Genetic-programming-based modeling of RC beam torsional strength

KSCE Journal of Civil Engineering (2010) 14(3):371-384DOI 10.1007/s12205-010-0371-6

− 371 −

www.springer.com/12205

Structural Engineering

Genetic-programming-based Modeling of RC Beam Torsional Strength

Abdulkadir Cevik*, Musa Hakan Arslan**, and Mehmet Alpaslan Köroglu***

Received April 20, 2009/Revised August 30, 2009/Accepted October 1, 2009

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Abstract

This study investigates the use of Genetic Programming (GP) to model Reinforced Concrete (RC) beam torsional strength.Experimental data of 76 rectangular RC beams from an existing database were used to develop the GP model. The following inputparameters, which affect torsional strength, were selected: beam cross-sectional area, closed stirrup dimensions, stirrup spacing,closed stirrup cross-sectional area of one leg, stirrup and longitudinal reinforcement yield strength, stirrup steel ratio, longitudinalreinforcement steel ratio and concrete compressive strength. Moreover, a short review of well-known building codes in relation to thedesign of RC beams under pure torsion is presented. The accuracy of the codes in predicting the RC beam torsional strength was alsocompared with the proposed GP model using the same test data. The study concludes that the proposed GP model predicts RC beamtorsional strength more accurately than building codesKeywords: reinforced concrete beam, genetic programming, torsional strength, building code

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1. Introduction

Monolithic Reinforced Concrete (RC) constructions aresubject to significant torsional moments that affect their strengthand cause deformation. In the literature, numerous analytical andexperimental studies have been reported regarding the torsionalbehavior of RC members subjected to pure tension or a com-bination of tension with other effects, such as axial load, shearand bending.

There are many variables that affect RC beam torsional strength,such as beam cross-sectional area, closed stirrup dimensions,stirrup spacing, closed stirrup cross-sectional area of one leg,stirrup and longitudinal reinforcement yield strength and con-crete compressive strength. The effect of these variables on RCbeam torsional strength has been extensively studied, and severalempirical approaches have been developed related to thesevariables. For instance, Victor and Muthukrishnan (1973) studiedthe effect of variations in stirrups on RC beam torsional capacity,and they proposed an empirical relationship to describe thestirrup contribution to torsional capacity. Rasmussen and Baker(1995) examined the behavior of reinforced normal concrete andhigh strength concrete beams subject to pure torsion. The testshowed that high strength concrete increases the beam torsionalcapacity and stiffness. McMullen and Rangan (1978) presentedthe results of a torsion test on rectangular RC beams with theaspect ratio and reinforcement as the main variables. The effectof high strength concrete on RC beam torsional behavior under

pure tension was investigated by Koutchoukali and Belarbi(2001) and Fang and Shiau (2004). According to their research,the torsional capacity of under-reinforced beams is independentof concrete strength. They also found that longitudinal reinforce-ment was more effective in controlling crack width than transversereinforcement. The torsional behavior of normal strength con-crete beams has also been reported by other researchers (Collinsand Mitchell, 1980; Hsu, 1968; Hsu and Mo, 1985).

Test data are often used to validate, calibrate or even developmodels. Even though RC beam torsional strength has been care-fully examined experimentally, estimation of torsional strength isstill a difficult task because of the complex behavior of RCbeams under torsion.

This paper first investigates and presents a short review of thetheories of RC member torsional strength. Next, some well-known building code torsional strength calculation approaches areexplained. This study evaluates the accuracy of building codes inpredicting the ultimate RC beam torsional strength. Thus, experi-mental data of 76 beams subjected to torsion were used from theexisting databases provided by Rasmussen and Baker (1995),Koutchoukali and Belarbi (2001), Fang and Shiau (2004), andHsu (1968), and the mentioned building code approaches (ACI-318, 2005; Eurocode-2, 2002; TBC-500-2000, 2000; CSA, 1994;BS8110,1985; AS3600,2001) are examined by comparing theirpredictions. The second aim of this study is to investigate theapplicability of Genetic Programming (GP) to propose a newmodel for RC beam torsional strength based on experimental

*Associate Professor, Dept. of Civil Engineering, University of Gaziantep, 27310, Turkey (E-mail: [email protected])**Assistant Professor, Dept. of Civil Engineering, Selcuk University, Konya 42075, Turkey (Corresponding Author, E-mail: [email protected])

***Research Assistant, Dept. of Civil Engineering, Selcuk University, Konya 42075, Turkey (E-mail: [email protected])

Abdulkadir Cevik, Musa Hakan Arslan, and Mehmet Alpaslan Köroglu

− 372 − KSCE Journal of Civil Engineering

results collected from the literature. The results obtained fromthe proposed GP model and building codes are compared.

2. Theories of Torsional Strength and Torsion inthe Building Standards

In the literature, even though several theoretical models havebeen proposed that are based on failure mechanisms, the torsionalstrength analysis methods can be approximately divided into twogroups: space truss analogy theory and the skew-bending theory.Rausch (1929) developed the space truss model, and somedesign codes are still based on this model. In the space trussmodel, the torsion is resisted by compression diagonals, made upof the concrete between cracks, which spiral around the beam ata constant angle. The theory was later extended by manyscholars. It is assumed, in this theory that the concrete beambehaves similarly to a thin-walled box with a constant shear flowin the wall cross-section in torsion, which produces a constanttorsional moment (Elfegren et al., 1974). The absence of a coredoes not affect the strength of the members in torsion; thus, theacceptability of the space truss analogy approach is based onhollow sections. Therefore, in the RC beam torsion design pro-cess, the beam can be considered to be an equivalent tubularmember.

