Data driven models for compressive strength prediction of concrete at high temperatures

Abstract

The use of data driven models has been shown to be useful for simulating complex engineering processes, when the only information available consists of the data of the process. In this study, four data-driven models, namely multiple linear regression, artificial neural network, adaptive neural fuzzy inference system, and K nearest neighbor models based on collection of 207 laboratory tests, are investigated for compressive strength prediction of concrete at high temperature. In addition for each model, two different sets of input variables are examined: a complete set and a parsimonious set of involved variables. The results obtained are compared with each other and also to the equations of NIST Technical Note standard and demonstrate the suitability of using the data driven models to predict the compressive strength at high temperature. In addition, the results show employing the parsimonious set of input variables is sufficient for the data driven models to make satisfactory results.

Keywords

data driven model / compressive strength / concrete / high temperature

Introduction

Concrete is a common material that is widely used in the construction of buildings, subway projects, and some other civil structures. Today there is an ever-increasing demand for structural concrete with respect to the rapid development of urbanization. Concrete as a key element of the structures may experience the abnormal effects like wearing, freezing, chemical attacks over the life of the structures. One of the abnormal effects is high temperature and fire. Airport aprons, industrial structures including industrial ground and chimney operating under high temperature and plants producing chemical materials with high danger of fire are some examples of concrete structures susceptible to the effect of high temperature [¹]. Fires in concrete structures lead to very high temperatures of concrete. It is found that if the surface of concrete can reach above 100 °C, heat transfers can increase the interior temperature of the concrete to 300°C–700°C [²].

Concrete is a non-combustible material, but when concrete exposed to high temperature, its chemical, physical and mechanical properties change [³]. In hot concrete, chemical and physical reactions such as dehydration, decomposition [⁴,⁵], rapid increase of vapor pressure and thermal stresses [⁶] can be happened which can be lead to cracking, spalling and perforation of concrete and subsequently the degradation of mechanical properties of concrete [⁶]. Tanyildizi [⁷] showed the crack widths and lengths values increased with increase in temperature of concrete. While the events such as fire and explosion may not have caused direct damage, it is probable that such incidents weaken the structure’s stiffness or strength for a long or short-term duration [⁸].

The loss of strengths of concrete with high temperature can be arisen from different sources. The difference between the coefficients of thermal expansion of the cement paste and the aggregates which is called the thermal incompatibility of the concrete components is the first one. Thermal stresses between the expanding aggregate and shrinking cement paste are then appeared as a result of thermal incongruity [⁹]. The thermal stresses can collapse the interfacial transition zone between the aggregate and the surrounding cement paste that can decrease the concrete strength [¹⁰]. In addition to the thermal incompatibility of the concrete components, building of internal-pressure due to the evaporation of water during the heating process, and also the chemical changes in the cement paste and aggregates may be some sources of the loss of strengths of concrete with high temperature [¹¹].

Many factors like the properties of constituent materials of concrete, rate of temperature increasing, and maximum temperature may effect on the actual behavior of concrete at high temperatures [¹²,¹³]. With respect to the importance of engineering structures subject to the high temperatures, the efficiency of structural concrete at high temperature need to be investigated. Ožbolt et al. [¹⁴] investigated 3D thermo-mechanical numerical analysis of reinforced concrete beams exposed to elevated temperature. Caggiano and Etse [¹⁵] used a coupled thermo-mechanical interface model for concrete failure analysis under high temperature. The coupled thermal-mechanical effect in the interface model was considered through the formulation of a temperature dependent maximum strength criterion and fracture energy-based softening or post-cracking rule. In this way, an elasto-thermo-plastic interface model was proposed to predict the concrete cracking and failure behavior. The surface exposed to high temperature gets heated quickly, whereas inner parts of the cross-section have significantly lower temperatures. These temperature gradients induce restrained stresses and cause the concrete to crack [¹⁴]. It is important to remark that strength criterion degradation due to temperature effects is strictly related to the concrete cracking. Depending on the selected modeling scale and dimension, various strategies exist in the literature to model concrete cracking [¹⁶–²¹].

