Prediction of Punching Capacity of Slab-Column Connections ...downloads.hindawi.com/journals/ace/2019/7904685.pdf · the theory referred to in these formulas for punching ca-pacity

Research ArticlePrediction of Punching Capacity of Slab-ColumnConnections without Transverse Reinforcement Based on aBackpropagation Neural Network

Jie Bu 1 Fanzhen Zhang 1 Meng Zhu1 Zhiyang He1 and Qigao Hu2

1College of Aerospace Science and Engineering National University of Defense Technology Changsha 410072 China2College of Military Education and Training National University of Defense Technology Changsha 410072 China

Correspondence should be addressed to Fanzhen Zhang zhangfanzhen7163com

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

Academic Editor Roman Wan-Wendner

Copyright copy 2019 Jie Bu et al -is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Punching shear failure of slab-column connections can cause the progressive collapse of a structure In this study a punching testdatabase is first established -en based on the LevenbergndashMarquardt (LM) algorithm and using the nonlinear function of thebackpropagation neural network (BPNN) a prediction model of the punching capacity of slab-column connections withouttransverse reinforcement is established Finally the proposedmodel is compared with the formulas of the Chinese American andEuropean standards using several methods -e statistical eigenvalue method shows that the BPNN model has the highestaccuracy and the lowest dispersion-e defect point counting method shows that the BPNNmodel had the fewest total number ofdefects and was the safest and most economical -e influencing factor analysis suggests that factors in the BPNN model had themost reasonable influence on the punching bearing capacity of slab-column connections Finally the model is verified using a casestudy and the Matlab program -e results show that the average error of the formulas in the Chinese American and Europeanstandards are 2108 3021 and 1147 respectively higher than that of the BPNN model

1 Introduction

Bullockrsquos Department Store collapsed in 1994 SanfengDepartment Store in South Korea toppled down in 1995 andMorbio Inferior Shopping Center in Switzerland fell down in2004 [1ndash3] -e above accidents show that slab-columnconnections without transverse reinforcement are prone topunching failure when they experience accidental loadswhich then triggers a chain reaction and finally leads to theprogressive collapse of the entire structure -erefore re-searchers have conducted a large number of trials andtheoretical analyses on the mechanism of slab-columnconnections without transverse reinforcement underpunching failure and this research study has led to acomprehensive cognition of the variables that can triggerpunching failure Some mechanical models have beenproposed that are based on the knowledge of the mechanismof slab-column connections to resist shearing Such models

include Muttonrsquos critical crack punching theory [4] Kin-nunen and Nylanderrsquos conical shell model [5] and Alex-ander and Simmondrsquos space truss model [6] However thesetheoretical equations are hard to unify and are in complexform hence they are not suitable for engineering applica-tions Currently the recommended calculation methods forpunching capacity in the national standards consist pri-marily of half-experience and half-theory formulas based onexperimental data fitting -ese methods include the cal-culation formulas in the GB 50010-2010 [7] ACI318-14 [8]and EC 2-04 [9] codes -ese formulas are concise and clearmaking them accessible to engineering designers Howeverthe theory referred to in these formulas for punching ca-pacity primarily considers one kind of resisting shearmechanism or two while ignoring other factors -us theapplicability of these formulas is limited [10ndash12] -e ac-curacy and reliability of the calculations also need to befurther improved

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 7904685 19 pageshttpsdoiorg10115520197904685

Due to the complexity of the mechanism of slab-col-umn connections without transverse reinforcement underpunching shearing failure it is difficult to predict thepunching capacity of slab-column connections by fittingthe test data taking all factors into consideration -is isespecially the case when it comes to influencing factorswith large nonlinearity and uncertainty as well as dis-creteness caused by brittle failure An artificial neuralnetwork is a data-processing model enlightened by a bi-ological neural network the most common of which iscalled a BPNN -e Kolmogorov theorem has proven thatthe BPNN has strong nonlinear mapping and general-ization abilities Each continuous function or mappingfunction can be realized by three levels of networks [13]-e BPNN has been widely used for solving many civilengineering problems including the shear performance ofconcrete members Kumar and Barai established a neuralnetwork that can be used to predict the extreme shearingforce of fiber concrete beams without transverse re-inforcement [14] Mansour predicted the extreme shearingstrength of an RC beam with transverse reinforcementusing an artificial neural network [15] Meanwhile Claderaand Mari used a neural network to establish a designprocedure for a normalhigh-strength concrete beam withand without stirrups [16] Using the nonlinear function ofa BPNN this study established a prediction model for thepunching capacity of slab-column connections withouttransverse reinforcement In addition the BPNN model isevaluated using the statistical eigenvalue method thedefect point method and the influencing factor analysismethod -e proposed model provides a new method forthe calculation of punching capacity

2 Establishment of the BPNN Model forPunching Capacity

-e BPNN is a forward network based on error backpropagation It is composed of trained artificial neuronsand is used to represent the optimal connection between agiven input and expected output -e establishment of theBPNN model for punching capacity primarily includes thefollowing steps -e first step is to establish trainingsamples As the research background for this study it wasnecessary to determine the input and output sample pa-rameters and establish the corresponding punching testdatabase -e second step is the design of the BPNNmodelIt primarily includes the setting of the number of networklayers number of input layer nodes number of hiddenlayer nodes number of output layer nodes activationfunction training method training parameters and otherfactors -e third step is centered on BPNN training -eLevenbergndashMarquardt (LM) algorithm is used to iterativelyupdate the weights and thresholds of the network so thatthe errors between the output value of the network and theexpected output are continuously reduced and thus theoptimal neural network is achieved -e fourth step is theevaluation of the training results -e performance of theneural network is then evaluated by cost function andregression analyses

21 Establishment of the Punching Test Database At thebeginning of the 20th century the progressive collapse ofplate structures caused by punching failures of slab-columnconnections attracted the attention of scholars [17]-erefore scholars around the world have conducted nu-merous studies in the past century on the punching failure ofslab-column connections -e research parameters haveprimarily included concrete strength longitudinal re-inforcement size effect column section shape and size shearspan to depth ratio boundary conditions punching shearreinforcement loading mode (axial or partial load) ag-gregate type and size and other factors Many researchresults have been obtained due to these studies

In this study most of the domestic and foreign litera-tures regarding punching tests of slab-column connectionswithout transverse reinforcement were collected [18ndash46]-e engineering application practice the integrity of keydata and the research purpose of this study were all con-sidered To select the quantifiable influencing parameters ofpunching capacity as comprehensively as possible it wasdetermined that concrete strength (fc) longitudinal re-inforcement ratio (ρ) yield strength of longitudinal re-inforcement (fy) effective height of the plate section (h0)shear span to depth ratio ((L1 minus c)2h0) and ratio of columnside length to plate effective height (ch0) would be used asthe control variables of punching capacity (Vu) of slab-column connections -is was done so as to keep otherinfluencing factors consistent and to screen the data Finallya set of BPNN training samples were established usingfc ρ fy h0 (L1 minus c)2h0 and ch0 as input values and Vu asthe output -e punching test database included 206 groupsof slab-column connections without transverse re-inforcement (see Table 1)

Due to differences in test designs the material perfor-mance indices loading modes and other aspects of the testdata in different literatures there is a need to define auniform description of key test data Hence the database isnow explained First the database was unified using theinternational system of units (SI) in the database Secondthe concrete strength was uniformly converted into axialcompressive strength fc and the conversion relationshipsof concrete strength (fc fcprime ft and fcu) are shown in Table 2-ird the slab-column connection in the database was allsquare cylinders and vertical loads were applied using a steelpad short column or along the support boundary -esimulation of the simply supported boundary (the forcecharacteristics of the reverse-bending line) was realizedusing the simply supported four sides

22 Design of the BPNN Model A neural network can bemultilayered Existing theories have proven that a networkwith a single hidden layer can achieve arbitrary nonlinearmapping by appropriately increasing the number of neuronnodes [10] A single hidden layer can meet the needs of theproblem background involved in this study -e number ofinput and output layer nodes depends on the dimension ofthe input and output vectors According to the backgroundof 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 Specimen number fc (MPa) ρ () fy (MPa) h0 (mm) (L1 minus c)2h0 (mm) ch0 (mm) Vu (kN)

1

Elstner [15]

A-1b 239 115 332 118 647 215 3652 A-1c 276 115 332 118 647 215 3563 A-1d 343 115 332 118 647 215 3514 A-1e 193 115 332 118 647 215 3565 A-2b 185 247 321 114 669 223 4006 A-2c 348 247 321 114 669 223 4677 A-7b 265 247 321 114 669 223 5128 A-3b 215 37 321 114 669 223 4459 A-3c 252 37 321 114 669 223 53410 A-3d 328 37 321 114 669 223 54711 A-4 248 115 332 118 603 302 40012 A-5 264 25 321 114 625 312 53413 A-6 238 37 321 114 625 312 49814 A-13 249 05 294 121 588 294 23615 B-2 425 05 321 114 669 223 20016 B-4 426 09 303 114 669 223 33417 B-9 395 2 341 114 669 223 50518 B-14 449 3 325 114 669 223 57819

Moe [16]

S1-60 221 11 399 114 669 223 38920 S2-60 21 15 399 114 669 223 35621 S3-60 215 2 399 114 669 223 36422 S4-60 226 26 399 114 669 223 33423 S1-70 233 11 483 114 669 223 39324 S3-70 241 2 483 114 669 223 37825 S4-70 33 26 483 114 669 223 37426 S4-70A 195 26 483 114 669 223 31227 S5-60 211 11 393 114 692 178 34328 S5-70 219 11 483 114 692 178 37829 H-1 248 11 328 114 665 232 37230 M1A 198 15 481 114 647 268 44331

Manterola [17]

P1-S1 243 092 304 107 1355 093 21632 P2-S1 319 092 304 107 1285 234 25733 P3-S1 282 092 304 107 1192 421 30134 P1-S2 23 092 324 107 1355 093 19635 P2-S2 313 092 324 107 1285 234 28336 P3-S2 303 092 324 107 1192 421 39737 P1-S3 365 092 324 107 1355 093 18438 P2-S3 335 122 324 107 1355 093 21139 P3-S3 362 057 324 107 1355 093 16540 Corley [18] AN-1 178 15 404 111 709 229 33441 AN-2 194 1 444 111 732 183 26642

