Modelling the High Strain Rate Tensile Behavior of Steel Fiber Reinforced Concrete Using Artificial Neural Network Approach

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Introduction
The experimental studies in the literature evidence that normal concrete shows low ductility and energy dissipation capacity under high strain rate loads, while introducing various types of fibers into the concrete mixtures, especially steel fibers, can significantly improve its behavior under high strain rate loads (Nili et al. 2010, Soufeiani et al. 2016).The crack bridging effects of steel fibers in concrete causes higher impact resistance and energy dissipation capacity of steel fiber reinforced concrete (SFRC) composite materials under high strain rate loads.In other words, the fiber bridging mechanism, mainly those of fiber pull-out and snubbing effect at the fiber exit point, limits crack propagation and enhances the energy dissipation capacity.In this regard, prediction of the behavior of SFRC materials under tensile impact tests is complex due to the nonlinear relationship between the impact force and the effective variables.However, some researchers developed the analytical models to predict the behavior of SFRC under impact tests (Soufeiani et al. 2016).In this area, proposing a formulation with a design framework to accurately predict the behavior of SFRC composite materials under impact loads considering the influence of effective parameters is an issue that needs to be addressed.Mathematical methodologies in the field of machine learning, for example, artificial neural network (ANN) can be helpful for the development of models to accurately predict the impact behaviour of SFRC materials.In this regard, the present study focuses on proposing an analytical model with design framework for predicting the tensile behavior of SFRC composite materials under high strain loads using ANN method and considering the effective parameters.

Architecture of the ANN models
Artificial neural networks are inspired by the architecture of the human central nervous system composing of an input layer including the variables and an output layer, and more layers are added between these input and output layers, called hidden layers (for more details see Cascardi et al. 2017, Pham et al. 2016, Pham et al. 2014, Ramezansefat et al. 2020).In this study, the proposed ANN-model was developed in Python programming language to estimate dynamic increase factor (DIF) of SFRC composite materials under high strain rate tensile loads.For the training and testing purposes of the ANN-model, a large database related to the experimental tests of SFRC materials under high strain tensile loads were collected from the literature.The analyzed database included 42 SFRC samples tested under strain rates of less than 10s -1 and 115 SFRC samples under strain rates of higher than 10s -1 , thus totally 157 SFRC samples was included in the database and utilized in this study.The effective parameters adopted in the proposed model including volume fraction of steel fibers ( f v ), aspect ratio of steel fibers ( LD ), tensile strength of steel fibers ( ts f ), and concrete tensile strength ( cs f ), were all reported for the specimens used in this database.80% of the specimens was randomly selected as the training data and the remaining 20% was the test data.Four neurons were adopted in the input layer for considering the four variables, while one neuron was defined in the output layer (Figure 1).For the sake of simplicity for proposing a closed form formulation deriving from the ANN-model, two neurons were adopted in the hidden layer.Transfer function in the hidden layer was sigmoid and in the output layer was linear.After designing the network and standardizing the value of the input parameters to improve the ANN-model and make the training faster, the network would be trained.

Proposed model for prediction of Dynamic Increase Factor (DIF)
Dynamic increase factor (DIF) is defined as the ratio of dynamic to static strength, to determine the effects of strain rate on the concrete properties.Accordingly, DIF parameter is usually considered as a function of strain rate of loading.In this regard, the CEB-FIP Model Code 1990 (MC1990) and CEB-FIP Model Code 2010 (MC2010) are proposed the design formulations for estimating the DIF of concrete under compression and tension as a function of strain rate of loading (CEB-FIP 1990, 2010).The authors already developed the design formulations for the prediction of DIF of SFRC materials under compression considering the effective parameters based on modifying the proposed formulations in MC2010.Figure 2 shows the comparison of the experimental compressive DIF with the corresponding DIF obtained from MC 1990, MC 2010 and the developed model, evidencing the good predictive performance of the developed model compared to the design codes recommendations.The details of this model can be found elsewhere (Ramezansefat et al. 2020).
Input Layer

Hidden Layer
Output Layer In the current section, the ANN-model is developed based on modifying the proposed formulation in MC2010, to predict the DIF of SFRC materials under tension, considering the steel fiber effects in concrete.The performance of the developed model is compared with the proposed formulations in MC1990 and MC2010 to predict the tensile DIF of SFRC materials.MC1990 proposed Eqs.1-2 to estimate the tensile strength of normal concrete under high rates of loading, established using appropriate underlying theory derived from thermodynamics and fracture mechanics analysis (CEB-FIP 1990).
  ).In a log (DIF) versus log(  ) the relationship is bilinear with a change in slope around 30 1 s  and 10 1 s  in MC1990 and MC2010, respectively.The formulations proposed in MC1990 and MC2010 are valid for normal concrete.However, these formulations need to be updated for the case of SFRC materials due to the steel fiber effects in concrete.In order to consider the steel fiber effects in concrete in the formulations proposed by MC2010 (Eqs.3-4), the alterations were conducted on the power of Eq. 3 proposed by MC2010 ( 1 k in Eq. 5) for the range of and on the constant coefficient of Eq. 4 proposed by MC2010 ( 2 k in Eq. 6) for the strain rates beyond the 10.In this regard, 1 k and 2 k parameters were derived from experimental database using Eqs.5-6, and were adopted as output variable in the ANN-model.It should be noted that the other possibilities for the modification of the formulation of MC2010 (e.g.power of Eq. 4) was investigated by the authors, and the proposed modification strategy was adopted due to providing the best performance for predicting the tensile DIF of SFRC materials.k and 2 k parameters were defined as the output layer.The architecture of the ANN-model was optimized in terms of the different numbers of neurons and the different transfer functions in the hidden layer, aiming to provide the highest coefficient of correlation (R 2 ) with experimental data.Since the main objective of this study is to propose a closed form design formulation derived from the ANN-model for the tensile DIF of SFRC materials, the neurons number adopted in the hidden layer was minimized.The performance of the proposed ANN-model was verified against the experimental results for two ranges of lower and higher than intermediate strain rates.The plot of the experimental tensile DIF versus the corresponding ANN-model predictions for the database is shown in Figures 3 and 4, and also, compared with the DIFs obtained from MC1990 and MC2010.However, the coefficient of correlation ( 2 R ) of DIFs obtained according to MC1990 and MC2010 using Eqs. 1 and 3 for strain rates lower than intermediate level are, respectively, 0.475 and 0.36 for training data and 0.97 and 0.93 for test data.In addition, for strain rates strain rates higher than intermediate level, 2 R of DIFs of MC1990 and MC2010 (Eqs. 2 and 4) are, respectively, 0.1468 and 0.004 for training data and 0.164 and 0.0064 for test data.This comparison evidences that the proposed ANNmodel can predict the tensile DIF of SFRC materials with better accuracy compared to the models proposed in MC1990 and MC2010.Moreover, the experimental tensile DIF is plotted versus strain rate in Figure 5

