Predicting the effect of stirrups on shear strengthof reinforced normal-strength concrete (NSC) andhigh-strength concrete (HSC) slender beams usingartificial intelligence

H. El Chabib, M. Nehdi, and A. Saïd

Abstract: The exact effect that each of the basic shear design parameters exerts on the shear capacity of reinforcedconcrete (RC) beams without shear reinforcement (Vc) is still unclear. Previous research on this subject often yieldedcontradictory results, especially for reinforced high-strength concrete (HSC) beams. Furthermore, by simply adding Vc

and the contribution of stirrups Vs to calculate the ultimate shear capacity Vu, current shear design practice assumesthat the addition of stirrups does not alter the effect of shear design parameters on Vc. This paper investigates the va-lidity of such a practice. Data on 656 reinforced concrete beams were used to train an artificial neural network modelto predict the shear capacity of reinforced concrete beams and evaluate the performance of several existing shearstrength calculation procedures. A parametric study revealed that the effect of shear reinforcement on the shear strengthof RC beams decreases at a higher reinforcement ratio. It was also observed that the concrete contribution to shear re-sistance, Vc, in RC beams with shear reinforcement is noticeably larger than that in beams without shear reinforcement,and therefore most current shear design procedures provide conservative predictions for the shear strength of RC beamswith shear reinforcement.

Key words: analysis, artificial intelligence, beam depth, compressive strength, modeling, shear span, shear strength.

Résumé : L’effet précis que chaque paramètre de conception de base du cisaillement exerce sur la capacité de résis-tance au cisaillement de poutres en béton armé sans renforcement en cisaillement (Vc) n’est pas encore bien connu. Lesrecherches antérieures sur ce sujet présentaient souvent des résultats de rendement contradictoires, particulièrement pourles poutres en béton à haute résistance. De plus, en ajoutant simplement Vc et Vs pour calculer Vu, la pratique couranteen conception contre le cisaillement présume que l’addition d’étriers ne modifie pas l’effet des paramètres de concep-tion contre le cisaillement sur Vc. Cet article examine la validité d’une telle pratique. Les données de 656 poutres enbéton armé ont été utilisées pour former un modèle de réseau neuronal artificiel à prédire la capacité de résistance aucisaillement de poutres en béton armé et à évaluer le rendement de plusieurs procédures existantes de calcul de la ré-sistance au cisaillement. Une étude paramétrique a indiqué que l’effet du renforcement en cisaillement sur la résistanceau cisaillement des poutres en béton armé diminue à un plus haut rapport de renforcement. Il a également été remarquéque la contribution du béton à la résistance au cisaillement, Vc, dans les poutres en béton armé avec renforcement encisaillement est plus importante que dans les poutres sans renforcement en cisaillement et, qu’ainsi, les plus récentesprocédures de conception contre le cisaillement fournissent des prévisions conservatrices de résistance au cisaillementdes poutres en béton armé avec renforcement en cisaillement.

Mots clés : analyse, intelligence artificielle, épaisseur des poutres, résistance en compression, modélisation, portée encisaillement, résistance au cisaillement.

[Traduit par la Rédaction] Chabib et al. 944

1. Introduction

Significant progress has been made in studying the shearstrength of reinforced concrete (RC) slender beams (shear

span to beam depth ratio of a/d ≥ 2.5); several shear predic-tion techniques have appeared in the literature since theoriginal method proposed by Ritter (1899) at the end of thenineteenth century and the subsequent work of Mörsch

Can. J. Civ. Eng. 33: 933–944 (2006) doi:10.1139/L06-033 © 2006 NRC Canada

933

Received 31 August 2005. Revision accepted 19 February 2006. Published on the NRC Research Press Web site at http://cjce.nrc.caon 22 September 2006. Reposted on the Web site with addition on 23 October 2006.

H. El Chabib, M. Nehdi,1 and A. Saïd.2 Department of Civil and Environmental Engineering, The University of Western Ontario,London, ON N6A 5B9, Canada.

Written discussion of this article is welcomed and will be received by the Editor until 30 November 2006.

1Corresponding author (e-mail: mnehdi@eng.uwo.ca).2Present address: Howard R. Hughes College of Engineering, Civil and Environmental Engineering, University of Nevada, LasVegas, NV 89154-4015, USA.

(1909) during the first decade of the twentieth century. De-spite the substantial amount of experimental work and abun-dant effort to analytically model the shear failure mechanismof RC slender beams (Bazant and Kim 1984; Vecchio andCollins 1986; Pang and Hsu 1996), there is still no simple,analytically derived formula that can accurately predict theshear strength of such beams (Zararis 2003). Thus, most cur-rent shear design procedures and standards adopt semi-empirical and statistically derived techniques.

