Advances in Civil Engineering
Volume 2019 (2019), Article ID 7904685, 19 pages
https://doi.org/10.1155/2019/7904685

Research Article

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

Jie Bu,¹ Fanzhen Zhang,¹ Meng Zhu,¹ Zhiyang He,¹ and Qigao Hu²

¹College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410072, China
²College of Military Education and Training, National University of Defense Technology, Changsha 410072, China

Correspondence should be addressed to Fanzhen Zhang; zhangfanzhen7@163.com

Received 23 July 2019; Revised 9 October 2019; Accepted 21 October 2019; Published 23 December 2019

Academic Editor: Roman Wan-Wendner

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.

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 [1–3]. 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 [10–12]. 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 [18–46]. 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).

Table. 1: The data of slab-column connections are provided in Table.

No.
Literature resources
Specimen number
(MPa)
(%)
(MPa)
(mm)
(mm)
(mm)
(kN)

1
Elstner [15]
A-1b
23.9
1.15
332
118
6.47
2.15
365

2
A-1c
27.6
1.15
332
118
6.47
2.15
356

3
A-1d
34.3
1.15
332
118
6.47
2.15
351

4
A-1e
19.3
1.15
332
118
6.47
2.15
356

5
A-2b
18.5
2.47
321
114
6.69
2.23
400

6
A-2c
34.8
2.47
321
114
6.69
2.23
467

7
A-7b
26.5
2.47
321
114
6.69
2.23
512

8
A-3b
21.5
3.7
321
114
6.69
2.23
445

9
A-3c
25.2
3.7
321
114
6.69
2.23
534

10
A-3d
32.8
3.7
321
114
6.69
2.23
547

11
A-4
24.8
1.15
332
118
6.03
3.02
400

12
A-5
26.4
2.5
321
114
6.25
3.12
534

13
A-6
23.8
3.7
321
114
6.25
3.12
498

14
A-13
24.9
0.5
294
121
5.88
2.94
236

15
B-2
42.5
0.5
321
114
6.69
2.23
200

16
B-4
42.6
0.9
303
114
6.69
2.23
334

17
B-9
39.5
2
341
114
6.69
2.23
505

18
B-14
44.9
3
325
114
6.69
2.23
578

19
Moe [16]
S1-60
22.1
1.1
399
114
6.69
2.23
389

20
S2-60
21
1.5
399
114
6.69
2.23
356

21
S3-60
21.5
2
399
114
6.69
2.23
364

22
S4-60
22.6
2.6
399
114
6.69
2.23
334

23
S1-70
23.3
1.1
483
114
6.69
2.23
393

24
S3-70
24.1
2
483
114
6.69
2.23
378

25
S4-70
33
2.6
483
114
6.69
2.23
374

26
S4-70A
19.5
2.6
483
114
6.69
2.23
312

27
S5-60
21.1
1.1
393
114
6.92
1.78
343

28
S5-70
21.9
1.1
483
114
6.92
1.78
378

29
H-1
24.8
1.1
328
114
6.65
2.32
372

30
M1A
19.8
1.5
481
114
6.47
2.68
443

31
Manterola [17]
P1-S1
24.3
0.92
304
107
13.55
0.93
216

32
P2-S1
31.9
0.92
304
107
12.85
2.34
257

33
P3-S1
28.2
0.92
304
107
11.92
4.21
301

34
P1-S2
23
0.92
324
107
13.55
0.93
196

35
P2-S2
31.3
0.92
324
107
12.85
2.34
283

36
P3-S2
30.3
0.92
324
107
11.92
4.21
397

37
P1-S3
36.5
0.92
324
107
13.55
0.93
184

38
P2-S3
33.5
1.22
324
107
13.55
0.93
211

39
P3-S3
36.2
0.57
324
107
13.55
0.93
165

40
Corley [18]
AN-1
17.8
1.5
404
111
7.09
2.29
334

41
AN-2
19.4
1
444
111
7.32
1.83
266

42
Hawkins [19]
S2150-1
28.1
1.54
331
127
7.00
2.00
463

43
