Prediction of Punching Capacity of Slab-Column Connections without Transverse Reinforcement Based on a Backpropagation Neural Network

Punching shear failure of slab-column connections can cause the progressive collapse of a structure. In this study, a punching test database is first established. -en, based on the Levenberg–Marquardt (LM) algorithm and using the nonlinear function of the backpropagation neural network (BPNN), a prediction model of the punching capacity of slab-column connections without transverse reinforcement is established. Finally, the proposedmodel is compared with the formulas of the Chinese, American, and European standards using several methods. -e statistical eigenvalue method shows that the BPNN model has the highest accuracy and the lowest dispersion.-e defect point counting method shows that the BPNNmodel had the fewest total number of defects and was the safest and most economical. -e influencing factor analysis suggests that factors in the BPNN model had the most reasonable influence on the punching bearing capacity of slab-column connections. Finally, the model is verified using a case study and the Matlab program. -e results show that the average error of the formulas in the Chinese, American, and European standards are 21.08%, 30.21%, and 11.47%, respectively, higher than that of the BPNN model.


Introduction
Bullock's Department Store collapsed in 1994, Sanfeng Department Store in South Korea toppled down in 1995, and Morbio Inferior Shopping Center in Switzerland fell down in 2004 [1][2][3].
e above accidents show that slab-column connections without transverse reinforcement are prone to punching failure when they experience accidental loads, which then triggers a chain reaction and finally leads to the progressive collapse of the entire structure. erefore, researchers have conducted a large number of trials and theoretical analyses on the mechanism of slab-column connections without transverse reinforcement under punching failure, and this research study has led to a comprehensive cognition of the variables that can trigger punching failure. Some mechanical models have been proposed that are based on the knowledge of the mechanism of slab-column connections to resist shearing. Such models include Mutton's critical crack punching theory [4], Kinnunen and Nylander's conical shell model [5], and Alexander and Simmond's space truss model [6]. However, these theoretical equations are hard to unify and are in complex form; hence, they are not suitable for engineering applications. Currently, the recommended calculation methods for punching capacity in the national standards consist primarily of half-experience and half-theory formulas based on experimental data fitting. ese methods include the calculation formulas in the GB 50010-2010 [7], ACI318-14 [8], and EC 2-04 [9] codes. ese formulas are concise and clear, making them accessible to engineering designers. However, the theory referred to in these formulas for punching capacity primarily considers one kind of resisting shear mechanism or two, while ignoring other factors. us, the applicability of these formulas is limited [10][11][12]. e accuracy and reliability of the calculations also need to be further improved.
Due to the complexity of the mechanism of slab-column connections without transverse reinforcement under punching shearing failure, it is difficult to predict the punching capacity of slab-column connections by fitting the test data taking all factors into consideration. is is especially the case when it comes to influencing factors with large nonlinearity and uncertainty, as well as discreteness caused by brittle failure. An artificial neural network is a data-processing model enlightened by a biological neural network, the most common of which is called a BPNN. e Kolmogorov theorem has proven that the BPNN has strong nonlinear mapping and generalization abilities. Each continuous function or mapping function can be realized by three levels of networks [13]. e BPNN has been widely used for solving many civil engineering problems, including the shear performance of concrete members. Kumar and Barai established a neural network that can be used to predict the extreme shearing force of fiber concrete beams without transverse reinforcement [14]. Mansour predicted the extreme shearing strength of an RC beam with transverse reinforcement using an artificial neural network [15]. Meanwhile, Cladera and Mari used a neural network to establish a design procedure for a normal/high-strength concrete beam with and without stirrups [16]. Using the nonlinear function of a BPNN, this study established a prediction model for the punching capacity of slab-column connections without transverse reinforcement. In addition, the BPNN model is evaluated using the statistical eigenvalue method, the defect point method, and the influencing factor analysis method. e proposed model provides a new method for the calculation of punching capacity.

