Introduction

One of the most complex issues in structural engineering is the investigation of the structural elements behaviors and estimation of final capacities. This issue is essential in determining the damages and the failures of elements under loading such as an earthquake. There are many efforts to investigate this topic which were published in the literature, and some of them are reviewed here by the authors. Panagiotako et al. [1] studied the effect of capacity design for Reinforced Concrete (RC) column under seismic loading and showed that in some cases, damage of the element could not be prevented by full capacity design. Hernandez et al. [2] investigated the effect of longitudinal reinforcement on the capacity of concrete columns and presented a method to determine a suitable combination of reinforcement. An experimental study on the compressive capacity of RC columns was done by Chen et al. [3]. They presented an analytical approach for their purpose and based on the experimental results, showed that their method could determine the considered behavior of special-shaped reinforced concrete columns. Some researchers studied the capacity of RC columns which made by recycled aggregate [4,5,6].

Today, soft computing (SC) has many applications in engineering problems [7,8,9,10]. There are numerous articles on the use of SC in civil engineering such as earthquakes [11, 12] dams [13], concrete [14] and structural control [15]. Also, these methods are considered to estimate the capacity of structural elements [16, 17] instead of finite element analysis which is a time-consuming approach [18, 19]. Liu et al. [20] studied the application of artificial neural networks to predict the shear strength of RC columns and verified their model with an experimental database. Jakubek [21] used fuzzy weight neural networks to predict the critical axial load of bulking tests in RC columns. Xu et al. [22] identified the seismic damages of RC columns by neural networks based on images. Their results indicated that these soft computing approaches could be used for damage detection of RC columns.

The current research investigated the application of a powerful soft computing approach namely ANFIS (adaptive neuro-fuzzy inference system) to determine the moment capacity in spiral-reinforced concrete columns. The presented model is trained and tested by a collection of the experimental database. Details of the proposed ANFIS structure are provided in the mathematical framework to increase the ability to use it by engineers.

ANFIS

Adaptive neuro-fuzzy inference system (ANFIS) is a fuzzy inference model in a neural network structure for function approximation [23]. It used a Sugeno-type fuzzy system in the five-layer network [23]. ANFIS contains an input vector with some Membership Function (MF) for each input. ANFIS used a hybrid approach, which is a combination of backpropagation and least squares methods, to find its unknown parameters. This type of soft computing method is widely considered as a powerful system because it has the ability of both artificial neural networks and fuzzy systems simultaneously [7, 23,24,25,26,27,28,29,30,31,31].

Database

Neuro-fuzzy inference system needs a database to determine its unknown coefficients, and in this paper, a collection of spiral-reinforced columns tests results, which were presented by other researchers [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65] and modified by PEER [66], was used. This database contains three types of cantilever column including octagonal, circular and square. More information can be seen in the PEER report. Also, five input variables which are described in Table 1 and presented in “Appendix” are used in this study. The two first parameters can also be defined by Eqs. 1 and 2:

$$ x_{1} = \frac{{\rho_{\text{l}} f_{\text{yl}} }}{{f_{\text{c}}^{'} }}, $$
(1)
$$ x_{2} = \frac{{\rho_{\text{s}} f_{\text{ys}} }}{{f_{\text{c}}^{'} }}, $$
(2)

where \( \rho_{\text{l}} , f_{\text{yl}} , \rho_{\text{s}} , f_{\text{ys}} , f_{\text{c}}^{'} \) are longitudinal reinforcement ratio (%), the yield stress of longitudinal reinforcement (MPa), volumetric transverse reinforcement ratio (%), the yield stress of transverse reinforcement (MPa) and also the compressive strength of concrete (MPa), respectively. Table 2 shows the details of the collected database.

Table 1 Description of the considered variables
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Table 2 Description of the considered parameters
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In this paper, the authors used Eq. 3 as a normalization relationship to convert all amounts of the database into a value between − 1 and + 1. In this equation, the parameters \( x_{n} , x_{\rm{min} } , x_{\rm{max} } \) are indicated to the normal, minimum and maximum values of \( x_{i} \).

$$ x_{n} = 2 \frac{{x_{i} - x_{{\rm min} } }}{{x_{{\rm max} } - x_{{\rm min} } }} - 1 $$
(3)

Based on Table 2 and also Eq. 3, the amount of the variables is normalized by Eqs. 4–9 before using in training and testing the proposed ANFIS model.

$$ X_{1} = 2 \frac{{x_{1} - 6.78}}{65.60} - 1 $$
(4)
$$ X_{2} = 2 \frac{{x_{2} - 1.05}}{36.99} - 1 $$
(5)
$$ X_{3} = 2 \frac{{x_{3} }}{6770} - 1 $$
(6)
$$ X_{4} = 2 \frac{{x_{4} - 0.07}}{0.6} - 1 $$
(7)
$$ X_{5} = 2 \frac{{x_{5} - 14}}{943} - 1 $$
(8)
$$ Y = 2 \frac{M - 22}{1278} - 1 $$
(9)

Proposed ANFIS model

As mentioned in the previous sections, ANFIS uses some membership functions for each input. In this research, four Gaussian membership functions (Eq. 10) were used for each of the five inputs in the proposed ANFIS structure (Fig. 1). The parameters of the membership functions for all inputs are presented in Table 3. These functions can be seen in Fig. 2:

$$ C_{i} \left( {x;\, \sigma ,c} \right) = {\text{e}}^{{\frac{{ - \left( {x - c} \right)^{2} }}{{2\sigma^{2} }}}} , $$
(10)

where c is the mean and σ is the variance for x.

