shearstrengthofinternalreinforcedconcretebeam-column...

Research ArticleShear Strength of Internal Reinforced Concrete Beam-ColumnJoints Intelligent Modeling Approach and Sensitivity Analysis

De-Cheng Feng 1 and Bo Fu 23

1Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education Southeast UniversityNanjing 211189 China2School of Civil Engineering Changrsquoan University Xirsquoan 710061 China3State Key Laboratory of Green Building inWestern China Xirsquoan University of Architecture and Technology Xirsquoan 710055 China

Correspondence should be addressed to Bo Fu 90_bofuchdeducn

Received 1 June 2020 Revised 6 July 2020 Accepted 8 August 2020 Published 25 August 2020

Academic Editor Moacir Kripka

Copyright copy 2020 De-Cheng Feng and Bo Fu-is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper an intelligent modeling approach is presented to predict the shear strength of the internal reinforced concrete (RC)beam-column joints and used to analyze the sensitivity of the influence factors on the shear strength -e proposed approach isestablished based on the famous boosting-family ensemble machine learning (ML) algorithms ie gradient boosting regressiontree (GBRT) which generates a strong predictive model by integrating several weak predictors which are obtained by the well-known individual ML algorithms eg DT ANN and SVM -e strong model is boosted as each weak predictor has its ownweight in the final combination according to the performance Compared with the conventional mechanical-driven shear strengthmodels eg the well-known modified compression field theory (MCFT) the proposed model can avoid the complicatedderivation process of shear mechanism and calibration of the involved empirical parameters thus it provides a more convenientfast and robust alternative way for predicting the shear strength of the internal RC joints To train and test the GBRTmodel a totalof 86 internal RC joint specimens are collected from the literatures and four traditional ML models and the MCFTmodel are alsoemployed as comparisons -e results indicate that the GBRTmodel is superior to both the traditional ML models and MCFTmodel as its degree-of-fitting is the highest and the predicting dispersion is the lowest Finally the model is used to investigate theinfluences of different parameters on the shear strength of the internal RC joint and the sensitivity and importance of thecorresponding parameters are obtained

1 Introduction

Reinforced concrete (RC) beam-column joint or connectionis one of the most critical and vulnerable components in RCstructures -e failure of the RC beam-column joints couldseriously affect the overall safety of the structures Especiallyit will suffer from the shear failure if there are insufficienttransverse reinforcements andor the material properties aredeteriorated due to the aging effects As it is known to allshear failure is a brittle failure type without any warnings-erefore it is vital to accurately predict the shear strengthof the RC beam-column joints to avoid shear failure indesign procedures in order to ensure the safety of thestructures

In general there are three commonly used approaches toassess shear strength of the RC joints ie experimentalstudy numerical simulation and theoretical analysis -eexperimental study is the most direct and classical waywhich can be traced back to 1970s [1] However it is costly inboth time and money and difficult to operate -e numericalsimulation eg finite element method (FEM) is also widelyadopted for its low cost [2 3] Nevertheless it usually hasseveral simplifications and some of the mechanisms are hardto be reflected in the FEM framework eg multistress statebehavior shear behavior and interfacial bond-slip behaviorApart from the experimental and numerical studies nu-merous theoretical models were also proposed to evaluatethe performance of the RC beam-column joints for instance

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8850417 19 pageshttpsdoiorg10115520208850417

the well-known modified compression field theory (MCFT)[4] the strut-and-tie method (STM) [5] etc -ese modelsare actually derived based on the shear mechanisms offundamental RC elements and can be widely used to evaluatethe behavior of any type of shear-dominated RC membersincluding the beam-column joints [6] A detailed review ofthe theoretical and empirical models for the RC joints can befound in [7]

In recent five years there are some latest development onRC joint models Eom et al [8] developed an energy-basedhysteresis model for RC beam-column joints by using theenergy function and the existing backbone curve of ASCESEI 41-06 [9] Hwang et al [10] proposed a shear strengthdegradation model for performance-based design of interiorbeam-column joints In the model all possible failuremechanisms of beams and joints including flexural yieldingof the beam end diagonal cracking and concrete crushing inthe joint panel bar bond-slip and bar elongation areconsidered Later Hwang and Park [11] developed designequations of the joint shear strength and hoop requirementfor the performance-based design of interior RC beam-column joints by considering the diagonal strut mechanismand truss mechanism -e target drift ratio and bar bondparameters are defined as the requirements of the joint shearstrength and hoop strength More recently Hwang and Park[12] modified the shear strength degradation model forinterior RC joints and applied it to exterior RC joints withstandard hooked bars Hwang et al [13] simplified thesoftened strut-and-tie model to facilitate design practice forthe strength prediction of discontinuity regions such as theRC beam-column joints -e shear-resisting mechanisms assuggested by the softened strut-and-tie model are consideredin the simplified model Similarly Huang and Kuang [14]proposed a shear strength model for exterior RC wide beam-column joints by introducing the softened strut-and-tieconcept Hassan and Moehle [15] collected a database ofexterior and corner beam-column joints without transversereinforcement Based on the database they evaluated severalexisting shear strength models and developed a strut-and-tiemodel based on the ACI 318 [16] strut-and-tie modelingprovisions and an empirical model by considering the effectsof joint aspect ratio column axial load and concretecompressive strength

Although the above empirical or theoretical approachesoffer simple and clear explanation of the shear mechanismthey also introduce empirical assumptions which will reducetheir accuracy Moreover the derivations seem to becomplicated since the iteration process is likely involved andsome of the parameters are empirical that needed to bedetermined according to the usersrsquo experience

In recent years with the flourishment of artificial in-telligence (AI) a brand new way is come to peoplersquos hori-zons ie using machine learning (ML) techniques to predictthe shear strength of the RC beam-column joints ML is atype of AI which has various functions eg classificationregression and clustering ML can learn the characteristicsof a certain type of data according to the existing databaseand then classify summarize and predict the things ofinterest Prediction of the shear strength of the RC joints is

essentially a regression problem -ere are already somesuccessful applications of prediction using ML in structuralengineering for instances evaluating the cement strengthvia fuzzy logic artificial neutral network (ANN) and geneexpression programming (GEP) [17 18] modeling theconcrete properties via ANN and support vector machine(SVM) [19ndash23] simulating the failure of brittle anisotropicmaterials such as masonry via ANN [24 25] predicting thestructural member capacities via hybrid ML algorithms[26 27] detecting the structural damage via GEP [28 29]etc A detailed state-of-art of the application of ML instructural engineering was summarized in [30]

However the majority of the ML algorithms used in theprevious studies were individual-type learning algorithmssuch as ANN family [31] SVM family [32] and decision tree(DT) family [33] -e disadvantages of the individual-typelearning algorithms are instable and with low accuracy Toimprove their performance a new type of learning algo-rithms known as ensemble learning algorithms has beenrecently proposed and successfully applied in various fields-e basic idea of the ensemble learning is to combine severalweak learners generated by individual learning algorithmsinto a strong one In brief the ensemble learning algorithmsare more stable and accurate compared to the individuallearning algorithms [34] -ere are mainly two categories ofensemble learning algorithms bagging and boosting For thebagging family the weak learners are produced in parallelwhile they are produced in sequence for the boosting family-eoretically bagging is more efficient and can effectivelyreduce the variance of the prediction and boosting is rel-atively less efficient in reducing the bias In practice boostingis superior to bagging in terms of accuracy for general cases-erefore one of the most typical boosting ensemblelearning algorithms referred to as gradient boosting re-gression tree (GBRT) [35] algorithm is used in this study

In this paper we aim to develop a GBRT-based intel-ligent method for predicting the shear strength of the RCbeam-column joints and make comparisons between theproposed data-driven model and some traditional ML-basedmodels as well as the conventional mechanical-drivenMCFTmodel Firstly some individual-type ML techniquesincluding linear regression (LR) SVM ANN and DT arebriefly reviewed -en the mathematical background andimplementation of GBRT are introduced Afterwards theshear strength data of 86 internal RC beam-column jointsare collected from the literature Based on the database theprediction results from the GBRT-based model are verifiedby a 10-fold validation test and compared with those fromthe individual-type ML models In addition one of therepresentative conventional mechanical-driven approachesie MCFT is briefly summarized and also used as com-parison with the GBRTmodel Finally sensitivity analysis ofinput variables is conducted for the GBRTmodel to quantifythe influences of different parameters

2 Review of the Traditional ML Techniques

21 Linear Regression (LR) Linear regression (LR) is one ofthe most widely used statistical analysis techniques in

2 Advances in Civil Engineering

determining the qualitative relationship between two ormore variables In general the least square method isadopted to solve the LR problem If only one independentvariable and one dependent variable are considered and therelationship between them is approximately linear then thistype of regression analysis is called simple linear regression(SLR) On the contrary if two ormore independent variablesare included and the relationship between the independentand dependent variables are approximately linear then thisregression analysis is called multiple linear regression(MLR) For the prediction problem considered in this studymore than two input parameters should be assigned as theindependent variables so it belongs to MLR

