# Firefly Algorithm-Based Artificial Neural Network to Predict the Shear Strength in FRP-Reinforced Concrete Beams

Authors: Mohammad Nikoo; Babak Aminnejad; Alireza Lork
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

The shear strength of fiber-reinforced polymer (FRP) reinforced concrete beams is often given a large safety margin by current construction requirements. Six characteristics are utilized as inputs to compute the shear strength of FRP-reinforced concrete beams. This study uses 198 samples from the literature to predict the shear strength of 139 training samples and 59 testing samples. Additionally, the ANN structure is optimized with the firefly algorithm. The FA-ANN model is also compared to ACI-440, CSA-S806, and BISE-99 codes, and the optimized model by Nehdi et al. Findings show that regarding the shear strength of FRP-reinforced concrete beams, the firefly algorithm-optimized model performs better than the other four models. Concerning accuracy, the coefficient of correlation,R2, was calculated as 0.961, while the average absolute error (AAE) is 0.22 for the shear strength of FRP-reinforced beams.


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## Body

## 1. Introduction

Corrosion problems in infrastructure impose huge expenses on rehabilitating structures worldwide [1]. Furthermore, exposing structures (e.g., water treatment facilities, marine structures, and bridges) to extreme corrosion compromises their structure and, therefore, reduces their service life tremendously [2]. Fiber-reinforced polymer (FRP) bars are a promising substitute for traditional reinforcing steel bars [3, 4]. Additionally, the greater strength, smaller weight, and higher axial stiffness-to-weight ratio of FRPs make them more attractive solutions. However, FRPs suffer from a few shortcomings compared to steel, including a smaller modulus of elasticity, brittleness, and anisotropy. Recent research has focused on predicting the shear strength of FRP-reinforced concrete beams in addition to other aspects [5]. According to reports, the modulus of elasticity, shear span-to-depth ratio, beam width, concrete compressive strength, and flexural reinforcement ratio are the primary factors affecting the shear strength of FRP-reinforced concrete beams without stirrups [6].The current shear provisions in building codes are extensions of their steel-reinforced concrete predecessors. The design codes differ significantly in their choice of influencing parameters on shear strength and their contributions. Researchers have assessed the shear provisions and concluded that they are exceedingly cautious or insufficient in some cases [5]. This overestimation results in an excess number of bars in the design, which causes reinforcing congestion and higher costs. Many provisions were devised at the time utilizing restricted experimental data [7]. The discrepancy between experimental shear test findings and code standards illuminates our lack of understanding of the shear mechanism in FRP-reinforced concrete members.Furthermore, concrete members such as footings, slabs, and bridge decks are constructed without stirrups. In addition, sudden and brittle failure may occur in these buildings without imminent warning [3, 8]. This highlights the significance of appropriately evaluating the shear strength mechanism in FRP-reinforced concrete members [2]. Also, research has been conducted on using basalt fiber-reinforced polymer (BFRP) in short beams so that ten beams with a length of 2.0 meters and a rectangular section with a width of 140 mm and variable height were tested under a four-point loading configuration. Using ABAQUS software, a finite element model was established to forecast the behavior of the tested beams to a level that accurately predicted the shear capabilities of the beams and recorded their failure mechanisms [9, 10]. Also, using basalt microfibers in the fiber-reinforced polymer is a disputed topic [11], and using optimization algorithms based on ANN to determine the torsional strength of a reinforced concrete beam [12, 13].Researchers have successfully utilized soft computing methods to increase the accuracy of shear strength prediction methods. Nehdi et al. used genetic algorithms to predict the shear strength of FRP-reinforced concrete beams [14]. Kara predicted the shear strength of FRP-reinforced beams without stirrups using genetic programming [15], while Gandomi et al. used a linear genetic programming approach for this purpose [16]. Bashir and Ashour proposed an artificial neural modeling approach, showed the feasibility of this approach, and determined the contribution of influencing factors using experimental data available at the time [17]. With more comprehensive experimental data available, Lee and Lee used the same approach for FRP-reinforced concrete beams without stirrups and proposed the relevant design equations [18]. The shear strength was predicted using a fuzzy inference technique by Nasrollahzadeh and Basiri [19]. Shahnewaz et al. optimized shear design equations using a genetic algorithm and reliability analysis for FRP-reinforced beams with stirrups [20]. Also, Golafshani and Ashour studied the feasibility of the biogeography-based optimization for FRP-reinforced concrete beams without stirrups [21]. Hasanzade-Inallu et al. estimated the shear strength of FRP-reinforced concrete beams without stirrups by amassing many experimental test results from the literature and modeling the shear strength with an ANN trained using a modified imperialist competitive optimization algorithm [7].Several opportunities exist within these soft computing tactics to propose new models applying novel techniques to reduce model uncertainty. Yang developed the firefly algorithm, a technique for metaheuristic optimization inspired by the flashing behavior of tropical fireflies. The algorithm has successfully performed over other metaheuristic algorithms [22]. This study focused on improving the predictive accuracy of a model by training an ANN with the firefly method. The proposed method considered all the relevant factors affecting shear strength and was validated using a database of 198 specimens from the literature. The second section gives background information on ANNs and the firefly algorithm. In Section 3, the setup and training of the model are discussed. Sections 4 and 5 present the findings and conclusion, respectively. Figure 1 illustrates the organization of this study.Figure 1
The article content outline.


