1 Introduction

Self-compacting concrete (SCC) was first developed in the year 1988, in Japan. Nowadays, SCC is one of the most efficient concrete mixes in the world. Self-compacting concrete is a type of concrete that can flow and fill the formwork without any external forces. Also, it has the ability to consolidate by its own weight. The design mix is mainly focused on two main criteria, namely the requirement of a large amount of finer particle and necessity of high-performing water-reducing admixture. Compared to other traditional concrete, it requires comparatively less human effort which is an added advantage. Also, it increases production rate and reduces noise disturbances. Lot of research studies are being carried out in the area of SCC technology; some of these researches propose better enhancement and durability. Besides all these advantages, SCC has some negative sides. The cost of production of SCC could be 2–3 times higher than ordinary concrete. Therefore, to minimize the cost, some different admixtures such as limestone filler, fly ash (FA), metakaolin, ground-granulated blast-furnace slag (GGBS) and ground clay bricks can be utilized. These ingredients which are used acts as an appropriate substitute for Portland cement [1,2,3,4,5,6,7]. To develop SCC three different criteria needed to be fulfilled, such as filling ability, passing ability, segregation resistance. So to meet out these requirements, several trial experiments need to be performed. Lot of questions arises whether SCC is economical and cost-effective. Another main hurdle is that it is time-consuming to determine the optimum mix. To overcome these limitations, the researchers use a different optimization technique to predict the fresh and hardened properties of concrete.

Sarıdemir [8] developed an artificial neural network (ANN) model to predict compressive strength of concrete containing silica fume and metakaolin. Similar to this, Bilim et al. [9] also proposed a prediction model to estimate the compressive strength of ground-granulated blast-furnace slag concrete. Ozcan et al. [10] performed a comparative study regarding two optimization techniques, namely ANN and fuzzy logic, to forecast compressive strength of silica fume concrete. For similar applications, some of the other researchers also proposed suitable models inspired by Fuzzy and adaptive neuro-fuzzy inference system (ANFIS) to calculate the compressive strength [11,12,13,14,15,16]. Besides all the advantages of ANN, it has some limitations too, such as poor generalizing performance, slow convergence speed and over-fitting problems. Moreover, to determine the number of hidden layers, there is no specific rule. To overcome these limitations, a new approach, namely support vector machine (SVM) was developed. It renovates and boosts the generalization performance to attain global minimum. Some of the applications using support vector machine technique have been discussed below.

Recently, Sonebi et al. [17] investigated the fresh properties of self-compacting concrete using support vector machine approach. The result was positive and encouraging, which shows better filling ability, flow ability and passing ability. Similar research work was also carried out by different researchers to predict compressive strength of concrete using support vector regression [18,19,20] and concluded that SVC would be a better and effective model for the forecasting compressive strength of all grades of concrete. Similar to above-mentioned studies, a numerical analysis was carried out by Yan et al. [21] to foreshow elastic modulus for the normal and high strength of concrete using SVM model. Naseri et al. [22] carried out an experimental analysis on SVM-based prediction technique to determine the hardened properties of self-compacting concrete using polypropylene fibre and nano-CuO. Liu [23] discussed the feasibility of using SVM model to determine autogenous shrinkage of concrete mixtures. The SVM model was compared with ANN model to compare its accuracy and efficacy, and the outcomes proved that SVM has the better predicting capability. In another study, Sobhani [24] commented the effectiveness of SVM model to prefigure the strength of no-slump concrete and finally results being contrasted with ANN in his studies. Dong et al. [25] developed a tool to predict acceleration response of nonlinear structure by using support vector machine. From the test results, it was verified that SVM-based model provides better remarkable performance for forecasting and simulation. Amir Saber Mahania et al. [26] employed two optimization techniques, namely particle swarm optimization (PSO) and ant colony optimization, thereafter it was hybridized and compared with weighted least squares support vector machine to determine the optimal shape of the double arch dam. Correspondingly, Yang [27] performed an experimental investigation on corroded reinforced concrete. The experimental result was examined with SVM output. It was noticed that SVM exhibits superior prediction efficiency. SVM can be utilized for wide applications in the field of civil engineering, namely nonlinear structural identification, flood stage forecasting, flood forecasting [28], fracture characteristics of concrete [29], downscaling of precipitation [30], soil improvement, forecasted stream flow and reservoir inflow [31], prediction of groundwater level (GWL) fluctuations [32].

