J. Cent. South Univ. (2019) 26: 3156-3174

DOI: https://doi.org/10.1007/s11771-019-4243-z

Modified imperialist competitive algorithm-based neural network to determine shear strength of concrete beams reinforced with FRP

Amir HASANZADE-INALLU¹, Panam ZARFAM², Mehdi NIKOO³

1. Department of Earthquake Engineering, Science and Research Branch, Islamic Azad University,Tehran, Iran;

2. Department of Structural Engineering, Science and Research Branch, Islamic Azad University,Tehran, Iran;

3. Young Researchers and Elite Club, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Fiber reinforced polymers (FRPs), unlike steel, are corrosion-resistant and therefore are of interest; however, their use is hindered because their brittle shear is formulated in most specifications using limited data available at the time. We aimed to predict the shear strength of concrete beams reinforced with FRP bars and without stirrups by compiling a relatively large database of 198 previously published test results (available in appendix). To model shear strength, an artificial neural network was trained by an ensemble of Levenberg-Marquardt and imperialist competitive algorithms. The results suggested superior accuracy of model compared to equations available in specifications and literature.

Key words: concrete shear strength; fiber reinforced polymer (FRP); artificial neural networks (ANNs); Levenberg-Marquardt algorithm; imperialist competitive algorithm (ICA)

Cite this article as: Amir HASANZADE-INALLU, Panam ZARFAM, Mehdi NIKOO. Modified imperialist competitive algorithm-based neural network to determine shear strength of concrete beams reinforced with FRP [J]. Journal of Central South University, 2019, 26(11): 3156-3174. DOI: https://doi.org/10.1007/s11771-019-4243-z.

1 Introduction

Corrosion significantly decreases the lifetime of concrete structures reinforced with steel [1]. Attempts to find a durable and corrosion-resistant material have drawn interest of researchers to fiber reinforced polymers (FRPs) [1-5]. Additionally, compared to steel, FRPs have other desirable features [6, 7] such as lower weight, higher strength, and higher axial stiffness-to-weight ratio [8-12].

Shear equations for concrete are generally semi-empirical [1, 13, 14]. Effective parameters on shear include shear friction, shear strength of uncracked compressive region of concrete, arch action, dowel action of longitudinal bars, and residual tensile stresses along cracks [1, 15]. Shear design equations for concrete members reinforced with FRP bars and without stirrups given by ACI 440.1R-15 [16], CAN/CSA S806-12 [17], BISE-99 [18], JSCE-97 [19], ISIS-M03-07 [20] and CNR-DT 203/2006 [21] are listed in Table 1. Table 1 also lists some of the proposed shear strength equations in Refs. [13, 22, 23].

Most current specifications and standards for concrete structures with FRP reinforcement are based on equations for steel-reinforced concrete members, and because of material differences between steel and FRP, some corrections are made in these equations [24]. However, these equations differ a lot in the choice of main parameters affecting shear strength [10]. For instance, to account for difference between modulus of elasticity of FRP (E_f) and that of steel (E_s), JSCE-97 [19], ISIS-M03-07 [20], CNR-DT 203/2006 [21], [24], NEHDI et al [13, 23] and NEHDI et al (Optimized equation) [13, 23] applied a correctional coefficient (E_f/E_s); however, this coefficient is raised to different powers in various equations [25, 26].

Table 1 Design equations and proposed models available in literature

Unlike steel, which yields, FRP has brittle rupture, which can be unsafe. As a result, current equations in specifications are cautious in predicting FRP shear strength, which can lead to higher bars in design and result in reinforcement congestion as well as higher cost for concrete elements reinforced with FRP. It’s worth noting that many of equations in specifications were formulated using limited available experimental data at the time [1, 23, 27].

Artificial neural networks (ANNs) are computational tools that are trained using collected data and can be utilized to predict output for new input [26, 29]. In recent decades, they have been successfully employed in numerous engineering problems [26, 29-39]. Some researchers have successfully used artificial intelligence methods to predict shear strength of concrete beams with FRP reinforcement [10, 25, 26, 40-42].