Current shear design procedures and standards estimatethe ultimate shear capacity of RC slender beams containingshear reinforcement (stirrups), Vu, by simply adding the con-tribution of concrete to shear resistance, Vc, to that of theshear reinforcement, Vs:

[1] V V Vu c s= +

Vs is simply determined based on the parallel truss modelwith 45° constant inclination diagonal shear cracks using thefollowing equation:

[2] V A f d s f b ds v yv v yv w= =/ ρ

where Av is the cross-sectional area of the stirrups, fyv is theyield strength of the stirrups, d is the effective depth of thebeam, s is the spacing between stirrups, ρv is the ratio ofshear reinforcement, and bw is the width of the beam.

Vc is the shear resistance of a similar RC beam but withoutshear reinforcement and is usually calculated using a semi-empirical equation. Experimental studies (Mphonde 1989;Russo and Puleri 1997) reported, however, that the addition ofstirrups significantly enhances the resistance of concrete toshear and usually leads to a higher Vc than that determinedfrom the equilibrium consideration of the 45° truss model.Moreover, Chana (1987) suggested that the shear failuremechanism of RC beams with shear reinforcement is funda-mentally different from that of beams without shear reinforce-ment, and therefore Vc and Vs mutually influence each other.Simply adding these parameters to determine the nominalshear resistance of RC beams with stirrups can induce signifi-cant error and has not yet been proven valid.

Furthermore, high-strength concrete (HSC) with compres-sive strength higher than 70 MPa has been employed in vari-ous challenging applications, and its use is growing rapidly.The mechanical properties of HSC are different from thoseof normal-strength concrete (NSC). Thus, empirical designequations, which are mostly based on experimental data de-veloped using NSC, need to be cautiously evaluated forHSC. For instance, the shear fracture surface in HSC mem-bers usually propagates across coarse aggregates rather thanaround them and is therefore smoother than that in NSCmembers. This behaviour tends to reduce shear resistancethrough aggregate interlock (a major shear transfer mecha-nism in RC beams not containing shear reinforcement),which may have serious implications in the shear design ofsuch beams.

Several shear prediction techniques have appeared in theliterature since the beginning of the last century, includingsemi-empirical, statistical, and analytical methods. Semi-empirical and statistical methods were often derived basedon observations from experimental studies on RC beamswith concrete compressive strength typically less than40 MPa. Conversely, analytical methods adopt more funda-

mental approaches, yet they often require extensive calcula-tions and need major simplifications to become suitable fordesign code practice. Despite the numerous studies investi-gating the behaviour of RC slender beams under shear, themechanism of shear failure is not yet fully understood. Theeffect of basic shear design parameters such as the longitudi-nal reinforcement and compressive strength of concrete onthe shear resistance of RC slender beams without stirrups isyet to be fully embedded in the shear design equations, andthe influence of stirrups on other shear-resisting mechanismsis simply not accounted for. Therefore, the ability of currentshear design methods to accurately predict the shear resis-tance of RC slender beams, especially those made of HSC,is still debatable (Russo and Puleri 1997).

Comparative studies (Sarsam and Al-Musawi 1992; Kim andPark 1994; Russo and Puleri 1997) on various shear strengthprediction techniques were conducted to evaluate their abilityto predict the shear strength of reinforced NSC and HSC slen-der beams with or without shear reinforcement. It was con-cluded that current shear design techniques are either veryconservative or simply not applicable for RC beams made ofHSC. This paper investigates the feasibility of using artificialintelligence to predict the shear strength of RC slender beamsand compares such predictions with results obtained from fivedifferent shear strength calculation methods, namely, the pro-visions of the American Concrete Institute (ACI Committee318 2004), the Australian Standards (AS 3600 2001), and theCanadian Standards Association (CSA CommitteeA23.3 1994) and the methods of Zsutty (1971) and Bazantand Kim (1984). A parametric study was also carried out toevaluate the effect of stirrups on other shear-resisting mecha-nisms of slender beams having shear reinforcement and theability of shear prediction techniques to quantitatively accountfor the effects of basic shear design parameters on the shearstrength of reinforced NSC and HSC slender beams with orwithout shear reinforcement.

2. Shear strength prediction methods

Most current RC design codes use semi-empirical or sta-tistically derived equations to calculate the shear capacity,Vc, of RC beams without shear reinforcement and simplyadd the contribution of stirrups, Vs, which is calculatedbased on the parallel truss model with 45° constant inclina-tion diagonal shear cracks, to obtain the shear capacity ofRC beams with stirrups. These codes vary considerably inevaluating the effect of basic shear design parameters on Vc.Moreover, it has been argued that stresses in stirrups areconsistently lower than those predicted by the 45° trussmodel (Duthinh and Carino 1996) and that other shear-resistance mechanisms interact with that of stirrups in differ-ent ways, leading to a variable contribution of the trussmechanism and to some enhancement of the beam actioncontribution (Russo and Puleri 1997), which leads to under-estimating the shear strength of RC slender beams. The vari-ous shear prediction methods used in this study are brieflydiscussed in the following sections.

