Advances in Civil Engineering

Advances in Civil Engineering / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 7904685 | https://doi.org/10.1155/2019/7904685

Jie Bu, Fanzhen Zhang, Meng Zhu, Zhiyang He, Qigao Hu, "Prediction of Punching Capacity of Slab-Column Connections without Transverse Reinforcement Based on a Backpropagation Neural Network", Advances in Civil Engineering, vol. 2019, Article ID 7904685, 19 pages, 2019. https://doi.org/10.1155/2019/7904685

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

Academic Editor: Roman Wan-Wendner
Received23 Jul 2019
Revised09 Oct 2019
Accepted21 Oct 2019
Published23 Dec 2019

Abstract

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. Then, 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 proposed model is compared with the formulas of the Chinese, American, and European standards using several methods. The statistical eigenvalue method shows that the BPNN model has the highest accuracy and the lowest dispersion. The defect point counting method shows that the BPNN model had the fewest total number of defects and was the safest and most economical. The 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. The 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.

1. 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 [13]. The 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. Therefore, 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. These methods include the calculation formulas in the GB 50010-2010 [7], ACI318-14 [8], and EC 2-04 [9] codes. These 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. Thus, the applicability of these formulas is limited [1012]. The 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. This 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. The 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]. The 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. The proposed model provides a new method for the calculation of punching capacity.

2. Establishment of the BPNN Model for Punching Capacity

The 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. The establishment of the BPNN model for punching capacity primarily includes the following steps. The 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. The 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. The third step is centered on BPNN training. The 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. The fourth step is the evaluation of the training results. The performance of the neural network is then evaluated by cost function and regression analyses.

2.1. 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]. Therefore, scholars around the world have conducted numerous studies in the past century on the punching failure of slab-column connections. The 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 [1846]. The 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 (), longitudinal reinforcement ratio (), yield strength of longitudinal reinforcement (), effective height of the plate section (), shear span to depth ratio (), and ratio of column side length to plate effective height () would be used as the control variables of punching capacity () of slab-column connections. This 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 as input values and as the output. The punching test database included 206 groups of slab-column connections without transverse reinforcement (see Table 1).


No.Literature resourcesSpecimen number (MPa) (%) (MPa) (mm) (mm) (mm) (kN)

