Survey Paper

A simple formulation for effective flexural stiffness of circularreinforced concrete columns

Naci Caglar n, Aydin Demir, Hakan Ozturk, Abdulhalim AkkayaDepartment of Civil Engineering, Sakarya University, 54187 Sakarya, Turkey

a r t i c l e i n f o

Article history:Received 29 April 2014Received in revised form6 August 2014Accepted 12 October 2014

Keywords:Effective flexural stiffnessMoment–curvatureReinforced concreteGenetic programmingEurocode-8TEC-2007

a b s t r a c t

Concrete cracking reduces flexural and shear stiffness of reinforced concrete (RC) members. Thereforeanalyzing RC structures without considering the cracking effect may not represent actual behavior.Effective flexural stiffness resulting from concrete cracking depends on some important parameters suchas confinement, axial load level, section dimensions and material properties of concrete andreinforcing steel.

In this study, a simple formula as a securer, quicker and more robust is proposed to determine theeffective flexural stiffness of cracked sections of circular RC columns. This formula is generated by geneticprogramming (GP). The generalization capabilities of the explicit formulations are compared by crosssectional analysis results and confirmed on a 3-D building model. Moreover the results from GP basedformulation are compared with EC-8 and TEC-2007. It is demonstrated that the GP based model is highlysuccessful to determine the effective flexural stiffness of circular RC columns.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

It is known that concrete cracking affects the behavior ofreinforced concrete (RC) members primarily by reducing theirflexural and shear stiffness. Linear elastic analysis of RC structureswithout considering the cracking effect or using constant coeffi-cients given in some current codes for cracking may not representactual behavior of RC members.

In current codes such as Eurocode and Turkish Earthquake Codefor the seismic design and performance evaluation of RC structures,effective flexural stiffness of RC members (beams, columns andshear walls etc.) stemming from concrete cracking can be calculatedusually by reducing their moment of inertia to specified values orusing some empirical formulations. These equations do not takeinto account the most important parameters affecting the inelasticbehavior of RC members such as confinement, axial load level,section dimensions and material properties of concrete and reinfor-cing steel (Dogangun, 2013).

The structural analyses not being inclusive of the effective flexuralstiffness with these major parameters will not represent actualbehavior of the structures. It can be observed as an importantdeviation in periods of structures and inelastic displacement demand

of earthquake. That will cause faulty determination of seismic perfor-mance of the structures.

A lot of studies have been performed to determine the effectiveflexural stiffness, EIef f , of RC members. While some of them haveignored the most important parameters affecting ductile and inelasticbehavior of RC elements, others have proposed very complex and notpractical formulations. Because of these reasons it is highly importantto propose a formula consisting of both features being simple to beused easily in practice by design engineers and including theparameters representing actual behavior of RC members.

Due to the low tensile strength of concrete, cracking, which isprimarily load dependent, may occur at service loads and reduce theflexural and shear stiffness of RC members. The effective flexuralstiffness of a structural concrete column is significantly affected bycracking along its length and by inelastic behavior of the concrete,reinforcing steel, and structural steel. EIef f is, therefore, a complexfunction of a number of variables that cannot be readily transformedinto a unique and simple analytical equation (Tikka and Mirza, 2008).

However, the analysis of reinforced concrete structures isusually carried out by linear elastic models which either neglectthe concrete cracking effect or consider it by reducing the stiffnessof members arbitrarily. It is also quite possible that the design oftall reinforced concrete structures on the basis of linear elastictheory may not satisfy serviceability requirements. For accuratedetermination of the deflections, cracked members in reinforcedconcrete structures need to be identified and their effectiveflexural and shear rigidities determined (Dundar and Kara, 2007).

Contents lists available at ScienceDirect

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

Engineering Applications of Artificial Intelligence

http://dx.doi.org/10.1016/j.engappai.2014.10.0110952-1976/& 2014 Elsevier Ltd. All rights reserved.

n Correspondence to: Sakarya University, Engineering Faculty, Department ofCivil Engineering, Esentepe Campus, 54187 Sakarya, Turkey.

E-mail address: caglar@sakarya.edu.tr (N. Caglar).

Engineering Applications of Artificial Intelligence 38 (2015) 79–87

In the design and seismic performance evaluation of RCstructures, the moments of inertia of the beams and columns areusually reduced at the specified ratios to compute the lateral driftby considering the cracking effects on the stiffness of the structuralframe. The gross moment of inertia of columns is generallyreduced to 80% of their uncracked values while the gross momentof inertia of beams is reduced to 50%, without considering thetype, history and magnitude of loading, and the reinforcementratios in the members (Stafford-Smith and Coull, 1991).