Fig. 1
figure 1

Proposed ANFIS structure

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Table 3 Details of the membership function
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Fig. 2
figure 2

Membership functions of the input parameters

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The proposed ANFIS uses linear-type functions (Eq. 5) as the output of each node with five coefficients and one constant value. Table 4 presents the details of these linear functions and their coefficients.

$$ f_{j} = a_{1j} X_{1} + a_{2j} X_{2} + a_{3j} X_{3} + a_{4j} X_{4} + a_{5j} X_{5} + C_{j} \;\quad j = 1, \ldots ,4, $$
(11)

where \( X_{i} \) is the normalized value of inputs and \( a_{1} , \ldots , a_{5} \) are coefficients of the linear function. C is also the constant value of the equation. In this equation, j denotes the number of linear functions.

Table 4 The parameters of Eq. 11
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There are four rules in the proposed ANFIS. The weight of each rule \( W_{i} \) (j = 1,…,4) is calculated by Eq. 12. In these equations, MF is a membership function value of each input, which can be calculated by Eq. 10 based on the presented amounts of Table 3.

$$ \begin{aligned} W_{1} & = {\text{MF}}_{1,X1} {\text{MF}}_{1,X2} {\text{MF}}_{1,X3} {\text{MF}}_{1,X4} {\text{MF}}_{1,X5} \\ W_{2} & = {\text{MF}}_{2,X1} {\text{MF}}_{2,X2} {\text{MF}}_{2,X3} {\text{MF}}_{2,X4} {\text{MF}}_{2,X5} \\ W_{3} & = {\text{MF}}_{3,X1} {\text{MF}}_{3,X2} {\text{MF}}_{3,X3} {\text{MF}}_{3,X4} {\text{MF}}_{3,X5} \\ W_{4} & = {\text{MF}}_{4,X1} {\text{MF}}_{4,X2} {\text{MF}}_{4,X3} {\text{MF}}_{4,X4} {\text{MF}}_{4,X5} \\ \end{aligned} $$
(12)

The final output of the proposed ANFIS model is determined by Eq. 13:

$$ - 1 \le \left( {Y = \frac{{\mathop \sum \nolimits_{j = 1}^{4} W_{j} f_{j} }}{{\mathop \sum \nolimits_{j = 1}^{4} W_{j} }}} \right) \le 1. $$
(13)

As mentioned in the previous section, the result of the ANFIS is a normal value between − 1 and 1, and therefore, it needs to convert to the corresponding real value (22–1300 kN m) by Eq. 14:

$$ M\left( {{\text{kN}}\;{\text{m}}} \right) = 1278\frac{Y + 1}{2} + 22. $$
(14)

Results of the ANFIS

The training of the proposed system was done by 70 datasets, and the results of this phase showed that the model was trained very well. Also, to validate the ANFIS, 12 datasets were applied to the trained system. The results of regression plots (Fig. 3) for the normal values showed the correlation coefficients equal to 0.99 and 0.98 for the train and test phases, respectively.

Fig. 3
figure 3

Regression plots of the results based on normal values

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Figure 4 shows the obtained results by ANFIS against the experimental values after converting the normalized values into their corresponding real values for the considered database. It is clear from the figure that the proposed ANFIS was able to estimate the moment capacity of spiral-reinforced concrete columns.

Fig. 4
figure 4

The results of the ANFIS vs. experimental values

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The amount of root means squared error for the train, and test phases (Figs. 5, 6) were 50.36 and 92.11 which was shown that the ANFIS could be used as a suitable tool for prediction. Figure 7 illustrated the results of ANFIS for all of 82 datasets.

Fig. 5
figure 5

Results for the train data

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Fig. 6
figure 6

Results for the test data

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Fig. 7
figure 7

Results for all data

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Figure 8 shows the effect of changes in input variables (X1, …, X5) on the output parameter (Y). In drawing each of these graphs, the values of the three variables from the five input variables are considered constant, which is equal to its corresponding median value (see Table 2), and the values of the other two variables have been varied between − 1 and 1. Then, the output value for this database is calculated and plotted.

Fig. 8
figure 8

The effect of input changes on the considered output

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Conclusion

This paper presents a neuro-fuzzy inference system namely ANFIS to predict the moment capacity of spiral-reinforced concrete columns which are failed in flexure. For this purpose, a collection of 82 datasets were used to train and test the model. The system created based on five input parameters including longitudinal reinforcement, transverse reinforcement index, axial force, diameter to length ratio and also a shear force to calculate the target (moment capacity). The proposed ANFIS used Fuzzy C-means approach to determine its unknown coefficients. Also, four clusters and Gaussian membership functions are applied to creating the neuro-fuzzy model. The results of the paper in both training and testing phases indicated that this type of soft computing methods with high accuracy could be considered for predicting the moment capacity of the considered RC columns. The model presented in this article has many applications in the design of concrete structures. Also, due to the proposed neuro-fuzzy model in a mathematical framework, it is an efficient and feasible model. Therefore, it is easy for engineers to understand the equations of this paper and to use them for their purposes. In the future works, other soft computing methods can be used to estimate the moment capacity of the RC columns.