S2150-2
28.6
1.56
331
127
7.00
2.00
440

44
S4150-1
33.2
1.52
331
127
7.00
4.00
579

45
S4150-2
33.4
1.52
336
127
7.00
4.00
581

46
Yoshio Kakuta [20]
SB1-S1
32.6
1.17
384.4
75
6.33
0.67
140

47
SB1-S2
35
1.17
384.4
75
6.33
0.67
130

48
SB2-S3
30.7
1.17
384.4
75
6.00
1.33
200

49
SB2-S4
33.2
1.17
384.4
75
6.00
1.33
160

50
SB3-S6
30.5
1.17
384.4
75
5.67
2.00
204

51
SB3-S7
32.2
1.17
384.4
75
5.67
2.00
220

52
SB4-S8
30.9
1.17
384.4
75
5.33
2.67
240

53
SB4-S9
28.9
1.17
384.4
75
5.33
2.67
235

54
SP1-S10
29.1
0.47
384.4
75
6.00
1.33
118

55
SP2-S11
31.2
0.47
384.4
75
6.00
1.33
132

56
SP3-S12
31.9
1.55
384.4
75
6.00
1.33
210

57
SP3-S13
30.9
1.86
384.4
75
6.00
1.33
200

58
SP4-S14
31.2
1.86
384.4
75
6.00
1.33
223

59
SP5-S15
31.2
1.86
384.4
75
6.00
1.33
220

60
SC1-S16
32.3
1.18
347.1
75
6.00
1.33
170

61
SC1-S17
31.1
1.18
347.1
75
6.00
1.33
190

62
SC4-S18
28
0.92
384.4
75
6.00
1.33
169

63
SC4-S19
28.8
0.92
384.4
75
6.00
1.33
160

64
SA3-S22
29.2
1.17
384.4
75
9.33
1.33
189

65
SA3-S23
29.4
1.17
384.4
75
9.33
1.33
192

66
SA4-S24
34.1
1.17
384.4
75
12.67
1.33
160

67
SA4-S25
31.6
1.17
384.4
75
12.67
1.33
150

68
SH3-S28
29
1.17
384.4
120
3.75
0.83
301

69
SH3-S29
31.1
1.17
384.4
120
3.75
0.83
292

70
S61
39.6
0.79
343
170
3.53
1.76
660

71
S62
41.8
0.79
343
170
3.53
1.76
660

72
S63
35.4
1.19
473.6
170
3.53
1.76
750

73
S64
39.3
1.19
473.6
170
3.53
1.76
750

74
S65
39.3
1.09
384.4
170
3.53
1.76
635

75
S66
39.3
1.09
384.4
170
3.53
1.76
600

76
S67
36.7
3.37
384.4
120
5.00
2.50
400

77
S68
41.3
3.37
384.4
120
5.00
2.50
660

78
S69
39.2
2.11
384.4
120
5.00
2.50
540

79
S70
39.7
2.11
384.4
120
5.00
2.50
710

80
S71
40.8
1.12
384.4
120
5.00
2.50
600

81
S72
44.7
1.12
384.4
120
5.00
2.50
600

82
S73
17.9
0.99
343.2
120
5.83
0.83
190

83
S74
22.4
0.99
343.2
120
5.83
0.83
240

84
S75
42.8
0.99
343.2
120
5.83
0.83
312

85
S76
41.8
0.99
343.2
120
5.83
0.83
341

86
S89
26.8
1.12
343.2
80
5.63
1.25
174

87
S90
25.5
1.12
343.2
80
5.63
1.25
165

88
Regan [21]
I/1
24.5
2.4
500
77
10.58
2.60
194

89
I/2
22.3
1.2
500
77
10.58
2.60
176

90
I/3
26.1
1.4
500
77
10.58
2.60
194

91
I/4
30.7
1.2
500
77
10.58
2.60
194

92
I/5
26.8
1.5
480
79
10.32
2.53
165

93
I/6
20.8
0.8
480
79
10.32
2.53
165

94
I/7
28.9
0.8
480
79
10.32
2.53
186

95
V/4
33.8
0.8
628
118
5.92
0.86
285

96
Jianlan Zheng [22]
J-1
26
1.06
298.2
125
3.80
2.00
450.4

97
J-3
26
0.96
298.2
125
3.80
2.00
427

98
J-4
43.1
1.32
289.1
80
5.94
3.13
302.4

99
J-5
27.5
1.32
289
95
5.00
2.63
307

100
J-6
27.5
1.06
278
95
5.00
2.63
306.4

101
Tomas-zewicz [23]
ND65-1-1
55.3
1.49
500
275
4.18
0.73
2050

102
ND65-2-1
60.1
1.75
500
200
5.13
0.75
1200

103
ND95-1-1
70.4
1.49
500
275
4.18
0.73
2250

104
ND95-1-3
75.7
2.55
500
275
4.18
0.73
2400

105
ND95-2-1
74.2
1.75
500
200
5.13
0.75
1100

106
ND95-2-1D
73
1.75
500
200
5.13
0.75
1300

107
ND95-2-3
75.3
2.62
500
200
5.13
0.75
1450

