its force (119860 sh119891yh) and thedirection of the shear force given by

sin120572119894= 1 minus

2119886119894

1198631015840 119886119894= [1199040+ (119894 minus 1) 119904] cot 120579 (9)

and 119899 is the number of the hoops that are crossed by the crack

119899 = roundup(1198631015840

119904tan 120579 minus

1199040

119904) (10)

where 119904 is the pitch 120579 is crackrsquos inclination angle and 1199040is

the distance from the end of the crack to the first hoop that iscrossed by it

Advances in Civil Engineering 5

Kowalsky and Priestley [11] discussed Gheersquos modelconsidering that in the derivation of the truss mechanismequation it was assumed that a diagonal crack is ableto mobilize transverse reinforcement along a crack lengthextending the full width of the confined core of the concreteIn the compression zone of the column however all cracksare by definition closed Therefore if the crack is closedshear cannot be transferred across it by tension strain inthe transverse reinforcement Therefore it is evident that areduced column width minus119863 minus 119888 minus cov minus (Figure 1) is appropri-ate for calculating the number of hoops mobilized by thecracks between the compression struts and then a revisedtruss component equation should be adopted The authorsobtained an approximated equation given by

119881119904=120587

2119860119904119908119891119910119908

119863 minus 119888 minus cov119904

cot 120579 (11)

where 119860119904119908

is the area of the hooprsquos one leg 119891119910119908

is the hooprsquosyield strength 119888 is the compression zonersquos depth and cov isthe concrete cover

With reference to the concrete mechanism the followingequation was discussed in the cited work

119881119888= 120572120573120574radic1198911015840

119888(08119860

119892) (12)

where the parameter 120572 accounts for the column aspectratio the factor 120573 is a modified factor that accounts forthe longitudinal steel ratio and 120574 represents the reductionin strength of the concrete shear resisting mechanism withincreasing ductility

Also Kim and Mander [36] reported a deficiency of themodel originally proposed by Ghee et al [8] and verifiedthat the applied integral averaging by the reinforcement termrestricts its use only to members with diameter at leastfour times the spacing of the shear reinforcement For allother ratios the formula could be even more than 50nonconservative For this reason they proposed a correctivecoefficient to convert the amount of shear implied by acontinuous truss model to the discrete model

The current ACI 318-M-08 [37] considers a portion ofthe design shear force to be carried by the concrete shearresisting mechanism 119881

119888 with the remainder carried by

trussmechanism119881119904 involving transverse reinforcementThe

code proposes the following equation to calculate 119881119888for

members subjected to combined shear moment and axialcompression

119881119888= 017(1 +

119875

14119860119892

)radic1198911015840119888119887119908119889 (13)

The transverse reinforcement contribution is also calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (14)

where the meaning of various terms has been explainedbefore

Merta [17] developed a simple shear capacitymodel of RCcircular sections It was based on the truss analogy by addingan empirical concrete contribution term to the capacity of theshear reinforcement obtaining finally the following relation

119881 = [37120588119897+ 018 + 008(

119875

119860119892

)03] 119896radic119891101584011988807119860119892

+ 119860119904119908119891119910119908

(18119899119905+ 120582119899119889minus 1)

(15)

where 119899119905represents the number of hoops across the crack

and is evaluated considering only the hoops outside thecompression zone

119899119905=119863 minus 119888 minus cov

119904cot 120579 (16)

while 120582 is assumed to be equal to 053 and 119899119889is given by

119899119889= INT [

1198632 minus cov119904

cot 120579] (17)

being 119888 cong 03119863 The influence of the main variables on theshear capacity such as the longitudinal reinforcement ratio120588

119897

the axial load level 119875119860119892 and the shear span-to-depth ratio

119886119863 has been determined empirically based on a total of44 pieces of data of circular cross-section specimens withoutshear reinforcement under monotonic load The term 07119860

119892

represents the sectionrsquos effective shear area while 119896 = 1 if119886119863 gt 25 and 125 if 119886119863 le 25

3 A Data-Driven Approach for Estimationof Shear Strength of Reinforced ConcreteCircular Columns The EvolutionaryPolynomial Regression Paradigm

Numerical Regression is the most powerful and commonlyapplied form of regression that provides a solution to theproblem of finding the best model to fit the observed data(eg fitting a line through a set of points) However thefunctional form (linear exponential logarithmic etc) hasto be selected before fitting commences On the other handgenetic pogramming uses simple but very powerful artificialintelligence tactics for computer learning inspired by naturalevolution to find the appropriate mathematical model to fita set of points The computer produces and evolves a wholepopulation of functional expressions based on how closelyeach of them fits the data The evolutionary polynomialregression (EPR) is a recently developed hybrid regressionmethod by Giustolisi and Savic [25] that integrates the bestfeatures of numerical regression with genetic programming[39 40] An example of the generalmodel structures that EPRcan manage is reported in

Y = 1198860+

119898

sum

119895=1

119886119895sdot (X1)ES(1198951)

sdot sdot (X119896)ES(119895119896)

sdot 119891 ((X1)ES(119895119896+1)

sdot sdot (X119896)ES(1198952119896)

)

(18)

6 Advances in Civil Engineering

where 119898 is the number of additive terms 119886119895are numerical

parameters to be estimated X119894are candidate explanatory

variables ES(119895 119911) (with 119911 = 1 2119896) is the exponentof the 119911th input within the 119895th term in (18) and 119891 is auser-selected function among a set of possible alternatives(including no function selection)The exponents ES(119895 119911) areselected from a user-defined set of candidate values Notethat it is recommended to include 0 among such exponentsin order to allow EPR to exclude some of the candidateexplanatory variables inX

119894from those present in the returned

models thus selecting themost influencing input variables onthe target prediction

Starting from a general structure such as that in (18)the final model expression is obtained through a populationbased strategy that mimics the evolution of the individuals innature it is a combination of a genetic algorithm (GA) [41]paradigm for finding the best exponents to be assigned toeach candidate input and a least squares (LS) regression [42]for the identification of the constant values 119886

119895[25]

Recently the EPR has been upgraded to MOGA EPRthrough the implementation of a search pattern of thebest models based on a multiobjective genetic algorithm(MOGA) [43] optimization scheme being able to find severalmodel expressions maximizing fitness to data and parsimonyof mathematical formulations simultaneously [29] MOGAEPR explores the space of 119898-terms formulas using at leastthree objectives among the following conflicting ones (i)maximization of model accuracy (ii) minimization of thenumber of model coefficients and (iii) minimization ofthe number of actually used model inputs (ie whoseexponent is not null in the resulting model structure)Based on the Pareto dominance criterion [44] MOGAEPR finally obtains a set of optimal solutions (ie thePareto front) which can be considered as trade-offs betweenstructural complexity and accuracy MOGA EPR displaysthe returned formulas within the Pareto front as rankedaccording to their structural complexity this way the com-parisons among formulas according to different criteria (egselected inputs structure etc) are facilitated as well as theselection of models suited for different modeling purposes[29]

In particular this possibility has been exploited in thispaper in order to find the best model for the estimationof shear strength of reinforced concrete circular columnsThus the first step was to define the MOGA EPR searchoptions (candidate model attributes candidate exponentsfor attributes maximum number of parameters objectivefunctions etc) then the set of optimal models with respectto the objective functions produced by MOGA EPR hasbeen analyzed by considering (i) the model structure withrespect to physical insight of the phenomenon at stake(ii) the similarities andor recurrent groups of variableswithin the mathematical structures of the Pareto models(iii) the generalization performance of models in terms ofstatistical indicators andmathematical parsimony Moreoverthe returnedmodels have been also evaluated looking at theirconsistency with existing codestheories about the physicalphenomenon as described above All these issues can lead toa quite robust model selection Figure 2 is illustrative of the

decision support framework allowed by MOGA EPR as hereimplemented

The present analyses have been performed using theEPR MOGA-XL v1 which is a Microsoft Office Excel add-infunction freely available at wwwhydroinformaticsit whichemploys a MOGA optimizer named optimized multiobjec-tive genetic algorithm (OPTIMOGA) [45]