In Eurocode 8 (EC-8), the stiffness of the load bearing elementsof RC buildings should be evaluated taking into account the effectof cracking corresponding to the initiation of yielding of thereinforcement. It is advised that the elastic flexural and shearstiffness properties of concrete and masonry elements may betaken to be equal to one-half of the corresponding stiffness of theuncracked elements (Eurocode-8, 2004). It can be said thatdimension, concrete compressive strength, reinforcement of sec-tions and axial force acting on the sections are ignored and justgiven a constant coefficient.

In Turkish Earthquake Code 2007 (TEC-2007), there are someempirical formulae to consider the effective flexural stiffness ofcracked sections in modeling of buildings. Parameters to calculateEIef f are related to the axial force acting on column, cross-sectionarea and compressive strength of the concrete. It should be notedthat reinforcing ratio of the section is ignored.

Several studies have been conducted to determine the effectiveflexural stiffness of RC columns. Avsar et al. (2012) have studiedeffective flexural rigidities for ordinary RC columns by performinga parametric analysis of the sectional response of a wide range ofcolumn sections. Using a multilinear regression analysis they haveproposed a parameter, αef f , representing the ratio of effectiveflexural stiffness to uncracked flexural stiffness of the section. Theproposed equations for αef f yield better results for the memberswith lower concrete strength and longitudinal reinforcement ratioat any axial load level. Thus, they are recommended to be used inthe evaluation of older-type structures having inadequate reinfor-cement with low strength concrete.

Tikka and Mirza (2008) have made a statistical research oneffective flexural stiffness of slender structural concrete columnsto develop a general equation that could be used to compute EIef fof both reinforced concrete and composite columns by using amultilinear regression analysis. With the proposed formulae theyhave tried to improve efficiency of EIef f equations given in TheCanadian Standards Association (CSA) code for the design ofconcrete structures.

The aim of this paper is to introduce a simple formula as asecurer, quicker and more robust to determine the effectiveflexural stiffness of cracked sections of circular RC columns. Thisformula has been generated by using genetic programming (GP).The advantages of GP based formulation are attributed to itssimplicity and usage for different kinds of structural engineeringproblems for which sufficient experimental results exist. Theresults from GP based formulation are compared with TEC-2007and EC-8. It is demonstrated that the GP based model is highlysuccessful to determine the effective flexural stiffness of circularRC columns.

GP is one of the soft-computing approaches and a relativelynew form of artificial intelligence. Since it was first proposed byKoza (1992), GP has garnered considerable attention due to itsability to model nonlinear relationships for input–output map-pings. In recent years, the GP has been effectively applied in manyengineering applications (Gandomi et al., 2010; Ashour et al.,2003; Cevik et al., 2010; Soh and Yang, 2000; Chen et al., 2012).

Gandomi et al. (2010) have proposed a novel approach for theformulation of elastic modulus of both normal-strength concrete(NSC) and high-strength concrete (HSC) using a variant of GP,

namely linear genetic programming (LGP). They have developedthe models based on experimental results collected from theliterature and carried out a subsequent parametric analysis toevaluate the sensitivity of the elastic modulus to the compressivestrength variations. They have exposed the LGP results to be moreaccurate than those obtained using the building codes and varioussolutions reported in the literature.

Ashour et al. (2003) have investigated the feasibility of usingGP to create an empirical model for the complicated non-linearrelationship between various input parameters associated with RCdeep beams and their ultimate shear strength. They have con-structed the GP model from a set of experimental results availablein the literature. They have showed the predictions obtained fromGP agree well with experimental observations.

Cevik et al. (2010) have studied the use of GP to model RC beamtorsional strength. They used experimental data of 76 rectangularRC beams from an existing database to develop the GP model.They have compared the accuracy of the codes in predicting the RCbeam torsional strength with the proposed GP model using thesame test data. They have concluded that the proposed GP modelpredicts RC beam torsional strength more accurately thanbuilding codes.