1Elstner [15]A-1b23.91.153321186.472.15365
2A-1c27.61.153321186.472.15356
3A-1d34.31.153321186.472.15351
4A-1e19.31.153321186.472.15356
5A-2b18.52.473211146.692.23400
6A-2c34.82.473211146.692.23467
7A-7b26.52.473211146.692.23512
8A-3b21.53.73211146.692.23445
9A-3c25.23.73211146.692.23534
10A-3d32.83.73211146.692.23547
11A-424.81.153321186.033.02400
12A-526.42.53211146.253.12534
13A-623.83.73211146.253.12498
14A-1324.90.52941215.882.94236
15B-242.50.53211146.692.23200
16B-442.60.93031146.692.23334
17B-939.523411146.692.23505
18B-1444.933251146.692.23578
19Moe [16]S1-6022.11.13991146.692.23389
20S2-60211.53991146.692.23356
21S3-6021.523991146.692.23364
22S4-6022.62.63991146.692.23334
23S1-7023.31.14831146.692.23393
24S3-7024.124831146.692.23378
25S4-70332.64831146.692.23374
26S4-70A19.52.64831146.692.23312
27S5-6021.11.13931146.921.78343
28S5-7021.91.14831146.921.78378
29H-124.81.13281146.652.32372
30M1A19.81.54811146.472.68443
31Manterola [17]P1-S124.30.9230410713.550.93216
32P2-S131.90.9230410712.852.34257
33P3-S128.20.9230410711.924.21301
34P1-S2230.9232410713.550.93196
35P2-S231.30.9232410712.852.34283
36P3-S230.30.9232410711.924.21397
37P1-S336.50.9232410713.550.93184
38P2-S333.51.2232410713.550.93211
39P3-S336.20.5732410713.550.93165
40Corley [18]AN-117.81.54041117.092.29334
41AN-219.414441117.321.83266
42Hawkins [19]S2150-128.11.543311277.002.00463
43S2150-228.61.563311277.002.00440
44S4150-133.21.523311277.004.00579
45S4150-233.41.523361277.004.00581
46Yoshio Kakuta [20]SB1-S132.61.17384.4756.330.67140
47SB1-S2351.17384.4756.330.67130
48SB2-S330.71.17384.4756.001.33200
49SB2-S433.21.17384.4756.001.33160
50SB3-S630.51.17384.4755.672.00204
51SB3-S732.21.17384.4755.672.00220
52SB4-S830.91.17384.4755.332.67240
53SB4-S928.91.17384.4755.332.67235
54SP1-S1029.10.47384.4756.001.33118
55SP2-S1131.20.47384.4756.001.33132
56SP3-S1231.91.55384.4756.001.33210
57SP3-S1330.91.86384.4756.001.33200
58SP4-S1431.21.86384.4756.001.33223
59SP5-S1531.21.86384.4756.001.33220
60SC1-S1632.31.18347.1756.001.33170
61SC1-S1731.11.18347.1756.001.33190
62SC4-S18280.92384.4756.001.33169
63SC4-S1928.80.92384.4756.001.33160
64SA3-S2229.21.17384.4759.331.33189
65SA3-S2329.41.17384.4759.331.33192
66SA4-S2434.11.17384.47512.671.33160
67SA4-S2531.61.17384.47512.671.33150
68SH3-S28291.17384.41203.750.83301
69SH3-S2931.11.17384.41203.750.83292
70S6139.60.793431703.531.76660
71S6241.80.793431703.531.76660
72S6335.41.19473.61703.531.76750
73S6439.31.19473.61703.531.76750
74S6539.31.09384.41703.531.76635
75S6639.31.09384.41703.531.76600
76S6736.73.37384.41205.002.50400
77S6841.33.37384.41205.002.50660
78S6939.22.11384.41205.002.50540
79S7039.72.11384.41205.002.50710
80S7140.81.12384.41205.002.50600
81S7244.71.12384.41205.002.50600
82S7317.90.99343.21205.830.83190
83S7422.40.99343.21205.830.83240
84S7542.80.99343.21205.830.83312
85S7641.80.99343.21205.830.83341
86S8926.81.12343.2805.631.25174
87S9025.51.12343.2805.631.25165
88Regan [21]I/124.52.45007710.582.60194
89I/222.31.25007710.582.60176
90I/326.11.45007710.582.60194
91I/430.71.25007710.582.60194
92I/526.81.54807910.322.53165
93I/620.80.84807910.322.53165
94I/728.90.84807910.322.53186
95V/433.80.86281185.920.86285
96Jianlan Zheng [22]J-1261.06298.21253.802.00450.4
97J-3260.96298.21253.802.00427
98J-443.11.32289.1805.943.13302.4
99J-527.51.32289955.002.63307
100J-627.51.06278955.002.63306.4
101Tomas-zewicz [23]ND65-1-155.31.495002754.180.732050
102ND65-2-160.11.755002005.130.751200
103ND95-1-170.41.495002754.180.732250
104ND95-1-375.72.555002754.180.732400
105ND95-2-174.21.755002005.130.751100
106ND95-2-1D731.755002005.130.751300