Hawkins [19]

S2150-1 281 154 331 127 700 200 46343 S2150-2 286 156 331 127 700 200 44044 S4150-1 332 152 331 127 700 400 57945 S4150-2 334 152 336 127 700 400 581

Advances in Civil Engineering 3

Table 1 Continued


46

Yoshio Kakuta [20]

SB1-S1 326 117 3844 75 633 067 14047 SB1-S2 35 117 3844 75 633 067 13048 SB2-S3 307 117 3844 75 600 133 20049 SB2-S4 332 117 3844 75 600 133 16050 SB3-S6 305 117 3844 75 567 200 20451 SB3-S7 322 117 3844 75 567 200 22052 SB4-S8 309 117 3844 75 533 267 24053 SB4-S9 289 117 3844 75 533 267 23554 SP1-S10 291 047 3844 75 600 133 11855 SP2-S11 312 047 3844 75 600 133 13256 SP3-S12 319 155 3844 75 600 133 21057 SP3-S13 309 186 3844 75 600 133 20058 SP4-S14 312 186 3844 75 600 133 22359 SP5-S15 312 186 3844 75 600 133 22060 SC1-S16 323 118 3471 75 600 133 17061 SC1-S17 311 118 3471 75 600 133 19062 SC4-S18 28 092 3844 75 600 133 16963 SC4-S19 288 092 3844 75 600 133 16064 SA3-S22 292 117 3844 75 933 133 18965 SA3-S23 294 117 3844 75 933 133 19266 SA4-S24 341 117 3844 75 1267 133 16067 SA4-S25 316 117 3844 75 1267 133 15068 SH3-S28 29 117 3844 120 375 083 30169 SH3-S29 311 117 3844 120 375 083 29270 S61 396 079 343 170 353 176 66071 S62 418 079 343 170 353 176 66072 S63 354 119 4736 170 353 176 75073 S64 393 119 4736 170 353 176 75074 S65 393 109 3844 170 353 176 63575 S66 393 109 3844 170 353 176 60076 S67 367 337 3844 120 500 250 40077 S68 413 337 3844 120 500 250 66078 S69 392 211 3844 120 500 250 54079 S70 397 211 3844 120 500 250 71080 S71 408 112 3844 120 500 250 60081 S72 447 112 3844 120 500 250 60082 S73 179 099 3432 120 583 083 19083 S74 224 099 3432 120 583 083 24084 S75 428 099 3432 120 583 083 31285 S76 418 099 3432 120 583 083 34186 S89 268 112 3432 80 563 125 17487 S90 255 112 3432 80 563 125 16588

Regan [21]

I1 245 24 500 77 1058 260 19489 I2 223 12 500 77 1058 260 17690 I3 261 14 500 77 1058 260 19491 I4 307 12 500 77 1058 260 19492 I5 268 15 480 79 1032 253 16593 I6 208 08 480 79 1032 253 16594 I7 289 08 480 79 1032 253 18695 V4 338 08 628 118 592 086 28596

Jianlan Zheng [22]

J-1 26 106 2982 125 380 200 450497 J-3 26 096 2982 125 380 200 42798 J-4 431 132 2891 80 594 313 302499 J-5 275 132 289 95 500 263 307100 J-6 275 106 278 95 500 263 3064


Table 1 Continued


101

Tomas-zewicz [23]

ND65-1-1 553 149 500 275 418 073 2050102 ND65-2-1 601 175 500 200 513 075 1200103 ND95-1-1 704 149 500 275 418 073 2250104 ND95-1-3 757 255 500 275 418 073 2400105 ND95-2-1 742 175 500 200 513 075 1100106 ND95-2-1D 73 175 500 200 513 075 1300107 ND95-2-3 753 262 500 200 513 075 1450108 ND95-2-3D 676 262 500 200 513 075 1250109 ND95-2-3D+ 825 262 500 200 513 075 1450110 ND95-3-1 716 184 500 88 568 114 330111 ND115-1-1 943 149 500 275 418 073 2450112 ND115-2-1 1002 175 500 200 513 075 1400113 ND115-2-3 91 262 500 200 513 075 1550114

-eodor-akopoulos [24]

FS-1 302 056 460 100 77 15 1735115 FS-8 312 056 460 100 795 1 1503116 FS-10 31 056 460 100 745 2 1914117 FS-19 295 037 460 100 77 15 1365118

Yujie An [25]

A-1 295 124 3255 93 538 215 274119 A-2 314 132 3255 93 538 215 277120 A-3 212 132 3255 93 538 215 249121 A-4 295 091 3255 93 538 215 240122 A-5 192 058 3255 93 538 215 170123

Sistonen [26]

L1 31 046 621 172 398 117 503124 L2 31 045 621 176 389 115 537125 L3 31 045 621 173 396 116 530126 L4 31 067 612 170 402 236 686127 L5 31 066 612 172 399 232 696128 L6 31 065 612 175 390 232 799129 L7 229 064 586 177 443 114 478130 L8 229 116 576 174 394 517 1111131 L9 229 117 576 172 399 522 1107132 L10 229 116 576 173 396 521 1079133

Ghannoum [27]S1-U 346 096 445 109 814 206 301

134 S2-U 485 096 445 109 814 206 363135 S3-U 55 096 445 109 814 206 443136 Krueger [28] P0A 352 1 480 121 1012 248 423137 Beutel [29] P1 219 081 572 190 526 211 615138 Ospina [30] SR-1 343 087 430 120 592 208 3651139

Park [31]S1 246 106 453 90 694 278 230

140 S2 26 2 466 90 694 278 3168141 S3 246 098 485 130 481 192 4432142

Birkle [32]1 329 154 488 124 706 202 483

143 7 32 13 531 190 711 158 825144 10 289 11 524 260 663 135 1046145

Xiaokun Huang [33]

BAN105 302 05 490 168 536 119 5866146 BAN113 298 13 507 164 549 122 8053147 BAN105(1) 283 05 490 168 536 119 5223148 BAN113(1) 267 13 507 164 549 122 7968149

Yuanwei Zhang [34]

A20-1 322 157 4584 98 485 255 356150 A20-2 329 12 4584 128 371 195 470151 A20-3 343 097 4584 158 301 158 646152 A20-4 279 175 4584 88 483 398 408153 A35-1 314 157 4584 98 485 255 360154 A35-2 378 114 371 99 480 253 357155 A35-3 402 079 4135 100 475 250 293156 A50-1 443 157 4584 98 485 255 412157 A50-2 447 114 371 99 480 253 354


Table 1 Continued


158

Guandalini [35]

PG-1 277 15 573 210 595 124 1023159 PG-2b 384 025 552 210 595 124 440160 PG-4 32 025 541 210 595 124 408161 PG-10 285 033 577 210 595 124 540162 PG-11 341 075 570 210 595 124 763163 PG-6 341 15 526 96 651 135 238164 PG-7 341 075 550 100 625 130 241165 PG-8 341 028 525 117 534 111 140166 PG-9 341 022 525 117 534 111 115167

Guidottiv [36]

PT22 575 082 552 196 638 133 989168 PT31 567 148 552 212 590 123 1433169 PG19 434 078 510 206 607 126 860170 PG20 465 156 551 201 622 129 1094171 PG23 389 081 510 199 628 131 839172 PG24 378 161 551 194 644 134 1102173

Rizk [37]

NSC1 312 052 435 1625 486 154 479174 NSC2 328 217 433 1575 502 159 678175 HSC1 507 065 435 1625 486 154 675176 HSC2 514 098 440 160 494 156 798177 HSC3 514 113 433 1575 502 159 811178 HSC4 55 167 433 1575 502 159 802179 HSC5 564 248 433 1375 575 182 788180 HSC6 571 268 433 1275 620 196 801181 NSC3 323 04 450 105 752 238 228182 HSC7 571 188 435 1125 702 222 481183

Lips [38]

PL1 353 163 583 193 681 067 682184 PV1 335 15 709 210 595 124 974185 PL3 355 159 583 197 569 264 1324186 PL4 305 158 550 267 453 127 1625187 PL5 317 15 580 353 329 125 2491188

Jian Peng [39]

C7-30-1 245 086 604 150 700 167 473189 C7-30-2 226 128 604 150 700 167 600190 C7-50-1 394 086 604 150 700 167 723191 C7-50-2 359 128 604 150 700 167 801192 C5-30-1 264 086 604 150 500 167 678193 C5-30-2 258 128 604 150 500 167 692194 C5-50-1 348 086 604 150 500 167 691195 C5-50-2 34 128 604 150 500 167 855196 Caldentey [40] 1 346 107 575 200 513 225 974197 2 349 107 575 200 513 225 956198

Inacio [41]

SNSC 336 1 523 105 633 190 2892199 SHSC1 1007 094 493 1042 638 192 4129200 SHSC2 1043 124 523 1016 655 197 429201 SHSC3 1039 148 523 1017 654 197 4609202

Bartolac [42]S2-1 403 15 560 96 714 135 3939

203 S2-2 378 15 560 96 714 135 3613204 S2-3 394 15 560 96 714 135 3852205 Einpaul [43] PE4 329 159 517 197 293 132 985206 PE3 322 154 517 204 809 127 961

Table 2 Conversion relationships of concrete strength (fc fcprime ft and fcu)

Conversion relationship Conversion factorsfc αc1αc2fcu

αc1

076 (leC50)

αc2

1 (leC40)

α

08 (C20～C40)

ft 0395αc2f055cu

Linear interpolation(C50simC80)


083 (C50)086 (C60)0875 (C70)089 (geC80)fcu αfcprime 082 (C80) 087 (C80)


study the number of nodes used for the input layer is six-e input layer nodes are fc ρ fy h0 (L1 minus c)2h0 ch0-e number of nodes in the output layer is one which is thenetwork output Vu -e number of hidden layer nodes has agreat influence on the performance of the BPNN modelGenerally more hidden layer nodes can contribute to betterperformance It was found that when the node of the hiddenlayer was 15 the network had the best performance and thesmallest error For the selection of the activation functionthe general hidden layer used a tan-sigmoid function (hy-perbolic tangent function) while the output layer adopted alinear function -e training algorithm chose the LM al-gorithm with a fast convergence speed and a small meansquare error -e BPNN model used an iterative updatingmethod to determine weights and thresholds -e initialweights and thresholds were defined as small nonzero valuesthat were randomly selected by the computer -e finalneural network structure is shown in Figure 1

