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A neural network model for predicting maximum

shear capacity of concrete beams without

transverse reinforcement

Ayman Ahmed Seleemah

Abstract: Different relationships have been proposed by codes and researchers for predicting the shear capacity of

members without transverse reinforcement. In this paper, the applicability of the artificial neural network (ANN) tech-

nique as an analytical alternative to existing methods for predicting this shear capacity is investigated using a critically

reviewed and agreed upon database of experimental work that serves as a basis of comparison and (or) assessment of

existing and new relationships. Both ANN and eight different codes and researchers predictions of the shear capacity

of the specimens of the database were compared. The ANN predictions are much superior to those of any of the cur-

rent available relationships.

Key words: artificial neural networks, shear capacity, transverse reinforcement, beams.

Rsum : Les codes et les chercheurs ont propos diverses relations pour prdire la rsistance au cisaillement de mem-brures sans armature transversale. Le prsent article tudie lapplicabilit de la technique des rseaux neuronaux artifi-

ciels comme alternative analytique aux mthodes existantes de prdiction de la rsistance au cisaillement. Nous avons

utilis pour cela une base de donnes du travail exprimental accepte et examine scrupuleusement pour servir de base

de comparaison et-ou dvaluation des relations existantes et nouvelles. Nous avons compar les prvisions mises par

les rseaux neuronaux artificiels, huit diffrents codes et des chercheurs concernant la rsistance au cisaillement

dchantillons de la base de donnes. Nous avons trouv que les prvisions des rseaux neuronaux artificiels sont gran-

dement suprieures celles de toute autre relation actuellement disponible.

Mots cls : rseaux neuronaux artificiels, rsistance au cisaillement, armature transversale, poutres.

[Traduit par la Rdaction] Seleemah 657

Introduction

Civil and structural engineers attempt to improve the anal-ysis, design, and control of the behavior of structural sys-tems. The behavior of structural systems, however, iscomplex and often governed by both known and unknownmultiple variables, with their interrelationship generally un-known, nonlinear, and sometimes very complicated. The tra-ditional approach used in most research in modeling startswith an assumed form of an empirical or analytical equationand is followed by a regression analysis using experimentaldata to determine unknown coefficients such that the equa-tion will fit the data.

In the last two decades, researchers explored the potentialof artificial neural networks (ANNs) as an analytical alterna-tive to conventional techniques, which are often limited bystrict assumptions of normality, linearity, homogeneity, vari-

able independence, etc. Researchers found ANNs particu-

larly useful for function approximation and mapping prob-lems, which are tolerant of some imprecision and have aconsiderable amount of experimental data available. In astrict mathematical sense, ANNs do not provide closed-formsolutions for modeling problems but offer a complex and ac-curate solution based on a representative set of historical ex-amples of the relationship. Advantages of ANNs include theability to learn and generalize from examples, producemeaningful solutions to problems even when input data con-tain errors or are incomplete, adapt solutions over time tocompensate for changing circumstances, process informationrapidly, and transfer readily between computing systems(Flood and Kartam 1994).

This paper focuses on the prediction of the ultimate shear

strength of reinforced concrete beams. This topic is still anactive area of research because of the complexity of theshear transfer mechanism and the large number of affectingparameters. While many efforts have been conducted to un-derstand the shear behavior of reinforced concrete beamsand (or) to derive equations for estimating such shear capac-ity, some researchers explored the application of ANNs forsuch predictions. For example, Sanad and Saka (2001)applied ANNs to predict the ultimate shear capacity of rein-forced concrete deep beams; Mansour et al. (2004) success-fully used ANNs to predict the shear capacity of reinforcedconcrete beams with transverse reinforcement; and Oreta

Can. J. Civ. Eng. 32: 644657 (2005) doi: 10.1139/L05-003 2005 NRC Canada

644

Received 22 April 2004. Revision accepted 4 January 2005.Published on the NRC Research Press Web site athttp://cjce.nrc.ca on 3 August 2005.

A.A. Seleemah. Department of Civil Engineering, BenhaHigh Institute of Technology, Benha, Egypt (e-mail:[email protected]).

Written discussion of this article is welcomed and will bereceived by the Editor until 31 December 2005.


