Abstract
This study investigates an adaptiveweighted instancedbased learning, for the prediction of the ultimate punching shear capacity (UPSC) of fiberreinforced polymer (FRP) reinforced slabs. The concept of the new method is to employ the Differential Evolution to construct an adaptive instancebased regression model. The performance of the proposed model is compared to those of Artificial Neural Network (ANN) and traditional formulabased methods. A dataset which contains the testing results of FRPreinforced concrete slabs has been collected to establish and verify new approach. This study shows that the investigated instancebased regression model is capable of delivering the prediction result which is far more accurate than traditional formulas and very competitive with the blackbox approach of ANN. Furthermore, the proposed adaptiveweighted instancedbased learning provides a means for quantifying the relevancy of each factor used for the prediction of UPSC of FRPreinforced slabs.
1. Introduction
In civil engineering, fiberreinforced polymer (FRP) composites have been increasingly employed due to their strength and stiffness, good thermomechanical properties, capacity for resisting corrosion, low weight, and outstanding durability [1–4]. It is proper to note that corrosion of steel reinforcement is the major factor which influences the deterioration and shortens the service life of reinforced concrete structures [5–7]. Thus, the utilizations of FRP composites have created the condition for enhancing the productivity of construction process, meliorating the performance of concrete structures, reduction of maintenance budgets spent for infrastructure, and possible elongation of structure service lives [8, 9].
Steelreinforced twoway flat slabs are popular structural systems that can simplify and accelerate onsite operations and facilitate flexible partitioning of space [10]. The twoway flat slabs are particularly efficient for constructing parking structures. Additionally, computing the slab’s resistance of shear stresses at supporting columns is a major concern in the design procedure of this structure [11]. Notably, the slabsupporting column connections have been shown to be vulnerable to high shear stresses and this might bring about brittle and sudden punching shear failures [10]. Especially when the steel reinforcements get corroded due to moisture and other hostile factors in the operational environment, punching shear failures may occur at these slabcolumn connections. Accordingly, these may lead to progressive collapse of the whole structure [12].
Due to such reasons, FRP bars are recently considered as effective substitutions to the traditional steel bars in concrete flat slabs [13]. Moreover, UPSC is critical factor which determines the design process of concrete slabs supported by columns. Thus, this problem is extensively studied in the literature [14]. As a consequence, various researches have been conducted to investigate the applicability of existing empirical approaches and modify them to predict the punching shear capacity of FRPreinforced slabs [1, 10, 15, 16].
Recently, machine learning has been proved to provide a feasible alternative for modeling the punching shear capacity of FRPreinforced slabs [6, 9, 15, 17]. The Artificial Neural Network (ANN) has been employed to predict the FRPreinforced slab punching shear capacity [6]. The research shows that the predictive result produced by ANN is considerably more accurate than those computed by the empirical formulas. Despite the fact that ANN is a powerful method for modeling nonlinear systems [18, 19], the learning process of ANN suffers from certain challenges [20].
One major difficulty of ANN is that its model establishment stage is accomplished via a gradient descent (GD) algorithm. GD algorithm is known to be very complex and may contain many local minima [20]. Another disadvantage of the ANN approach lies in their knowledge representation [21]; the blackbox nature of the method makes it difficult for structural engineers to comprehend how the ANN predicts the punching shear capacity of FRPreinforced slabs. In addition, the ANN approach intrinsically provides no means of measuring the contribution of each input factors to the model performance.
This research aims at extending the body of knowledge by investigating a new learning alternative for modeling the punching shear capacity of FRPreinforced slabs. The proposed approach is hybridization of an improved kernel regression and the Differential Evolution (DE) [22]. The research objective is to establish a prediction model with transparent structure, selfadaptive learning, and the capability to express the relevancy of input variables.
Notably, the kernel regression belongs to the class of instancebased regression, which is also called nonparametric regression. Nonparametric regression offers a flexible and effective way of approximating the regression function especially when the form of the regression function is inherently complex [23, 24]. The reason is that, in such circumstance, it can be difficult to construct a universal parametric model based on a limited amount of training samples. Since good performances of the nonparametric approach have been observed throughout the literature [21, 25–28], the nonparametric regression is worth being investigated in solving the problem of interest.
Furthermore, learning based on instances or examples is a common practice in construction engineering. Thus, an instancebased model for punching shear capacity of FRPreinforced slabs can be easily perceived by practical engineers; this may facilitate the applicability of the new approach. The rest of the article is organized as follows. The second section introduces the research method. The third section describes the proposed hybrid instancebased learning, followed by the experimental results. The final section summarizes our research with several conclusions.
