ANN based Performance Evaluation of Fiber Reinforced Concrete Beams Incorporating Nano Fillers and Micro Fillers

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ANN based Performance Evaluation of Fiber Reinforced Concrete Beams Incorporating Nano Fillers and Micro Fillers

  1. Meenatchi, P. N. Raghunath, K. Suguna

    Department of Civil and Structural Engineering, Annamalai University, India

    Abstract:- An attempt has been made in this study to evaluate the performance of fiber reinforced concrete beams incorporating nano fillers and micro fillers using the neural computational tool ANN. The test data necessary for this exercise were collected from published literature. MATLAB software has been used for this purpose. Back propagation network with Levenberg Marquardt Algorithm, Bayesian Regularization Algorithm and Scaled Conjugate Gradient Algorithm was chosen for the proposed study .The length, breadth, depth, Ast, Asv, fck and fy were considered as input parameters and first crack load, deflection at first crack load, yield load, deflection at yield load, ultimate load and deflection at ultimate load were considered as target parameters. In the present study comparison has been made between the test results from published literature and results predicted from ANN using different algorithms. Statistical indicators such as RMSE, R² and MAPE were found to estimate the accuracy of results predicted through ANN modeling. The results predicted through ANN modeling exhibit better convergence with the experimental results collected through published literature.

    Key Words: ANN, Deflection, Fiber Reinforced Concrete, MATLAB.

    1. INTRODUCTION

      Fiber reinforcement is introduced into concrete with a view to enhance the ductility of concrete and also to impart increased resistance to fatigue and impact. The intent of adding nano-fillers and micro-fillers in concrete is to improve the performance of concrete under a wide variety of environmental conditions. Fiber reinforced concrete (FRC) is concrete containing fibrous material which increases its structural integrity. It contains short discrete fibers that are uniformly distributed and randomly oriented. Fibers include steel fibers, glass fibers, synthetic fibers and natural fibers. Fiber-reinforced normal concrete are mostly used for on-ground floors and pavements, but can be considered for a wide range of applications (beams, pliers, foundations etc). some of the important factors of fiber reinforced concrete are (1).increases the tensile strength of the concrete, improved structural strength(2).increases the durability of the concrete (3). closely spaced and uniformly dispersed fibers to concrete would act as crack arrester (4). Improved freeze-thaw resistance.

    2. LITERATURE REVIEW

      Manar A.Ahmed et al (2017) investigated the structural performance of reinforced concrete beams with nano-metakaolin in shear. Seven beam specimens of size 2000mmx100mmx200mm were cast and tested at 28 days. Nanometakaolin was incorporated at 10%. The fracture mechanism was studied by focusing on crack and deflection behavior for beams under static loading. Different values of shear reinforcement ratio, shear span to depth ratio and characteristic strength were considered. The experimental result showed that the ultimate load, cracking load and toughness of the beam with 10% NMK as portion of cement increased by 7.4%, 31.5% and 17.6% respectively. The addition of nano- metakaolin significantly improved the performance of concrete mixes.

      Stephen Jebamalai Raj.J (2013) investigated the static and fatigue response of high strength fiber reinforced concrete beam with FRP laminates. An analytical model was proposed for predicting the performance parameters of HSFRC beams. Modeling and analysis were done using the MSC/NASTRAN – PATRAN software. The analysis results showed that the deflection of HSFRC beams was lesser than the experimental results. The presence of steel fibers reduced the deflection more in FEA analysis.

      In Hwan Yang et al (2010) investigated the structural performance of ultra high performance concrete beams subjected to bending. UHPC does not include coarse aggregate and had steel fibers with volumetric ratio of 2%. Amount of rebar and placing method of UHPC were considered. The experimental results showed the ability of UHPC to redistribute stresses and undergo multiple cracks before fiber pullout. Placing UHPC at end of beams provides better structural performance than placing UHPC at mid-span. In addition UHPC beams exhibit a better post – cracking behavior.

      Doo Yeol Yoo et al (2015) studied the structural performance of ultra high performance concrete beams with different steel fibers. UHPC containing no coarse aggregate with two different reinforcement ratios. Two different fibers (smooth and twisted) and three different length (Lf = 13,19.5 and 30mm) were used. Control specimen made up of UHPC matrix without fiber were also fabricated and tested. The experimental results showed that addition of steel fibers slightly increased the compressive

      strength, elastic modulus and significantly improved the flexural performance. The beams without fibers failed by crushing of concrete at compressive zone whereas beams with steel fibers failed by rupture of steel rebars.

      S.Syed Ibrahim et al (2016) investigated the structural performance of glass fiber reinforced polymer laminated steel fiber reinforced concrete beams. Strengthening of RC beams was done by introducing short steel fibers and externally bonded GFRP laminates on tension face of the beam. Two groups of beams were cast and tested. Group A had RC beams with and without externally bonded GFRP laminates and in group B had SFRC beams with and without externally bonded GFRP laminates. The experimental results showed that the maximum increase in ultimate load was upto 130%, 27.8% and 91.9% and reduction in deflection to the tune of 27.40% when compared to SFRC beams.

      Chao Xu et al (2019) studied the behavior of composite steel – concrete beams containing different amounts of steel fibers and conventional reinforcement. The steel cages in steel reinforcement concrete beams were replaced by steel fibers. Several steel fiber volume were designed and used to improve the mechanical properties of concrete. The combination of proper amount of steel fiber and steel had better bearing capacity and deformability of composite members. Shear span to depth ratio, web reinforcement, with or without longitudinal reinforcement were considered. The experimental results showed that addition of steel fibers improved the mechanical response, flexural and shear strength and ductility. The stitching effect of steel fibers between cracks lead to stable bearing capacity of specimens. Steel fibers as alternative for shear reinforcement and longitudinal reinforcement greatly enhanced the bearing capacity. Increasing the content of steel fibers delayed the bond failure.

      Venkatesan.K.R et al (2015) investigated the flexural behavior of high strength steel fiber reinforced concrete beams. Beam specimens of size 3000mmx 150mmx250mm were constructed and tested. The authors attempted different steel fiber volume fractions of 0.5%, 1.0% and 1.5%. The stress transfer mechanism of failure caused significant improvement in ductility, toughness, impact resistance, tensile and flexural strength, shrinkage, fatigue life, abrasion resistance, and durability and cavitation resistance. The optimum fiber volume fraction was 1% and the beams failed in flexure mode only. SFRC beams with 1% fiber volume fraction has increased ultimate load by 30.3% and increase in ductility was 41.34%.

      Ravichandran.R.S (2012) investigated the flexural behavior of composite high strength concrete fiber reinforced polymer beams. The beam specimens f size 3000x150x250mm were tested. Reinforcement ratio, GFRP laminate material and their thickness were considered. The beams were strengthened with chopped strand mat glass fiber reinforced polymer and uni- directional cloth glass fiber reinforced polymer of 3mm and 5mm thickness. The experimental results showed an increase in ultimate load by about 23.51%, 88.22%, and increase in deflection at ultimate stage by about 15.85% to 211.59% for GFRP strengthened beams. GFRP strengthened beams failed in flexure modes only.

      Mohamed K.Ismail et al (2017) investigated an experimental study on flexural behavior of large scale concrete beams incorporating crumb rubber and steel fibers. Percentage of CR (0 -35% by volume of sand), volume of SFs (0%, 0.35% and 1%) and length of SFs (35mm and 60mm) were considered. The experimental results showed that increase in CR appeared to narrow the crack widths, reduce self weight and improve deformability at given loads. Increase in CR content lead to higher reduction in ultimate load, ductility and toughness of SCRC and VRC beams.

      Deepthi Dennison et al (2014) investigated the effect of metakaolin on the structural behavior of normal and steel fiber reinforced concrete beams. Compressive strength, split tensile strength, flexural strength and modulus of elasticity of concrete with various percentage of metakaolin 0%,5%, 7.5% by weight of cement were considered. The optimum metakaolin crimped steel fiber reinforced concrete 1.5%, 2% and 2.5% by volume of cement were also considered. The experimental results showed that load deflection pattern of metakaolin crimped steel fiber reinforced concrete beams showed greater load carrying capacity and deflection than that of control specimens in flexure and shear due to the effective bridging action of CSF. RC beams strengthened with MK and CSF showed a better structural performance in terms of first crack load, load deflection response, ultimate load carrying capacity, midspan deflection, and energy absorption and ductility index.

      Halit Cenan Mertol at al (2015) investigated the flexural behavior of lightly and heavily reinforced steel fiber concrete beams. The beams specimen of size 3500mmx180mmx250mm were constructed and tested. Ten different longitudinal reinforcement ratios with minimum 0.2% and maximum 2.5% covering the range from under reinforced beam to over reinforced beams were used for testing. Two specimens were cast for each longitudinal ratio, one specimen using conventional concrete (CC) and another specimen using steel fiber reinforced concrete (SFRC). Load deflection behaviors were obtained and evaluated in terms of ultimate load, ultimate deflection, service stiffness, post peak stiffness, and flexural toughness. The results indicate that the use of SFRC increased ultimate load and service stiffness of beams slightly compared to CC specimens. As reinforcement increased the ultimate deflection of SFRC specimens has significantly improved than that of CC specimens. For over reinforced sections, the post peak stiffness of SFRC specimens was observed to be significantly lower than that of CC specimens. The flexural toughness of SFRC specimens was greater than that of CC specimens with the difference being significantly larger over reinforced sections.