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(2004) applied ANNs to a set of 155 experimental tests tosimulate the size effect on the shear strength of reinforcedconcrete beams without transverse reinforcement. In the cur-rent paper, one of the largest and most confident databases isutilized to investigate the applicability of the ANN tech-nique to predict the concrete contribution to shear resistanceof beams. Moreover, a comparison of both ANN and differ-

ent codes and researchers predictions for this importantstructural criteria is conducted.

Shear capacity of beams without transverse

reinforcement

Different relationships have been proposed by researchersall over the world for predicting the shear capacity of beamswithout transverse reinforcement. The importance of theserelationships is that they represent the concrete contributionto the overall shear resistance of a reinforced concrete beamwith transverse reinforcement, which is a common designcase.

Unfortunately, these proposed relationships are usuallyempirical and designed to fit a limited set of shear test re-sults that is most familiar to the researcher. Moreover, therelationships differ quite considerably in their selected pa-rameters because there is no generally accepted model forthe load transfer and the ultimate capacity of members with-out transverse reinforcement. This fact was discussed in thetwo state-of-the-art reports by the Joint ASCEACI Com-mittee 445 on Shear and Torsion (1998) and the CEB(1997). Furthermore, the test data used for comparing pro-posals with tests differ very much from one researcher to an-other in the amount and quality of selected data.

Therefore, it is essential to collect a unique and criticallyreviewed database of experimental work that would serve asa basis of comparison and (or) assessment of assumed rela-tionships. Moreover, it is important also to explore othermethods for predicting the shear capacity that would give re-sults superior to those suggested relationships. In this paper,the capability of ANNs as an analytical alternative to con-ventional techniques for predicting the ultimate shear capac-ity of concrete beams is investigated.

Shear database

A databank established by Reineck (1999) and checked bya group of experts was used to verify the empirical equationfor members without transverse reinforcement proposed forthe German standard DIN 1045-1 (DIN 2001). Kuchma

(2000) conducted a wide-range collection of test data onmembers both with and without transverse reinforcement.Recently, both authors have merged their databanks into acollection shear databank (CSDB) that contains mainlybeams with a rectangular cross section (933 beams, 231 ofwhich have a shear span to beam depth ratio of less than2.40).

A subcommittee was then established within the JointASCEACI Committee 445 on Shear and Torsion to come toa consensus on criteria that must be satisfied if a specific testresult is to be used for assessing the capability of empiricaldesign rules or expressions derived from models for struc-

tural behavior. A sanctioned set of criteria was developedfor accepting a test result into an evaluation-level database.Examples of these criteria include a minimum compressivestrength, a minimum overall height and width, and a check

against flexural failure. A databank of members that satisfythese agreed-upon criteria was established and called theevaluation shear databank (ESDB) (Reineck et al. 2003).This databank contained 439 shear tests collected from 64references. All beams in this database have a rectangularcross section, do not contain shear reinforcement, and weresubjected to point loads. Extensive discussions and reviewon the ESDB were then conducted by the American Con-crete Institute (ACI) Subcommittee 445-F on Beam Shear,which led to some changes in the database, and a total of 41tests were excluded for several reasons. The revised version(398 tests) was intended to serve as a basis for any codechanges.

This database is utilized in this study to evaluate and dem-

onstrate the capability of the ANN technique for predictingthe shear capacity of members without transverse reinforce-ment. Moreover, it was also used to evaluate eight differentcodes and formulas that exist for predicting such shear ca-pacity, and several comparisons of the predictions of thesemethods and those of the ANNs are conducted.

It should be pointed out that the aforementioned databasecovers a very wide range of beam depth, beam breadth,shear span to beam depth ratio, maximum aggregate sizeused in concrete and its tensile and compressive strengths,main reinforcement percentage, and yield stress. Table 1summarizes the ranges of the parameters in the database.

Existing models for shear capacity

prediction

Eight different major codes and methods of estimation ofthe shear capacity of reinforced concrete beams withoutshear reinforcement were collected and are as follows: ACICommittee 318 (1999), BSI (1990), ECS (1992), NZS(1995), Zsutty (1971), Collins and Kuchma (1999), Niwa etal. (1986), and Reineck (1991). The expressions utilized byeach of these methods are summarized in Table 2. Thesemethods were applied to the whole database (398 speci-mens) to achieve the prediction of each method for the shearcapacity of each specimen.