## 2. Background

### 2.1. Artificial Neural Networks (ANNs)

ANNs were modeled based on how the human brain performs tasks and contains data processing units called neurons arranged in layers. Feedforward (FF) is a class of ANNs consisting of one input layer, one hidden layer, and one output layer. In a fully integrated FF, each layer’s neurons are connected to neurons in preceding and subsequent layers. The function of a neuron is first to apply weights to its inputs to reflect the relevance of each input on the output, then add a constant called bias to the result, and then apply a function, called the activation function, to the resulting sum. The hyperbolic tangent function is a typical activation function applied for regression problems [23–25].Typically, ANN weights are initialized randomly, causing the network’s output to deviate from the desired values. A training method should modify the network weights and biases to minimize the model’s error (i.e., the difference between the output and target values) [23, 24].Training an ANN is an optimization process, and the approaches for tackling this optimization problem are divided into gradient-based and metaheuristic methods. Gradient-based techniques are fast, but they can become trapped in local minima. In contrast to gradient-based methods, metaheuristic methods are not trapped in local minima. The response provided by metaheuristic approaches is not always the global minimum. Nevertheless, these approaches often aim to explore and exploit a substantial amount of the solution space to attain the correct answers [23, 24].ANNs are exposed to the overfitting problem. An overfitted model can reliably predict outputs for the range of inputs observed during training, but it lacks the generality necessary to predict outputs for inputs not received during training. Typically, this effect is countered by separating data into training and testing sets. The training set alters the network’s weights, while the testing set is used to choose more extensible networks [24, 26].


### 2.2. Firefly Algorithm (FA)

FA is based on the flashing pattern of tropical fireflies proposed by Yang. Researchers believe these bioluminescent signals’ primary functions are to attract potential mates and prey. Also, they might act as a defensive alerting mechanism to predators. Firefly algorithm idealized this flashing behavior by using the following three simplifying assumptions:(i)
All fireflies are unisex and can absorb other fireflies regardless of gender [22].(ii)
A firefly’s attractiveness is proportional to its luminosity; if two fireflies are present, the dimmer one will approach the brighter one. Due to the inverse relationship between light intensity and distance from the light source, fireflies become less attractive as their distance from the light source increases. If a firefly cannot find a firefly that is brighter than itself, it will move randomly [22].(iii)
The architecture of the objective function governs the luminance and, thus, the attractiveness of a firefly. For our implementation, the lower the cost of a firefly, the higher its brightness will be [22].When training an ANN using the firefly algorithm, the network’s weights and biases were specified as firefly locations, and the network’s prediction error was defined as its cost function. As the final weights and biases of the network, the position of the firefly with the lowest cost (most brightness) was selected. Figure2 [22, 27] shows the flowchart of the firefly algorithm.Figure 2
Flowchart of firefly algorithm.


## 2.1. Artificial Neural Networks (ANNs)

ANNs were modeled based on how the human brain performs tasks and contains data processing units called neurons arranged in layers. Feedforward (FF) is a class of ANNs consisting of one input layer, one hidden layer, and one output layer. In a fully integrated FF, each layer’s neurons are connected to neurons in preceding and subsequent layers. The function of a neuron is first to apply weights to its inputs to reflect the relevance of each input on the output, then add a constant called bias to the result, and then apply a function, called the activation function, to the resulting sum. The hyperbolic tangent function is a typical activation function applied for regression problems [23–25].Typically, ANN weights are initialized randomly, causing the network’s output to deviate from the desired values. A training method should modify the network weights and biases to minimize the model’s error (i.e., the difference between the output and target values) [23, 24].Training an ANN is an optimization process, and the approaches for tackling this optimization problem are divided into gradient-based and metaheuristic methods. Gradient-based techniques are fast, but they can become trapped in local minima. In contrast to gradient-based methods, metaheuristic methods are not trapped in local minima. The response provided by metaheuristic approaches is not always the global minimum. Nevertheless, these approaches often aim to explore and exploit a substantial amount of the solution space to attain the correct answers [23, 24].ANNs are exposed to the overfitting problem. An overfitted model can reliably predict outputs for the range of inputs observed during training, but it lacks the generality necessary to predict outputs for inputs not received during training. Typically, this effect is countered by separating data into training and testing sets. The training set alters the network’s weights, while the testing set is used to choose more extensible networks [24, 26].