In 1958, the skew-bending theory, which considers the internaldeformational behavior of a series of transverse warped surfacesalong the beam in detail, was proposed by Lessig (1959). Themodel was further refined by Collins et al. (1965), as well as Hsu(1968). In particular, Hsu made a major experimental contribu-tion to the development of current skew-bending theory. Thebasic approach of the theory is that the failure of a rectangularsection in torsion occurs because of bending about an axis that isparallel to the wide face of the section and inclined at about 45o

to the longitudinal axis of the beam. In previous versions of ACIcode (from 1971 to 1989) (ACI 318, 1971; ACI 318, 1989; ACI318, 1995) beam torsional strength was calculated using thistheory. According to these codes, the beam torsional strength, Tn,was considered to consist of two parts: one part is contributed byconcrete, Tc, while the other part is contributed by web reinforce-ment, Ts. Hsu (1968) examined hollow and solid rectangularbeams and observed that the concrete core does not contribute tothe ultimate torsional strength. Later, Hsu concluded that theconcrete contribution, Tc, was mainly due to the shear resistanceof the diagonal concrete struts.

In 1995, a new ACI code (ACI 318, 1995) was proposed witha radically different design procedure based on the thin-walledtube, space truss analogy, which is considerably simpler to un-derstand and apply and is equally accurate. The torsion provi-sions in ACI 318 have been revised using the thin walled tubeanalogy.

According to the current torsion provision of ACI (ACI 318,2005), meaningful additional RC beam torsional strength, Tn, canbe achieved only by using both closed stirrups and longitudinalsteel bars, while the torsion moment resisted by the concrete

compression struts, Tc, is assumed to be zero. Thus, the concretecontribution is ignored; there is no advantage in using higherconcrete strengths to resist ultimate torsion. The torsionalstrength, Tn , is given as follows:

(1)

In Eq. (1), cotθ can be assumed to be:

(2)

In Eqs. (1) and (2), Ao is the gross area enclosed by the shearflow path that is equal to 0.85 Ash, where Ash is the area enclosedby the centre of stirrups; θ is angle of the compression diagonals;fyl is the longitudinal torsional reinforcement yield strength; fyv isthe closed stirrup yield strength; Al is the total area of longitudinaltorsional reinforcement; ph is the perimeter of the centerline ofthe outmost closed transverse torsional reinforcement; s is thestirrup spacing, and At is the cross sectional area of one leg of aclosed stirrup.

In the Australian Standard AS3600 (2001) and CanadianStandard CSA (1994), RC beam design subject to pure torsion isbased on the space truss model, and the Tn value is determinedusing the same equation as that in ACI-318 (2005). Unlike ACI318 (2005), CSA (1994) and AS3600 (2001), The British Stand-ard BS8110 (1985) for RC structures calculates the torsionalstrength using Eq. (3):

(3)

where Asv is the area of two legs of the stirrups in a section, and x1

and y1 are the center-to-center distances of the shorter and longerstirrup legs given in Fig. 1. The torsional strength, Tn, is describ-

Tn2AOAtfyv

s--------------------cotθ=

cotθ Alfylstffyvph-------------=

Tn0.8x1y1 0.87fys( )Asv

s--------------------------------------------=

Fig. 1. The Cross Section of a Rectangular Reinforced ConcreteBeam


Vol. 14, No. 3 / May 2010 − 373 −

ed as Eq. (4) in the Turkish Building Code TBC-500-2000(2000).

(4)

In Eq. (4), Ae is the area enclosed by the lines that connect thecentroids of the reinforcing bars at the corner of the section, asseen in Fig. 1.

According to the European Standard Eurocode-2 (2002), tor-sional strength is calculated three different ways, and the mini-mum result is chosen for use in this study.

(5)

(6)

(7)

where Ak is the area enclosed by the center-lines of the effectivewall thickness. The effective wall thickness, tef, can be calculatedas A/u, where A is the total area and u is the perimeter of thecross-section; fc is the compressive strength of concrete.

3. Selection of Database (Description of data)

The experimental database considered here (Table A.1) was

collected from various studies (Rasmussen and Baker, 1995;Koutchoukali and Belarbi, 2001; Fang and Shiau, 2004; Hsu,1968; Tang, 2006). The test specimens were solid rectangularbeams subject to pure tension, and none of them was a deepbeam. The concrete compressive strength ranged from 25.58MPa to 109.8 MPa; the stirrup percentage ranged from 0.40% to2.56%; the longitudinal reinforcement yielding stress rangedfrom 314 MPa to 560 MPa; and the stirrup yielding stress rangedfrom 320 MPa to 672 MPa. The experimental database consistsof a total of 76 tests (details in Table A.1). Beams are identifiedusing the notations in the first row with the first letter of theresearcher’s name. The same series of tests has been used byseveral authors. Tang (2006) developed radial basis function neural(RBFN) networks to predict the ultimate torsional strength of RCbeams; Zhang (2002) and Hossain et al. (2006) improved theanalytical methods for predicting the RC beam nonlinear responseusing the same database.