2.1. ACI, CSA, and AS standardsThe shear design provision of the ACI building code

(ACI Committee 318 2004) and the simplified method of

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the CSA (CSA Committee A23.3 1994) assume that theshear capacity of RC slender beams with shear reinforce-ment is simply equal to the superposition of the contribu-tion of concrete (Vc) and that of stirrups (Vs). Vs is

calculated using eq. [2], and Vc is assumed to be equal tothe shear capacity of a similar beam that has no shear rein-forcement and is calculated using the following expres-sions:

[3] V fVdM

b d f b dc c l w c w0.158 17.2 0.3 ACI e= ′ +

≤ ′ρ quation 11-5

[4] V f b d dc c w0.2 for 300 mm CSA simplied method= ′ ≤λ

[5] Vd

f b d f b d dc c w c w0.1 for mm CSA s=+

′ ′ >260

1000300λ λ� implied method

where d is the effective depth of the beam (mm); bw is thewidth of the beam (mm); fc′ is the compressive strength ofthe concrete cylinder (MPa); ρl is the longitudinal reinforce-ment ratio; λ is a factor to account for the density of con-crete; Vc is the shear capacity at failure (N); and M and V arethe moment and shear force, respectively, at a critical sec-tion.

The ACI method (equation 11-5 in ACI Committee318 2004) is recommended only for concrete with compres-sive strength fc′ < 70 MPa and limits the shear contributionof stirrups, Vs, to 0.66(fc′)1/2bwd, and the CSA simplifiedmethod (CSA Committee A23.3 1994) requires a minimumshear reinforcement of Av = 0.06(fc′)1/2bws/fyv.

Shear design equations provided by the Australian Stan-dards (AS 3600 2001) are based on a variable-angle trussmodel and calculate the shear capacity contributed by con-crete Vuc and that contributed by the shear reinforcement Vususing eqs. [6a]–[6d] and [7], respectively:

[6a] V b d A f b dc 1 v o st c v o= ′β β β2 31 3( / ) / AS

[6b] β1 1000= −1.1 1.6 o( / )d ≥ 1.1

[6c] β2 = 1.0 for beams not subjected to axial forces

[6d] β3 = 2do/a 1.0 ≤ β3 ≤ 2.0

[7] V A f d s f b dus sv sy,f o v t sy,f v o v AS= =( cot )/ cotθ ρ θ

where bv is the effective width of the beam; do is the dis-tance from the extreme compression fibre to the centroid ofthe outmost layer of longitudinal reinforcement; a is theshear span; Ast is the cross-sectional area of the longitudinaltensile reinforcement; θv is the angle of inclination of theconcrete compression strut with the horizontal axis of thebeam; and Asv, fsy,f, ρt, and s are the cross-sectional area,yield stress, ratio, and spacing of the vertical shear rein-forcement, respectively.

The Australian Standards also provide limits for the ulti-mate shear capacity of RC slender beams, Vu, and the mini-mum shear reinforcement required according to eqs. [8] and[9], respectively:

[8] Vu = Vuc + Vus ≤ 0.2fc′bvd0 AS

[9] A f b fsv,min c vs sy,f0.06 AS= ′ /

2.2. Empirical and other methodsEmpirically derived shear equations developed by Zsutty

(1971) and equations developed by Bazant and Kim (1984)based on fracture mechanics were also used in this study tocalculate the shear strength of reinforced NSC and HSCslender beams. They are expressed (in N) as follows:

[10] Vu = Vc + Vs = 2.2(fc′ρld/a)1/3bwd + Avfyvd/s

Zsutty

[11a] V f f b du l3

c v yv w0.83= + ′[ . / ]ξχ ρ ρ χ167

Bazant and Kim

where ξ and χ are functions to account for the size effect andeffectiveness of stirrups, respectively, and are calculated asfollows:

[11b] ξ = +1 1 25/ /d da

[11c] χ ρ= ′ +f d ac l250 5( / )

and da is the maximum size of coarse aggregate (in mm).

3. Artificial neural network approach

Artificial neural networks (ANNs) are highly adaptive,data-driven trainable systems capable of capturing hiddenand complex behaviour through learning from training ex-amples. The multilayer perceptron networks (MLP) are themost widely used ANNs in engineering applications becauseof their ability to implement nonlinear transformations forfunctional approximation problems and map a given input(s)into a desired output(s). MLP networks consist of an inputlayer, an output layer, and one or more hidden layers. Eachlayer contains a number of processing elements (units) par-tially or fully connected; the strength of each connection isrepresented by a numerical value called a weight. The mainobjective in building an ANN-based model is to train a spe-cific network architecture using a comprehensive database tosearch for an optimum set of weights for the connections be-tween its processing units for which the trained ANN canpredict accurate values of outputs for a given set of inputsfrom within the range of the training data. Explaining thefundamental basis of ANNs is beyond the scope of this pa-per, and details on how to build a neural network model canbe found elsewhere (Rumelhart et al. 1986; Haykin 1994).