## 2.2. Firefly Algorithm (FA)

FA is based on the flashing pattern of tropical fireflies proposed by Yang. Researchers believe these bioluminescent signals’ primary functions are to attract potential mates and prey. Also, they might act as a defensive alerting mechanism to predators. Firefly algorithm idealized this flashing behavior by using the following three simplifying assumptions:(i)
All fireflies are unisex and can absorb other fireflies regardless of gender [22].(ii)
A firefly’s attractiveness is proportional to its luminosity; if two fireflies are present, the dimmer one will approach the brighter one. Due to the inverse relationship between light intensity and distance from the light source, fireflies become less attractive as their distance from the light source increases. If a firefly cannot find a firefly that is brighter than itself, it will move randomly [22].(iii)
The architecture of the objective function governs the luminance and, thus, the attractiveness of a firefly. For our implementation, the lower the cost of a firefly, the higher its brightness will be [22].When training an ANN using the firefly algorithm, the network’s weights and biases were specified as firefly locations, and the network’s prediction error was defined as its cost function. As the final weights and biases of the network, the position of the firefly with the lowest cost (most brightness) was selected. Figure2 [22, 27] shows the flowchart of the firefly algorithm.Figure 2
Flowchart of firefly algorithm.


## 3. Methods and Materials

### 3.1. Dataset

The experimental data required to train and test artificial neural networks were collected from the dataset by Hasanzade–Inallu et al. [7]. It contains 198 test cases of shear strength of FRP-reinforced concrete bars without stirrups. Table 1 provides descriptive statistics regarding the findings of the experimental test.Table 1
Descriptive statistics of the experimental data [7].
fc′ (MPa)ρf (%)Ef (GPa)a/dbw (mm)d (mm)Vcf (kN)Mean41.571.07583003.32284.40336.7992.97Standard deviation13.080.6442.471.44190.08210.90112.77Min22.700.1823.201.0089.00104.009.80Max88.303.43192.0012.501000.001097.00953.00Here,fc′ = concrete compressive strength; ρf = FRP longitudinal reinforcement ratio; Ef = FRP bar modulus of elasticity; a/d = shear span-to-depth ratio; bw = web thickness; d = effective depth; and Vcf = shear strength of the concrete beam.


### 3.2. Model Setup

The selected parameters for training artificial neural networks (ANNs), drawn from past research, are web thickness (bw), effective depth (d), shear span-to-depth ratio (a/d), FRP longitudinal reinforcement ratio (ρf), FRP bar modulus of elasticity (Ef), and concrete compressive strength (fc′). These six parameters served as inputs to neural networks; the network’s output was the shear strength.The ANNs’ input variables have varying ranges, which can increase the training time or cause the optimization algorithm to diverge [24]. Each variable was normalized to a range of [−1, 1] using the following equation to bring different input and output variables to similar ranges:(1)Yn=2Y−YminYmax−Ymix−1,where Yn is the normalized variable value, Ymax is the maximum value, and Ymin is the minimum value. Y is the original variable value (un-normalized). Each variable’s minimum and maximum values are presented in Table 1. Since the ANN will be trained using the normalized values of the variables, it is necessary to normalize any future input into a trained network and un-normalize the network’s output into its original range [7].The ANN architecture is problem-dependent [23]. The optimal architecture, including the number of neurons and hidden layers, was thus determined by trial and error (i.e., best representing the data). Forty-five architectures with two or three hidden layers were trained, and the best-performing architectures were chosen. This research used ANN models and the firefly algorithm in MATLAB software [28].