      Daniel C.T.Cardoso et al (2019) studied the influence of steel fibers on the flexural behavior of RC beams with low reinforcing ratios. The beam specimens size of 1200mm x150mmx150mm are cast and tested. Hooked end steel fibers with aspect ratios of 45or 80 were used to produce matrices with fiber content ranging from 0 to 2 % in volume and high compressive strength. Beams with reinforcing ratios 0.28, 0.44 and 0.70% were analyzed using digital image correlation to monitor displacement field and cracking distribution throughout the constant moment region during loading. The experimental results showed that the capacity ranges from 21 to 109% of SFRC with respect to plain RC beams and significant reduction in crack openings for given rebar stress. SFRC beams exhibited sufficient plastic rotation capacity, but with reduction of plastic hinge length as a consequence of crack localization.

      Masoud Ghahremannejad et al (2018) studied the experimental investigation and identification of single and multiple cracks in synthetic fibers concrete beams. Synthetic fiber volume fractions of 0.5% and 1% were used to prepare the specimen and results compared with concrete beams with no synthetic fibers. The beams specimen size of 1143mmx152mmx229mm were cast and tested. Six reinforced and unreinforced beams with flexure behavior were considered. The digital image correlation method was utilized to record the width, spacing, number and locations of the cracks for all the specimens during displacement control type loading. The results indicated that using 1% synthetic fiber increased the failure load of the reinforced concrete beams and improved serviceability by reduced the number of cracks and width of cracks. When the cracks opened significantly, the beams with 1% fiber dosage were able to carry higher loads due to bridging action of the fibers at the cracks locations.

      Alberto Meda et al (2012) investigated the flexural behavior of RC beams in fiber reinforced concrete . Seven beams specimen with size of 3600mmX200mmX300mm were cast and tested at 28 days. The three fiber contents, two longitudinal reinforcement ratios, either bonded or unbounded were considered. Experimental results proved that fibers, when provided in sufficient amount, are able to move the beam failure from concrete crushing to steel rupture. On the other hand, in all cases the addition of fibers determines a stiffer and in general enhanced post- cracking behavior in service conditions.

      Holschemacher.K et al (2010) investigated the effect of steel fibers on mechanical properties of high strength concrete. SFRC is tough and high residual strength after appearing first crack. The beams specimens with size of 700mmX150mmX150mm were cast and tested at 28 days. The different bar reinforcements (26mm and 212mm) and three types of fibers configurations ( two straight with end hooks with different ultimate tensile strength and one corrugated) were used. Three different fibers contents were applied. The selected fibers content a more ductile behavior and higher load levels in the post cracking range were obtained.

      Hamid Pesaran Behbahani et al(2012) investigated the flexural behavior of steel fiber added RC beams with C30 and C50 classes of concrete. The discrete and short steel fibers with optimum percentage of hooked end and the dimensions of 0.75mm in diameter, 50mm length were added in RC beams with two different classes of concrete. The experimental results proved that SFARC beams with 1% by volume of SFs have higher first cracking strength, ultimate flexural strength, stiffness and ductility. Samir A.Ashour et al (2000) investigated the effect of the concrete compressive strength and tensile reinforcement ratio on the flexural behavior of fibrous concrete beams. A total twenty seven beams specimen with size of 200mmX250mmX3400mm were cast and tested after 28 days. Concrete compressive strength of 49, 79 and 102 MPa and tensile reinforcement ratios of 1.18, 1.77 and 2.37% were used. The fibers contents were 0.0, 0.5 and 1.0% by volume. The results proved that the additional moment strength provided by fibers were not affected by the amount of tensile reinforcement ratio. The flexural rigidity increases as the concrete compressive strength and steel fiber content increases.

      Olivito R.S et al (2010) investigated an experimental study on the tensile strength of steel fiber reinforced concrete. Different mixtures were prepared varying both mix design and fiber length. Fibers contents in volume were 1% and 2% . Mechanical characterization were performed by means of uni axial compression test with the aim of deriving the ultimate compressive strength of fiber concrete. Four point bending test on notched specimens were carried out to derive the first crack strength and ductiliy indexes.

      Lampropoulos.A.P et al (2016) investigated the strengthening of reinforced concrete beams using ultra high performance fiber reinforced concrete. Numerical model were developed by using finite element method. The reliability of numerical model has been conducted on full scale beams strengthened with ultra high performance fiber reinforced concrete layers and jackets and these results were compared to respective results of beams strengthened with conventional RC layers and with combination of ultra high performance fiber reinforced concrete and steel reinforcing bars.

      Zealakshmi.D et al (2016) investigated the flexural behavior of confined hybrid fiber in the plastic hinging region of the hi gh strength concrete beams. Steel and polypropylene fiber with varying volume fractions only at the critical sections along with stirrup confinement to improve the flexural behavior and ductility of a high strength concrete beam. A total 13 beam specimens with size of 120mmX200mmX2100mm were designed as an over reinforced section by providing high percentage of steel more than balanced state. Out of 13 beams one beam kept as reference beam without fiber reinforcement, four beams were designed as steel fiber and remaining with hybrid fibers. All the beams are tested under three point loadings and deflections were measured at respective places using dial gauges. The experimental results concluded that beams improves the flexural behavior, load carrying capacity and ductile manner.

      Soulioti D.V et al (2011) investigated the Effect of fiber geometry and volume fraction on the flexural behaviour of steel fiber reinforced concrete. The flexural toughness, flexural strength and residual strength factors of the beam specimens were evaluated in accordance with ASTM standard. The experimental results proved that the improvement in mechanical properties in particular the toughness, were observed with the increase of the volume fraction of steel fibers in the concrete.

      Goran H.Mahmud et al (2013) investigated the Experimental and numerical studies of size effects of ultra high performance steel fiber reinforced concrete beams. The size effects on flexural strength of similar notched UHPFRC beams under three point bending tests were conducted. The experimental results found that the size effects on the beam nominal strength are little due to high ductility of UHPFRC.

      Mehmet Ozcan et al (2009) investigated the experimental and finite element analysis on the steel fibre reinforced concrete beams ultimate behavior. Three SFRC beams with 250mmX300mmX2000mm dimensions are produced using a concrete class of C20 with 30kg/m3 dosage of steel fibres and steel class S420 with shear stirrups. SFRC beams are subjected to bending by a four point loading setup in certified beam loading frame, exactly after having been moist cured for 28 days. The test are with control of loads. The beam are loaded until they are broken and the loadings are stopped when the tensile bars are broken into two pieces. Applied loads and mid section deflections are carefully recorded at every 5kN loads increment from the beginning till the ultimate failure.

      Siva Chidambaram.R et al (2019) investigated the Flexural behavior of reinforced concrete beams with high performance fibre reinforced cementitious composites. Beam specimens with moderate confinement were used and tested under monotonic loading. Seven diverse types of FRCC including hybrid composites using fibres in different profiles and in different volumes. The moment curvature, stiffness, behavior, ductility, crack pattern and modified flexural damage ratio were considered. The experimental outputs demonstrates the improved post yield behavior with less rate of stiffness degradation and better damage tolerance capacity than conventional technique.

      Su Tae Kang (2012) investigated the flexural behavior of UHPCC considering the effect of fibre orientation distribution. A three point bending test were carried out and the fibre orientation were quantitatively estimated by the help of image analysis process. The measured fibre orientation distributions for two different flexural performance confirmed that the fibre orientation distribution have strong impact on the deflection hardening behavior in bending. Finite element analysis were performed to predict flexural behavior of UHPCC considering the difference in fibre bridging behavior depending on the fibre orientation distribution.

      Lokeshwaran.N et al (2017) investigated the flexural resistance of self compacted concrete beams by partial replacement of OPC with GGBS. Self compacting concrete were considered using GGBS by replacing the cement with 10%, 20%,30%,40% and 50% by weight. The mechanical properties of SCG (GGBS incorporated SCC) were found to be increased compared to concrete. Six reinforced concrete beams of shear span to depth ratio were tested for flexural capacity and ductile behavior. The experimental cracking moment of SCGB beams were found to be more than the theoretical cracking moment enhancing its flexural resistance. SCGB beams with higher percentage of GGBS exhibits higher ductility.