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4. Experimental database

Although several parameters contribute to the success oftraining an MLP network, the learning material provided forthe training remains the most important factor that affectsthe performance and generalization of the network. Morecomprehensive learning material is likely to provide betternetwork generalization and assist in capturing the embeddedrelationships between inputs and corresponding outputs.Therefore, it is essential to train a neural network model ona comprehensive and sufficiently large shear database tocapture the embedded relationships between the most influ-ential parameters of RC slender beams and their correspond-ing shear capacity. In this study, shear strength results for656 reinforced NSC and HSC slender beams with or withoutshear reinforcement were collected from the literature. Onlysimply supported beams with a shear span to beam depth ra-tio of a/d ≥ 2.4 that exhibited shear failure were considered.About 60% of the database represent shear results obtainedon NSC beams (36% of which are with shear reinforcementand 64% are without shear reinforcement). The remaining40% of the database represent results obtained on HSCbeams (53% of which are shear reinforced and 47% arewithout shear reinforcement). The database was compiled ina patterned format. Each pattern consists of an input vectorcontaining the geometrical and mechanical properties of RC

beams and an output vector containing the correspondingshear capacity of the beam. Table 1 provides the range andaverage values of all parameters used in the database.

5. ANN model

Several ANN architectures were tested in this study to de-velop a feed-forward back-propagation MLP network thatcan accurately predict the shear strength of RC slenderbeams. The architecture of the network that was adoptedconsists of an input layer containing six variables represent-ing the commonly known shear design parameters (d, bw,a/d, ρv yvf , ρl, and fc′), an output layer with one unit repre-senting the shear capacity (Vu), and a hidden layer of 10 pro-cessing units (Fig. 1). All beams considered herein failed inshear, and the longitudinal reinforcement did not reach thecorresponding yield stress. Thus, such a parameter was notconsidered in the input vector to reduce the number of con-nections in the network and enhance its performance. TheANN was trained using 542 data patterns and tested on theremaining 114 patterns. The testing patterns were randomlyselected from the original database and were not used in thetraining process. Among the testing patterns, 58% were NSCbeams (32% without shear reinforcement and 26% withshear reinforcement) and 42% were HSC beams (19% with-

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Training data (542 beams) Testing data (114 beams)

Parameter Min. Max. Avg. Min. Max. Avg.

d (mm) 40.60 1097.28 332.56 95.00 930.00 366.85bw (mm) 38.10 457.00 196.71 38.10 457.00 215.61

a/d 2.40 6.05 3.26 2.40 5.00 3.12ρl (%) 0.48 5.80 2.35 0.50 5.04 2.43

ρv yvf (MPa) 0.00 2.24 0.36 0.00 2.53 0.46

fc′ (MPa) 10.50 125.30 46.26 13.70 125.00 49.05

Vu (kN) 2.69 749.30 134.34 8.38 787.55 162.32

Table 1. Statistical data for shear design parameters and shear capacity of beams used in the database.

Fig. 1. Architecture of the neural network model.

out shear reinforcement and 23% with shear reinforcement).Variable learning rate and momentum were used to avoidlengthy training and ensure global convergence of the net-work (Haykin 1994).

6. Results and discussion

The performance of the ANN model thus developed andtrained was evaluated using the testing database describedearlier. The network was presented with the input vectors ofthe testing patterns and asked to predict the correspondingshear capacities. The predicted shear capacities were subse-quently compared with the experimentally measured values,and the performance of the network was evaluated based onthe average absolute error (AAE) and a performance factor(PF) calculated using eqs. [12] and [13], respectively:

[12] AAEm p

m

=−

∑1100

n

V V

V

[13] PF = Vm/Vp

where Vm and Vp are the measured and predicted shear ca-pacity, respectively.

The shear capacities of all beams in the testing database,which was used to evaluate the performance of the ANNmodel, were also calculated using the five shear strength cal-culation procedures discussed earlier. The average, standarddeviation (SD), and coefficient of variation (COV) for thePF and AAE of all shear design methods investigated arelisted in Table 2 for both NSC and HSC beams with shearreinforcement. It is shown that the ACI method (equation11-5 in ACI Committee 318 2004) and the CSA simplifiedmethod (CSA Committee A23.3 1994) are fairly conserva-tive in predicting the shear capacity of RC slender beams, asindicated by an AAE varying between 20% and 27% and anaverage PF reaching as high as 1.36 in the case of HSCbeams. The equations of Zsutty (1971) provided reasonablyaccurate predictions and tended to underestimate the mea-sured shear capacity by 10% in the case of NSC beams. Itsaccuracy is clearly lower for HSC beams, however, with anAAE of 17% and average PF of 1.13. Table 2 also showsthat the ANN model and the equations developed by Bazantand Kim (1984) and those of AS 3600 (2001) provided themost accurate shear capacity predictions in the case of NSCbeams, with an AAE of 12% and average PF of 1.00. Theaccuracy of the equations of Bazant and Kim, however, and

those of AS 3600 in predicting the shear capacity of HSCbeams is lower than that for NSC beams, and that of theANN was similar for both NSC and HSC beams.

