## 4 Computational and Mathematical Aspects
### Failure Criteria
#### 4.4.3 Failure Criterion Based on Artificial Neural Networks  
This section summarizes the basic concepts of artificial neural networks (ANNs) as well as the architecture of the optimum ANN model developed for the modeling of masonry compressive strength. ANNs are information processing models configured for a specific application through a training process. A trained ANN has learned to rapidly map a given input into the desired output quantities (similar to curve fitting procedures) and thereby can be used as a meta-model enhancing the computational efficiency of a numerical analysis process. This major advantage of a trained ANN over conventional numerical analysis procedures, such as regression analysis, under the condition that the training and validation data cover the entire range of input parameters values, is that the results can be produced with much less computational effort [93; 177; 178; 179; 180; 181; 182; 183; 184; 185; 186].  
In the present study, using the available in the literature experimental results conducted by [PERSON] and [PERSON] (1973) [160] concerning the failure of concrete under biaxial stresses, the Back-Propagation Neural Network (BPNN) technique was used for the development of a reliable and robust ANN that can predict the failure of concrete. To this end, for each failure point (experimental data), the polar angle was used as input parameter, while the polar radius was selected as output parameter.  
In light of the above, 36,900 BPNN models (900 with one hidden layer and 36,000 with two hidden layers) were studied. The parameters used for the training of NN models are summarized in Table 2. All developed ANN models were ranked based on the Root Mean Square Error (RMSE). Based on the results, the optimum BPNN model is that of 1-3-5-1 structure. This network corresponds to the case of architecture with two hidden layers of 3 and 5 neurons, respectively (Figure 11). Using this optimum ANN model, the concrete failure criterion under biaxial stress state both in dimensional principal stresses terms (Figure 12a) and in polar coordinates (Figure 12b) was produced.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Training Algorithm & Levenberg–Marquardt Algorithm \\ \hline Number of Hidden Layers & 1; 2 \\ \hline Number of Neurons per Hidden Layer & 1 to 30 by step 1 \\ \hline Cost Function & Mean Square Error (MSE); Sum Square Error (SSE) \\ \hline \multirow{3}{*}{Transfer Functions} & Hyperbolic Tangent Sigmoid transfer function \\  & (tansig); Log-sigmoid transfer function (logsig); \\ \cline{1-1}  & Linear transfer function (purelin) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Training Parameters of BPNN models.  
Figure 10: Dimensionless failure surface of masonry material using the simplified failure criterion: (**a**) non-dimensionless normal stress terms; and (**b**) non-dimensionless Principal stress terms.  
Figure 11: Architecture of the optimum 1-3-5-1 BPNN model.  
Figure 12: _Cont._  
It is common practice in published literature on NN models for the authors to present the architecture of the optimum NN model without providing any information regarding the final values of NN weights. Admittedly, this practice has very little value for other researchers and practicing engineers. To be useful, a proposed NN architecture should be accompanied by the (quantitative) values of weights [186]. In such a case, the NN model can be readily implemented in an MS-Excel spread sheet, thus available to anyone interested in the issue of modeling.  
In this study, the final values of weights and biases of the optimum proposed BPNN 1-3-5-1 model are explicitly reported in Figure 13 and Table 3. By using the reported values of weights and biases, as well as the architecture of the NN (Figure 11), one can easily build an ANN-based estimator for the failure criterion of concrete under biaxial stress state.