To minimize prediction errors, ANNs must be trained using an optimization algorithm. The Levenberg-Marquardt algorithm is a fast and efficient optimization algorithm; however, one of its disadvantages is its inability to pass local minimums [43]. Imperialist competitive algorithm is another optimization algorithm inspired by the socio-political evolution of humans. In this algorithm, some imperialist countries along with their colonies try to solve the optimization problem [30, 44-46].

In this study, we aimed to predict the shear strength of concrete beams reinforced with longitudinal FRP bars and without stirrups by compiling a database of 198 test results published in the literature and by using an ensemble of Levenberg-Marquardt and imperialist competitive algorithms. Section 2 summarizes ANNs and imperialist competitive algorithm. Section 3 details the methodology used to train ANNs using ensemble algorithm. Results and discussion are provided in Section 4 followed by conclusion in Section 5.

2 Artificial neural networks and imperialist competitive algorithm

2.1 Artificial neural networks and Levenberg- Marquardt algorithm

Artificial neural networks (ANNs) are mathematical models inspired by the configuration of biological nervous systems. ANNs predict output (s) based on input, and consist of information processing units called neurons. One of the commonly used types of ANNs is feedforward networks in which neurons are arranged in layers that include one input layer, one or more hidden layers, and one output layer. Neurons in each layer are connected to all neurons in layers immediately before and after them. The network architecture is determined by the number of layers and the number of neurons in each layer [47-52].

Connection links between neurons have weights associated with them. The net input of a neuron is the sum of weighted outputs from neurons of the previous layer. Each neuron applies a function, called the activation function, to its net input to determine its output. In a feedforward ANN, information flows through the network by feeding the inputs in the input layer to the neurons in the first hidden layer to determine their outputs. These outputs are then fed to the next layer and this process is repeated until the output (s) of the ANN is determined. The weights of an ANN are usually initialized randomly and therefore the network output will have an associated error. To minimize the output error, the optimum weights of the ANN must be determined using the training data. This optimization process is called training the network [47-52].

If the error measure is mean squared error (MSE), the Levenberg-Marquardt algorithm, which is a fast optimization algorithm, can be used to train the network; however, this algorithm has the disadvantage of getting stuck at local minima and slow rate of convergence [43, 44, 48]. The implementation of this algorithm is included in the MATLAB [53] software and is used in this study.

2.2 Imperialist competitive algorithm

Imperialist competitive algorithm (ICA) is an evolutionary optimization technique inspired by socio-political evolution of humans. It starts with an initial population (countries in the world) and then best (lower cost) countries are selected as imperialists and the rest form the colonies. In an n-dimensional optimization problem, a country is a 1×n array, the cost of which is found by evaluating cost function at that 1×n array. The outline of this algorithm is shown in Figure 1. The main components of the initially proposed algorithm are initialization, assimilation, and imperialist competition [46]. Revolution is a component that was later proposed in code implementation [54] and in an enhanced version of the algorithm [55].

Initialization begins by creating some random solutions as initial countries and then sorting them based on ascending values of cost function. A number of low-cost solutions are selected as imperialists and the rest form the colonies. The colonies are then divided among imperialists proportional to imperialists’ power, which is obtained from imperialist’s normalized cost. Assimilation is the movement of a colony towards its imperialist, which is modeled by a vector from colony to imperialist. In imperialist competition, some (usually one) of the weakest colonies of the weakest empires will be possessed by more powerful empires, and empires with no colonies left will collapse. The algorithm iterates through this cycle until all the empires, except the most powerful (lowest cost) one, are collapsed. The imperialist of that empire becomes the solution of the optimization problem [46, 55].