4 Results and Comparison with Literature andTechnical Code Models

The proposed model has been verified by comparing it toexisting ones and to experimental tests The experimentaldata herein considered have been obtained from Merta [17]Capon and Cossio [19] Khalifa and Collins [20] Clarke andBirjandi [18] and Kim [38] and include the results of severaltests performed by different authorsThe test specimens referto circular columns characterized by six different diametersand having both shear and longitudinal reinforcementsLongitudinal reinforcement consistsmainly of high yield barsplaced evenly at eight locations round the pitch circle Shearreinforcement is generally in the form of 6mm or 8mmmild steel links The compressive strength of concrete rangesfrom 132MPa to 493MPa the stirrup percentage rangesfrom 01 to 045 the yielding stress of shear reinforcementranges from 250MPa to 1728MPa and the longitudinal steelpercentage ranges from 22 to 56

A total of 61 circular columns have been selected withdifferent geometric and mechanical parameters such asdimension of the section diameter (119863) area of concrete(119860119892) total cross-section of links at the neutral axis of the

section (119860 sh) spacing of links along themember (119904) concretecompressive strength (119891

119888) yield strength of closed links (119891yh)

total area of longitudinal reinforcement (119860 sl) yield strengthof longitudinal reinforcement (119891yl) concrete cover (cov)effective depth (119889) and axial compression load (119875)The rangeof datasets is listed in Table 1

The main objective of this study is to investigate theapplication of EPR MOGA technique to get possible newformulations of shear strength of circular RC columns able toreproduce the experimental data better than the existing codeexpressions In fact the correlation coefficient 1198772 betweenexperimental data and previsions of ACI 318M-08 [37] andBS 8100 [33] are 9015 and 7692 respectively thus thereis some room for possible improvements

Since most of experimental data refer to pure-shear teststhe dependence of shear strength from the axial force 119875 isnot investigated It can be accounted for by adding a termrepresentative of the axial force contribution as the oneproposed in Merta or in technical code models

41 EPR MOGA Search Settings Bearing in mind whatabove mentioned about MOGA EPR search paradigm somecompact expressions will be sought for in terms of bothnumber of parameters and explanatory variables involvedThis in turn would facilitate their physical interpretationand technical soundness for practical purposes The base

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Advances in Civil Engineering

displacements Ghee et al [8] proposed a model for the shearstrength of circular sections under cyclic load by adopting anadditive approach based on the contribution from concreteand transverse reinforcement The concrete contribution forlow flexural ductility (120583 lt 2) was defined as the ldquoinitialconcrete strengthrdquo and calculated as a strength at maximumlateral load as follows

119881119888= 037120572(1 +

3119875

1198911015840119888119860119892

)radic1198911015840119888119860119890

(in N) (2)

where 119875 is the axial load 119860119892is the cross-sectional area

and 1198911015840

119888is the concrete compressive strength The effective

area 119860119890is proposed as 08 times the cross-sectional area 119860

119892

which approximately corresponds to the area of the confinedconcrete core

For low aspect ratios (119886119863 lt 2) the shear capacityenhancement factor 120572 was proposed as

120572 =2

119886119863ge 10 (3)

where 119886 is the shear span and 119863 is the sectionrsquos diameterThe transverse reinforcement capacity was calculated from45-degree truss mechanisms as follows

119881119904=120587

2

119860 sh119891yh1198631015840

119904 (4)

where 119860 sh is the area of the shear reinforcement 119904 is thehooprsquos spacing 119891yh is their yield strength and 119863

1015840 is the corediameter measured at the center line of the transverse hoopor spiral It was assumed that all transverse reinforcementsexposed by a presumed 45∘ diagonal crack are at yield Anintegral averaging was imposed assuming that the spacingof the shear reinforcement is sufficiently small comparedto the diameter It was noted that for low 119863

1015840

119904 ratios theequation can be up to 10 nonconservative At higherflexural ductilities (120583 gt 2) the degradation of the concreteproduces a reduced concrete contribution and then a ldquofinalconcrete strengthrdquo is proposed One should point out thatthis formulation such as other ones implicitly assumes thatall hoops exposed by a crack are at yield and also consider aldquosmearedrdquo distribution of transverse reinforcement whereasin reality transverse reinforcement consists of a discretenumber of hoops that traverse a diagonal crack plane asshown in Figure 1 The later assumption is valid just if thepitch is sufficiently small compared to the diameter1198631015840

Clarke and Birjandi [18] proposed to use for circular sec-tions the same shear design approach as given by the Britishcodes of practice BS 5400 [34] for rectangular sections

However a modification was suggested that is that thesectionrsquos effective depth should be considered as the distancefrom the extreme compression fiber to the centroid of thetension reinforcement

Following this the effective shear area represents the areacorresponding to the effective depth

The design approach for rectangular section in Britishcodes of practice [34] consists in adding together the concrete

c

d

cov

0 1

D

x

y

Vd

Aswfyw

i ntmiddot middot middotmiddot middot middot

120579

(D minus c minus cov)cot120579

Figure 1 Geometrical and mechanical characteristics affecting theshear strength of a circular column

and transverse reinforcement contributions to the shearcapacity The concrete term is defined as

119881119888= [027120572(

100119860 sl119887119908119889

)

13

(500

119889)

14

(119891cu)13

] 119887119908119889 (5)

where 119860 sl is the area of longitudinal steel 119887119908 is the sectionrsquoswidth 119889 is the effective depth and 119891cu is the concrete cubestrength For loads applied at a distance119886V closer than 2119889 fromthe support the concrete shear capacity is increased by

120572 =2119889

119886Vge 10 (6)

The shear force carried by transverse reinforcement is calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (7)

where 119860 sh is the cross-sectional area of the linkrsquos both legs atthe sectionrsquos neutral axis In amember with axial compressionload 119875 the shear capacity should be multiplied by (1 +

005119875119860119892)

Dancygier [35] modified Gheersquos model by adopting adiscrete computation of tension-force components developedby reinforcing hoops in the direction of the shear force givenby the following expression

119881119904= 2119860 sh119891yh (

119899

sum

119894=1

cos120572119894) (8)

where 119894 is the index of the hoop that is crossed by the inclinedcrack 120572 is the angle between its force (119860 sh119891yh) and thedirection of the shear force given by

sin120572119894= 1 minus

2119886119894

1198631015840 119886119894= [1199040+ (119894 minus 1) 119904] cot 120579 (9)

and 119899 is the number of the hoops that are crossed by the crack

119899 = roundup(1198631015840

119904tan 120579 minus

1199040

119904) (10)

where 119904 is the pitch 120579 is crackrsquos inclination angle and 1199040is

the distance from the end of the crack to the first hoop that iscrossed by it

Advances in Civil Engineering 5

Kowalsky and Priestley [11] discussed Gheersquos modelconsidering that in the derivation of the truss mechanismequation it was assumed that a diagonal crack is ableto mobilize transverse reinforcement along a crack lengthextending the full width of the confined core of the concreteIn the compression zone of the column however all cracksare by definition closed Therefore if the crack is closedshear cannot be transferred across it by tension strain inthe transverse reinforcement Therefore it is evident that areduced column width minus119863 minus 119888 minus cov minus (Figure 1) is appropri-ate for calculating the number of hoops mobilized by thecracks between the compression struts and then a revisedtruss component equation should be adopted The authorsobtained an approximated equation given by

119881119904=120587

2119860119904119908119891119910119908

119863 minus 119888 minus cov119904

cot 120579 (11)

where 119860119904119908

is the area of the hooprsquos one leg 119891119910119908

is the hooprsquosyield strength 119888 is the compression zonersquos depth and cov isthe concrete cover

With reference to the concrete mechanism the followingequation was discussed in the cited work

119881119888= 120572120573120574radic1198911015840

119888(08119860

119892) (12)

where the parameter 120572 accounts for the column aspectratio the factor 120573 is a modified factor that accounts forthe longitudinal steel ratio and 120574 represents the reductionin strength of the concrete shear resisting mechanism withincreasing ductility