23 Training of the BPNN Model -e input vector (fc

ρ fy h0 (L1 minus c)h0 and ch0) is constructed using theparameters in the punching database and Vu is the expectedoutput value -ere are 206 groups of inputs and expectedoutputs We assume that the input vector of the kth sampleis x(k) (x1

(k) x6(k)) (k 1 206) -e corre-

sponding output of the hidden layer is h(k) (h1(k)

h15(k)) (k 1 206) -e output of the output layer is

y(k) (k 1 206) -e expected output is d(k) (k

1 206) -e connection weight between the input layerand the hidden layer is then set to be ω(h)

ij (i 1 6

j 1 15) -e connection weight between the hiddenlayer and the output layer is ωj

(0) (j 1 15) -ethreshold vector of neurons in the hidden layer is a

(a1 a15) -e threshold of the output layer neuron is b-e activation function of the hidden layer neurons is f(middot)

and the activation function of the output layer neuron isg(middot)

-en the jth output in the hidden layer is as follows

hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)

-e output in the output layer is as follows

y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)

-e error of the kth sample is as follows

ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)

-e cost function (the average error of all samples) is asfollows

E 1

2 times 2061113944

206

k1ek (4)

All the weights (15times 6 + 15) and thresholds (15 + 1) wereput into a weight matrix (121times 1) It was assumed that theweight matrix after the nth iteration was W(n) -e LMalgorithm was then applied to update W(n) iteratively -eJacobian matrix of the cost function was then defined -ematrix uses the partial derivative of the error with respect tothe parameter as the element

Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)

-e expression of the iterative update of the weightmatrix is as follows

W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)

where e (e1 e2 em)T and μ is a tentative parameterFor a given parameter μ if the cost function E can be re-duced by a change in the threshold ∆W μ is reducedOtherwise μ increases

According to the above calculation results the trainingalgorithm flow of the BPNN model is shown in Figure 2

24 Result of the BPNN Model -e sample data of 206groups are divided randomly into three groups trainingsamples (144) validation samples (31) and test samples (31)Training samples are used for error adjustment in thenetwork training Test samples are used to measure thenetwork generalization -e training is stopped when thegeneralization stops improving Test samples are used tomeasure the performance of network after training -etraining samples are then substituted into the above trainingalgorithm for iterative calculation to obtain an optimizedBPNN model to determine the final weight and thresholdvalues and to obtain the neural network code for thepunching capacity prediction model (see Appendix)

Figure 3 shows the variation of the cost function of thetraining samples verification samples and test samples withthe number of iterations It can be seen from the figure thatwhen the number of iterations reaches 20 the cost functionvalue is the minimum and the training stops at this timeFigure 4 shows the training process of the neural networkFigure 5 shows the linear regression analysis between theoutput values of the training samples verification samplestest samples and population samples and the expectedoutput values According to this figure the correlationbetween the neural network output and the expected outputis very high reaching 099663 098419 098922 and099342 respectively indicating that the neural network hasvery good performance and a good fitting effect

3 Assessment of the BPNN Model

Since the BPNN model for predicting the punching ca-pacity of slab-column connections without transverse re-inforcement established in this study was directly based onresults of existing test data the predicted value of the modelwas given a safety reserve of 5 -is means that the


predicted value of the model would be 95 of the outputvalue of the model Furthermore the value of a model isdetermined by its application in engineering practice-erefore the present study established a set of relativelyperfect evaluation system to evaluate the performance ofthe model More specifically the statistical eigenvaluemethod was used to evaluate the precision and discretedegree of the model and the defect point counting methodwas used to evaluate the safety and economy of the modelIn addition the influencing factor analysis method wasused to evaluate whether the model can reasonably reflect

the impact of various factors on the punching bearingcapacity

31 0e Statistical Eigenvalue Method Based on thepunching test database of the slab-column connectionswithout transverse reinforcement the predicted results ofthis model were compared with the calculated results of theChinese concrete structural design code GB 50010-2010 theAmerican concrete code ACI 318-14 and the Europeanmodel code EC 2-04 For the sake of description the fol-lowing variable is defined

fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn

The connection weight of the ith

neuron

Input of theith neuron

sum

Figure 1 Topological structure of the BPNN model

EndMake

k = k + 1 and micro = micro β

Initialize network weights thresholds error margin ε and constant micro β (0 lt β lt 1)

make N = 0 and micro = micron

Calculate the network output and the cost function

Compute Jacobianmatrices

Compute ΔW micro = microβ

If E(n) lt ε

Compute W(n + 1) = W(n) + ΔW andcompute E(n + 1)

If E(n + 1) lt E(n)

Figure 2 -e BPNN model algorithm flowchart

105

Gra

dien

t

Gradient = 3931721 at epoch 26

100

mu

Mu = 1 at epoch 26

Validation checks = 6 at epoch 26

5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs

Figure 3 Cost functions


TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs

Best validation performance is 35006758 at epoch 20

Figure 4 Training process

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)

where Vtest is the value of the punching capacity obtained inthe test and Vcal is the value of the punching capacity cal-culated using various methods as shown in Table 3

As shown in Table 3 the mean and median of the BPNNmodel are 105834 and 105486 which are the minimum infour formulas followed by EC 2-04 and GB 50010-2010 -eAmerican standard is a little conservative And the mean andmedian of ACI 318-14 are 131155 and 132631 -ereforethe BPNN model is the most accurate in four predictionmethods

According to the standard deviation and variable co-efficient of X the discretization degree of the BPNNmodel isthe smallest followed by EC 2-04 while discretization de-grees of GB 50010-2010 and ACI 318-14 are relatively largeMoreover P5 and P95 represent the 5th and 95th percen-tiles respectively P5 means that 5 of the data are less thanthis value and P95 means that 5 data are more than thisvalue -us 90 of the data are within the range of (P5P95) By comparing the interval of (minimum maximum)and the interval of (P5 P95) it can be seen that the dis-tribution interval (located near 1) of the BPNN modelprediction results is far better than that of the other stan-dards -is also indicates that compared with the threestandards the dispersion degree of the BPNN model pre-diction results is smaller

320eDefect PointCountingMethod Drawing on the ideasof the defect point deduction method [47] and the pointcounting method [48] in management science the defect

point countingmethod is proposed to evaluate the safety andeconomy of the model -e basic idea is as follows-e ratioX of the test value (Vtest) and the predicted value of eachpunching capacity calculation method (Vcal) is used toclassify the severity of the defects and to determine theweight and number of defect points of each defect grade-en the total number of defects is obtained using theweighted sum -e fewer the total number of points thebetter the method can predict the punching capacity of slab-column connections For the ratio of the experimental valueand the predicted value when X is less than 1 the slab-column connection is not safe When it is greater than 1 it issafe However if X is too large it is too conservative anduneconomical Based on the severity classification principleof bell system defects and combined with the actual situa-tion the X value was divided into six severity levels ofdefects Based on the test data the number of defect points ineach punching design method was counted Table 4 showsthe number of defect points in the slab-column connectionsIt can be seen from Table 4 that the total number of defectpoints in the BPNN model was 129 which is far fewer thanthe total number of defect points in the other three stan-dards -e number of defect points was 246 in the Chinesestandard 242 in the American standard and 194 in theEuropean standard these were 9069 876 and 5039respectively higher than that of the BPNN model -isshows that the BPNN model had the best comprehensiveperformance for safety and economy

330e Influencing Factor Analysis Method -e ratio of thetest value and the predicted value is used as the ordinate-econcrete strength (fc) ratio of longitudinal reinforcement

DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)

Figure 5 Linear regression between the neural network output and the expected output (a) Training R 099663 (b) ValidationR 098419 (c) Test R 098922 (d) All R 099342


(ρ) yield strength of longitudinal reinforcement (fy) ef-fective height of plate section (h0) shear span to depth ratio((L1 minus c)2h0) and ratio of column side length to plateeffective height (ch0) are used as the abscissa-e upper andlower limits of 95 andmean value of each group of data aregiven and the influences of various factors on the predictionresults of the punching capacity are then analyzed

Figures 6ndash8 respectively depict the influence of theeffective height of the plate section (h0) the ratio of thecolumn side length to the plate effective height (ch0) andthe concrete strength (fc) on the ratio of the test value to thepredicted values

It can be seen from Figure 6 that when the effectiveheight of the plate is less than 100mm the dispersion ofprediction results of each formula is relatively large Whenthe effective height of the plate is greater than 100mm theaccuracy and dispersion of the BPNN model predictionresults are significantly better than those of the other threeformulas From Figure 7 we can see that when the parameterch0 is less than 2 the prediction results of Chinese andAmerican formulas are not ideal Meanwhile the BPNNmodel and the European formula predict better results overthe entire interval Figure 8 shows that the Chinese andAmerican formulas have an obvious trend distribution withthe change in concrete strength while the BPNN model andthe European formula show better prediction results and lessdispersion

Although Chinese American and European standardsdid not consider the influence of the shear span to depthratio ((L1 minus c)2h0) the yield strength of the longitudinalreinforcement (fy) and the ratio of the longitudinal re-inforcement (ρ) on the punching capacity the BPNNmodel was still used to evaluate the influence of these threevariables in this study as shown in Figures 9ndash11 -ecalculation results show that with changes in the threevariables the ratio of the test value to the predicted value of

the BPNN model is uniformly distributed on both sides ofthe mean and there is no trend distribution or largedispersion degree

In general Chinese and American formulas are con-servative and their prediction results are highly dispersed-e European formula predicts better accuracy and has asmaller dispersion Meanwhile the BPNN model considersthe most comprehensive influencing factors and does notshow a trend distribution with changes in fc ρ fy h0(L1 minus c)2h0 and ch0 -e above results indicate that theBPNN model reasonably reflected the impact of variousfactors on the punching capacity of slab-column connec-tions without transverse reinforcement