22 Support Vector Machine (SVM) Based on the statisticallearning theory proposed by Vapnik [36] the support vectormachine (SVM) is an effective optimizing tool to improvethe generalization performance and obtain the globallyoptimal and unique solution In implementing the SVMregression the primary goal is to minimize an upper boundof the generalization error based on the structural riskminimization -e essence of the SVM regression is to mapthe input variables into a high-dimensional feature space bya nonlinear mapping and then conduct linear regression inthe space

23 Artificial Neural Networks (ANN) -e artificial neuralnetwork (ANN) is a complex information processing systemcomposed of a huge number of interconnected processingelements (neurons) arranged in layers It is the abstractionsimplification and simulation of the structure and mech-anism of biological nervous systems such as human brainsJust as the learning process in biological systems the ANNinvolves adjustments to the synaptic connections betweenthe neurons When it is applied to solve engineeringproblems a neural network can be a vector mapper whichmaps input vector(s) to an output one(s)

24 Decision Trees (DT) Decision tree (DT) is one of thebasic classification and regression methods -e DT re-gression approach mainly refers to one of the binary treestructures ie classification and regression tree (CART)algorithm in which the characteristic values of internalnodes are only yes or no -e main task for CART is todivide the characteristic space into several units Everyunit has a certain output As each node is judged by yes orno the divided boundary is parallel to the coordinateaxis Any testing data can be fallen into a unit accordingto its characteristic and thus obtain its correspondingoutput

3 Boosted ML Approach Gradient BoostingRegression Tree (GBRT)

-ough the abovementioned traditional ML methods havealready been applied in several aspects of structural engi-neering including predicting the behavior of structural

members there still exist some drawbacks For some cases aldquobestrdquo model may not be easily obtained using those algo-rithms Meanwhile models by different algorithms will havetheir own hypotheses which may lead to great model un-certainty -erefore this paper employs the ensemblelearning algorithms to generate the predictive model for thejoint shear strength Specifically the boosting family gra-dient boosting regression tree (GBRT) is adopted -e en-semble learning method is superior to the individuallearning method since it offers a powerful framework toobtain a strong estimator (or learner) by integrating severalweak estimators (or learners) produced by the individuallearning method so the accuracy and robustness are bothenhanced-e boosting idea is reflected in the weights of theweak learners the one with higher score will get higherweight in the final strong learner -e fundamental andtheoretical backgrounds as well as the implementationprocedure are all presented herein

31 Gradient Boosting Framework of Ensemble ML Asmentioned before ensemble learning is not an individual-type ML method It is accomplished by integrating multipleweak learners into a strong one Boosting is a major group ofensemble learning algorithms which generates the weaklearners subsequently and can be interpreted as an opti-mization algorithm on a suitable cost function -e basicidea of boosting is to update the weight of each weak learnerby its learning error If a weak learner has a large learningerror it will be assigned a large weight so that it could be paidmore attention in the subsequent training process Likeother boosting methods the gradient boosting integratesseveral weak learners into a single strong learner in an it-erative way

Supposing it requires M steps to find out the final stronglearner and at the m-th(m isin [1 M]) step we have an im-perfect modelfm(x)which is the sum of weak learners in theprevious steps

fm(x) 1113944mminus1

i1αihi(x) (1)

where x is the vector containing the input variables hi(x)

and αi are the weak learner and the corresponding weight atstep i isin [1 m]

-e imperfect model can be improved by adding a newweak learner hm(x) as fm+1(x) fm(x) + hm(x) -en theoptimization problem becomes how to find hm(x) -esolution of gradient boosting starts with the observation thata perfect hm(x) would imply

fm+1(x) fm(x) + hm(x) y (2)

where y is the target output or the tested value of the outputEquation (2) can be equivalently expressed as

hm(x) y minus fm(x) (3)

-erefore in the following gradient boosting algorithmfits hm(x) with the residual y minus fm(x) Like other membersof the boosting algorithms fm+1(x) is attempted to correct

Advances in Civil Engineering 3

the errors of its predecessor fm(x) It is observed that theresidual y minus fm(x) is the negative gradient of the squaredloss function 12[y minus fm(x)]2 so the negative gradient canbe extended to other kinds of loss functions In other wordsthe gradient boosting algorithm is a gradient descent al-gorithm which can be generalized by varying the lossfunction and the gradient

32 Gradient Boosting Regression Tree (GBRT) As can beseen in the previous section gradient boosting is actually aframework for ensembling numerous weak learners ratherthan a specific learning algorithm -eoretically any indi-vidual algorithms from the ANN SVM and DTfamilies canbe used to train the weak learners However unlike otherboosting algorithms the individual algorithm for trainingthe weak learners in gradient boosting is restricted to the DTalgorithms thus it is called as GBRT In each step (or it-eration) a new DT is established by fitting the negativegradient of the loss function -e number of DT is deter-mined by the iteration number

-e GBRT model superimposes multiple DTs and isexpressed as

fM(x) 1113944M

m1T xΘm( 1113857 (4)

where T(xΘm) represents the weak learner by DT Θm

denotes the parameters of m-th DTmodel M is the numberof DTs respectively

For a dataset D (x1 y1) (x2 y2) (xN yN)1113864 1113865

where N denotes the number of the samples the essence oftraining the boosting DT model is selecting the optimalparameters of DTs Θ Θ1Θ2 ΘM1113864 1113865 to minimize theloss function 1113936

Ni1 L[yi fM(xi)] ie

arg minΘ

1113944

N

i1L yi fM xi( 11138571113858 1113859 arg min

Θ1113944

N

i1L yi 1113944

M

m1T xΘm( 1113857⎡⎣ ⎤⎦

(5)

Here the loss function is used to reflect the differencebetween the sample real value yi and the output of the GBRTfM(xi)

Note that the GBRT model in equation (4) can also bewritten in a forward step way and expressed as

fm(x) fmminus1(x) + T xΘm( 1113857 m 1 2 M (6)

-erefore training of the GBRTmodel can be achievedby M iteration steps Specifically at the initial step we definef1(x) 0 and for the m-th iteration step a new T(xΘm) isgenerated -e parameters Θm of T(xΘm) should be ob-tained to minimize the loss function

1113954Θm arg minΘm

1113944

N

i1L yi fmminus1 xi( 1113857 + T xiΘm( 11138571113858 1113859 (7)

where 1113954Θm are the optimal DT parametersIf the squared loss function is used then one obtains

L yi fmminus1 xi( 1113857 + T xiΘm( 11138571113858 1113859 yi minus fmminus 1 xi( 1113857 minus T xiΘm( 11138571113858 11138592

rmi minus T xiΘm( 11138571113960 11139612

(8)

where rmi yi minus fmminus1(xi) represents the residual of themodel fmminus1(x)

-erefore the solution of equation (8) converts to theselection of appropriateΘm to minimize the difference of theresidual rmi of the DT and the output T(xΘm) orequivalently (xi rmi)1113966 1113967

i12Ncan be used as the sample set

of the decision tree T(xΘm) and the optimal parameters1113954Θm are obtained according to the conventional DT gener-ation process

Moreover in a more generalized sense the negativegradient of the loss function can be used to represent theresidual of the model ie

rmi minuszL yi f xi( 1113857( 1113857

zf xi( 11138571113890 1113891

f(x)fmminus1(x)

(9)

With (xi rmi)1113966 1113967i12N

we can fit the m-th DT hmwhose leaf nodes can be represented by Rmj j 1 2 Jwhere J indicates the number of leaf nodes of the DT Foreach leaf node of the regression tree hm calculate the optimalfitting value cmj

cmj argminc

1113944xiisinRmj

L yi fmminus1 xi( 1113857 + c1113858 1113859 j 1 2 J

(10)

-en the weak learner for this step can be written as

Tm(x) 1113944

J

j1cmjI x isin Rmj1113872 1113873 (11)

and the updated strong learner till this step is

fm(x) fmminus1(x) + Tm(x) fmminus1(x) + 1113944

J

j1cmjI x isin Rmj1113872 1113873

(12)

After M steps the strong learner is finally obtained by

fM(x) 1113944M

m11113944

J

j1cmjI x isin Rmj1113872 1113873 (13)

-e procedure of the GBRT algorithm can be summa-rized as follows

(1) Initialization of the function for the weak learnerf0(x) arg min

c1113936

Ni1 L(yi c)

(2) For the m-th iteration (m 1 2 M)

(a) For each sample (xi yi) i 1 2 N thenegative gradient is calculated using equation (9)

(b) Train the m-th DT hm by using(xi rmi)1113966 1113967

i12N and the corresponding areas


of the i minus th leaf nodes are denoted as Rmj j

1 2 J

(c) For each leaf node of the regression tree hmcalculate the optimal fitting value cmj usingequation (10)