Studies on the strength deterioration of concrete exposed to high temperatures since the 1950s, have been carried out worldwide [²²–²⁹] and still undergoing up to day. Abrams [³⁰], Malhotra [³¹], and Schneider [²⁵] have analyzed the relation between temperature increase and concrete strength decrease. The effect of cement paste, mortar, concrete samples and reinforced concrete elements at high temperature was investigated in Ref. [²⁹]. Malhotra [³¹] investigated the effect of many various parameters on the compressive strength of concrete at different high temperatures. Based on the results of this study, the aggregate/cement ratio was a main parameter on the residual strength of the concrete at elevated temperatures.

Based on the predictions and recommendations of the previous studies, some codes have been created in the form of curves and equations. For design purposes, design curves advised in the codes [³¹–³⁷] may be used for mechanical properties of concrete at high temperature. In some codes such as ACI 216.1 [³⁷] and BS EN 1992-1-2:2004 [³⁸], is given in the form of curves and only dependent on aggregate type.

In addition to the design curves, prediction models, such as regression were also developed [³⁹–⁴²]. In NIST Technical Note 1681 [⁴²] a direct relationship between the temperature increase and decrease in the compressive strength of concrete in the form of Eqs. (1) and (2) were proposed. The Eq. (1) is for normal strength concrete (NSC) and the Eq. (2) is for high strength concrete (HSC). In this guideline, concrete with a compressive strength less than 83 MPa is known as NSC, and concrete with a compressive strength greater than 83 MPa is known as HSC.

(1){{\displaystyle F}}_{\mathrm{c},T}=\{\begin{array}{ll}{{\displaystyle F}}_{\mathrm{c},\mathrm{0}},\hfill & T\le \mathrm{450}°\mathrm{C,}\hfill \\ {{\displaystyle F}}_{\mathrm{c},\mathrm{0}}\left[\mathrm{2.011}-\mathrm{2.353}\left({\displaystyle \frac{T-\mathrm{20}}{\mathrm{1000}}}\right)\right],\hfill & T>\mathrm{450}°\mathrm{C,}\hfill \end{array}

(2)

where {{\displaystyle F}}_{\mathrm{c},\mathrm{0}} stands for the compressive strength at ordinary temperature, and {{\displaystyle F}}_{\mathrm{c},T} is the compressive strength at the temperature of T. For engineers, such simple equations would be very useful to estimate the strength of concrete structure when exposed to high temperature.

The results of the previous studies on the concrete exposed to high temperatures has provided a large pool of data which is a valuable potential for the development of data driven models. Data driven models extract relationships from input-output data without demanding the complete conceptual knowledge of the process behind the data. Universally, data driven models can be used as the suitable alternatives of analytical or conceptual models. The basic idea of the data driven models is to establish a model according to the relationship between available input-output data, and by using this relation to simulate and predict the output values for unseen data.

Data driven models such as multiple linear regression (MLR), artificial neural network (ANN), adaptive neural fuzzy inference system (ANFIS), and K nearest neighbor (KNN) were as developed the prediction models in different fields. Duan et al. [⁴³] developed ANN for predicting the elastic modulus of recycled aggregate concrete. Khademi et al. [⁴⁴] investigated the capability of ANN for predicting the concrete compressive strength based on the results of ultrasonic pulse velocity test. Sadrmomtazi et al. [⁴⁵] compared the results of regression, ANN and ANFIS for modeling compressive strength of EPS lightweight concrete. Tayfur et al. [⁴⁶] used fuzzy logic and ANN for strength prediction of high-strength concrete. Akbari et al. [⁴⁷,⁴⁸] used KNN for real time daily inflow forecasting. Khademi et al. [⁴⁹] used MLR, ANN and ANFIS models for determining the displacement in reinforced concrete buildings. Hamdia et al. [⁵⁰] investigated ANN and ANFIS models for predicting the fracture toughness of polymer nanocomposites.

The main aim of this research is to develop four data driven models (MLR, ANN, ANFIS, and KNN) for compressive strength prediction of concrete exposed to high temperatures. In this research, the results of total 207 compressive tests available in the literature are used as the experimental database of the data driven models. For each model, two combinations of input variables are used and the results are presented. In addition, outputs of the data driven models are compared with those of NIST Technical Note 1681 and conclusions are presented.