Based on the above calculation and analysis it is clearthat the BPNN model established in this study is superior tothe calculation formulas in the Chinese American andEuropean standards

In order to illustrate the necessity of the parametersconsidered in the BPNN model the American formula isused to illustrate the influence of these parameters It can beseen from Figure 12 that as the reinforcement ratio in-creases the result also increases which is obviously un-reasonable -e main reason for this is that the Americanformula does not consider the effect of the reinforcementratio As shown in Figures 13 and 14 the prediction resultof the American formula has a large dispersion withchanges in fy and (L1 minus c)2h0 Furthermore there aremany unsafe results which are not allowed in the actualapplication -erefore the parameters selected in thispaper have a significant impact on the punching capacityEmpirically ACI 318-14 uses the shear force of criticalsection as the punching capacity of slab-column connec-tions -e parameters used in ACI 318-19 to calculate theshear strength include critical section perimeter effectiveheight of plate section and concrete strength Fromcomparison results of this study parameters used in the

Table 3 Statistical eigenvalue

X GB 50010-2010 ACI 318-14 EC 2-04 BPNN modelMean 11979 131155 114481 105834Median 119218 132631 113721 105486Standard deviation 029515 032284 019464 012292Variable coefficient 024639 024615 017002 011614Minimum 044833 049899 068027 07239Maximum 199606 224406 163209 180068P5 078062 086058 082005 088283P95 172043 192329 153279 125306

Table 4 Number of defect points

X Defect level Weight value GB 50010-2010 ACI 318-14 EC 2-04 BPNN modellt05 Very dangerous 10 4 1 0 005ndash075 Dangerous 5 6 6 4 2075ndash1 Low degree of safety 2 45 33 42 541ndash125 Safe and reasonable 0 76 48 116 150125ndash2 Conservative 1 86 122 55 11gt2 Too conservative 2 0 7 0 0Total defect count 246 242 194 129


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)

Figure 6 Influence of h0 on the ratio of the test value to the predicted value (a) BPNNmodel (b) GB 50010-2010 (c) ACI 318-14 (d) EC 2-04

0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)

Figure 7 Influence of ch0 on the ratio of the test value to the predicted value (a) BPNNmodel (b) GB 50010-2010 (c) ACI 318-14 (d) EC 2-04

0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)

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


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()

Figure 11 ρ influence on the BPNN model

0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20

Figure 9 (L1 minus c)2h0 influence on the BPNN model

0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)

Figure 10 fy influence on the BPNN model

ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()

Figure 12 ρ influence on ACI 318-14

ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)

Figure 13 fy influence on ACI 318-14

ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0

Figure 14 (L1 minus c)2h0 influence on ACI 318-14


paper are perhaps more appropriate than the code -epaper suggests that ACI 318-14 should also take theseparameters into account to improve the reliability of theformula -e same advice applies to GB 50010-2010 and EC2-04

4 Application Example

To verify the reliability and accuracy of the model presentedin this study further the punching test of slab-columnconnections without transverse reinforcement [42 49]completed by Hunan University was selected as the calcu-lation example-e relevant parameters of the punching testof slab-column connections are shown in Table 5

To run this calculation example the following actionswere performed first we ran the Matlab working envi-ronment and edited the neural network code in Appendixand opened its root directory Next we typed the NeuralNetwork Function (x1) on the command line x1 is[fc ρ fy h0 (L1 minus c)2h0 ch0] -e output was the value ofthe punching capacity of slab-column connections predictedin this study Finally predicted results were compared withcalculated results of Chinese American and Europeanstandards in Table 6

Vtest is the value of the punching capacity measured bythe test VBP VGB VACI andVEC are the values of thepunching capacity calculated by the BPNN model theChinese American and European standards respectivelyFrom the table above we can observe that the maximum of

VTestVBP is 11208 the minimum is 10172 the mean is1072253 and the error is within 10 In addition the meanvalues of VTestVGB VTestVACI andVTestVEC are 12982881396167 and 119527 respectively -e calculation errors ofthe Chinese American and European standards are 21083021 and 1147 higher than that of the BPNN modelrespectively -us it can be seen that using the Matlab tooland the BPNN model prediction code established in thisstudy can calculate the punching capacity of slab-columnconnections more quickly and accurately

5 Conclusion

In general shear failure is induced by the transfer betweenthe column and slab of a shearing force combined with amoment -e research considers the transfer of a shearingforce (without eccentricity with respect to the centroid of theshear critical section) In this study a BPNN was used tostudy the punching capacity calculation of slab-columnconnections without transverse reinforcement -e mainconclusions are as follows

First the quantifiable impact parameters of the punchingcapacity of slab-column connections without transversereinforcement were considered comprehensively Apunching test database (including 206 groups) of slab-col-umn connections without transverse reinforcement wasestablished -is included influencing factors of concretestrength fc the ratio of longitudinal reinforcement ρ theyield strength of longitudinal reinforcement fy the effective

Table 5 Punching test parameters for the slab-column connections

Specimen number fc (MPa) ρ () fy (MPa) h0 (mm) (L1 minus c)2h0 ch0 Vu (kN)

S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860

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

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

VBP (kN) VTestVBP VGB (kN) VTestVGB VACI (kN) VTestVACI VEC (kN) VTestVEC

S5-1 4748092 108886 44016 117457 3960000 130556 4357664 118642S5-2 5629783 108708 45192 135422 4053926 150965 5058992 120973S7-1 4773357 11208 43344 123431 3904166 137033 4320096 123840S7-2 5465992 10172 41664 133449 3765125 147671 4810487 115581C7-30-3 6680036 103293 47376 145643 4301661 160403 5726001 120503C7-50-3 7426983 108389 54768 146984 5052687 159321 6369424 126385C7-70-1 5738016 106309 60984 100026 598995 101837 5608018 108773C7-70-2 7333379 107045 61656 127319 6088621 128929 6475931 121218C7-70-3 7919113 108598 61992 138728 6150124 139835 7176964 119828Mean 107225 129829 139617 119527Error 703 298 396 195Relative error 0 2108 3021 1147


height of the plate section h0 the shear span to depth ratio(L1 minus c)2h0 and the ratio of column side length to the plateeffective height ch0 -e BPNN model was designed andtrained and then was evaluated by means of mean squareerror and regression analyses -e BPNN model showedgood performance

Second the predicted value of the BPNNmodel of 95was used as the model output In this study the accuracyand discreteness of the BPNN model were evaluated usingthe statistical eigenvalue method and the safety andeconomy of the BPNN model were evaluated using thedefect point counting method Whether the BPNN modelcould reasonably reflect the impact of various factors onthe punching capacity was evaluated using the influencingfactor analysis method Research results suggested that themean median standard deviation and variation co-efficient of VtestVcal of the BPNN model were optimal thenumber of defect points were fewest and the predictedresults did not show a trend distribution and large dis-persion of relevant factors -is means that the BPNNmodel can meet relevant requirements well and is superiorto the formulas in the Chinese American and Europeanstandards

-ird based on the BPNN model established in thisstudy the code was run in the Matlab working environmentParameters of the test model of Hunan University werecalculated and they were compared with the correspondingresults of the Chinese American and European standards-e results indicate that the BPNN model had the bestcalculation results with an average error within 10Compared with the BPNN model the calculation errors ofthe Chinese American and European standards were in-creased by 2108 3021 and 1147 respectively

Appendix

Neural Network Code for Prediction Model ofPunching Capacity

function [y1]myNeuralNetworkFunction (x1)MYNEURALNETWORKFUNCTION neural net-work simulation function Generated by Neural Network Toolbox functiongenFunction 13-Feb-2019 17 51 35 [y1]myNeuralNetworkFunction (x1) takes thesearguments xQx6 matrix input 1 and returns yQx1 matrix output 1 where Q is the number of samplesokltlowastRPMT0gt NEURAL NETWORK CONSTANTS

Input 1x1_step1middotxoffset [75 022 278 178 29263959390666666667]x1_step1middotgain [000719424460431655 0574712643678161 000464037122969838 0023121387283237018823518904255 0439710268466749] x1_step1middotymin minus 1 Layer 1b1 [minus 067013086397926890925 020485204467667655903 minus 39823333846859974017 026809837709864853261 17519871170795724424 099641617199102761049 49763998448198938362 11368201646791966652minus 0025928389261043835456 minus 092100364806924983618 036880965470930515693 1119473168509247607minus 57906209524835876579 18210861550491890259 10562373148061163786]IW1_1 [054241403693654410834 024796025994322995478 minus 0014337576347544853453 018764574836693678761 minus 002829727644609779727 014145204455730814308 minus 24509908541198388576 28901071801540454409 minus 20586515897176820289 1068335537227783183minus 46183794460269416149 minus 002484542256670946303minus 76917611610029394953 30974401276965424223 minus 37294875430272114158 44642135026415905585 73171434926831198098 minus 12328913326533490036 35595775453552365164 33699076853018046229 15200070719384131124 minus 34388910985329763648 minus 62360139697241718792 minus 26537507383974756614 minus 20904525084619258735 43392922147544492617 59324277695856784121 minus 85047866210874421 minus 27523798326730455344 40482637507758116868 minus 022774737124199373461 minus 076951597973573482303 69372813682084002451 minus 016830700683789390215 5845781338065211940634576020408617522683 minus 11252702180792377717 minus 22694352690532371142 1564784156023274253 minus 49174176477441013944 minus 45068636359880498432 minus 56545914419692682173 10059197750593041221 minus 017002386664310031872 minus 89848474105867879302 10203632176534829235 49221652171394971731 minus 75710047392480142037 minus 14840366841073464599 38623959556322895459 16833086970154664819 14067932842560260109 12210393313963610051 minus 100187503973542280110548272447130120355 17632488982764233931 minus 032313056811567220095 minus 13903358650695212262 10460118017996697937 25325861994828038348 78613465153517045891 74271278087830125969 minus 28222326515948368275 minus 17772430280884681242 minus 3905950647510644469 minus 17387776097206195658 69442041861601406794 022091386915732519336 589654429623693499 19608851775534088535 8065055313758417554517641288972052406869 minus 64259520951425530555 21257615458185741275 063700368732926271331 047411663304245349471 076026839039380778029 minus 61951688429695117577 34323263734923310508 minus 12974701439316240759 2757336565240562809 minus 29415561788931592346 minus 47208935670603473511 74539269787059


