Nuclear Engineering and Design 178 (1997) 1–11

Artificial neural networks in prediction of mechanical behaviorof concrete at high temperature

Abhijit Mukherjee *, Sudip Nag BiswasDepartment of Ci6il Engineering, Indian Institute of Technology, Bombay 400076, India

Received 19 February 1996; received in revised form 13 May 1997; accepted 10 June 1997

Abstract

The behavior of concrete structures that are exposed to extreme thermo-mechanical loading is an issue of greatimportance in nuclear engineering. The mechanical behavior of concrete at high temperature is non-linear. Theproperties that regulate its response are highly temperature dependent and extremely complex. In addition, theconstituent materials, e.g. aggregates, influence the response significantly. Attempts have been made to trace thestress–strain curve through mathematical models and rheological models. However, it has been difficult to include allthe contributing factors in the mathematical model. This paper examines a new programming paradigm, artificialneural networks, for the problem. Implementing a feedforward network and backpropagation algorithm thestress–strain relationship of the material is captured. The neural networks for the prediction of uniaxial behavior ofconcrete at high temperature has been presented here. The results of the present investigation are very encouraging.© 1997 Elsevier Science S.A.

1. Introduction

Modeling of the mechanical behavior of con-crete at temperatures above ambient is necessaryin the analysis of hypothetical accident situationsin a nuclear reactor or other accidental fire situa-tions. Design of fire resistant structural elementsrequire realistic knowledge of the behavior ofconcrete at high temperatures. The response ofconcrete at high temperature and pressure is non-linear. The parameters that regulate its behavior,e.g. modulus of elasticity, comprehensive strength,ultimate strain, thermal conductivity, etc., arenon-linear functions of temperature. Moreover,

the properties of constituent materials, e.g. aggre-gates, influence the behavior of concrete signifi-cantly (Diederichs et al., 1987). Influence oftemperature on the properties of concrete havebeen studied through many experiments. Most ofthe tests have been directed toward studies of thetemperature dependence of various properties ofconcrete concerned with fracture, strength anddeformation behavior at high temperatures underuniaxial conditions and in a steady state. Theseresults are useful for the prediction of behavior of1-D structural members such as beams andcolumns but constitutive equations and failurecriterion for concrete which are derived from suchtests cannot be applied to complex concrete struc-tures like pressure vessels, slabs or other concrete* Corresponding author.

0029-5493/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved.

PII S0029 -5493 (97 )00152 -0

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plate and shell elements directly, as they havemainly biaxial stress conditions and are subjectedto considerable temperature variations. Recently,some sophisticated test instruments have been re-ported (Thienel, 1993) which allow the study offracture and stress–strain behavior of concretesubjected to biaxial stresses at high temperatures.Experimental results under transient conditions,that are very realistic in the case of fire hazards,are now available. However, the analytical modelsdeveloped so far are limited to uniaxial behavioronly.

Simulating a reliable model is a mountainoustask when the material in question experienceselastic and inelastic deformations, thermal dilata-tion, transient creep and cracking as well asdegradation of its material properties with in-creasing temperature. Concrete is one such mate-rial and the difficulties encountered duringmodeling its response at an elevated temperaturehas been acknowledged by various researchers(Diederichs et al., 1989). Simulation with mathe-matical models is based on a semi-empirical rela-tionship between the parameters that must beexplicitly stated for computation. The exact rela-tionships in some cases are yet to be ascertained.

Recently, a new programming paradigm, artifi-cial neural networks (ANN) has been proposed tomodel complicated multiparameter material be-havior (Ghaboussi et al., 1991; Mukherjee et al.,1995; Mukherjee and Rao, 1996). The basic ad-vantage of artificial neural networks lies in thefact that it is able to automatically map a rela-tionship from the supplied input and outputparameters. Therefore, complicated relationshipsbetween various parameters can be found out bythe network. Moreover, ANN is a model freeestimator. Therefore, the parameters that aredifficult to measure experimentally can beavoided. For example, the hardening parameter ofconcrete which is a function of temperature isdifficult to measure. An ANN can be trained withthe input of temperature only and the networkshould map the hardening parameter itself.