107ND95-2-375.32.625002005.130.751450
108ND95-2-3D67.62.625002005.130.751250
109ND95-2-3D+82.52.625002005.130.751450
110ND95-3-171.61.84500885.681.14330
111ND115-1-194.31.495002754.180.732450
112ND115-2-1100.21.755002005.130.751400
113ND115-2-3912.625002005.130.751550
114Theodor-akopoulos [24]FS-130.20.564601007.71.5173.5
115FS-831.20.564601007.951150.3
116FS-10310.564601007.452191.4
117FS-1929.50.374601007.71.5136.5
118Yujie An [25]A-129.51.24325.5935.382.15274
119A-231.41.32325.5935.382.15277
120A-321.21.32325.5935.382.15249
121A-429.50.91325.5935.382.15240
122A-519.20.58325.5935.382.15170
123Sistonen [26]L1310.466211723.981.17503
124L2310.456211763.891.15537
125L3310.456211733.961.16530
126L4310.676121704.022.36686
127L5310.666121723.992.32696
128L6310.656121753.902.32799
129L722.90.645861774.431.14478
130L822.91.165761743.945.171111
131L922.91.175761723.995.221107
132L1022.91.165761733.965.211079
133Ghannoum [27]S1-U34.60.964451098.142.06301
134S2-U48.50.964451098.142.06363
135S3-U550.964451098.142.06443
136Krueger [28]P0A35.2148012110.122.48423
137Beutel [29]P121.90.815721905.262.11615
138Ospina [30]SR-134.30.874301205.922.08365.1
139Park [31]S124.61.06453906.942.78230
140S2262466906.942.78316.8
141S324.60.984851304.811.92443.2
142Birkle [32]132.91.544881247.062.02483
1437321.35311907.111.58825
1441028.91.15242606.631.351046
145Xiaokun Huang [33]BAN10530.20.54901685.361.19586.6
146BAN11329.81.35071645.491.22805.3
147BAN105(1)28.30.54901685.361.19522.3
148BAN113(1)26.71.35071645.491.22796.8
149Yuanwei Zhang [34]A20-132.21.57458.4984.852.55356
150A20-232.91.2458.41283.711.95470
151A20-334.30.97458.41583.011.58646
152A20-427.91.75458.4884.833.98408
153A35-131.41.57458.4984.852.55360
154A35-237.81.14371994.802.53357
155A35-340.20.79413.51004.752.50293
156A50-144.31.57458.4984.852.55412
157A50-244.71.14371994.802.53354
158Guandalini [35]PG-127.71.55732105.951.241023
159PG-2b38.40.255522105.951.24440
160PG-4320.255412105.951.24408
161PG-1028.50.335772105.951.24540
162PG-1134.10.755702105.951.24763
163PG-634.11.5526966.511.35238
164PG-734.10.755501006.251.30241
165PG-834.10.285251175.341.11140
166PG-934.10.225251175.341.11115
167Guidottiv [36]PT2257.50.825521966.381.33989
168PT3156.71.485522125.901.231433
169PG1943.40.785102066.071.26860
170PG2046.51.565512016.221.291094
171PG2338.90.815101996.281.31839
172PG2437.81.615511946.441.341102
173Rizk [37]NSC131.20.52435162.54.861.54479
174NSC232.82.17433157.55.021.59678
175HSC150.70.65435162.54.861.54675
176HSC251.40.984401604.941.56798
177HSC351.41.13433157.55.021.59811
178HSC4551.67433157.55.021.59802
179HSC556.42.48433137.55.751.82788
180HSC657.12.68433127.56.201.96801
181NSC332.30.44501057.522.38228
182HSC757.11.88435112.57.022.22481
183Lips [38]PL135.31.635831936.810.67682
184PV133.51.57092105.951.24974
185PL335.51.595831975.692.641324
186PL430.51.585502674.531.271625
187PL531.71.55803533.291.252491
188Jian Peng [39]C7-30-124.50.866041507.001.67473
189C7-30-222.61.286041507.001.67600
190C7-50-139.40.866041507.001.67723
191C7-50-235.91.286041507.001.67801
192C5-30-126.40.866041505.001.67678
193C5-30-225.81.286041505.001.67692
194C5-50-134.80.866041505.001.67691
195C5-50-2341.286041505.001.67855
196Caldentey [40]134.61.075752005.132.25974
197234.91.075752005.132.25956
198Inacio [41]SNSC33.615231056.331.90289.2
199SHSC1100.70.94493104.26.381.92412.9
200SHSC2104.31.24523101.66.551.97429
201SHSC3103.91.48523101.76.541.97460.9
202Bartolac [42]S2-140.31.5560967.141.35393.9
203S2-237.81.5560967.141.35361.3
204S2-339.41.5560967.141.35385.2
205Einpaul [43]PE432.91.595171972.931.32985
206PE332.21.545172048.091.27961