4. Genetic Programming

Genetic Programming (GP) is an extension of Genetic Algori-thms proposed by Koza (1992). Koza defines GP as a domain-independent problem-solving approach in which computer pro-grams evolve to solve, or approximately solve, problems based

Tn2AlAefyv

2 x1 y1+( )----------------------=

Tn fys Asw s⁄( )2Akcotθ=

Tn fy As uk⁄( )2Aktanθ=

Tn 1.2 1 f– c 250⁄( )fcAktefsinθcosθ=

Fig. 2. Genetic Programming Flowchart (Koza, 1992)



on the Darwinian principle of reproduction and survival of thefittest and analogs of naturally occurring genetic operations, suchas crossover (sexual recombination) and mutation. GP repro-duces computer programs to solve problems by executing thefollowing steps (Fig. 2): 1) Generate an initial population of random compositions of the

functions and terminals of the problem (computer programs).2) Execute each program in the population and assign it a fitness

value 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 pro-gramming result (Koza, 1992).

Gene Expression Programming (GEP) software, which is usedin this study, is an extension of GP that evolves computer pro-grams of different sizes and shapes encoded in linear chromo-somes of fixed length. The chromosomes are composed of multi-ple genes; each gene encodes a smaller sub-program. Further-more, the structural and functional organization of the linearchromosomes allows unconstrained operation of important geneticoperators, such as mutation, transposition, and recombination.One strength of the GEP approach is that the creation of geneticdiversity is extremely simplified as genetic operators work at thechromosome level. Another strength of GEP is its unique, multi-genic nature, which allows more complex programs composedof several sub-programs to evolve. As a result, GEP surpassesthe old GP system 100-10,000 times.

The phenotype of GEP individuals consists of the same kind ofdiagram representations used by GP. However, these complexentities are encoded in simpler, linear structures of fixed length -the chromosomes. Thus, there are two main players in GEP: thechromosomes and the ramified structures or Expression Trees(ETs); the latter expresses the genetic information encoded in theformer. The information decoding process (from the chromosomesto the ETs) is called translation, this translation implies a type ofcode and a set of rules. The genetic code is very simple: a one-to-one relationship between the symbols of the chromosome andthe functions or terminals they represent. The rules are also verysimple; they determine the spatial organization of the functionsand terminals in the ETs, and the type of interaction betweensub-ETs in multigenic systems (Ferreira, 2001, 2002; Ireland etal., 2002).

In GEP, there are two languages: the language of the genes andthe language of ETs. In this simple replicator/phenotype system,knowing the sequence or structure of one implies that the other isknown. In nature, although it is possible to infer the sequence ofproteins given the sequence of genes and vice versa, almost

nothing is known about the rules that determine the three-dimen-sional structure of proteins. However, in GEP, thanks to thesimple rules that determine ET structure and their interactions, itis possible to infer the phenotype exactly given the gene sequ-ence, and vice versa. This bilingual and unequivocal system iscalled the Karva language.

4.1 Solving a Simple Problem with GEPFor each problem, the type of linking function, as well as the

number of genes and the length of each gene, are chosen a priorifor each problem. When attempting to solve a problem, one canalways start by using a single-gene chromosome and then pro-ceed by increasing the length of the head. If it becomes verylarge, one can increase the number of genes and choose a func-tion to link the sub-ETs. One can start with addition for algebraicexpressions or for Boolean expressions, but in some cases anotherlinking function might be more appropriate (e.g., multiplicationor IF). The goal is to find a good solution, and GEP provides themeans of finding one very efficiently.

As an illustrative example, consider the following case in whichthe objective is to show how GEP can be used to model complexrealities with high accuracy. Suppose one is given a sampling ofthe numerical values from the following curve (remember,however, that in real-world problems the function is unknown):

y=3a2 +2a+1 (8)

Ten randomly chosen points in the real interval [-10, +10] areprovided, and the goal is to find a function that fits those valueswithin a certain error. In this case, sample data in the form of tenpairs (ai, yi) is given, where ai is the value of the independentvariable in the given interval and yi is the dependent variable (ai

values: -4.2605, -2.0437, -9.8317, -8.6491, 0.7328, -3.6101,2.7429, -1.8999, -4.8852, 7.3998; the corresponding yi valuescan be easily evaluated). These ten pairs are the fitness cases (theinput) that will be used as the adaptation environment. The fitnessof a particular program will depend on how well it performs inthis environment (Ferreira, 2002; Ferreira, 2001; Ireland et al.,2002).

There are five major steps in preparing to use gene expressionprogramming. The first is to choose the fitness function. For thisproblem, one can measure the fitness fi of an individual programi with the following expression:

(9)

where M is the range of selection; C(i, j) is the value returned bythe individual chromosome i for fitness case j (out of Ct fitnesscases); and Tj is the target value for fitness case j. If, for all j, theprecision, |C(i,j)−Tj|, is less than or equal to 0.01, then theprecision is equal to zero, and fi = fmax =Ct*M. For this problem,we use M =100 and, therefore, fmax =1000. The advantage of thistype of fitness function is that the system can find the optimalsolution itself. However, there are other fitness functions availablethat can be appropriate for different problem types (Ferreira,

fi M C i j,( ) Tj––( )j 1=

Ct

∑=


Vol. 14, No. 3 / May 2010 − 375 −

2002; Ferreira, 2001; Ireland et al., 2002).The second step is choosing the set of terminals T and the set

of functions F to create the chromosomes. In this problem, theterminal set consists of the independent variable, i.e., T={a}.The choice of the appropriate function set is not so obvious, but agood guess can always be made in order to include all thenecessary functions. In this case, to make things simple, we usethe four basic arithmetic operators. Thus, F={+, −, *, /}. Itshould be noted that there many other functions that can be used.The third step is to choose the chromosomal architecture, i.e., thelength of the head and the number of genes.