### 3.3. Model Training

Data were separated randomly into two groups to prevent overfitting. Seventy percent (139 test cases) were used to train the networks, while thirty percent (59 test cases) were used to test the network on data not seen during training to identify the networks with the most excellent generalization capabilities. The hyperbolic tangent function was employed as the activation function for hidden layers, while the identity function was used for the output layer.Training a neural network aims to optimize its weights and biases (i.e., parameters) to reduce the network’s prediction error. The firefly algorithm (FA) was used to train the networks. For every network architecture, the network parameters were defined as fireflies, and by generating a population of fireflies, FA generated the initial possible network parameters. As the iterations of FA progressed, the network parameters (firefly positions) were updated to reflect the behavior of the training dataset and, therefore, minimize the prediction error. The final solution of the FA was chosen as the optimum network parameter for the given network architecture. The parameters chosen for FA were inspired by the recommendations given by Yang [22] and were somewhat altered to best train the ANN model. The parameters used are given in Table 2, and the description of the parameters is given in Section 2.2.Table 2
Firefly algorithm parameters [29].
ParameterValuePopulation size100Mutation coefficient0.25Light absorption coefficient1Attraction coefficient base value2Mutation coefficient damping ratio0.99M (exponent of distance term)2As recommended by most researchers, the chosen error measure for ANN models was the mean squared error (MSE) function given in the following equation:(2)MSE=1n∑i=1nVpredicted,i−Vexperiment,i2,where n is the number of training samples (139 in our case), Vpredicted,i is the neural network output for the ith sample, and Vexperiment,i is the shear strength of the ith test case reported from experiments.


## 3.1. Dataset

The experimental data required to train and test artificial neural networks were collected from the dataset by Hasanzade–Inallu et al. [7]. It contains 198 test cases of shear strength of FRP-reinforced concrete bars without stirrups. Table 1 provides descriptive statistics regarding the findings of the experimental test.Table 1
Descriptive statistics of the experimental data [7].
fc′ (MPa)ρf (%)Ef (GPa)a/dbw (mm)d (mm)Vcf (kN)Mean41.571.07583003.32284.40336.7992.97Standard deviation13.080.6442.471.44190.08210.90112.77Min22.700.1823.201.0089.00104.009.80Max88.303.43192.0012.501000.001097.00953.00Here,fc′ = concrete compressive strength; ρf = FRP longitudinal reinforcement ratio; Ef = FRP bar modulus of elasticity; a/d = shear span-to-depth ratio; bw = web thickness; d = effective depth; and Vcf = shear strength of the concrete beam.


## 3.2. Model Setup

The selected parameters for training artificial neural networks (ANNs), drawn from past research, are web thickness (bw), effective depth (d), shear span-to-depth ratio (a/d), FRP longitudinal reinforcement ratio (ρf), FRP bar modulus of elasticity (Ef), and concrete compressive strength (fc′). These six parameters served as inputs to neural networks; the network’s output was the shear strength.The ANNs’ input variables have varying ranges, which can increase the training time or cause the optimization algorithm to diverge [24]. Each variable was normalized to a range of [−1, 1] using the following equation to bring different input and output variables to similar ranges:(1)Yn=2Y−YminYmax−Ymix−1,where Yn is the normalized variable value, Ymax is the maximum value, and Ymin is the minimum value. Y is the original variable value (un-normalized). Each variable’s minimum and maximum values are presented in Table 1. Since the ANN will be trained using the normalized values of the variables, it is necessary to normalize any future input into a trained network and un-normalize the network’s output into its original range [7].The ANN architecture is problem-dependent [23]. The optimal architecture, including the number of neurons and hidden layers, was thus determined by trial and error (i.e., best representing the data). Forty-five architectures with two or three hidden layers were trained, and the best-performing architectures were chosen. This research used ANN models and the firefly algorithm in MATLAB software [28].


## 3.3. Model Training

Data were separated randomly into two groups to prevent overfitting. Seventy percent (139 test cases) were used to train the networks, while thirty percent (59 test cases) were used to test the network on data not seen during training to identify the networks with the most excellent generalization capabilities. The hyperbolic tangent function was employed as the activation function for hidden layers, while the identity function was used for the output layer.Training a neural network aims to optimize its weights and biases (i.e., parameters) to reduce the network’s prediction error. The firefly algorithm (FA) was used to train the networks. For every network architecture, the network parameters were defined as fireflies, and by generating a population of fireflies, FA generated the initial possible network parameters. As the iterations of FA progressed, the network parameters (firefly positions) were updated to reflect the behavior of the training dataset and, therefore, minimize the prediction error. The final solution of the FA was chosen as the optimum network parameter for the given network architecture. The parameters chosen for FA were inspired by the recommendations given by Yang [22] and were somewhat altered to best train the ANN model. The parameters used are given in Table 2, and the description of the parameters is given in Section 2.2.Table 2
Firefly algorithm parameters [29].
ParameterValuePopulation size100Mutation coefficient0.25Light absorption coefficient1Attraction coefficient base value2Mutation coefficient damping ratio0.99M (exponent of distance term)2As recommended by most researchers, the chosen error measure for ANN models was the mean squared error (MSE) function given in the following equation:(2)MSE=1n∑i=1nVpredicted,i−Vexperiment,i2,where n is the number of training samples (139 in our case), Vpredicted,i is the neural network output for the ith sample, and Vexperiment,i is the shear strength of the ith test case reported from experiments.