108
ND95-2-3D
67.6
2.62
500
200
5.13
0.75
1250

109
ND95-2-3D+
82.5
2.62
500
200
5.13
0.75
1450

110
ND95-3-1
71.6
1.84
500
88
5.68
1.14
330

111
ND115-1-1
94.3
1.49
500
275
4.18
0.73
2450

112
ND115-2-1
100.2
1.75
500
200
5.13
0.75
1400

113
ND115-2-3
91
2.62
500
200
5.13
0.75
1550

114
Theodor-akopoulos [24]
FS-1
30.2
0.56
460
100
7.7
1.5
173.5

115
FS-8
31.2
0.56
460
100
7.95
1
150.3

116
FS-10
31
0.56
460
100
7.45
2
191.4

117
FS-19
29.5
0.37
460
100
7.7
1.5
136.5

118
Yujie An [25]
A-1
29.5
1.24
325.5
93
5.38
2.15
274

119
A-2
31.4
1.32
325.5
93
5.38
2.15
277

120
A-3
21.2
1.32
325.5
93
5.38
2.15
249

121
A-4
29.5
0.91
325.5
93
5.38
2.15
240

122
A-5
19.2
0.58
325.5
93
5.38
2.15
170

123
Sistonen [26]
L1
31
0.46
621
172
3.98
1.17
503

124
L2
31
0.45
621
176
3.89
1.15
537

125
L3
31
0.45
621
173
3.96
1.16
530

126
L4
31
0.67
612
170
4.02
2.36
686

127
L5
31
0.66
612
172
3.99
2.32
696

128
L6
31
0.65
612
175
3.90
2.32
799

129
L7
22.9
0.64
586
177
4.43
1.14
478

130
L8
22.9
1.16
576
174
3.94
5.17
1111

131
L9
22.9
1.17
576
172
3.99
5.22
1107

132
L10
22.9
1.16
576
173
3.96
5.21
1079

133
Ghannoum [27]
S1-U
34.6
0.96
445
109
8.14
2.06
301

134
S2-U
48.5
0.96
445
109
8.14
2.06
363

135
S3-U
55
0.96
445
109
8.14
2.06
443

136
Krueger [28]
P0A
35.2
1
480
121
10.12
2.48
423

137
Beutel [29]
P1
21.9
0.81
572
190
5.26
2.11
615

138
Ospina [30]
SR-1
34.3
0.87
430
120
5.92
2.08
365.1

139
Park [31]
S1
24.6
1.06
453
90
6.94
2.78
230

140
S2
26
2
466
90
6.94
2.78
316.8

141
S3
24.6
0.98
485
130
4.81
1.92
443.2

142
Birkle [32]
1
32.9
1.54
488
124
7.06
2.02
483

143
7
32
1.3
531
190
7.11
1.58
825

144
10
28.9
1.1
524
260
6.63
1.35
1046

145
Xiaokun Huang [33]
BAN105
30.2
0.5
490
168
5.36
1.19
586.6

146
BAN113
29.8
1.3
507
164
5.49
1.22
805.3

147
BAN105(1)
28.3
0.5
490
168
5.36
1.19
522.3

148
BAN113(1)
26.7
1.3
507
164
5.49
1.22
796.8

149
Yuanwei Zhang [34]
A20-1
32.2
1.57
458.4
98
4.85
2.55
356

150
A20-2
32.9
1.2
458.4
128
3.71
1.95
470

151
A20-3
34.3
0.97
458.4
158
3.01
1.58
646

152
A20-4
27.9
1.75
458.4
88
4.83
3.98
408

153
A35-1
31.4
1.57
458.4
98
4.85
2.55
360

154
A35-2
37.8
1.14
371
99
4.80
2.53
357

155
A35-3
40.2
0.79
413.5
100
4.75
2.50
293

156
A50-1
44.3
1.57
458.4
98
4.85
2.55
412

157
A50-2
44.7
1.14
371
99
4.80
2.53
354

158
Guandalini [35]
PG-1
27.7
1.5
573
210
5.95
1.24
1023

159
PG-2b
38.4
0.25
552
210
5.95
1.24
440

160
PG-4
32
0.25
541
210
5.95
1.24
408

161
PG-10
28.5
0.33
577
210
5.95
1.24
540

162
PG-11
34.1
0.75
570
210
5.95
1.24
763

163
PG-6
34.1
1.5
526
96
6.51
1.35
238

164
PG-7
34.1
0.75
550
100
6.25
1.30
241

165
PG-8
34.1
0.28
525
117
5.34
1.11
140

166
PG-9
34.1
0.22
525
117
5.34
1.11
115

167
Guidottiv [36]
PT22
57.5
0.82
552
196
6.38
1.33
989

168
PT31
56.7
1.48
552
212
5.90
1.23
1433

169
PG19
43.4
0.78
510
206
6.07
1.26
860

170
PG20
46.5
1.56
551
201
6.22
1.29
1094

171
PG23
38.9