039274 032595830157754457357 33624573492982667844 14011897034149485286 085463835267573085019 28748323421137191858 minus 19927818384202813373] Layer 2b2 078597773687780803087LW2_1 [34772906528603062526 012639460740666358385 minus 023439286264789302439 0020693346676863687977 minus 00060410169541254875661 0080236891899916409909 12792025383667153804 minus 0029240579066094837118 minus 021752104710756975381 minus 0040404470314744492243 004108162773976927612 0050369066295788737786 minus 0035981188002320835251 minus 024956065236877369995 012526282041160077818] Output 1y1_step1middotymin minus 1y1_step1middotgain 0000841750841750842y1_step1middotxoffset 115 SIMULATION

DimensionsQ size(x1 1) samples Input 1x1 x1primexp1mapminmax_apply (x1 x1_step1) Layer 1a1 tansig_apply (repmat (b1 1 Q) + IW1_1lowast xp1) Layer 2a2 repmat (b2 1 Q) + LW2_1lowast a1 Output 1y1mapminmax_reverse (a2 y1_step1)y1 y1primeend MODULE FUNCTIONS

Map Minimum and Maximum Input ProcessingFunctionfunction ymapminmax_apply (x settings)y bsxfun (minus x settingsmiddotxoffset)y bsxfun (times y settingsmiddotgain)y bsxfun (plus y settingsmiddotymin)end Sigmoid Symmetric Transfer Functionfunction a tansig_apply(n sim)a 2(1 + exp(minus 2lowast n)) minus 1end Map Minimum and Maximum Output Reverse-Processing Functionfunction xmapminmax_reverse (y settings)x bsxfun (minus y settingsmiddotymin)x bsxfun (rdivide x settingsmiddotgain)

x bsxfun (plus x settingsmiddotxoffset)end

Data Availability

-e punching test data used to support the findings of thisstudy have been included in Table 1 and the data can also beobtained from the papers in ldquoReferencesrdquo -e neural net-work codes for predicting punching capacity of slab-columnconnections without transverse reinforcement are listed inAppendix

Disclosure

Qigao Hu is the co-author

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-ework presented in this study was funded by the NationalNatural Science Foundation of China ldquoResearch on pro-gressive collapse mechanism of reinforced concrete flat platestructure under blast loading (Grant no 51608525)rdquo -eauthors thank LetPub (httpwwwletpubcom) for its lin-guistic assistance during the preparation of this manuscript

References

[1] D Mitchell R H DeVall M Saatcioglu R SimpsonR Tinawi and R Tremblay ldquoDamage to concrete structuresdue to the 1994 northridge earthquakerdquo Canadian Journal ofCivil Engineering vol 22 no 4 pp 361ndash377 1995

[2] T W Park ldquoInspection of collapse cause of sampoong de-partment Storerdquo Forensic Science International vol 217no 1ndash3 pp 119ndash121 2012

[3] M Yaser Post-Punching Behavior of Reinforced ConcreteSlabs EPFL Lausanne Switzerland 2010

[4] A Muttoni ldquoPunching shear strength of reinforced concreteslabs without transverse reinforcementrdquo ACI StructuralJournal vol 105 no 42 pp 440ndash445 2008

[5] S Kinnunen and H Nylander Punching of Concrete SlabsWithout Shear Reinforcement Transactions of the Royal In-stitute of Technology Stockholm Sweden 1960

[6] S D B Alexander and S H Simmonds ldquoBond model forconcentric punching shearrdquo ACI Structural Journal vol 89no 3 pp 325ndash328 1992

[7] China Architecture amp Building PressMinistry of Housing andUrban-Rural Development of the Peoplersquos Republic of ChinaCode for Design of Concrete Structures GB 50010-2010 ChinaArchitecture amp Building Press Beijing China 2010

[8] American Concrete Institute Building code requirements forstructural concrete (ACI 318-14) American Concrete In-stitute Farmington Hills MI USA 2014

[9] British Standards Institution Eurocode 2 Design of ConcreteStructures Part 1ndash1 General Rules and Rules for BuildingsBritish Standards Institution London UK 2004

[10] N Venkata Ramana ldquoReview on punching shear strength ofslabsrdquo International Journal of Engineering Research andDevelopment vol 13 no 10 pp 1ndash25 2017


[11] M Lapi M Orlando F Angotti and P Spinelli ldquoComparisonof recent code provisions for punching shear capacity of RCslabs without shear reinforcementrdquo Lecture Notes in CivilEngineering vol 10 pp 113ndash132 2018

[12] L Dang R Pang S Liang et al ldquoUltimate bearing capacitycalculation of slab-column connections and comparativestudy on different codes at home and abroadrdquo ArchitectureScience vol 35 no 1 pp 32ndash37 2019

[13] H Wang and J Sun ldquoClassification method based on BPneural network and SVMrdquo Software vol 36 no 11 pp 96ndash992015

[14] S Kumar and S V Barai ldquoNeural networks modeling of shearstrength of SFRC corbels without stirrupsrdquo Applied SoftComputing vol 10 no 1 pp 135ndash148 2010

[15] M Y Mansour M Dicleli J Y Lee and J Zhang ldquoPredictingthe shear strength of reinforced concrete beams using artificialneural networksrdquo Engineering Structures vol 26 no 6pp 781ndash799 2004

[16] A Cladera and A R Marı ldquoShear design procedure forreinforced normal and high-strength concrete beams usingartificial neural networks Part I beams without stirrupsrdquoEngineering Structures vol 26 no 7 pp 917ndash926 2004

[17] A N Talbot Reinforced Concrete Wall Footings and ColumnFootings University of Illinois at Urbana Champaign UrbanaIL USA 1925

[18] R C Elstner and E Hognestad ldquoShearing strength of rein-forced concrete slabsrdquo ACI Journal Proceedings vol 53 no 7pp 51ndash53 1956

[19] J Moe Shearing Strength of Reinforced Concrete Slabs andFootings under Concentrated Loads Portland Cement Asso-ciation Research and Development Laboratories Skokie ILUSA 1961

[20] M Manterola Poinccedilonnement de Dalles Sans Armaturedrsquoeffort Trenchant Dalles Structures Planes CEB-Bull ParisFrance 1966

[21] W G Corley and N M Hawkins ldquoShearhead reinforcementfor slabsrdquo ACI Journal Proceedings vol 65 no 10pp 811ndash815 1968

[22] N M Hawkins M E Criswell and F Roll Shear Strength ofSlabs without Shear Reinforcement vol 42 ACI Special-Publication Naples FL USA 1974

[23] Y Kakuta A Itoh and Y Fujita ldquoExperimental study onpunching strength of reinforced concrete slabsrdquo Proceedingsof the Japan Society of Civil Engineers vol 1974 no 229pp 105ndash115 1974

[24] P E Regan and M W Braestrup Punching Shear in Rein-forced Concrete a State of Art Report CEB Secretariat Per-manent Arlington VI USA 1985

[25] J Zheng and Z Zheng ldquoExperimental study on punchingstrength of reinforced concrete slabsrdquo Journal of FuzhouUniversity (Natural Science Edition) vol 20 no 2 pp 65ndash691992

[26] A Tomaszewicz High Strength Concrete SP2 Plates andShells Report 23 Punching Shear Capacity of ReinforcedConcrete Slabs SINTEF Structures and Concrete TrondheimNorway 1993

[27] D D -eodorakopoulos and N Swamy ldquoContribution ofsteel fibers to the strength characteristics of lightweightconcrete slabndashcolumn connections failing in punching shearrdquoACI Structural Journal vol 90 no 4 pp 342ndash350 1993

[28] Y An and G Zhao ldquoExperimental study on punching re-sistance of reinforced concrete slab mixed with steel fibersrdquoJournal of Building Structures vol 15 no 2 pp 11ndash14 1994

[29] E Sistonen M Lydman and S Huovinen 0e GeometricalModel of the Calculation Formula of the Punching HearCapasity of the Reinforced Concrete Slab Helsinki Universityof Technology Laboratory of Structural Engineering andBuilding Physics Espoo Finland 1997

[30] C M Ghannoum Effect of HighndashStrength Concrete on thePerformance of Slab-Column Specimens MC Gill UniversityMontreal Canada 1998

[31] G Krueger Resistance au Poinconnement Excentre desPlancers Dalles Ecole Polytechnique Federale de LausannLausanne Switzerland 1999

[32] R R K Beutel Durchstanzen Schubbewehrter Flachdecken imBereich von Innenstutzen RWTHAachen University AachenGermany 2002

[33] C E Ospina S D B Alexander and J J R Cheng ldquoPunchingof twondashway concrete slabs with fiberndashreinforced polymerreinforcing bars or gridsrdquo ACI Structural Journal vol 100no 61 pp 589ndash595 2001

[34] H G Park K K Choi and J K Wight ldquoStrain-based shearstrength model for slender beams without web re-inforcementrdquo ACI Structural Journal vol 103 no 6pp 783ndash793 2006

[35] G Birkle and W H Dilger ldquoInfluence of slab thickness onpunching shear strengthrdquo ACI Structural Journal vol 105no 2 pp 180ndash185 2008

[36] X Huang L Liu and X Tao ldquoConsidering the influence oflongitudinal reinforcement ratio and the length of sideloading surface than plate punching by experimental re-searchrdquo Journal of Civil Engineering vol 41 no 7 pp 21ndash252008

[37] Y Zhang ldquoResearch on impact shear test of reinforced concreteslabrdquo MS thesis College of Civil Engineering HunanUniversity Changsha China 2009

[38] S Guandalini O L Burdet and A Muttoni ldquoPunching testsof slabs with low reinforcement ratiosrdquo ACI StructuralJournal vol 106 no 1 pp 87ndash93 2009