Soh and Yang (2000) have described a GP-based approach forsimultaneous sizing, geometry, and topology optimization ofstructures. They have presented an encoding strategy to mapbetween the real structures and the GP parse trees. They haverevealed that the proposed approach is capable of producing thetopology and shape of the desired trusses and the sizing of all thestructural components. They have also discovered that thisapproach can potentially be a powerful search and optimizationtechnique in solving civil engineering problems.

Chen et al. (2012) developed a weighted GP system to constructthe relation models between the aseismic capacity of schoolbuildings, and their basic design parameters. They constructednumerical models to obtain aseismic abilities of buildings andsimulated stress responses and behaviors of the numerical modelsbased on the structural configuration and material properties ofbuildings. They suggested this system to predict the aseismiccapacity of the school buildings.

In this study, a simple formulation is proposed for ηef f , to becalled as effective flexural stiffness ratio which is the proportion ofeffective flexural stiffness (EIef f ) to gross sectional flexural stiffness(EI0) of RC columns through the following steps: (i) Moment–curvature relationship of several commonly used circular rein-forced concrete sections is obtained by using a cross sectionalanalysis program, XTRACT, (ii) the data to perform GP analysis areproduced by taking into account the most important parametersaffecting EIef f of RC columns, (iii) a GP-based analysis is conductedto estimate ηef f accurately, (iv) the GP-based estimates are com-pared with the numerical study results, EC-8 and TEC-2007 andlater the performance of the proposed formulae is verified on a3-D building by a pushover analysis, and (v) finally the results arepresented in graphical form.

2. Overview of moment–curvature relationship

The behavior of an RC member subjected to bending or com-bined bending and axial load can be understood if the moment–curvature relationship is available. By studying this relationship onecan predict the strength and the stiffness, as well as the ductilitycharacteristics of the cross-sections (Ersoy et al., 2008).

The curvature which is one of the geometrical parametersrepresenting deformation is defined as unit rotation angle. It isthe derivative of the inclination of the tangent with respect to

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–8780

arc length (Fig. 1).

Curvature : ϕ¼ dθdx

¼ d2ydx2

¼ 1ρ

ð1Þ

ϕ¼MEI

ð2Þ

where EI is the flexural stiffness.Moment–curvature relationship can be obtained either experi-

mentally or analytically. Since it is not feasible to test elementswhenever moment–curvature relationship is needed, realisticanalytical methods are of primary importance.

Moment–curvature relationship for RC sections can be gener-ated using equilibrium and compatibility equations and materialmodels. With the aid of computers it is possible to use morerealistic material models, and consider strain hardening in steeland step by step cover crushing (Ersoy and Ozcebe, 1998).

3. A cross sectional analysis program for moment–curvaturerelationship

The realistic EIef f can be best derived from moment–curvaturerelationship. Moment–curvature relationships of the sections usedin this study are obtained by using a cross sectional analysisprogram named XTRACT which is a fully interactive program forthe analysis of any cross section. It can generate moment curva-tures, axial force–moment interactions, and moment–momentinteractions for concrete, steel, prestressed and composite struc-tural cross sections. It can handle the input of any arbitrary crosssection (even with holes) made up of any material input from theavailable nonlinear material models (XTRACT and User Manual).

Moment–curvature relationship of a circular reinforced con-crete section derived from XTRACT and its bilinearized curve arefigured in Fig. 2.

4. Genetic programming (GP)

Genetic programming (GP) is a branch of genetic algorithmsand its basis is the same Darwinian concept of survival of thefittest. GP creates a population of solutions whose genotypes areprograms symbolized by tree structures. The main goal of GP is tosolve a problem by searching highly fit computer programs in thespace of all possible programs. The aim of this approach is to

obtain global optimum solutions by keeping many solutions thatmay potentially be close to minima (Ashour et al., 2003).

Koza (1992) describes GP as a domain-independent problem-solving approach in which computer programs are evolved tosolve, or approximately solve, problems based on the Darwinianprinciple of reproduction and survival of the fittest and analogs ofnaturally occurring genetic operations such as crossover (sexualrecombination) and mutation. GP reproduces computer programsto solve problems by completing the steps in Fig. 3. As can be seenfrom Fig. 3, GP augments computer programs by executing thefollowing steps:

1) Generate an initial population of random compositions of thefunctions and terminals of the problem (computer programs).

2) Execute each program in the population and assign it a fitnessvalue according to how well it solves the problem.