## 4. Results

### 4.1. Model Evaluation

Test data’s mean squared error (MSE) was chosen as the error metric to select the best-performing artificial neural network (ANN) among 45 trained architectures. The top four performing ANNs are given in Table3. The networks are sorted in the order of increasing values of the MSE value of test data. ANN identities are labeled with ANN mL (n1-n2-n3), where m is the number of hidden network layers, and n1, n2, and n3 are the number of neurons in the first, second, and third hidden layers, respectively.Table 3
Top four ANN statistics.
NumTopologyTrainTestR2y = ax + bRMSEAAER2y = ax + bRMSEAAE1FA-ANN 3L (2-6-2)0.972y = 0.9426x + 3.939420.180.210.952y = 1.0783x + 0.968326.370.252FA-ANN 3L (8-2-4)0.970y = 0.9554x + 3.968320.370.240.942y = 1.1022x + 1.685830.980.323FA-ANN 3L (3-7-3)0.955y = 0.9303x + 5.074925.140.260.928y = 1.1101x - 0.487333.840.304FA-ANN 3L (3-8-2)0.965y = 0.9696x + 2.850221.950.250.943y = 1.1669x - 4.126134.730.31Figure3 provides a visual representation of the performance of the top four networks by plotting the values predicted by the networks against their experimental values from the database.Figure 3
Experimental vs. predicted values of shear strength for (a) ANN 3L (2-6-2) model, (b) ANN 3L (8-2-4), (c) ANN 3L (3-7-3) model, and (d) ANN 3L (3-8-2) model.
(a)(b)(c)(d)Since the points in Figure3 are near the y = x line, the projected shear strength values correspond to the observed values. Figures 4 and 5 depict the training and testing performance of the leading network, ANN 3L (2-6-2).Figure 4
Comparison between ANN 3L (2-6-2) calculated and observation data used in training.Figure 5
Comparison between ANN 3L (2-6-2) calculated and observation data used in testing.As demonstrated in Figures4 and 5, the best-performing network’s projected values for both the training and testing phases closely match their experimental values. It is noteworthy that the ANN 3L (2-6-2) performs well even for one test case where the shear strength is about 810 kN, which is much higher than the rest of the test cases. The ANN 3L (2-6-2) model is chosen for further analysis.


### 4.2. Comparison with Existing Equations

ACI-440.1R-15 [30], CAN/CSA-S806-12 [31], BISE-99 [32], and Nehdi et al. (optimized equation method) provide the shear design requirements for FRE-reinforced concrete beams without stirrups.) [33, 34]. Thus, they were used on experimental test data. Their predictions were calculated to evaluate the relative accuracy of the selected ANN 3L (2-6-2) model compared to other models recommended by code provisions and some published models (strength reduction coefficient assumed to be φ = 1). For a visual indication of the accuracy of equations given by ACI-440.1R-15 [30], CAN/CSA-S806-12 [31], BISE-99 [32], and the optimized equation by Nehdi et al. [14, 33], their predictive values are plotted against the experimental values and are illustrated in Figure 6. The three considered codes underestimate the shear strength, confirming the reports from other researchers [7, 17].Figure 6
Experimental vs. predicted values of shear strength for (a) ACI-440.1R-15 equation, (b) CSA-S806-12 equation, (c) BISE-99 equation, and (d) Nehdi et al. (optimized) equation.
(a)(b)(c)(d)In addition, the mean, standard deviation (SD), and coefficient of variation (CV) of the equations’ forecasted shear resistance were assessed. The values provided by the top four ANN models are given in Table4. The model is considered accurate when the mean value is near one and SD returns a low value. Compared to other ANN models, although ANN 3L (2-6-2) does not have a mean value closest to one, it is identified as the best model since its CV value is the lowest. Therefore, the model’s statistics are more reliable. The model’s coefficient of variation (R2) is also calculated and depicted in Figure 7. The ANN 3L (2-6-2) model has the most significant variation coefficient, and best describes the variation in the data.Table 4
Statistical index of experimental to the predicted shear strength of FRP-reinforced beams.
NumTopologyAllR2y = ax + bRMSEAAE1FA-ANN 3L (2-6-2)0.961y = 0.9734x + 3.83822.210.222FA-ANN 3L (8-2-4)0.956y = 0.9887x + 4.141524.030.263FA-ANN 3L (3-7-3)0.939y = 0.9713x + 4.446528.020.274FA-ANN 3L (3-8-2)0.950y = 1.0145x + 1.895626.410.275ACI-4400.315y = 0.166x + 23.551111.690.536CSA-S8060.451y = 0.3358x + 40.06288.280.227BISE-990.308y = 0.196x + 33.473104.760.368Nehdi et al0.881y = 0.9748x + 13.56841.910.25Figure 7
Coefficient of determination values for ANN models, multiple regression models, and models published in literature and codes.The Taylor diagram (Figure8) provides an additional visual comparison of the performance of the FA-ANN model and that of the other code models. It presents a graphical representation of each model’s applicability based on the root mean square-centered difference, correlation coefficient, and standard deviation. The results of the study indicate that the FA-ANN model better estimates the total deflection closest to the experimental shear strength of FRP-reinforced beams, followed by the model developed by Nehdi et al. Compared to the ACI-440 and BISE-99 code models, the CSA-S806 model produced higher root mean square-centered difference and SD values, indicating a model with less precision when approximating experimental data.Figure 8
Taylor diagram depiction of the various models’ projections.