However, most of the research studies on concrete are limited only on predicting the hardened concrete properties. It is also observed from the literature survey that there are no such articles available focusing on prediction of both fresh and hardened properties of self-compacting concrete using support vector regression (SVR) method. Therefore, an effort has been made to predict both fresh and hardened properties of SCC using SVR technique. Also, the efficiency of SVR model is compared to artificial neural network (ANN) and multivariable regression analysis (MVR). An experimental database of 115 samples was collected from the various literatures to develop the SVR model. This study is mainly concentrated on estimating the fresh properties (L-box test, V-funnel test, slump flow)and hardened properties (compressive strength) of SCC from binder content, fly ash, water–powder ratio, fine aggregate, coarse aggregate and superplasticizer as input parameters.

2 Data collection

The main objective of this study is to predict the output parameters related to fresh and hardened properties of SCC. Most of the previous research works are predominately limited to evaluating a single output characteristic of concrete by considering a large number of input variables. Therefore, in this study four output parameters are considered for prediction. The datasets of 115 SCC concrete mix proportions considered for modelling the SVR are presented in Appendix.

The SVR model is designed with six input parameters, namely binder content (B), fly ash percentage (P), water–powder ratio (W/B), fine aggregate (F), coarse aggregate (C) and super plasticizer dosage (SP). In the previous studies, modelling was concentrated only on developing the database from the experimental work; therefore, their database is restricted to a particular variable. In this present study, data are collected from diverse and distinctive source; therefore, it can be applied over to wide applications. In the present study, out of 115 experimental data points, 80% of the data are used to train the SVR model, and the remaining 20% of the data are used to test the model [33]. Statistical parameters of input and output variables used to develop SVR models are enlisted in Table 1.

Table 1 Statistical parameters of input and output variable
Full size table

3 Support vector machine

The support vector machine was first introduced by Vapnik et al. [34]. During the initial stage, it was limited to solve the classification related problems, later it was upgraded to solve regression-related problems also. SVM regression follows the principle of structural risk minimization (SRM) which is much more superior compared to conventional Empirical Risk Minimization (ERM) principle. SRM principle is employed to decline the upper bound generalization error which is very crucial for any statistical learning process. Support vector regression (SVR) is an extension of SVM to solve the prediction- and regression-related problems. Both SVR and SVM use very similar algorithms, but predict different types of variables. The main difference comes in the slack variables used in the two techniques. SVM for classification involves assigning one slack variable to each training data point, whereas in SVM for regression, there are two slack variables for each training data point. In addition to that the main characteristic that differentiates SVR from other model is its ability to improvise generalization performance and to obtain an optimal global solution at the minimum time period. In this paper, Vapnik’s ε-insensitive loss function has been used to solve nonlinear regression estimation, and the brief description of SVM regression can be found in Ref. [35, 36].

4 Linear support vector regression

Considering a training dataset {(xi, yi), i = 1, 2, 3 … n}, where n is the size of training dataset, xi is the input vector and yi is the output vector, respectively. The general linear regression form of SVR can be written as

$$f\left( {{\mathbf{x}} \cdot {\mathbf{w}}} \right) = {\mathbf{w}} \cdot {\mathbf{x}} + b$$
(1)

where \(\left( {{\mathbf{w}} \cdot {\mathbf{x}}} \right)\) indicates the dot product, w is the weight vector, b is the bias and x is the test pattern in normalized form. The SRM theory can be performed by reducing empirical risk Remp(w, b) described as equation, and empirical risk can be defined by using ε-insensitive loss function \(L _{\epsilon } (y_{i} ,f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}})} \right)\) indicate as Eq. (3) [34]