Also Kim and Mander [36] reported a deficiency of themodel originally proposed by Ghee et al [8] and verifiedthat the applied integral averaging by the reinforcement termrestricts its use only to members with diameter at leastfour times the spacing of the shear reinforcement For allother ratios the formula could be even more than 50nonconservative For this reason they proposed a correctivecoefficient to convert the amount of shear implied by acontinuous truss model to the discrete model

The current ACI 318-M-08 [37] considers a portion ofthe design shear force to be carried by the concrete shearresisting mechanism 119881

119888 with the remainder carried by

trussmechanism119881119904 involving transverse reinforcementThe

code proposes the following equation to calculate 119881119888for

members subjected to combined shear moment and axialcompression

119881119888= 017(1 +

119875

14119860119892

)radic1198911015840119888119887119908119889 (13)

The transverse reinforcement contribution is also calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (14)

where the meaning of various terms has been explainedbefore

Merta [17] developed a simple shear capacitymodel of RCcircular sections It was based on the truss analogy by addingan empirical concrete contribution term to the capacity of theshear reinforcement obtaining finally the following relation

119881 = [37120588119897+ 018 + 008(

119875

119860119892

)03] 119896radic119891101584011988807119860119892

+ 119860119904119908119891119910119908

(18119899119905+ 120582119899119889minus 1)

(15)

where 119899119905represents the number of hoops across the crack

and is evaluated considering only the hoops outside thecompression zone

119899119905=119863 minus 119888 minus cov

119904cot 120579 (16)

while 120582 is assumed to be equal to 053 and 119899119889is given by

119899119889= INT [

1198632 minus cov119904

cot 120579] (17)

being 119888 cong 03119863 The influence of the main variables on theshear capacity such as the longitudinal reinforcement ratio120588

119897

the axial load level 119875119860119892 and the shear span-to-depth ratio

119886119863 has been determined empirically based on a total of44 pieces of data of circular cross-section specimens withoutshear reinforcement under monotonic load The term 07119860

119892

represents the sectionrsquos effective shear area while 119896 = 1 if119886119863 gt 25 and 125 if 119886119863 le 25

3 A Data-Driven Approach for Estimationof Shear Strength of Reinforced ConcreteCircular Columns The EvolutionaryPolynomial Regression Paradigm

Numerical Regression is the most powerful and commonlyapplied form of regression that provides a solution to theproblem of finding the best model to fit the observed data(eg fitting a line through a set of points) However thefunctional form (linear exponential logarithmic etc) hasto be selected before fitting commences On the other handgenetic pogramming uses simple but very powerful artificialintelligence tactics for computer learning inspired by naturalevolution to find the appropriate mathematical model to fita set of points The computer produces and evolves a wholepopulation of functional expressions based on how closelyeach of them fits the data The evolutionary polynomialregression (EPR) is a recently developed hybrid regressionmethod by Giustolisi and Savic [25] that integrates the bestfeatures of numerical regression with genetic programming[39 40] An example of the generalmodel structures that EPRcan manage is reported in

Y = 1198860+

119898

sum

119895=1

119886119895sdot (X1)ES(1198951)

sdot sdot (X119896)ES(119895119896)

sdot 119891 ((X1)ES(119895119896+1)

sdot sdot (X119896)ES(1198952119896)

)

(18)

6 Advances in Civil Engineering

where 119898 is the number of additive terms 119886119895are numerical

parameters to be estimated X119894are candidate explanatory

variables ES(119895 119911) (with 119911 = 1 2119896) is the exponentof the 119911th input within the 119895th term in (18) and 119891 is auser-selected function among a set of possible alternatives(including no function selection)The exponents ES(119895 119911) areselected from a user-defined set of candidate values Notethat it is recommended to include 0 among such exponentsin order to allow EPR to exclude some of the candidateexplanatory variables inX

119894from those present in the returned

models thus selecting themost influencing input variables onthe target prediction

Starting from a general structure such as that in (18)the final model expression is obtained through a populationbased strategy that mimics the evolution of the individuals innature it is a combination of a genetic algorithm (GA) [41]paradigm for finding the best exponents to be assigned toeach candidate input and a least squares (LS) regression [42]for the identification of the constant values 119886

119895[25]

Recently the EPR has been upgraded to MOGA EPRthrough the implementation of a search pattern of thebest models based on a multiobjective genetic algorithm(MOGA) [43] optimization scheme being able to find severalmodel expressions maximizing fitness to data and parsimonyof mathematical formulations simultaneously [29] MOGAEPR explores the space of 119898-terms formulas using at leastthree objectives among the following conflicting ones (i)maximization of model accuracy (ii) minimization of thenumber of model coefficients and (iii) minimization ofthe number of actually used model inputs (ie whoseexponent is not null in the resulting model structure)Based on the Pareto dominance criterion [44] MOGAEPR finally obtains a set of optimal solutions (ie thePareto front) which can be considered as trade-offs betweenstructural complexity and accuracy MOGA EPR displaysthe returned formulas within the Pareto front as rankedaccording to their structural complexity this way the com-parisons among formulas according to different criteria (egselected inputs structure etc) are facilitated as well as theselection of models suited for different modeling purposes[29]

In particular this possibility has been exploited in thispaper in order to find the best model for the estimationof shear strength of reinforced concrete circular columnsThus the first step was to define the MOGA EPR searchoptions (candidate model attributes candidate exponentsfor attributes maximum number of parameters objectivefunctions etc) then the set of optimal models with respectto the objective functions produced by MOGA EPR hasbeen analyzed by considering (i) the model structure withrespect to physical insight of the phenomenon at stake(ii) the similarities andor recurrent groups of variableswithin the mathematical structures of the Pareto models(iii) the generalization performance of models in terms ofstatistical indicators andmathematical parsimony Moreoverthe returnedmodels have been also evaluated looking at theirconsistency with existing codestheories about the physicalphenomenon as described above All these issues can lead toa quite robust model selection Figure 2 is illustrative of the

decision support framework allowed by MOGA EPR as hereimplemented

The present analyses have been performed using theEPR MOGA-XL v1 which is a Microsoft Office Excel add-infunction freely available at wwwhydroinformaticsit whichemploys a MOGA optimizer named optimized multiobjec-tive genetic algorithm (OPTIMOGA) [45]

4 Results and Comparison with Literature andTechnical Code Models

The proposed model has been verified by comparing it toexisting ones and to experimental tests The experimentaldata herein considered have been obtained from Merta [17]Capon and Cossio [19] Khalifa and Collins [20] Clarke andBirjandi [18] and Kim [38] and include the results of severaltests performed by different authorsThe test specimens referto circular columns characterized by six different diametersand having both shear and longitudinal reinforcementsLongitudinal reinforcement consistsmainly of high yield barsplaced evenly at eight locations round the pitch circle Shearreinforcement is generally in the form of 6mm or 8mmmild steel links The compressive strength of concrete rangesfrom 132MPa to 493MPa the stirrup percentage rangesfrom 01 to 045 the yielding stress of shear reinforcementranges from 250MPa to 1728MPa and the longitudinal steelpercentage ranges from 22 to 56

A total of 61 circular columns have been selected withdifferent geometric and mechanical parameters such asdimension of the section diameter (119863) area of concrete(119860119892) total cross-section of links at the neutral axis of the

section (119860 sh) spacing of links along themember (119904) concretecompressive strength (119891

119888) yield strength of closed links (119891yh)

total area of longitudinal reinforcement (119860 sl) yield strengthof longitudinal reinforcement (119891yl) concrete cover (cov)effective depth (119889) and axial compression load (119875)The rangeof datasets is listed in Table 1

The main objective of this study is to investigate theapplication of EPR MOGA technique to get possible newformulations of shear strength of circular RC columns able toreproduce the experimental data better than the existing codeexpressions In fact the correlation coefficient 1198772 betweenexperimental data and previsions of ACI 318M-08 [37] andBS 8100 [33] are 9015 and 7692 respectively thus thereis some room for possible improvements