The fourth major step in preparing to use gene expressionprogramming is to choose the linking function. In this case, wewill link the sub-ETs by addition. Other linking functions arealso available, such as subtraction, multiplication and division.

Finally, the fifth step is to choose the set of genetic operatorsthat cause variation and their rates. In this case, we use a com-bination of all genetic operators (mutation at pm=0.051; IS andRIS transposition at rates of 0.1 and three transposes of lengths 1,2, and 3; one-point and two-point recombination at rates of 0.3;and gene transposition and gene recombination at rates of 0.1).

To solve this problem, we choose an evolutionary time of 50generations and a small population of 20 individuals in order tosimplify the analysis of the evolutionary process and not fill thistext with pages of encoded individuals. However, one of theadvantages of GEP is that it is capable of solving relatively com-plex problems using small population sizes and, thanks to thecompact Karva notation, it is possible to fully analyze the evolu-tionary history of a run. A perfect solution can be found ingeneration 3, which has the maximum value (1000) of fitness.The sub-ETs codified by each gene are given in Fig. 3. Note thatit corresponds exactly to the same test function given in Equation8 (Ferreira, 2001, 2002; Ireland et al., 2002).

Thus expressions for each corresponding Sub-ET can be givenas follows:

y =(a2 +a)+(a+1)+(2a2)=3a2 +2a+1 (10)

5. Numerical Application

In this study, the GeneXproTools 4.0 (www.gepsoft.com) soft-ware package is used for GP modeling of RC beam torsional

strength. Among the experimental databases, 15 tests were usedas a testing set and the remaining 61 tests as the training set forGP training. The proposed GP formula is an empirical equationbased on the experimental database given in section 3. In theproposed GP model, the input parameters were selected based onpreviously published studies by Fang and Shiau (2004), Hsu(1968), Koutchoukali and Belarbi (2001) and Rasmussen andBaker (1995) which are cross section (x, y), closed stirrup dimen-sions (x1, y1), concrete compressive strength (fc), stirrup spacing(s), cross-sectional area of one leg of a closed stirrup (At), closedstirrup yield strength (fyv), total area of the longitudinal torsionalreinforcement (Al), longitudinal torsional reinforcement yieldstrength (fyl), stirrup steel ratio (ρ t) and longitudinal reinforce-ment steel ratio (ρ l). The ranges of variables in the experimentaldatabase where the proposed GP model will be valid are given inTable 1. Related GP training parameters are presented in Table 2.The statistical parameters of the proposed GP model are given inTable 3, while the GP model performance is compared with thetest results in Fig. 4. The entire database with corresponding ex-perimental and GP results are given in Table A.1. The expressiontree of the GP model is presented in Fig. 5, where d0, d1, d2, d3,d4 and d5 correspond to x1, y1, fc, ρ t, ρ l and cot(θ) respectively.The constants shown in Fig. 5 are -8715, 7765 and -6405, re-spectively. After substituting the constants, the final formulation

Fig. 3. ET for the Problem of Eq. (8)

Table 1. Ranges of Variables of the Database

Minimum Maximum Increment

x (mm) 160 350 Variable

y (mm) 275 508 Variable

x1 (mm) 130 300 Variable

y1 (mm) 216 469 Variable

fc (MPa) 26 110 Variable

s (mm) 50 215 Variable

At (mm2) 71 127 Variable

fyv (MPa) 319 672 Variable

Al (mm2) 381 3438 Variable

fyl (MPa) 310 638 Variable

ρ t (%) 0.22 2.56 Variable

ρ l (%) 0.30 3.51 Variable



for RC beam torsional strength is obtained as follows:

Tu =1.61*10-7x12(y1+ fc) cot(θ ) (11)

6. Discussion

6.1 Code ApproachesThe prediction accuracy of various standards of building codes

related to beam torsional strength for the tested 76 specimens arepresented in Table 4. As seen from Table 4, ACI-318-(2005),AS3600 (2001) and CSA (1994) torsional strength expressionshave the most powerful estimation capacity. There are severalreasons for the differences between the code and test results:

In all code approaches, except Eurocode-2-03 (2002), given inEq. (7), the concrete contribution is ignored after torsional crack-ing, which makes no distinction between the behavior of normaland high strength concretes. Therefore, there is no advantage inusing higher concrete strength in resisting ultimate torsion.However, the test series have shown that the ultimate RC beamtorsional strength increases as the concrete quality increases.According to Rasmussen and Baker (1995), when concrete strengthwas the only varied parameter (cross-sectional dimensions, strengthand reinforcement dimensions were kept constant for all beams),the test series showed that the ultimate RC beam torsional strengthincreases as concrete strength increases. However, Koutchoukaliand Belarbi (1997) found that torsional strength of under-rein-forced concrete is independent of concrete strength.• When calculating torsional strength, the main parameter is the

shear flow area, which is determined differently in the buildingcodes. Taking the centers of longitudinal bars or center-to-center of stirrups for this calculation creates a considerabledifference in the total result.