Establishment of the BPNN Model for
Punching Capacity e BPNN is a forward network based on error back propagation. It is composed of trained artificial neurons and is used to represent the optimal connection between a given input and expected output. e establishment of the BPNN model for punching capacity primarily includes the following steps. e first step is to establish training samples. As the research background for this study, it was necessary to determine the input and output sample parameters and establish the corresponding punching test database. e second step is the design of the BPNN model. It primarily includes the setting of the number of network layers, number of input layer nodes, number of hidden layer nodes, number of output layer nodes, activation function, training method, training parameters, and other factors. e third step is centered on BPNN training. e Levenberg-Marquardt (LM) algorithm is used to iteratively update the weights and thresholds of the network so that the errors between the output value of the network and the expected output are continuously reduced, and thus the optimal neural network is achieved. e fourth step is the evaluation of the training results. e performance of the neural network is then evaluated by cost function and regression analyses.

Establishment of the Punching Test Database.
At the beginning of the 20th century, the progressive collapse of plate structures caused by punching failures of slab-column connections attracted the attention of scholars [17]. erefore, scholars around the world have conducted numerous studies in the past century on the punching failure of slab-column connections. e research parameters have primarily included concrete strength, longitudinal reinforcement, size effect, column section shape and size, shear span to depth ratio, boundary conditions, punching shear reinforcement, loading mode (axial or partial load), aggregate type and size, and other factors. Many research results have been obtained due to these studies.
In this study, most of the domestic and foreign literatures regarding punching tests of slab-column connections without transverse reinforcement were collected . e engineering application practice, the integrity of key data, and the research purpose of this study were all considered. To select the quantifiable influencing parameters of punching capacity as comprehensively as possible, it was determined that concrete strength (f c ), longitudinal reinforcement ratio (ρ), yield strength of longitudinal reinforcement (f y ), effective height of the plate section (h 0 ), shear span to depth ratio ((L 1 − c)/2h 0 ), and ratio of column side length to plate effective height (c/h 0 ) would be used as the control variables of punching capacity (V u ) of slabcolumn connections.
is was done so as to keep other influencing factors consistent and to screen the data. Finally, a set of BPNN training samples were established using f c , ρ, f y , h 0 , (L 1 − c)/2h 0 , and c/h 0 as input values and V u as the output. e punching test database included 206 groups of slab-column connections without transverse reinforcement (see Table 1).
Due to differences in test designs, the material performance indices, loading modes, and other aspects of the test data in different literatures, there is a need to define a uniform description of key test data. Hence, the database is now explained. First, the database was unified using the international system of units (SI) in the database. Second, the concrete strength was uniformly converted into axial compressive strength f c , and the conversion relationships of concrete strength (f c , f c ′ , f t , and f cu ) are shown in Table 2. ird, the slab-column connection in the database was all square cylinders, and vertical loads were applied using a steel pad, short column, or along the support boundary. e simulation of the simply supported boundary (the force characteristics of the reverse-bending line) was realized using the simply supported four sides.

Design of the BPNN Model.
A neural network can be multilayered. Existing theories have proven that a network with a single hidden layer can achieve arbitrary nonlinear mapping by appropriately increasing the number of neuron nodes [10]. A single hidden layer can meet the needs of the problem background involved in this study. e number of input and output layer nodes depends on the dimension of the input and output vectors. According to the background of punching capacity of the BPNN model proposed in this 2 Advances in Civil Engineering Table. 1: e data of slab-column connections are provided in Table. No.
Literature resources Elstner [15] A-1b 23 Advances in Civil Engineering 3 Guandalini [   Advances in Civil Engineering study, the number of nodes used for the input layer is six. e input layer nodes are f c , ρ, f y , h 0 , (L 1 − c)/2h 0 , c/h 0 . e number of nodes in the output layer is one, which is the network output V u . e number of hidden layer nodes has a great influence on the performance of the BPNN model. Generally, more hidden layer nodes can contribute to better performance. It was found that when the node of the hidden layer was 15, the network had the best performance and the smallest error. For the selection of the activation function, the general hidden layer used a tan-sigmoid function (hyperbolic tangent function), while the output layer adopted a linear function. e training algorithm chose the LM algorithm with a fast convergence speed and a small mean square error. e BPNN model used an iterative updating method to determine weights and thresholds. e initial weights and thresholds were defined as small nonzero values that were randomly selected by the computer. e final neural network structure is shown in Figure 1.
en, the jth output in the hidden layer is as follows: e output in the output layer is as follows: e error of the kth sample is as follows: e cost function (the average error of all samples) is as follows: All the weights (15 × 6 + 15) and thresholds (15 + 1) were put into a weight matrix (121 × 1). It was assumed that the weight matrix after the nth iteration was W(n).
e LM algorithm was then applied to update W(n) iteratively. e Jacobian matrix of the cost function was then defined. e matrix uses the partial derivative of the error with respect to the parameter as the element: � 1, . . . , 206; j � 1, . . . , 121). (5) e expression of the iterative update of the weight matrix is as follows: where e � (e 1 , e 2 , . . . , e m ) T and μ is a tentative parameter. For a given parameter μ, if the cost function E can be reduced by a change in the threshold ∆W, μ is reduced. Otherwise, μ increases.
According to the above calculation results, the training algorithm flow of the BPNN model is shown in Figure 2.