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bw (mm) 76.21000.0

d (mm) 1102000

a/d 2.418.03

d/bw 0.257.20

M/Vd 1.417.03

f1c

(MPa) 12.6110.9

f1t (MPa) 1.275.29

fy (MPa) 275.91779.3

l (%) 0.1386.635Max. aggregate size (mm) 6.3538.00

Note: M, maximum bending moment; f1t, uniaxial con-crete tensile strength; l, percentage of main steel. All otherterms as defined in the text.

Table 1. Ranges of parameters in the database.


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It should be mentioned that the original values of the ba-sic design shear strength Rd of the Eurocode EC2 (ECS1992) is given by the code in a tabular form in terms of thecharacteristic cylinder strength fck (in MPa). Appropriateconversions were conducted to relate this factor to the uniax-ial concrete compressive strength f1c. Moreover, for the Brit-ish Standard BS5400-4 (BSI 1990), the characteristicconcrete cube strength, fcu, was also converted in terms off1c. Generally speaking, appropriate conversions for the unit

system and (or) the compressive strength were conductedwhenever such conversions were essential. The equationsmost frequently used in these conversions are as follows:

[1] f fck c 1.60=

[2] f f1c c0.95=

[3] f f1c c,cube0.75=

where fc is the cylinder strength, and fc,cube is the cubestrength of concrete.

Neural network based modeling

A neural network is a nonlinear dynamic system consist-ing of a large number of highly interconnected processingunits, called artificial neurons. Its architecture and opera-tions are inspired by the biological structure of neurons andthe internal operation of the human brain. The main compu-tational characteristics of neural networks are their ability tolearn functional relationships from examples and to discoverpatterns and regularities in data through self-organization. Inthis study, the multilayer feed-forward neural networks areutilized because they are very suitable for modeling the non-linear mapping type of problems. Figure 1 shows a typical

architecture of multilayer feed-forward neural networks withan input layer, an output layer, and one hidden layer.

As shown in Fig. 1, the artificial neurons are arranged inlayers, and all the neurons in each layer have connections toall the neurons in the next layer. Associated with each con-nection between these artificial neurons, a weight value (wi)is defined to represent the connection weight. The connec-

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Reference Equation Remarks

ACI Committee 318 1999 V F b d c c w= ( / )6

BSI 1990V

b dA b d fc

w

ms w cu

0.27=

( / ) ( )/ /100 1 3 1 3fcu 40.0 MPa

ECS 1992 Vrd1 = [Rdk(1.2 + 40l)]bwd k = (1.6 d/1000) 1.0;l = As/bwd 0.02NZS 1995 V f b d c l c w0.07= + ( )10 a/d 2

Zsutty 1971 V f d a b d u c l w2.2= ( / ) / 1 3 a/d 2.5

Collins and Kuchma 1999V

Sf b dc

Ec w= +245

1275where S S aE x= +35 16/( ) Sx 0.9d

Niwa et al. 1986V b d

f A b d

dc w

c s w0.2 0.751.4=

+

( ) ( / )

( / )

/ /

/

1 3 1 3

1 4

100

1000 a d/

Reineck 1991 Vu = (0.4bwdfct + Vdu)/(1 + 0.54); fct = 0.246(fc)2/3;

Vdu/bw dfc = 1.338/9/(fc)2/3; = fcd/Eswu , wu = 0.09mm

Es = 2 000 000 MPa

Note: Es, modulus of elasticity of steel; fct, axial tensile strength of concrete; k, constant relating to section depth and curtailment; SE,crack spacing parameter; Sx, crack spacing; Vc, nominal shear strength; Vdu, dowel force; Vrd1, shear resistance; Vu, ultimate shear force; wu,limiting crack width; m, partial safety factor for strength; , dimension-free value for the crack width, which determines the friction capac-ity. All other terms as defined in the text.

Table 2. Summary of the expressions to calculate the concrete contribution to shear resistance.

Fig. 1. Typical architecture of multilayer ANN.