Figure 1 Flowchart of imperialist competitive algorithm [46]

3 Methodology

3.1 Introduction of parameters influencing shear strength

Effective parameters on shear strength of concrete beams with FRP reinforcement bars are: concrete compressive strength (f′_c), FRP longitudinal reinforcement ratio (ρ_f), FRP bar modulus of elasticity (E_f), shear span to depth ratio (a/d), web thickness (b_w) and effective depth (d) [22, 25, 56-61]. These six parameters were selected as the input of the neural network and shear strength of beam as the output. Accuracy of ANN model, to a great extent, depends on the number of samples used for training [25], so a relatively large database was compiled consisting of 198 test results published in literature for concrete beams reinforced with FRP bars and without stirrups. These data are included in Appendix for reference and their statistics are listed in Table 2.

3.2 Training of ANN by ensemble of Levenberg- Marquardt and imperialist competitive algorithms

The number of hidden layers and neurons of an ANN is problem-specific [1, 10], so, a total of 351 different architectures having one, two, or three hidden layers were trained. The total sum of neurons in hidden layers ranged from 6 to 13. For architectures with one hidden layer, networks with 6 to 13 neurons were trained, making 8 different architectures. For networks with two hidden layers, first ANN had 1 neuron in the first hidden layer and 5 in the second, making a total of six neurons. The next ANN had 2 neurons in the first hidden layer and 4 in second, and so on. The last two hidden layers architectures had 12 neurons in the first hidden layer and 1 in the second, totaling 13 neurons. The total number of two hidden layer architectures were 68. For three hidden layers, first ANN had 1 neuron in the first hidden layer, 1 in the second, and 4 in the third. The next ANN had 1 neuron in the first hidden layer, 2 in second and 3 in the third, and so on. A total of 275 different architectures with three hidden layers were trained with the last one having 11 neurons in the first hidden layer, 1 in the second, and 1 in the third, totaling thirteen neurons.

Table 2 Statistics of experimental data (198 beams)

To avoid overfitting, a known phenomenon in training an ANN, data were randomly divided into 3 groups, 70% for training, 15% for evaluation, and 15% for testing [48, 49]. Hyperbolic tangent function was chosen as the activation function of neurons in hidden and output layers.

In modeling, scaling is always useful [25] and its goal is to set the input data into a specific range. If scaling is not used, it is possible for a training algorithm to converge slowly or even diverge [49]. To scale data into the range of -1 to 1, each of the input parameters and the output parameters was substituted in the following relation [25]:

                       (1)

where X_n is the scaled (normalized) value of the parameter; X_max is the maximum value of the parameter; X_min is the minimum value of the parameter; X is the un-scaled value of the parameter [25]. The scaling process is a pre- processing step on data. To utilize the trained ANN for new data not encountered in the training step, the new data should also be scaled using Eq. (1), and the maximum and minimum values used in training should be substituted into the equation. After obtaining the result from the ANN, it is necessary to un-scale the output into its original range used in training [25].

The coding was done in MATLAB R2016a [53] software. First, each of the inputs was normalized into a range of -1 to 1 using Eq. (1) and then for each of the 351 architectures mentioned before, the ANN was trained using Levenberg-Marquardt algorithm, and if the trained network had an MSE of 0.02 or less, the training data and the network were saved and next step commenced. Otherwise, the network was trained again, up to 10 times, and if an MSE of 0.02 or less could not be reached, this threshold was increased by 0.01 to reach a threshold of 0.03. The iterations continued until reaching a defined threshold in 10 epochs or increasing the threshold by 0.01 again and continuing the iterations.

In training an ANN, the solutions of the optimization algorithm are the weights that minimize the network error. The original ICA uses random numbers as the initial countries (solutions to the optimization problem) and then chooses a predefined number of countries as imperialists based on their cost being low. Instead of using random low-cost solutions as the initial imperialists, we initialized the predefined number of imperialists with the weights of the trained network from the previous step. This enables us to use the best solution obtained by LM as the starting point of ICA. Training data were the same previously saved training data and testing data were the remaining data. The parameters used in training with imperialist competitive algorithm are listed in Table 3. See Figure 2 for an outline of this ensemble algorithm. The inspiration for using an ensemble of ICA and LM was that other ensemble methods published in Refs. [62–64] have reported satisfactory results.