      Julita Krassowska et al (2019) investigated the shear behavior of two span fibre reinforced concrete beams. The beams had varied stirrup spacing and two sorts of fibre were used as dispersed reinforcement. The steel fibre content were 78.5kg/m3 and the basalt fibre content were 5kg/m3. Concrete beams without addition of fibres were also examined as reference one. The effectiveness of both sorts of fibres as shear reinforcement were assessed on the basis of strain development and crack pattern analysis. The digital image correlation technique were used to monitor the development of cracks around the central supports of beam. It were shown that fibres control the cracking process and deformation in reinforced concrete beams and they can be effectively used as additional or the only shear reinforcement.

      Faith Altun et al (2013) investigated the reinforced concrete beams behavior of steel fibre added lightweight concrete. Light weight concrete having a lower module of elasticity, a faster rate of crack development in RC members. The steel fibres are employed as an additive to the concrete in order to increase the energy absorption capacity and to control the crack development. Light weight concrete and reinforced concrete specimens were produced with the addition of steel fibres in different strength and ratios. The specimens were tested on four points through loading experiment. The experimental results found that the addition of steel fibre increased the toughness capacity, ductility and bearing strength.

      Praveenkumar.S (2019) investigated the behavior of high performance fibre reinforced concrete composite beams in flexure. The blend mix included both free STF (Q series) and PPF (R series) and the hybridization of STF and PPF (S series) at a total volume fraction of 1% by volume of concrete with 10% bagasse ash as a substitution of cement. Structural behavior of eleven bagasse ash blended high performance concrete beams reinforced with steel, polypropylene and hybrid fibres were examined. The behavior of each beam were assessed with respect to initial crack, ultimate load, ultimate deflection, flexural strength, ductility and toughness. The inclusion of fibres increased the failure load and ensure the ductility behavior of beams. The experimental results found that adding hybrid fibres enhanced the mechanical properties as well as structural behavior of the beams.

      Doo Yeol Yoo et al(2014) investigated the material and bond properties of ultra high performance fibre reinforced concrete with micro steel fibres. Four different volume ratios of micro steel fibres (Vf = 1%, 2%, 3% and 4%) were used within an identical mortar matrix. The experimental results showed that 3% steel fibre by volume yielded the best performance in terms of compressive strength, elastic modulus, shrinkage behavior, and interfacial bond strength. These parameters improved as the fibre content were increased upto 3%. Flexural behavior such as flexural strength, deflection and crack mouth opening displacement at peak load have pseudo linear relationships with the fibre content.

      Silva.J.V et al (2016) investigated the influence of nano sio2 and nano Al2o3 additions on the shear strength and the bending moment capacity of RC beams. Concrete mixtures with additions of either silica nano particles( nano sio2)or aluminium nano particles (nano Al2o3) were produced. An experimental program were organized in two different sets: eight beams tested until shear failure were reached and eight beams tested until failure in bending were observed. These beams were produced with

      eight distinct concrete mixtures, with and without nano particles and with and without steel fibres. The experimental results found that the addition of nano particles, mainly nano Al2o3 to the concrete mixtures can lead to a strength increase, both in shear and in bending and also concluded that the interaction between the nano particles and steel fibres have a negative effect on the beam strength.

    3. MATERIALS AND METHODS

        1. Artificial Neural Network(ANN)

          Artificial Neural Networks are software implementations of the neural structures of human brain. ANN is a computational system influenced from the structure, processing capability and learning ability of a human brain. Let us take a look at the structure of our brain. Human brain contains billions of neurons that act as organic switches. All these neurons are interconnected to form a huge and complex structure called Neural Network. The output of a single neuron is dependent on inputs from thousands of interconnected neurons. the Learning of a human brain is simply repeated activation of certain neural connections and this repetition strengthens the connection. So, for a specified input, the neural connections make sure that output is always a desired one. A simple feedback from the outcome helps the learning process as it strengthens the neural connections. ANN are composed of multiple nodes, which imitate biological neurons of human brain. The neurons are connected by links and they interact with each other. The nodes can take input data and perform simple operations on the data. The result of these operations is passed to other neurons. The output at each node is called its activation or node value.Each link is associated with weight. ANN are capable of learning, which takes place by altering weight values. The simple ANN modeling shown in fig.1.

          Figure 1. Simple Artificial Neural Network

        2. Working of Artificial Neural Networks

          A neural network combines several processing layers, using simple elements operating in parallel and inspired by biological nervous systems. It consists of an input layer, one or more hidden layers, and an output layer. The layers are interconnected via nodes, or neurons, with each layer using the output of the previous layer as its input. The working diagram of ANN shown in fig.2.

          Figure 2. Neural Networks

        3. Elements of Artificial Neuron Networks

          The basic elements of any Artificial Neural Network as follows:

          • Processing Elements

          • Topology

          • Learning Algorithm

          Processing Elements

          ANN is a simplified computational model of a biological neural network, an ANN consists of basic processing units or elements similar to that of neurons of a brain in general, a processing unit is made up of summing unit followed by an output unit. The function of a summing unit is to taken input values, weight each input value and calculate the weighted sum of those values. The processing elements shown in fig.3

          Figure 3. Processing Elements

          Topology

          Any Artificial Neural Network will become useful only when all the processing elements are organized in an appropriate manner so that they can accomplish the task of pattern recognition. This organization or arrangement of the processing elements, their interconnections, inputs and outputs is simply known as Topology. Generally, in an ANN, the processing units are arranged into layers and all the units in a particular layer have the same activation values and output values. Connection can be made between layers in multiple ways like processing unit of one layer connected to a unit of another layer, processing unit of a layer connected to a unit of same layer, etc. The toplogy diagram shown in fig.4

          Figure 4. Topology

          Learning Algorithm

          The final and important elements of any ANN are Learning Algorithms or Laws. The operation any neural network is governed by Neural Dynamics consisting of both activation state dynamics and synaptic weight dynamics. Learning Algorithms or Laws are implementations of synaptic dynamics and are described in terms of first derivative of the weights. These leaning laws can be supervised, unsupervised or a hybrid of both.

        4. Data used in ANN modeling

          The geometrical properties of beam such as length (L), breadth (B) and depth (D), characteristic compressive strength of concrete (fck), yield stress (fy), diameter of reinforcement (Ast) and diameter of stirrups (Asv) were considered as the input parameters. The experimental results of all the test beams such as first crack load, deflection at first crack load, yield load, deflection at yield load, ultimate load and deflection at ultimate load were considered as the target parameters. The input and target parameters for ANN modeling are presented through Table 1 and 2.

        5. Steps involved in ANN modeling

          In this stage, the input data are divided into three groups which are train data, validate data and test data. The step wise procedure for ANN modeling is presented through the fig

          1. Designing of ANN modeling

            The basic flowchart for the designing of neural network. The flow chart are shown in fig.5.

            Access and prepare the data

            Create the neural network

            Configure the networks inputs and outputs

            Tune the network parameters

            Train the network

            Validate the networks results

            Integrate the network

            Figure 5. Flow chart for ANN mdeling

          2. Building the network

            This stage specifies the number of hidden layers, neurons in each layers, transfer function in each layers, training function, learning function and performance function as shown in fig.6

            Figure.6 Architecture of neural network

          3. Training the network

            The training process are shown in fig 7 and 8

            Figure 7. Training Networks Wizard

            Figure 8. Neural Network Testing

          4. Test performance of model

      The fitness of the developed model is show through fig 9 to 12.

      Figure 9. Neural Network Regression Plot

      Figure.10 Neural Network Training Performance Plot

      Figure 11. Neural Network Training State

      Figure 12. Neural Network Error Histogram State

    4. REGRESSION ANALYSIS

      Regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Regression analysis is also used to understand which among the independent variables are related to the dependent variables, and to explore the forms of these relationships. In restricted circumstances, regression analysis can be used to infer causal relationship between the independent and dependent variables. Many techniques for carrying out regression analysis have been developed. Familiar methods such as linear regression and ordinary least squares regression are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data. Non parametric regression refers to techniques that allows the regression function to lie in a specified set of functions, which may be infinite dimensional.

      The performance of regression analysis methods in practice depends on the form of the data generating process, and how it relates to the regression approach bing used. Since the truth form of the data generating process is generally not known, regression analysis often depends to some

      extent on making assumptions about this process. Regression models for prediction are often useful even when the assumptions are moderately violated, although they may not perform optimally. However, in many applications, especially with small effects or questions of causality based on observational data, regression methods can give misleading results.

        1. Regression equation for fibre reinforced concrete beams incorporated micro fillers and nano fibres

          The coefficient of regression parameter, statistics and ANOVA for fibre reinforced concrete beams incorporating nano fillers and micro fillers calculated through ORGIN PRO are presented in Tables 3(a),3(b) and 3(c) to 8(a),8(b) and 8(c).