Result of the BPNN Model.
e sample data of 206 groups are divided randomly into three groups: training samples (144), validation samples (31), and test samples (31). Training samples are used for error adjustment in the network training. Test samples are used to measure the network generalization. e training is stopped when the generalization stops improving. Test samples are used to measure the performance of network after training. e training samples are then substituted into the above training algorithm for iterative calculation to obtain an optimized BPNN model, to determine the final weight and threshold values and to obtain the neural network code for the punching capacity prediction model (see Appendix). Figure 3 shows the variation of the cost function of the training samples, verification samples, and test samples with the number of iterations. It can be seen from the figure that when the number of iterations reaches 20, the cost function value is the minimum, and the training stops at this time. Figure 4 shows the training process of the neural network. Figure 5 shows the linear regression analysis between the output values of the training samples, verification samples, test samples, and population samples and the expected output values. According to this figure, the correlation between the neural network output and the expected output is very high, reaching 0.99663, 0.98419, 0.98922, and 0.99342, respectively, indicating that the neural network has very good performance and a good fitting effect.

Assessment of the BPNN Model
Since the BPNN model for predicting the punching capacity of slab-column connections without transverse reinforcement established in this study was directly based on results of existing test data, the predicted value of the model was given a safety reserve of 5%.
is means that the Advances in Civil Engineering predicted value of the model would be 95% of the output value of the model. Furthermore, the value of a model is determined by its application in engineering practice. erefore, the present study established a set of relatively perfect evaluation system to evaluate the performance of the model. More specifically, the statistical eigenvalue method was used to evaluate the precision and discrete degree of the model, and the defect point counting method was used to evaluate the safety and economy of the model. In addition, the influencing factor analysis method was used to evaluate whether the model can reasonably reflect the impact of various factors on the punching bearing capacity.

3.1.
e Statistical Eigenvalue Method. Based on the punching test database of the slab-column connections without transverse reinforcement, the predicted results of this model were compared with the calculated results of the Chinese concrete structural design code GB 50010-2010, the American concrete code ACI 318-14, and the European model code EC 2-04. For the sake of description, the following variable is defined:

Advances in Civil Engineering
where V test is the value of the punching capacity obtained in the test and V cal is the value of the punching capacity calculated using various methods, as shown in Table 3. As shown in Table 3, the mean and median of the BPNN model are 1.05834 and 1.05486, which are the minimum in four formulas, followed by EC 2-04 and GB 50010-2010. e American standard is a little conservative. And the mean and median of ACI 318-14 are 1.31155 and 1.32631. erefore, the BPNN model is the most accurate in four prediction methods.
According to the standard deviation and variable coefficient of X, the discretization degree of the BPNN model is the smallest, followed by EC 2-04, while discretization degrees of GB 50010-2010 and ACI 318-14 are relatively large. Moreover, P5 and P95 represent the 5th and 95th percentiles, respectively. P5 means that 5% of the data are less than this value, and P95 means that 5% data are more than this value. us, 90% of the data are within the range of (P5, P95). By comparing the interval of (minimum, maximum) and the interval of (P5, P95), it can be seen that the distribution interval (located near 1) of the BPNN model prediction results is far better than that of the other standards.
is also indicates that, compared with the three standards, the dispersion degree of the BPNN model prediction results is smaller.