y13 ρt fcρl

5

Table 2. Parameters of GP Model

P1 Function Set + , - , * , / , , ln

P2 Chromosomes 30-200

P3 Head Size: 2-6

P4 Number of Genes: 1-4

P5 Linking Function: Addition, Multiplication

P6 Fitness Function Error Type: MAE, RMSE, Custom Function

P7 Mutation Rate: 0,044

P8 Inversion Rate: 0,1

P9 One-Point Recombination Rate: 0,3

P10 Two-Point Recombination Rate: 0,3

P11 Gene Recombination Rate: 0,1

P12 Gene Transposition Rate: 0,1

Table 3. Statistical Parameters of the Proposed GP Model

Mean COV RMSE R2

GP Testing Set 1.25 0.17 12.9 0.95

GP Training Set 1.2 0.13 13.9 0.97

GP Total Set 1.21 0.14 13.7 0.96

Fig. 4. Performance of Test and GP Results

Fig. 5. Expression Tree for Torsional Strength of RC Beams


Vol. 14, No. 3 / May 2010 − 377 −

• The building codes assume that the longitudinal bars andstirrups yield. However, in the experiment that represents thereal conditions, which is more realistic than analyticalapproaches, neither the longitudinal bars nor the stirrups yielded.In particular, high values of yield stress, larger reinforcementsizes and weaker concrete prevent the longitudinal bars andstirrups from yielding.

• In the TBC-500-2000 (2000) and BS8110 (1985), the angle ofcracks are neglected (or assumed to be 45o). This assumptioninduces important differences between the code approachesand test results.

• The theoretical values computed by using code formulationsare generally higher than the experimental torsional strength.This can be explained by the fact that the thin-walled tube andspace truss analogies deviate in the isolated case of over-rein-forced beams with low concrete strength.

• The comparison suggests that most equations overestimate thestrength, especially in the case of beams with low concretestrength. This is expected because most of the methods do nottake concrete strength into account when calculating the tor-sional strength.

6.2 Genetic Programming (GP)Predicted values achieved through the proposed GP formula-

tions are compared with the experimental results of beam tor-sional strength in Table A1, and the test set and training setstatistical parameters from the GP formulations are given inTable 3. Based on the findings of the GP, the following compari-sons can be drawn:• The results of the proposed GP formulation performed better

than the building code results. The torsional moment calcula-tion is especially effective for beams and columns because ofthe monolithic structure of reinforced concrete structures. Tocalculate torsional strength and torsional reinforcement, dif-ferent equations and approaches are used in the existing codes.

In practice, the reinforcement placed in the cross-section toresist the bending and shearing bearing effect of the torsionmoment does not move in a safe manner. Therefore, theformulas for torsional reinforcement and torsional strength aresimple. As can be seen in this study, this simple approach in thecodes remains very limited when predicting torsional strength.

• To show the overall correlation trend, the theoretical line withTutest /TuGP is drawn on the graph along with the plotted datapoints. The overall predictions from the GP model are found tobe better than those of the ACI-318 (2005), AS3600 (2001) andCSA (1994) equations.

• The error between the test and GP model is quite small for thementioned parameters. However, when comparing the codeand test data, especially for over reinforced concrete, the codeprediction capability diminishes. The building codes assumethat the longitudinal bars and stirrups yield.

• According to the final torsional strength formulation, the ulti-mate RC beam torsional strength, Tu, under pure torsion can befairly accurately estimated using only five input variables: x1,y1, fc, Al and ρ t (i.e., the short dimension of the closed stirrup,the long dimension of the closed stirrup, concrete compres-sive strength, longitudinal torsional reinforcement and stirrupsteel ratio). This is consistent with most existing analyticalmethods.

• The GP outcomes offer original contributions in addition to itshigh estimation capacity.

• In the recent literature, Tang (2006) used a radial basis functionneural network (RBFN) to predict the ultimate torsionalstrength of RC beams, which selected the five most importantparameters as x1, y1, fc, Al and ρ t (i.e., the short dimension of theclosed stirrup, the long dimension of the closed stirrup,concrete compressive strength, longitudinal torsional reinforce-ment and stirrup steel ratio) as inputs to the neural network. Inhis study, a RFBN using all available variables as the inputvariables was developed. From the study, it can be seen that the

Table 4. Prediction Accuracy of Existing Building Codes

Building Standards Expression for torsional strength COV RMSE R2

(%)

ACI-318-2005 0.29 23.5 85.93

BS8110 0.29 22.7 81.76

TBC-500-2000 0.4 38.1 71.07

AS3600 0.29 23.5 85.93

Eurocode-2-01 0.35 49.1 73.44

Eurocode-2-02 0.30 23.5 85.93

Eurocode-2-03 0.56 57.9 61.88

CSA 0.29 23.5 85.93

Tn2AOAtfyv

s--------------------cotθ=

Tn0.8x1y1 0.87fys( )Asv

s--------------------------------------------=

Tn2AlAefyv

2 x1 y1+( )----------------------=

Tn2AOAtfyv

s--------------------cotθ=

Tn fys Asw s⁄( )2Akcotθ=

Tn fy As uk⁄( )2Aktanθ=

Tn 1.2 1 f– ck 250⁄( )fckAktefsinθcosθ=

Tn2AOAtfyv

s--------------------cotθ=



developed RFBN models performance well in terms of the R2

value. The R2 values are all greater than 0.98.