Figure 2 illustrates the predicted shear capacities of rein-forced NSC and HSC beams with shear reinforcement ver-sus the corresponding experimentally measured shearcapacities. It can be observed that the data points for theANN, AS 3600 (2001), and Bazant and Kim (1984) methodsare the closest to the equity line, whereas those for the othermethods are scattered over a relatively wider range, espe-cially for beams having shear capacities larger than 200 kN(beams with large effective depth, large amount of longitudi-nal reinforcement, and (or) large amount of transverse rein-forcement). Figure 2 also shows that in most cases the ACImethod (equation 11-5 in ACI Committee 318 2004) and thesimplified CSA method (CSA Committee A23.3 1994) un-derestimate the shear capacity of RC beams with shear rein-forcement, and that their accuracy in estimating the shearcapacity of HSC beams is slightly lower than that for NSCbeams. Despite its relatively accurate predictions in the caseof NSC beams without shear reinforcement (Mphonde1989), the method of Zsutty (1971) provided conservativepredictions in the case of RC beams containing shear rein-forcement. The conservative predictions provided by theCSA simplified method, ACI method, and Zsutty equationsare partially due to the fact that all three methods assumethat no interactions occur between stirrups and other sheartransfer mechanisms and simply calculate the shear capacityof RC beams containing stirrups by superimposing the con-tribution of stirrups on that of concrete. This can be furtherillustrated by comparing the performance of the Zsutty equa-tions with that of AS 3600, in which Vc is calculated usingequations similar to those of Zsutty but differ in calculatingVs. AS 3600 considers a variable-angle truss model, whereasZsutty assumes a 45° constant-angle truss model.

6.1. Effect of stirrups on shear strengthMost current shear design codes generally assume that

adding shear reinforcement to a RC slender beam will onlyenhance its shear strength by the shear capacity of stirrupscalculated using eq. [2]. Such a practice ignores the influ-ence of stirrups on the contribution of other shear design pa-rameters and assumes that the presence of stirrups does notinterfere with other shear-resisting mechanisms, which pre-sumes a linear relationship between the amount of shear re-inforcement and the nominal shear strength. A sensitivity

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NSC (30 beams) HSC (26 beams)

Vmeasured/Vpredicted Vmeasured/Vpredicted

Prediction methoda AAE (%) Avg. SD COV AAE (%) Avg. SD COV

ACI 23.11 1.29 0.30 23.56 26.70 1.36 0.36 26.77CSA simplified 19.63 1.24 0.28 22.82 22.12 1.28 0.33 25.66Zsutty 14.74 1.10 0.23 21.06 17.44 1.13 0.27 23.90AS 3600 12.70 1.03 0.21 20.26 15.92 1.10 0.27 24.05Bazant–Kim 12.07 1.00 0.18 17.69 16.11 0.96 0.20 20.59ANN 12.33 1.00 0.19 19.09 12.50 1.00 0.18 18.45

aACI, ACI Committee 318 (2004); ANN, artificial neural network; AS 3600, AS 3600 (2001); Bazant–Kim, Bazant and Kim(1984); CSA simplified, CSA Committee A23.3 (1994); Zsutty, Zsutty (1971).

Table 2. Performance of shear design methods (beams with shear reinforcement).

analysis was conducted in this study to investigate the effectof stirrups on the shear strength of RC slender beams usingthe various shear calculation procedures discussed earlier. Aset of NSC beams was generated from a single beam ran-domly selected from the database, in which all beams share

the same geometrical and mechanical properties, except forthe amount of shear reinforcement.

The shear strength of the generated beams (calculated us-ing the various shear design methods) along with the experi-mentally measured shear strength of beams tested by Placas

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Fig. 2. Measured versus predicted shear capacity for HSC and NSC beams with stirrups (only testing points are shown; training pointsare not shown). ACI, ACI Committee 318 (2004); ANN, artificial neural network; AS 3600, AS 3600 (2001); Bazant–Kim, Bazant andKim (1984); CSA simplified, CSA Committee A23.3 (1994); Zsutty, Zsutty (1971).