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 , and the conversion relationships of concrete strength (fc, , ft, and fcu) are shown in Table 2. Third, 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. The simulation of the simply supported boundary (the force characteristics of the reverse-bending line) was realized using the simply supported four sides.


Conversion relationshipConversion factors

0.76 (≤C50)1 (≤C40)0.8 (C20~C40)
Linear interpolation
(C50∼C80)
Linear interpolation
(C40∼C80)
0.83 (C50)
0.86 (C60)
0.875 (C70)
0.89 (≥C80)
0.82 (C80)0.87 (C80)

2.2. 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. The 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 study, the number of nodes used for the input layer is six. The input layer nodes are . The number of nodes in the output layer is one, which is the network output . The 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. The training algorithm chose the LM algorithm with a fast convergence speed and a small mean square error. The BPNN model used an iterative updating method to determine weights and thresholds. The initial weights and thresholds were defined as small nonzero values that were randomly selected by the computer. The final neural network structure is shown in Figure 1.

2.3. Training of the BPNN Model

The input vector () is constructed using the parameters in the punching database, and is the expected output value. There are 206 groups of inputs and expected outputs. We assume that the input vector of the kth sample is . The corresponding output of the hidden layer is . The output of the output layer is . The expected output is . The connection weight between the input layer and the hidden layer is then set to be . The connection weight between the hidden layer and the output layer is . The threshold vector of neurons in the hidden layer is . The threshold of the output layer neuron is . The activation function of the hidden layer neurons is and the activation function of the output layer neuron is .

Then, the jth output in the hidden layer is as follows:

The output in the output layer is as follows:

The error of the kth sample is as follows:

The 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 . The LM algorithm was then applied to update W(n) iteratively. The Jacobian matrix of the cost function was then defined. The matrix uses the partial derivative of the error with respect to the parameter as the element:

The expression of the iterative update of the weight matrix is as follows:where is a tentative parameter. For a given parameter , if the cost function can be reduced by a change in the threshold ∆W, is reduced. Otherwise, increases.

According to the above calculation results, the training algorithm flow of the BPNN model is shown in Figure 2.

2.4. Result of the BPNN Model

The 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. The training is stopped when the generalization stops improving. Test samples are used to measure the performance of network after training. The 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.

3. 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%. This means that the 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. Therefore, 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. The 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:where is the value of the punching capacity obtained in the test and is the value of the punching capacity calculated using various methods, as shown in Table 3.


GB 50010-2010ACI 318-14EC 2-04BPNN model

Mean1.19791.311551.144811.05834
Median1.192181.326311.137211.05486
Standard deviation0.295150.322840.194640.12292
Variable coefficient0.246390.246150.170020.11614
Minimum0.448330.498990.680270.7239
Maximum1.996062.244061.632091.80068
P50.780620.860580.820050.88283
P951.720431.923291.532791.25306

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. The American standard is a little conservative. And the mean and median of ACI 318-14 are 1.31155 and 1.32631. Therefore, 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. Thus, 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. This also indicates that, compared with the three standards, the dispersion degree of the BPNN model prediction results is smaller.

3.2. The 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. The basic idea is as follows. The ratio, X, of the test value () and the predicted value of each punching capacity calculation method () is used to classify the severity of the defects and to determine the weight and number of defect points of each defect grade. Then, the total number of defects is obtained using the weighted sum. The fewer the total number of points, the better the method can predict the punching capacity of slab-column connections. For the ratio of the experimental value and the predicted value, when X is less than 1, the slab-column 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. The 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. This shows that the BPNN model had the best comprehensive performance for safety and economy.


Defect levelWeight valueGB 50010-2010ACI 318-14EC 2-04BPNN model

<0.5Very dangerous104100
0.5–0.75Dangerous56642
0.75–1Low degree of safety245334254
1–1.25Safe and reasonable07648116150
1.25–2Conservative1861225511
>2Too conservative20700
Total defect count246242194129

3.3. The Influencing Factor Analysis Method

The ratio of the test value and the predicted value is used as the ordinate. The concrete strength (), ratio of longitudinal reinforcement (), yield strength of longitudinal reinforcement (), effective height of plate section (), shear span to depth ratio (), and ratio of column side length to plate effective height () are used as the abscissa. The upper and lower limits of 95% and mean value of each group of data are given, and the influences of various factors on the prediction results of the punching capacity are then analyzed.

Figures 68, respectively, depict the influence of the effective height of the plate section (), the ratio of the column side length to the plate effective height (), and the concrete strength () on the ratio of the test value to the predicted values.

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 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 (), the yield strength of the longitudinal reinforcement (), and the ratio of the longitudinal reinforcement () 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 911. The 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. The 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 , , , , , and . The 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. The 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 and . Furthermore, there are many unsafe results, which are not allowed in the actual application. Therefore, 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. The 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 paper are perhaps more appropriate than the code. The paper suggests that ACI 318-14 should also take these parameters into account to improve the reliability of the formula. The same advice applies to GB 50010-2010 and EC 2-04.