0.81
510
199
6.28
1.31
839

172
PG24
37.8
1.61
551
194
6.44
1.34
1102

173
Rizk [37]
NSC1
31.2
0.52
435
162.5
4.86
1.54
479

174
NSC2
32.8
2.17
433
157.5
5.02
1.59
678

175
HSC1
50.7
0.65
435
162.5
4.86
1.54
675

176
HSC2
51.4
0.98
440
160
4.94
1.56
798

177
HSC3
51.4
1.13
433
157.5
5.02
1.59
811

178
HSC4
55
1.67
433
157.5
5.02
1.59
802

179
HSC5
56.4
2.48
433
137.5
5.75
1.82
788

180
HSC6
57.1
2.68
433
127.5
6.20
1.96
801

181
NSC3
32.3
0.4
450
105
7.52
2.38
228

182
HSC7
57.1
1.88
435
112.5
7.02
2.22
481

183
Lips [38]
PL1
35.3
1.63
583
193
6.81
0.67
682

184
PV1
33.5
1.5
709
210
5.95
1.24
974

185
PL3
35.5
1.59
583
197
5.69
2.64
1324

186
PL4
30.5
1.58
550
267
4.53
1.27
1625

187
PL5
31.7
1.5
580
353
3.29
1.25
2491

188
Jian Peng [39]
C7-30-1
24.5
0.86
604
150
7.00
1.67
473

189
C7-30-2
22.6
1.28
604
150
7.00
1.67
600

190
C7-50-1
39.4
0.86
604
150
7.00
1.67
723

191
C7-50-2
35.9
1.28
604
150
7.00
1.67
801

192
C5-30-1
26.4
0.86
604
150
5.00
1.67
678

193
C5-30-2
25.8
1.28
604
150
5.00
1.67
692

194
C5-50-1
34.8
0.86
604
150
5.00
1.67
691

195
C5-50-2
34
1.28
604
150
5.00
1.67
855

196
Caldentey [40]
1
34.6
1.07
575
200
5.13
2.25
974

197
2
34.9
1.07
575
200
5.13
2.25
956

198
Inacio [41]
SNSC
33.6
1
523
105
6.33
1.90
289.2

199
SHSC1
100.7
0.94
493
104.2
6.38
1.92
412.9

200
SHSC2
104.3
1.24
523
101.6
6.55
1.97
429

201
SHSC3
103.9
1.48
523
101.7
6.54
1.97
460.9

202
Bartolac [42]
S2-1
40.3
1.5
560
96
7.14
1.35
393.9

203
S2-2
37.8
1.5
560
96
7.14
1.35
361.3

204
S2-3
39.4
1.5
560
96
7.14
1.35
385.2

205
Einpaul [43]
PE4
32.9
1.59
517
197
2.93
1.32
985

206
PE3
32.2
1.54
517
204
8.09
1.27
961

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.

Table 2: Conversion relationships of concrete strength (fc, , ft, and fcu).

Conversion relationship
Conversion 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.

Figure 1: Topological structure of the BPNN model.

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.

Figure 2: The BPNN model algorithm flowchart.

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.

Figure 3: Cost functions.

Figure 4: Training process.

(a)

(b)

(c)

(d)

Figure 5: Linear regression between the neural network output and the expected output. (a) Training: R = 0.99663. (b) Validation: R = 0.98419. (c) Test: R = 0.98922. (d) All: R = 0.99342.

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.

Table 3: Statistical eigenvalue.

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

Mean
1.1979
1.31155
1.14481
1.05834

Median
1.19218
1.32631
1.13721
1.05486

Standard deviation
0.29515
0.32284
0.19464
0.12292

Variable coefficient
0.24639
0.24615
0.17002
0.11614

Minimum
0.44833
0.49899
0.68027
0.7239

Maximum
1.99606
2.24406
1.63209
1.80068

P5
0.78062
0.86058
0.82005
0.88283

P95
1.72043
1.92329
1.53279
1.25306

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.