$$R_{emp} \left( {{\mathbf{w}},{\mathbf{b}}} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} L_{\epsilon } (y_{i,} f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}}} \right))$$
(2)
$$L _{\epsilon } \left( {y_{i} ,f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}}} \right)} \right) = \left\{ {\begin{array}{*{20}l} {\epsilon , \quad if\,\, \left| {y_{i} - f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}}} \right)} \right| \le \epsilon } \hfill \\ {\left| {y_{i} - f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}})} \right)} \right| - \epsilon ,\, otherwise } \hfill \\ \end{array} } \right.$$
(3)

\(L _{\epsilon } \left( {y_{i} ,f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}}} \right)} \right)\) is the ε-insensitive loss function, or the tolerance error between the target output (yi) and the estimated output values \(f\left( {{\mathbf{x}}_{\varvec{i}} ,{\mathbf{w}}} \right)\) in optimization process, and \({\mathbf{x}}_{\varvec{i}} \varvec{ }\) is defined as training pattern. By using ε-insensitive loss function in linear regression problem, the complexity of SVR model can be solved by minimizing \(||w||\)2. The deviation of training data outside the \(\epsilon\)-zone can be estimated by using non-negative slack variable \(\left( { \xi_{i}^{*} \xi_{i} } \right)\).

$$\mathop {\lim }\limits_{{w,b,\xi ,\xi^{*} }} \left[ {\frac{1}{2}{\mathbf{w}} \cdot {\mathbf{w}} + C\left( {\mathop \sum \limits_{i = 1}^{n} \xi_{i}^{*} + \mathop \sum \limits_{i = 1}^{n} \xi_{i} } \right)} \right]$$
(4)
$${\text{Subjected to,}}\;\left\{ {\begin{array}{*{20}l} {y_{i} - w \cdot {\mathbf{x}}_{\text{i}} - b \le \epsilon + \xi_{i}^{*} } \hfill \\ {w \cdot {\mathbf{x}}_{\text{i}} + b - y_{i} \le \epsilon + \xi_{i,} } \hfill \\ {\xi_{i}^{*} , \xi_{i} \ge 0} \hfill \\ \end{array} } \right.\quad i = 1, \ldots ,n$$

To solve the above-mentioned problem, a saddle point of Lagrange function has to be found

$$\begin{aligned} & L({\mathbf{w}}, \xi^{*} ,\xi , \alpha^{*} ,\alpha ,C,\gamma^{*} ,\gamma \\ & = \frac{1}{2}{\mathbf{w}} \cdot {\mathbf{w}} + c\left( {\mathop \sum \limits_{i = 1}^{n} \xi_{i}^{*} + \mathop \sum \limits_{i = 1}^{n} \xi_{i} } \right) - \mathop \sum \limits_{i = 1}^{n} \alpha_{i} \left[ {y_{i} - {\mathbf{w}} \cdot {\mathbf{x}}_{i} - b + \epsilon + \xi_{i} } \right] \\ &\quad-\, \mathop \sum \limits_{i = 1}^{n} \alpha_{i}^{*} \left[ {{\mathbf{w}} \cdot {\mathbf{x}}_{i} + b - y_{i} + \epsilon + \xi_{i}^{*} } \right] \\ & \quad - \mathop \sum \limits_{i}^{n} (\gamma_{i}^{*} \xi_{i}^{*} + \gamma_{i} \xi_{i} ) \\ \end{aligned}$$
(5)