7. Main Effects of Variables on Torsional Strength

The Main Effect plot is an important graphical tool to visualizethe independent impact of each variable on torsional strength.This graphical tool enables a better and simple picture of theoverall importance of variable effects on the torsional strength.The slope of the line for each variable reflects the degree of itseffect on the output. To obtain the main effect plot, a wide rangeof parametric studies have been performed using the proposedGP model. From the main effect plot in Fig. 6, it can be con-cluded that all variables used for GP modeling given in the ex-perimental database have significant effects on torsional strength.Variables that are observed to be directly proportional from Fig.6 are x1, y1, fc , ρ t, ρ l and cot(θ). The evaluation of separateinteraction effects between any two variables is also performedand shown in Figs. 7-21. These figures demonstrate the followingresults of this study:• All figures show that the torsional strength is significantly in-

fluenced with increasing x1, y1, fc, ρ t, ρ l and cot(θ ). Inparticular, the angle of cracks (θ) affects the torsional strengthof reinforced concrete beams. Among the standards that arealso considered in this study, some building codes do not haveany particular procedure to evaluate the crack angle. In thecalculation of the torsional strength, θ is accepted as 45o. In theliterature, Compression Field Theory (CFT) proposed byCollins and Mitchell (1980) is sometimes used to determine theultimate RC beam torsional capacity. CFT assumes that theangle of cracks and strain are related. The ultimate torquescalculated using the CFT analytical method demonstrate much

better agreement with the experiments. Ameli and Rnagh(2007) tested CFT theory for various tested specimens and

Fig. 6. Main Effect Plot for Tu

Fig. 7. Interaction Effects for Different ρ t Values

Fig. 8. Interaction Effects for Different ρ t Values


Vol. 14, No. 3 / May 2010 − 379 −

found that none of the codes successfully predicted the ultimatetorque accurately; moreover, for some tested beams, the codeapproaches were too conservative, and for some approaches,they were too risky.

• Concrete compressive strength cannot be ignored, which iscommon in building codes. Figs. 9-11 show that the ultimatetorsional strength increases as the concrete quality increases. Inparticular, high longitudinal steel ratios and high stirrup steel

ratios (ρ l and ρ t) affected the concrete contributions.• As shown in Fig. 7 and 8, as the stirrup steel ratio (ρ t) incre-

ases, the torsional strength (Tu) increases.• It can be clearly understood from Fig. 12 through 20 that

increasing either the short dimension (x1) or the long dimension(y1) of the closed stirrup causes the torsional strength (Tu) toincrease. The cross section of the beam plays a major role inevaluating the torsional strength.

Fig. 9. Interaction Effects for Different fc Values



Fig. 12. Interaction Effects for Different y1 Values





According to Figs. 17, 18 and 19, the stirrup steel ratio (ρ t),longitudinal steel ratio (ρ l) and concrete compressive strength(fc) are more effective in big cross sections. Tang (2006) reportedthat the relationship between torsional strength and concretecompressive strength is almost linear. In the other words, theultimate RC beam torsional strength increases as the concretestrength increases.

8. Conclusions

This study is a pioneer work that addresses the feasibility ofGP as an alternative approach for the empirical formulation ofRC beam torsional strength. The use of GP provides an alterna-tive way to estimate RC beam torsional strength. The proposedGP model is based on a large experimental database collectedfrom the literature. The results of the proposed GP model are


Fig. 16. Interaction Effects for Different x1 Values






Vol. 14, No. 3 / May 2010 − 381 −

seen to be far more accurate than those of current design codesand equations available in literature. Most of the available designcodes and equations are based on regression analysis of pre-defined functions. However, in the GP approach presented in thisstudy, there is no predefined function. The GP approach gener-ates various formulations and optimizes the one that best fits theexperimental database. The outcomes of this study are satisfac-tory and suggest that GP approaches should be used in furtherRC structures applications.

Notations

A : Total areaAe : Area enclosed by lines connecting the centroids of

the reinforcing bars at the corner of the sectionAk : Area enclosed by the centre-lines of the effective

wall thicknessAl : Total area of longitudinal torsional reinforcementAo : Gross area enclosed by the shear flow pathAsh : Area enclosed by the centre of stirrupsAsv : Area of the two legs of stirrups at a section (=2At)At : Cross sectional area of one-leg of closed stirrupai : Outputs of neural networkfc : Compressive strength of concretefyl : Vyield strength of longitudinal torsional reinforce-

mentfyv : Yield strength of closed stirrupsk : Number of samples in training or test data

m : Number of segments in training or test datan : Number of outputs of neural network for training

and test proceduresph : Perimeter of centerline of outmost closed trans-

verse torsional reinforcementR2 : Correlation coefficient

s : Spacing of stirrupssx : Normalized value of variableTc : Torsion moment resisted by the concrete compres-

sion strutsTn : Nominal torsional strength

Tu(estimated) : Predicted ultimate torsional strengthTu(experimental) : Measured ultimate torsional strength

tef : The effective wall thicknessti : Desired outputsu : Perimeter of the cross-sectionx : Short dimension of the cross sectiony : Long dimension of the cross section

x1 : Center-to-center of the shorter and longer legs ofstirrups

y1 : Center-to-center of the longer legs of stirrupsz : Variable values

zmin : Variable minimum values zmax : Variable maximum valuesθ : Angle of compression diagonalsρl : Steel ratio of longitudinal reinforcementρt : Steel ratio of stirrups

Acknowledgements

This research was supported by Gaziantep University Re-search project Unit and Selcuk University BAP Office (SU-BAP).