and Regan (1971) are plotted versus the capacity of stirrupsin Fig. 3. It can be observed that the ANN response was theclosest to the experimentally measured data points, followedby those of the AS 3600 (2001) and Bazant and Kim (1984)equations. The ANN model and AS 3600 were the onlymethods to capture a nonlinear relationship for the effect ofstirrups on shear strength and show that such an effect wasgreater at a low shear reinforcement ratio than that at a rela-tively higher ratio. This behaviour was also reported byMphonde (1989) based on experimental studies conductedby Bresler and Scordelis (cited in Mphonde 1989) andHaddadine et al. (cited in Mphonde 1989), in which theyfound that stirrups were more efficient at a low to moderate

shear reinforcement ratio than at a high reinforcement ratio.Figure 3 also shows that shear strengths predicted by theANN are 65%–75% higher than those calculated by the ACImethod (equation 11-5 in ACI Committee 318 2004). Thesevalues are similar to findings of Bresler and Scordelis andHaddadine et al. (cited in Mphonde 1989), who reported thatthe shear strengths of RC beams having moderate shear rein-forcement are 75%–80% higher than corresponding valuescalculated using the provisions of the ACI method.

Similar behaviour was observed in the case of HSC beams(Fig. 4). Again, the ANN response and shear strength valuescalculated using the equations of AS 3600 (2001) were theclosest to the experimental shear strength of beams having

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Fig. 3. Effect of shear reinforcement ρv yvf on shear strength of reinforced NSC beams.

Fig. 4. Effect of shear reinforcement ρv yvf on shear strength of reinforced HSC beams.

similar geometrical and mechanical properties tested byKong and Rangan (1998). Figure 4 also shows that theBazant and Kim (1984) equation slightly overestimates theshear strength of HSC beams, in agreement with the data inTable 2. This appears to be due to an overestimation of theeffect of the concrete compressive strength on the shear ca-pacity at high strength values.

6.2. Influence of stirrups on effect of concretecompressive strength

It is generally accepted that an increase in the compres-sive strength of concrete increases the shear strength of RCslender beams. It was found, however, that this is particu-larly true for NSC beams and that the shear strength of RCbeams without shear reinforcement slightly decreases withan increase in fc′ beyond 70 MPa due to loss in the shear-resisting mechanism of aggregate interlock (Thorenfeldt andDrangsholt 1990; Duthinh and Carino 1996). Furthermore,most current shear design techniques either do not acknowl-edge such a variation in the effect of concrete compressivestrength on the shear capacity of beams or simply do not ac-count for the influence of adding shear reinforcement toother shear transfer mechanisms.

Prior to investigating the effect of stirrups on the contribu-tion of compressive strength to the shear strength of RCslender beams, the success of the ANN model in capturingthe relationship between the compressive strength of con-crete and the shear strength of RC beams should be evalu-ated. The shear strengths of a set of RC beams generatedfrom a single beam randomly selected from the databasewere calculated using all shear calculation methods adoptedin this study, including the ANN method. All beams sharethe same geometrical and mechanical properties, except forfc′, which varies from 25 to 90 MPa.

The shear strengths of the generated beams along with theexperimentally measured shear strengths of three beams withsimilar properties and tested by Mphonde (1989) are plottedversus the compressive strength of concrete in Fig. 5. The

figure shows that, although all methods considered hereincapture the trend of the effect of compressive strength on theshear strength of RC beams relatively well (regardless oftheir accuracy), the predictions provided by the ANN modelare the closest to the experimentally measured shear strengthvalues, followed by those of the Bazant and Kim (1984)equations and AS 3600 (2001). Moreover, results from anexperimental investigation conducted by Johnson andRamirez (1989) indicate that for a constant low amount ofshear reinforcement, the overall reserve shear strength afterdiagonal cracking diminishes with an increase in concretecompressive strength. Only the ANN model response cap-tured such a behaviour, whereas all other methods failed toshow a decrease in the rate of growth of shear strength withhigher fc′. (Note that equation 11-5 in the ACI method (ACICommittee 318 2004) does not capture this effect, but sim-ply imposes a limit on fc′ < 70 MPa.)

To investigate the influence of stirrups on the effect ofconcrete compressive strength on shear capacity, the shearstrengths of three sets of beams predicted using the Bazantand Kim (1984) equation and the ANN model (the twomethods that best captured the relationship between fc′ andthe shear strength of RC beams) were plotted versus fc′ inFig. 6. Beams in each set share the same properties exceptfc′, and beams in different sets with similar fc′ share the sameproperties except the amount of shear reinforcement, ρv yvf .Each set also contains experimental shear strengths for threebeams having similar properties tested by Mphonde (1989).

Experimental results in Fig. 6 show that increasing ρv yvfby 100% from 0.344 MPa (set 1) to 0.692 MPa (set 2) leadsto an average increase in shear strength of about 60%, andincreasing ρv yvf by 50% from 0.692 MPa (set 2) to1.034 MPa (set 3) leads to an average increase in shearstrength of about 12%. This is in agreement with previousobservations from the ANN response that the effect of stir-rups is less significant at high values of ρv yvf . Figure 6 alsoshows that the contribution of fc′ to shear strength of NSCslender beams is not significantly affected by the amount of

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Fig. 5. Effect of concrete compressive strength fc′ on shear strength of RC beams with stirrups.

shear reinforcement, ρv yvf . Such a contribution in the case ofHSC slender beams decreases at higher values of ρv yvf ,however.