          First Crack Load

          Table 3(a) Parameters of First Crack Load

          First Crack Load

          Values

          Standard Error

          Intercept

          48.51111

          9.7267

          b in mm

          -0.09309

          0.03302

          d in mm

          0.02161

          0.02704

          l in mm

          0.00966

          0.00188

          Fck

          0.05932

          0.02795

          Fy

          -0.02651

          0.01873

          Ast in mm2

          0.026

          0.00588

          Asv in mm2

          -0.55841

          0.10228

          Table 3 (b ) Statistics for First Crack Load

          First Crack Load

          Number of Points

          379

          Degree of Freedom

          371

          Residual Sum of Squares

          195805.04422

          Adj. R-Square

          0.21525

          Table 3 (c) ANOVA Table for First Crack Load

          First Crack Load

          DF

          Sum of Squares

          Mean Square

          F Value

          Prob >F

          Model

          7

          58416.43285

          8345.20469

          15.81201

          0

          Error

          371

          195805.04422

          527.7764

          Total

          378

          254221.47707

          Deflection at First Crack Load

          Table 4 (a) Parameters of Deflection at First Crack Load

          Deflection at First Crack Load

          Values

          Standard Error

          Intercept

          88.97371

          80.91538

          b in mm

          -0.12249

          0.27468

          d in mm

          0.4449

          0.22493

          l in mm

          -0.02356

          0.01563

          Fck

          0.17406

          0.23255

          Fy

          -0.10067

          0.15577

          Ast in mm2

          0.03318

          0.04894

          Asv in mm2

          -1.35944

          0.85082

          Table 4 (b ) Statistics for Deflection at First Crack Load

          Deflection at First Crack Load

          Number of Points

          379

          Degree of Freedom

          371

          Residual Sum of Squares

          1.35505E7

          Adj. R-Square

          -0.00309

          Table 4 (c) ANOVA Table for Deflection at First Crack Load

          Deflection at First Crack Load

          DF

          Sum of Squares

          Mean Square

          F Value

          Prob >F

          Model

          7

          213159.58759

          30451.36966

          0.83373

          0.55978

          Error

          371

          1.35505E7

          36524.25616

          Total

          378

          1.37637E7

          Yield Load

          Table 5 (a) Parameters of Yield Load

          Yield Load

          Values

          Standard Error

          Intercept

          76.40433

          14.66513

          b in mm

          -0.37179

          0.04978

          d in mm

          0.46918

          0.04077

          l in mm

          -0.01128

          0.00283

          Fck

          0.22926

          0.04215

          Fy

          -0.10195

          0.02823

          Ast in mm2

          0.03677

          0.00887

          Asv in mm2

          -0.26065

          0.1542

          Table 5 (b ) Statistics for Yield Load

          Yield Load

          Number of Points

          379

          Degree of Freedom

          371

          Residual Sum of Squares

          445107.76998

          Adj. R-Square

          0.33189

          Table 5 (c) ANOVA Table for Yield Load

          First Crack Load

          DF

          Sum of Squares

          Mean Square

          F Value

          Prob >F

          Model

          7

          233683.77525

          33383.39646

          27.82526

          0

          Error

          371

          445107.76998

          1199.7514

          Total

          378

          678791.54522

          Deflection at Yield Load

          Table 6 (a) Parameters of Deflection at Yield Load

          Deflection at Yield Load

          Values

          Standard Error

          Intercept

          6.41907

          6.26723

          b in mm

          -0.05275

          0.02128

          d in mm

          0.02298

          0.01742

          l in mm

          0.0039

          0.00121

          Fck

          0.01455

          0.01801

          Fy

          0.00967

          0.01207

          Ast in mm2

          0.0171

          0.00379

          Asv in mm2

          -0.09324

          0.0659

          Table 6 (b ) Statistics for Deflection at Yield Load

          Deflection at Yield Load

          Number of Points

          379

          Degree of Freedom

          371

          Residual Sum of Squares

          81291.30095

          Adj. R-Square

          0.13423

          Table 6 (c) ANOVA Table for Deflection at Yield Load

          Deflection at Yield Load

          DF

          Sum of Squares

          Mean Square

          F Value

          Prob >F

          Model

          7

          14374.99422

          2053.5706

          9.37216

          1.04021E-10

          Error

          371

          81291.30095

          219.11402

          Total

          378

          95666.29518

          Ultimate Load

          Table 7(a) Parameters of Ultimate Load

          Ultimate Load

          Values

          Standard Error

          Intercept

          65.16612

          21.38557

          b in mm

          -0.1173

          0.0726

          d in mm

          0.44043

          0.05945

          l in mm

          -0.02201

          0.00413

          Fck

          0.23504

          0.06146

          Fy

          -0.1468

          0.04117

          Ast in mm2

          0.01257

          0.01294

          Asv in mm2

          0.91816

          0.22487

          Table 7 (b ) Statistics for Ultimate Load

          Ultimate Load

          Number of Points

          379

          Degree of Freedom

          371

          Residual Sum of Squares

          946530.48175

          Adj. R-Square

          0.27466

          Table 7(c) ANOVA Table for Ultimate Load

          Ultimate Load

          DF

          Sum of Squares

          Mean Square

          F Value

          Prob >F

          Model

          7

          383041.68221

          54720.24032

          21.44802

          0

          Error

          371

          946530.48175

          2551.2951

          Total

          378

          1.32957E6

          Deflection at Ultimate Load

          Table 8(a) Parameters of Deflection at Ultimate Load

          Deflection at Ultimate Load

          Values

          Standard Error

          Intercept

          27.16152

          10.88859

          b in mm

          0.22081

          0.03698

          d in mm

          -0.08007

          0.03032

          l in mm

          0.02168

          0.00212

          Fck

          0.09975

          0.03129

          Fy

          -0.08273

          0.02098

          Ast in mm2

          -0.00939

          0.00661

          Asv in mm2

          -0.37328

          0.11445

          Table 8(b ) Statistics for Deflection at Ultimate Load

          Deflection at Ultimate Load

          Number of Points

          377

          Degree of Freedom

          369

          Residual Sum of Squares

          243873.84226

          Adj. R-Square

          0.34536

          Table 8 (c) ANOVA Table for Deflection at Ultimate Load

          Deflection at Ultimate Load

          DF

          Sum of Squares

          Mean Square

          F Value

          Prob >F

          Model

          7

          135724.96018

          19389.28003

          29.33748

          0

          Error

          369

          243873.84226

          660.90472

          Total

          376

          379598.80244

          INPUT PARAMETERS AND TARGET PARAMETRS FOR FIBER REINFORCED CONCRETE BEAMS

          Table 4.1 Input Parameters for ANN based Modeling of Fiber Reinforced Concrete Beams with Incorporating Nano fillers and Micro fillers

          Ref

          Test Specimen

          b in mm

          d in mm

          l in mm

          fck N/mm²

          Fy

          Ast mm²

          Asv mm²

          [27]

          B1

          100

          200

          2000

          25

          415

          226.19

          50.265

          B2

          100

          200

          2000

          25

          415

          226.19

          50.265

          B3

          100

          200

          2000

          25

          415

          226.19

          50.265

          B4

          100

          200

          2000

          25

          415

          226.19

          50.265

          B5

          100

          200

          2000

          25

          415

          226.19

          50.265

          B6

          100

          200

          2000

          25

          415

          226.19

          50.265

          B7

          100

          200

          2000

          25

          415

          226.19

          50.265

          [6]

          NF

          150

          220

          2500

          200.9

          510

          198.6

          71.3

          NF

          150

          220

          2500

          200.9

          510

          126.7

          71.3

          S13

          150

          220

          2500

          211.8

          495

          198.6

          71.3

          S13

          150

          220

          2500

          211.8

          495

          126.7

          71.3

          S19.5

          150

          220

          2500

          209.7

          491

          198.6

          71.3

          S19..5

          150

          220

          2500

          209.7

          491

          126.7

          71.3

          S30

          150

          220

          2500

          209.7

          491

          198.6

          71.3

          S30

          150

          220

          2500

          209.7

          491

          126.7

          71.3

          T30

          150

          220

          2500

          232.1

          491

          198.6

          71.3

          T30

          150

          220

          2500

          232.1

          491

          126.7

          71.3

          [18]

          CC

          230

          250

          2300

          75.9

          477

          64

          71

          CC-SN

          230

          250

          2300

          89.3

          477

          64

          71

          CC-ST

          230

          250

          2300

          104.4

          477

          64

          71

          GG

          230

          250

          2300

          75.9

          477

          127

          71

          GG-SN

          230

          250

          2300

          89.3

          477

          127

          71

          GG-ST

          230

          250

          2300

          104.4

          477

          127

          71

          [44]

          S0L0

          150

          250

          3000

          26.65

          500

          226.194

          50.265

          S0L5

          150

          250

          3000

          26.65

          500

          226.194

          50.265

          S1L0

          150

          250

          3000

          26.65

          500

          226.194

          50.265

          S1L5

          150

          250

          3000

          26.65

          500

          226.194

          50.265

          [4]

