Shantaram Parab,Shreya Srivastava,Pijush Samui and A.Ramachandra Murthy

1 Introduction

Concrete is of high excellence in terms of strength and long term performance are considered to be ideal requirements for special applications.Concretes of strengths exceeding 80 MPa are now commonly used in the construction of high-rise buildings,long span bridges and offshore structures.The major problems being faced by civil engineers are maintenance,retro fitting and preservation of these structures.Ultra High Strength Concrete(UHSC)is a highly engineered material with several chemical and mineral admixtures materials.It has been successfully applied in the field for the construction of Sherbrook Pedestrian Bridge,Canada,The Glenmore/Legs by Pedestrian,Alberta,Canada and P shaped UHPC beams installed in footbridges in Auckland,New Zealand[Seibert(2008);Rebentrost and Wight(2009)].Since UHSC is a relatively new material,the fracture behavior of this material is not well understood[Richard and Cheyrezy(1994,1995);Mingzhe et al.(2010);Goltermann et al.(1997)].

Concrete being a quasi-brittle materials exhibit a nonlinear region before the peak of the stress–strain relationship and substantial post-peak strain softening.Linear elastic fracture mechanics cannot be applied directly to the quasi-brittle materials[7].Due to high heterogeneity nature in concrete,cracks follow the weakest matrixlinks in the material. They lead their way through the weak bonds,voids,mortar and get arrested on encountering a hard aggregate,forming crack face bridges.Micro cracking,crack bridging and aggregate interlocking are a few of many specific mechanisms that absorb energy during fracture process.These mechanisms contribute to the tendency of the main crack to follow a tortuous path[Bazant(2000);Barenblatt(1959);Dugdale(1960)].This tortuous nature of the crack causes difficulty in computing the fracture energy.Therefore,modeling the exact nature of the fracture surface poses a new challenge to the researchers.In these days,most theoretical works in fracture mechanics are based on the fundamental assumption that cracks have smooth surfaces.This assumption is helpful to use analytical models in the field of fracture mechanics.

Over the past few years,researchers have used different statistical modelling methods such as Artificial Neural Network Support Vector Regression,Multivariate Adaptive Regression Splines and Relevance Vector Machine for prediction of fracture characteristics of concrete.Yuvaraj et al.(2013)used Support vector regression,Artificial Neural Network[Yuvaraj et al.(2012)]and Multivariate Adaptive Regression Splines[Yuvaraj et al.(2013)]to predict the fracture characteristics of concrete beams.Though the performance of ANN is acceptable,its results are hard to interpret.Support vector machines do not directly provide probability estimates and in the case of MARS,parameter confidence intervals and other checks on the model cannot be calculated directly,unlike linear regression models.This study uses Gaussian Process Regression(GPR)and Least Squares Support Vector Machines(LSSVM)for the prediction of fracture characteristics of concrete.GPR(Gaussian Process Regression)gives a non-parametric modelling approach and probabilistic Bayesian framework which can be applied to various engineering problems.The probabilistic GP chooses hyper parameters directly from the training The probabilistic GP chooses hyper parameters directly from the training data,gives a probabilistic measure of the uncertainty of the model prediction and obtain a relatively good model when only a small set of training data is available[Azman and Kocijan(2007);Pal and Deswal(2010);Likar and Kocijan(2007);Yuan et al.(2008)].In comparison to back-propagation neural networks,Gaussian processes are conceptually simpler to understand and implement in practice.The LSSVM is a statistical learning theory which adopts a least squares linear system as a loss functions instead of the quadratic program in original support vector machine(SVM)[Suykans and Vandewalle(1999);Baesens et al.(2000);Espinoza et al.(2003);Lu et al.(2003);Mitra et al.(2007)].It is closely related to Gaussian processes and regularization networks.It requires solving a set of only linear equations(linear programming),which is much easier and computationally very simple.Both GPR and LSSVM have a strong potential for predicting the facture characteristics with high correlation and precision to the experimental value.They differ from most of the other black-box identification approaches as it does not try to approximate the modeled system by fitting the parameters of the selected basis functions but rather searches for the relationship among measured data.