2. Research Methodology
2.1. A Review of Formulas for Estimating Punching Shear Capacity of FRPReinforced Slabs
The shear resistance of the concrete () is known to influence the punching shear capacity of twoway reinforced concrete flat slabs. acts over the area proportional to the length of a critical perimeter () multiplied by the effective depth of the section (). Currently, there is a critical need to investigate design equations and prediction models for determining the punching shear strength of concrete slabs reinforced with FRP composite bars. The reason is that existing design equations applied for FRPreinforced concrete sections are originated from those previously applied for steelreinforced counterparts with certain adjustments for considering the replacement of steel by FRP. The following section reviews formulabased methods for the prediction of UPSC of FRPreinforced slabs; it is noted that the system of units for all equations is SI.
The American Concrete Institute (ACI) Code (ACI 31811) introduces a design formula to account for the shear transfer in twoway steelreinforced concrete slabs:where denotes the specified compressive strength of the concrete (MPa), represents the perimeter of the critical section for slabs and footings at a distance of away from the column face, and is the average flexural depth of the slab.
In addition, The British Standard (BS 811097) suggests a formula to attain the punching shear capacity for steelreinforced slabs as shown below:where denotes the characteristic concrete cube compressive strength (N/mm^{2}), represents the steel reinforcement ratio, is the perimeter of the critical section for slabs and footings at a distance of 1.5d/2 away from the loaded area (mm), and represents the average flexural depth of the slab.
On the basis of experiments, ElGhandour et al. [29] suggested a modification to the ACI’s equation by multiplying it by the term to account for the use of FRP bars as follows: where and are Young’s modulus of the FRPreinforced slab and Young’s modulus of the steel reinforcement, respectively.
ElGhandour et al. [30] modified the BS 811097’s design equation and suggested an alternative formula to obtain the shear strength of FRPreinforced concrete slabs as follows:
Matthys and Taerwe [31] put forward an enhancement of the BS 811097 as follows:
Ospina et al. [32] introduced an improved version of the equation proposed by Matthys and Taerwe [31]; in this revision, the cube root of the modular ration is replaced by the square root. This design formula is shown as follows:
A design equation has been proposed by the subcommittee ACI440H [16] for calculation of steelreinforced twoway concrete slabs. This equation has considered the influence of reinforcement stiffness to account for the shear transfer in twoway concrete slabs as follows:where is the perimeter of the critical section for slabs and footings at a distance of away. And is the cracked transformed section neutral axis depth (mm) and is calculated as follows:where is defined in the following equation:
In (9), it is noted that = / denotes modular ratio and is the concrete modulus of elasticity.
2.2. The Collected Dataset of Punching Shear Tests
The dataset of FRPreinforced concrete flat slab used in this study consists of 82 tests recorded in previous research works [6, 15, 33]. As the previous work of Vu and Hoang [17], the type of column section (), section area of column (), effective flexural depth of slab (), compressive strength of concrete (), Young’s modulus of the FRPreinforced slab (), and reinforcement ratio () are considered as input factors that determine the ultimate punching capacity of the FRPreinforced concrete flat slab. The shape of the column section includes three forms: square (), circle (), and rectangle (). In the dataset, the numbers of square, circle, and rectangle sections are 50, 13, and 19, respectively. The range of the punching capacity varies from 61 kN to 1600 kN. Table 1 shows the variables and their statistical descriptions. The whole dataset is provided in the Appendix.
2.3. InstanceBased Regression
Given a set of independent observations , where and are called predictor variable and response variable, respectively, and denotes the number of data points, the task of regression analysis is to construct a function that provides the mapping relation between the predictor variable and the response variable:where represents a random noise. is generally called the regression function or regression of on X, which can be employed to predict the value of based on an unknown input variable X.
The approaches of regression analysis can be categorized into two major classes: parametric model and nonparametric model [34]. In the former class, the underlying functional form of the model is known and the model’s free parameters are estimated directly from the dataset at hand. Linear/nonlinear regression models are the typically parametric and the Artificial Neural Network (ANN) can be considered as an advanced form of nonlinear parametric model. In the latter class, there is no restriction on the functional form of the regression model; the model generally stores all the collected data instances in its memory and utilizes such memory to predict unknown input data [24, 35]. Therefore, the approach of nonparametric learning is also called instancebased learning.