6.3. Influence of stirrups on effect of longitudinalreinforcement ratio

A analysis similar to that used in investigating the influ-ence of stirrups on how compressive strength of concretecontributes to the shear capacity of beams was carried out todetermine the effect of the longitudinal reinforcement ratioon the shear strength of RC beams, and the influence of stir-rups on such an effect. Figure 7 shows the relationship be-tween the longitudinal reinforcement ratio and the predictedshear strength of a set of HSC beams using all shear design

methods considered in this study. The figure also includesexperimental shear strength values for three similar beamstested by Sarsam and Al-Musawi (1992). It can be observedthat the ANN model provided the most accurate shearstrength predictions among all shear design methods. Fur-thermore, Fig. 7 shows that only the equations proposed byZsutty (1971), Bazant and Kim (1984), and AS 3600 (2001)captured the trend of the effect of longitudinal steel ratio onthe shear strength of HSC slender beams with shear rein-forcement, but their quantitative predictions were less accu-rate than those of the ANN model.

Although AS 3600 (2001) and Zsutty (1971) adopted asimilar effect for the longitudinal tensile reinforcement onthe shear strength of RC slender beams, shear values calcu-

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Fig. 6. Influence of stirrups on the effect of fc′ on shear strength.

Fig. 7. Effect of longitudinal steel ratio ρl on shear strength of RC beams with stirrups.

lated using the equations of Zsutty were considerably lowerthan both experimental values and predictions of the ANNmodel, Bazant and Kim (1984) equations, and AS 3600.This is again due to the fact that equations proposed byZsutty do not account for the influence of stirrups on othershear-resisting mechanisms and, like most current designcodes, simply superimposes the capacity of stirrups on theshear capacity of a similar beam not containing shear rein-forcement in calculating the shear capacity of RC beamshaving stirrups. On the other hand, Fig. 7 shows that not ad-equately considering the effect of the longitudinal steel ratioon shear-resisting mechanisms (e.g., ACI method (equation11-5 in ACI Committee 318 (2004) and the CSA simplifiedmethod (CSA Committee A23.3 1994)) leads to even moreconservative shear strength estimates for RC slender beamswith shear reinforcement.

The ANN model and the equation of Bazant and Kim(1984) show a similar effect of the longitudinal reinforce-ment ratio, ρl, on the shear strength of typically reinforcedslender beams (ρl < 2.5). For highly reinforced beams, how-ever, such an effect is noticeably larger for the ANN re-sponse than for the equation of Bazant and Kim. The ANNresponse is in agreement with results obtained from an ex-perimental investigation conducted by Kong and Rangan(1998) in which they stated that increasing the longitudinalreinforcement ratio from 1.66% to 2.79% leads to a small in-crease in shear strength, yet a sharp increase in the shearstrength was reported when increasing the longitudinal steelratio from 2.79% to 3.69%.

The influence of stirrups on the effect of ρl on the shearstrength of HSC slender beams is shown in Fig. 8. The fig-ure illustrates the shear strength of three sets of beams ver-sus ρl for different values of ρv yvf . All beams share the sameproperties except for ρl and ρv yvf . Beams in the first set arewithout shear reinforcement (ρv yvf = 0), and those in thesecond and third sets have ρv yvf = 0.75 and 1.14 MPa, re-spectively. Figure 8 also shows the measured shear strengthof four similar beams tested by Sarsam and Al-Musawi

(1992). All tested beams contain shear reinforcement; threehave ρv yvf = 0.74 MPa, and the other three have ρv yvf =1.14 MPa. Although the ANN model and the equation ofBazant and Kim (1984) offered a nearly identical responsein the case of RC slender beams without shear reinforce-ment, their response was different for beams with shear rein-forcement. The equation of Bazant and Kim assumes thatthe effect of ρl on the shear strength is independent of ρv yvf ,whereas the ANN response shows a significant enhancementof shear strength with higher ρl for ρl > 2.5%. Such an en-hancement in the effect of ρl due to the addition of stirrups,as predicted by the ANN model, is in agreement with exper-imental results reported by Sarsam and Al-Musawi (1992)and Kong and Rangan (1998) as mentioned earlier. The im-provement in the effect of longitudinal steel reinforcementon the shear capacity of RC beams with stirrups is believedto be due to the fact that stirrups help confine the longitudi-nal steel bars in place, thus preventing shear cracks fromwidening, and therefore allowing an increase in dowel ac-tion.