Since most of experimental data refer to pure-shear teststhe dependence of shear strength from the axial force 119875 isnot investigated It can be accounted for by adding a termrepresentative of the axial force contribution as the oneproposed in Merta or in technical code models

41 EPR MOGA Search Settings Bearing in mind whatabove mentioned about MOGA EPR search paradigm somecompact expressions will be sought for in terms of bothnumber of parameters and explanatory variables involvedThis in turn would facilitate their physical interpretationand technical soundness for practical purposes The base

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 5

Kowalsky and Priestley [11] discussed Gheersquos modelconsidering that in the derivation of the truss mechanismequation it was assumed that a diagonal crack is ableto mobilize transverse reinforcement along a crack lengthextending the full width of the confined core of the concreteIn the compression zone of the column however all cracksare by definition closed Therefore if the crack is closedshear cannot be transferred across it by tension strain inthe transverse reinforcement Therefore it is evident that areduced column width minus119863 minus 119888 minus cov minus (Figure 1) is appropri-ate for calculating the number of hoops mobilized by thecracks between the compression struts and then a revisedtruss component equation should be adopted The authorsobtained an approximated equation given by

119881119904=120587

2119860119904119908119891119910119908

119863 minus 119888 minus cov119904

cot 120579 (11)

where 119860119904119908

is the area of the hooprsquos one leg 119891119910119908

is the hooprsquosyield strength 119888 is the compression zonersquos depth and cov isthe concrete cover

With reference to the concrete mechanism the followingequation was discussed in the cited work

119881119888= 120572120573120574radic1198911015840

119888(08119860

119892) (12)

where the parameter 120572 accounts for the column aspectratio the factor 120573 is a modified factor that accounts forthe longitudinal steel ratio and 120574 represents the reductionin strength of the concrete shear resisting mechanism withincreasing ductility

Also Kim and Mander [36] reported a deficiency of themodel originally proposed by Ghee et al [8] and verifiedthat the applied integral averaging by the reinforcement termrestricts its use only to members with diameter at leastfour times the spacing of the shear reinforcement For allother ratios the formula could be even more than 50nonconservative For this reason they proposed a correctivecoefficient to convert the amount of shear implied by acontinuous truss model to the discrete model

The current ACI 318-M-08 [37] considers a portion ofthe design shear force to be carried by the concrete shearresisting mechanism 119881

119888 with the remainder carried by

trussmechanism119881119904 involving transverse reinforcementThe

code proposes the following equation to calculate 119881119888for

members subjected to combined shear moment and axialcompression

119881119888= 017(1 +

119875

14119860119892

)radic1198911015840119888119887119908119889 (13)

The transverse reinforcement contribution is also calcu-lated as

119881119904=

119860 sh119891yh119889

119904 (14)

where the meaning of various terms has been explainedbefore

Merta [17] developed a simple shear capacitymodel of RCcircular sections It was based on the truss analogy by addingan empirical concrete contribution term to the capacity of theshear reinforcement obtaining finally the following relation

119881 = [37120588119897+ 018 + 008(

119875

119860119892

)03] 119896radic119891101584011988807119860119892

+ 119860119904119908119891119910119908

(18119899119905+ 120582119899119889minus 1)

(15)

where 119899119905represents the number of hoops across the crack

and is evaluated considering only the hoops outside thecompression zone

119899119905=119863 minus 119888 minus cov

119904cot 120579 (16)

while 120582 is assumed to be equal to 053 and 119899119889is given by

119899119889= INT [

1198632 minus cov119904

cot 120579] (17)

being 119888 cong 03119863 The influence of the main variables on theshear capacity such as the longitudinal reinforcement ratio120588

119897

the axial load level 119875119860119892 and the shear span-to-depth ratio

119886119863 has been determined empirically based on a total of44 pieces of data of circular cross-section specimens withoutshear reinforcement under monotonic load The term 07119860

119892

represents the sectionrsquos effective shear area while 119896 = 1 if119886119863 gt 25 and 125 if 119886119863 le 25

3 A Data-Driven Approach for Estimationof Shear Strength of Reinforced ConcreteCircular Columns The EvolutionaryPolynomial Regression Paradigm

Numerical Regression is the most powerful and commonlyapplied form of regression that provides a solution to theproblem of finding the best model to fit the observed data(eg fitting a line through a set of points) However thefunctional form (linear exponential logarithmic etc) hasto be selected before fitting commences On the other handgenetic pogramming uses simple but very powerful artificialintelligence tactics for computer learning inspired by naturalevolution to find the appropriate mathematical model to fita set of points The computer produces and evolves a wholepopulation of functional expressions based on how closelyeach of them fits the data The evolutionary polynomialregression (EPR) is a recently developed hybrid regressionmethod by Giustolisi and Savic [25] that integrates the bestfeatures of numerical regression with genetic programming[39 40] An example of the generalmodel structures that EPRcan manage is reported in

Y = 1198860+

119898

sum

119895=1

119886119895sdot (X1)ES(1198951)

sdot sdot (X119896)ES(119895119896)

sdot 119891 ((X1)ES(119895119896+1)

sdot sdot (X119896)ES(1198952119896)

)

(18)

6 Advances in Civil Engineering

where 119898 is the number of additive terms 119886119895are numerical

parameters to be estimated X119894are candidate explanatory

variables ES(119895 119911) (with 119911 = 1 2119896) is the exponentof the 119911th input within the 119895th term in (18) and 119891 is auser-selected function among a set of possible alternatives(including no function selection)The exponents ES(119895 119911) areselected from a user-defined set of candidate values Notethat it is recommended to include 0 among such exponentsin order to allow EPR to exclude some of the candidateexplanatory variables inX

119894from those present in the returned

models thus selecting themost influencing input variables onthe target prediction

Starting from a general structure such as that in (18)the final model expression is obtained through a populationbased strategy that mimics the evolution of the individuals innature it is a combination of a genetic algorithm (GA) [41]paradigm for finding the best exponents to be assigned toeach candidate input and a least squares (LS) regression [42]for the identification of the constant values 119886

119895[25]

Recently the EPR has been upgraded to MOGA EPRthrough the implementation of a search pattern of thebest models based on a multiobjective genetic algorithm(MOGA) [43] optimization scheme being able to find severalmodel expressions maximizing fitness to data and parsimonyof mathematical formulations simultaneously [29] MOGAEPR explores the space of 119898-terms formulas using at leastthree objectives among the following conflicting ones (i)maximization of model accuracy (ii) minimization of thenumber of model coefficients and (iii) minimization ofthe number of actually used model inputs (ie whoseexponent is not null in the resulting model structure)Based on the Pareto dominance criterion [44] MOGAEPR finally obtains a set of optimal solutions (ie thePareto front) which can be considered as trade-offs betweenstructural complexity and accuracy MOGA EPR displaysthe returned formulas within the Pareto front as rankedaccording to their structural complexity this way the com-parisons among formulas according to different criteria (egselected inputs structure etc) are facilitated as well as theselection of models suited for different modeling purposes[29]

In particular this possibility has been exploited in thispaper in order to find the best model for the estimationof shear strength of reinforced concrete circular columnsThus the first step was to define the MOGA EPR searchoptions (candidate model attributes candidate exponentsfor attributes maximum number of parameters objectivefunctions etc) then the set of optimal models with respectto the objective functions produced by MOGA EPR hasbeen analyzed by considering (i) the model structure withrespect to physical insight of the phenomenon at stake(ii) the similarities andor recurrent groups of variableswithin the mathematical structures of the Pareto models(iii) the generalization performance of models in terms ofstatistical indicators andmathematical parsimony Moreoverthe returnedmodels have been also evaluated looking at theirconsistency with existing codestheories about the physicalphenomenon as described above All these issues can lead toa quite robust model selection Figure 2 is illustrative of the

decision support framework allowed by MOGA EPR as hereimplemented

The present analyses have been performed using theEPR MOGA-XL v1 which is a Microsoft Office Excel add-infunction freely available at wwwhydroinformaticsit whichemploys a MOGA optimizer named optimized multiobjec-tive genetic algorithm (OPTIMOGA) [45]