          BY-0-2.5

          140

          180

          1600

          27.6

          486

          157.079

          33.183

          BY-1-2.5

          140

          180

          1600

          27.5

          486

          157.079

          33.183

          BY-2-2.5

          140

          180

          1600

          28.4

          486

          157.079

          33.183

          BX-1-2.5

          140

          180

          1600

          27.5

          486

          157.079

          33.183

          BX-2-2.5

          140

          180

          1600

          28.4

          486

          157.079

          33.183

          BX-1-2.5-360

          140

          180

          1600

          27.5

          486

          157.079

          33.183

          BX-2-2.5-360

          140

          180

          1600

          28.4

          486

          157.079

          33.183

          BX-1-2.5-180

          140

          180

          1600

          27.5

          486

          157.079

          33.183

          BX-1-2.5-180

          140

          180

          1600

          28.4

          486

          157.079

          33.183

          BY-0-3.5

          140

          180

          2000

          27.6

          486

          157.079

          33.183

          BY-1-3.5

          140

          180

          2000

          27.5

          486

          157.079

          33.183

          BY-2-3.5

          140

          180

          2000

          28.4

          486

          157.079

          33.183

          BX-1-3.5

          140

          180

          2000

          27.5

          486

          157.079

          33.183

          BX-2-3.5

          140

          180

          2000

          28.4

          486

          157.079

          33.183

          BX-1-2.5-360

          140

          180

          2000

          27.5

          486

          157.079

          33.183

          BX-2-2.5-360

          140

          180

          2000

          28.4

          486

          157.079

          33.183

          BX-1-3.5-180

          140

          180

          2000

          27.5

          486

          157.079

          33.183

          BX-2-3.5-180

          140

          180

          2000

          28.4

          486

          157.079

          33.183

          [46]

          M60

          150

          250

          3000

          67

          537

          339.29

          50.265

          M60-0.5%

          150

          250

          3000

          67

          537

          339.29

          50.265

          M60-1.0%

          150

          250

          3000

          67

          537

          339.29

          50.265

          M60-1.5%

          150

          250

          3000

          67

          537

          339.29

          50.265

          [40]

          RA

          150

          250

          3000

          64

          456

          157.079

          50.265

          RAC3

          150

          250

          3000

          64

          456

          157.079

          50.265

          RAC5

          150

          250

          3000

          64

          456

          157.079

          50.265

          RAU3

          150

          250

          3000

          64

          456

          157.079

          50.265

          RAU5

          150

          250

          3000

          64

          456

          157.079

          50.265

          RB

          150

          250

          3000

          64

          456

          235.619

          50.265

          RBC3

          150

          250

          3000

          64

          456

          235.619

          50.265

          RBC5

          150

          250

          3000

          64

          456

          235.619

          50.265

          RBU3

          150

          250

          3000

          64

          456

          235.619

          50.265

          RBU5

          150

          250

          3000

          64

          456

          235.619

          50.265

          [28]

          B1

          250

          250

          2440

          65.61

          417

          981.747

          78.539

          B2

          250

          250

          2440

          58.44

          417

          981.747

          78.539

          B3

          250

          250

          2440

          48.35

          417

          981.747

          78.539

          B4

          250

          250

          2440

          38.35

          417

          981.747

          78.539

          B5

          250

          250

          2440

          59.15

          417

          981.747

          78.539

          B6

          250

          250

          2440

          49.45

          417

          981.747

          78.539

          B7

          250

          250

          2440

          40.26

          417

          981.747

          78.539

          B8

          250

          250

          2440

          29.73

          417

          981.747

          78.539

          B9

          250

          250

          2440

          31.1

          417

          981.747

          78.539

          B10

          250

          250

          2440

          32.38

          417

          981.747

          78.539

          B11

          250

          250

          2440

          30.71

          417

          981.747

          78.539

          B12

          250

          250

          2440

          31.51

          417

          981.747

          78.539

          [29]

          CB1

          150

          200

          2400

          35.7

          382

          226.194

          28.274

          CB2

          150

          200

          2400

          35.7

          382

          508.938

          28.274

          CB3

          150

          200

          2400

          35.7

          382

          628.318

          28.274

          HT1

          150

          200

          2400

          56.08

          418

          226.194

          28.274

          HT2

          150

          200

          2400

          56.08

          418

          508.938

          28.274

          HT3

          150

          200

          2400

          56.08

          418

          628.318

          28.274

          HC1

          150

          200

          2400

          56.08

          480

          226.194

          28.274

          HC2

          150

          200

          2400

          56.08

          480

          508.938

          28.274

          HC3

          150

          200

          2400

          56.08

          480

          628.318

          28.274

          RC1

          150

          200

          2400

          106.5

          410

          226.194

          28.274

          RC2

          150

          200

          2400

          106.5

          410

          508.938

          28.274

          RC3

          150

          200

          2400

          106.5

          410

          628.318

          28.274

          [13]

          CCO.20

          180

          250

          3500

          32

          420

          78.5

          50.265

          SFRCO.20

          180

          250

          3500

          34.7

          420

          78.5

          50.265

          CCO.30

          180

          250

          3500

          34.2

          420

          113.1

          50.265

          SFRCO.30

          180

          250

          3500

          40.6

          420

          113.1

          50.265

          CCO.40

          180

          250

          3500

          33.8

          420

          153.9

          50.265

          SFRCO.40

          180

          250

          3500

          31.1

          420

          153.9

          50.265

          CCO.53

          180

          250

          3500

          37.8

          420

          201.1

          50.265

          SFRCO.53

          180

          250

          3500

          41.7

          420

          201.1

          50.265

          CCO.81

          180

          250

          3500

          31.4

          420

          307.9

          50.265

          SFRCO.81

          180

          250

          3500

          29.6

          420

          307.9

          50.265

          CC1.06

          180

          250

          3500

          35

          420

          402.1

          50.265

          SFRC1.06

          180

          250

          3500

          43.9

          420

          402.1

          50.265

          CCI.60

          180

          250

          3500

          36.7

          420

          603.2

          50.265

          SFRC1.60

          180

          250

          3500

          31.9

          420

          603.2

          50.265

          CC2.02

          180

          250

          3500

          21.5

          420

          763.4

          50.265

          SFRC2.02

          180

          250

          3500

          25

          420

          763.4

          50.265

          CC2.13

          180

          250

          3500

          25.7

          420

          804

          50.265

          SFRC2.13

          180

          250

          3500

          31.8

          420

          804

          50.265

          CC2.50

          180

          250

          3500

          24.7

          420

          942.5

          50.265

          SFRC2.50

          180

          250

          3500

          22.3

          420

          942.5

          50.265

          [8]

          B1 0 -6

          150

          150

          1200

          76.3

          575

          31.172

          19.634

          B2 0-8

          150

          150

          1200

          76.3

          574

          50.265

          19.634

          B3 0-10

          150

          150

          1200

          76.3

          553

          78.539

          19.634

          B4-0.5/45-10

          150

          150

          1200

          95.2

          553

          78.539

          19.634

          B5-1/45-6

          150

          150

          1200

          80.2

          553

          78.539

          19.634

          B6-2/45-6

          150

          150

          1200

          81.3

          575

          31.172

          19.634

          B7-2/45-8

          150

          150

          1200

          81.3

          574

          50.265

          19.634

          B8-2/45-10

          150

          150

          1200

          81.3

          553

          78.539

          19.634

          B9-2/80-10

          150

          150

          1200

          81.3

          553

          78.539

          19.634

          [30]

          B1-0.0F

          152

          229

          1144

          27.6

          414

          235.619

          71

          B2-0.0F

          152

          229

          1144

          27.6

          414

          235.619

          71

          B1-0.5F

          152

          229

          1144

          27.6

          414

          235.619

          71

          B2-0.5F

          152

          229

          1144

          27.6

          414

          235.619

          71

          B1-1.0F

          152

          229

          1144

          27.6

          414

          235.619

          71

          B2-1.0F

          152

          229

          1144

          27.6

          414

          235.619

          71

          Table 4.2 Target Parameters for ANN based Modeling of Fiber Reinforced Concrete Beams with Incorporating Nano fillers and Micro fillers

          Ref

          Test Specimen

          first crack load (kN)

          Deflection (mm)

          yield load (kN)

          Deflection (mm)

          ultimate load (kN)

          Deflection (mm)

          [27]

          B1

          19

          14.1

          0

          0

          53

          21.02

          B2

          25

          19

          6

          22

          56.8

          24.07

          B3

          32

          20

          15

          30

          62.1

          22.02

          B4

          31

          19.5

          25

          32

          60.2

          20.07

          B5

          9

          10

          30

          36

          43.3

          14.49

          B6

          27

          22

          35

          39

          59.8

          20.01

          B7

          18

          15

          40

          42

          46.3

          16.04

          [6]