e Defect Point Counting Method.
Drawing on the ideas of the defect point deduction method [47] and the point counting method [48] in management science, the defect point counting method is proposed to evaluate the safety and economy of the model. e basic idea is as follows. e ratio, X, of the test value (V test ) and the predicted value of each punching capacity calculation method (V cal ) is used to classify the severity of the defects and to determine the weight and number of defect points of each defect grade. en, the total number of defects is obtained using the weighted sum. e fewer the total number of points, the better the method can predict the punching capacity of slabcolumn connections. For the ratio of the experimental value and the predicted value, when X is less than 1, the slabcolumn connection is not safe. When it is greater than 1, it is safe. However, if X is too large, it is too conservative and uneconomical. Based on the severity classification principle of bell system defects and combined with the actual situation, the X value was divided into six severity levels of defects. Based on the test data, the number of defect points in each punching design method was counted. Table 4 shows the number of defect points in the slab-column connections. It can be seen from Table 4 that the total number of defect points in the BPNN model was 129, which is far fewer than the total number of defect points in the other three standards. e number of defect points was 246 in the Chinese standard, 242 in the American standard, and 194 in the European standard; these were 90.69%, 87.6%, and 50.39%, respectively, higher than that of the BPNN model. is shows that the BPNN model had the best comprehensive performance for safety and economy.

e Influencing Factor Analysis Method.
e ratio of the test value and the predicted value is used as the ordinate. e concrete strength (f c ), ratio of longitudinal reinforcement It can be seen from Figure 6 that when the effective height of the plate is less than 100 mm, the dispersion of prediction results of each formula is relatively large. When the effective height of the plate is greater than 100 mm, the accuracy and dispersion of the BPNN model prediction results are significantly better than those of the other three formulas. From Figure 7, we can see that when the parameter c/h 0 is less than 2, the prediction results of Chinese and American formulas are not ideal. Meanwhile, the BPNN model and the European formula predict better results over the entire interval. Figure 8 shows that the Chinese and American formulas have an obvious trend distribution with the change in concrete strength, while the BPNN model and the European formula show better prediction results and less dispersion.
Although Chinese, American, and European standards did not consider the influence of the shear span to depth ratio ((L 1 − c)/2h 0 ), the yield strength of the longitudinal reinforcement (f y ), and the ratio of the longitudinal reinforcement (ρ) on the punching capacity, the BPNN model was still used to evaluate the influence of these three variables in this study, as shown in Figures 9-11. e calculation results show that, with changes in the three variables, the ratio of the test value to the predicted value of the BPNN model is uniformly distributed on both sides of the mean, and there is no trend distribution or large dispersion degree.
In general, Chinese and American formulas are conservative, and their prediction results are highly dispersed.
e European formula predicts better accuracy and has a smaller dispersion. Meanwhile, the BPNN model considers the most comprehensive influencing factors and does not show a trend distribution with changes in f c , ρ, f y , h 0 , (L 1 − c)/2h 0 , and c/h 0 . e above results indicate that the BPNN model reasonably reflected the impact of various factors on the punching capacity of slab-column connections without transverse reinforcement.
Based on the above calculation and analysis, it is clear that the BPNN model established in this study is superior to the calculation formulas in the Chinese, American, and European standards.
In order to illustrate the necessity of the parameters considered in the BPNN model, the American formula is used to illustrate the influence of these parameters. It can be seen from Figure 12 that, as the reinforcement ratio increases, the result also increases, which is obviously unreasonable. e main reason for this is that the American formula does not consider the effect of the reinforcement ratio. As shown in Figures 13 and 14, the prediction result of the American formula has a large dispersion with changes in f y and (L 1 − c)/2h 0 . Furthermore, there are many unsafe results, which are not allowed in the actual application.
erefore, the parameters selected in this paper have a significant impact on the punching capacity. Empirically, ACI 318-14 uses the shear force of critical section as the punching capacity of slab-column connections. e parameters used in ACI 318-19 to calculate the shear strength include critical section perimeter, effective height of plate section, and concrete strength. From comparison results of this study, parameters used in the    14 Advances in Civil Engineering paper are perhaps more appropriate than the code. e paper suggests that ACI 318-14 should also take these parameters into account to improve the reliability of the formula. e same advice applies to GB 50010-2010 and EC 2-04.