4 Results and Comparison with Literature andTechnical Code Models

The proposed model has been verified by comparing it toexisting ones and to experimental tests The experimentaldata herein considered have been obtained from Merta [17]Capon and Cossio [19] Khalifa and Collins [20] Clarke andBirjandi [18] and Kim [38] and include the results of severaltests performed by different authorsThe test specimens referto circular columns characterized by six different diametersand having both shear and longitudinal reinforcementsLongitudinal reinforcement consistsmainly of high yield barsplaced evenly at eight locations round the pitch circle Shearreinforcement is generally in the form of 6mm or 8mmmild steel links The compressive strength of concrete rangesfrom 132MPa to 493MPa the stirrup percentage rangesfrom 01 to 045 the yielding stress of shear reinforcementranges from 250MPa to 1728MPa and the longitudinal steelpercentage ranges from 22 to 56

A total of 61 circular columns have been selected withdifferent geometric and mechanical parameters such asdimension of the section diameter (119863) area of concrete(119860119892) total cross-section of links at the neutral axis of the

section (119860 sh) spacing of links along themember (119904) concretecompressive strength (119891

119888) yield strength of closed links (119891yh)

total area of longitudinal reinforcement (119860 sl) yield strengthof longitudinal reinforcement (119891yl) concrete cover (cov)effective depth (119889) and axial compression load (119875)The rangeof datasets is listed in Table 1

The main objective of this study is to investigate theapplication of EPR MOGA technique to get possible newformulations of shear strength of circular RC columns able toreproduce the experimental data better than the existing codeexpressions In fact the correlation coefficient 1198772 betweenexperimental data and previsions of ACI 318M-08 [37] andBS 8100 [33] are 9015 and 7692 respectively thus thereis some room for possible improvements

Since most of experimental data refer to pure-shear teststhe dependence of shear strength from the axial force 119875 isnot investigated It can be accounted for by adding a termrepresentative of the axial force contribution as the oneproposed in Merta or in technical code models

41 EPR MOGA Search Settings Bearing in mind whatabove mentioned about MOGA EPR search paradigm somecompact expressions will be sought for in terms of bothnumber of parameters and explanatory variables involvedThis in turn would facilitate their physical interpretationand technical soundness for practical purposes The base

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Advances in Civil Engineering

where 119898 is the number of additive terms 119886119895are numerical

parameters to be estimated X119894are candidate explanatory

variables ES(119895 119911) (with 119911 = 1 2119896) is the exponentof the 119911th input within the 119895th term in (18) and 119891 is auser-selected function among a set of possible alternatives(including no function selection)The exponents ES(119895 119911) areselected from a user-defined set of candidate values Notethat it is recommended to include 0 among such exponentsin order to allow EPR to exclude some of the candidateexplanatory variables inX

119894from those present in the returned

models thus selecting themost influencing input variables onthe target prediction

Starting from a general structure such as that in (18)the final model expression is obtained through a populationbased strategy that mimics the evolution of the individuals innature it is a combination of a genetic algorithm (GA) [41]paradigm for finding the best exponents to be assigned toeach candidate input and a least squares (LS) regression [42]for the identification of the constant values 119886

119895[25]

Recently the EPR has been upgraded to MOGA EPRthrough the implementation of a search pattern of thebest models based on a multiobjective genetic algorithm(MOGA) [43] optimization scheme being able to find severalmodel expressions maximizing fitness to data and parsimonyof mathematical formulations simultaneously [29] MOGAEPR explores the space of 119898-terms formulas using at leastthree objectives among the following conflicting ones (i)maximization of model accuracy (ii) minimization of thenumber of model coefficients and (iii) minimization ofthe number of actually used model inputs (ie whoseexponent is not null in the resulting model structure)Based on the Pareto dominance criterion [44] MOGAEPR finally obtains a set of optimal solutions (ie thePareto front) which can be considered as trade-offs betweenstructural complexity and accuracy MOGA EPR displaysthe returned formulas within the Pareto front as rankedaccording to their structural complexity this way the com-parisons among formulas according to different criteria (egselected inputs structure etc) are facilitated as well as theselection of models suited for different modeling purposes[29]

In particular this possibility has been exploited in thispaper in order to find the best model for the estimationof shear strength of reinforced concrete circular columnsThus the first step was to define the MOGA EPR searchoptions (candidate model attributes candidate exponentsfor attributes maximum number of parameters objectivefunctions etc) then the set of optimal models with respectto the objective functions produced by MOGA EPR hasbeen analyzed by considering (i) the model structure withrespect to physical insight of the phenomenon at stake(ii) the similarities andor recurrent groups of variableswithin the mathematical structures of the Pareto models(iii) the generalization performance of models in terms ofstatistical indicators andmathematical parsimony Moreoverthe returnedmodels have been also evaluated looking at theirconsistency with existing codestheories about the physicalphenomenon as described above All these issues can lead toa quite robust model selection Figure 2 is illustrative of the

decision support framework allowed by MOGA EPR as hereimplemented

The present analyses have been performed using theEPR MOGA-XL v1 which is a Microsoft Office Excel add-infunction freely available at wwwhydroinformaticsit whichemploys a MOGA optimizer named optimized multiobjec-tive genetic algorithm (OPTIMOGA) [45]

4 Results and Comparison with Literature andTechnical Code Models

The proposed model has been verified by comparing it toexisting ones and to experimental tests The experimentaldata herein considered have been obtained from Merta [17]Capon and Cossio [19] Khalifa and Collins [20] Clarke andBirjandi [18] and Kim [38] and include the results of severaltests performed by different authorsThe test specimens referto circular columns characterized by six different diametersand having both shear and longitudinal reinforcementsLongitudinal reinforcement consistsmainly of high yield barsplaced evenly at eight locations round the pitch circle Shearreinforcement is generally in the form of 6mm or 8mmmild steel links The compressive strength of concrete rangesfrom 132MPa to 493MPa the stirrup percentage rangesfrom 01 to 045 the yielding stress of shear reinforcementranges from 250MPa to 1728MPa and the longitudinal steelpercentage ranges from 22 to 56

A total of 61 circular columns have been selected withdifferent geometric and mechanical parameters such asdimension of the section diameter (119863) area of concrete(119860119892) total cross-section of links at the neutral axis of the

section (119860 sh) spacing of links along themember (119904) concretecompressive strength (119891

119888) yield strength of closed links (119891yh)

total area of longitudinal reinforcement (119860 sl) yield strengthof longitudinal reinforcement (119891yl) concrete cover (cov)effective depth (119889) and axial compression load (119875)The rangeof datasets is listed in Table 1

The main objective of this study is to investigate theapplication of EPR MOGA technique to get possible newformulations of shear strength of circular RC columns able toreproduce the experimental data better than the existing codeexpressions In fact the correlation coefficient 1198772 betweenexperimental data and previsions of ACI 318M-08 [37] andBS 8100 [33] are 9015 and 7692 respectively thus thereis some room for possible improvements

Since most of experimental data refer to pure-shear teststhe dependence of shear strength from the axial force 119875 isnot investigated It can be accounted for by adding a termrepresentative of the axial force contribution as the oneproposed in Merta or in technical code models

41 EPR MOGA Search Settings Bearing in mind whatabove mentioned about MOGA EPR search paradigm somecompact expressions will be sought for in terms of bothnumber of parameters and explanatory variables involvedThis in turn would facilitate their physical interpretationand technical soundness for practical purposes The base