By performing partial differential of Eq. (5) with respect to \({\mathbf{w}}\), \(b\), \(\xi_{i}^{*} {\text{and }}\xi_{i}\), Lagrange function can be minimized by applying Karush–Kuhn–Tucke (KKT) conditions

$$\frac{\partial L}{\partial w} = {\mathbf{w}} + \mathop \sum \limits_{i = 1}^{n} \alpha_{i} {\mathbf{x}}_{i} - \mathop \sum \limits_{i = 1}^{n} \alpha_{i}^{*} {\mathbf{x}}_{i} = 0; {\mathbf{w}} = \mathop \sum \limits_{i = 1}^{n} (\alpha_{i}^{*} - \alpha_{i} ){\mathbf{x}}_{i}$$
(6a)
$$\frac{\partial L}{\partial b} = \mathop \sum \limits_{i = 1}^{n} \alpha_{i} - \mathop \sum \limits_{i = 1}^{n} \alpha_{i}^{*} = 0; \mathop \sum \limits_{i = 1}^{n} \alpha_{i} = \mathop \sum \limits_{i = 1}^{n} \alpha_{i}^{*}$$
(6b)
$$\frac{\partial L}{{\partial \xi^{*} }} = C - \mathop \sum \limits_{i = 1}^{n} \gamma_{i}^{*} - \mathop \sum \limits_{i = 1}^{n} \alpha_{i}^{*} = 0; \mathop \sum \limits_{i = 1}^{n} \gamma_{i}^{*} = C - \mathop \sum \limits_{i = 1}^{n} \alpha_{i}^{*}$$
(6c)
$$\frac{\partial L}{\partial \xi } = C - \mathop \sum \limits_{i = 1}^{n} \gamma_{i} - \mathop \sum \limits_{i = 1}^{n} \alpha_{i} = 0; \mathop \sum \limits_{i = 1}^{n} \gamma_{i} = C - \mathop \sum \limits_{i = 1}^{n} \alpha_{i}$$
(6d)

where the parameter w of Eq. (6a) is related to parameter w of Eq. (1). After that, putting Eq. (6) into the Lagrange function (5), the dual optimization function can be expressed as

$$\mathop {\hbox{max} }\limits_{{\alpha ,\alpha^{ *} }} \left[ {{\mathbf{w}}\left( {{\varvec{\upalpha}},{\varvec{\upalpha}}^{*} } \right)} \right] = \mathop {\hbox{max} }\limits_{{\alpha ,\alpha^{ *} }} \left[ {\mathop \sum \limits_{i = 1}^{n} y_{i} \left( {\alpha_{i}^{*} - \alpha_{i} } \right) - \epsilon \mathop \sum \limits_{i = 1}^{n} (\alpha_{i}^{*} - \alpha_{i} ) - \frac{1}{2}\mathop \sum \limits_{ij = 1}^{n} (\alpha_{i}^{*} - \alpha_{i} )\left( {\alpha_{i}^{*} - \alpha_{i} } \right)\left( {{\mathbf{x}}_{i} \cdot {\mathbf{x}}_{j} } \right)} \right]$$
(7)
$${\text{subjected to}}\left\{ {\begin{array}{*{20}c} {\mathop \sum \limits_{i = 1}^{n} \left( {\alpha_{i}^{*} - \alpha_{i} } \right) = 0,} \\ {0 \le \alpha_{i}^{*} , \alpha_{i} \le 0 } \\ \end{array} } \right.\quad i = 1, \ldots \ldots .n$$

where \(\alpha_{i}^{*}\) and \(\alpha_{i}\) are defined as Lagrange multiplier [37]. After Solving Eq. (7) with constrains in Eq. (8), the final linear regression function can be expressed as

$$f\left( {{\mathbf{x}},{\varvec{\upalpha}}^{*} , {\varvec{\upalpha}}} \right) = \mathop \sum \limits_{i = 1}^{n} \left( {a_{i}^{*} - a_{i} } \right)\left( {{\mathbf{x}}_{i} \cdot {\mathbf{x}}} \right) + b$$
(8)

5 Nonlinear support vector regression

To solve a complex real-world problem, the linear SVR is not suitable. To perform nonlinear SVR, mapping of input data into high-dimensional feature space is required where linear regression is possible. The input training pattern \({\mathbf{x}}_{i}\) is reformed into feature space \(\varphi \left( {{\mathbf{x}}_{i} } \right)\) by a nonlinear function. After that, the optimization algorithm is applied in the same way as linear SVR. Correspondingly, the nonlinear support vector regression can be expressed as follows

$$f\left( {{\mathbf{x}},{\mathbf{w}}} \right) = {\mathbf{w}} \cdot \varphi \left( {\mathbf{x}} \right) + b$$
(9)

where w and b denote parameter vector and \(\varphi \left( {\mathbf{x}} \right)\) is used as mapping function from input features to a high-dimensional feature space.