The aim of this study is to predict the fracture characteristics of high strength concrete beams(HSC)and Ultra high strength concrete beams(UHSC).This paper presents development and validation of models based on concept of Gaussian Process Regression(GPR)and Least Squares Support Vector Machines(LSSVM)to predict fracture parameters and failure load(Pmax)of high strength and ultra-high strength concrete beams.

2 Gaussian Process Regression

This study incorporates Gaussian Process Regression(GPR)for prediction of Failure load(Pmax),fracture energy(GF),critical stress intensity factor(KIC)and critical crack tip opening displacement(CTODC).In GPR,the learning of data is modeled as Bayesian estimation problem.It is assumed that the parameters of GPR are random variables.GPR has been successfully adopted for solving different problems in engineering[Yuan et al.(2008);Pal and Deswal(2010)]

Let us consider the following set of samples

Where x is input variable,y is output,RNis N-dimensional vector space and R is one dimensional vector space.This article uses Beam length(L),Cross-section area(A),Notch depth(a),water-cement ratio(w/c),compressive strength(fck),split tensile strength(σt)and modulus of elasticity(E)as input variables.The output of GPR is failure load(Pmax),fracture energy(GF),critical stress intensity factor(KIC)and critical crack tip opening displacement(CTODC).So,x=[L,A,a,w/c,fck,σt,E]and y=[Pmax,GF,KIC,CTODC].

GPR uses the following expression for prediction of y.

where f(xi)is latent function andεiis Gaussian noise.GPR treats f(xi)as random variable.

The joint distribution of y is given by the following equation.

Where K(x,x)is kernel function and I is identity matrix.

The predictive distribution of yD+1corresponding to a new given input xD+1is given by the following expression.

Where KD+1is covariance matrix and its expression is given by

The distribution of yD+1is Gaussian with mean and variance(Williams,1998):

To develop GPR,a covariance function is required.The details of GPR is given by Williams and Rasmussen(1996).Radial basis function has been used a covariance function.

3 Least Squares Support Vector Machine(LSSVM)

SVM is a novel machine tool and especially useful for the classification and prediction with small sample cases.This novel approach motivated by statistical learning theory led to a class of algorithms characterized by the use of nonlinear kernels,high generalization of abilities and the sparseness of solution.Unlike the classical neural network approach the SVM formulation of the learning problem leads to quadratic programming with linear constraints.However the size of the matrix involved in the QP problem is directly proportional to the number of training points.Hence to reduce the complexity of optimization processes,a modified version called LS-SVM is proposed by taking equality instead of inequality constraints to obtain a linear set of equations instead of a QP problem in dual space.Instead of solving a QP problem by SVM,LS-SVM can obtain the solution of a set of linear equations.The formation of LS-SVM introduced is as follows.The following regression model can be constructed by using non-linear mapping function?(x).

where w is the weight vector and b is the bias term.By mapping the original input data into a high dimensional space,the nonlinear separable problem becomes linearly separable in space.Then the following cost function is formulated in the framework of empirical risk minimization

subject to equality constraints

where ekis the random error and gamma is a regularization parameter in determining the trade-off between minimizing the training error and minimizing the model complexity.To solve this optimization problem Lagrange function is constructed as:

where akare Lagrange multiplier.The solution of equation(9)can be obtained by partially differentiating it with respect to w,b,ekand ak

where K(x,xk)is the nonlinear kernel function.In comparison with some other feasible kernel functions,the RBF function a more compact supported kernel and is able to reduce a more computational complexity of the training process and improve generalization performance of LS-SVM.As a result RBF kernel was selected as kernel function as:

whereσis the scale factor for tuning.

4 Development of GPR and LSSVM Models

Out of the 87 data sets which are available,61 datasets are used to train the models and 26 datasets are used to test the accuracy of the models[Yuvaraj et al.(2013)].Tables 1 and 2 show the training and testing data-sets respectively.The data was normalized between 0 and 1 before being used in the model as following:

The assessment of the model is done on the basis of coefficient of regression value R which is calculated using the formula:

whereEaiandEpiare the actual and predicted values,respectively,are mean of actual and predicted E values.For an effective and good model the R value should be close to one.Also while comparing the models the values of R is compared and the model with R value closer to one and higher than the other is considered better and used.