Artificial neural networks learn through exam-ples. The examples are presented in the form ofinput and corresponding output parameters. Thenetwork then attempts to learn, or in other words

map the relationship between the input and out-put from the examples presented to it. The input–output parameter can be generated fromexperimental results. As a result, the materialbehavior is captured directly from the experimentswithout describing the exact relationship betweenthe parameters. The ‘trained’ network would con-tain adequate information about the material be-havior to qualify as a material model. In a neuralnetwork the knowledge is stored in a distributedfashion. Therefore it is able to generalize theknowledge and predict reliably even for problemsthat are not a part of the training examples.

In a mathematical model the empirical rulesand expressions are stated to constrain the funda-mental laws of mechanics (like conservation law,symmetry requirements, invariance, etc.) relatingvarious material properties like elastic behavior,yielding, strain hardening, etc. On the contrary,trained neural networks predict the material be-havior ‘intuitively’. Consequently a neural net-work based model cannot guarantee to prove thatthe internal mechanism of the model is followingthe laws of mechanics. However, the fact that thebackpropagation networks are trained with anexhaustive set of experimental data means thatthe model should approximate the laws of me-chanics.

In the present paper we show the strength ofartificial neural networks in capturing the behav-ior of concrete at high temperature. The resultfrom experimentation have been used directly totrain the network. Therefore, all the complicatingeffects have been included directly in the model.Predictions of the neural network have been com-pared with those of the existing mathematicalmodels. The results have been very encouraging.

2. Concrete under high temperature and highpressure

The behavior of concrete structures that areexposed to thermal loading is an issue of greatpractical importance. By the use of high strengthconcrete the economy of structures can be en-hanced considerably in conventional house-build-ing as well as in construction for energy

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technology. This involves knowledge of the hightemperature behavior of concrete because thestructures may either in normal service or undercatastrophic conditions (fire, reactor accident,etc.) be subjected to high temperature and theymust be correspondingly designed.

During the heating of a concrete specimen sev-eral non-liner physical phenomena are observed.First the material expands. The thermal dilatationis governed by the coefficient of thermal expan-sion which is a non-linear function of tempera-ture. Another effect is the decrease in stiffness inthe form of Young’s modulus. The damage (The-landersson, 1982) is mainly caused by (i) thethermal incompatibility between the aggregateand the cement paste and (ii) chemical decomposi-tion of the cement paste. It is common to taketemperature as an internal state variable (damagemeasure) and formulate a mathematical model onthe basis of isothermal test data only. The draw-back of this approach is that the material re-sponse depends strongly on the path in thestress–temperature space. This is known as the‘temperature change effect’ observed in pure tor-sion and uniaxial compression tests. To explainthis behavior some researchers have introduced anextra component of strain termed ‘transitionalcreep’ or ‘transitional strain’. It is not yet clearwhich mechanism causes transient creep, but itseems that between 100 and 200°C drying of thecement matrix is an important factor (dryingcreep), while at higher temperatures a change ofthe chemical structure of the cement matrix isprimarily responsible for the observed strain(transitional thermal strain). Transitional thermalcreep results in the relaxation and redistributionof the thermal stress, rendering the elastic stressanalysis inappropriate for the structure that isheated for the first time (Khoury et al., 1985).

Diederichs et al. (1987) conducted an investiga-tion at high temperature up to 600°C on HTR-concrete under uniaxial conditions. Theinvestigation was carried out with different mixproportions and with different aggregates. Theyconcluded that higher loss of strength was wit-nessed in the Rhine gravel concrete than with thebasalt concrete when exposed to thermal condi-tions. The main difference in the deformation

behavior of the two types of concrete was foundin the thermal expansion, which depends mainlyon the thermal expansion of the coarse aggre-gates.