Methodology

As stated earlier, four data-driven models namely MLR, ANN, ANFIS, and KNN models are used for the purpose of this study. The models are described in the following.

MLR

MLR is a statistical relation between some of variables by fitting a linear equation to obtained data. This model keeps one variable as an output variable, and others are considered to be as input variables.

A MLR model is generally given in Eq. (3):

(3)\widehat{y}={{\displaystyle a}}_{\mathrm{0}}+\underset{i=\mathrm{1}}{\overset{m}{{\displaystyle \Sigma }}}{{\displaystyle a}}_i{{\displaystyle x}}_i,

where \widehat{y}is the model’s output, x_i is input variable to the model, mis the number of input variables, and a_{\mathrm{0}}, a_{\mathrm{1}}, a_mare regression coefficients or parameters of the model. The parameters are determined in such a way that an error function like mean square error (MSE) on a known data set is to be minimized.

In this study, the MLR model is devised by using the regression toolbox in Excel software.

ANN

ANNs consists of a network of nonlinear information processing elements (artificial neurons) which mimic the behavior of biological learning. ANN model is a data-driven model which can learn the nonlinear relationships involved in the process.

Multi-layer feed-forward neural network is the most commonly used configuration of the neural network in engineering applications. The multi-layer feed-forward network consists of a number of layers of neurons (nodes). The layers consist of an input layer, one or more hidden layers and an output layer. Each neuron with getting the help from activation function receives weighted inputs from other neurons and communicates its outputs to other neurons.

Hornik et al. [⁵¹] proved that a network with a single hidden layer containing an adequately large number of neurons can be used for function approximation to any desired accuracy. So, this study uses a single hidden layer network as well as many other studies. Figure 1 [⁵²] shows a schematic diagram for the multi-layer feed-forward neural network used in this study.

The output is calculated using the Eq. (4):

(4)\widehat{y}=f\left({\displaystyle \underset{j=\mathrm{1}}{\overset{q}{\mathrm{\Sigma }}}{w^{\prime }}_{j}g\left({\displaystyle \underset{i=\mathrm{1}}{\overset{m}{\mathrm{\Sigma }}}{x}_i{w}_{ij}+b_j}\right)+b^{\prime }}\right),

where fand gare transfer functions for output and hidden layers, respectively, w_{ij} and {w^{\prime }}_j are the neuron weights of the input to hidden and hidden to output layers, respectively, b_jand b^{\prime }are the neuron biases of the hidden and output layers, respectively, and qis the number of neurons in the hidden layer. Other variables or parameters were defined before.

The number of neurons in the hidden layer is determined through a trial and error process for each ANN model. In fact, for developing the model, several structures of network with different number of hidden neurons are usually tested in terms of accuracy to select an appropriate architecture of the network.

Training the network that means the adjustment of the weights and biases of the networks, is done in such a way that the ANN’s output matches with measured output as much as possible.

In this study, Matlab software and its corresponding neural network toolbox are used for constructing and training the proposed ANN model.

ANFIS

Fuzzy inference system which is based on Zadeh’ theory of fuzzy sets may be considered as a mathematical model in the form of fuzzy if-then rules. There are two common types of fuzzy inference systems: Mamdani and Assilian [⁵³]; Takagi and Sugeno [⁵⁴]. The antecedent of rules are in the form of fuzzy sets in both types of fuzzy inference systems while the consequences are different. Mamdani system uses fuzzy consequence and Takagi and Sugeno system employs crisp function as the consequence. Since ANFIS is a Takagi and Sugeno system, this system is described in more details in the following.