          NF

          36.6

          1.12

          46

          9.15

          62.6

          94.53

          NF

          30.6

          1.09

          77.9

          12.06

          97.9

          73.03

          S13

          26.6

          0.75

          80.6

          11.96

          87.3

          28.41

          S13

          23.3

          0.67

          109.9

          12.73

          124.1

          20.3

          S19.5

          18

          0.82

          78

          11.54

          93.3

          30.51

          S19.5

          16.7

          0.63

          103.3

          12.29

          125.2

          43.35

          S30

          21.3

          1.12

          79.9

          11.33

          95.9

          30.46

          S30

          18.7

          0.61

          105.3

          13.01

          124.6

          45.28

          T30

          18

          0.78

          77.9

          11.03

          96.6

          36.22

          T30

          14.7

          0.51

          111.9

          13.22

          133.9

          43.64

          [18]

          CC

          21

          0.68

          20

          10.5

          212

          43.82

          CC-SN

          27

          0.52

          35

          15.2

          218

          36.59

          CC-ST

          42

          0.83

          40

          20

          225

          36.81

          GG

          23

          0.67

          50

          22.2

          207

          38.85

          GG-SN

          35

          0.65

          60

          21.5

          230

          50.88

          GG-ST

          39

          0.79

          70

          25

          259

          56.19

          [44]

          S0L0

          19.62

          3.48

          39.24

          8.83

          49.05

          30.25

          S0L5

          29.43

          3.7

          63.77

          10.42

          88.29

          21.22

          S1L0

          22

          5.82

          39.6

          11.2

          58.8

          48.16

          S1L5

          49.05

          10

          93.2

          21.1

          112.82

          34.92

          [4]

          BY-0-2.5

          0

          0

          41.46

          6.99

          48.45

          31.2

          BY-1-2.5

          10

          5

          43.52

          5.98

          50.2

          14

          BY-2-2.5

          20

          15

          47.67

          5.93

          50.8

          18.95

          BX-1-2.5

          30

          20

          57.5

          7.85

          63.6

          12.87

          BX-2-2.5

          40

          20.2

          59.82

          6.82

          67.9

          21.08

          BX-1-2.5-360

          50

          25

          58.95

          9.31

          64.95

          20.64

          BX-2-2.5-360

          60

          30

          61.57

          8.87

          70.2

          21.55

          BX-1-2.5-180

          70

          35

          61.37

          6.77

          70.4

          19.7

          BX-1-2.5-180

          80

          41

          64.46

          8.38

          74.4

          24.2

          BY-0-3.5

          15

          10

          26.41

          9.8

          30.8

          25.73

          BY-1-3.5

          20

          12

          26.2

          10.05

          30.2

          24.75

          BY-2-3.5

          25

          30

          26.99

          7.48

          30.9

          16.15

          BX-1-3.5

          30

          22

          37.96

          10.47

          41.8

          16.82

          BX-2-3.5

          35

          22

          38.2

          10.01

          42.7

          20.8

          BX-1-2.5-360

          40

          28

          39.84

          10.64

          44.1

          18.55

          BX-2-2.5-360

          45

          30

          41.19

          10.77

          45.55

          48.64

          BX-1-3.5-180

          50

          32

          42.22

          10.9

          45.95

          17.1

          BX-2-3.5-180

          60

          33.3

          42.27

          11.03

          46.9

          27.1

          [46]

          M60

          3.5

          3.63

          6

          8.46

          8.25

          13.44

          M60-0.5%

          3.75

          3.58

          6.25

          8.29

          8.29

          14.81

          M60-1.0%

          4.25

          3.44

          7

          7.75

          7.75

          15.5

          M60-1.5%

          4.25

          3.55

          6.75

          6.75

          8.06

          16.1

          [40]

          RA

          14.39

          1.26

          29.42

          7.91

          41.68

          21.05

          RAC3

          16.52

          1.41

          36.77

          9.02

          51.48

          33.46

          RAC5

          21.28

          3.67

          46.58

          10.1

          66.19

          46.81

          RAU3

          32.94

          7.98

          51.48

          11.42

          71.09

          53.26

          RAU5

          36.81

          9.23

          53.7

          10.74

          78.45

          57.21

          RB

          28.32

          3.68

          39.22

          8.11

          53.93

          31.28

          RBC3

          30.95

          4.71

          51.48

          11.35

          61.29

          36.23

          RBC5

          32.17

          4.97

          53.24

          12.41

          63.74

          56.91

          RBU3

          33.69

          9.35

          58.8

          12.85

          88.25

          61.04

          RBU5

          39.41

          11.14

          63

          12.69

          100.51

          65.59

          [28]

          B1

          11.18

          20

          11.6

          15

          100.68

          36.9

          B2

          10.13

          7

          11.5

          15.1

          97.55

          38

          B3

          8.6

          5

          11.5

          14

          90.7

          37.2

          B4

          7.02

          5.6

          11.7

          14.1

          84.26

          34.1

          B5

          11.39

          22

          11.8

          15.3

          101.64

          41.65

          B6

          9.49

          10

          12

          20.2

          94.51

          40.4

          B7

          7.44

          11

          11.7

          16

          86.8

          31.4

          B8

          6.16

          10

          12.7

          21

          78.63

          22.5

          B9

          6.79

          9.95

          12.5

          22.2

          82.66

          28

          B10

          8.2

          14

          12.8

          19

          89.45

          38.75

          B11

          6.65

          11

          12.7

          14.5

          82.11

          27.2

          B12

          7.91

          13

          12.9

          13

          86.68

          36.95

          [29]

          CB1

          12

          1.15

          43

          12.1

          48

          31.5

          CB2

          13

          1.29

          90.5

          13.2

          94.5

          25.2

          CB3

          20

          1.38

          117.5

          12.2

          117.5

          12.2

          HT1

          35

          1.73

          124

          10.8

          130

          17.1

          HT2

          45

          2.23

          135

          9.6

          176

          14.7

          HT3

          45

          1.63

          178

          13.2

          194

          17.4

          HC1

          11

          1.01

          40

          9.5

          53

          74

          HC2

          15

          0.88

          90.5

          12.6

          103

          40.9

          HC3

          15

          0.74

          112.5

          12.7

          126

          40.9

          RC1

          10

          0.74

          43

          11.1

          59

          86.7

          RC2

          10

          0.34

          90

          10.1

          104

          39.1

          RC3

          15

          0.75

          110

          10.6

          131

          27.3

          [13]

          CCO.20

          0

          0

          0

          0

          11.81

          110.9

          SFRCO.20

          5

          3

          10

          5

          18.72

          64.5

          CCO.30

          10.5

          12.5

          20

          18

          18.48

          109.5

          SFRCO.30

          13

          14

          30

          11

          21.29

          86.2

          CCO.40

          20

          20.3

          40

          31

          25.39

          107.9

          SFRCO.40

          26

          22

          50

          35

          27.54

          95.3

          CCO.53

          32

          24

          55

          42

          31.56

          104..3

          SFRCO.53

          40.5

          39.9

          60

          59

          34.85

          157

          CCO.81

          55

          41

          65

          48

          47.9

          92.1

          SFRCO.81

          68

          40

          70.5

          53

          47.01

          176.5

          CC1.06

          74

          35

          76

          65

          58.11

          56.7

          SFRC1.06

          83

          22

          80

          78

          61.77

          175.7

          CCI.60

          97

          39

          84

          36

          81.35

          41.5

          SFRC1.60

          102

          10.5

          99

          21

          82.4

          164.2

          CC2.02

          107

          15

          101

          17

          87.85

          59.4

          SFRC2.02

          110

          25

          105

          14

          97.3

          126

          CC2.13

          113

          34

          110

          70

          93.74

          70.7

          SFRC2.13

          118

          28

          117

          63

          100.2

          101.2

          CC2.50

          120

          14

          115

          58

          100.82

          70

          SFRC2.50

          122

          15.9

          125

          78.5

          108.15

          133

          [8]

          B1 0 -6

          12

          14.1

          4.8

          5.9

          4.14

          57.8

          B2 0-8

          15

          15

          7.1

          9.1

          6.05

          38

          B3 0-10

          18

          15.9

          9.6

          11.1

          6.48

          31.8

          B4-0.5/45-10

          20

          19

          11.6

          11.5

          4.78

          19.8

          B5-1/45-6

          25

          20.6

          13.2

          12.3

          7.82

          26.9

          B6-2/45-6

          29

          28

          10.1

          6.5

          4.48

          23.2

          B7-2/45-8

          34

          31

          11.2

          7.9

          4.21

          24.9

          B8-2/45-10

          36

          33

          14.2

          13.4

          7.71

          24.8

          B9-2/80-10

          40

          37

          14.9

          14.2

          7.85

          16.3

          [30]

          B1-0.0F

          5

          5

          156

          6

          188.2

          13.56

          B2-0.0F

          13

          12.5

          178

          9.74

          206.4

          14.78

          B1-0.5F

          15

          15

          156

          3.91

          194.8

          12.65

          B2-0.5F

          17

          17.5

          178

          6.64

          198.8

          10.82

          B1-1.0F

          25

          20.1

          156

          2.94

          211.7

          9.65

          B2-1.0F

          34

          23.5

          178

          4.23

          234.4

          13.23

          Table 4.3 Experiment and ANN results (Levenberg Marquardt Algorithm) for Fiber Reinforced Concrete Beams with Incorporating Nano fillers and Micro fillers