Previous studies found that the restriction of to a certain parametric form can limit the flexibility of the learning capability. A particularly undesirable situation happens when a parametric model that is inappropriate for the dataset is chosen. In such case, the model in no way can reflect the nature of the dataset regardless of how accurately its free parameters are estimated. Furthermore, this shortcoming of parametric models can be circumvented by abandoning the restriction on the functional form of . This leads to the method of nonparametric learning or instancebased learning.
Instancebased learning offers an approach for constructing the mapping function with a greater degree of flexibility [24]. This approach is particularly useful when the underlying form of the regression function is very complex. In such a case, it is much more efficient to construct a series of simple local regression models instead of a complicated global regression model. Furthermore, a significant advantage of instancebased learning is that its prediction process can be easily comprehended by practical engineers; this is in contrast to the blackbox learning approach used in the ANN.
In this research, the instancebased learning approach of interest is the kernel regression with the utilization of kernel smoothing. In the kernel regression, the estimated value of the response variable is computed by a weighted average of all the data stored in the model’s memory. The predicted value of the predictor variable , denoted as , is computed as follows [36–38]:where represents the th data point in the memory which consists of data points.
The Gaussian kernel is commonly selected to calculate the weight as follows:where the parameter denotes the kernel width and is the similarity between the two data points and .
Herein, the Minkowski distance, a generalization of both the Euclidean distance and the Manhattan distance, is used to measure the similarity between data points:where is the data dimension and is a scalar. It is noted that when and , the Minkowski distance is equivalent to the Euclidean distance and Manhattan distance, respectively.
In addition, it is noted that the relevance of each input factor, or feature, can be quantified by associating it with a weighting value which varies from 0 to 1. The higher the weighting value is, the more relevant the input attribute is. The feature weights can be embedded into the distance function to measure the similarity between two data instances. The incorporation of feature weight into the instancebased regression model and more detailed explanation are discussed in Section 3 of the article.
2.4. Differential Evolution (DE)
DE is an evolutionary algorithm developed by Storn and Price [22]. This metaheuristic is a populationbased stochastic approach which is very effective for global optimization in continuous domains [39, 40]. The DE algorithm relies on a novel crossovermutation operator which is a linear combination of three different individuals and one subjecttoreplacement parent (also called target vector) [41, 42]. This crossovermutation operator of the DE algorithm aims at producing a trial vector (also called child vector) which must compete with its parent through the selection process. The selection operator of DE is a fitness based comparison between the parent and the corresponding offspring [43].
The DE algorithm [22] is unquestionably one of the most powerful approaches for solving complex optimization problems [44, 45]. Successful applications of the DE optimization algorithm as well as other metaheuristic approaches for solving complex or illdefined engineering problems have been observed in various fields [46–51]. Given that the problem of interest is to minimize a cost function , where the number of decision variables is D, the DE algorithm can be described in Algorithm 1. In Algorithm 1, it is noted that denotes a random integer.

Initially, a population consisting of Ddimensional parameter vectors is randomly generated, where and denotes the current generation number. Accordingly, a mutant vector is computed for each target vector in the following manner:where , , and are random indexes lying between 1 and NP. NP represents the population size. F denotes the mutation scale factor, which controls the amplification of the differential variation. represents the newly created mutant vector.
The crossover operator of DE aims at increasing the diversity of the current population by exchanging components of the target and mutant vectors. Within the crossover phase, a trial vector is generated in the following way:where represents the trial vector. j denotes the index of element for any vector. rand_{j} is a uniform random number lying between 0 and 1. Cr denotes the crossover probability. represents a randomly chosen index of which guarantees that at least one parameter from the mutant vector () is copied to the trial vector ().
The selection stage compares the trial vector with the target vector to identify a better solution for the optimization problem at hand. The selection operator is demonstrated as follows:
2.5. Artificial Neural Network (ANN)
ANN is a soft computing method which is inspired from biological neural networks. This technique simulates the knowledge acquisition and inference processes in the human brain. The advantages of the ANN can be realized by its flexibility and universal learning capability. Nevertheless, primary drawbacks of the ANN are the trialanderror process for properly determining a network configuration and its blackbox learning nature.
The learning task is to train a function , where denotes the number of input attributes. An ANN model, which includes the input, hidden, and output layers, is demonstrated in Figure 1. and represent weight matrices of the hidden layer and the output layer, respectively; N denotes the number of neurons in the hidden layer; represents a bias vector of the hidden layer; is a bias vector of the output layer; denotes an activation function (e.g., logsigmoid).