6.4. Influence of stirrups on effect of shear span tobeam depth ratio

The effect of the shear span to beam depth ratio (a/d) onthe shear strength of HSC slender beams and the influenceof stirrups on such an effect are illustrated in Figs. 9 and 10,respectively. Figure 9 shows the variation in shear strengthof an RC beam with variable shear span to beam depth ratioas calculated by the various shear calculation methods con-sidered in this study. While some of these methods disregardthe effect of a/d on the shear strength of RC beams (the sim-plified method of CSA), others incorporate a slight variationin shear strength for 2.5 < a/d < 3.5 that becomes negligiblefor a/d > 3.5. The largest variation in shear strength versusa/d was captured by the ANN model and equations providedby Zsutty (1971) and AS 3600 (2001) (Fig. 9), yet such avariation is relatively minor, especially for a/d > 3.5. On theother hand, Fig. 10 shows the influence of adding stirrups on

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942 Can. J. Civ. Eng. Vol. 33, 2006

Fig. 8. Influence of stirrups on effect of longitudinal steel ratio ρl on shear strength.

the effect of a/d on the shear strength of HSC slenderbeams. The figure presents variation in shear strength witha/d for different values of ρv yvf (0.0, 0.5, and 1.29) as pre-dicted by the ANN model, the Bazant and Kim (1984) equa-tion, and the Zsutty equation. Responses of all threemethods show a similar relationship between the shearstrength of RC beams and a/d, regardless of the amount ofshear reinforcement. Therefore, increasing the number ofstirrups has a limited influence on an already insignificanteffect of a/d on the shear strength of RC slender beams.

7. Conclusions

This study investigated using artificial neural networks(ANNs) for predicting the ultimate shear capacity of rein-

forced NSC and HSC slender beams with or without shearreinforcement and comparing the predictions with those ofseveral existing shear prediction methods. Furthermore, aparametric study was carried out to evaluate the effects ofbasic shear design parameters on the shear strength of RCslender beams and the influence of stirrups on such effects.The following conclusions are drawn:(1) A successfully trained ANN model can be used as an ef-

fective tool for predicting the shear capacity of rein-forced NSC and HSC slender beams. The ANNapproach outperformed all other shear capacity predic-tion methods considered in this study. The ANN modelproved to be fast, reliable, accurate, and easy to use. Itshould be noted that the AS 3600 standard, whichadopts a variable-angle truss crack model, outperformed

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Fig. 9. Effect of shear span to beam depth ratio a/d on shear strength of RC beams with stirrups.

Fig. 10. Influence of stirrups on effect of shear span to beam depth ratio a/d on shear strength.

traditional design equations (ACI method and CSA sim-plified method) that adopt a 45° constant-inclination di-agonal shear crack model.

(2) The ANN approach adequately captured the effect ofshear reinforcement on the shear capacity of RC beams.As expected, it showed that shear strength increaseswith an increase in the amount of shear reinforcement.Unlike existing shear prediction methods, however, theANN model predicted that such an increase diminishesat higher values of shear reinforcement, which is inagreement with experimental results.

(3) Experimental results and results obtained using theANN model and the equation of Bazant and Kim (1984)showed that the shear strength of RC slender beamswith shear reinforcement is more than 50% higher thanvalues calculated using the ACI method and the CSAsimplified method. It was shown that calculating theshear strength of RC beams containing shear reinforce-ment by simply superimposing the capacity of stirrupson the shear capacity of a similar beam without shearreinforcement is not accurate and leads to conservativeresults.

(4) The ANN approach adequately captured the influenceof compressive strength (fc′) on the shear capacity of RCslender beams with shear reinforcement. It showed thatshear strength increases with an increase in fc′ up to70 MPa, but such an increase tends to diminish abovethat value. It also showed that increasing the capacity ofstirrups does not impact the effect of fc′ on the shearstrength of NSC beams. For HSC beams, however, theeffect of fc′ on shear strength tends to decrease with anincrease in the capacity of the stirrups.

(5) The ANN model also showed that the amount of longi-tudinal tensile reinforcement influences the ultimateshear strength of HSC beams with shear reinforcementin general, and that this influence is more pronouncedfor higher values of the longitudinal reinforcement ratio.This observation is also supported to a lesser extent bythe equations of Bazant and Kim (1984) and Zsutty(1971). Conversely, equation 11-5 in the ACI method(ACI Committee 318 2004) underestimates such an ef-fect, and the CSA simplified method (CSA CommitteeA23.3 1994) does not account for it.

(6) Lastly, the ANN analysis showed that the ratio of shearspan to beam depth, a/d, slightly affects the shearstrength of HSC slender beams with shear reinforce-ment, and that such an effect diminishes at higher valuesof a/d. This behaviour is somewhat consistent for allmethods considered in this study that account for the ef-fect of a/d on shear strength, regardless of the amountof shear reinforcement.

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