          Ref

          Test Specimens

          first crack load (kN)

          Deflection (mm)

          yield load (kN)

          Deflection (mm)

          ultimate load (kN)

          Deflection (mm)

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          [27]

          B1

          19

          26.209

          14.1

          18.333

          0

          23.351

          0

          30.799

          53

          55.720

          21.02

          20.446

          B2

          25

          26.209

          19

          18.333

          6

          23.351

          22

          30.799

          56.8

          55.720

          24.07

          20.446

          B3

          32

          26.209

          20

          18.333

          15

          23.351

          30

          30.799

          62.1

          55.720

          22.02

          20.446

          B4

          31

          26.209

          19.5

          18.333

          25

          23.351

          32

          30.799

          60.2

          55.720

          20.07

          20.446

          B5

          9

          26.209

          10

          18.333

          30

          23.351

          36

          30.799

          43.3

          55.720

          14.49

          20.446

          B6

          27

          26.209

          22

          18.333

          35

          23.351

          39

          30.799

          59.8

          55.720

          20.01

          20.446

          B7

          18

          26.209

          15

          18.333

          40

          23.351

          42

          30.799

          46.3

          55.720

          16.04

          20.446

          [6]

          NF

          36.6

          23.366

          1.12

          5.958

          46

          46.453

          9.15

          19.011

          62.6

          60.685

          94.53

          79.910

          NF

          30.6

          30.890

          1.09

          6.026

          77.9

          75.381

          12.06

          22.683

          97.9

          99.141

          73.03

          77.862

          S13

          26.6

          19.693

          0.75

          0.406

          80.6

          82.931

          11.96

          8.785

          87.3

          95.879

          28.41

          36.408

          S13

          23.3

          27.838

          0.67

          1.696

          109.9

          101.134

          12.73

          14.580

          124.1

          125.777

          20.3

          40.724

          S19.5

          18

          14.067

          0.82

          1.326

          78

          81.644

          11.54

          6.217

          93.3

          95.369

          30.51

          31.571

          S19..5

          16.7

          22.543

          0.63

          0.362

          103.3

          97.320

          12.29

          12.505

          125.2

          123.162

          43.35

          37.245

          S30

          21.3

          14.067

          1.12

          1.326

          79.9

          81.644

          11.33

          6.217

          95.9

          95.369

          30.46

          31.571

          S30

          18.7

          22.543

          0.61

          0.362

          105.3

          97.320

          13.01

          12.505

          124.6

          123.162

          45.28

          37.245

          T30

          18

          19.624

          0.78

          1.261

          77.9

          79.373

          11.03

          7.853

          96.6

          96.852

          36.22

          26.962

          T30

          14.7

          27.384

          0.51

          2.674

          111.9

          93.373

          13.22

          14.630

          133.9

          124.067

          43.64

          34.600

          [18]

          CC

          21

          28.569

          0.68

          2.624

          20

          31.471

          10.5

          15.845

          212

          194.017

          43.82

          42.812

          CC-SN

          27

          33.194

          0.52

          0.202

          35

          42.744

          15.2

          17.891

          218

          217.811

          36.59

          41.832

          CC-ST

          42

          36.540

          0.83

          0.930

          40

          51.256

          20

          20.234

          225

          234.219

          36.81

          44.825

          GG

          23

          36.203

          0.67

          3.283

          50

          52.817

          22.2

          20.701

          207

          217.510

          38.85

          48.531

          GG-SN

          35

          42.982

          0.65

          2.159

          60

          66.674

          21.5

          24.409

          230

          243.192

          50.88

          51.181

          GG-ST

          39

          48.822

          0.79

          2.422

          70

          78.526

          25

          28.643

          259

          262.266

          56.19

          58.145

          [44]

          S0L0

          19.62

          35.234

          3.48

          8.413

          39.24

          50.569

          8.83

          14.729

          49.05

          75.132

          30.25

          34.916

          S0L5

          29.43

          35.234

          3.7

          8.413

          63.77

          50.569

          10.42

          14.729

          88.29

          75.132

          21.22

          34.916

          S1L0

          22

          35.234

          5.82

          8.413

          39.6

          50.569

          11.2

          14.729

          58.8

          75.132

          48.16

          34.916

          S1L5

          49.05

          35.234

          10

          8.413

          93.2

          50.569

          21.1

          14.729

          112.82

          75.132

          34.92

          34.916

          [4]

          BY-0-2.5

          0

          40.878

          0

          22.552

          41.46

          54.784

          6.99

          9.316

          48.45

          61.516

          31.2

          19.772

          BY-1-2.5

          10

          40.916

          5

          22.553

          43.52

          54.861

          5.98

          9.323

          50.2

          61.616

          14

          19.739

          BY-2-2.5

          20

          40.575

          15

          22.545

          47.67

          54.174

          5.93

          9.262

          50.8

          60.721

          18.95

          20.036

          BX-1-2.5

          30

          40.916

          20

          22.553

          57.5

          54.861

          7.85

          9.323

          63.6

          61.616

          12.87

          19.739

          BX-2-2.5

          40

          40.575

          20.2

          22.545

          59.82

          54.174

          6.82

          9.262

          67.9

          60.721

          21.08

          20.036

          BX-1-2.5-360

          50

          40.916

          25

          22.553

          58.95

          54.861

          9.31

          9.323

          64.95

          61.616

          20.64

          19.739

          BX-2-2.5-360

          60

          40.575

          30

          22.545

          61.57

          54.174

          8.87

          9.262

          70.2

          60.721

          21.55

          20.036

          BX-1-2.5-180

          70

          40.916

          35

          22.553

          61.37

          54.861

          6.77

          9.323

          70.4

          61.616

          19.7

          19.739

          BX-1-2.5-180

          80

          40.575

          41

          22.545

          64.46

          54.174

          8.38

          9.262

          74.4

          60.721

          24.2

          20.036

          BY-0-3.5

          15

          36.268

          10

          21.003

          26.41

          33.662

          9.8

          11.818

          30.8

          40.577

          25.73

          22.502

          BY-1-3.5

          20

          36.252

          12

          20.995

          26.2

          33.680

          10.05

          11.802

          30.2

          40.579

          24.75

          22.469

          BY-2-3.5

          25

          36.396

          30

          21.072

          26.99

          33.521

          7.48

          11.947

          30.9

          40.561

          16.15

          22.763

          BX-1-3.5

          30

          36.252

          22

          20.995

          37.96

          33.680

          10.47

          11.802

          41.8

          40.579

          16.82

          22.469

          BX-2-3.5

          35

          36.396

          22

          21.072

          38.2

          33.521

          10.01

          11.947

          42.7

          40.561

          20.8

          22.763

          BX-1-2.5-360

          40

          36.252

          28

          20.995

          39.84

          33.680

          10.64

          11.802

          44.1

          40.579

          18.55

          22.469

          BX-2-2.5-360

          45

          36.396

          30

          21.072

          41.19

          33.521

          10.77

          11.947

          45.55

          40.561

          48.64

          22.763

          BX-1-3.5-180

          50

          36.252

          32

          20.995

          42.22

          33.680

          10.9

          11.802

          45.95

          40.579

          17.1

          22.469

          BX-2-3.5-180

          60

          36.396

          33.3

          21.072

          42.27

          33.521

          11.03

          11.947

          46.9

          40.561

          27.1

          22.763

          [46]

          M60

          3.5

          8.850

          3.63

          5.270

          6

          7.979

          8.46

          0.046

          8.25

          8.949

          13.44

          21.913

          M60-0.5%

          3.75

          8.850

          3.58

          5.270

          6.25

          7.979

          8.29

          0.046

          8.29

          8.949

          14.81

          21.913

          M60-1.0%

          4.25

          8.850

          3.44

          5.270

          7

          7.979

          7.75

          0.046

          7.75

          8.949

          15.5

          21.913

          M60-1.5%

          4.25

          8.850

          3.55

          5.270

          6.75

          7.979

          6.75

          0.046

          8.06

          8.949

          16.1

          21.913

          [40]