[39] R Guidotti Poinccedilonnement des Planchers-dalles AvecColonnes Superposees Fortement Sollicitees Ecole Poly-technique Federale de Lausann Lausanne Switzerland 2010

[40] E Rizk H Marzouk A Hussein et al ldquoEffect of re-inforcement ratio on punching capacity of RC platesrdquo Ca-nadian Journal of Civil Engineering vol 38 no 7pp 729ndash735 2011

[41] S Lips Punching of Flat Slabs with Large Amounts of ShearReinforcement Ecole Polytechnique Federale de LausannLausanne Switzerland 2012

[42] W Yi F Hong and J Peng ldquoExperimental study on punchingshear failure of reinforced concrete slab-column connec-tionsrdquo Building Structure vol 46 no 15 pp 11ndash18 2016

[43] A P Caldentey P P Lavaselli H G Peiretti et al ldquoInfluenceof stirrup detailing on punching shear strength of flat slabsrdquoEngineering Structures vol 49 pp 855ndash860 2013

[44] M Inacio A Ramos V Lucio and D Faria ldquoPunching ofhigh strength concrete flat slabsndashexperimental investigationrdquoin Proceedings of the Fib Symposium pp 1ndash5 TelndashAviv IsraelApril 2013

[45] M Bartolac D Damjanovic and I Duvnjak ldquoPunchingstrength of flat slabs with and without shear reinforcementrdquoGraCevinar vol 67 no 8 pp 771ndash779 2015

[46] J Einpaul J Bujnak M F Ruiz et al ldquoStudy on influence ofcolumn size and slab slenderness on punching strengthrdquo ACIStructural Journal vol 113 no 1 pp 135ndash138 2016

[47] S Hu and H Wang ldquoComparison of GIS product qualityevaluation based on fuzzy set and defect deduction methodrdquo


Journal of Earth Science and Environment vol 28 no 2pp 96ndash99 2006

[48] G Liu Quality Management pp 258ndash260 Tsinghua Uni-versity Press Beijing China 2003

[49] J Zhao W Yi and Z Zhu ldquoExperimental research on post-punching behavior of slab-column connectionsrdquo Journal ofBuilding Structures vol 36 no 7 pp 35ndash42 2015


Due to the complexity of the mechanism of slab-col-umn connections without transverse reinforcement underpunching shearing failure it is difficult to predict thepunching capacity of slab-column connections by fittingthe test data taking all factors into consideration -is isespecially the case when it comes to influencing factorswith large nonlinearity and uncertainty as well as dis-creteness caused by brittle failure An artificial neuralnetwork is a data-processing model enlightened by a bi-ological neural network the most common of which iscalled a BPNN -e Kolmogorov theorem has proven thatthe BPNN has strong nonlinear mapping and general-ization abilities Each continuous function or mappingfunction can be realized by three levels of networks [13]-e BPNN has been widely used for solving many civilengineering problems including the shear performance ofconcrete members Kumar and Barai established a neuralnetwork that can be used to predict the extreme shearingforce of fiber concrete beams without transverse re-inforcement [14] Mansour predicted the extreme shearingstrength of an RC beam with transverse reinforcementusing an artificial neural network [15] Meanwhile Claderaand Mari used a neural network to establish a designprocedure for a normalhigh-strength concrete beam withand without stirrups [16] Using the nonlinear function ofa BPNN this study established a prediction model for thepunching capacity of slab-column connections withouttransverse reinforcement In addition the BPNN model isevaluated using the statistical eigenvalue method thedefect point method and the influencing factor analysismethod -e proposed model provides a new method forthe calculation of punching capacity

2 Establishment of the BPNN Model forPunching Capacity

-e BPNN is a forward network based on error backpropagation It is composed of trained artificial neuronsand is used to represent the optimal connection between agiven input and expected output -e establishment of theBPNN model for punching capacity primarily includes thefollowing steps -e first step is to establish trainingsamples As the research background for this study it wasnecessary to determine the input and output sample pa-rameters and establish the corresponding punching testdatabase -e second step is the design of the BPNNmodelIt primarily includes the setting of the number of networklayers number of input layer nodes number of hiddenlayer nodes number of output layer nodes activationfunction training method training parameters and otherfactors -e third step is centered on BPNN training -eLevenbergndashMarquardt (LM) algorithm is used to iterativelyupdate the weights and thresholds of the network so thatthe errors between the output value of the network and theexpected output are continuously reduced and thus theoptimal neural network is achieved -e fourth step is theevaluation of the training results -e performance of theneural network is then evaluated by cost function andregression analyses

21 Establishment of the Punching Test Database At thebeginning of the 20th century the progressive collapse ofplate structures caused by punching failures of slab-columnconnections attracted the attention of scholars [17]-erefore scholars around the world have conducted nu-merous studies in the past century on the punching failure ofslab-column connections -e research parameters haveprimarily included concrete strength longitudinal re-inforcement size effect column section shape and size shearspan to depth ratio boundary conditions punching shearreinforcement loading mode (axial or partial load) ag-gregate type and size and other factors Many researchresults have been obtained due to these studies

In this study most of the domestic and foreign litera-tures regarding punching tests of slab-column connectionswithout transverse reinforcement were collected [18ndash46]-e engineering application practice the integrity of keydata and the research purpose of this study were all con-sidered To select the quantifiable influencing parameters ofpunching capacity as comprehensively as possible it wasdetermined that concrete strength (fc) longitudinal re-inforcement ratio (ρ) yield strength of longitudinal re-inforcement (fy) effective height of the plate section (h0)shear span to depth ratio ((L1 minus c)2h0) and ratio of columnside length to plate effective height (ch0) would be used asthe control variables of punching capacity (Vu) of slab-column connections -is was done so as to keep otherinfluencing factors consistent and to screen the data Finallya set of BPNN training samples were established usingfc ρ fy h0 (L1 minus c)2h0 and ch0 as input values and Vu asthe output -e punching test database included 206 groupsof slab-column connections without transverse re-inforcement (see Table 1)

Due to differences in test designs the material perfor-mance indices loading modes and other aspects of the testdata in different literatures there is a need to define auniform description of key test data Hence the database isnow explained First the database was unified using theinternational system of units (SI) in the database Secondthe concrete strength was uniformly converted into axialcompressive strength fc and the conversion relationshipsof concrete strength (fc fcprime ft and fcu) are shown in Table 2-ird the slab-column connection in the database was allsquare cylinders and vertical loads were applied using a steelpad short column or along the support boundary -esimulation of the simply supported boundary (the forcecharacteristics of the reverse-bending line) was realizedusing the simply supported four sides

22 Design of the BPNN Model A neural network can bemultilayered Existing theories have proven that a networkwith a single hidden layer can achieve arbitrary nonlinearmapping by appropriately increasing the number of neuronnodes [10] A single hidden layer can meet the needs of theproblem background involved in this study -e number ofinput and output layer nodes depends on the dimension ofthe input and output vectors According to the backgroundof punching capacity of the BPNN model proposed in this




1

Elstner [15]


Moe [16]

S1-60 221 11 399 114 669 223 38920 S2-60 21 15 399 114 669 223 35621 S3-60 215 2 399 114 669 223 36422 S4-60 226 26 399 114 669 223 33423 S1-70 233 11 483 114 669 223 39324 S3-70 241 2 483 114 669 223 37825 S4-70 33 26 483 114 669 223 37426 S4-70A 195 26 483 114 669 223 31227 S5-60 211 11 393 114 692 178 34328 S5-70 219 11 483 114 692 178 37829 H-1 248 11 328 114 665 232 37230 M1A 198 15 481 114 647 268 44331

Manterola [17]


Hawkins [19]

S2150-1 281 154 331 127 700 200 46343 S2150-2 286 156 331 127 700 200 44044 S4150-1 332 152 331 127 700 400 57945 S4150-2 334 152 336 127 700 400 581


Table 1 Continued


46

Yoshio Kakuta [20]


Regan [21]

I1 245 24 500 77 1058 260 19489 I2 223 12 500 77 1058 260 17690 I3 261 14 500 77 1058 260 19491 I4 307 12 500 77 1058 260 19492 I5 268 15 480 79 1032 253 16593 I6 208 08 480 79 1032 253 16594 I7 289 08 480 79 1032 253 18695 V4 338 08 628 118 592 086 28596

Jianlan Zheng [22]

J-1 26 106 2982 125 380 200 450497 J-3 26 096 2982 125 380 200 42798 J-4 431 132 2891 80 594 313 302499 J-5 275 132 289 95 500 263 307100 J-6 275 106 278 95 500 263 3064


Table 1 Continued


101

Tomas-zewicz [23]



FS-1 302 056 460 100 77 15 1735115 FS-8 312 056 460 100 795 1 1503116 FS-10 31 056 460 100 745 2 1914117 FS-19 295 037 460 100 77 15 1365118

Yujie An [25]

A-1 295 124 3255 93 538 215 274119 A-2 314 132 3255 93 538 215 277120 A-3 212 132 3255 93 538 215 249121 A-4 295 091 3255 93 538 215 240122 A-5 192 058 3255 93 538 215 170123

Sistonen [26]

L1 31 046 621 172 398 117 503124 L2 31 045 621 176 389 115 537125 L3 31 045 621 173 396 116 530126 L4 31 067 612 170 402 236 686127 L5 31 066 612 172 399 232 696128 L6 31 065 612 175 390 232 799129 L7 229 064 586 177 443 114 478130 L8 229 116 576 174 394 517 1111131 L9 229 117 576 172 399 522 1107132 L10 229 116 576 173 396 521 1079133

Ghannoum [27]S1-U 346 096 445 109 814 206 301


Park [31]S1 246 106 453 90 694 278 230

140 S2 26 2 466 90 694 278 3168141 S3 246 098 485 130 481 192 4432142

Birkle [32]1 329 154 488 124 706 202 483

143 7 32 13 531 190 711 158 825144 10 289 11 524 260 663 135 1046145

Xiaokun Huang [33]

BAN105 302 05 490 168 536 119 5866146 BAN113 298 13 507 164 549 122 8053147 BAN105(1) 283 05 490 168 536 119 5223148 BAN113(1) 267 13 507 164 549 122 7968149

Yuanwei Zhang [34]