(d) Update the learner fm(x) fmminus1(x) + 1113936Jj1

cmjI(x isin Rmj)

(3) After M iterations the strong learner is obtainedusing equation (13)

33 Implementation of GBRT In this study one of the mostwidely used DT ie CART is employed as the individuallearning algorithm-e implementation of the GBRTcan besummarized as the following four steps

(1) Collect and process the data such as the setting ofinputoutput variables and the grouping of thetrainingtesting datasets

(2) Train the regression model using the GBRTwith thetraining dataset

(3) Validate the trained model with the testing dataset(4) Apply the model to the realistic problems

-e flowchart of the abovementioned procedure isdepicted in Figure 1

Another important issue associated with the imple-mentation of GBRT is the determination of model param-eters which have two levels ie the framework level and thelevel for the individual learning algorithm At the frameworklevel there are two main parameters ie the number ofiteration (number of weak learners) and the learning ratewhich is used to avoid the overfitting problem At the singlelearning algorithm level there are four primary parametersie the maximum depth of the tree the minimum samplesfor split the minimum samples of leaf node and theminimum change in impurity -e selected values of theseparameters are determined based on previous studies inliterature and practical modeling experience as shown inTable 1

4 Collection of Experimental Data for ShearStrength of Internal RC Beam-Column Joints

In implementing the ML techniques for prediction of theshear strength for RC joints an experimental database isrequired to train the predictive model and validate themodel -erefore a database including the experimentalresults of 86 internal RC beam-column joints was col-lected for this purpose in this study In the database thereare 10 input parameters covering material properties andgeometric dimensions and reinforcing details of the testspecimens ie the concrete strength fc the section widthof column bc the section height of column hc the sectionwidth of beam bb the section height of beam hb theyielding strength of beam longitudinal bar fyb theyielding strength of column longitudinal bar fyc theyielding strength of joint transverse bar fyv the transversebar ratio ρv and the axial load ratio n -e only output is

the joint shear strength τ -e statistical information ofthese parameters eg mean and standard deviation(StD) and the distributions of the aforementioned pa-rameters are illustrated in Table 2 and Figure 2 -e detailsof the tested specimens in the database are given inTable 3

5 Results and Discussion

51 10-Fold Cross-Validation Results To validate the pro-posed method the 10-fold cross-validation method isfirstly used to evaluate the modelrsquos performance -e 10-fold cross-validation method is developed to minimize thebias associated with random sampling of the training andtesting datasets It divides the experimental data samplesinto 10 subsets and for each run 9 are set as trainingsubsets and 1 is set as validating subset It is believed thatrepeating this for 10 times is able to represent the gen-eralization and reliability of the predictive modelMoreover three commonly used indicators are intro-duced to assess the prediction performance which arerespectively defined as

Coefficient of determination R-squared (R2)

R2

1 minus1113944

N

i1 Pi minus Ti( 11138572

1113944N

i1 Ti minus T( 11138572 (14)

Root mean squared error (RMSE)

RMSE

1113944N


N

1113971

(15)

Mean absolute error (MAE)

MAE 1113944

N

i1 Pi minus Ti

11138681113868111386811138681113868111386811138681113868

N (16)

where Pi and Ti are the predicted and tested values re-spectively T is the mean value of all the tested values N isthe total number of the samples in the dataset

Among the three indicators R2 indicates the degree ofthe linear correlation between the predicted and testedvalues RMSE shows the deviation between the predictedand tested values MAE reflects the ratio of the predictionerror to the tested values -e closer the R2 to 1 the smallerthe RMSE or MAE the better performance the predictionmodel possesses Table 4 shows the 10-fold cross-validationstatistic results of the GBRT model

It can be drawn from Table 4 that the average de-termination coefficient R2 for the 10-fold results is 0875which is close to 1 the average RMSE and MAE are0948MPa and 0722MPa respectively which are small-e standard derivations (StD) for R2 RMSE and MAE


are 0082 0347 and 0245 respectively which means theprediction performance has low variance All of theseindices demonstrate that the proposed method has ex-cellent performance in predicting the shear strength ofinternal RC joints

52 Prediction Results of Different ML Models To demon-strate the prediction performance of the GBRT model thefour general ML models ie LR SVM ANN and DT arealso used to predict the shear strength of the 86 specimens-e optimized parameters of the fourmodels are determined

Collect data

Database

bullbullbull

Testing dataset

Training dataset

Weak learner

Learning

Weight

Weak learner

Learning

Weight

Weak learner

Learning

WeightUpdate Update

Validate

Strong learner

sum

Figure 1 Flowchart of implementation of GBRT

Table 1 Model parameters of GBRT

Level Parameter Setting

Framework Maximum iteration number 100Learning rate 003

Single learning algorithm

Maximum depth of the tree 10Maximum leaf nodes 200

Minimum samples for split 6Minimum samples of leaf node 5

Table 2 Statistics of parameters for the internal RC beam-column joints

Parameter Unit Maximum Minimum Mean StD Typef c MPa 11690 1657 4331 1907 Inputb c mm 45800 20000 26812 6603 Inputh c mm 45800 20000 27723 8172 Inputb b mm 40600 15000 19474 6460 Inputh b mm 50000 25000 32259 5696 Inputf yb MPa 145600 31400 45748 17987 Inputf yc MPa 145600 31400 45809 15046 Inputfyv MPa 137400 27600 45173 26208 Inputρv 200 018 066 038 Inputn mdash 118 000 025 023 Inputτ MPa 1487 285 730 281 Output


03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)

Figure 2 Statistic distribution of the input and output parameters (a) Concrete strength (MPa) (b) Section width of column (mm)(c) Section height of column (mm) (d) Section width of beam (mm) (e) Section height of beam (mm) (f ) Yielding strength of beamlongitudinal bar (MPa) (g) Yielding strength of column longitudinal bar (MPa) (h) Yielding strength of joint transverse bar (MPa)(i) Transverse bar ratio () (j) Axial load ratio (k) Joint shear strength (MPa)


Table 3 Main data of the internal RC beam-column joint tests

References Specimen fc (MPa) bc times hc (mmtimesmm) bb times hb (mmtimesmm) fyb (MPa) fyc (MPa) fyv (MPa) ρv n τ (MPa)

[37]

I 2470 250times 250 150times 350 3911 3538 2775 036 0226 415I-1 2720 250times 250 150times 350 3794 3660 3011 036 0204 416I-2 3195 250times 250 150times 350 3794 3660 3011 036 0173 510SJ1-1 3178 250times 250 150times 350 3663 3490 3009 042 0173 553SJ1-2 3448 250times 250 150times 350 3663 3490 3009 042 1180 582SJ1-4 2980 250times 250 150times 350 3803 3803 3009 042 0440 732SJ2-2 2840 250times 250 150times 350 3663 3490 3009 150 0174 617SJ5-1 3788 250times 250 150times 350 3663 3490 3009 042 0178 647SJ-3 3448 250times 250 150times 350 4620 4046 3400 126 0125 641SJ-B 2700 450times 450 250times 500 3260 3880 3840 076 0167 549

[38]

J3-50 4612 200times 200 150times 300 3900 3850 3070 083 0285 714J4-50 4740 200times 200 150times 300 3900 3850 3070 083 0361 783J4-30 4674 200times 200 150times 300 3900 3850 3070 133 0374 792J5-80 4671 200times 200 150times 300 3280 4050 3070 066 0467 880J5-50 4816 200times 200 150times 300 3280 4050 3070 083 0449 898J6-80 4694 200times 200 150times 300 3280 4050 3070 066 0558 926J6-50 4360 200times 200 150times 300 3280 4050 3070 083 0514 829J6-30 4812 200times 200 150times 300 3280 4050 3070 133 0540 907J7-80 4360 200times 200 150times 300 3280 4050 3070 066 0716 791J7-50 4842 200times 200 150times 300 3280 4050 3070 083 0636 898J7-30 4381 200times 200 150times 300 3280 4050 3070 133 0712 884J8-50 4945 200times 200 150times 300 3280 4050 3070 083 0744 922J8-30 4381 200times 200 150times 300 3280 4050 3070 133 0814 838

[39]ZHJ4 2736 200times 200 150times 250 3602 3602 3500 100 0330 691ZHJ5 2736 200times 200 150times 250 3602 3602 3500 121 0330 693ZHJ6 2736 200times 200 150times 250 3602 3602 3500 181 0330 707

[40]

J-1 2736 200times 200 150times 250 3800 3800 3500 104 0330 678J-2 2736 200times 200 150times 250 3800 3800 3500 112 0330 684J-3 1657 200times 200 150times 250 4730 4730 3750 130 0270 649J-4 1953 200times 200 150times 250 4730 4730 3750 200 023 655JL1 2143 250times 250 150times 300 3140 3140 2760 058 021 532

[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688

[42]J4 257 300times 300 200times 300 401 401 368 028 030 422J5 287 300times 300 200times 300 401 401 368 028 007 504C2 256 300times 300 200times 300 324 422 368 090 008 536