4. 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. The relevant parameters of the punching test of slab-column connections are shown in Table 5.


Specimen number (MPa) (%) (MPa) (mm) (kN)

S5-123.71.5100445915051.66667517
S5-224.91.5397145915051.66667612
S7-123.11.576245915071.66667535
S7-221.41.4710845915071.66667556
C7-30-3271.53373453.615071.66667690
C7-50-336.11.6086453.615071.66667805
C7-70-148.61.38443453.615071.66667610
C7-70-250.11.54283453.615071.66667785
C7-70-350.91.52514453.615071.66667860

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 []. The 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.


Specimen numberThe BPNN modelGB 50010-2010ACI 318-14EC 2-04
(kN) (kN) (kN) (kN)

S5-1474.80921.08886440.161.17457396.00001.30556435.76641.18642
S5-2562.97831.08708451.921.35422405.39261.50965505.89921.20973
S7-1477.33571.1208433.441.23431390.41661.37033432.00961.23840
S7-2546.59921.0172416.641.33449376.51251.47671481.04871.15581
C7-30-3668.00361.03293473.761.45643430.16611.60403572.60011.20503
C7-50-3742.69831.08389547.681.46984505.26871.59321636.94241.26385
C7-70-1573.80161.06309609.841.00026598.9951.01837560.80181.08773
C7-70-2733.33791.07045616.561.27319608.86211.28929647.59311.21218
C7-70-3791.91131.08598619.921.38728615.01241.39835717.69641.19828
Mean1.072251.298291.396171.19527
Error7.03%29.8%39.6%19.5%
Relative error021.08%30.21%11.47%

is the value of the punching capacity measured by the test. 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 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 are 1.298288, 1.396167, and 1.19527, respectively. The 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. Thus, 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.

5. Conclusion

In general, shear failure is induced by the transfer between the column and slab of a shearing force combined with a moment. The 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. The 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. This included influencing factors of concrete strength , the ratio of longitudinal reinforcement , the yield strength of longitudinal reinforcement , the effective height of the plate section , the shear span to depth ratio , and the ratio of column side length to the plate effective height . The BPNN model was designed and trained and then was evaluated by means of mean square error and regression analyses. The 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 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. This means that the BPNN model can meet relevant requirements well and is superior to the formulas in the Chinese, American, and European standards.

Third, 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. The 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.