3) Create a new population of computer programs.i. Copy the best existing programs (reproduction).ii. Create new computer programs by mutation.iii. Create new computer programs by crossover (sexual

reproduction).iv. Select an architecture-altering operation from the programs

stored so far.4) The best computer program that appeared in any generation,

the best-so-far solution, is designated as the genetic program-ming result (Cevik et al., 2010).

More details can be found in reference Koza (1992). In thisstudy, GeneXproTools program which has been developed byCandida Ferreira is used.

5. Numerical study

In this study, the GP-based model was applied to estimate ηef f ofcircular reinforced concrete columns which have different dimen-sions and/or reinforcing configurations (Fig. 4). To train and test theGP model, training and testing sets are generated. For this purpose;370 circular reinforced concrete column sections which havedifferent geometrical properties, longitudinal and transverse rein-forcing configurations, concrete compressive strength and subjectedto different axial loads were composed and they were designed

Fig. 1. Moment–curvature relationship.

Fig. 2. Moment–curvature relationship of a circular RC section obtained fromXTRACT.

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–87 81

according to both Turkish Earthquake Code (TEC-2007) (TEC-2007,2007) and Turkish design and construction code for reinforcedconcrete structures (TS-500) (TS-500, 2002). Different axial loadcombinations were selected by following requirements of thosecodes (Eqs. 3 and 4):

Ndmr0:50Acf ck ðTEC 2007Þ ð3Þ

Ndr0;6Acf ck ðTS 500Þ ð4Þ

where Nd and Ndm are the axial force acting on the section, Ac isarea of the section and f ck is compressive strength of concrete.

The compressive strength of concrete sections used in thisstudy is selected as a variable from 10 MPa to 30 MPa. Additionallyminimum yield and rapture strength of reinforcing bars are420 MPa and 550 MPa respectively. Mander concrete model(Mander et al., 1988) was used for confined and unconfinedconcrete in compression. The stress–strain relationship of reinfor-cing steel is taken from TEC-2007 and that is widely used inliterature. It is assumed that plane sections remain plain afterbending and the bonding between reinforcing steel and concrete isperfect. The shear deformation of the section is neglected.

Moment–curvature relationship of those sections was deter-mined by using XTRACT and it is defined as a bilinear curve with4 parameters illustrated in Fig. 4. Consequently EIef f of sectionswas obtained and the data for GP analysis were created.

370 data in total are randomly divided into two parts as thetraining and testing sets. 333 data from database are selected astraining set and employed to train GP based model. Remaining 37data, which are not used in the training process, are selected as thetesting set and used to validate the generalization capability of GPbased model. The maximum and minimum values of input andoutput parameters are given in Table 1. The testing set is tabulatedin Table 2.

Fig. 3. Flow chart of standard genetic programming algorithm (Koza, 1992).

Fig. 4. The general form of a circular RC column section and its bilinear moment–curvature curve.

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–8782

where d is the diameter of the section, Nd is the axial load andf cm is 28-day compressive strength of concrete. ρt is the long-itudinal reinforcing steel ratio and ρw is the transverse (volu-metric) reinforcing steel ratio.

GP algorithm parameters are given in Table 3. In order toevaluate how well the GP optimizer performs convergence graphsfor both training and testing sets are given in Figs. 5 and 6. Thebest fitness for testing sets corresponding to 0.0014 of meansquared error, 0.8896 of R-square and 0.0380 of standard deviationvalues are obtained around 700 (Fig. 6).

GeneXproTools which is used to generate the proposed formulain this study has three different kinds of crossover which aretabulated in Table 4.

6. Results and discussions

Having performed the numerical and empirical analyses, twounique formulae have been proposed. The first main one wasobtained from GP analysis and called as proposed formula (PF), the

Table 1Range of parameters in the database.

Min Max

Input parametersD ðmÞ 0.30 3.00Nd ðkNÞ 0 88390ρt ð%Þ 0.01048 0.03840ρw ð%Þ 0.007178 0.01608f cm ðkN=m2Þ 10,000 30,000

Output parametersηef f ¼ EIef f =EI0 0.204 0.733

Table 2Testing set.