Table 3 Input parameters used in training

Figure 2 Flowchart of proposed ensemble training algorithm

After training ANN using ICA, error metrics for training and testing data were calculated and saved. Also, the trained ANN computer file was saved for future use. The process described in this subsection was then applied to the remaining architectures.

4 Results and discussion

From the 351 trained artificial neural networks (ANNs), five were selected that had the least mean squared error (MSE) of normalized test data. Low MSE of test data means that the ANN can generalize better and is less prone to overfitting. Table 4 shows the error metrics for these five ANNs. Network names are designated with ANN-mL (n₁-n₂-n₃) where m designates number of hidden layers, and n₁, n₂, and n₃ designate number of neurons in the first, second, and third hidden layers respectively.

To analyze the effect of using the ensemble method, models were also trained using ICA and LM algorithms alone. The training parameters were the values given in Table 3. The MSE of models trained using LM, ICA, and ensemble algorithm are given in Table 5. It is obvious that, training using the ensemble method improves the accuracy of the network.

Table 6 shows the statistics of experimental to predicted shear strength for code equations, models in literature (calculated with strength reduction coefficient φ=1), and ANN models. Looking at the means, it is obvious that ACI 440.1R-15 [16], BISE-99 [18], JSCE-97 [19], and ISIS-M03-07 [20] equations underestimate the shear strength of FRP reinforced beams. Figures 3 depicts experimental versus predicted shear strength of ANN-3L (4-2-7), ACI 440.1R-15 [16], CNR-DT 203/2006 [21, 24], and NEHDI et al [13, 23] (optimized equations) models. Among the previously published equations, NEHDI et al (optimized equation) is selected because it has a mean closest to 1 and therefore predicts shear strength more accurately. Shear strength values predicted by NEHDI et al (optimized equation) are included in Appendix. By comparing the means, it can be concluded that the ANN models predict the shear strength more accurately than previously published equations.

To analyze the sensitivity of predictions on variation of the input parameters, the graph of (V_c^exp./V_c^predict.) versus each of the six input parameters is depicted in Figures 4 and 5 for ANN models and equations in literature respectively. Among the ANN models, ANN-3L (4-2-7) shows less sensitivity with respect to variation of the input parameters and therefore is selected for further analysis. Shear strength values predicted by ANN-3L (4-2-7) are included in Appendix. Referring to Table 6, the mean, standard deviation, and coefficient of variation on experimental to predicted shear strength of ANN-3L (4-2-7) model are 1.0032, 0.1918, and 0.1912 respectively.

Table 4 Statistics of top five ANNs

Table 5 Statistics of top five ANNs trained with LM, ICA, or ensemble algorithm

ANN-3L (4-2-7) model is least sensitive to changes of web thickness (b_w), effective depth (d), and FRP longitudinal reinforcement ratio (ρ_f), i.e., there is no considerable difference between experimental and predicted values of shear strength for the ranges of these parameters.

If we wish to keep (V_c^exp./V_c^predict.) between 0.95 and 1.05, ANN-3L (4-2-7) satisfies this requirement for shear span to depth (a/d) values less than 5.5. We believe that this is attributed to having only 25 (13%) beams having a/d more than 5.5, and therefore more samples in this range may be needed. For values of concrete compressive strength (f′_c) between about 27 (MPa) and 52 (MPa), the value of (V_c^exp./V_c^predict.) is between 0.95 and 1.05 which may be because 22 (11%) beams have f′_c more than 52 (MPa). Although 40 (20%) beams have a value of FRP bar modulus of elasticity (E_f) more than about 113000 MPa, the value of (V_c^exp./V_c^predict.) is not between 0.95 and 1.05 for this range of E_f.

To make use of ANN-3L (4-2-7), weights and biases were extracted from its trained model. Before using input data, they must be scaled using Eq. (1), by substituting the max and min values from Table 2 for each of the six inputs. The input is denoted by a 6×1 vector called a⁽¹⁾. To calculate shear strength estimate, Eqs. (2)-(6) must be used.