The ANN structure used for regression analysis is shown as follows [52]:
Generally, the weight matrices and the bias vectors of an ANN are trained via a process that employs the framework of error backpropagation [19]. Moreover, the Mean Square Error (MSE) is employed as the objective function for training an ANN structure for function approximation tasks [53]:where represents the number of data samples and is an output error. ( and denote the predicted and actual outputs, resp.).
3. The Proposed AdaptiveWeighted InstanceBased Regression Model for the Prediction of Punching Shear Capacity of FRPReinforced Concrete Slabs
This section of the article describes the proposed instancebased regression model for the prediction of punching shear capacity of FRPreinforced concrete slabs, denoted as IRMPSC in detail. The main idea of the IRMPSC is to employ the DE optimization algorithm to select an appropriate set of the instancebased regression model’s parameters including the kernel width (h), the Minkowski distance scalar (q), and the feature weights (). The proposed model consists of five major steps: data processing, model construction and model prediction, Differential Evolution optimization, model evaluation, and optimized model. Figure 2 illustrates the IRMPSC’s flowchart. In this figure, and represent the current generation and the maximum generation of the DE algorithm, respectively.
3.1. Data Processing
The dataset including 82 punching tests of FRPreinforced concrete slab collected from the literature is employed to construct and verify the prediction model. The type of column section, section area of column, effective flexural depth of slab, compressive strength of concrete, Young’s modulus of the FRPreinforced slab, and reinforcement ratio are considered as influencing factors that determine the ultimate punching capacity of the FRPreinforced concrete flat slab. The whole dataset is randomly separated into three sets: the training, validating, and testing sets. The training set is used to establish the model memory; the other two datasets are employed to validate and test the model performance. It is noted that, before being separated, the whole dataset has been normalized into the range of . This data normalization aspires to prevent the situation in which features with greater numeric magnitudes dominate those with smaller magnitudes.
3.2. Model Construction and Model Prediction
At this phase, the kernel regression is utilized as the instancebased learning approach to model the mapping relation between the ultimate punching capacity of the FRPreinforced concrete flat slab and its six influencing input variables. The model simply stores all data samples of the training set in its memory. When new input data is available, the model computes the kernel weights using this new sample and other existing samples. Accordingly, the estimated punching capacity of the new input data is calculated as a weighted average of all punching capacities of all the data in the memory.
Furthermore, it is noted that, in this study, the relevance of the input feature is quantified by its feature weight varying from 0 to 1. As mentioned earlier, a higher feature weight means that the feature of interest is more relevant for estimating the output. The feature weights are embedded into the distance measurement between two data instances. Accordingly, the modified Minkowski distance is provided as follows:
It is worth noticing that each data feature represents an axis in the input space. For the problem at hand, the input space is a sixdimensional space. As can be seen from (19), the introduction of the feature weight can elongate or truncate the axis in the input space. A high elongates the axis k; this makes the feature very significant for the prediction process. The reason is that when the axis is elongated, even a small difference of can result in a large kernel weight. The reversed result occurs when an axis is truncated. In the extreme case when , the quantity imposes no influence on the final kernel weight between two data instances. It is noted that the feature weight used in this section is different from the Gaussian kernel weight () defined in (11) and (12). The feature weight expresses the relevancy of input variable; meanwhile, the Gaussian kernel weight exhibits the data similarity.
3.3. Differential Evolution Optimization
The construction of the model necessitates the kernel width (h), the Minkowski distance scalar (q), and the feature weights () as hyperparameters. At each iteration, the DE algorithm guides the population of vectors to search for better solutions. Gradually, inferior solutions characterized with worse values of fitness are discarded; superior solutions with better values of fitness survive to the next iterations. The DE’s search progresses until the current generation () exceeds the maximum generation ().
3.4. Model Evaluation
To identify an appropriate set of hyperparameters, it is necessary to define an objective function that quantifies the model performance. In this study, the model’s prediction accuracy in terms of Root Mean Squared Error (RMSE) when estimating the validating dataset is employed for model evaluation:where and denote predicted and actual value for output th. In addition, N is the number of data samples in the validating set.
3.5. Optimized Model
When the DE based searching process terminates, the optimized prediction model, characterized by an appropriate set of hyperparameters, has been identified. The proposed IRMPSC is ready for the prediction of UPSC of the FRPreinforced concrete flat slab.