The general form of the Takagi-Sugeno fuzzy rules is given in the following statement [⁵⁵]:

(5)R_i:\mathrm{I}\mathrm{f}\hspace{0.17em}{{\displaystyle x}}_{\mathrm{1}}\hspace{0.17em}\mathrm{i}\mathrm{s}\hspace{0.17em}{{\displaystyle A}}_{i\mathrm{1}}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}{{\displaystyle x}}_{\mathrm{2}}\hspace{0.17em}\mathrm{i}\mathrm{s}\hspace{0.17em}\hspace{0.17em}{{\displaystyle A}}_{i\mathrm{2}}\hspace{0.17em}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}\hspace{0.17em}{{\displaystyle x}}_m\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{s}\hspace{0.17em}\hspace{0.17em}{{\displaystyle A}}_{im},\hspace{1em}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\hspace{0.17em}\hspace{0.17em}{y}_i=a_{i\mathrm{1}}{x}_{\mathrm{1}}+a_{i\mathrm{2}}{x}_{\mathrm{2}}+\cdots +a_{im}{x}_m+a_{i\mathrm{0}},

where R_i is the ith fuzzy rule, x_j is jth antecedent (input) variable, y_i is consequent (output) variable of ith rule, A_{ij} is fuzzy set of jth antecedent variable and ith rule, and a_{ij} is the model output coefficients.

For construction the Takagi and Sugeno fuzzy model, in the first step, input/output (I/O) space should be partitioned. Several partitioning methods are known from which the two most commonly used are grid partitioning and fuzzy subtractive clustering. In the grid partitioning, the domains of the input variables can simply be divided into a number of equally spaced and equally shaped membership functions. In this method, the membership functions for every input variable are built independently of each other and the relationship between the variables is ignored. Other drawback is that increasing the number of antecedent variables will result in an exponential increase of the number of rules for the model. In the fuzzy subtractive clustering, the database is divided into fuzzy clusters. In this way, each cluster matches to one rule of the model and so, the number of rules is equal to the number of clusters [⁵⁶].

In the next step, the parameters of the model including the antecedent membership functions of the fuzzy sets and the model output coefficients of each rule are also determined. These parameters can be determined such that the outputs of the model becomes similar to the observation data. For this purpose, Adaptive neurofuzzy inference system (ANFIS) can be employed. ANFIS makes use of the back propagation gradient descent algorithm to learn and determine the antecedent and the consequent. Recently, a hybrid-learning method which takes advantages of both the gradient descent and the least-squares method is also used for the purpose of learning.

Having the constructed the model, output of the model is calculated by Eq. (6) [⁵⁴]:

(6)\stackrel{\wedge }{y}={\displaystyle \frac{{\displaystyle \underset{i=\mathrm{1}}{\overset{c}{\mathrm{\Sigma }}}{w}_i}{y}_i}{{\displaystyle \underset{i=\mathrm{1}}{\overset{c}{\mathrm{\Sigma }}}{w}_i}}},

where c is the number of rules, and w_i is degree of fulfillment of the ith fuzzy rule for a given input crisp vector X={(x_{\mathrm{1}},x_{\mathrm{2}},\mathrm{...},x_m)}^{\mathrm{T}}which can be determined by Eq. (7) using product conjunction operator.

(7)w_i={\mu }_{A_{i\mathrm{1}}}(x_{\mathrm{1}})\cdot {\mu }_{A_{i\mathrm{2}}}(x_{\mathrm{2}})\mathrm{...}{\mu }_{A_{im}}(x_m),

where {\mu }_{A_{ij}}(x_j)is membership degree of x_j in the fuzzy setA_{ij}.

In this study, Matlab software with using fuzzy logic toolbox is employed for constructing the ANFIS model.

KNN

KNN learning, as an instance based learning algorithm, predict the target value of a given test pattern based on a combination of the target values of K selected neighbors as follows:

(8)\widehat{y}={\displaystyle \frac{{\displaystyle \underset{i=\mathrm{1}}{\overset{K}{\mathrm{\Sigma }}}{w}_i{y}_i}}{{\displaystyle \underset{i=\mathrm{1}}{\overset{K}{\mathrm{\Sigma }}}{w}_i}}},

where w_idenotes the weight of each neighbor, y_i is the output values of each neighbor and K is the number of nearest neighbors. To determine K nearest neighbors, a weighted Euclidean norm is usually used to evaluate the closeness (similarity) of the feature (input) vector of query instance and any feature vector of training data set. The number of neighbors is often selected empirically by cross-validation or domain experts in practice [⁵⁷].