Application Example
To verify the reliability and accuracy of the model presented in this study further, the punching test of slab-column connections without transverse reinforcement [42,49] completed by Hunan University was selected as the calculation example. e relevant parameters of the punching test of slab-column connections are shown in Table 5.
To run this calculation example, the following actions were performed: first, we ran the Matlab working environment and edited the neural network code in Appendix and opened its root directory. Next, we typed the Neural Network Function (x1) on the command line; x1 is [f c , ρ, f y , h 0 , (L 1 − c)/2h 0 , c/h 0 ]. e output was the value of the punching capacity of slab-column connections predicted in this study. Finally, predicted results were compared with calculated results of Chinese, American, and European standards in Table 6.
V test is the value of the punching capacity measured by the test. V BP , V GB , V ACI , and V EC are the values of the punching capacity calculated by the BPNN model, the Chinese, American, and European standards, respectively. From the table above, we can observe that the maximum of V Test /V BP is 1.1208, the minimum is 1.0172, the mean is 1.072253, and the error is within 10%. In addition, the mean values of V Test /V GB , V Test /V ACI , and V Test /V EC are 1.298288, 1.396167, and 1.19527, respectively. e calculation errors of the Chinese, American, and European standards are 21.08%, 30.21%, and 11.47% higher than that of the BPNN model, respectively. us, it can be seen that, using the Matlab tool and the BPNN model, prediction code established in this study can calculate the punching capacity of slab-column connections more quickly and accurately.

Conclusion
In general, shear failure is induced by the transfer between the column and slab of a shearing force combined with a moment. e research considers the transfer of a shearing force (without eccentricity with respect to the centroid of the shear critical section). In this study, a BPNN was used to study the punching capacity calculation of slab-column connections without transverse reinforcement. e main conclusions are as follows.
First, the quantifiable impact parameters of the punching capacity of slab-column connections without transverse reinforcement were considered comprehensively. A punching test database (including 206 groups) of slab-column connections without transverse reinforcement was established.
is included influencing factors of concrete strength f c , the ratio of longitudinal reinforcement ρ, the yield strength of longitudinal reinforcement f y , the effective  height of the plate section h 0 , the shear span to depth ratio (L 1 − c)/2h 0 , and the ratio of column side length to the plate effective height c/h 0 . e BPNN model was designed and trained and then was evaluated by means of mean square error and regression analyses. e BPNN model showed good performance.
Second, the predicted value of the BPNN model of 95% was used as the model output. In this study, the accuracy and discreteness of the BPNN model were evaluated using the statistical eigenvalue method, and the safety and economy of the BPNN model were evaluated using the defect point counting method. Whether the BPNN model could reasonably reflect the impact of various factors on the punching capacity was evaluated using the influencing factor analysis method. Research results suggested that the mean, median, standard deviation, and variation coefficient of V test /V cal of the BPNN model were optimal; the number of defect points were fewest; and the predicted results did not show a trend distribution and large dispersion of relevant factors.
is means that the BPNN model can meet relevant requirements well and is superior to the formulas in the Chinese, American, and European standards.
ird, based on the BPNN model established in this study, the code was run in the Matlab working environment. Parameters of the test model of Hunan University were calculated, and they were compared with the corresponding results of the Chinese, American, and European standards.
e results indicate that the BPNN model had the best calculation results, with an average error within 10%. Compared with the BPNN model, the calculation errors of the Chinese, American, and European standards were increased by 21.08%, 30.21%, and 11.47%, respectively.