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Fig. 21. Interaction Effects for Different ρ l Values



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www.gepsoft.com

AppendixTable A1. Experimental Database (Rasmussen and Baker, 1995; Koutchoukali and Belarbi, 2001; Fang and Shiau, 2004; Hsu, 1968;

Tang, 2006)

Ref. no x y x1 y1 fc s At fy Al fyl rt rl Tu (Test)

Tu(GP)

Tu(GP)/Tu(Test)

Fang and Shiau, 2004

FS-1 350 500 300 450 78.5 100 71.33 440 1196.6 440 0.61 0.68 92.00 103.27 1.12F2-2 350 500 300 450 78.5 100 71.33 440 2027.2 410 0.61 1.16 115.10 122.91 1.07FS-3 350 500 300 450 78.5 50 71.33 440 2027.2 410 1.22 1.16 155.30 150.47 0.97FS-4 350 500 300 450 78.5 50 71.33 440 2865 520 1.22 1.64 196.00 170.34 0.87FS-5 350 500 300 450 78.5 55 126.7 440 3438 560 1.97 1.96 239.00 209.03 0.87FS-6 350 500 300 450 68.4 90 71.33 420 1719 500 0.68 0.98 126.70 115.90 0.91FS-7 350 500 300 450 68.4 80 126.7 360 1719 500 1.36 0.98 135.20 142.46 1.05FS-8 350 500 300 450 68.4 90 71.33 440 2865 500 0.68 1.64 144.50 137.22 0.95FS-9 350 500 300 450 35.5 100 71.33 440 1191.6 440 0.61 0.68 79.70 79.27 0.99FS-10 350 500 300 450 35.5 100 71.33 440 2027.2 410 0.61 1.16 95.20 94.37 0.99FS-11 350 500 300 450 35.5 50 71.33 440 2027.2 410 1.22 1.16 116.80 115.53 0.99FS-12 350 500 300 450 35.5 50 71.33 440 2865 520 1.22 1.64 138.00 130.79 0.95FS-13 350 500 300 450 35.5 55 126.7 440 3438 560 1.97 1.96 158.00 160.49 1.02FS-14 350 500 300 450 35.5 90 71.33 420 1719 500 0.68 0.98 111.70 93.16 0.83FS-15 350 500 300 450 35.5 80 126.7 360 1719 500 1.36 0.98 125.00 114.51 0.92FS-16 350 500 300 450 35.5 90 71.33 420 2865 500 0.68 1.64 117.30 110.56 0.94

Koutch-oukali and

Belarbi, 2001

KB-1 203 305 165 267 39.6 108 71.33 373 506.8 386 0.92 0.82 19.40 18.11 0.93KB-2 203 305 165 267 64.6 108 71.33 399 506.8 386 0.92 0.82 18.90 21.24 1.12KB-3 203 305 165 267 75 108 71.33 373 506.8 386 0.92 0.82 21.10 22.40 1.06


Vol. 14, No. 3 / May 2010 − 383 −

Table A1. (Continued)


Tu(GP)

Tu(GP)/Tu(Test)