### 4.3. Sensitivity Analysis

A sensitivity analysis determined the effect of inputs on the outputs [35]. In this work, the Gevrey et al. [35, 36] profile approach was implemented in MATLAB software [28]. This technique evaluates each input variable individually while keeping the others constant. During execution, the scale separated the range of each input variable into many equal intervals. The remaining variables were assigned to n distinct constant values, and the network output was calculated across the whole range of the chosen variable, resulting in n distinct output groups. Finding the median output for each input scenario was the last step in combining the n output groups. The minimum, first quartile, median, third quartile, and maximum were utilized as constant values for each variable. Lek offers the details of the method [35, 36]. The chosen scale was 192, as suggested by Gevrey et al. [35, 36].Figure9 depicts the explanatory variables’ relative influence and contribution (six inputs) on the response variable (shear strength).Figure 9
Relative contribution of six input parameters to the prediction of shear strength.The two most essential criteria are beam width and effective depth. Since average stress equals shear force divided by cross-sectional area, width increase directly affects shear strength. Since higher effective depth results in longer diagonal shear cracks at maximum loading, beams with greater effective depth have greater shear strength. Shear strength has the most negligible effect on concrete compressive strength.


### 4.4. Predictive Model and ANN Weights

Since the generated ANN 3L (2-6-2) would be useless without its source file, the weights and biases of the trained network are given in this study. The input data must be normalized with equation (1) using the minimum and maximum values from Table 1, and the output from the network must be denormalized using equation (3). The input is the 6×1 vector, a1. The shear strength is determined by the following equations [7, 37]:(3)a2=tanhϑ1×a1+b1,a3=tanhϑ2×a2+b2,a4=tanhϑ3×a3+b3,Vcpredictnormalized=tanhϑ4×a4+b4,Vcpredict=Vcpredictnormalized+12×Vmax−Vmin+Vmin,where tanh = hyperbolic tangent function; Vcpredict = predicted value of shear strength, and Vmax and Vmin = minimum and maximum shear strengths provided in Table 1. The weight (θ) and bias (b) matrices are provided in the following:(4)ϑ1=10−1×2.2164−0.68838.6643−10.0000−9.94213.37550.2448−2.89902.679210.0000−8.3346−2.2653,ϑ2=10−1×−10.00006.01456.7892−4.32312.17428.77698.36498.6428−6.4922−9.6582−5.2643−10.0000T,ϑ3=10−1×−3.9723−5.31907.23724.0631−0.68536.15898.4195−4.6722−9.4349−7.1096−2.5752−6.2628,ϑ4=10−1×9.5694−8.6457,b1=10−1×−3.29277.0137T,b2=10−1×9.8089−2.7279−9.96239.1751−3.7018−9.3895T,b3=10−1×1.2541−7.1203T,b4=0.6549.