Thelandersson (1982) proposed an analyticalmodel in which the basic assumption is that mois-ture related effects are considered only indirectlyand the model is less suitable in situations wherethe moisture is confined in the material. The-landersson assumed the existence of a plastic anda viscoplastic yield surface. The viscoplastic yieldsurface is used to describe the time dependenteffects, while the plastic surface is used to describethe instantaneous non-linear behavior. Both thesurfaces change their sizes with the temperature,but they keep the same shape which contradictsthe experimental data obtained by Ehm andSchneider (1985) and Kordina et al. (1986). The-landersson (1987) presented an improved model inwhich the theory is greatly simplified (linearity instress and creep neglected). The results deviatefrom the experimental results, for combinations ofstress and temperature that approach failure. The-landersson argues that to describe the variousphenomena occurring when concrete is subjectedto high temperature, it is necessary to adopt anapproach in which the hygral and the thermo-chemical response are interdependent. A methodto describe the interdependence at the linear levelwas presented, but the non-linear one is not intro-duced. An improvement was suggested by DeBorst and Peeters (1989). The main proposal ofthis model was to outline the consistent approachto smeared crack analysis of concrete structuresthat are exposed to extreme thermal conditions.In that model the total strain rate is decomposedinto a crack and concrete strain rate. Each strainrate is governed by a separate constitutive lawwhich opens the possibility of constructing a con-sistent set of rate equations. The model showedthe difficulties associated with incorporating acrack at higher temperatures. They argue that theclassical smeared crack approach is ill suited, andinstead they proposed a strain rate formulation inwhich the total strain was divided into individualcomponents. Their algorithm is limited to thelinear range only.

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Khennane and Backer (1993) have suggested amodel to capture the uniaxial behavior of con-crete under high temperature. The basic formula-tion used in the model assumes that the totalstrain rate of concrete under first time heating canbe decomposed into additive components of elas-tic strain rate, plastic strain rate, transient creepstrain rate and the thermal strain rate. The elasticstrain is governed by the Hooke’s law, where themodulus of elasticity (ET) is a function of temper-ature (T). The plastic strain rate and the harden-ing parameter are path dependent. Therefore, theydepend on the stress at any point in the stress–strain curve. The estimate of the hardeningparameter is made by assuming the stress–straincurve in the plastic region to be a quarter of anellipse in shape. The free thermal strain is ob-tained as a product of the temperature and thecoefficient of free thermal expansion of the con-crete (V) which varies with temperature. Howeverthe coefficient of free thermal expansion is ex-pressed as a piece-wise linear function (constantfor a certain range of temperature) due to its verycomplex behavior.

To validate the model three combinations ofload and temperature that can occur in a realsituation have been used. The first combinationconsists of a variable load and a constant temper-ature (isothermal condition). The second combi-nation consists of variable temperature and aconstant loading and the third consists of a com-bined variation of both load and temperature.The model provided better results than the previ-ous models. In the isothermal condition case, thesimulation is satisfactory in the elastic zone, but itgradually deviated from the experimental resultsin the plastic zone with the increase in tempera-ture. In the constant load case the simulatedresponse calculated by considering the load effectson the properties give more accurate results thanby not considering the effects. However, bothresults deviate widely from the experimental resultwith increasing temperature. In the restrainedcase, the results of this model maintained thenature of actual experimental result but they devi-ated widely in value.

In this paper, we demonstrate the power ofartificial neural networks (ANN) in the prediction

of stress–strain relationships in concrete at hightemperature. ANN has produced far superior re-sults than the previous semi-analytical expres-sions. A comparison between the existingsemi-analytical predictions and those by ANN ispresented. Moreover, using the methodology de-scribed here an ANN for biaxial behavior can bedeveloped, for which no analytical model exists.The present work is a step towards that goal.

3. Artificial neural networks

Artificial neural networks were developed tomodel the human brain. The concept of neuralnetworks is discussed in detail elsewhere(Mukherjee et al., 1996). A brief description isincluded in the following section.

3.1. Artificial neurons

An artificial neuron is a very approximatemodel of the biological neuron. It can carry out asimple mathematical operation and/or can com-pare two values. An artificial neuron gets inputfrom other neurons or directly from the environ-ment. The path connecting two neurons is associ-ated with a certain variable weight whichrepresents the synaptic strength of the connection.The input to a neuron from another neuron isobtained by multiplying the output of the con-nected neuron by the synaptic strength of theconnection between them. The artificial neuronthen sums up all the weighted inputs coming to it.

xj= %m

i=1

wijoi (1)

where xj is the summation of all the inputs forneuron j, wij is the synaptic strength betweenneurons i and j, oi is the output of neuron i, andm is the total number of neurons sending input toneuron j.