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 7

Table 1 Details and test results of specimens with circular cross-section

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]M12 152 18145839 28 10 5624 300 100 1216 500 39920846 0 450M13 152 18145839 28 10 5624 300 100 1216 500 39920846 0 460M14 152 18145839 28 10 5624 300 100 1216 500 39920846 0 38011-1 300 70685835 241 20 9900 300 150 240 500 39584067 0 186011-2 300 70685835 241 20 9900 300 150 240 500 39584067 0 188012-1 300 70685835 238 20 10125 300 75 240 500 39584067 0 211012-2 300 70685835 238 20 10125 300 75 240 500 39584067 0 239013-1 300 70685835 484 20 9900 300 150 240 500 39584067 0 227013-2 300 70685835 484 20 9900 300 150 240 500 39584067 0 228014-1 300 70685835 505 20 10125 300 75 240 500 39584067 0 279014-2 300 70685835 505 20 10125 300 75 240 500 39584067 0 288015-1 300 70685835 243 20 9900 300 150 240 500 254469 0 145015-2 300 70685835 243 20 9900 300 150 240 500 254469 0 148016-1 300 70685835 467 20 9900 300 150 240 500 254469 0 185016-2 300 70685835 467 20 9900 300 150 240 500 254469 0 186017-1 300 70685835 237 20 5850 300 150 240 500 16257742 0 117017-2 300 70685835 237 20 5850 300 150 240 500 16257742 0 115019-1 300 70685835 266 20 5850 300 150 240 500 254469 0 113019-2 300 70685835 266 20 5850 300 150 240 500 254469 0 129020-1 300 70685835 493 20 5850 300 150 240 500 254469 0 149020-2 300 70685835 493 20 5850 300 150 240 500 254469 0 137021-1 300 70685835 222 20 5850 300 150 240 500 39584067 0 131021-2 300 70685835 222 20 5850 300 150 240 500 39584067 0 151022-1 300 70685835 455 20 5850 300 150 240 500 39584067 0 1630

Clarke and Birjandi [18] 22-2 300 70685835 455 20 5850 300 150 240 500 39584067 0 164023-1 300 70685835 251 20 5850 300 150 240 500 16257742 0 101023-2 300 70685835 251 20 5850 300 150 240 500 16257742 0 103024-1 300 70685835 489 20 5850 300 150 240 500 16257742 0 114024-2 300 70685835 489 20 5850 300 150 240 500 16257742 0 128025-1 300 70685835 243 20 5850 300 150 240 500 254469 0 98025-2 300 70685835 243 20 5850 300 150 240 500 254469 0 122026-1 300 70685835 471 20 5850 300 150 240 500 254469 0 114026-2 300 70685835 471 20 5850 300 150 240 500 254469 0 150027-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 125027-2 300 70685835 228 20 5850 300 150 240 500 39584067 0 134028-1 300 70685835 228 20 5850 300 150 240 500 39584067 0 158028-2 300 70685835 453 20 5850 300 150 240 500 39584067 0 175037-1 300 70685835 439 20 9900 300 150 240 500 39584067 2709 232037-2 300 70685835 439 20 9900 300 150 240 500 39584067 0 218038-1 300 70685835 361 20 9900 300 150 240 500 39584067 2709 209038-2 300 70685835 361 20 9900 300 150 240 500 39584067 0 206039-1 300 70685835 363 20 9900 300 150 240 500 39584067 2706 217239-2 300 70685835 363 20 9900 300 150 240 500 39584067 0 197040-1 300 70685835 341 20 9900 300 150 240 500 39584067 2741 225040-2 300 70685835 341 20 9900 300 150 240 500 39584067 0 183043-1 500 19634954 378 20 9800 300 140 400 500 51050881 0 313043-2 500 19634954 378 20 9800 300 140 400 500 51050881 0 366044-1 500 19634954 329 20 9800 300 140 400 500 51050881 0 301044-2 500 19634954 329 20 9800 300 140 400 500 51050881 0 3290

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Civil Engineering

Table 1 Continued

References Number 119863 119860119892

119891119888

cov 119860 sh 119891yh 119904 119889 119891yl 119860 sl 119875 119881test

[mm] [mm2] [MPa] [mm] [mm2] [MPa] [mm] [mm] [MPa] [mm2] [kN] [kN]

Khalifa and Collins [20]

SC1 445 15552847 193 23 6675 410 150 356 516 58945291 1017 3240SC2 445 15552847 23 23 20025 510 150 356 516 58945291 1083 4780SC3 445 15552847 245 23 300375 510 150 356 516 58945291 1085 5780SC4 445 15552847 265 23 20025 430 150 356 516 58945291 1050 45601 400 12566371 422 0 5760 700 60 320 700 4712389 0 4300

Merta [17] 2 400 12566371 422 0 5760 700 60 320 700 4712389 0 4320Capon and Cossio [19] F125 251 4948087 132 15 6275 250 250 2008 400 15240108 0 595

F125 251 4948087 132 15 6275 250 125 2008 400 15240108 0 820YJC200R 445 15552847 404 25 14240 445 200 356 460 6003399 0 3230YJC150R 445 15552847 36 25 140175 445 150 356 460 6003399 0 4110

Kim [38] YJC100R 445 15552847 36 25 14240 445 100 356 460 6003399 0 4790YJC200W 445 15552847 332 25 445 1728 200 356 460 6003399 0 3150YJC100W 445 15552847 36 25 0 1728 100 356 460 0 0 4340

∙ Data preprocessing

∙ Similarities between terms in different

∙ Selected attributes∙ Complexity of expressions

∙ Consistency with existing theories∙ Physical insight∙ Selection of candidate model attributes

∙ Selection of candidate set of exponents∙ Definition of objective functions Evaluation of

generalizationperformance Other criteria

Model selection process

Accuracy

Com

plex

ity Selected model(s)

models

y = f(X1 Xk)

middot middot middot

middot middot middot

Model r

Model hModel 2

Model 1

Figure 2 Decision support framework for model selection based on EPR MOGA paradigm

model structure reported in (18) is adopted with no function119891 selected accordingly each additive monomial term isassumed to be a combination of the input variables raised torelevant exponents

The candidate explanatory variables (ie model inputssee Table 1) are 119863 119889 119860

119892 119860 sh 119904 119860 sl 119891119888 and 119891yh they

have been selected by both looking at the existing literatureand performing some preliminary sensitivity analyses Thecandidate exponents are [minus1 minus05 minus23 0 05 23 1] in orderto represent both linear and nonlinear relationships as wellas directinverse dependence between candidate input(s) andoutput (ie model target)Themaximum number of additiveterm in the final expressions is assumed to be 119898 = 3 so thatcould be easily compared with the existing formulations

42 Selection of EPR MOGA Models Two EPR MOGAruns were finally selected characterized by different setsof exponents and by the same set of input variables (119863119889 119860119892 119860 sh 119904 119860 sl 119891119888 119891yh) Models obtained from EPR

MOGAruns showdifferent number of terms and explanatoryvariables as well as different accuracy The model selectionprocedure takes into account some key aspects agreementwith experimental data model parsimony and consistencywith both physical insight about the shear behavior of circularRC columns and previous building code formulations Fromamong all models obtained the following formulas have beenselected (the unit for 119881Rd is [N] and all the mechanical andgeometrical quantities are expressed in [N] and [mm] resp)

119881Rd = 41525119889120588ℎ119891yh + 013552119860

119892radic119891119888radic120588119897

(1198772

= 9561)

(19)

119881Rd = + 7121radic119860119892119891yh + 0043393120588

ℎ119860119892radic119891yh119891119888

+ 0013495120588119897119860119892

radic119863 (1198772

= 9784)

(20)

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 9

119881Rd = 10737119889119860 sh119904

119891yh + 025274119863119889radic119891119888

(1198772

= 9153)

(21)

119881Rd = 098243119889119860 sh119904

119891yh + 0086185119863119889radic119891119888

times (1 + 562119860 sl119863119889

) (1198772

= 9598)

(22)

119881Rd = 10072119889119860 sh119904

119891yh + 23931119889119863(119860 sl119860119892

)11989123

119888

+ 0098766119863119889radic119891119888 (119877

2

= 9539)

(23)

where 120588ℎ= (119860 sh(119904119863)) sdot 100 and 120588

119897= (119860 sl119860119892) sdot 100

Although the EPR MOGA search for models is based onthe coefficient of determination (COD) as defined in [25]model performances are reported in terms of 1198772 in order toallow a straight comparison with code provisions [33 37] andalternative formulations [17 18]

Expressions (19)ndash(22) are obtained by considering the setof exponents [minus1 minus05 0 05 1] while in expression (23)exponents belonging to the set [minus1 minus23 minus05 0 05 23 1]are assumed