# D (m) Nd (kN) ρt (%) ρw (%) fcm (kN/m2) ηeff

XTRACT PF SPF TEC-2007 EC-8

1 0.30 880 0.01309 0.01608 25,000 0.492 0.475 0.590 0.800 0.502 0.30 700 0.02160 0.01608 25,000 0.463 0.513 0.569 0.795 0.503 0.30 700 0.02667 0.01608 25,000 0.501 0.541 0.575 0.795 0.504 0.30 250 0.03840 0.01608 25,000 0.463 0.519 0.490 0.455 0.505 0.50 1800 0.01098 0.00873 25,000 0.415 0.455 0.554 0.756 0.506 0.50 2100 0.01434 0.00873 25,000 0.482 0.500 0.579 0.800 0.507 0.50 2450 0.01814 0.00873 25,000 0.551 0.542 0.605 0.800 0.508 0.50 50 0.02710 0.00873 25,000 0.423 0.443 0.380 0.400 0.509 0.50 200 0.03226 0.00873 25,000 0.477 0.483 0.429 0.400 0.5010 0.50 50 0.01098 0.00873 12,000 0.268 0.286 0.295 0.400 0.5011 0.50 50 0.01098 0.00873 30,000 0.237 0.271 0.257 0.400 0.5012 0.50 50 0.03226 0.00873 18,000 0.498 0.472 0.406 0.400 0.5013 0.75 2000 0.01320 0.00748 25,000 0.369 0.424 0.475 0.508 0.5014 0.75 1100 0.01670 0.00748 25,000 0.373 0.421 0.430 0.400 0.5015 0.75 100 0.02062 0.00748 25,000 0.412 0.408 0.352 0.400 0.5016 0.75 2050 0.02495 0.00748 25,000 0.493 0.504 0.503 0.514 0.5017 0.75 3000 0.02972 0.00748 25,000 0.562 0.550 0.548 0.629 0.5018 0.75 2900 0.03485 0.00748 25,000 0.609 0.568 0.552 0.617 0.5019 1.00 2000 0.01941 0.00802 25,000 0.412 0.439 0.438 0.403 0.5020 1.00 9820 0.02310 0.00802 25,000 0.626 0.571 0.612 0.800 0.5021 1.00 6000 0.03144 0.00802 25,000 0.603 0.564 0.562 0.674 0.5022 1.50 5000 0.01294 0.00718 25,000 0.349 0.395 0.430 0.418 0.5023 1.50 22090 0.01540 0.00718 25,000 0.571 0.529 0.604 0.800 0.5024 1.50 15000 0.02096 0.00718 25,000 0.528 0.526 0.561 0.720 0.5025 1.50 5000 0.02738 0.00718 25,000 0.523 0.493 0.469 0.418 0.5026 1.50 22090 0.01294 0.00718 14,000 0.512 0.580 0.692 0.800 0.5027 1.50 22090 0.02738 0.00718 10,000 0.664 0.705 0.760 0.800 0.5028 1.50 22090 0.02738 0.00718 20,000 0.690 0.616 0.648 0.800 0.5029 2.00 8000 0.01356 0.00838 25,000 0.355 0.386 0.418 0.403 0.5030 2.00 39280 0.01572 0.00838 25,000 0.581 0.527 0.603 0.800 0.5031 2.00 24000 0.02053 0.00838 25,000 0.513 0.511 0.546 0.674 0.5032 2.50 13000 0.01084 0.00848 25,000 0.321 0.354 0.412 0.408 0.5033 2.50 61380 0.01258 0.00848 25,000 0.554 0.504 0.600 0.800 0.5034 2.50 39000 0.01643 0.00848 25,000 0.474 0.488 0.544 0.691 0.5035 3.00 20000 0.01048 0.00722 25,000 0.324 0.367 0.422 0.418 0.5036 3.00 88390 0.01203 0.00722 25,000 0.551 0.504 0.600 0.800 0.5037 3.00 60000 0.01733 0.00722 25,000 0.502 0.505 0.555 0.720 0.50

Table 3GP algorithm parameters.

Population size 370Maximum number of generations 1000Probability of mutation 0.044

Fig. 5. Convergence graph for training set.

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–87 83

other was produced empirically by effectively simplifying the firstone and named as simplified proposed formula (SPF).

In literature it is difficult to find a formulation to compute ηef fdirectly. Also in TEC-2007, instead of calculating ηef f directly, it isgiven different relations with other formulae and necessary tomake linear interpolations for intermediate values. Therefore thisis a pioneering study that ηef f can directly be calculated by PF andSPF formulae. Moreover if it is necessary to calculate EIef f , it can beeasily computed by multiplying ηef f with EI0 already known fromsection properties.