The diagram of nonlinear support vector regression with ε-insensitive loss function is shown in Fig. 1. In the figure, the support vectors are marked with bold points, which have the largest difference from the decision boundary. On the right-hand side of the diagram indicates ε-insensitive loss function, it has an error tolerance ε, upper bound and lower bound are calculated by slack variables \(\left( {\xi_{i}^{*} ,\xi_{i} } \right)\) . Finally, nonlinear support vector regression can be expressed as

Fig. 1
figure 1

Nonlinear SVR with insensitive loss function

Full size image
$$\mathop {\hbox{max} }\limits_{{\alpha ,\alpha^{ *} }} \left[ {{\mathbf{w}}\left( {{\varvec{\upalpha}},{\varvec{\upalpha}}^{*} } \right)} \right] = \mathop {\hbox{max} }\limits_{{\alpha ,\alpha^{ *} }} \left[ {\mathop \sum \limits_{i = 1}^{n} y_{i} \left( {\alpha_{i}^{*} - \alpha_{i} } \right) - \epsilon \mathop \sum \limits_{i = 1}^{n} (\alpha_{i}^{*} + \alpha_{i} ) - \frac{1}{2}\mathop \sum \limits_{ij = 1}^{n} (\alpha_{i}^{*} - \alpha_{i} )\left( {\alpha_{i}^{*} - \alpha_{i} } \right)\left\{ {{\mathbf{\varphi }}\left( {{\mathbf{x}}_{i} } \right) \cdot {\mathbf{\varphi }}\left( {{\mathbf{x}}_{j} } \right)} \right\}} \right]$$
(10)
$${\text{subjected to}}\left\{ {\begin{array}{*{20}l} {\mathop \sum \limits_{i = 1}^{n} \left( {\alpha_{i}^{*} - \alpha_{i} } \right) = 0,} \hfill \\ {0 \le \alpha_{i}^{*} , \alpha_{i} \le 0 } \hfill \\ \end{array} } \right.\quad i = 1, \ldots n$$

As the inner product \({\mathbf{\varphi }}\left( {{\mathbf{x}}_{i} } \right) \cdot {\mathbf{\varphi }}\left( {{\mathbf{x}}_{j} } \right)\) is complex, by using Mercer’s condition [38], the inner product can be replaced by using kernel function \({\mathbf{\varphi }}\left( {{\mathbf{x}}_{i} } \right) \cdot {\mathbf{\varphi }}\left( {{\mathbf{x}}_{j} } \right) = {\mathbf{K}}\left( {{\mathbf{x}}_{i} , {\mathbf{x}}_{j} } \right)\).Therefore, Eq. (11) can be written as

$$\mathop { \hbox{max} }\limits_{{\alpha ,\alpha^{ *} }} \left[ {{\mathbf{w}}\left( {{\varvec{\upalpha}},{\varvec{\upalpha}}^{*} } \right)} \right] = \mathop {\hbox{max} }\limits_{{\alpha ,\alpha^{ *} }} \left[ {\mathop \sum \limits_{i = 1}^{n} y_{i} \left( {\alpha_{i}^{*} - \alpha_{i} } \right) - \epsilon \mathop \sum \limits_{i = 1}^{n} (\alpha_{i}^{*} + \alpha_{i} ) - \frac{1}{2}\mathop \sum \limits_{ij = 1}^{n} (\alpha_{i}^{*} - \alpha_{i} )\left( {\alpha_{i}^{*} - \alpha_{i} } \right){\mathbf{K}}\left( {{\mathbf{x}}_{i} , {\mathbf{x}}_{j} } \right)\} } \right]$$
(11)
$${\text{subjected to}}\left\{ {\begin{array}{*{20}c} {\mathop \sum \limits_{i = 1}^{n} \left( {\alpha_{i}^{*} - \alpha_{i} } \right) = 0,} \\ {0 \le \alpha_{i}^{*} , \alpha_{i} \le 0 } \\ \end{array} } \right. \quad i = 1, \ldots n$$