## 4.1. Model Evaluation

Test data’s mean squared error (MSE) was chosen as the error metric to select the best-performing artificial neural network (ANN) among 45 trained architectures. The top four performing ANNs are given in Table3. The networks are sorted in the order of increasing values of the MSE value of test data. ANN identities are labeled with ANN mL (n1-n2-n3), where m is the number of hidden network layers, and n1, n2, and n3 are the number of neurons in the first, second, and third hidden layers, respectively.Table 3
Top four ANN statistics.
NumTopologyTrainTestR2y = ax + bRMSEAAER2y = ax + bRMSEAAE1FA-ANN 3L (2-6-2)0.972y = 0.9426x + 3.939420.180.210.952y = 1.0783x + 0.968326.370.252FA-ANN 3L (8-2-4)0.970y = 0.9554x + 3.968320.370.240.942y = 1.1022x + 1.685830.980.323FA-ANN 3L (3-7-3)0.955y = 0.9303x + 5.074925.140.260.928y = 1.1101x - 0.487333.840.304FA-ANN 3L (3-8-2)0.965y = 0.9696x + 2.850221.950.250.943y = 1.1669x - 4.126134.730.31Figure3 provides a visual representation of the performance of the top four networks by plotting the values predicted by the networks against their experimental values from the database.Figure 3
Experimental vs. predicted values of shear strength for (a) ANN 3L (2-6-2) model, (b) ANN 3L (8-2-4), (c) ANN 3L (3-7-3) model, and (d) ANN 3L (3-8-2) model.
(a)(b)(c)(d)Since the points in Figure3 are near the y = x line, the projected shear strength values correspond to the observed values. Figures 4 and 5 depict the training and testing performance of the leading network, ANN 3L (2-6-2).Figure 4
Comparison between ANN 3L (2-6-2) calculated and observation data used in training.Figure 5
Comparison between ANN 3L (2-6-2) calculated and observation data used in testing.As demonstrated in Figures4 and 5, the best-performing network’s projected values for both the training and testing phases closely match their experimental values. It is noteworthy that the ANN 3L (2-6-2) performs well even for one test case where the shear strength is about 810 kN, which is much higher than the rest of the test cases. The ANN 3L (2-6-2) model is chosen for further analysis.


## 4.2. Comparison with Existing Equations

ACI-440.1R-15 [30], CAN/CSA-S806-12 [31], BISE-99 [32], and Nehdi et al. (optimized equation method) provide the shear design requirements for FRE-reinforced concrete beams without stirrups.) [33, 34]. Thus, they were used on experimental test data. Their predictions were calculated to evaluate the relative accuracy of the selected ANN 3L (2-6-2) model compared to other models recommended by code provisions and some published models (strength reduction coefficient assumed to be φ = 1). For a visual indication of the accuracy of equations given by ACI-440.1R-15 [30], CAN/CSA-S806-12 [31], BISE-99 [32], and the optimized equation by Nehdi et al. [14, 33], their predictive values are plotted against the experimental values and are illustrated in Figure 6. The three considered codes underestimate the shear strength, confirming the reports from other researchers [7, 17].Figure 6
Experimental vs. predicted values of shear strength for (a) ACI-440.1R-15 equation, (b) CSA-S806-12 equation, (c) BISE-99 equation, and (d) Nehdi et al. (optimized) equation.
(a)(b)(c)(d)In addition, the mean, standard deviation (SD), and coefficient of variation (CV) of the equations’ forecasted shear resistance were assessed. The values provided by the top four ANN models are given in Table4. The model is considered accurate when the mean value is near one and SD returns a low value. Compared to other ANN models, although ANN 3L (2-6-2) does not have a mean value closest to one, it is identified as the best model since its CV value is the lowest. Therefore, the model’s statistics are more reliable. The model’s coefficient of variation (R2) is also calculated and depicted in Figure 7. The ANN 3L (2-6-2) model has the most significant variation coefficient, and best describes the variation in the data.Table 4
Statistical index of experimental to the predicted shear strength of FRP-reinforced beams.
NumTopologyAllR2y = ax + bRMSEAAE1FA-ANN 3L (2-6-2)0.961y = 0.9734x + 3.83822.210.222FA-ANN 3L (8-2-4)0.956y = 0.9887x + 4.141524.030.263FA-ANN 3L (3-7-3)0.939y = 0.9713x + 4.446528.020.274FA-ANN 3L (3-8-2)0.950y = 1.0145x + 1.895626.410.275ACI-4400.315y = 0.166x + 23.551111.690.536CSA-S8060.451y = 0.3358x + 40.06288.280.227BISE-990.308y = 0.196x + 33.473104.760.368Nehdi et al0.881y = 0.9748x + 13.56841.910.25Figure 7
Coefficient of determination values for ANN models, multiple regression models, and models published in literature and codes.The Taylor diagram (Figure8) provides an additional visual comparison of the performance of the FA-ANN model and that of the other code models. It presents a graphical representation of each model’s applicability based on the root mean square-centered difference, correlation coefficient, and standard deviation. The results of the study indicate that the FA-ANN model better estimates the total deflection closest to the experimental shear strength of FRP-reinforced beams, followed by the model developed by Nehdi et al. Compared to the ACI-440 and BISE-99 code models, the CSA-S806 model produced higher root mean square-centered difference and SD values, indicating a model with less precision when approximating experimental data.Figure 8
Taylor diagram depiction of the various models’ projections.