Design formulation based on the ANN-model
In the previous section, it was confirmed that the DIF of SFRC materials under tension obtained using the ANN-Model have a good agreement with the relevant experimental data.However, it is inconvenient for engineers to use the networks for engineering design purposes, since they need to have the fundamental knowledge of ANN and Python to be able to use the proposed model.Hence, in this section, a simplified closed form formulation derived from the developed ANN-model based on modifying the MC2010 formulation is proposed to predict the DIF of SFRC materials under tension.The simplified formulation is derived from the trained networks by using input parameters and transfer functions and combining the weight matrix and the bias matrix (Figure 6), and more details about the adopted strategy can be found elsewhere (Pham et al. 2016,Pham et al. 2014, Yousif DS, 2013).The sigmoid transfer function was used in the hidden layer (see Eq. 7) and linear transfer function (see Eq. 8) was used in the output layer.The procedure to develop the user-friendly equations based on the ANN-model to determine 1 k and 2 k parameters is represented in Figure 6.

Conclusion
The current study develops a model based on artificial neural network (ANN) approach to predict the dynamic increase factor (DIF) of SFRC composite materials under high strain rate tensile loads.The developed model is based on modifying the proposed formulation in fib model code 2010 (MC2010) and considers the steel fiber effects in concrete by adopting the effective parameters in the model (i.e.: volume fraction of steel fibers ( f v ), aspect ratio of steel fibers ( LD), tensile strength of steel fibers ( ts f ), and concrete tensile strength ( cs f )).Moreover, a simplified closed form formulation derived from the developed ANN-model is proposed with a design framework to predict the DIF of SFRC materials under tension.In addition of the evaluation of the predictive performance of the developed model by comparing with the relevant experimental data, the proposed models in the design codes of MC2010 and MC1990 were assessed for the prediction of the tensile DIF of SFRC materials.The proposed formulations in MC1990 and MC2010 for the tensile DIF of concrete cannot predict well the tensile DIF of SFRC material, due to the lack of parameters to consider the impact of steel fibers in concrete in these formulations.However, the simplified closed form formulations developed in this study can provide good estimations of SFRC strength at high strain rate tensile loads.The models developed in this study will be used for designing the impact resistance of SFRC elements subjected to high strain rate loads such as blast and impact, representative of terrorist attacks and accidents which is an ongoing research project in the University of Minho.For the next step, the authors aims to extend the developed model to achieve a higher degree of accuracy in predicting the tensile DIF of SFRC materials by adding more important parameters in the ANN-model, i.e. shape of steel fibers, fiber orientation factor, and fiber efficiency factor.

Figure 1 .
Figure 1.Architecture of the comprehensive version of the proposed network.

Figure 2 .
Figure 2. Comparison of experimental DIF with DIF obtained from MC 1990, MC 2010 and proposed ANN model (Ramezansefat et al. 2020).
dynamic tensile strength under high rates of loading, cs f is static tensile strength, regard, MC2010 proposed the following equations for the tensile DIF of normal concrete: in MC1990 and MC2010 were proposed for two domains of strains, the first ranging from low to intermediate (

1 k and 2 k
parameters were considered as a function of volume fraction of steel fibers ( f v ), aspect ratio of steel fibers ( LD), tensile strength of steel fibers ( adopted in the input layer of the ANN-model.Two neurons were introduced in the hidden layer, and 1

Figure 3 .
Figure 3.Comparison of DIFs obtained from MC1990 and MC2010 and ANN-model with corresponding experimental values for strain rates lower than intermediate level

Figure 4 .
Figure 4. Comparison of DIF obtained from MC 1990 and MC 2010 models and ANN withcorresponding experimental values strain rates higher than intermediate level.
and compared with the tensile DIF obtained from MC1990, MC2010 and ANN-model.This figure also evidences the good predictive performance of the developed ANN-model for the tensile DIF of SFRC materials in comparison with the proposed formulations in MC1990 and MC2010.

Figure 5 .
Figure 5.Comparison of experimental DIF with DIF obtained from MC 1990, MC 2010 and proposed ANN model.

Figure 6 :
Figure 6: Architecture of the proposed ANN equation