A20-1 322 157 4584 98 485 255 356150 A20-2 329 12 4584 128 371 195 470151 A20-3 343 097 4584 158 301 158 646152 A20-4 279 175 4584 88 483 398 408153 A35-1 314 157 4584 98 485 255 360154 A35-2 378 114 371 99 480 253 357155 A35-3 402 079 4135 100 475 250 293156 A50-1 443 157 4584 98 485 255 412157 A50-2 447 114 371 99 480 253 354


Table 1 Continued


158

Guandalini [35]


Guidottiv [36]

PT22 575 082 552 196 638 133 989168 PT31 567 148 552 212 590 123 1433169 PG19 434 078 510 206 607 126 860170 PG20 465 156 551 201 622 129 1094171 PG23 389 081 510 199 628 131 839172 PG24 378 161 551 194 644 134 1102173

Rizk [37]


Lips [38]

PL1 353 163 583 193 681 067 682184 PV1 335 15 709 210 595 124 974185 PL3 355 159 583 197 569 264 1324186 PL4 305 158 550 267 453 127 1625187 PL5 317 15 580 353 329 125 2491188

Jian Peng [39]


Inacio [41]


Bartolac [42]S2-1 403 15 560 96 714 135 3939

203 S2-2 378 15 560 96 714 135 3613204 S2-3 394 15 560 96 714 135 3852205 Einpaul [43] PE4 329 159 517 197 293 132 985206 PE3 322 154 517 204 809 127 961



αc1

076 (leC50)

αc2

1 (leC40)

α

08 (C20～C40)

ft 0395αc2f055cu








(k) x6(k)) (k 1 206) -e corre-





ij (i 1 6






hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)


y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)


ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)


E 1

2 times 2061113944

206

k1ek (4)


Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)


W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)











fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References
























































1

Elstner [15]


Moe [16]

S1-60 221 11 399 114 669 223 38920 S2-60 21 15 399 114 669 223 35621 S3-60 215 2 399 114 669 223 36422 S4-60 226 26 399 114 669 223 33423 S1-70 233 11 483 114 669 223 39324 S3-70 241 2 483 114 669 223 37825 S4-70 33 26 483 114 669 223 37426 S4-70A 195 26 483 114 669 223 31227 S5-60 211 11 393 114 692 178 34328 S5-70 219 11 483 114 692 178 37829 H-1 248 11 328 114 665 232 37230 M1A 198 15 481 114 647 268 44331

Manterola [17]


Hawkins [19]

S2150-1 281 154 331 127 700 200 46343 S2150-2 286 156 331 127 700 200 44044 S4150-1 332 152 331 127 700 400 57945 S4150-2 334 152 336 127 700 400 581


Table 1 Continued


46

Yoshio Kakuta [20]


Regan [21]

I1 245 24 500 77 1058 260 19489 I2 223 12 500 77 1058 260 17690 I3 261 14 500 77 1058 260 19491 I4 307 12 500 77 1058 260 19492 I5 268 15 480 79 1032 253 16593 I6 208 08 480 79 1032 253 16594 I7 289 08 480 79 1032 253 18695 V4 338 08 628 118 592 086 28596

Jianlan Zheng [22]

J-1 26 106 2982 125 380 200 450497 J-3 26 096 2982 125 380 200 42798 J-4 431 132 2891 80 594 313 302499 J-5 275 132 289 95 500 263 307100 J-6 275 106 278 95 500 263 3064


Table 1 Continued


101

Tomas-zewicz [23]



FS-1 302 056 460 100 77 15 1735115 FS-8 312 056 460 100 795 1 1503116 FS-10 31 056 460 100 745 2 1914117 FS-19 295 037 460 100 77 15 1365118

Yujie An [25]

A-1 295 124 3255 93 538 215 274119 A-2 314 132 3255 93 538 215 277120 A-3 212 132 3255 93 538 215 249121 A-4 295 091 3255 93 538 215 240122 A-5 192 058 3255 93 538 215 170123

Sistonen [26]

L1 31 046 621 172 398 117 503124 L2 31 045 621 176 389 115 537125 L3 31 045 621 173 396 116 530126 L4 31 067 612 170 402 236 686127 L5 31 066 612 172 399 232 696128 L6 31 065 612 175 390 232 799129 L7 229 064 586 177 443 114 478130 L8 229 116 576 174 394 517 1111131 L9 229 117 576 172 399 522 1107132 L10 229 116 576 173 396 521 1079133

Ghannoum [27]S1-U 346 096 445 109 814 206 301


Park [31]S1 246 106 453 90 694 278 230

140 S2 26 2 466 90 694 278 3168141 S3 246 098 485 130 481 192 4432142

Birkle [32]1 329 154 488 124 706 202 483

143 7 32 13 531 190 711 158 825144 10 289 11 524 260 663 135 1046145

Xiaokun Huang [33]

BAN105 302 05 490 168 536 119 5866146 BAN113 298 13 507 164 549 122 8053147 BAN105(1) 283 05 490 168 536 119 5223148 BAN113(1) 267 13 507 164 549 122 7968149

Yuanwei Zhang [34]

A20-1 322 157 4584 98 485 255 356150 A20-2 329 12 4584 128 371 195 470151 A20-3 343 097 4584 158 301 158 646152 A20-4 279 175 4584 88 483 398 408153 A35-1 314 157 4584 98 485 255 360154 A35-2 378 114 371 99 480 253 357155 A35-3 402 079 4135 100 475 250 293156 A50-1 443 157 4584 98 485 255 412157 A50-2 447 114 371 99 480 253 354


Table 1 Continued


158

Guandalini [35]


Guidottiv [36]

PT22 575 082 552 196 638 133 989168 PT31 567 148 552 212 590 123 1433169 PG19 434 078 510 206 607 126 860170 PG20 465 156 551 201 622 129 1094171 PG23 389 081 510 199 628 131 839172 PG24 378 161 551 194 644 134 1102173

Rizk [37]


Lips [38]

PL1 353 163 583 193 681 067 682184 PV1 335 15 709 210 595 124 974185 PL3 355 159 583 197 569 264 1324186 PL4 305 158 550 267 453 127 1625187 PL5 317 15 580 353 329 125 2491188

Jian Peng [39]


Inacio [41]


Bartolac [42]S2-1 403 15 560 96 714 135 3939

203 S2-2 378 15 560 96 714 135 3613204 S2-3 394 15 560 96 714 135 3852205 Einpaul [43] PE4 329 159 517 197 293 132 985206 PE3 322 154 517 204 809 127 961



αc1

076 (leC50)

αc2

1 (leC40)

α

08 (C20～C40)

ft 0395αc2f055cu








(k) x6(k)) (k 1 206) -e corre-





ij (i 1 6






hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)


y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)


ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)


E 1

2 times 2061113944

206

k1ek (4)


Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)


W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)











fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































Table 1 Continued


46

Yoshio Kakuta [20]


Regan [21]

I1 245 24 500 77 1058 260 19489 I2 223 12 500 77 1058 260 17690 I3 261 14 500 77 1058 260 19491 I4 307 12 500 77 1058 260 19492 I5 268 15 480 79 1032 253 16593 I6 208 08 480 79 1032 253 16594 I7 289 08 480 79 1032 253 18695 V4 338 08 628 118 592 086 28596

Jianlan Zheng [22]

J-1 26 106 2982 125 380 200 450497 J-3 26 096 2982 125 380 200 42798 J-4 431 132 2891 80 594 313 302499 J-5 275 132 289 95 500 263 307100 J-6 275 106 278 95 500 263 3064


Table 1 Continued


101

Tomas-zewicz [23]



FS-1 302 056 460 100 77 15 1735115 FS-8 312 056 460 100 795 1 1503116 FS-10 31 056 460 100 745 2 1914117 FS-19 295 037 460 100 77 15 1365118

Yujie An [25]

A-1 295 124 3255 93 538 215 274119 A-2 314 132 3255 93 538 215 277120 A-3 212 132 3255 93 538 215 249121 A-4 295 091 3255 93 538 215 240122 A-5 192 058 3255 93 538 215 170123

Sistonen [26]

L1 31 046 621 172 398 117 503124 L2 31 045 621 176 389 115 537125 L3 31 045 621 173 396 116 530126 L4 31 067 612 170 402 236 686127 L5 31 066 612 172 399 232 696128 L6 31 065 612 175 390 232 799129 L7 229 064 586 177 443 114 478130 L8 229 116 576 174 394 517 1111131 L9 229 117 576 172 399 522 1107132 L10 229 116 576 173 396 521 1079133

Ghannoum [27]S1-U 346 096 445 109 814 206 301


Park [31]S1 246 106 453 90 694 278 230

140 S2 26 2 466 90 694 278 3168141 S3 246 098 485 130 481 192 4432142

Birkle [32]1 329 154 488 124 706 202 483

143 7 32 13 531 190 711 158 825144 10 289 11 524 260 663 135 1046145

Xiaokun Huang [33]

BAN105 302 05 490 168 536 119 5866146 BAN113 298 13 507 164 549 122 8053147 BAN105(1) 283 05 490 168 536 119 5223148 BAN113(1) 267 13 507 164 549 122 7968149

Yuanwei Zhang [34]

A20-1 322 157 4584 98 485 255 356150 A20-2 329 12 4584 128 371 195 470151 A20-3 343 097 4584 158 301 158 646152 A20-4 279 175 4584 88 483 398 408153 A35-1 314 157 4584 98 485 255 360154 A35-2 378 114 371 99 480 253 357155 A35-3 402 079 4135 100 475 250 293156 A50-1 443 157 4584 98 485 255 412157 A50-2 447 114 371 99 480 253 354


Table 1 Continued


158

Guandalini [35]


Guidottiv [36]

PT22 575 082 552 196 638 133 989168 PT31 567 148 552 212 590 123 1433169 PG19 434 078 510 206 607 126 860170 PG20 465 156 551 201 622 129 1094171 PG23 389 081 510 199 628 131 839172 PG24 378 161 551 194 644 134 1102173

Rizk [37]


Lips [38]