4. Experimental Results
When the IRMPSC model has been constructed, the proposed model can be employed to predict the punching shear capacity of FRPreinforced flat slab. Besides the proposed instancebased regression, previously used methods including ANN [6] and the empirical formulas are utilized as benchmark approaches. The ANN consists of 1 hidden layer with 6 neurons. The applied learning rate was 0.001 together with a number of 5000 learning epochs. The LevenbergMarquardt algorithm is used to train the ANN model [54]. Additionally, to quantify model performance, Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE), and Coefficient of Determination () are computed.
In the first experiment, the dataset has been randomly divided into two sets; the first set (Set 1) was used for model construction, while the second set (Set 2) was used for model testing. The numbers of data samples in Set 1 and Set 2 are 72 and 10, respectively. Set 1 is further divided into two subsets: subset 1 which constitutes the model memory (80%) and subset 2 (20%) which is reserved for model validating. The testing results of the IRMPSC are illustrated in Figure 3. This figure demonstrates that the proposed approach has obtained a good fit to a straight line.
Furthermore, in this experiment, the prediction performance of the proposed instancebased model is compared to that of the ANN. The results comparison is reported in Table 2. The RMSE, MAPE, and in the testing process obtained from the IRMPSC are 63.34, 12.70, and 0.96. The ANN approach yields the following results: RMSE = 52.43, MAPE = 15.98, and . Observably, in terms of the MAPE, the new method has achieved 3.28% improvement compared with the ANN. The two prediction models obtain the same value of . Nevertheless, the ANN shows better performance in terms of RMSE.
In the second experiment, a tenfold cross validation process is conducted. Because all of the subsamples are mutually exclusive, the cross validation process can accurately evaluate the model performance. The average prediction performances (measured in RMSE, MAPE, and ) of the IRMPSC and the ANN are reported in Table 3. Additionally, the prediction results obtained from formulabased approaches are also provided in this table. The results of the empirical formulas are calculated using the equations (from (1) to (7)) mentioned previously in this article.
From Table 3, it is recognizable that the equation proposed by Ospina et al. [32] is the best formulabased method, and the RMSE, MAPE, and of this method are 117.51, 15.48, and 0.91, respectively. This outcome is shown to be far worse than the two machine learning approaches. RMSE, MAPE, and of the IRMPSC are 65.99, 11.81, and 0.97, respectively. The three measurements of the ANN are 62.29, 12.86, and 0.96. Thus, the newly proposed method overcomes the ANN as measured by MAPE and ; meanwhile the ANN is better than the IRMPSC in terms of RMSE. In summary, compared with the best empirical formula, the IRMPSC has attained roughly 43% and 24% improvements in RMSE and MAPE, respectively.
Furthermore, it is beneficial to investigate the relevance of the six input features employed in this study. As mentioned earlier, the relevance of input features can be quantified by the feature weight . The average feature weights of the six input attributes, obtained from the tenfold cross validation process, are as follows: , , , , , and . Thus, it can be seen that the type of column section (), the effective flexural depth of slab (), and the reinforcement ratio () are highly relevant for the prediction of UPSC of the FRPreinforced concrete slabs. The reason is that these three features obtain the highest feature weights. Meanwhile, the other input attributes, including the section area of column (), the compressive strength of concrete (), and Young’s modulus of the FRP slab (), obtain lower values of feature weights. Nevertheless, since all weights are greater than zero, these three features (, , and ) are still relevant for the prediction process.
5. Conclusion
This research proposed an alternative, named as IRMPSC, for calculating the punching shear capacity of the FRPreinforced twoway slabs. The new approach employs the kernel regression as an instancebased learning technique to infer the mapping function between the punching shear capacity and its input features. To construct the instancebased learning model, the DE is employed.
Experimental results have proved that the IRMPSC can deliver prediction performance which is far more accurate than commonly used empirical formulas, reflected in low prediction errors (RMSE and MAPE) and a high Coefficient of Determination (). The new method has also demonstrated a competitive capability compared with the ANN approach. However, as an instancebased learning method, the frameworks of learning and making predictions of the IRMPSC are much more transparent than the blackbox ANN. Furthermore, another advantage of the new method is that it can help to reveal the relevance of input features quantified by the feature weight. Thus, the new model can be better understood and easily applied by structural engineers, who may not be familiar with machine learning, to design FRPreinforced slabs.
Thus, the applicability of the new approach may be limited to the ranges of input features recorded in the dataset. Accordingly, more slab test results should be collected both by actual experiments and by analyses via nonlinear Finite Element Method packages; these data can be incorporated into the current dataset to enhance the generalization and applicability of the IRMPSC. These tasks can be a future direction of the current study.
Appendix
See Table 4.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.