Each neuron is associated with a thresholdvalue and a squashing function. The squashingfunction is used to compare the weighted sum ofinputs and the threshold value of that neuron. Ifthe threshold value is exceeded by the weightedsum the neuron goes to a higher state, i.e. the

A. Mukherjee, S. Nag Biswas / Nuclear Engineering and Design 178 (1997) 1–11 5

output of the neuron becomes ‘high’. Many differ-ent squashing functions are used in different ap-plications. In the present work a backpropagationlearning algorithm has been used. This algorithmnecessitates the use of a continuous, differentiableweighting function. Therefore, a sigmoidalsquashing function has been used here as follows:

oj=1

1+e−a(xj−uj ) (2)

where oj is the output of neuron j, xj is thesummation of all the weighted sums of the inputsfor neuron j, uj is the threshold value of neuron j,and a is a parameter which controls the slope ofthe squashing function.

The output of the neuron for a given input canbe controlled to a desired value by adjusting thesynaptic strengths and the threshold values of theneuron. In an ANN several neurons can be con-nected in a variety of ways. Many different typesof neural networks have already been developed.The network architecture has to be selected keep-ing the problem at hand in mind. The presentwork requires training with a set of examples in asupervised manner. Therefore, a feedforward net-work is most suitable. A brief description of thefeedforward network follows.

3.2. Feedforward networks

In a feedforward network the neural units areclassified into different layers. The network con-sists of one input layer, one or two hidden layers

and one output layer of neurons. Fig. 1 presents atypical feedforward network. It may be noted thatall the neurons between two successive layers arefully connected, i.e. each neuron of a layer isconnected to each neuron of the neighboringlayer. However, there is no connection betweenneurons of the same layer or the neurons whichare not in successive layers. The input layer re-ceives input information and passes it onto theneurons of the hidden layer(s), which in turn passthe information to the output layer. The outputfrom the output layer is the prediction of the netfor the corresponding input supplied at the inputnodes. Each neuron in the network behaves in thesame way as discussed in Eqs. (1) and (2). Thereis no reliable method for deciding the number ofneural units required for a particular problem.This is decided based on experience and a fewtrials are required to determine the best configura-tion of the network.

In a feedforward network the knowledge isstored in a distributed manner in the form ofsynaptic strengths and thresholds. Thus it can begeneralized, i.e. it may be used for situations forwhich the network has not been trained. Initially,the synaptic strengths and the threshold valuesare allocated randomly. To train the network forspecific knowledge a set of training examples isprepared. A training example consists of a set ofvalues for the input neurons and the correspond-ing values for the output neurons. Several of suchinput–output pairs are prepared carefully toreflect all the aspects that the network needs tolearn. All the training examples together form thetraining set. In the beginning of the training pro-cess, as the synaptic strengths and thresholds areselected randomly the output predicted by thenetwork for a particular input and the outputsupplied in the corresponding training examplesmay not match. However, the synaptic strengthsand the thresholds can be adjusted so that thenetwork predicts the output correctly. As severalexamples are to be learnt by the network theremust be a sufficient number of neural units in thenetwork. The adjustments in the synapticstrengths and thresholds are carried out followinga ‘learning algorithm’. The backpropagation al-gorithm has been used in the present work for thispurpose.Fig. 1. A feeforward neural network.

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3.3. The backpropagation algorithm

The backpropagation algorithm is a generalizedform of the least mean square training algorithmfor perceptron learning (Lippmann, 1987; Hecht-Nilsen, 1989). It uses the gradient search methodto minimize the error function which is the meansquare difference between the desired and thepredicted output. The error for pth example isgiven by

Ep=%j

(dj−oj)2 (3)

where dj is the output desired at neuron j and oj isthe actual output of neuron j,

As presented in Eqs. (1) and (2) the output oj isa function of synaptic strengths and outputs ofthe previous layer.

oj= f(bj)= f�%

i

wijoi�

(4)

The error can be minimized by moving along thesteepest descent direction on the error surface

(E(wij

=(E(bj

(bj

(wij

=(E(bj

oi=djoi (5)

where, dj for a neuron is

dj= f %(bj) %k

dkwkj (6)

and f % indicates the first order derivative of thefunction and k indicates a neuron in the layerwhich is successive to the layer which containsneuron j.