The weight of each neighbor is often determined based on the distance between the neighbor and the quarry instance. So a smaller weight is assigned to a farther neighbor and a larger weight is assigned to a closer neighbor. In this way, some kernel functions such as linear kernel, inversion kernel, exponential kernel, and Gaussian kernel which inversely change to distance can be employed [⁴⁷].

In this study, a computer code is provided to implement the KNN model.

Application

Data collection

To apply the data driven models for prediction of compressive strength of the concrete at high temperature, respective data should be provided. For this purpose, a total of 207 input-output data pairs from 10 published sources [3, 9, ⁵⁸–⁶⁴] were gathered. It should be noted that the samples which were chosen from the mentioned references were taken at the age of 28 days. For training and testing of the proposed models, 165 and 42 samples were randomly selected respectively. For ANN and ANFIS models, a checking (validation) data set is also used to monitor the training process. The idea for the checking data might be utilized to avoid over-fitting. For this purpose, 31 data were selected randomly from the training data. The random splitting of the data into different data sets has been done so that the data sets have the relatively similar statistics. To fairly compare the efficiency of the models, the data set used for the testing data in all models was the same.

The input variables selection

In data driven models, an important step is the choice of the input variables representing the process to be modeled. The input variables of a data driven model should comprise all relevant information on the target output. On the other hand, they are largely dependent on the available information in the forms of input-output data pairs. The data available from the literature about the concrete at high temperature comprise the mix proportions, type of concrete specimen used by different researchers, temperature and compressive strength related to the temperature. So for predicting the 28-days compressive strength of concrete at any temperature, a proper choice of the input variables could be the mix proportions (The amount of different components, such as water, cement, fine and coarse aggregates, and admixtures), type of concrete specimen and temperature.

An alternative choice of the input variables could be the compressive strength at ordinary temperature along with the temperature. The latter parsimonious combination of the input variables may be useful especially where the mix design is not given.

The both mentioned combinations of the input variables are investigated in this study and the structures of the input-output of the models are schematically shown in Fig. 2. In this figure, for the set 1, the input variables are cement (C), water (W), fine aggregates (FA), coarse aggregates (CA), silica fume (SF), Nano silica (NS), fly ash (F), superplasticizer (SP), type of concrete specimen (TS), and temperature (T); and for the set 2, they includes compressive strength at ordinary temperature (F_c,0) and temperature (T). For both sets, the output variable is compressive strength of concrete at the temperature T (F_c,T).

Table 1 presents the ranges of input and output of the total data used for this study. As the type of concrete specimen is a qualitative input, a transform is necessary into quantitative value. The data collected in this study include five different types of specimens (two sizes of cubic and three sizes of cylindrical molds). This study assigns an integer value of 1 to 5 for each type of specimen.

The models application

As stated earlier, training data set is used to construct the data driven models. The optimal parameters of the data driven models were found by minimizing Mean Square Error (MSE) function on the training data. In addition to the parameters, there are various training options to achieve satisfying results for each model except the MLR model. The options like the number of neurons in the hidden layer and the type of transfer functions in the hidden and output layers, in the ANN model; the identification method, the number of rules and the type of fuzzy membership functions, in the ANFIS model; and the number of nearest neighbors and the type of kernel function in the KNN model were determined when the final configuration of each model outperformed other tested configurations in terms of accuracy. Tables 2, 3, and 4 show the property types and their values considered in the final configuration of the ANN, ANFIS, and KNN models, respectively for both combinations of the input variables.

Performance criteria

In this study, the coefficient of determination (R^{\mathrm{2}}), Mean Absolute Percentage Error (MAPE) and Relative Root Mean Square Error (RRMSE) as more common criteria in the literature are used for evaluating data driven modeling results.

(9)R^{\mathrm{2}}=\mathrm{1}-{\displaystyle \frac{{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{(y_i-{\widehat{y}}_i)}^{\mathrm{2}}}}{{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{(y_i-\overline{y})}^{\mathrm{2}}}}},

(10)MAPE\hspace{0.17em}={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|}\hspace{0.17em}\times \mathrm{100}\%\hspace{0.17em}\hspace{0.17em},

(11)RRMSE\hspace{0.17em}=\sqrt{{\displaystyle \frac{{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}(y_i}-{\widehat{y}}_i{)}^{\mathrm{2}}}{n{(\overline{y})}^{\mathrm{2}}}}}\times \mathrm{100}\%.