## 4.3. Sensitivity Analysis

A sensitivity analysis determined the effect of inputs on the outputs [35]. In this work, the Gevrey et al. [35, 36] profile approach was implemented in MATLAB software [28]. This technique evaluates each input variable individually while keeping the others constant. During execution, the scale separated the range of each input variable into many equal intervals. The remaining variables were assigned to n distinct constant values, and the network output was calculated across the whole range of the chosen variable, resulting in n distinct output groups. Finding the median output for each input scenario was the last step in combining the n output groups. The minimum, first quartile, median, third quartile, and maximum were utilized as constant values for each variable. Lek offers the details of the method [35, 36]. The chosen scale was 192, as suggested by Gevrey et al. [35, 36].Figure9 depicts the explanatory variables’ relative influence and contribution (six inputs) on the response variable (shear strength).Figure 9
Relative contribution of six input parameters to the prediction of shear strength.The two most essential criteria are beam width and effective depth. Since average stress equals shear force divided by cross-sectional area, width increase directly affects shear strength. Since higher effective depth results in longer diagonal shear cracks at maximum loading, beams with greater effective depth have greater shear strength. Shear strength has the most negligible effect on concrete compressive strength.


## 4.4. Predictive Model and ANN Weights

Since the generated ANN 3L (2-6-2) would be useless without its source file, the weights and biases of the trained network are given in this study. The input data must be normalized with equation (1) using the minimum and maximum values from Table 1, and the output from the network must be denormalized using equation (3). The input is the 6×1 vector, a1. The shear strength is determined by the following equations [7, 37]:(3)a2=tanhϑ1×a1+b1,a3=tanhϑ2×a2+b2,a4=tanhϑ3×a3+b3,Vcpredictnormalized=tanhϑ4×a4+b4,Vcpredict=Vcpredictnormalized+12×Vmax−Vmin+Vmin,where tanh = hyperbolic tangent function; Vcpredict = predicted value of shear strength, and Vmax and Vmin = minimum and maximum shear strengths provided in Table 1. The weight (θ) and bias (b) matrices are provided in the following:(4)ϑ1=10−1×2.2164−0.68838.6643−10.0000−9.94213.37550.2448−2.89902.679210.0000−8.3346−2.2653,ϑ2=10−1×−10.00006.01456.7892−4.32312.17428.77698.36498.6428−6.4922−9.6582−5.2643−10.0000T,ϑ3=10−1×−3.9723−5.31907.23724.0631−0.68536.15898.4195−4.6722−9.4349−7.1096−2.5752−6.2628,ϑ4=10−1×9.5694−8.6457,b1=10−1×−3.29277.0137T,b2=10−1×9.8089−2.7279−9.96239.1751−3.7018−9.3895T,b3=10−1×1.2541−7.1203T,b4=0.6549.


## 5. Conclusion

One hundred ninety-eight published experimental test results were compiled to solve the fiber-reinforced polymer (FRP) reinforced concrete beam shear strength prediction problem with longitudinal bars and without stirrups. The firefly algorithm (FA) was used to train the ANN models. The results analysis suggests the following:(1)
The trained FA-ANN 3L model (2-6-2) is more accurate than other ANNs with similar topologies for estimating concrete beam shear strength. This model’s RMSE and AAE for all available data were 22.21 and 0.22, respectively.(2)
Nehdi et al. introduced a straightforward experimental model for estimating the shear strength of concrete beams. It offered acceptable results; however, these results are less accurate than the proposed FA-ANN 3L (2-6-2) model.(3)
The trained ANN 3L (2-6-2) model can predict the shear strength more accurately than the three code provisions and two models suggested in the literature. The mean value and standard deviation of experimental to predicted values are 1.0342 and 0.3073, respectively.(4)
According to the sensitivity analysis results, the beam width and effective depth are the most influential parameters on the shear strength of FRP-reinforced beams. The least influential factor is concrete compressive strength.(5)
Based on the weights and biases of the ANN 3L (2-6-2) model (top-trained model), a predictive model was developed to enable access to the trained model without the source file.(6)
Findings confirm literature concerns that code standards underestimate the shear strength of FRP-reinforced concrete beams, which could increase costs and reinforce bar congestion.



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*Source: 4062587-2023-02-21.xml*