Therefore, the weight matrix can be adjustedrecursively for each example

wij(t+1)=wij(t)+hdjxi (7)

where h is an adjustable gain term which controlsthe rate of convergence.

The above operation is repeated for each exam-ple and for all the neurons until a satisfactoryconvergence is achieved for all the examplespresent in the training set.

For the present investigations an artificial neu-ral network simulator (Mukherjee et al., 1996) isused. The networks have been trained using anIBM compatible PC 486 machine. A feedforward

network is adopted for training purposes. Theerror is reduced using a backpropagation al-gorithm. A sigmoid function is used as thethreshold function. The function produces an out-put between 0 and 1. Therefore, the range inputand output is scaled between these two values.

4. ANN for uniaxial mechanical behavior ofconcrete under high temperature and pressure

The behavior of concrete at high temperatureand pressure has been modeled using the feedfor-ward artificial neural network. Extensive experi-mental work has been reported earlier (Anderbergand Thelandersson, 1976). These have been usedhere for neural training. The behavior of concreteunder three different conditions namely� varying load under isothermal conditions� varying temperature under constant load� varying temperature under total restrainthave been considered.

4.1. Varying load under isothermal conditions

The behavior of concrete under isothermal con-ditions is relatively simple. Under this condition,concrete experiences instantaneous mechanicalstrain only. For the isothermal stress–strain rela-tionship without sustained loading, creep does nottake place. Therefore, only the properties thatconstitute the elastic and the plastic response inthe material are considered in the model. In thiscase a strain driven model is used. The straindriven model has strain controlled input andstress is obtained as the output. It may be notedthat the neural networks can be trained fromexperimental results. Therefore, only those inputparameters should be chosen that can be mea-sured easily experimentally. In this case, the inputnodes are provided with current strain (o), tem-perature (T), elastic modulus (ET) at that temper-ature, compressive strength ( fcT) and ultimatestrain (oulT). At the output node current stress (s)was obtained (Fig. 2). A configuration of twohidden layers with five nodes at each layer wastaken. The training set consisted of the experi-mentally observed stress–strain relationships at

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Fig. 2. Network configuration for training isothermal curve(strain driven-uniaxial).

Fig. 3. (a) Compressive strength of concrete. (b) Ultimatestrain of concrete. (c) Modulus of elasticity of concrete. (d)Coefficient of thermal expansion of concrete.

temperatures 20, 265, 500 and 650°C (Anderbergand Thelandersson, 1976). The network was ini-tialized with random weights and thresholds. Thedifference between the output required and theactual output at the output node is backpropa-gated to train the network. This process is re-peated for all the examples that the net is to learn.The examples are presented repeatedly. Thepresent network was trained for a number cyclesand at the end of training the RMS differencebetween the desired and the obtained output overall the examples obtained was 0.000484. Thecurves produced by the ANN after training gave agood agreement with the experimental data. Thetrained network was further asked to predict therelationship at 165°C and 400°C. Although thenetwork predicted a behavior very close to theexperimental curves the predictions of the net-work showed an undulation that is not seen inpublished experimental results (Anderberg andThelandersson, 1976).

The experimental curves, for various propertiesof concrete under varying temperatures (Fig. 3)are extremely irregular and seems to be anoma-lous in some local regions (Fig. 3b). Therefore,when the experimentally observed input parame-ters were used these irregularities were mouldedinto the network. Therefore, the consistency inbehavior at different temperatures was lost. Khen-nane and Backer (1993) in the mathematicalmodel have used idealized curves avoiding localaberrations. The idealized curve is obtained byassuming piece-wise linear functions within two

A. Mukherjee, S. Nag Biswas / Nuclear Engineering and Design 178 (1997) 1–118

Fig. 4. Stress strain curves under isothermal conditions. Fig. 6. Comparison of temperature–strain curves.