In the equations, n is the number of data, y_i and {\widehat{y}}_i are the actual and predicted output of ith sample of the data, respectively, \bar{y} and \bar{\widehat{y}} are the averaged actual and predicted outputs of the data.

The R^{\mathrm{2}}coefficient ranges from 0 to 1 and the model with higher quantity of R^{\mathrm{2}} has more efficiency. Two other criteria measure percentage error of the model in two different forms which range from 0 to 100. It is obvious the less values of these two criteria, the better the model.

Results and discussions

Set 1 of the input variables

The results of the models are depicted in Table 5 for the set 1 of the input variables. The table shows the coefficient of determination (R^{\mathrm{2}}) between the results obtained from the each model and experimentally actual data, together with mean absolute percentage error (MAPE) and the relative root mean square error (RRMSE). From the overall observation of the table, it can be seen the performance of the ANN, ANFIS, and KNN models in the training and testing data are relatively satisfactory. But, the results of the MLR are not successful and clearly show that the relationship between the variables of the model is not linear.

Moreover, the comparison of R^{\mathrm{2}}, MAPE, and RRMSE revealed that the ANN model has higher prediction ability than other successful models. For the testing data sets, the coefficient of determination (R^{\mathrm{2}}) was higher for ANN compared to those of ANFIS and KNN (0.92 versus 0.89 and 0.84, respectively). The better performance of ANN can also be observed since MAPE, and RRMSE values are lower for ANN than those of ANFIS and KNN. It is worth to note that MAPE, and RRMSE values were as low as 12% and 12%, respectively. This superiority may be arisen from this fact the nonlinear relations involved between the mix proportions of concrete and temperature with the compressive strength can be captured better by the ANN model.

Set 2 of the input variables

The results of the models including the performance criteria R^{\mathrm{2}} together with MAPE and RRMSE are presented in Table 6 for the set 2 of the input variables. In addition to the results of four data driven models implemented in this research, the results of NIST Technical Note 1681 are also included in Table 6. Comparison of the results indicates that the ANFIS model has a higher value of R^{\mathrm{2}}and a lower value of MAPE and RRMSE, so that its performance is more accurate. Comparison of the results implies that R^{\mathrm{2}}is higher for ANFIS than KNN as the second best model, showing a better prediction by ANFIS than by KNN. The error indexes appeared in Table 6 are also slightly lower for ANFIS.

Although the results obtained by MLR model was not successful as well as other data driven models, but the efficiency of the model is relatively close to that of other data driven models. The R^{\mathrm{2}}value of the MLR model is 0.77 which is just only 0.11 less than the R^{\mathrm{2}}value of the best model for testing data set. Also the MAPE and RRMSE values of the MLR model are near to those of other data driven models. With respect to the simple applicability of the MLR model, relatively high efficiency of the model may be valuable. By applying the MLR model on the training data set, the following equation was obtained:

(12){{\displaystyle F}}_{\mathrm{c},T}=\mathrm{0.65}{{\displaystyle F}}_{\mathrm{c},\mathrm{0}}-\mathrm{0.06}T+\mathrm{28.47.}

Equation (12) can be especially useful for practical applications. As the MLR model and NIST equations are based on the linear relations, a comparison between the results of them may be interesting. The R^{\mathrm{2}}value of the MLR model is considerably higher than R^{\mathrm{2}}of the NIST equations. Also other performance criteria of the MLR model are better than those of the NIST equations. To draw a better perspective of the performance of the MLR model and NIST equations, scatter plots between the experimental and computed outputs by two tools over the testing data are shown in Fig. 3. The equality (diagonal) line is also included in Fig. 3. As it can be seen from the figure, congregation of data points around near the equality line is slightly more for the MLR as compared to the NIST which indicates the MLR model is more effective than NIST equations.