　　　　　　　　　　(2)

                   (3)

                   (4)

             (5)

    (6)

where V_max and V_min are their indicated values in Table 2 and

θ⁽¹⁾=;

;

;

;

;

;

;

.

Table 6 Ratio of experimental to predicted shear strength of FRP reinforced beams without stirrups

Figure 3 Experimental versus predicted value of shear strength for ANN-3L(4-2-7) model (a), ACI 440.1R-15 (b), CNR-DT 203/2006 (c) and NEHDI et al (Optimized) (d)

5 Conclusions

Using the dataset of 198 experimental results collected from literature, different architectures of ANNs were trained using an ensemble algorithm to predict the shear strength of concrete beams with longitudinal FRP reinforcement and without stirrups. The following conclusions were drawn:

Figure 4 Sensitivity of ANN shear strength models with respect to a/d (a), f′_c (b), ρ_f (c), E_f (d), b_w (e) and d (f)

1) Equations provided by ACI 440.1R-15 [16], CAN/CSA S806-12 [17], BISE-99 [18], JSCE-97 [19], ISIS-M03-07 [20] and CNR-DT 203/2006 [21], NEHDI et al [13, 23], and EL-SAYED et al [22] underestimate the shear strength of concrete beams reinforced FRP bars.

2) The ANN-3L (4-2-7) model showed superior accuracy in comparison to other top ANN models and existing equations. It had a mean of 1.0032, standard deviation of 0.1918, and coefficient of variation of 0.1912 for experimental to predicted values of shear strength.

Figure 5 Sensitivity of existing shear strength equations with respect to a/d (a), f　′_c(b), ρ_f (c), E_f (d), b_w (e) and d (f)

3) The ANN-3L (4-2-7) model showed to be least sensitive to changes of web thickness (b_w), effective depth (d), and FRP longitudinal reinforcement ratio in comparison to other ANN models and existing equations. For a/d, f′_c, and E_f, the requirement of (V_c^exp./V_c^predict.) being between 0.95 and 1.05 was satisfied for values 900 of a/d less than about 5.5, f′_c between about 27 MPa and 52 MPa, and E_f less than about 113000 MPa.

4) To predict shear strength with values of a/d more than 5.5, f′_c larger than 52 MPa, and E_f larger about 113000 MPa more accurately, additional experiments are needed.

5) To utilize the ANN-3L (4-2-7) model, the weights and biases of the model were extracted and used to develop a series of equations to enable the computation of shear strength of concrete beams reinforced with FRP bars and without stirrups.

6) Training the ANN using the ensemble of Levenberg-Marquardt and Imperialist Competitive algorithms led to more accurate results compared to training the ANN using each algorithm separately.

Appendix

Experimental data previously published in literature

to be continued

Continues

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Continues

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Continues

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(Edited by YANG Hua)

中文导读

基于改进的帝国主义竞争算法的神经网络FRP加固混凝土梁的抗剪强度

摘要：纤维增强聚合物(FRPS)与钢不同，是耐腐蚀的，因此引起人们的兴趣；然而，它们的使用受到限制，因为它们的脆性剪切在大多数规范中可使用的数据有限。我们的目的是通过编制一个比较大的数据库来预测FRP筋和无箍加固混凝土梁的抗剪强度，该数据库包含198份以前公布的试验结果(见附录)。为了建立抗剪强度模型，利用Levenberg-Marquardt和帝国主义竞争算法的集合训练了一个人工神经网络。结果表明，与规范和文献中的方程相比，模型具有更高的精度。

关键词：混凝土抗剪强度；纤维增强聚合物(FRP)；人工神经网络(ANNS)；Levenberg-Marquardt算法；帝国主义竞争算法(ICA)

Received date: 2018-08-03; Accepted date: 2019-06-17

Corresponding author: Panam ZARFAM, PhD; Tel: +98-912-134-2896; E-mail: zarfam@srbiau.ac.ir; ORCID: 0000-0002-7951-0630