For nonlinear regression, numerous kernel functions have been discussed in the above-mentioned literature. The most commonly used kernel function and their corresponding equations are presented in Table 2. However, in the present study, radial basis and exponential radial basis kernel function have been used. SVR model has been implemented in MATLAB 2013a environment with a support vector machine code [36]. Parameters σ and d are user-defined kernel function. By trial and error, the optimal values of C, ε and kernel parameters are determined [39].

Table 2 Different types of kernel function
Full size table

6 Predictive model development

6.1 SVR models development

In order to predict the fresh (L-box ratio, V-funnel, slump flow) and hardened properties of concrete (compressive strength of concrete), nonlinear regression technique was developed through SVR toolbox. Experimental studies indicate that properties of concrete are mostly influenced by binder content (B), fly ash (P), water–powder ratio (W/B), fine aggregate (F), coarse aggregate (C) and superplasticiser dose (SP). Thus, these above-mentioned parameters are taken as an input variable for SVR model, and slump flow (D), L-box ratio, V-funnel, compressive strength (Fc28) are taken as the output variable. These input parameters have different units; hence, the data have to be normalized. After the normalization, the data values lie between the range 0 and 1. To develop the SVR algorithm, several parameters are needed to be determined, namely kernel specific parameters d and σ, error insensitive loss function ε and penalty parameters C. However, the choice of ε and C can be estimated by following some guidelines. The large value of penalty parameter C indicates to minimize the empirical risk which makes the learning algorithm more complicated, on the other hand, the smaller value of penalty parameter C indicates minimization of error within the margin, and this allows learning algorithm with a poor approximation.

The complexity or smoothness of the approximation is influenced by ε parameter. Moreover, it also decides the number of support vector used in final prediction operation. So in order to develop regression function, a smaller value of ε leads to complexities in the learning process having a large volume of support vector, still it would be effective in prediction. On the other hand, the greater value of ε may lead to less number of support vector, resulting in data destruction, which creates flattening in the regression function. However, in the present study, SVR modelling kernel parameter σ and d, loss function parameter ε and penalty parameter C are finalized by trial and error method.

6.2 ANN model development

ANN has been used in solving various civil engineering-related problems. For the sake of conciseness, it is restrained to the short discussion of ANN in the present study and can be found in the literature. The input parameter of ANN are considered as dosage of binder content, fly ash, water–powder ratio, fine aggregate, coarse aggregate and superplasticiser, whereas, slump flow value, L-box ratio, V-funnel time and compressive strength have been considered as output variables. The ANN modelling was implemented in MATLAB 2013 software with neural network toolbox. Total 80% experimental data were used for training, and remaining 20% data used for testing the trained model. A feed-forward multilayer perceptron neural network with one hidden layer was adopted. the number of neurons in hidden layer was varied to find the optimum architecture. Optimum architecture of ANN model was characterized by the number of neurons in hidden layer with tan-sigmoid (hyperbolic tangent) transfer function and a pure linear transfer function at output layer. Bayesian regularization back-propagation training algorithm is used for its better generalization to the training data.

6.3 MVR model development

In the present study for the prediction of slump flow value, L-box ratio, V-funnel time and compressive strength of concrete, multivariable regression analysis is also conducted. Here, 80% data are used to develop the MVR model as it was used in ANN model, and rest of 20% data is used to predict the efficiency of the model. A relationship between dependable variable and independable is shown by the following equation:

$$Y = a_{0} + a_{1} x_{1} + a_{2} x_{2} + \ldots + a_{p} x_{p} \pm e$$

where Y is the dependent variable, \(a_{0}\) is an intercept. \(a_{1}\), \(a_{2}\) and \(a_{p}\) are the slopes \(x_{1} ,x_{2}\) and \(x _{p}\) are independent variables and e is the error. ‘‘a” values are obtained via least square optimization of error.