Proposedmodels are compared to ACI 318-M-08 [37] BS8100 [33] predictions and the two existing models of Clarkeand Birjandi [18] and Merta [17] the corresponding corre-lation coefficients 1198772 with respect to the experimental dataherein considered are respectively equal to 9015 76929623 and 9529 Figure 3 depicts the shear strength aspredicted by the EPR models (19)ndash(23) versus the exper-imental results similarly Figure 4 shows the same graphswith reference to the code formulations and the literaturemodels mentioned above As expected from the values of thecoefficients 1198772 expression (20) provides the smallest scatterin the plot of the experimentaltheoretical ultimate shearstrength of specimens however the complexity of expression(20) discourages its use for practical purposes Formulas (19)(22) and (23) provide predictions similar to the Clarke andBirjandi ones and are anyway slightly better than the Mertaones

Formula (23) predictions are more conservative thanClarke and Birjandi ones with a similar or often closercorrelation with experimental data It is also evident that theACI 318-M-08 and BS 8100 predictions which neglect theinfluence of the longitudinal reinforcement contribution arehighly conservative In fact BS 8100 predictions in Figure 4are calculated by adopting cot 120579 = 25 otherwise by puttingcot 120579 = 1 more conservative values of shear strength areobtained

In expressions (19) and (20) the physical consistencyof the problem is not easily recognizable while on thecontrary models (21)ndash(23) are similar to the building codeformulations and thus preserve the physical insight into theproblem In particular (21) differs from the ACI 318-M-08[37] prediction only for the multiplying factor 025274 of theconcrete contribution which is higher than the code one

allowing a better agreement with the experimental values ofshear capacity

Equation (21) implies that the shear capacity of themember is the sum of the shear carried by concrete andtransverse reinforcement nevertheless according to thisapproach which is consistent with code formulation a maxi-mum correlation coefficient equal to 9153 can be obtainedOn the contrary in the other proposed models also thecontribution of longitudinal reinforcement is accounted forallowing gaining higher values of the correlation coefficient

In particular in (22) and (23) the concrete and transversereinforcement contributions preserve the same shapes as ACI318-M-08 while the longitudinal reinforcement contributionis function of the concrete tensile strength which in factis proportional to radic119891

119888according to ACI 318-M-08 [37] or

Vecchio and Collins [13] and to11989123119888

according to Italian code[46] Moreover the longitudinal reinforcement contributionin (23) is similar to the Italian code expression valid inthe case of rectangular sections in the absence of stirrups[46] These results suggest that the third contribution tothe shear capacity of circular sections is due to the doweleffect longitudinal reinforcement affects the strain state ofthe element and the crack widths influencing the concretecontribution by aggregate interlock to the shear strength

One of the most significant advantages of EPR is that thistechnique is capable of representing the relationship betweenthe parameters in the form of a polynomial expressionso parametric studies can be conducted to investigate thecontribution of different parameters involved and to assesswhether the developed model is able to capture the physicalrelationships between various parameters of the system Inthe context of the present study this also gives an insightto the user about the degree of influence that differentparameters can have on the shear of circular sections Toachieve this a parametric study is carried out where allparameters are set to their mean values except one thatis changed from its minimum to maximum values in thetesting set The results of the parametric study for the shearstrength model of circular elements are shown in Figure 5with reference to code provisions (ACI 318-M-08 BS 8100)and (20) (22) By adopting formula (23) graphs quite similarto formula (22) ones are obtained which are omitted here forsake of brevity

The results show that the shear strength increases asthe concrete compressive strength the amount of transversereinforcement the amount of longitudinal reinforcementand the section diameter increase In particular the sectiondiameter has a significant influence on the value of theshear strength This is consistent with the expected shearbehavior of circular sections It emerges that the EPRmodelsdirectly developed using experimental data are able tocapture the various aspects of shear behavior of circularsection correctly and with higher accuracy than exitingtechnical codes Moreover differently from ACI 318-M-08and BS 8100 predictions the proposed EPR models can alsocapture the influence of longitudinal reinforcement on theshear strength In fact according to the theoretical modelsshear strength should be proportional to the longitudinal

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Advances in Civil Engineering

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 19 (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

Vformula 20 (kN) Vformula 21 (kN)

Vformula 22 (kN) Vformula 23 (kN)

Figure 3 Shear strength of circular section members with shear reinforcement EPR models

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 11

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VClarke and Birjandi (kN)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Vte

st(k

N)

VMerta (kN)

VACI 318 (kN) VBS 8100 (kN)

Figure 4 Shear strength of circular section members with shear reinforcement code and literature models

reinforcement ratio this is due to the circumstance that as thelongitudinal reinforcement ratio decreases flexural stressesand strains in concrete increase thus leading to larger crackwidths in addition dowel action is weakened due to the lowlongitudinal reinforcement ratio So the proposedmodels arecoherent with this mechanical behavior

The above considerations clearly indicate that the pro-posed models (see (22) and (23)) improve the prediction ofthe shear capacity preserving the physical insight into theproblem moreover they represent sufficiently conservativedesign equations and their simplicitymakes them suitable forincorporation in technical codes

5 Conclusions

In the present paper new formulas for calculating theshear strength of RC circular columns have been proposedDifferently from standard approaches they are obtained

by EPR In the EPR approach no preprocessing of thedata is required and there is no need for normalization orscaling of the data The efficiency of such approach has beentested by using experimental data of 61 circular RC elementsreported in technical literature The considered inputs aredimension of the section diameter area of concrete totalcross-section of links at the neutral axis of the section spacingof links along the member concrete compressive strengthyield strength of closed links total area of longitudinalreinforcement yield strength of longitudinal reinforcementand effective depth Models obtained from EPRMOGA runsshow different number of terms and explanatory variables aswell as different accuracy Among all the proposed modelsthe selected ones provide good predictions particularly withrespect to the code ones The increased accuracy descendsfrom including the contribution of the dowel effect exercisedby the longitudinal reinforcement in the expression of theshear strength

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Advances in Civil Engineering

0

50

100

150

200

250

300

350

400

0 01 02 03 04 05 06 07 08 09 1 11 12 13

050

100150200250300350400450500

100 150 200 250 300 350 400 450 500 550 6000

306090

120150180210240270300

0 700 1400 2100 2800 3500 4200 4900 5600 6300 7000

0306090

120150180210240270300

10 15 20 25 30 35 40 45 50 55

ACI 318 BS 8100Formula 22 Formula 20

ACI 318 BS 8100Formula 22 Formula 20

V(k

N)

V(k

N)

V(k

N)

V(k

N)

fc (MPa) Ashs (mm)

D (mm) Asl (mm2)

Figure 5 Results of the parametric study conducted on the EPR and code models

A parametric study was also conducted to evaluate theeffects of different parameters on the shear strength ofcircular elements and to verify if the developed models canrepresent the physical relationships between the contribut-ing parameters Comparison of the results shows that thedeveloped EPR models provide very accurate predictions forthe shear strength of circular sections They can capture andrepresent various aspects of shear behavior directly fromexperimental data The developed models present a struc-tured and transparent representation of the system allowinga physical interpretation of the problem that gives the user aninsight into the relationship between themechanical behaviorand various contributing parameters From a practical pointof view the EPR models presented in this paper provide veryaccurate results and are easy to use

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Fiore D Foti P Monaco D Raffaele and G Uva ldquoAnapproximate solution for the rheological behavior of non-homogeneous structures changing the structural system during

the construction processrdquo Engineering Structures vol 46 pp631ndash642 2013

[2] A Fiore G Marano and P Monaco ldquoEarthquake-inducedlateral-torsional pounding between two equal height multi-storey buildings undermultiple bi-directional groundmotionsrdquoAdvances in Structural Engineering vol 16 no 5 pp 845ndash8652013

[3] A Fiore P Monaco and D Raffaele ldquoViscoelastic behaviourof non-homogeneous variable-section beams with post-ponedrestraintsrdquo Computers and Concrete vol 9 no 5 pp 357ndash3742012