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tions between the neurons are individually weighed so thatthe total of i inputs (Xi) to the single neuron is

[4] input = w X bii

i +where b is the connection weight associated with a bias nodethat has an input value of 1.0.

The weights may be positive or negative such that some

inputs will be excitatory and others will be inhibitory. Thisinput passes through an activation function to produce thevalues of Yi of the hidden layer(s) or Oi of the output layer.The activation function may have many forms. The most fa-miliar and effective form in our case is the sigmoid function,defined as

[5] output = 1/{[1 + exp[ (input)]}

where is a constant that typically varies between 0.01 and1.00.

Signals are received at the input layer, pass through thehidden layers, and reach the output layer, producing the out-put of the network. The learning process primarily involves

the determination of connection weight and bias matricesand the pattern of connections. It is through the presentationof examples, or training cases, and application of the learn-ing rule that the neural network obtains the relationship em-bedded in the data.

In applications of multilayer feed-forward neural networksto engineering modeling, there are four important issues to beaddressed: (i) representation of the problem in which thecomposition of the input and output layers must be deter-mined, (ii) determination of the configuration of hiddenlayer(s) through trial and error process, (iii) training of theANN, and (iv) generalization verification of the trained ANN.

The representation scheme depends on the nature of theproblem and the intended use of the neural network model.

Experience shows that there is no unique solution for repre-sentation schemes; different neural networks can producesimilar results with the same set of training data.

Input and output layers of ANNIn this study, the neural networks were designed to have

an input layer that consists of six input nodes representingthe most important parameters that affect the shear capacityof reinforced concrete beams. Based on careful study of re-cent approaches for the shear phenomena in concrete mem-bers, it was decided to design the input layer to consist of(i) beam depth (d), (ii) beam breadth (bw), (iii) ratio of shearspan (a) to beam depth (a/d), (iv) uniaxial concrete compres-sive strength (f1c), (v) percentage of main reinforcement (l ),

and (vi) steel yield stress (fy). The output layer consisted ofone node representing the ultimate shear capacity of thebeam.

Training of the networkIn a multilayer feed-forward neural network, training re-

fers to the iterative process involving the presentation oftraining data to the network, the invocation of learning rulesto modify the connection weights, and, usually, the evolutionof the network architecture, such that the knowledge embed-ded in the training data is appropriately captured by theweight structure of the network. During the training phase,

the training data consist of input and associated output pairsrepresenting the problem that we want the network to learn.

Training and testing patternsAn important factor that can significantly influence the

ability of a network to learn and generalize is the number ofpatterns in the training set. Although it increases the time re-

quired to train a network, increasing the number of trainingpatterns provides more information about the shape of thesolution surface and thus increases the potential level of ac-curacy that can be achieved by the network. Another impor-tant condition is that the training data should be welldistributed within the problem domain.

Carpenter and Barthelemy (1994) stated that a necessarycondition for obtaining a unique approximation is to havethe number of training pairs equal to or greater than thenumber of weights and biases associated with the network.In our case this number ranges between nine and 161 fornetworks with one and 20 nodes in a single hidden layer, re-spectively. It also ranges between 11 and 191 for networkswith two hidden layers having one and 10 neurons per hid-

den layer, respectively. Oreta and Kawashima (2003) statedthat the minimum number (NT) of training data pairs, forANNs with bias terms at the input and hidden layers, shouldbe NT = J(I + 1) + K(J + 1), where I is the number of inputnodes, J is the number of hidden layer nodes, and K is thenumber of output nodes. Some ANN experts suggest that NTshould be in the range 10(I + J + K), which yields a numberbetween 80 and 270. Applying all suggestions to our case,we reach a number of training data pairs between nine and270. Carpenter and Hoffman (1995) suggested that an ANNmodel that is 20%50% overdetermined tends to producereasonably good approximations. Compromising these sug-gestions and having in mind that we have a total of 398 datapatterns, it was decided to use 50% of the data (199 speci-mens) for training and save the other 50% for testing or vali-dation. Training data were first selected randomly, and thenthey were checked to make sure that they satisfy a good dis-tribution within the problem domain.