[43]

U1 262 331times 458 280times 458 449 457 409 050 040 538U2 418 331times 458 280times 458 449 449 409 050 025 787U3 266 331times 458 280times 458 449 402 409 050 039 603U4 361 458times 331 406times 458 449 449 409 050 029 717U5 359 331times 458 280times 458 449 449 409 050 004 757U6 368 331times 458 280times 458 449 449 409 050 048 815U7 372 458times 331 406times 458 449 449 409 050 046 724

[44]

J1 700 300times 300 200times 300 718 718 955 075 012 1084J3 1070 300times 300 200times 300 718 718 955 075 012 1375J4 700 300times 300 200times 300 718 718 955 075 012 1144J5 700 300times 300 200times 300 718 718 955 075 012 1126J6 535 300times 300 200times 300 718 718 955 075 012 998

[45]



by using the grid search after setting the initial values -etotal dataset is divided into for training and testing as 8-2ie 80 of the data is used for training and 20 of the data isused for testing Figure 3 shows the prediction results of theGBRTmodel and the four general ML models for the testing

dataset It is clear that the GBRTmodel has stronger linearcorrelation compared with other four ML models -ereason is that the GBRT is an ensemble learning algorithmwith strong learner while other four models use individual-type learning algorithms with weak learners

Table 4 10-fold cross-validation statistic results

Fold R2 RMSE (MPa) MAE (MPa)1 0827 1217 09372 0926 0816 07043 0937 0726 06194 0682 1621 10085 0819 0967 08006 0910 0873 05797 0952 0531 03948 0962 0637 04609 0905 0649 052510 0833 1443 1192Average 0875 0948 0722StD 0082 0347 0245

Table 3 Continued


[46]

Aa-4 304 200times 200 150times 300 419 419 350 022 006 319Aa-7 381 200times 200 150times 300 400 400 350 022 005 308Aa-8 381 200times 200 150times 300 400 400 350 022 010 319Aa-1 411 200times 200 150times 300 400 400 350 022 005 310Aa-2 411 200times 200 150times 300 400 400 350 022 010 297Ab-1 411 200times 200 150times 300 400 400 350 022 005 306Ab-2 411 200times 200 150times 300 400 400 350 022 010 285

[47]HNO3 887 400times 400 300times 400 442 442 681 086 017 1301HNO4 887 400times 400 300times 400 605 610 681 086 017 1487HNO5 1169 400times 400 300times 400 623 610 681 086 013 1239

[48] S3 24 300times 300 200times 300 465 450 390 036 005 364

[49]A1 402 220times 220 160times 250 1069 644 291 043 008 525A2 402 220times 220 160times 250 409 388 291 043 008 487A3 402 220times 220 160times 250 1069 644 291 043 023 525

[50]

LIJ1 255 343times 457 343times 343 440 470 400 061 0 379LIJ2 328 343times 457 343times 343 440 470 400 061 0 486LIJ3 311 343times 457 343times 343 440 470 400 061 0 461LIJ4 343 343times 457 343times 343 440 470 400 061 0 503

[51]BCJ2 303 254times 254 203times 305 414 448 414 052 0 555BCJ3 274 305times 254 203times 305 414 448 414 043 0 508BCJ4 272 356times 254 203times 305 414 448 414 037 0 511

[52]

HSLCJ-1 486 260times 260 150times 300 435 421 449 018 015 725HSLCJ-2 489 260times 260 150times 300 435 421 449 022 015 75HSLCJ-3 495 260times 260 150times 300 435 421 449 018 025 776HSLCJ-4 482 260times 260 150times 300 435 421 449 022 025 782

[53]

JD1 419 250times 250 150times 300 427 443 318 033 0147 728JD2 4395 250times 250 150times 300 427 443 318 084 0147 738JD3 4334 250times 250 150times 300 427 443 318 144 0147 746JD5 4106 250times 250 150times 300 427 443 318 084 0059 635JD6 4435 250times 250 150times 300 427 443 318 084 0235 805

[54] MZJD-1 372 300times 300 200times 400 320 331 318 067 01 585MZJD-2 372 300times 300 200times 400 320 331 318 067 03 729

[55] 8-3 5417 250times 250 200times 300 4796 4796 4796 049 01 9868-5 5534 250times 250 200times 300 4796 4796 4796 049 01 1129


20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20Tested value (MPa)

Training setTesting sety = x

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)

Figure 3 Prediction results of different ML models (a) GBRT (b) LR (c) SVM (d) ANN (e) DT


Table 5 exhibits the prediction performance of the five MLmodels by providing the average statistical indices of the 10-fold cross-validation results Obviously the GBRT model hasthe closest R2 to 1 and smallest values of RMSE and MAEamong the five MLmodels It further verifies the superiority ofthe GBRTmodel over the general individual-type ML models

6 Comparison with Conventional Mechanical-Driven Approach

61 Typical Mechanical-Driven Approach MCFT In thissection the derivation of MCFT is briefly summarized as it is arepresentative conventional mechanical-driven shear strengthprediction method A basic assumption for MCFT is that thecrack direction of a RC plane element is in accordance with theprincipal compressive stress and varies accordingly -e def-initions of stress strain rotational angle and principal di-rection are illustrated in Figure 4 where the x-y coordinatesystem is the local system and the 1-2 coordinate system in-dicates the principal tensile strain-principal compressive strainsystem-e strain vector and stress vector of the RC element inthe local system are denoted as εx εy cxy1113960 1113961

Tand

fx fy vxy1113960 1113961T respectively

-e derivation of the MCFT includes three parts iecompatibility equations equilibrium equations and constitu-tive equations -e detailed formulations are given as follows

611 Compatibility Equations According to Mohrrsquos circleof strain the principal tensile strain ε1 and the principalcompressive strain ε2 of the element are calculated as

ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Accordingly the rotational angle θ from the principalstrain direction to the x-axis can be obtained by

tan2θ εx minus ε2εy minus ε2

ε1 minus εy

ε1 minus εx

ε1 minus εy

εy minus ε2εx minus ε2ε1 minus εx

(18)

612 Equilibrium Equations -e basic element consists of asteel bar and concrete such that its equilibrium conditioncan be derived from the stress state as shown in Figure 4which can be expressed as

fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)

where σcx and σcy are the normal stresses of concrete in the xand y directions respectively τcxy is the shear stress ofconcrete ρsx and ρsy denote the reinforcement ratios in the x

and y directions respectively σsx and σsy are the normalstresses of the steel bar in the x and y directions respectively

Considering the condition of Mohrrsquo circle of stress thenormal stresses and shear stress of concrete are obtained by

σcx σc1 minus τcxy

tan θ

σcy σc1 minus τcxy tan θ

τcxy σc1 minus σc2( 1113857sin θ cos θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

where σc1 and σc2 are the principal stresses in the 1 and 2directions

613 Constitutive Equations With equations (19) and (20)it is found that the stress vector of the RC element can beobtained by the stress states of concrete and steel -ere-fore the constitutive stress-strain relations of these twomaterials are necessary for the state determination of theelement Especially the steel bars are assumed in uniaxialstress state and the concrete is subjected to biaxial stressstate which can be described in the two principaldirections

For reinforcement steel the uniaxial elastic perfectly-plastic model is adopted which is given by

σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)

where Esx εsx and fyx are the elastic modulus strain andyielding strength of the steel bar in the x direction re-spectively Esy εsy and fyy are the elastic modulus strainand yielding strength of the steel bar in the y directionrespectively

For concrete the shear stress state is distinctly differentfrom the uniaxial stress state In consideration of the tensilestress perpendicular to the principal compressive directionhaving influences on the compressive behavior of concrete itis recommended using the modified uniaxial stress-strainrelationships to represent the stress-strain relationship of theRC plane element subjected to combined stress state whichare

Stress-strain relationship in the tensile principaldirection

σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)

Stress-strain relationship in the compressive principaldirection


σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)

where Ec is the elastic modulus of concrete ft and fcprime are the

tensile and compressive strengths of concrete respectivelyεcr and ε0 are the strains corresponding to the tensilestrength and the compressive strength respectively fc2maxis the maximum compressive stress in the principal com-pressive direction It is clear that the modification equation(24) considers the reduction of concrete compressivestrength due to the existence of tensile stress

614 Crack Check Note that the abovementioned equa-tions handle with the global behavior of the element in anaverage sense while it cannot provide the local behaviordescription -e local equilibrium across a crack should alsobe satisfied say

ρsx σsxcr minus σsx( 1113857 σc1 + fci + vci

tan θ

ρsy σsycr minus σsy1113872 1113873 σc1 + fci + vci tan θ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)

where σsxcr and σsycr are the steel stress at the crack fci andvci are the local compressive and shear stresses at the crack

respectively -e abovementioned equation can be satisfiedif there are no local compressive and shear stresses say