Note:

L-length,A-c/s area,a0-Notch depth,w/c-Water-cementations material ratio,fck-compressive strength,σt-Split tensile strength,E-modulus of elasticity,Pmax-Ultimate load,GF-Fracture energy,KIC-critical stress intensity factor,CTODCCritical crack tip opening displacement.

The success of GPR depends on the value ofεand s.The design values ofεand s have been determined by trial and error approach.The best values ofεand s for each of the GPR models has been given in Table 3.

To achieve a high level performance with LS-SVM models,some parameters have to be tuned including regularization parametersγand the kernel parameter corresponding to the kernel type,i.e.σ.These parameters have been determined using trial and error approach.The best values ofγandσfor each of the LSSVM models has been given in Table 4.

5 Results and Discussions

The GPR and LSSVM models have been developed using the MATLAB software for training and prediction of the fracture characteristics of high strength and ultrahigh strength beams.The models have been trained using 61 data sets and 26 datasets are used to validate the model.The performance of GPR models for Pmax,GF,KICand CTODChas been respectively shown in Figures 1,2,3 and 4,and the same for LSSVM models has been shown in Figures 5,6,7 and 8.The values of R for training and testing data-sets of each models,and the corresponding Root mean squared error(RMSE)values has been shown in Table 5.

Table 1:Training data-sets

The developed LSSVM models give the following equations for the Prediction of

Figure 1:Performance of GPR(PmaxModel).

Figure 2:Performance of GPR(GFModel).

Figure 3:Performance of GPR(KICModel).

Figure 4:Performance of GPR(CTODCModel).

Figure 5:Performance of LSSVM(PmaxModel).

Figure 6:Performance of LSSVM(GFModel).

Figure 7:Performance of LSSVM(KICModel).

Figure 8:Performance of LSSVM(CTODCModel).

Table 2:Testing data-sets.

Table 3:Values of ε and s for the GPR models.

Pmax,GF,KICand CTODCrespectively:

Table 4:Values of γ and σ for the LSSVM models.

Table 5:Values of R and RMSE.

The correspondingαkvalues for each of the LSSVM models are shown in Figures 9,10,11 and 12.The comparison between Artificial Neural Network(ANN)Support Vector Regression(SVR),Multivariate Adaptive Regression Splines(MARS),GPR and LSSVM models for prediction of fracture parameters in terms of Correlation Coefficient(R)is shown in Figure 13.

These results prove that the GPR and LSSVM models are more accurate and reliable for the prediction of the fracture parameters of high strength and Ultra-high strength concretes.

6 Summary and Conclusions

Figure 9:Values of αkfor LSSVM(Pmaxmodel).

Figure 10:Values of αkfor LSSVM(GFmodel).

Figure 11:Values of αkfor LSSVM(KICmodel).

Figure 12:Values of αkfor LSSVM(CTODCmodel).

Figure 13:Comparison of ANN,SVR,MARS,GPR and LSSVM in terms of R.

This study shows the efficient and feasible use of GPR and LSSVM based approach for the prediction of fracture parameters of high strength and Ultra-high strength concrete mixes.Brief description has been outlines for GPR and LSSVM.Experimental data of high strength concrete and ultra high strength concrete has been used to develop and validate the GPR and LSSVM based models.Performance of the models has been verified with other popular models such as ANN,SVR and MARS models and it is found that LSSVM is one of the efficient models due to its better coefficient of correlation(R).The developed equations by LSSVM can be used by the users for determination of fracture parameters of high strength and ultra high strength concrete mixes,which in turn will be useful for remaining life prediction and residual strength of concrete structural components.

Azman,K.;Kocijan,J.(2007):Application of Gaussian processes for black-box modelling of biosystems.ISA Transactions,vol.46,pp.443–457.

Baesens,B.;Viaenel,S.;Geste,T.V.;Suykens,J.A.K.;Dedene,G.;Moor,B.D.;Vanthienen,J.(2000):An Empirical Assessment of Kernel Type Performance for Least Squares Support Vector Machine Classifiers.Fourth International Conference on knowledge-based intelligent Engineering Systems&Allied Technologies.