Back-propagation learning algorithmIn this study, the training phase of ANNs is implemented

by using the back-propagation learning algorithm. A back-propagation network typically starts out with a random set ofweights. The network adjusts its weights each time it sees aninputoutput pair. Each pair requires two stages: a forwardpass and a backward pass. The forward pass involves pre-senting a sample input to the network and letting activations

flow until they reach the output layer. During the backwardpass, the networks actual output (from the forward pass) iscompared with the target output and error estimates are com-puted for the output units. The weights connected to the out-put units can be adjusted to reduce those errors. We can thenuse the error estimates of the output units to derive error es-timates for the units in the hidden layers. Lastly, errors arepropagated back to the connections stemming from the inputunits.

The back-propagation algorithm updates its weightsincrementally, after seeing each inputoutput pair. After ithas seen all the inputoutput pairs (and adjusted its weights

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many times), it is said that one epoch has been completed.Training a back-propagation network usually requires manythousands of epochs.

An error criteria for the network output is usually chosenand the maximum number of iterations is set to provide acondition for terminating the learning process. The perfor-mance of ANNs can be monitored by monitoring the train-ing error with respect to the number of iterations. If thenetwork learns, the error will approach a minimum value.After the training phase, the ANNs can be tested for theother set of patterns, which the network has never seen,where the final values of the weights obtained in the trainingphase are used. No weight modification is involved in thetesting phase.

Normalization of input and output dataNeural networks usually provide improved performance

when the data lie within the range (0, 1). Because a sigmoidfunction is used, a slow rate of learning occurs near the endpoints of the sigmoid function. To avoid this, all input valuesand associated outputs are transformed to values rangingfrom 0.1 to 0.9 rather than from 0.0 to 1.0. The following

equation is used:

[6] XT = 0.1 + 0.8[(X Xmin)/(Xmax Xmin )]

where XT is the normalized value, X is the original value,Xmax is the maximum value of the attribute or output, andXmin is the minimum value of the attribute or output.

Network validation and error analysisAfter an ANN model has been trained, validation of the

network on the test patterns should be undertaken. The usualpractice for ANN model validation is to evaluate the net-work performance measure using a selected error metricbased on both training and testing data. The evaluation and

validation of an ANN prediction model can be done by us -ing common error metrics such as the root mean squared er-ror (RMS error). The definition for RMS error is given inthe following equation:

[7] RMS = ( ) /O Y nmijj

m

i

n

ij

==

11

2

where n is the number of patterns in the validation set, m isthe number of components in the output vector, O is the out-put of a single neuron, and Y is the target output for the sin-gle neuron.

Network topologySince there is no direct and precise way of determining

the most appropriate number of hidden layers and number ofneurons to include in each hidden layer for a problem, a trialand error procedure is typically used to approach the bestnetwork topology for a particular problem. In this paper,several network topologies were examined. These includednetworks with one hidden layer containing 1, 2, 4, 6, 8, 10,12, 14, 16, 18, and 20 neurons in the single hidden layer;and networks with two hidden layers containing 1, 2, 4, 5, 6,8, and 10 neurons in each hidden layer. The target networkwould be the smallest sized network that not only produces aminimum error for the training pattern but also gives a gen-eralized solution that performs well for the testing pattern.

Results and discussion

Error traceFigure 2 shows the RMS error trace with the number of it-

erations for networks with a single hidden layer (ANN 6-12-1) and those with two hidden layers (ANN 6-4-4-1). Clearly,the RMS training error decreases with an increase in the

number of iterations, whereas the RMS testing error de-creases to a minimum and then begins to increase again.During the first part of the training, performance on thetraining set improves as the network adjusts its weightsthrough back-propagation. Performance on the test set(dataset that the network is not allowed to learn on) also im-proves. After a while, the weights shift around, looking for apath for further improvement. When this path is found, per-formance on the training set improves but performance onthe test set gets worse due to the fact that the network beginsto memorize the individual inputoutput pairs rather thansettling for weights that generally describe the mapping forall cases. Therefore, it is not always beneficial to increasethe maximum number of allowed iterations. In this study,

this number was set to 40 000 iterations, which was fairlysuitable for most of the studied cases.