ρsx σsxcr minus σsx( 1113857 ρsy σsycr minus σsy1113872 1113873 σc1 (26)

However a constrain should be ensured that the steelstresses at the crack should not exceed the yield strength ofthe steel ie (σsxcrσsycr)lt (fyxfyy) -erefore if thiscondition is not satisfied the local stresses should be cal-culated iteratively -e expressions for the local stresses are

vci 018vcimax + 164fci minus 082f2ci

vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)

where w is the crack width a is the maximum aggregate sizefci is calculated according to ref [4]

-e whole process of using MCFT applied to the shearstrength of internal RC joints can be depicted in Figure 5More details can also be found in [4]

62 Comparison between GBRT and MCFT To furtherevaluate the performance of the GBRT model the con-ventional MCFT is also used to predict the shear strength ofthe 86 RC internal beam-column joints -e statistic resultsfrom the MCFTmodel are compared with the GBRTmodeland shown in Table 6 Note that to fairly compare theperformance of the two models the prediction results in the10 testing sets of the 10-fold cross-validation process areused for the GBRT model

Table 5 Comparisons of predictive performance between GBRT and general ML models

Model R2 RMSE (MPa) MAE (MPa)GBRT 0875 0948 0722LR 0747 1443 1038SVM 0798 1257 0813ANN 0626 1729 1019DT 0802 1285 0840

vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)

Figure 4 -e basic RC element state (a) Stress state (b) Strain state (c) Principal direction


As can be seen from Table 6 the determination coefficientof the GBRTmodel has been improved by 254 and closer to1 compared to the mechanical-driven MCFT model and allthe other two indicators have been dropped more than 50In other words the ML-based method has obviously betterperformance than the MCFT-based method Furthermorethe predicted and tested values are also plotted in Figure 6Evidently the GBRT results match the experimental resultsmuch better than those of the MCFT model

Table 7 gives the statistic results of predicted valuetestedvalue ratios for the MCFT and GBRT models It can beconcluded from Table 7 that the GBRT model statisticallyunderestimates the shear strength because the mean value isless than 1 while the MCFTmodel slightly overestimates theshear strength Apparently the mean predicted valuetestedvalue ratios for the GBRT approach is closer to 1 with lessdispersion (StD) compared to the MCFT method

Figure 7 further illustrates the predicted valuetestedvalue ratios for the GBRTandMCFTmodels -e solid linethe top dotted line and the bottom line represent the meanvalue mean value plus StD and mean value minus StDrespectively Evidently better prediction performance isachieved by the GBRT model

7 Model Sensitivity Analysis

71 Sensitivity of Input Parameters With the developedGBRT model it is convenient for us to investigate the in-fluences of different parameters on the shear strength of theinternal RC joint and even quantify the influences In thisstudy 10 input variables with different value ranges areadopted to conduct a comprehensive parametric analysis Inthe parametric analysis the control variable method is usedie one control parameter varies while other parameters arefixed-e specimen J6 of [44] is used as the reference model-e numerical ranges of the 10 inputs are shown in Table 8

Figure 8 shows the predicted shear strength of theinternal RC joints with different input variables by usingthe GBRT model It can be drawn from Figure 8 thatamong all the input variables concrete strength fc is themost significant parameter affecting the shear strengthWith the increasing of concrete strength fc beam width bbcolumn width bc column height hc yielding strength ofcolumn longitudinal bar fyc yielding strength of jointtransverse bar fyv transverse bar ratio ρv or axial loadratio n the shear strength has a general ascending trendOn the contrary yielding strength of beam longitudinal

Given the strain state of the joint(εx εy γxy)T

Calculate the principal direction θusing Eq (18)

Calculate the principal strains(ε1 ε2 γ12)T using Eq (17)

Calculate the concrete stress(σc1 σc2)T

Calculate the steel stress(σsx σsy)T

Concrete modelEqs (22)ndash(24)

Steel modelEq (21)

Calculate local stress at the crack(fci vci)T using Eq (27)

Check ifσsxcrσsycr lt fyxfyy

Integrate the stress state(σx σy τxy)T using Eq (19)

Calculate steel stress at the crack(σsxcr σsycr)T using Eq (25)

End

Yes

No

Figure 5 Flowchart of implementation of MCFT in calculating shear strength of internal RC joints

Table 6 Comparisons of predictive performance between GBRT and MCFT models

Model R2 RMSE (MPa) MAE (MPa)MCFT 0765 1355 1066GBRT 0960 0562 0380(GBRT-MCFT)MCFT () 254 minus585 minus643


bar fyb has negative effects on the shear strength -einfluence of beam height hb on the shear strength isnegligible

72 Feature Importance Feature importance which is usedto quantify the importance of the input variables (or fea-tures) is conducted to further investigate the sensitivity of

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)

Figure 7 Comparisons of the predicted valuetested value for the GBRT and MCFT models (a) GBRT results (b) MCFT results

Table 7 Statistic results of predicted valuetested value ratios for the MCFT and GBRT models

Model Mean value StDMCFT 0961 0180GBRT 1012 0092

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)

Figure 6 Comparisons of shear strength of RC joints by the GBRTpredictive model and MCFTmodel (a) GBRTresults (b) MCFTresults

Table 8 Numerical ranges of input variables

f c (MPa) b c (mm) h c (mm) b b (mm) h b (mm)[20 20 100] [200 100 500] [200 100 500] [150 100 450] [200 100 500]f yb (MPa) f yc (MPa) fyv (MPa) ρv () n[300 140 1000] [300 140 1000] [300 140 1000] [02 02 10] [0 01 06]


13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued


each input variable on the shear strength of the internal RCjoints -e calculation of feature importance can be summa-rized as follows Firstly some out-of-bag (OOB) samples areselected Secondly the values of the target input variable arerandomly shuffledwhile other inputs remain unchanged-enthe feature importance can be calculated as the accuracy dif-ference of the two predictions using the GBRTmodel Figure 9shows the relative feature importance of all input variables It isclear that concrete strength fc is the key feature determining theshear strength of the internal RC joints which is in accordancewith the conclusion obtained from the previous subsection-einfluences of the yielding strength of joint transverse bar fyvtransverse bar ratio ρv and axial load ratio n on shear strengthare subdominant -e remaining input variables are insig-nificant features -e feature importance results are also inaccordance with the sensitivity results performed before

991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)

Figure 8 Prediction results of parametric analysis (a) Concrete strength (MPa) (b) Section width of column (mm) (c) Section height ofcolumn (mm) (d) Section width of beam (mm) (e) Section height of beam (mm) (f ) Yielding strength of beam longitudinal bar (MPa) (g)Yielding strength of column longitudinal bar (MPa) (h) Yielding strength of joint transverse bar (MPa) (i) Transverse bar ratio () (j)Axial load ratio

Variable importance

0 20 40 60 80 100Relative importance ()

fcfyv

fybfycbbbchchb

nρv

Figure 9 Relative feature importance of all input variables


8 Conclusions

-is paper presents a ML-based approach to predict theshear strength of internal RC beam-column joints One ofthe famous ensemble learning methods GBRT is employedto solve the prediction problem A database of 86 sets ofinternal RC joint tests is collected from the literature Someindividual-type ML methods and the conventional MCFTmethod are adopted for comparisons of the developed GBRTprediction model -e model sensitivity analysis of inputparameters is conducted for the proposed GBRT-basedmodel Based on the results the following conclusions can bedrawn

(1) -e GBRT model can accurately and efficientlypredict the shear strength of internal RC beam-column joints with given input variables

(2) If 80 of the whole dataset is used to train the GBRTmodel the average determination coefficient R2 ofthe 10-fold cross-validation is 0875 which meansthat the prediction error is low Meanwhile theaverage RMSE and MAE are 0948MPa and0722MPa indicating that the prediction model hasa low prediction deviation

(3) Among all the ML-based prediction models used inthis study the GBRT model performs best with theclosest R2 to 1 and smallest values of RMSE andMAE It indicates that the GBRTmodel is superior tothe individual-type ML algorithms

(4) -e GBRTmodel has better prediction performancecompared with the conventional MCFT model inboth average sense and variance sense and exhibits asignificant superiority in terms of the three perfor-mance indicators

(5) Among all the input variables concrete strength fc isthe most critical feature affecting the shear strengthof the internal RC joints With the increasing of theconcrete strength the shear strength significantlyincreases Other input variables are relatively sub-ordinate or even unimportant

Data Availability

-e data will be available from the corresponding authorupon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-e authors greatly appreciate the financial supports fromthe Natural Science Foundation of Jiangsu Province (Grantno BK20170680) the National Natural Science Foundationof China (Grant nos 51708106 and 51908048) the NaturalScience Foundation of Shaanxi Province (Grant nos2019JQ-021) the Fundamental Research Funds for the

Central Universities CHD (Grant no 300102289301) andthe Open Project of State Key Laboratory of Green Buildingin Western China (Grant no LSKF202007)