[4] A Fiore and P Monaco ldquoAnalysis of the seismic vulnerabilityof the ldquoQuinto Orazio Flaccordquo school in Bari (Italy)rdquo IngegneriaSismica vol 2011 no 1 pp 43ndash62 2011

[5] A Fiore and P Monaco ldquoEarthquake-induced poundingbetween the main buildings of the ldquoQuinto Orazio Flaccordquoschoolrdquo Earthquake and Structures vol 1 no 4 pp 371ndash3902010

[6] A Fiore and P Monaco ldquoPOD-based representation of thealongwind equivalent static force for long-span bridgesrdquo Windand Structures vol 12 no 3 pp 239ndash257 2009

[7] M M Tahir P N Shek A Sulaiman and C S Tan ldquoExperi-mental investigation of short cruciform columns using univer-sal beam sectionsrdquo Construction and Building Materials vol 23no 3 pp 1354ndash1364 2009

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 13

[8] A B Ghee M J N Priestley and T Paulay ldquoSeismic shearstrength of circular reinforced concrete columnsrdquo ACI Struc-tural Journal vol 86 no 1 pp 45ndash59 1989

[9] M J N Priestley R Verma and Y Xiao ldquoSeismic shear strengthof reinforced concrete columnsrdquo ASCE Journal of StructuralEngineering vol 120 no 8 pp 2310ndash2329 1994

[10] Y-L Wong T Paulay and M J N Priestley ldquoResponseof circular reinforced concrete columns to multi-directionalseismic attackrdquo ACI Structural Journal vol 90 no 2 pp 180ndash191 1993

[11] M J Kowalsky and M J N Priestley ldquoImproved analyti-cal model for shear strength of circular reinforced concretecolumns in seismic regionsrdquo ACI Structural Journal vol 97 no3 pp 388ndash396 2000

[12] M P Collins E C Bentz and Y J Kim ldquoShear strengthof circular reinforced concrete columnsrdquo in S M UzumeriSymposium Behavior and Design of Concrete Structures forSeismic Performance pp 45ndash86 ACI Farmington Hills MichUSA 2002

[13] F J Vecchio and M P Collins ldquoThe modified compression-field theory for reinforced concrete elements subjected to shearrdquoJournal of the American Concrete Institute vol 83 no 2 pp 219ndash231 1986

[14] E C Bentz F J Vecchio andM P Collins ldquoSimplifiedmodifiedcompression field theory for calculating shear strength ofreinforced concrete elementsrdquo ACI Structural Journal vol 103no 4 pp 614ndash624 2006

[15] American Association of State Highway and TransportationOfficial AASHTO LRFD Bridge Design Specifications 4th edi-tion 2007

[16] J TurmoG Ramos andA C Aparicio ldquoShear truss analogy forconcrete members of solid and hollow circular cross sectionrdquoEngineering Structures vol 31 no 2 pp 455ndash465 2009

[17] I Merta ldquoShear strength model of reinforced concrete circularcross-section membersrdquo in Vortrag Third International Con-ference on Structural Engineering Mechanics and ComputationCape Town South Africa Recent Developments in StructuralEngineeringMechanics and Computation A Zingoni EdMill-press Science Publishers Rotterdam The Netherlands 2007

[18] J L Clarke and F K Birjandi ldquoThe behaviour of reinforcedconcrete circular sections in shearrdquoThe Structural Engineer vol71 no 5 pp 73ndash81 1993

[19] M J F Capon and R D Cossio ldquoDiagonal tension in concretemembers of circular sectionrdquo Tech Rep 466 Portland CementAssociation Foreign Literature Study 1965

[20] J U Khalifa and M P Collins ldquoCircular reinforced concretemembers subjected to shearrdquo Publication 81-08 Department ofCivil Engineering University of Toronto 1981

[21] Y Nagato Shear Strength of ReinforcedConcrete Members withCircular Cross-Section Federal University of Rio de Janeiro

[22] M J N Priestley F Seible and M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

[23] J H Lee S H Ko and J H Choi ldquoShear strength and capacityprotection of RC bridge columnsrdquo in Proceedings of ANCER2004 Hawaii Hawaii USA 2004

[24] U G Jensen and L C Hoang ldquoShear strength prediction ofcircular RC members by the crack sliding modelrdquo Magazine ofConcrete Research vol 61 no 9 pp 691ndash703 2009

[25] O Giustolisi and D A Savic ldquoA symbolic data-driven tech-nique based on evolutionary polynomial regressionrdquo Journal ofHydroinformatics vol 8 no 3 pp 207ndash222 2006

[26] L Berardi Z Kapelan O Giustolisi and D A Savic ldquoDevel-opment of pipe deterioration models for water distributionsystems using EPRrdquo Journal of Hydroinformatics vol 10 no 2pp 113ndash126 2008

[27] A Doglioni D Mancarella V Simeone and O GiustolisildquoInferring groundwater system dynamics from hydrologicaltime-series datardquo Hydrological Sciences Journal vol 55 no 4pp 593ndash608 2010

[28] A Fiore L Berardi and G C Marano ldquoPredicting torsionalstrength of RC beams by using Evolutionary PolynomialRegressionrdquo Advances in Engineering Software vol 47 no 1 pp178ndash187 2012

[29] O Giustolisi and D A Savic ldquoAdvances in data-driven analysesandmodelling using EPR-MOGArdquo Journal ofHydroinformaticsvol 11 no 3-4 pp 225ndash236 2009

[30] E Morsch Der EisenbetonbaumdashSeine Theorie und AnwendungKonrad Wittwer Stuttgart Germany 1922

[31] M P Nielsen Limit Analysis and Concrete Plasticity Prentice-Hall Series in Civil Engineering Prentice-Hall EnglewoodCliffs NJ USA 1984

[32] ldquoEurocode 2 design of concrete structures prEN 1992-1-1 draftfor stage 49rdquo European Standard Commission of the EuropeanCommunities European Committee for Standardization 2002

[33] BS 8100 Structural Use of Concrete Part I Code of Practicefor Design and Construction British Standards InstitutionLondon UK 1985

[34] ldquoSteel concrete and composite bridges part 4 code of practicefor design of concrete bridgesrdquo BS 5400 British StandardsInstitution London UK 1990

[35] A N Dancygier ldquoShear carried by transverse reinforcement incircular RC elementsrdquo Journal of Structural Engineering vol 127no 1 pp 81ndash83 2001

[36] J H Kim and J B Mander ldquoTheoretical shear strength ofconcrete columns due to transverse steelrdquo Journal of StructuralEngineering vol 131 no 1 pp 197ndash199 2005

[37] ACI 318-M-08 Building Code Requirements for Structural Con-crete (ACI 318M-08) and Commentary American ConcreteInstitute 2008

[38] Y J Kim The shear response of circular columns reinforcedwith high strength spirals [MS thesis] Department of CivilEngineering University of Toronto 2000

[39] G C Marano R Greco G Quaranta A Fiore J Avakian andD Cascella ldquoParametric identification of nonlinear devices forseismic protection using soft computing techniquesrdquo AdvancedMaterials Research vol 639-640 no 1 pp 118ndash129 2013

[40] G Quaranta A Fiore and G C Marano ldquoOptimum designof prestressed concrete beams using constrained differentialevolution algorithmrdquo Structural and Multidisciplinary Opti-mization vol 49 no 3 pp 441ndash453 2014

[41] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[42] N R Draper and H Smith Applied Regression Analysis JohnWiley amp Sons New York NY USA 1998

[43] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley London UK 1989

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Advances in Civil Engineering

[44] V ParetoCours DrsquoEconomie Politique Rouge and Cic Vol I andII Lausanne Switzerland 1896

[45] D Laucelli andOGiustolisi ldquoScour depthmodelling by amulti-objective evolutionary paradigmrdquo EnvironmentalModelling andSoftware vol 26 no 4 pp 498ndash509 2011

[46] D M LL PP 9 Gennaio 1996 Norme tecniche per il calcololrsquoesecuzione ed il collaudo delle strutture in cemento armatonormale e precompresso e per le strutture metalliche 1996(Italian)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of