Koutch-oukali and

Belarbi, 2001

KB-4 203 305 165 267 80.6 108 71.33 399 506.8 386 0.92 0.82 19.40 22.87 1.18

KB-5 203 305 165 267 93.9 108 71.33 386 506.8 386 0.92 0.82 21.00 24.10 1.15

KB-6 203 305 165 267 76.2 102 71.33 386 506.8 386 0.98 0.82 18.40 22.89 1.24

KB-7 203 305 165 267 72.9 95 71.33 386 649.46 373 1.05 1.05 22.50 24.94 1.11

KB-8 203 305 165 267 75.9 90 71.33 386 760.2 373 1.11 1.23 23.70 27.07 1.14

KB-9 203 305 165 267 76.7 70 71.33 386 794.4 380 1.42 1.28 24.00 29.58 1.23

Rasmussen and Baker,

1995

RB-1 160 275 130 245 41.7 90 78.54 665 1543.9 620 1.49 3.51 16.60 19.50 1.17

RB-2 160 275 130 245 38.2 90 78.54 669 1543.9 638 1.49 3.51 15.30 18.95 1.24

RB-3 160 275 130 245 36.3 90 78.54 672 1543.9 605 1.49 3.51 15.30 18.59 1.22

RB-4 160 275 130 245 61.8 90 78.54 665 1543.9 612 1.49 3.51 20.00 22.21 1.11

RB-5 160 275 130 245 57.1 90 78.54 665 1543.9 614 1.49 3.51 18.50 21.64 1.17

RB-6 160 275 130 245 61.7 90 78.54 665 1543.9 612 1.49 3.51 19.10 22.20 1.16

RB-7 160 275 130 245 77.3 90 78.54 658 1543.9 617 1.49 3.51 20.10 23.95 1.19

RB-8 160 275 130 245 76.9 90 78.54 656 1543.9 614 1.49 3.51 20.70 23.91 1.15

RB-9 160 275 130 245 76.2 90 78.54 663 1543.9 617 1.49 3.51 21.00 23.83 1.13

RB-10 160 275 130 245 109.8 90 78.54 655 1526.8 618 1.49 3.51 24.70 26.92 1.09

RB-11 160 275 130 245 105 90 78.54 660 1526.8 634 1.49 3.51 23.60 26.54 1.12

RB-12 160 275 130 245 105.1 90 78.54 655 1543.9 629 1.49 3.51 24.80 26.56 1.07

Hsu, 1968

HS-1 254 381 215.9 342.9 27.58 152.4 71.33 341.29 508 313.71 0.54 0.52 22.30 25.66 1.15

HS-2 254 381 215.9 342.9 28.61 181.1 126.7 319.92 635 316.47 0.81 0.66 29.30 31.61 1.08

HS-3 254 381 215.9 342.9 28.06 127 126.7 319.92 762 327.5 1.15 0.79 37.50 36.94 0.98

HS-4 254 381 215.9 342.9 30.54 92.2 126.7 323.56 889 319.92 1.59 0.92 47.30 43.92 0.93

HS-5 254 381 215.9 342.9 29.03 69.9 126.7 321.3 1016 332.33 2.09 1.05 56.20 49.04 0.87

HS-6 254 381 215.9 342.9 28.82 57.2 126.7 322.67 1143 331.64 2.56 1.18 61.70 54.08 0.88

HS-7 254 381 215.9 342.9 25.99 127 126.7 318.54 508 319.92 1.15 0.52 26.90 31.32 1.16

HS-8 254 381 215.9 342.9 26.75 57.2 126.7 319.92 508 321.99 2.56 0.52 32.50 40.14 1.24

HS-9 254 381 215.9 342.9 28.82 152.4 126.7 342.67 762 319.23 0.96 0.79 29.80 35.23 1.18

HS-10 254 381 215.9 342.9 26.48 152.4 126.7 341.98 1143 334.4 0.96 1.18 34.40 39.23 1.14

HS-11 254 381 215.9 342.9 26.61 152.4 71.33 337.84 508 333.02 0.54 0.52 22.40 25.46 1.14

HS-12 254 381 215.9 342.9 25.58 181.1 126.7 330.95 635 322.67 0.81 0.66 27.70 30.43 1.10

HS-13 254 381 215.9 342.9 28.41 127 126.7 333.02 762 341.67 1.15 0.79 40.20 37.09 0.92

HS-14 254 381 215.9 342.9 30.61 92.2 126.7 333.02 889 330.26 1.59 0.92 47.90 43.96 0.92

HS-15 254 381 215.9 342.9 29.85 149.4 71.33 353.01 635 326.12 0.55 0.66 30.40 28.65 0.94

HS-16 254 381 215.9 342.9 30.54 104.9 71.33 357.15 762 328.88 0.79 0.79 40.60 33.95 0.84

HS-17 254 381 215.9 342.9 26.75 139.7 126.7 326.12 889 321.99 1.05 0.92 43.80 37.20 0.85

HS-18 254 381 215.9 342.9 26.54 104.9 126.7 326.81 1016 318.54 1.39 1.05 49.60 42.06 0.85

HS-19 254 381 215.9 342.9 27.99 82.6 126.7 330.95 1143 335.09 1.77 1.18 55.70 47.89 0.86

HS-20 254 381 215.9 342.9 29.37 69.9 126.7 340.6 2288 317.85 2.09 2.36 60.10 64.27 1.07

HS-21 254 381 215.9 342.9 45.23 98.6 71.33 348.87 635 325.43 0.84 0.66 36.00 37.10 1.03

HS-22 254 381 215.9 342.9 44.75 127 126.7 333.71 762 343.36 1.15 0.79 45.60 43.16 0.95

HS-23 254 381 215.9 342.9 44.95 92.2 126.7 326.12 889 315.09 1.59 0.92 58.10 49.91 0.86




Tu(GP)

Tu(GP)/Tu(Test)

Hsu, 1968

HS-24 254 381 215.9 342.9 45.02 69.9 126.7 325.43 1016 310.26 2.09 1.05 70.70 56.61 0.80

HS-25 254 381 215.9 342.9 45.78 57.2 126.7 328.88 1143 325.43 2.56 1.18 76.70 63.02 0.82

HS-26 254 508 215.9 469.9 29.79 187.5 71.33 339.22 508 321.99 0.4 0.39 26.80 29.50 1.10

HS-27 254 508 215.9 469.9 30.89 120.7 71.33 333.71 635 322.67 0.63 0.49 40.30 36.69 0.91

HS-28 254 508 215.9 469.9 26.82 155.7 126.7 327.5 762 338.53 0.87 0.59 49.60 41.02 0.83

HS-29 254 508 215.9 469.9 28.27 114.3 126.7 341.98 889 325.43 1.18 0.69 64.90 47.99 0.74

HS-30 254 508 215.9 469.9 26.89 85.9 126.7 327.5 1016 330.95 1.57 0.79 72.00 53.91 0.75

HS-31 254 508 215.9 469.9 29.92 127 126.7 349.56 1144 334.4 1.06 0.89 39.10 51.55 1.32

HS-32 254 508 215.9 469.9 30.96 146.1 126.7 322.67 1430 319.23 0.92 1.11 52.70 53.87 1.02

HS-33 254 508 215.9 469.9 28.34 104.9 126.7 328.88 1716 321.99 1.28 1.33 63.30 61.28 0.97

HS-34 254 508 215.9 215.9 27.03 215.9 71.33 341.29 381 341.29 0.22 0.3 11.30 11.31 1.00

HS-35 254 508 215.9 215.9 26.54 117.6 71.33 344.74 508 334.4 0.41 0.39 15.30 14.20 0.93

HS-36 254 508 215.9 215.9 26.89 139.7 126.7 329.57 635 330.95 0.61 0.49 20.00 17.05 0.85

HS-37 254 508 215.9 215.9 27.17 98.6 126.7 327.5 762 336.46 0.86 0.59 25.30 19.94 0.79

HS-38 254 508 215.9 215.9 27.23 73.2 126.7 328.88 889 328.19 1.16 0.69 29.70 22.77 0.77

HS-39 254 508 215.9 215.9 27.58 54.1 126.7 327.5 1016 315.78 1.57 0.79 34.20 26.00 0.76

*Bold sets are used testing set.

Table A1. (Continued)

Genetic-programming-based modeling of RC beam torsional strength

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