References

[1] N W Hanson and H W Connor ldquoSeismic resistance ofreinforced concrete beamndashcolumn jointsrdquo Journal of Struc-tural Division ASCE vol 11 no 56 pp 533ndash559 1967

[2] H Dabiri A Kheyroddin and A Kaviani ldquoA numerical studyon the seismic response of RC wide column-beam jointsrdquoInternational Journal of Civil Engineering vol 17 no 3pp 377ndash395 2019

[3] R Costa P Providencia and A Dias ldquoComponent-basedreinforced concrete beam-column joint modelrdquo StructuralConcrete vol 18 no 1 pp 164ndash176 2017

[4] F J Vecchio and M P Collins ldquo-e modified compression-field theory for reinforced concrete elements subjected toshearrdquo ACI Structural Journal vol 83 no 2 pp 219ndash2311986

[5] T Paulay and M J N Priestley Seismic Design of ReinforcedConcrete and Masonry Building Wiley New York NY USA1992

[6] X Long and C K Lee ldquoImproved strut-and-tie method for2D RC beam-column joints under monotonic loadingrdquoComputers and Concrete vol 15 no 5 pp 807ndash831 2015

[7] C Lima E Martinelli and C Faella ldquoCapacity models forshear strength of exterior joints in RC frames state-of-the-artand synoptic examinationrdquo Bulletin of Earthquake Engi-neering vol 10 no 3 pp 967ndash983 2012

[8] T S Eom H J Hwang and H G Park ldquoEnergy-basedhysteresis model for reinforced concrete beam-column con-nectionsrdquo ACI Structural Journal vol 112 no 2 pp 157ndash1662015

[9] ASCESEI 41 Seismic Rehabilitation of Existing BuildingsAmerican Society of Civil Engineers Reston VA USA 2007

[10] H-J Hwang T-S Eom and H-G Park ldquoShear strengthdegradation model for performance-based design of interiorbeam-column jointsrdquo ACI Structural Journal vol 114 no 5pp 1143ndash1154 2017

[11] H J Hwang and H G Park ldquoRequirements of shear strengthand hoops for performance-based design of interior beam-column jointsrdquo ACI Structural Journal vol 116 no 2pp 245ndash256 2019

[12] H-J Hwang and H-G Park ldquoPerformance-based sheardesign of exterior beam-column joints with standard hookedbarsrdquo ACI Structural Journal vol 117 no 2 pp 67ndash80 2020

[13] S-J Hwang R-J Tsai W-K Lam and J P Moehle ldquoSim-plification of softened strut-and-tie model for strength pre-diction of discontinuity regionsrdquo ACI Structural Journalvol 114 no 5 pp 1239ndash1248 2017

[14] R Y C Huang and J S Kuang ldquoPredicting strength of ex-terior wide beam-column joints for seismic resistancerdquoJournal of Structural Engineering vol 146 no 2 Article ID04019209 2020

[15] W M Hassan and J P Moehle ldquoShear strength of exteriorand corner beam-column joints without transverse rein-forcementrdquo ACI Structural Journal vol 115 no 6pp 1719ndash1727 2018

[16] ACI Committee 318 Building Code Requirements for Struc-tural Concrete (ACI 318-14) and Commentary (ACI 318R-14)American Concrete Institute Farmington Hills MI USA2014


[17] S Akkurt G Tayfur and S Can ldquoFuzzy logic model for theprediction of cement compressive strengthrdquo Cement andConcrete Research vol 34 no 8 pp 1429ndash1433 2004

[18] A Baykasoglu T Dereli and S Tanıs ldquoPrediction of cementstrength using soft computing techniquesrdquo Cement andConcrete Research vol 34 no 11 pp 2083ndash2090 2004

[19] G Trtnik F Kavcic and G Turk ldquoPrediction of concretestrength using ultrasonic pulse velocity and artificial neuralnetworksrdquo Ultrasonics vol 49 no 1 pp 53ndash60 2009

[20] R Siddique P Aggarwal and Y Aggarwal ldquoPrediction ofcompressive strength of self-compacting concrete containingbottom ash using artificial neural networksrdquo Advances inEngineering Software vol 42 no 10 pp 780ndash786 2011

[21] A-D Pham N-D Hoang and Q-T Nguyen ldquoPredictingcompressive strength of high performance concrete usingmetaheuristic-optimized least squares support vector re-gressionrdquo Journal of Computing in Civil Engineering vol 30no 3 Article ID 06015002 2015

[22] B A Omran Q Chen and R Jin ldquoComparison of datamining techniques for predicting compressive strength ofenvironmentally friendly concreterdquo Journal of Computing inCivil Engineering vol 30 no 6 Article ID 04016029 2016

[23] P G Asteris and V G Mokos ldquoConcrete compressivestrength using artificial neural networksrdquo Neural Computingand Applications vol 32 no 15 pp 11807ndash11826 2019

[24] V Plevris and P G Asteris ldquoModeling of masonry failuresurface under biaxial compressive stress using Neural Net-worksrdquo Construction and Building Materials vol 55pp 447ndash461 2014

[25] P G Asteris and V Plevris ldquoAnisotropic masonry failurecriterion using artificial neural networksrdquo Neural Computingand Applications vol 28 no 8 pp 2207ndash2229 2017

[26] J-S Jeon A Shafieezadeh and R DesRoches ldquoStatisticalmodels for shear strength of RC beam-column joints usingmachine-learning techniquesrdquo Earthquake Engineering ampStructural Dynamics vol 43 no 14 pp 2075ndash2095 2014

[27] D-T Vu and N-D Hoang ldquoPunching shear capacity esti-mation of FRP-reinforced concrete slabs using a hybridmachine learning approachrdquo Structure and InfrastructureEngineering vol 12 no 9 pp 1153ndash1161 2016

[28] W Zheng and F Qian ldquoPromptly assessing probability ofbarge-bridge collision damage of piers through probabilistic-based classification of machine learningrdquo Journal of CivilStructural Health Monitoring vol 7 no 1 pp 57ndash78 2017

[29] A Santos E Figueiredo M Silva R Santos C Sales andJ C Costa ldquoGenetic-based EM algorithm to improve therobustness of Gaussian mixture models for damage detectionin bridgesrdquo Structural Control and Health Monitoring vol 24no 3 p e1886 2017

[30] H Salehi and R Burguentildeo ldquoEmerging artificial intelligencemethods in structural engineeringrdquo Engineering Structuresvol 171 pp 170ndash189 2018

[31] R J Schalkoff Artificial Neural Networks McGraw-Hill NewYork NY USA 1997

[32] M A Hearst S T Dumais E Osuna J Platt andB Scholkopf ldquoSupport vector machinesrdquo IEEE IntelligentSystems andJeir Applications vol 13 no 4 pp 18ndash28 1998

[33] S R Safavian and D Landgrebe ldquoA survey of decision treeclassifier methodologyrdquo IEEE Transactions on Systems Manand Cybernetics vol 21 no 3 pp 660ndash674 1991

[34] Z-H Zhou ldquoEnsemble learningrdquo Encyclopedia of BiometricsS Z Li Ed pp 411ndash416 Springer Press New York NY USA2015

[35] K P Murphy Machine Learning A Probabilistic PerspectiveMIT Press Cambridge MA USA 2012

[36] V N Vapnik ldquoAn overview of statistical learning theoryrdquoIEEE Transactions on Neural Networks vol 10 no 5pp 988ndash999 1999

[37] Research Group on Frame Joints ldquoShear strength of rein-forced concrete beam-column joints under low reversed cyclicloadingrdquo Journal of Building Structures vol 4 no 6 pp 1ndash171983 in Chinese

[38] C Zhao D Zhang T Wang and D Chen ldquoExperimentalstudy on the seismic property of the beam-column joints inhigh strength concrete frame under alternating loadrdquo Journalof Shenyang Architectural and Civil Engineering Institutevol 9 no 3 pp 260ndash268 1993 in Chinese

[39] X Lu Z Guo and Y Wang ldquoExperimental study on seismicbehavior of beam-column sub-assemblages in RC framerdquoJournal of Building Structures vol 22 no 1 pp 2ndash7 2001 inChinese

[40] Q Yu and S Li ldquoResearch on framersquos joint that concretestrength of core is inferior to that of columnrdquo Journal of TongjiUniversity vol 32 no 12 pp 1583ndash1588 2004 in Chinese

[41] B Xu M Cheng M Zhang and J Qian ldquoExperimental studyon behavior of reinforced concrete beam-column joint withlower core concrete strengthrdquo Building Structures vol 36no 6 pp 18ndash23 2006 in Chinese

[42] S Otani K Kitayama and H Aoyama ldquoReinforced concreteinterior beam-column joints under simulated earthquakeloadingrdquo in Proceedings of the US-New Zealand-Japan Sem-inar on Design of Reinforced Concrete Beam-Column JointsMonterey Canada 1984