EC-8 and TEC-2007 do not consider some of the most impor-tant parameters affecting flexural stiffness of RC members. Whilein EC-8 it is just given a constant coefficient; ηef f ¼ 0:5, a formula-tion consisting of axial force acting on column, cross-section areaand compressive strength of concrete is given to calculate ηef f inTEC-2007. Both of codes do not take into account the effect ofreinforcement. However PF and SPF formulae proposed in thisstudy consider all of parameters that have been mentioned above.

According to EC-8; the elastic flexural and shear stiffnessproperties of concrete elements may be taken to be equal toone-half of the corresponding stiffness of the uncracked elements.

In TEC 2007, ηef f can be calculated by using following equa-tions.

ηef f ¼EIef fEIo

¼ 0:40 ifNd

Acf cmr0:01 ð5Þ

ηef f ¼EIef fEIo

¼ 0:80 ifNd

Acf cmZ0:04 ð6Þ

where EI0 is uncracked flexural stiffness of the cross section. Alinear interpolation should be made for intermediate values of Nd.

Proposed formula (PF):

ηef f ¼EIef fEI0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρt

Nd

Acf cm

� �þ2 ρt�ρw

� �svuut ð7Þ

Simplified proposed formula (SPF):

ηef f ¼EIef fEI0

¼ Nd

4Acf cmþ ρt�ρw

� �0:25 ð8Þ

Variables given in Eqs. (7) and (8) are as follows: Ac is the area ofthe section, Nd is axial load acting on the section and f cm is 28-daycompressive strength of concrete.

ρt is the longitudinal reinforcing steel ratio. It can be calculated asdivision of total area of longitudinal reinforcing bars by section area.

ρw is the transverse (volumetric) reinforcing steel ratio taken asthe (volume of transverse reinforcement)/(volume of concrete). Itis also the reinforcing ratio in the two principal orthogonaldirections added together (XTRACT and User Manual).

In this study, in order to determine the fitness of the givenspecimen for PF, statistical results of mean squared error (MSE),R-square (R2) and standard deviation (SD) are calculated. Theseresults are tabulated for the proposed and current code basedformulae in Table 5. The closeness of the standard deviation andMSE to zero and R-square to 1 shows the fitness of the formula(GeneXproTools).

It can be clearly seen in Table 5 that PF gives more accurateresults than both TEC-2007 and EC-8. The convergence differenceof SPF to PF can be ignored since as well as PF, SPF yields moreaccurate results than the design codes. R2 of EC-8 cannot becalculated because it cannot be taken other than 0.5 value. There-fore it is given as not a number (NaN).

Performance of ηef f computed via PF, SPF, EC-8 and TEC-2007have been demonstrated by numerical XTRACT analysis resultsthrough Figs. 7–10 in which the closer circles to the diagonal linemean the better result. It can clearly be seen that performance ofthe proposed formulae in this study are highly effective than EC-8and TEC-2007 in terms of convergence to exact ηef f values.

In addition, the results are compared with the results ofnumerical study performed by using XTRACT and illustrated fortesting set in Fig. 11. As can be seen from Fig. 11, PF and SPF results

Fig. 6. Convergence graph for testing set.

Table 4GP algorithm parameters for crossover (GeneXproTools).

One-point recombination 0.30Two-point recombination 0.30Gene recombination 0.10

Table 5Statistical results of proposed and current code based formulae for testing set.

PF SPF TEC-2007 EC-8

MSE 0.0014 0.0051 0.0330 0.0114R2 0.8896 0.6894 0.4916 NaNSt. dev. 0.0380 0.0623 0.1267 0.1063

Fig. 7. Performance of testing set for PF.

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–8784

agree well with the XTRACT results and it can be said that they canrepresent effectively actual behavior of RC columns.

7. Case study

In order to see performance of the proposed formula, on a real6-story RC building, a 3D modal analysis along with a staticpushover analysis have been performed by using ηef f computedby EC-8, TEC-2007, PF, SPF and analytically determined EIef f ofXTRACT. Plan view of 6-story RC building with 3 m story heights isgiven in Fig. 12. There is a concrete shear wall core area at themiddle of the building and diameters of circular columns are50 cm. Additionally the compressive strength of concrete sectionsis 20 MPa and minimum yield and rapture strength of reinforcing

bars are 420 MPa and 550 MPa respectively. ηef f of beams andshear walls were determined by cross sectional moment–curva-ture relationship and taken the same for all cases in order to seethe real performance of the proposed formulae on the circularcolumns.