Table 4: Number of defect points.

Defect level
Weight value
GB 50010-2010
ACI 318-14
EC 2-04
BPNN model

<0.5
Very dangerous
10
4
1
0
0

0.5–0.75
Dangerous
5
6
6
4
2

0.75–1
Low degree of safety
2
45
33
42
54

1–1.25
Safe and reasonable
0
76
48
116
150

1.25–2
Conservative
1
86
122
55
11

>2
Too conservative
2
0
7
0
0

Total defect count
246
242
194
129

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 6–8, 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.

(a)

(b)

(c)

(d)

Figure 6: Influence of on the ratio of the test value to the predicted value. (a) BPNN model. (b) GB 50010-2010. (c) ACI 318-14. (d) EC 2-04.

(a)

(b)

(c)

(d)

Figure 7: Influence of on the ratio of the test value to the predicted value. (a) BPNN model. (b) GB 50010-2010. (c) ACI 318-14. (d) EC 2-04.

(a)

(b)

(c)

(d)

Figure 8: Influence of on the ratio of the test value to the predicted value. (a) BPNN model. (b) GB 50010-2010. (c) ACI 318-14. (d) EC 2-04.

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 9–11. 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.

Figure 9: influence on the BPNN model.

Figure 10: fy influence on the BPNN model.

Figure 11: influence on the BPNN model.

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.

Figure 12: influence on ACI 318-14.

Figure 13: fy influence on ACI 318-14.

Figure 14: influence on ACI 318-14.

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.

Table 5: Punching test parameters for the slab-column connections.

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

(kN)

S5-1
23.7
1.51004
459
150
5
1.66667
517

S5-2
24.9
1.53971
459
150
5
1.66667
612

S7-1
23.1
1.5762
459
150
7
1.66667
535

S7-2
21.4
1.47108
459
150
7
1.66667
556

C7-30-3
27
1.53373
453.6
150
7
1.66667
690

C7-50-3
36.1
1.6086
453.6
150
7
1.66667
805

C7-70-1
48.6
1.38443
453.6
150
7
1.66667
610

C7-70-2
50.1
1.54283
453.6
150
7
1.66667
785

C7-70-3
50.9
1.52514
453.6
150
7
1.66667
860

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.

Table 6: Calculated values of the punching capacity of slab-column connections.

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

(kN)

(kN)

(kN)

(kN)

S5-1
474.8092
1.08886
440.16
1.17457
396.0000
1.30556
435.7664
1.18642

S5-2
562.9783
1.08708
451.92
1.35422
405.3926
1.50965
505.8992
1.20973

S7-1
477.3357
1.1208
433.44
1.23431
390.4166
1.37033
432.0096
1.23840

S7-2
546.5992
1.0172
416.64
1.33449
376.5125
1.47671
481.0487
1.15581

C7-30-3
668.0036
1.03293
473.76
1.45643
430.1661
1.60403
572.6001
1.20503

C7-50-3
742.6983
1.08389
547.68
1.46984
505.2687
1.59321
636.9424
1.26385

C7-70-1
573.8016
1.06309
609.84
1.00026
598.995
1.01837
560.8018
1.08773

C7-70-2
733.3379
1.07045
616.56
1.27319
608.8621
1.28929
647.5931
1.21218

C7-70-3
791.9113
1.08598
619.92
1.38728
615.0124
1.39835
717.6964
1.19828

Mean

1.07225

1.29829

1.39617

1.19527

Error

7.03%

29.8%

39.6%

19.5%

Relative error

0

21.08%

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 1 x1_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 1 b1 = [−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 2 b2 = 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 1 y1_step1·ymin = −1; y1_step1·gain = 0.000841750841750842; y1_step1·xoffset = 115; % =====SIMULATION======== % Dimensions Q = size(x1, 1); % samples % Input 1 x1 = x1′; xp1 = mapminmax_apply (x1, x1_step1); % Layer 1 a1 = tansig_apply (repmat (b1, 1, Q) + IW1_1 ∗ xp1); % Layer 2 a2 = repmat (b2, 1, Q) + LW2_1 ∗ a1; % Output 1 y1 = mapminmax_reverse (a2, y1_step1); y1 = y1′; end % =====MODULE FUNCTIONS======== % Map Minimum and Maximum Input Processing Function function 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 Function function a = tansig_apply(n, ∼) a = 2./(1 + exp(−2 ∗ n)) − 1; end % Map Minimum and Maximum Output Reverse-Processing Function function 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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