Barenblatt,G.I.(1959):The formation of equilibrium cracks during brittle fracture:general ideas and hypotheses,axially symmetric cracks.Appl Math Mech,vol.23,pp.622–36.

Bazant,Z.P.(2000):Size effect.Int J Solid Struct,vol.3,pp.69–80.

Dugdale,D.S.(1960):Yielding of steel sheets containing slits.J.Mech Phys Solid,vol.8,pp.100–4.

Espinoza,M.;Suykens,J.A.K.;Moor,B.D.(2003):Least Squares Support Vector Machines and Primal Space Estimation.IEEE Conference on Decision and Control Maui.

Goltermann,P.;Johansen,V.;Palbol,L.(1997):Packing of aggregates:an alternate tool to determine the optimal aggregate mix.ACI Mater J,pp.435–43.

Likar,B.;Kocijan,J.(2007):Predictive control of a gas-liquid separation plant based on a Gaussian Process model.Comp and Chem Engg,vol.31,pp.142–152

Lu,C.;Gestela,T.V.;Suykens,J.A.K.;Huffel,S.V.;Vergote,I.;Timmerman,D.(2003):Preoperative prediction of malignancy of ovarian tumors using least squares support vector machines.Artificial Intelligence in Medicine,vol.28,pp.281–306.

Mitra,V.;Wang,C.J.;Banerjee,S.(2007):Text classification:A least square support vector machine approach.Applied Soft Computing,vol.7,pp.908–914.

Pal,M.;Deswal,S.(2010):Modelling pile capacity using Gaussian process regression.Comp and Geotech,vol.37,pp.942–947.

Richard,P.;Cheyrezy,M.H.(1994):Reactive powder concretes with high ductility and 200–800 MPa compressive strength.ACI SP144-24(1),vol.144,pp.507–18.

Rebentrost,M.;Wight,G.(2009):UHPC Perspective from a Specialist Construction Company.UHPFRC 2009–November 17th&18th–Marseille,France,http://www.ductal-lafarge.com/SLib/21_Rebentrost_Wight.pdf.

Richard,P.;Cheyrezy,M.H.(1995):Composition of reactive powder concretes.Cem Concr Res,vol.25,no.7,pp.1501–11.

Seibert Perry.(2008):The use of UHPFRC(Ductalr)for bridges in North America:the technology,applications and challenges facing commercialization.In:International symposium on UHPC,Kassel,Germany;.

Suykans,J.A.K.;Vandewalle,J.(1999):Least Squares Support Vector Machine Classi fi ers.Neu Proc Let,vol.9,pp.293-300.

Yu,Z.;Sun,M.;Zheng,S.;Liang,L.(2010):Fatigue properties of RPC under cyclic loads of single-stage and multi-level amplitude.J.Wuhan Univ Technol,pp.167–73.

Yuan,J.;Wang,K.;Yu,T.;Fang,M.(2008):Reliable multi-objective optimization of high-speed WEDM process based on Gaussian process regression.Intl J Mach Tools&Manuf,vol.48,pp.47–60

Yuvaraj,P.;Murthy,A.R.;Iyer,N.R.;Samui,P.;Sekar,S.K.(2013):Mul-tivariate Adaptive Regression Splines Model to Predict Fracture Characteristics of High Strength and Ultra High Strength Concrete Beams.CMC:Comp Mat Cont,vol.36,no.1,pp.73-97.

Yuvaraj,P.;Ramachandra Murthy,A.;Iyer,N.R.;Sekar,S.K.;Samui,P.(2013):Support vector regression based models to predict fracture characteristics of high strength and ultra high strength concrete beams.Engineering Fracture Mechanics,vol.98,pp.29-43.

Yuvraj,P.;Ramachandra Murthy,A.;Nagesh R.Iyer,Sekar,S.K.and Pijush Samui.(2012):Application of Artifical Neural Network to predict fracture characteristicsn of high strength and ultra high strength concrete beams.2nd International Conference on Advances in Mechanical,Manufacturing and Building Sciences(ICAMB),pp.1129-1136.