SAP2000 Ultimate V.15 program is used for modal and push-over analysis. Periods and pushover curves of the building inprincipal directions X and Y are compared with each otherseparately (mode shapes of X and Y directions are determined as

Fig. 8. Performance of testing set for SPF.

Fig. 9. Performance of testing set for TEC-2007.

Fig. 10. Performance of testing set for EC-8.

Fig. 11. The ratio of EC-8, TEC-2007, PF and SPF versus numerical study results.

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–87 85

effective modes by providing modal participating mass ratiohigher than 70% of total mass of the building). Convergencepercentage performances of the proposed formulae and otherdesign codes with respect to analytically determined ηef f are givenin Tables 6 and 7 for the principal directions.

In order to acquire pushover curves of the building for each ηef fcase, a pushover analysis is performed by applying the steps givenin TEC-2007. For each case, nonlinear spectral displacements areobtained by modal displacement demand. Following, a new push-over analysis is performed until the nonlinear spectral displace-ment and finally the real pushover curves are obtained. Thepushover curves in principal directions are compared in Fig. 13and obtained nonlinear spectral displacement results and theirsconvergence percentages are tabulated in Table 8. It can be seenfrom the Fig. 13 and Table 8 that the proposed formulae representalmost actual behavior of the building.

8. Conclusion

This paper deals with the validation of the formulae which arebased on GP to calculate the effective flexural stiffness ratio, ηef f , ofcircular RC columns under axial force and bending moment. It is apioneering study in this field in terms of taking into account the mostimportant parameters affecting flexural stiffness of cracked sections.The proposed formulae consider the confinement, axial load level,section dimensions and material properties of concrete and reinfor-cing steel to determine effective flexural stiffness of circular RCcolumns. In this respect it will also close the current codes deficit.

The sections designed according to both TEC-2007 and TS-500are analyzed by XTRACT to generate moment–curvature relation-ship of the sections. In so doing the data for GP analysis arecreated. Following the analysis an explicit formulation for theeffective flexural stiffness ratio of circular RC columns is proposed.To acquire a simpler formula to be able to use in preliminarydesign for hand calculation the proposed formula is effectivelysimplified as a SPF. The results obtained from the formulations aretruly competent. The generalization capabilities of the explicitformulae are confirmed and compared by the numerical studyresults. In addition they are verified on a 3-D building model to seeefficiency of them. The proposed explicit formulations are verysimple, easy to use and can predict effective flexural stiffness ratio,ηef f , of circular RC columns in an effective manner. Moreover, oneof the advantages of PF and SPF formulae is that they do not haveof any complex operations. Thus the new proposed PF and SPFformulae can be easily used in practice and employed to anyprogramming language to determine effective flexural stiffness ofcircular RC columns.

This paper also demonstrates how robust GP is. Therefore it canbe used conveniently for an analytical formulation could not be

Fig. 12. General plan view of the 6-story RC building and section view of columns.

Table 6Modal periods.

Periods (s) XTRACT PF SPF TEC-2007 EC-8

X direction 0.441 0.410 0.411 0.364 0.330Y direction 0.497 0.466 0.467 0.407 0.368

Table 7Convergence percentage of modal periods for each ηef f case.

Direction PFXTRACT

SPFXTRACT

TEC�2007XTRACT

EC�8XTRACT

X 93% 93% 83% 75%Y 94% 94% 82% 74%

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–8786

obtained from the mathematical models and the results of expe-rimental and numerical studies.

Because all of the selected sections used in GP analysis aredesigned with respect to both TEC-2007 and TS-500 codesrequirements there is no limitation for the input variables of PFand SPF formulae. As a result the proposed PF and SPF formulaecan be conveniently used in design codes for the seismic designand performance analysis of existing and new structures.

References

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Fig. 13. Pushover curves in X and Y directions.

Table 8Nonlinear spectral displacements and convergence percentages.

Nonlinear spectraldisplacements

Convergence percentage ofnonlinear spectraldisplacements

X direction (cm) Y direction (cm) X direction (%) Y direction (%)

XTRACT 8.58 9.19 – –

PF 7.77 8.41 90.6 91.5SPF 7.82 8.46 91.0 92.0TEC-2007 6.58 7.25 76.7 78.9EC-8 5.77 6.33 67.3 68.9

N. Caglar et al. / Engineering Applications of Artificial Intelligence 38 (2015) 79–87 87