# An Adaptive Neuro-Fuzzy Inference Model to Predict Punching Shear Strength of Flat Concrete Slabs
## 5 ANFIS: This Study  
In this study, an ANFIS model was developed using MATLAB R2013a [39] with five input parameters: The slab effective depth \((d)\), compressive strength of concrete \((f_{c}^{\prime})\), reinforcement ratio \((\rho)\), yield strength of reinforcement \(\big{(}f_{y}\big{)}\), and width of square loaded area \((c)\). The output variable is punching shear strength of a two-way slab \((V)\). A set of 207 experimental data points, collected from several sources [40; 41; 42; 43; 44; 45; 46], was used to develop the model. The experimental data were randomly divided into two sets: The first one, with 164 data points, was used for training the model, and the  
Figure 1: The basic flow diagram for computations in an adaptive neuro-fuzzy interface system (ANFIS). second one, with 43 data points, was used for testing. A subtractive clustering technique produced by [PERSON] (1980) was used to generate the ANFIS model with the (genfis2) function in MATLAB. Genfis2 is used to help in the creation of the initial set of membership functions for sets of input and output data. Genfis2 preforms this model by extracting a set of rules. The rule extraction method first uses the subclust function to determine the number of rules and antecedent membership functions. The type and the number of membership functions were evaluated when the training and testing datasets were giving good predictions according to the root mean square error (RMSE). After experimenting with different learning algorithms with a number of different epochs, the best correlations were found through a hybrid learning algorithm (a combination of least squares and back-propagation algorithms for membership function parameter estimations). The final errors of the model for training and testing were 0.45 and 0.52, respectively, and were achieved after 200 epochs. The structure of the ANFIS model is illustrated in Figure 2. In the model, 10 of the Gaussian membership functions (gaussmf) are selected for each input, and 10 rules define the relationship between inputs and outputs. A Gaussian membership function has two parameters: \(c\), responsible for its center, and \(\sigma\), responsible for its width, and the equation for this type is ([PERSON], 1980; [PERSON], 1980)  
\[A(x)_{Gauss}=\exp\left[-\left(\frac{x-c}{2\sigma}\right)^{2}\right]. \tag{7}\]  
Readers are referred to [PERSON] (1980) for more details on this type of membership function. The numerical range of input parameters of the current study is listed in Table 1. The data used to build the ANFIS model are summarized in Table A1 in Appendix A. After the training procedure, the model was tested using the remaining data not used for the training. Figure 3 shows the performance for training and testing datasets. Figures 4 and 5 show the matching of the experimental results with the results of the ANFIS model for both training and testing sets, respectively. Figure 6 shows a comparison between the experimental results of punching shear and the results predicted by the ANFIS model for all samples used in the model (training and testing sets). The adequacy of the developed ANFIS was evaluated by considering the coefficient of correlation (\(R\)), the average and standard deviation of the ratio of predicted to experimental punching shear strength, and the root mean square error (RMSE). The equations of the statistical parameter RMSE and the coefficient of correlation (\(R\)) that were used to compare the performance of each method are  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{N}\left(V_{ne}-V_{ni}\right)^{2}}{N}}\, \tag{8}\]  
\[R=1-\sqrt{\frac{\sum_{i=1}^{N}\left(V_{ne}-V_{ni}\right)^{2}}{\sum_{i=1}^{N} \left(V_{ne}\right)^{2}}}\, \tag{9}\]  
where \(V_{ne}\) and \(V_{ni}\) are the experimental and prediction nominal punching shear strength (\(V_{n}\)) of two-way flat slabs, respectively, and \(N\) is the total number of samples considered. Figure 4: Experimental and predicted punching shear strength (training dataset). Figure 3: Convergence of the ANFIS for training and testing sets.