The predictions of the net along with the experi-mental results are presented in Fig. 4. The perfor-mance of the network is a very good for the wholerange of input and it was marginally better thanthat of the semi-analytical expression. To examinethe generalization achieved by the network a newtemperature, which was not a part of the trainingexamples was input. The predictions of the net-work showed very good agreement with the ex-perimental observations. However, the predictionsdiverged slightly from the experimental values inthe extreme parts of the curve. This divergencecan be attributed to target values close to 0 or 1.It may be noted that all input and output valuesare scaled between 0 and 1. Towards the extremi-ties of this range the slope of the sigmoidal func-tion is very low. Therefore, the rate of learning forthese values is very slow. It is desirable to keepthe range of target values between 0.2 and 0.8.This can be achieved by adding a scalar term withthe desired values and selecting a suitable factorso that the maximum and minimum values fitwithin 0.2 and 0.8. The performance of the net-work showed a marked improvement after this.The predictions of the network at temperatures165, 300 and 450°C have been presented in Fig. 5.The experimental results for 165°C are presentedtoo. The agreement between the ANN predictionand the experimental work is very good. Experi-mental results are not available for any othertemperature. However, the predictions of theANN seem to be realistic, and they agree verywell with the experimental results in the vicinity.

points in the experimental observation. In orderto find out the reason for the observed undulationthe network was trained once again with theidealized curves. It was felt that the input datasupplied to the network was deficient in informa-tion. Therefore, it was decided to use the experi-mental results at 20, 265, 400, 500 and 650°C fortraining. The use of the idealized curves alleviatesthe local anomalies and should give a betterglobal prediction for new problems (i.e. problemsthat are new to the network).

The training data set for the network was takenat temperatures 20, 265, 400, 500 and 650°C onsame configuration of network (Fig. 2). The inputremained the same as the previous case. Thenumber of nodes in each hidden layer was five.

Fig. 5. Protection ANN for new examples.

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4.2. Varying temperature under constant load

The behavior of concrete under constant loadand varying temperature has been studied. Whenthe temperature of concrete varies, it is subjectedto thermal expansion. The effect of sustainedloading causes transient creep. Therefore, the ef-fect of thermal expansion and transient creepmust be taken into account for modeling theresponse of concrete under a sustained load.Therefore, an additional parameter, the coefficientof thermal expansion, is included in the input.The coefficient of thermal expansion can be mea-sured easily. Therefore, this term has been in-cluded in the network. The network on its own isexpected to model the creep deformation and noadditional term is included for creep. The inputnodes for training this network consist of currenttemperature (T) (temperature at a point in thetemperature–strain curve), load level (h) (con-stant load expressed in terms of percentage oforiginal compressive strength), modulus of elastic-ity at that temperature (ET), compressive strength( fT), ultimate strain (oulT) and the coefficient ofthe thermal expansion (VT). The output is strain(o) corresponding to the given load level andtemperature. The idealized values of the proper-ties of concrete as mentioned earlier have beenused.

The next step is to select a configuration for thehidden layers of the network. The method of this

Fig. 8. Temperature–stress curves for restrained condition.

selection is not formalized yet. Therefore, sometrials have been taken to evolve a good configura-tion. A systematic study was conducted to find asuitable network configuration. Finally, a networkconsisting six nodes at the input, two hiddenlayers, each containing 18 nodes and an outputcontaining one node was chosen. The networkwas trained until the RMS difference between thepredicted and the desired output over all theexamples was within an acceptable limit. Theresults of the network after training is discussedbelow.

The predictions of the network along with ex-perimental and theoretical investigations (Khen-nane and Backer, 1993) are presented in Fig. 6. Itcan be observed that the network’s predictionfollows the experimental results very closely. Onlyat the 67.5% load level does the extreme valuehave a slight deviation. This may be due to thevery small value of the output in that region.