7 Result and discussion

All models, i.e. SVR, ANN, MVR, has been designed from a dataset of 115 SCC mix proportions. The dosages of binder content, water–powder ratio, fly ash percentage, volume of fine aggregate, volume of coarse aggregate and superplasticiser were varied from 370 to 733 kg/m3, 0.26 to 0.45, 0 to 60%, 656 to 1038 kg/m3, 590 to 935 kg/m3 and 0.74 to 21.84 kg/m3 respectively. To develop the SVR model, two different kernel function, namely, exponential radial basis kernel and radial basis kernel were considered. Different combinations of d, σ, C and ε were tried on training dataset, and for each combination of these parameters, performance of testing dataset was recorded. Table 3 shows SVR model performance for slump flow prediction with varying C for a constant kernel value and C. Effect of variation in σ on model performance for a constant kernel value and ε is presented in Table 3, and ε variation in models on performance for a constant kernel value and C is presented in Table 3. But due to space limitation, variation in kernel function on the performance of slump flow prediction is only presented here. The dataset of optimum kernel functions of the SVR model for different output parameters is detailed in Table 4.

Table 3 (a) SVR model performance with varying C for a constant kernel value and ε. (b) SVR model performance with varying σ for a constant kernel value and ε. (c) SVR model performance with varying ε for a constant kernel value and C
Full size table
Table 4 Value of support vector regression parameters
Full size table

To evaluate the performance of SVR, ANN, MVR model in predicting the response, different statistical parameters were used, namely coefficient of determination (R2), mean absolute deviation (MAD), mean square error (MSE), root mean square error (RMSE) and mean absolute percentage error (MAPE). Statistical performance of developed SVR, ANN and MVR model is summarized in Table 5. For predicting the fresh and hardened properties of SCC, exponential radial basis function shows higher degree of accuracy than other prediction model. Due to space limitation, only the best prediction model is discussed.

Table 5 Statistical errors of proposed SVR models
Full size table

The slump flow prediction for SVR–ERBF model is presented in Fig. 2. In the Figure, it is clearly demonstrated that all data points are lying within 90% of prediction interval, which confirms that SVR–ERBF model can efficiently use to predict the slump flow. A coefficient of determination 0.931 (MAD = 9.136, MSE = 136.370, RMSE = 11.678 and MAPE = 1.458) was obtained. And also, it suggest that SVR–ERBF model provides better result in slump flow prediction compared to SVR–RBF model (R2 = 0.590, MAD = 23.040, MSE = 889.769, RMSE = 29.829 and MAPE = 3.494).

Fig. 2
figure 2

Experimental versus predicted of slump flow (mm) using SVR–ERBF

Full size image

The L-box prediction for SVR–ERBF model is shown in Fig. 3. Similarly, for the prediction of L-box ratio, all the predicted data points are lying within 90% interval and coefficient of determination of 0.910 (MAD = 0.018, MSE = 0.001, RMSE = 0.025 and MAPE = 2.105) was achieved for SVR–ERBF model while SVR–RBF model shows lower performance (R2 = 0.595, MAD = 0.037, MSE = 0.003, RMSE = 0.057 and MAPE = 4.187).

Fig. 3
figure 3

Experimental versus predicted of L-box using SVR–ERBF

Full size image

Figure 4 presents the plot of experimental versus predicted values of V-funnel test. For SVR–ERBF model, all the data points are lying within 90% interval with coefficient of determination 0.958 (MAD = 0.488, MSE = 0.523, RMSE = 0.723 and MAPE = 9.381). The SVR–RBF model-based approach shows lower performance (R2 = 0.595, MAD = 1.402, MSE = 5.581, RMSE = 2.362 and MAPE = 29.437).