[43] D E Meinheit and J O Jirsa ldquo-e shear strength ofreinforced concrete beam-column jointsrdquo CESRL ReportNo 77-l University of Texas Austin TX USA 1977

[44] H Noguchi and T Kashiwazaki ldquoTest on high-strengthconcrete interior beam-column jointsrdquo in Proceedings of theTenth World Conference on Earthquake Engineering MadridSpain 1992

[45] K Oka and H Shiohara ldquoTest on high strength concreteinterior beam-column sub-assemblagerdquo in Proceedings of JeTenth World Conference on Earthquake Engineering MadridSpain 1992

[46] Y Higashi and Y Ohwada ldquoFailing behaviors of reinforcedconcrete beam-column connections subjected to lateral loadrdquoMemories of Faculty of Technology Tokyo Metropolitan Uni-versity vol 19 pp 91ndash101 University of Virginia Charlot-tesville VA USA 1969

[47] M Teraoka Y Kanoh K Tanaka and I C Hayashi ldquoStrengthand deformation behavior of RC interior beam-and-columnjoints using high strength concreterdquo in Proceedings of theSecond US-Japan-New Zealand-Canada Multilateral Meetingon Structural Performance of High Strength Concrete inSeismic Regions Honolulu HI USA 1994

[48] S S Zaid ldquoBehavior of reinforced concrete beam-columnconnections under earthquake loadingrdquo Doctoral disserta-tion University of Tokyo Tokyo Japan 2001

[49] S Fujii and S Morita ldquoComparison between interior andexterior RC beam-column joint behaviorrdquo Design of Beam-Column Joints for Seismic Resistance SP-123 pp 145ndash165ACI Detroit MI USA 1991

[50] D E Abrams ldquoScale relations for reinforced concrete beam-column jointsrdquo ACI Structural Journal vol 84 no 6pp 502ndash512 1987


[51] R T Leon ldquoShear strength and hysteretic behavior of interiorbeam-column jointsrdquo ACI Structural Journal vol 87 no 1pp 3ndash11 1990

[52] T Wu X Liu and H Wei ldquoExperimental study on seismicbehavior of frame interior joints with high-strength light-weight aggregate reinforced concreterdquo China Civil Engi-neering Journal vol 51 no 6 pp 32ndash42 2018 in Chinese

[53] H H Zhang Z L Lu and S Z Su ldquoExperimental study onseismic behavior of high-strength ceramic concrete framejointsrdquo Building Science vol 32 no 1 pp 81ndash87 2016 inChinese

[54] B Qu ldquoExperimental study on seismic performance of fulllightweight aggregate concrete beam-column jointsrdquoMasterrsquosthesis Jilin Jianzhu University Changchun China 2015 inChinese

[55] C L Decker M A Issa and K F Meyer ldquoSeismic investi-gation of interior reinforced concrete sand-lightweight con-crete beam-column jointsrdquo ACI Structural Journal vol 112no 3 pp 287ndash297 2015


the well-known modified compression field theory (MCFT)[4] the strut-and-tie method (STM) [5] etc -ese modelsare actually derived based on the shear mechanisms offundamental RC elements and can be widely used to evaluatethe behavior of any type of shear-dominated RC membersincluding the beam-column joints [6] A detailed review ofthe theoretical and empirical models for the RC joints can befound in [7]

In recent five years there are some latest development onRC joint models Eom et al [8] developed an energy-basedhysteresis model for RC beam-column joints by using theenergy function and the existing backbone curve of ASCESEI 41-06 [9] Hwang et al [10] proposed a shear strengthdegradation model for performance-based design of interiorbeam-column joints In the model all possible failuremechanisms of beams and joints including flexural yieldingof the beam end diagonal cracking and concrete crushing inthe joint panel bar bond-slip and bar elongation areconsidered Later Hwang and Park [11] developed designequations of the joint shear strength and hoop requirementfor the performance-based design of interior RC beam-column joints by considering the diagonal strut mechanismand truss mechanism -e target drift ratio and bar bondparameters are defined as the requirements of the joint shearstrength and hoop strength More recently Hwang and Park[12] modified the shear strength degradation model forinterior RC joints and applied it to exterior RC joints withstandard hooked bars Hwang et al [13] simplified thesoftened strut-and-tie model to facilitate design practice forthe strength prediction of discontinuity regions such as theRC beam-column joints -e shear-resisting mechanisms assuggested by the softened strut-and-tie model are consideredin the simplified model Similarly Huang and Kuang [14]proposed a shear strength model for exterior RC wide beam-column joints by introducing the softened strut-and-tieconcept Hassan and Moehle [15] collected a database ofexterior and corner beam-column joints without transversereinforcement Based on the database they evaluated severalexisting shear strength models and developed a strut-and-tiemodel based on the ACI 318 [16] strut-and-tie modelingprovisions and an empirical model by considering the effectsof joint aspect ratio column axial load and concretecompressive strength

Although the above empirical or theoretical approachesoffer simple and clear explanation of the shear mechanismthey also introduce empirical assumptions which will reducetheir accuracy Moreover the derivations seem to becomplicated since the iteration process is likely involved andsome of the parameters are empirical that needed to bedetermined according to the usersrsquo experience

In recent years with the flourishment of artificial in-telligence (AI) a brand new way is come to peoplersquos hori-zons ie using machine learning (ML) techniques to predictthe shear strength of the RC beam-column joints ML is atype of AI which has various functions eg classificationregression and clustering ML can learn the characteristicsof a certain type of data according to the existing databaseand then classify summarize and predict the things ofinterest Prediction of the shear strength of the RC joints is

essentially a regression problem -ere are already somesuccessful applications of prediction using ML in structuralengineering for instances evaluating the cement strengthvia fuzzy logic artificial neutral network (ANN) and geneexpression programming (GEP) [17 18] modeling theconcrete properties via ANN and support vector machine(SVM) [19ndash23] simulating the failure of brittle anisotropicmaterials such as masonry via ANN [24 25] predicting thestructural member capacities via hybrid ML algorithms[26 27] detecting the structural damage via GEP [28 29]etc A detailed state-of-art of the application of ML instructural engineering was summarized in [30]

However the majority of the ML algorithms used in theprevious studies were individual-type learning algorithmssuch as ANN family [31] SVM family [32] and decision tree(DT) family [33] -e disadvantages of the individual-typelearning algorithms are instable and with low accuracy Toimprove their performance a new type of learning algo-rithms known as ensemble learning algorithms has beenrecently proposed and successfully applied in various fields-e basic idea of the ensemble learning is to combine severalweak learners generated by individual learning algorithmsinto a strong one In brief the ensemble learning algorithmsare more stable and accurate compared to the individuallearning algorithms [34] -ere are mainly two categories ofensemble learning algorithms bagging and boosting For thebagging family the weak learners are produced in parallelwhile they are produced in sequence for the boosting family-eoretically bagging is more efficient and can effectivelyreduce the variance of the prediction and boosting is rel-atively less efficient in reducing the bias In practice boostingis superior to bagging in terms of accuracy for general cases-erefore one of the most typical boosting ensemblelearning algorithms referred to as gradient boosting re-gression tree (GBRT) [35] algorithm is used in this study

In this paper we aim to develop a GBRT-based intel-ligent method for predicting the shear strength of the RCbeam-column joints and make comparisons between theproposed data-driven model and some traditional ML-basedmodels as well as the conventional mechanical-drivenMCFTmodel Firstly some individual-type ML techniquesincluding linear regression (LR) SVM ANN and DT arebriefly reviewed -en the mathematical background andimplementation of GBRT are introduced Afterwards theshear strength data of 86 internal RC beam-column jointsare collected from the literature Based on the database theprediction results from the GBRT-based model are verifiedby a 10-fold validation test and compared with those fromthe individual-type ML models In addition one of therepresentative conventional mechanical-driven approachesie MCFT is briefly summarized and also used as com-parison with the GBRTmodel Finally sensitivity analysis ofinput variables is conducted for the GBRTmodel to quantifythe influences of different parameters

2 Review of the Traditional ML Techniques

21 Linear Regression (LR) Linear regression (LR) is one ofthe most widely used statistical analysis techniques in












i1αihi(x) (1)




fm+1(x) fm(x) + hm(x) y (2)








fM(x) 1113944M

m1T xΘm( 1113857 (4)





Ni1 L[yi fM(xi)] ie

arg minΘ

1113944

N

i1L yi fM xi( 11138571113858 1113859 arg min

Θ1113944

N

i1L yi 1113944

M

m1T xΘm( 1113857⎡⎣ ⎤⎦

(5)






1113944

N




rmi minus T xiΘm( 11138571113960 11139612

(8)







zf xi( 11138571113890 1113891

f(x)fmminus1(x)

(9)

With (xi rmi)1113966 1113967i12N


cmj argminc

1113944xiisinRmj


(10)


Tm(x) 1113944

J

j1cmjI x isin Rmj1113872 1113873 (11)