The existing analytical model (Khennane andBacker, 1993), agrees well with the experimentalobservations at lower temperatures but it deviatessharply from the experimental results as the tem-perature increases. The ANN, on the other hand,was in close agreement with the experimentalresults through out the range. This demonstratesthe power of the ANNs to capture a very complexrelationship which is usually intractable even byvery sophisticated analytical models. Predictionsof the ANN for new examples were obtained forload levels of 27.5, 30, 38, 42, 50 and 60% (Fig. 7)which fall in between the available experimental

Fig. 7. Comparison of ANN prediction with experimentalresults.

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results. The predictions obtained from the ANNagreed very well with the trend of the experimen-tal results.

It may be pointed out, that the neural networktraining was not provided with any additionalinformation about the difference in the values ofvarious parametric properties under varying tem-perature due to different sustained load levels.These values are difficult to measure at differentload levels. Therefore, the network is designed insuch a fashion that the user is not required toinput those values. The load levels (h) had beenprovided in the input node and the propertiesunder the load free (h=0) condition were pro-vided. It is assumed that the neural networks,being model free estimators will be capable ofmapping the relationship of the parameters underthe loaded condition from the values at the loadfree state.

4.3. Varying temperature under totally restrainedconditions

The neural network modeling was explored tosimulate the response of concrete under variationof temperature when subjected to heating underrestrained conditions. The stress developed undersuch a condition is due to restraining the elastic,plastic and thermal strains of the specimen. Itconsists of six nodes in the input, two hiddenlayers each containing 12 nodes and an output.The input contains temperature (T), the modulusof elasticity at that temperature and loading con-dition (ET), compressive strength ( fcT), ultimatestrength (oulT), the coefficient of thermal expan-sion (VT) and the rate of heating (l) and theoutput was the restrained stress (s). The experi-mental results available for a rate of heating of5°C min−1 was utilised for training the network.The ANNs prediction obtained after training isshown in Fig. 8 along with the experimental andanalytical results. The ANN showed close agree-ment with the experimental observations. The an-alytical results, however, were far from theexperimental results. This reinforces the claim ofeffectiveness of the ANNs in capturing complexmaterial behavior intuitively which is impossiblehitherto even by sophisticated analytical models.

The response for a heating rate of 4°C min−1 hasbeen predicted with the help of the network. Theexperimental results for this heating rate is notavailable. The ANNs prediction maintains thegeneral trend of the response under such condi-tions.

5. Concluding remarks

The response of concrete under high tempera-ture and pressure is highly non-linear. Theparameters that influence its behavior are alsonon-linear. Hence, a large number of parametersare required for modeling of the behavior of suchmaterial with a mathematical equation. In addi-tion, reliable values of these parameters aredifficult to measure. Therefore, the mathematicalmodels are often erratic in their prediction. TheANNs, with their capability to learn directly fromexperimental observations are an attractive alter-native. The ANNs are model free estimators.Therefore, the input of difficult-to-measureparameters can be avoided in an ANN. However,the ability of the ANN to learn and generalizecomplex relationships must be tested against reli-able experimental observations. The main hin-drance faced while modeling with a artificialneural network is the non-availability of sufficientexperimental results. However, with the limiteddata available for the present problem the predic-tion of the network is extremely satisfactory.

Guidelines on the configuration of ANNs arenot well established. Therefore a trial and errorapproach is adopted in the selection of networksize, training examples and test problems. Pastexperience plays an important role for selection ofthe various attributes of the net.

The sigmoidal threshold function has a verysmall slope towards its extremities. Therefore, thenetwork faces difficulty in learning the points withoutput values close to 0 or 1. The input datashould be scaled so that the output values shouldlie between 0.2 and 0.8 to facilitate fast learning.

In this paper, three different situations of load-ing on concrete have been investigated: varyingload under isothermal conditions; varying temper-ature under constant load; and varying tempera-

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ture under total restraint. In all these situationsthe ANN has been very effective in predicting thestress–strain behavior of concrete. The perfor-mance of the ANN was remarkably superior tothe existing mathematical models in the secondand third conditions. This has encouraged us toattempt the prediction of the stress–strain behav-ior of concrete at high temperature subjected tobiaxial stresses. Due to the complexity of theproblem mathematical models of this case arescant. Investigation by neural networks of theproblem is already in progress and will be re-ported in the future.

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