Fig. 4
figure 4

Experimental versus predicted of V-funnel using SVR–ERBF

Full size image

The same SVR model was used to predict compressive strength of concrete. After the training process, the model was used to predict the compressive strength. Figure 5 shows the experimental versus predicted compressive strength values with SVR–ERBF model. It can be observed that most of the data points are lying with its bound and its correlation of determination equal to 0.955 (MAD = 2.939, MSE = 14.312, RMSE = 3.783 and MAPE = 6.419) achieved with SVR–ERBF model. Similarly, from the SVR–RBF model, lower performance (MAD = 11.289, MSE = 228.700, RMSE = 15.123 and MAPE = 23.461) is observed.

Fig. 5
figure 5

Experimental versus predicted of compressive strength using SVR–ERBF

Full size image

7.1 Sensitivity analysis results and discussion

A sensitive analysis has been also carried out to obtain the effect the input parameters on output. Therefore, to estimate the effect of input parameters, a particular input parameter is varied within its range with the other parameters being fixed at reference value. Similar procedure is applied for other input parameters to investigate the effect of input parameters on output value. Therefore, the input parameters, i.e. the amount of cement, fly ash, water–powder ratio, fine aggregate, coarse aggregate and Superplasticiser, were varied from 370 to 550 kg/m3, 0 to 30%, 656 to 846 kg/m3, 676 to 848 kg/m3, 600 to 1000 kg/m3 and 0.74 to 11.24 kg/m3, respectively.

From Fig. 6, it is clear that the higher concentration of cement, fly ash and fine aggregate content absorbs water causing declination of slump flow diameter. However, increasing the amount of coarse aggregate, water–powder proportion and superplasticiser percentage results in the improvement of slump flow diameter.

Fig. 6
figure 6

Sensitivity of mix composition versus slump flow

Full size image

From Fig. 7, it can be observed that the increasing water–powder ratio, superplasticiser, fine aggregate, coarse aggregate increases L-box ratio, which reflects better passing ability of SCC. Similarly, an increase in the dosage of powder and fly ash reduces the L-box ratio.

Fig. 7
figure 7

Sensitivity of mix composition versus L-box ratio

Full size image

Similarly, to predict the V-funnel time, the same approach has been adopted. Influence of mix composition in predicted V-funnel value is presented in Fig. 8, and it indicates that increase in the amount of water–powder ratio, cement, fly ash, fine aggregate and superplasticiser dosage reduces V-funnel time. Likewise, increase in coarse content increases the V-funnel time.

Fig. 8
figure 8

Sensitivity of mix composition versus V-funnel

Full size image

In Fig. 9, it is illustrated that increasing the amount of cement content enhances compressive strength, and at the same time increase in fly ash percentage reduces the compressive strength. The relationship between water–powder ratio and compressive strength of SCC is also verified from the graph, as decrease in w/p ratio increases compressive strength. Also, it suggests that increase in superplasticiser dosage improves the compressive strength due to the fact that at higher concentration of superplasticiser dosage, requirement of water is less. The plot also indicates that increase in coarse aggregate content improves the hardened property of SCC.

Fig. 9
figure 9

Sensitivity of mix composition versus compressive strength

Full size image

8 Conclusions

In this paper, SVR, ANN, MVR approach was used to predict the fresh and hardened properties of self-compacting concrete. Input parameters such as the mix ingredient, namely binder content, fly ash percentage, water–powder ratio, fine aggregate content, coarse aggregate content and superplasticiser, are considered. Based on the study, following conclusions can be drawn.

  1. 1.

    Out of the three prediction model, SVR model with exponential radial basis function yields the best performance based on the highest value of coefficient of correlation of the training and testing data and lowest value of statistical error.

  2. 2.

    Sensitivity analysis shows a clear picture of effect of various input parameters on different outputs parameters, namely slump flow, L-box ratio,V-funnel and compressive strength.

  3. 3.

    This present study represents that the support vector regression technique with exponential radial basis function can be used as a powerful and reliable alternative to solve highly nonlinear problems such as prediction of SCC properties with a high degree of accuracy [40].