Comparison of the results of set 1 and set 2 of the input variables

As stated earlier, the best models are ANN and ANFIS for set 1 and set 2 of the input variables, respectively. Scatter plots between the experimental and computed outputs by the ANN and ANFIS models over the testing data are shown in Figs. 4 and 5, respectively. From the figures, it is found that the models show a relatively good match between the actual experimental and the predicted output values of the testing data and representative points are generally closely congregated to the line of equality. The equality (diagonal) lines are also included in Figs. 4–5. The distribution of the data points in Figs. 4–5 above and below the equality lines is almost uniform for each figure. This implies that models have no dominant bias in which neither overestimate nor underestimate experimental data.

A comparison between the results of appeared in Tables 5 and 6 shows for ANN, ANFIS, and KNN, the set 1 of input variables with ten input variables gives more accurate results as compared to those of the set 2 with two input variables. Specifically, the R^{\mathrm{2}} values for set 1 are 0.11, 0.01, and 0.01 more than those of set 2 for ANN, ANFIS, and KNN, respectively. The deterioration of the results from the set 1 to set 2 is the most value for ANN model where the MAPE and RRMSE values also increased 13% and 7%, respectively. It seems that ANN model have more capability where the number of input variables is greater. With respect to the more parsimonious set of input variables used in the set 2, sacrificing a little more accuracy could be justifiable. This allows more compact models and less computational burden. In the set of MLR model, the results are adverse in which the performance of model for set 2 is higher than that of set 2. It seems the nonlinearity involved between the mix proportions of concrete imbedded at compressive strength as the input variable of the linear regression model and resulted in better outputs.

Conclusions

From the data collected from a review of 207 experimental data, this study used four data driven models, namely MLR, ANN, ANFIS, and KNN models for compressive strength prediction of concrete at high temperature. For each model, two sets of input variables were used: a complete set including the mix proportions, type of concrete specimen and concrete temperature; a parsimonious set including the compressive strength at ordinary temperature along with the concrete temperature. The following conclusions can be drawn from this research study:

1) ANN, ANFIS, and KNN models had adequately extrapolation capabilities in predicting compressive strength of concrete at high temperature in both sets of input variables combinations.

2) Although, there was no significant difference between the results of the ANN, ANFIS, and KNN models, the ANN model was superior to other models in the set 1 while the ANFIS model was the best model for the set 2. It seems that ANN model have more capability where the number of input variables is high.

3) KNN model was ranked as the third best model for set 1 after ANN and ANFIS models, and the second best model for set 2 after ANFIS model. In contrast to ANN and ANFIS models, KNN model is a nonparametric model which does not require any learning effort which makes it simple in terms of implementation.

4) The capability of the regression model was low in predicting the compressive strength in the set 1 where the mix proportions along with temperature were as the input variables. However, the results were acceptable for the set 2 of input variables. It seems the nonlinearity involved between the mix proportions of concrete imbedded at compressive strength as the input variable of the linear regression model in the set 2, resulted in better outputs. A tradeoff between accuracy and simplicity may lead to using this model, especially in practical applications.

5) Employing the set 2 of input variables combination including two input variables (compressive strength at ordinary temperature and concrete temperature) is sufficient for the data driven models to make satisfactory compressive strength prediction of concrete at high temperature. In fact, in spite of sacrificing a little more accuracy, use of a more parsimonious set of input variables could be really justifiable. This can be useful for practical and engineering applications.

6) The results obtained from this study were compared to the equations of NIST Technical Note 1681. These comparisons showed that the equations of NIST do not closely agree with the experimental results, which demonstrates the suitability of applying data driven models such as ANN, ANFIS, KNN, and even MLR, which have shown to achieve good agreement with experimental results.

As a final point of this research, it should be recognized that the data driven models were used to predict the compressive strength of concrete expose to high temperature without dealing with complete conceptual knowledge of the physical behavior of concrete exposed to high temperature. In contrast, there are some studies in the literature in which the numerical analysis of concrete exposed to elevated temperature was investigated [¹⁴,¹⁵].The data driven models are easier while the numerical models are more understandable. Use of combination of simplicity of the data driven models and the conceptuality of the numerical models may lead to developing the more efficient models in the future.

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