PL1 353 163 583 193 681 067 682184 PV1 335 15 709 210 595 124 974185 PL3 355 159 583 197 569 264 1324186 PL4 305 158 550 267 453 127 1625187 PL5 317 15 580 353 329 125 2491188

Jian Peng [39]


Inacio [41]


Bartolac [42]S2-1 403 15 560 96 714 135 3939

203 S2-2 378 15 560 96 714 135 3613204 S2-3 394 15 560 96 714 135 3852205 Einpaul [43] PE4 329 159 517 197 293 132 985206 PE3 322 154 517 204 809 127 961



αc1

076 (leC50)

αc2

1 (leC40)

α

08 (C20～C40)

ft 0395αc2f055cu








(k) x6(k)) (k 1 206) -e corre-





ij (i 1 6






hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)


y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)


ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)


E 1

2 times 2061113944

206

k1ek (4)


Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)


W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)











fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































Table 1 Continued


101

Tomas-zewicz [23]



FS-1 302 056 460 100 77 15 1735115 FS-8 312 056 460 100 795 1 1503116 FS-10 31 056 460 100 745 2 1914117 FS-19 295 037 460 100 77 15 1365118

Yujie An [25]

A-1 295 124 3255 93 538 215 274119 A-2 314 132 3255 93 538 215 277120 A-3 212 132 3255 93 538 215 249121 A-4 295 091 3255 93 538 215 240122 A-5 192 058 3255 93 538 215 170123

Sistonen [26]

L1 31 046 621 172 398 117 503124 L2 31 045 621 176 389 115 537125 L3 31 045 621 173 396 116 530126 L4 31 067 612 170 402 236 686127 L5 31 066 612 172 399 232 696128 L6 31 065 612 175 390 232 799129 L7 229 064 586 177 443 114 478130 L8 229 116 576 174 394 517 1111131 L9 229 117 576 172 399 522 1107132 L10 229 116 576 173 396 521 1079133

Ghannoum [27]S1-U 346 096 445 109 814 206 301


Park [31]S1 246 106 453 90 694 278 230

140 S2 26 2 466 90 694 278 3168141 S3 246 098 485 130 481 192 4432142

Birkle [32]1 329 154 488 124 706 202 483

143 7 32 13 531 190 711 158 825144 10 289 11 524 260 663 135 1046145

Xiaokun Huang [33]

BAN105 302 05 490 168 536 119 5866146 BAN113 298 13 507 164 549 122 8053147 BAN105(1) 283 05 490 168 536 119 5223148 BAN113(1) 267 13 507 164 549 122 7968149

Yuanwei Zhang [34]

A20-1 322 157 4584 98 485 255 356150 A20-2 329 12 4584 128 371 195 470151 A20-3 343 097 4584 158 301 158 646152 A20-4 279 175 4584 88 483 398 408153 A35-1 314 157 4584 98 485 255 360154 A35-2 378 114 371 99 480 253 357155 A35-3 402 079 4135 100 475 250 293156 A50-1 443 157 4584 98 485 255 412157 A50-2 447 114 371 99 480 253 354


Table 1 Continued


158

Guandalini [35]


Guidottiv [36]

PT22 575 082 552 196 638 133 989168 PT31 567 148 552 212 590 123 1433169 PG19 434 078 510 206 607 126 860170 PG20 465 156 551 201 622 129 1094171 PG23 389 081 510 199 628 131 839172 PG24 378 161 551 194 644 134 1102173

Rizk [37]


Lips [38]

PL1 353 163 583 193 681 067 682184 PV1 335 15 709 210 595 124 974185 PL3 355 159 583 197 569 264 1324186 PL4 305 158 550 267 453 127 1625187 PL5 317 15 580 353 329 125 2491188

Jian Peng [39]


Inacio [41]


Bartolac [42]S2-1 403 15 560 96 714 135 3939

203 S2-2 378 15 560 96 714 135 3613204 S2-3 394 15 560 96 714 135 3852205 Einpaul [43] PE4 329 159 517 197 293 132 985206 PE3 322 154 517 204 809 127 961



αc1

076 (leC50)

αc2

1 (leC40)

α

08 (C20～C40)

ft 0395αc2f055cu








(k) x6(k)) (k 1 206) -e corre-





ij (i 1 6






hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)


y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)


ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)


E 1

2 times 2061113944

206

k1ek (4)


Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)


W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)











fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































Table 1 Continued


158

Guandalini [35]


Guidottiv [36]

PT22 575 082 552 196 638 133 989168 PT31 567 148 552 212 590 123 1433169 PG19 434 078 510 206 607 126 860170 PG20 465 156 551 201 622 129 1094171 PG23 389 081 510 199 628 131 839172 PG24 378 161 551 194 644 134 1102173

Rizk [37]


Lips [38]

PL1 353 163 583 193 681 067 682184 PV1 335 15 709 210 595 124 974185 PL3 355 159 583 197 569 264 1324186 PL4 305 158 550 267 453 127 1625187 PL5 317 15 580 353 329 125 2491188

Jian Peng [39]


Inacio [41]


Bartolac [42]S2-1 403 15 560 96 714 135 3939

203 S2-2 378 15 560 96 714 135 3613204 S2-3 394 15 560 96 714 135 3852205 Einpaul [43] PE4 329 159 517 197 293 132 985206 PE3 322 154 517 204 809 127 961



αc1

076 (leC50)

αc2

1 (leC40)

α

08 (C20～C40)

ft 0395αc2f055cu








(k) x6(k)) (k 1 206) -e corre-





ij (i 1 6






hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)


y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)


ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)


E 1

2 times 2061113944

206

k1ek (4)


Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)


W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)











fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References

























































(k) x6(k)) (k 1 206) -e corre-





ij (i 1 6






hj(k)

f 11139446

i1xiωij

(h)+ aj

⎛⎝ ⎞⎠ (k 1 206) (1)


y(k)

g 111394415

j1hjωj

(o)+ b⎛⎝ ⎞⎠ (k 1 206) (2)


ek 12

y(k)

minus d(k)

1113872 11138732 (k 1 206) (3)


E 1

2 times 2061113944

206

k1ek (4)


Jkj(W) zek

zWj

(k 1 206 j 1 121) (5)


W(n + 1) W(n) + ΔW

ΔW minus JTJ + μI1113960 1113961

minus 1J

Te

(6)











fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References

























































fc

ρ

fy

h0

(L1 ndash c)2h0

ch0

1

2

3

13

14

15

Punching capacity

Input layer

Hidden layer

Output layer

1

2

3

4

5

6

1

X2

θyi

Threshold

Current neuronX1

Xi

Xn

ω1ω2

ωi

ωn


neuron


sum


EndMake







If E(n) lt ε


If E(n + 1) lt E(n)


105

Gra

dien

t


100

mu

Mu = 1 at epoch 26


5 10 15 20 25026 epochs

0

2

4

6

Val

idat

ion

failu

re

5 10 15 20 25026 epochs

5 10 15 20 25026 epochs



TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































TrainValidation

TestBest

102

103

104

105

106

107

Mea

n sq

uare

d er

ror (

MSE

)

5 10 15 20 25026 epochs



DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

54

(a)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

94 lowast

targ

et +

27

(b)

Figure 5 Continued


X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































X Vtest

Vcal (7)







DataFitY = T

500

1000

1500

2000

Out

put ~

= 1 lowast

targ

et +

48

1000 1500 2000500Target

(c)

DataFitY = T

1000 1500 2000500Target

500

1000

1500

2000

Out

put ~

= 0

99 lowast

targ

et +

53

(d)
















0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References



































































0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

100 200 300 4000h0 (mm)

(a)

V tes

tVG

B

Mean

95 upper limit

95 lower limit0619

1776

100 200 300 4000h0 (mm)

00

05

10

15

20

(b)

0679

1945

V tes

tVA

CI

Mean

95 lower limit

95 upper limit

00

05

10

15

20

25

100 200 300 4000h0 (mm)

(c)

0763

1526V t

estV

EC

Mean

95 upper limit

95 lower limit

100 200 300 4000h0 (mm)

00

05

10

15

20

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

2 4 60ch0

(a)

062

1776

V tes

tVG

B

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(b)

Figure 7 Continued


0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

00

05

10

15

20

25

2 4 60ch0

(c)

0763

1526

V tes

tVEC

Mean

95 lower limit

95 upper limit

00

05

10

15

20

2 4 60ch0

(d)


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(a)

062

1776

V tes

tVG

B

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(b)

0679

1945

V tes

tVAC

I

Mean

95 upper limit

95 lower limit

20 40 60 80 100 1200fc (MPa)

00

05

10

15

20

25

(c)

V tes

tVEC

Mean

95 upper limit

95 lower limit

00

05

10

15

20

20 40 60 80 100 1200fc (MPa)

(d)



0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References






















































0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

05 10 15 2000ρ ()


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

5 10 150(L1 ndash c)2h0

00

05

10

15

20


0817

1299

V tes

tVBP

Mean

95 upper limit

95 lower limit

00

05

10

15

20

200 400 600 8000fy (MPa)


ACI

05

10

15

20

25

V tes

tVA

CI

10 15 2005ρ ()


ACI

0763

05

10

15

20

25

V tes

tVA

CI

400 600 800200fy (MPa)


ACI

1

05

10

15

20

25

V tes

tVAC

I

5 10 150(L1 ndash c)2h0









5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References




























































5 Conclusion





S5-1 237 151004 459 150 5 166667 517S5-2 249 153971 459 150 5 166667 612S7-1 231 15762 459 150 7 166667 535S7-2 214 147108 459 150 7 166667 556C7-30-3 27 153373 4536 150 7 166667 690C7-50-3 361 16086 4536 150 7 166667 805C7-70-1 486 138443 4536 150 7 166667 610C7-70-2 501 154283 4536 150 7 166667 785C7-70-3 509 152514 4536 150 7 166667 860









Appendix









Data Availability


Disclosure




Acknowledgments


References

























































Appendix









Data Availability


Disclosure




Acknowledgments


References


























































Data Availability


Disclosure




Acknowledgments


References




































































































Documents

Prediction of Punching Capacity of Slab-Column Connections ...downloads.hindawi.com/journals/ace/2019/7904685.pdf · the theory referred to in these formulas for punching ca-pacity