Achieving best networkThe performance of different networks in terms of RMS

error of both training and testing patterns is shown in Fig. 3.Increasing the number of neurons causes the RMS trainingerror to decrease. In other words, networks with a largenumber of neurons in the hidden layers achieve good learn-ing in terms of RMS training error. On the other hand, theygive worse results in terms of RMS testing error. This indi-cates that too much power of the network (large number of

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Fig. 2. Training and testing root mean square (RMS) error trace: (a) ANN 6-12-1; (b) ANN 6-4-4-1.


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neurons in the hidden layers) allows the network to memo-rize the training pattern rather than extract the noticeablefeatures that map the input data to the output data.

To judge which network performs better, a statistical errorof each network was calculated for all patterns, that is, 398pattern sets. The results are shown in Fig. 4, in which the av-erage error, median, standard deviation (SD), and standard

variation (SV) are shown for each of the tested networks.For the networks with a single hidden layer, the networkswith six and 10 neurons in the hidden layer are better thanall other networks. For the networks with two hidden layers,the one with six neurons in each hidden layer can be consid-ered best.

These three networks were selected and allowed to run200 000 more iterations to reach a total of 240 000 itera-tions. The maximum, minimum, average, SD, and SV of theratio of experimental to model-predicted shear capacity(mod = Vexp/Vpred) for the three networks with 40 000 itera-tions and their counterparts with 240 000 iterations areshown in Fig. 5. Note that the networks with 240 000 itera-tions are denoted ANN#. Clearly the ANN 6-6-1 network

with 40 000 iterations was superior to all other networks. Itis concluded that too much increase in the number of itera-tions, although consuming a large amount of computer run-ning time, did not improve the performance of the networksat all.

A comparison of the experimental and predicted shear ca-pacity by the aforementioned six networks for all 398 datapatterns is shown in Fig. 6. The line of equality and the linesof plus or minus 10% error are also plotted to facilitate visu-alization and judgment of the results. The network that givesresults closer to the equality line is of course better. ANN 6-6-1 is obviously the best network and has the greatest num-

ber of predictions lying between the plus and minus 10%error lines. Figure 7 shows a histogram of the ratio betweenexperimental and predicted shear capacity for ANN 6-6-1.The histogram is slightly skewed to the left-hand side andhas its maximum occurring at mod = 0.96.

Comparison with different modelsThe ANN 6-6-1 is employed to predict the shear capacity

of all 398 beam specimens, together with the eight differentcodes and methods listed in Table 2. The results are pre-sented in Fig. 8, which shows, in separate plots, comparisonsbetween experimental and predicted shear capacity by eachof the methods. It is clear from Fig. 8 that the ACI Commit-tee 318 (1999), BSI (1990), ECS (1992), Collins and

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Fig. 3. RMS error for different network topologies: (a) networks with a single hidden layer; (b) networks with two hidden layers.

Fig. 4. Statistical errors for all patterns (398 specimens): (a) networks with a single hidden layer; (b) networks with two hidden layers.

Fig. 5. Statistical data for the ratio between experimental and

predicted shear capacity (mod = Vexp/Vpred) for different net-works.


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Kuchma (1999), and Niwa et al. (1986) methods give widevariations on either side of the equality line. This means thatthese methods underestimate the shear capacity for somespecimens and overestimate the shear capacity for otherspecimens. The NZS (1995) and Zsutty (1971) methodsoverestimate the shear capacity. The Reineck (1991) methodis the only conservative method, in which most of the pre-dictions are underestimates. In contrast, the predictions ofthe ANN are extremely impressive, with most results lyingon or very close to the equality line. This accuracy suggeststhat the most critical variables controlling the shear capacityof concrete beams without transverse reinforcement are thesix that have been used as input data to the ANN model.

The ratio of the experimental to predicted shear capacityby specific model (mod = Vexp/Vpred) can be interpreted asthe additional factor of safety implied by this method, sinceno strength reduction factor was applied during the calcula-tion of shear capacity by any of the methods. For economicconsiderations, this additional safety factor should not be

very large because there is a concrete strength reduction fac-tor, m, that is applied during any design process. The ratiomod also should not be much lower than unity, since thismeans that the method overestimates the shear capacity ofthe beam and may lead to unsafe design.