Appendix

Neural Network Code for Prediction Model of Punching Capacity

function [y1] = myNeuralNetworkFunction (x1)%MYNEURALNETWORKFUNCTION neural network simulation function.%% Generated by Neural Network Toolbox function genFunction, 13-Feb-2019 17 : 51 : 35.%% [y1] = myNeuralNetworkFunction (x1) takes these arguments:% x = Qx6 matrix, input #1% and returns:% y = Qx1 matrix, output #1% where Q is the number of samples.%#ok< ∗ RPMT0>% =====NEURAL NETWORK CONSTANTS=====% Input 1x1_step1·xoffset = [75; 0.22; 278; 17.8; 2.926395939; 0.666666667];x1_step1·gain = [0.00719424460431655; 0.574712643678161; 0.00464037122969838; 0.023121387283237; 0.18823518904255; 0.439710268466749]; x1_step1·ymin = −1;% Layer 1b1 = [−0.67013086397926890925; 0.20485204467667655903; −3.9823333846859974017; 0.26809837709864853261; 1.7519871170795724424; 0.99641617199102761049; 4.9763998448198938362; 1.1368201646791966652; −0.025928389261043835456; −0.92100364806924983618; 0.36880965470930515693; 11.19473168509247607; −5.7906209524835876579; 1.8210861550491890259; 1.0562373148061163786];IW1_1 = [0.54241403693654410834 0.24796025994322995478 −0.014337576347544853453 0.18764574836693678761 −0.02829727644609779727 0.14145204455730814308; −2.4509908541198388576 2.8901071801540454409 −2.0586515897176820289 1.068335537227783183 −4.6183794460269416149 −0.02484542256670946303; −7.6917611610029394953 3.0974401276965424223 −3.7294875430272114158 4.4642135026415905585 7.3171434926831198098 −12.328913326533490036; 3.5595775453552365164 3.3699076853018046229 1.5200070719384131124 −3.4388910985329763648 −6.2360139697241718792 −2.6537507383974756614; −2.0904525084619258735 4.3392922147544492617 5.9324277695856784121 −8.5047866210874421 −2.7523798326730455344 4.0482637507758116868; −0.22774737124199373461 −0.76951597973573482303 6.9372813682084002451 −0.16830700683789390215 5.8457813380652119406 3.4576020408617522683; −11.252702180792377717 −2.2694352690532371142 1.564784156023274253 −4.9174176477441013944 −4.5068636359880498432 −5.6545914419692682173; 10.059197750593041221 −0.17002386664310031872 −8.9848474105867879302 1.0203632176534829235 4.9221652171394971731 −7.5710047392480142037; −1.4840366841073464599 3.8623959556322895459 1.6833086970154664819 1.4067932842560260109 1.2210393313963610051 −1.001875039735422801; 1.0548272447130120355 1.7632488982764233931 −0.32313056811567220095 −1.3903358650695212262 1.0460118017996697937 2.5325861994828038348; 7.8613465153517045891 7.4271278087830125969 −2.8222326515948368275 −1.7772430280884681242 −3.905950647510644469 −17.387776097206195658; 6.9442041861601406794 0.22091386915732519336 5.89654429623693499 1.9608851775534088535 8.0650553137584175545 1.7641288972052406869; −6.4259520951425530555 2.1257615458185741275 0.63700368732926271331 0.47411663304245349471 0.76026839039380778029 −6.1951688429695117577; 3.4323263734923310508 −1.2974701439316240759 2.757336565240562809 −2.9415561788931592346 −4.7208935670603473511 7.4539269787059039274; 0.32595830157754457357 3.3624573492982667844 1.4011897034149485286 0.85463835267573085019 2.8748323421137191858 −1.9927818384202813373];% Layer 2b2 = 0.78597773687780803087;LW2_1 = [3.4772906528603062526 0.12639460740666358385 −0.23439286264789302439 0.020693346676863687977 −0.0060410169541254875661 0.080236891899916409909 1.2792025383667153804 −0.029240579066094837118 −0.21752104710756975381 −0.040404470314744492243 0.04108162773976927612 0.050369066295788737786 −0.035981188002320835251 −0.24956065236877369995 0.12526282041160077818];% Output 1y1_step1·ymin = −1;y1_step1·gain = 0.000841750841750842;y1_step1·xoffset = 115;% =====SIMULATION========% DimensionsQ = size(x1, 1); % samples% Input 1x1 = x1′;xp1 = mapminmax_apply (x1, x1_step1);% Layer 1a1 = tansig_apply (repmat (b1, 1, Q) + IW1_1 ∗ xp1);% Layer 2a2 = repmat (b2, 1, Q) + LW2_1 ∗ a1;% Output 1y1 = mapminmax_reverse (a2, y1_step1);y1 = y1′;end% =====MODULE FUNCTIONS========% Map Minimum and Maximum Input Processing Functionfunction y = mapminmax_apply (x, settings)y = bsxfun (@minus, x, settings·xoffset);y = bsxfun (@times, y, settings·gain);y = bsxfun (@plus, y, settings·ymin);end% Sigmoid Symmetric Transfer Functionfunction a = tansig_apply(n, ∼)a = 2./(1 + exp(−2 ∗ n)) − 1;end% Map Minimum and Maximum Output Reverse-Processing Functionfunction x = mapminmax_reverse (y, settings)x = bsxfun (@minus, y, settings·ymin);x = bsxfun (@rdivide, x, settings·gain);x = bsxfun (@plus, x, settings·xoffset);end

Data Availability

The punching test data used to support the findings of this study have been included in Table 1, and the data can also be obtained from the papers in “References”. The neural network codes for predicting punching capacity of slab-column connections without transverse reinforcement are listed in Appendix.

Disclosure

Qigao Hu is the co-author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work presented in this study was funded by the National Natural Science Foundation of China, “Research on progressive collapse mechanism of reinforced concrete flat plate structure under blast loading (Grant no. 51608525).” The authors thank LetPub (http://www.letpub.com) for its linguistic assistance during the preparation of this manuscript.

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Copyright © 2019 Jie Bu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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