J


(12)


fM(x) 1113944M

m11113944

J

j1cmjI x isin Rmj1113872 1113873 (13)



c1113936

Ni1 L(yi c)







1 2 J



cmjI(x isin Rmj)














R2

1 minus1113944

N


1113944N

i1 Ti minus T( 11138572 (14)


RMSE

1113944N


N

1113971

(15)


MAE 1113944

N

i1 Pi minus Ti

11138681113868111386811138681113868111386811138681113868

N (16)







Collect data

Database

bullbullbull

Testing dataset

Training dataset

Weak learner

Learning

Weight

Weak learner

Learning

Weight

Weak learner

Learning

WeightUpdate Update

Validate

Strong learner

sum











03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)





[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References





































































i1αihi(x) (1)




fm+1(x) fm(x) + hm(x) y (2)








fM(x) 1113944M

m1T xΘm( 1113857 (4)





Ni1 L[yi fM(xi)] ie

arg minΘ

1113944

N

i1L yi fM xi( 11138571113858 1113859 arg min

Θ1113944

N

i1L yi 1113944

M

m1T xΘm( 1113857⎡⎣ ⎤⎦

(5)






1113944

N




rmi minus T xiΘm( 11138571113960 11139612

(8)







zf xi( 11138571113890 1113891

f(x)fmminus1(x)

(9)

With (xi rmi)1113966 1113967i12N


cmj argminc

1113944xiisinRmj


(10)


Tm(x) 1113944

J

j1cmjI x isin Rmj1113872 1113873 (11)



J


(12)


fM(x) 1113944M

m11113944

J

j1cmjI x isin Rmj1113872 1113873 (13)



c1113936

Ni1 L(yi c)







1 2 J



cmjI(x isin Rmj)














R2

1 minus1113944

N


1113944N

i1 Ti minus T( 11138572 (14)


RMSE

1113944N


N

1113971

(15)


MAE 1113944

N

i1 Pi minus Ti

11138681113868111386811138681113868111386811138681113868

N (16)







Collect data

Database

bullbullbull

Testing dataset

Training dataset

Weak learner

Learning

Weight

Weak learner

Learning

Weight

Weak learner

Learning

WeightUpdate Update

Validate

Strong learner

sum











03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)





[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References






























































fM(x) 1113944M

m1T xΘm( 1113857 (4)





Ni1 L[yi fM(xi)] ie

arg minΘ

1113944

N

i1L yi fM xi( 11138571113858 1113859 arg min

Θ1113944

N

i1L yi 1113944

M

m1T xΘm( 1113857⎡⎣ ⎤⎦

(5)






1113944

N




rmi minus T xiΘm( 11138571113960 11139612

(8)







zf xi( 11138571113890 1113891

f(x)fmminus1(x)

(9)

With (xi rmi)1113966 1113967i12N


cmj argminc

1113944xiisinRmj


(10)


Tm(x) 1113944

J

j1cmjI x isin Rmj1113872 1113873 (11)



J


(12)


fM(x) 1113944M

m11113944

J

j1cmjI x isin Rmj1113872 1113873 (13)



c1113936

Ni1 L(yi c)







1 2 J



cmjI(x isin Rmj)














R2

1 minus1113944

N


1113944N

i1 Ti minus T( 11138572 (14)


RMSE

1113944N


N

1113971

(15)


MAE 1113944

N

i1 Pi minus Ti

11138681113868111386811138681113868111386811138681113868

N (16)







Collect data

Database

bullbullbull

Testing dataset

Training dataset

Weak learner

Learning

Weight

Weak learner

Learning

Weight

Weak learner

Learning

WeightUpdate Update

Validate

Strong learner

sum











03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)





[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References




























































1 2 J



cmjI(x isin Rmj)














R2

1 minus1113944

N


1113944N

i1 Ti minus T( 11138572 (14)


RMSE

1113944N


N

1113971

(15)


MAE 1113944

N

i1 Pi minus Ti

11138681113868111386811138681113868111386811138681113868

N (16)







Collect data

Database

bullbullbull

Testing dataset

Training dataset

Weak learner

Learning

Weight

Weak learner

Learning

Weight

Weak learner

Learning

WeightUpdate Update

Validate

Strong learner

sum











03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)





[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References





























































Collect data

Database

bullbullbull

Testing dataset

Training dataset

Weak learner

Learning

Weight

Weak learner

Learning

Weight

Weak learner

Learning

WeightUpdate Update

Validate

Strong learner

sum











03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)





[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References



























































03

025

02

015

01

005

0

Relat

ive f

requ

enci

es

0 20 40 60 80 100 120Input 1

(a)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 2200 300 400 500

(b)

03025

02015

01005

0

Relat

ive f

requ

enci

es

Input 3200 300 400 500

(c)

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

150 200 250 300 350 400 450Input 4

(d)

Relat

ive f

requ

enci

es

250 300 350 400 450 500Input 5

06

05

04

03

02

01

0

(e)

Relat

ive f

requ

enci

es

06

05

04

03

02

01

00 500 1000 1500

Input 6

(f )

06

05

04

03

02

01

0

Relat

ive f

requ

enci

es

0 500 1000 1500Input 7

(g)

060504030201

0

Relat

ive f

requ

enci

es

200 400 600 800 1000 1200 1400Input 8

(h)

Relat

ive f

requ

enci

es

03

025

02

015

01

005

00 05 1 15 2

Input 9

(i)

Relat

ive f

requ

enci

es 03025

02015

01005

00 05 1

Input 10

(j)

Relat

ive f

requ

enci

es

025

02

015

01

005

00 5 10 15

Output

(k)





[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References





























































[37]


[38]



[40]


[41] SJ-3 2934 350times 350 175times 350 3547 3480 3815 050 106 688


[43]


[44]


[45]







Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References































































Table 3 Continued


[46]



[48] S3 24 300times 300 200times 300 465 450 390 036 005 364


[50]



[52]


[53]





20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References



























































20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(b)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(c)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)

0 5 10 15 20

Tested value (MPa)


(d)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)



(e)






Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References






























































Tand




ε1 εx + εy1113872 1113873

2+

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

ε2 εx + εy1113872 1113873

2minus

εx minus εy1113872 11138732

4+

cxy1113872 11138732

4

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)



ε1 minus εy

ε1 minus εx

ε1 minus εy


(18)


fx σcx + ρsxσsx

fy σcy + ρsyσsy

vxy τcxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





tan θ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)




σsx Esxεsx lefyx

σsy Esyεsy lefyy

⎧⎪⎨

⎪⎩(21)




σc1

Ecε1 ε1 le εcr

ft

1 +200ε1

1113968 ε1 gt εcr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References



























































σc2 fc2max 2ε2ε0

1113888 1113889 minusε2ε0

1113888 1113889

2⎡⎣ ⎤⎦ (23)

with

fc2max

fcprime

108 minus 034 ε1ε0( 1113857

le 1 (24)





tan θ


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)






vcimax

vcimax

minusfc

1113968

(031 + 24w)(a + 16)

(27)






vxy

fy

fx

y

fx

fy

vxy

vxy

vxy

x

(a)

1

1

εy

εx

γxy

(b)

y

x

2 1

θ

(c)















Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References







































































Steel modelEq (21)





End

Yes

No







2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References





























































2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(a)

2

15

1

05

00 20 40 60 80

Sample number

Pred

icte

d va

lue

teste

d va

lue

Mean plusmn StDMean

(b)




20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


15

ndash15

(a)

20

15

10

5

0

Pred

icte

d va

lue (

MPa

)


ndash30

30

(b)





13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References



























































13

12

11

10

9

8

7

6

τ (M

Pa)

20 40 60 80 100fc (MPa)

(a)

τ (M

Pa)

991

99

989

988

987

986200 300 400 500

bc (mm)

(b)

991

9905

99

9895

989

τ (M

Pa)

200 300 400 500hc (mm)

(c)

150 200 250 300 350 400 450

τ (M

Pa)

bb (mm)

10

998

996

994

992

99

(d)

11

105

10

95

9

85

τ (M

Pa)

hb (mm)200 300 400 500

(e)

τ (M

Pa)

10

998

996

994

992

99

fyb (MPa)400 600 800 1000

(f )

Figure 8 Continued



991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References




























































991

9905

99

9895

989

9885

988

9875

τ (M

Pa)

fyc (MPa)400 600 800 1000

(g)

τ (M

Pa)

fyv (MPa)

10

98

96

94

92

9300 400 500 600 700 800 900 1000

(h)

995

99

985

98

975

97

τ (M

Pa)

02 04 06 08 1ρv ()

(i)

τ (M

Pa)

101

10

99

98

97

96

95

940 02 04 06

n

(j)


Variable importance


fcfyv

fybfycbbbchchb

nρv



8 Conclusions







Data Availability




Acknowledgments



References



























































8 Conclusions







Data Availability




Acknowledgments



References



























































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