A comparison of statistical calculations of the ratio of theexperimental to predicted shear capacity for all models(mod ) and for all data patterns was conducted and the re-sults are shown in Fig. 9. As observed earlier, the method ofReineck (1991) gives the most conservative results, with amaximum ratio of 4.40, a minimum ratio of 0.87, and an av-erage ratio of 1.43. The ANN method gives the best results,with a maximum ratio of 1.26, a minimum ratio of 0.74, andan average ratio of 0.94. All other methods have a maximum

ratio lying between 2.00 and 3.30, a minimum ratio between0.32 and 0.58, and an average ratio between 0.87 and 1.47.Both the SD and SV are minimum for the ANN method.

To describe the influence of dominant parameters on thepredictions of each model, the ratio of actual to predictedshear capacity (mod ) is plotted versus the primary parame-ters in Figs. 1013. A common feature of all the plots is thevery large scatter in the predictions by different models, inwhich mod ranged from 0.33 to 4.41.

Figure 10 shows the plot of mod versus the ratio of shearspan to beam depth (a/d). Most tests were conducted for lowa/d ratios, for example, 81% of all tests (323 of 398 tests)

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Fig. 6. Comparison of experimental and predicted shear capacity for different networks. The broken lines represent 10% error on either

side of the line of equality.

Fig. 7. Histogram of the ratio of experimental to predicted shear

capacity for ANN 6-6-1.


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were carried out for beams with an a/d ratio of less than orequal to 4.0. A large scatter is noticed in the predictionsmade by all methods for a/d ratios between 2.4 and 4.0. Forthe ACI Committee 318 (1999), BSI (1990), ECS (1992),NZS (1995), Collins and Kuchma (1999), and Reineck(1991) models, the additional safety factor, mod , increaseswith a decrease in the a/d ratio, indicating a beneficial influ-

ence due to direct load transfer that is not captured by thesemodels. The results obtained from the neural network indi-cate consistent accuracy in all ranges of the a/d ratio, indi-cating that the shear phenomena were well captured by thisnetwork.

Variation of the additional safety factor, mod , with theconcrete uniaxial compressive strength (f1c) for the modelsconsidered in this study is shown in Fig. 11. Most of thetests in the database (73%) were carried out for normal-strength concrete (NSC) having f1c less than 45.0 MPa (289of 398 tests). The scatter of the predictions by the ACI Com-mittee 318 (1999), BSI (1990), and Collins and Kuchma

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Fig. 8. Comparison of experimental and predicted shear capacity for 398 specimens. The broken lines represent 10% error on either

side of the line of equality.

Fig. 9. Statistical data for the ratio between experimental and

predicted shear capacity for different shear proposals.


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Fig. 10. Ratio of experimental to predicted shear capacity versus the ratio a/d for different shear proposals.


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Fig. 11. Ratio of experimental to predicted shear capacity versus uniaxial concrete compressive strength (f1c) for different shear proposals.


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Fig. 12. Ratio of experimental to predicted shear capacity versus percentage of main steel (1 = As/bwd) for different shear proposals.


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Fig. 13. Ratio of experimental to predicted shear capacity versus beam depth (d) for different shear proposals.


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(1999) methods for the specimens with high-strength con-crete (HSC) having f1c greater than 45.0 MPa is very appar-ent. The results of the NZS (1995), Zsutty (1971), and Niwaet al. (1986) methods are mostly unaffected by the variationof f1c, indicating better recognition of the effect of f1c on theshear capacity of beams without shear reinforcement.Among all of the models shown, the results obtained from

the neural network are the most consistent, with values closeto unity for a wide variation in concrete uniaxial compres-sive strength.

Figure 12 shows a plot of mod versus the longitudinal re-inforcement percentage l (= As/bwd, where As is the area oflongitudinal reinforcement). Fifty-five percent of all tests(219 of 398 tests) were conducted for a relatively high rein-forcement ratio of1 > 2%. Only 14.6% of the tests (58 of398 tests) were conducted for 1 < 1%. The results obtainedusing the ACI Committee 318 (1999), BSI (1990), ECS(1992), Zsutty (1971), and Collins and Kuchma (1999)methods show a general increase in the models additionalsafety factor, mod , with the increase in 1 indicating a pro-nounced beneficial effect of the reinforcement ratio that is

not properly captured by any of these methods. Some ofthese methods, namely those of ECS and Zsutty, includedthe reinforcement ratio in their equations, but it seems thatthe beneficial effect of reinforcement exceeds what was in-cluded in the relationships. The models of Niwa et al. (1986)and Reineck (1991) are nearly unaffected by the change inthe reinforcement ratio, indicating a reasonable capturing ofthe effect of reinforcement ratio. The performance of theneural network model is superior to that of all other models.It not only is unaffected by the change in reinforcement ra-tio, but also gives values of mod near or equal to unity.

Lastly, the variation of the additional safety factor, mod ,with beam depth, d, is shown in Fig. 13; 87% of all the spec-imens (346 of 398 specimens) had a beam depth of less than

500 mm. Once again, most methods yield large scatter in theresults, especially for specimens with beam depths in therange 140300 mm. The additional safety factor, mod , pre-dicted using the ACI Committee 318 (1999), BSI (1990),NZS (1995), Zsutty (1971), and Collins and Kuchma (1999)models decreases with an increase in depth, indicating im-proper capturing of the effect of depth on the overall shearcapacity of beams. In contradiction to all other methods, thepredictions of the ANN are very consistent and very close tounity.

Summary and conclusions

Different relationships have been proposed by researchersall over the world for predicting the shear capacity of con-crete members without transverse reinforcement. Unfortu-nately, these proposed relationships are usually empiricaland designed to fit a limited set of shear test results that ismost familiar to the researcher. Accordingly, the relation-ships differ quite considerably in their selected parameters,since there is no generally accepted model for the load trans-fer and ultimate shear capacity of members without trans-verse reinforcement.

A unique and critically reviewed database of experimentalwork conducted on beams without transverse reinforcementwas collected from 64 different references. This database

contained a total of 398 tests and was intended, by the JointASCEACI Committee 445 on Shear and Torsion (1998)and the ACI Subcommittee 445-F on Beam Shear, to serveas a basis for any code changes regarding the shear phenom-ena. The database covered a very wide range of beamparameters, including their dimensions, concrete strengths,reinforcement ratios, and shear span to beam depth ratios.

This database was utilized in this study to evaluate the fol-lowing eight different existing codes and formulas for pre-dicting the shear capacity of members without transversereinforcement: ACI Committee 318 (1999), BSI (1990),ECS (1992), NZS (1995), Zsutty (1971), Collins andKuchma (1999), Niwa et al. (1986), and Reineck (1991).Moreover, the database was used to evaluate and demon-strate the capability of the feed-forward back-propagationartificial neural networks (ANNs) for predicting such shearcapacity.

Based on careful study of recent approaches for the shearphenomena in concrete members, it was decided to designthe ANN to have six nodes in the input layer containing dataregarding beam depth (d), beam breadth (bw), shear span to

beam depth ratio (a/d), uniaxial concrete compressivestrength (f1c), percentage of main reinforcement (1), andsteel yield stress (fy). The output layer consisted of one noderepresenting the ultimate shear capacity of the beam.

Several network topologies were examined. These in-cluded networks with one hidden layer containing 1, 2, 4, 6,8, 10, 12, 14, 16, 18, and 20 neurons in the single hiddenlayer; and networks with two hidden layers containing 1, 2,4, 5, 6, 8, and 10 neurons in each hidden layer. All the net-works were trained on 199 shear tests, and the performanceof the networks on both 199 training datasets and 199 testingdatasets was compared; the network with the best predictionwas selected for comparison with the eight models.

In comparing the predictions of the ANN with those from

the eight models, it was found that the results obtained fromthe ANN were the most accurate, giving values of maximumshear capacity very close to the experimental values. More-over, the results obtained using the ANN were very consis-tent and covered a very wide range of variation of any of theinput parameters. This accuracy suggests that the most criti-cal variables controlling the shear capacity of concretebeams without transverse reinforcement are the six variablesused as input data to the ANN model.

Acknowledgment

The author would like to thank Professor Amr W. Sadekof Cairo University and the Kuwait Institute for ScientificResearch (KISR) for his valuable suggestions regarding de-termination of the most appropriate number of training andtesting patterns.

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