          RA

          14.39

          20.048

          1.26

          9.784

          29.42

          48.463

          7.91

          10.961

          41.68

          61.774

          21.05

          45.565

          RAC3

          16.52

          20.048

          1.41

          9.784

          36.77

          48.463

          9.02

          10.961

          51.48

          61.774

          33.46

          45.565

          RAC5

          21.28

          20.048

          3.67

          9.784

          46.58

          48.463

          10.1

          10.961

          66.19

          61.774

          46.81

          45.565

          RAU3

          32.94

          20.048

          7.98

          9.784

          51.48

          48.463

          11.42

          10.961

          71.09

          61.774

          53.26

          45.565

          RAU5

          36.81

          20.048

          9.23

          9.784

          53.7

          48.463

          10.74

          10.961

          78.45

          61.774

          57.21

          45.565

          RB

          28.32

          29.347

          3.68

          9.281

          39.22

          55.377

          8.11

          16.594

          53.93

          78.439

          31.28

          51.229

          RBC3

          30.95

          29.347

          4.71

          9.281

          51.48

          55.377

          11.35

          16.594

          61.29

          78.439

          36.23

          51.229

          RBC5

          32.17

          29.347

          4.97

          9.281

          53.24

          55.377

          12.41

          16.594

          63.74

          78.439

          56.91

          51.229

          RBU3

          33.69

          29.347

          9.35

          9.281

          58.8

          55.377

          12.85

          16.594

          88.25

          78.439

          61.04

          51.229

          RBU5

          39.41

          29.347

          11.14

          9.281

          63

          55.377

          12.69

          16.594

          100.51

          78.439

          65.59

          51.229

          [28]

          B1

          11.18

          4.041

          20

          8.313

          11.6

          6.820

          15

          19.542

          100.68

          99.905

          36.9

          40.503

          B2

          10.13

          4.874

          7

          9.661

          11.5

          6.360

          15.1

          19.036

          97.55

          92.903

          38

          40.623

          B3

          8.6

          6.150

          5

          10.734

          11.5

          6.794

          14

          18.035

          90.7

          86.294

          37.2

          39.028

          B4

          7.02

          7.431

          5.6

          10.976

          11.7

          8.115

          14.1

          16.721

          84.26

          82.745

          34.1

          35.675

          B5

          11.39

          4.788

          22

          9.550

          11.8

          6.375

          15.3

          19.093

          101.64

          93.507

          41.65

          40.659

          B6

          9.49

          6.008

          10

          10.661

          12

          6.694

          20.2

          18.160

          94.51

          86.849

          40.4

          39.296

          B7

          7.44

          7.191

          11

          10.984

          11.7

          7.815

          16

          16.995

          86.8

          83.239

          31.4

          36.434

          B8

          6.16

          8.462

          10

          10.676

          12.7

          9.618

          21

          15.356

          78.63

          81.330

          22.5

          31.674

          B9

          6.79

          8.305

          9.95

          10.750

          12.5

          9.368

          22.2

          15.586

          82.66

          81.480

          28

          32.367

          B10

          8.2

          8.156

          14

          10.810

          12.8

          9.138

          19

          15.796

          89.45

          81.643

          38.75

          32.996

          B11

          6.65

          8.350

          11

          10.730

          12.7

          9.439

          14.5

          15.521

          82.11

          81.435

          27.2

          32.172

          B12

          7.91

          8.258

          13

          10.770

          12.9

          9.294

          13

          15.654

          86.68

          81.530

          36.95

          32.571

          [29]

          CB1

          12

          5.638

          1.15

          10.421

          43

          45.538

          12.1

          8.461

          48

          52.989

          31.5

          27.244

          CB2

          13

          22.071

          1.29

          9.510

          90.5

          93.425

          13.2

          12.122

          94.5

          99.652

          25.2

          25.723

          CB3

          20

          25.947

          1.38

          6.755

          117.5

          106.512

          12.2

          11.459

          117.5

          119.282

          12.2

          17.233

          HT1

          35

          29.556

          1.73

          2.119

          124

          113.494

          10.8

          14.260

          130

          130.866

          17.1

          33.496

          HT2

          45

          38.695

          2.23

          2.993

          135

          153.102

          9.6

          12.649

          176

          176.037

          14.7

          15.046

          HT3

          45

          45.349

          1.63

          4.195

          178

          168.879

          13.2

          13.149

          194

          196.493

          17.4

          5.926

          HC1

          11

          8.037

          1.01

          9.014

          40

          51.078

          9.5

          7.448

          53

          42.313

          74

          64.872

          HC2

          15

          14.687

          0.88

          1.010

          90.5

          91.138

          12.6

          9.641

          103

          92.074

          40.9

          56.505

          HC3

          15

          23.985

          0.74

          1.154

          112.5

          111.029

          12.7

          11.842

          126

          120.261

          40.9

          46.721

          RC1

          10

          7.396

          0.74

          6.911

          43

          42.553

          11.1

          17.276

          59

          60.909

          86.7

          78.768

          RC2

          10

          11.704

          0.34

          1.084

          90

          87.490

          10.1

          8.641

          104

          103.330

          39.1

          41.527

          RC3

          15

          19.044

          0.75

          3.359

          110

          109.519

          10.6

          8.030

          131

          129.544

          27.3

          26.268

          [13]

          CCO.20

          0

          0.024

          0

          9.951

          0

          8.294

          0

          12.465

          11.81

          12.705

          110.9

          82.289

          SFRCO.20

          5

          0.052

          3

          10.031

          10

          9.990

          5

          12.752

          18.72

          13.068

          64.5

          83.542

          CCO.30

          10.5

          11.720

          12.5

          14.099

          20

          20.784

          18

          21.245

          18.48

          20.830

          109.5

          99.604

          SFRCO.30

          13

          11.780

          14

          14.129

          30

          24.667

          11

          21.802

          21.29

          22.052

          86.2

          101.980

          CCO.40

          20

          25.224

          20.3

          18.496

          40

          33.163

          31

          30.126

          25.39

          29.218

          107.9

          114.510

          SFRCO.40

          26

          25.161

          22

          18.453

          50

          31.861

          35

          29.845

          27.54

          28.847

          95.3

          113.411

          CCO.53

          32

          39.607

          24

          22.583

          55

          47.308

          42

          38.591

          31.56

          37.746

          104..3

          126.536

          SFRCO.53

          40.5

          39.433

          39.9

          22.334

          60

          48.720

          59

          38.620

          34.85

          38.373

          157

          126.920

          CCO.81

          55

          65.867

          41

          28.694

          65

          66.024

          48

          49.689

          47.9

          49.284

          92.1

          131.080

          SFRCO.81

          68

          65.991

          40

          28.833

          70.5

          65.852

          53

          49.733

          47.01

          49.211

          176.5

          131.098

          CC1.06

          74

          81.219

          35

          30.071

          76

          78.103

          65

          53.658

          58.11

          58.034

          56.7

          123.238

          SFRC1.06

          83

          80.165

          22

          28.402

          80

          78.903

          78

          52.726

          61.77

          60.779

          175.7

          120.433

          CCI.60

          97

          100.775

          39

          28.917

          84

          97.850

          36

          56.276

          81.35

          80.445

          41.5

          102.080

          SFRC1.60

          102

          101.099

          10.5

          30.014

          99

          96.901

          21

          56.749

          82.4

          77.311

          164.2

          104.091

          CC2.02

          107

          109.789

          15

          30.467

          101

          106.353

          17

          58.215

          87.85

          86.437

          59.4

          96.517

          SFRC2.02

          110

          109.698

          25

          29.760

          105

          107.171

          14

          57.956

          97.3

          88.853

          126

          95.179

          CC2.13

          113

          111.186

          34

          29.111

          110

          109.924

          70

          57.973

          93.74

          93.276

          70.7

          92.492

          SFRC2.13

          118

          111.117

          28

          27.499

          117

          112.102

          63

          57.434

          100.2

          99.276

          101.2

          89.422

          CC2.50

          120

          115.208

          14

          27.444

          115

          117.681

          58

          58.123

          100.82

          105.678

          70

          85.678

          SFRC2.50

          122

          115.166

          15.9

          28.112

          125

          116.636

          78.5

          58.321

          108.15

          102.963

          133

          86.974

          [8]

          B1 0 -6

          12

          23.436

          14.1

          22.990

          4.8

          7.788

          5.9

          7.068

          4.14

          1.208

          57.8

          34.625

          B2 0-8

          15

          20.911

          15

          22.709

          7.1

          5.907

          9.1

          5.684

          6.05

          2.961

          38

          34.966

          B3 0-10

          18

          27.642

          15.9

          23.326

          9.6

          13.032

          11.1

          8.913

          6.48

          9.976

          31.8

          33.480

          B4-0.5/45-10

          20

          29.615

          19

          23.625

          11.6

          13.323

          11.5

          10.405

          4.78

          12.303

          19.8

          34.064

          B5-1/45-6

          25

          27.991

          20.6

          23.383

          13.2

          13.012

          12.3

          9.199

          7.82

          10.331

          26.9

          33.624

          B6-2/45-6

          29

          24.044

          28

          23.070

          10.1

          8.055

          6.5

          7.464

          4.48

          2.060

          23.2

          34.665

          B7-2/45-8

          34

          21.451

          31

          22.782

          11.2

          6.101

          7.9

          6.049

          4.21

          2.238

          24.9

          35.029

          B8-2/45-10

          36

          28.095

          33

          23.400

          14.2

          13.015

          13.4

          9.281

          7.71

          10.443

          24.8

          33.662

          B9-2/80-10

          40

          28.095

          37

          23.400

          14.9

          13.015

          14.2

          9.281

          7.85

          10.443

          16.3

          33.662

          [30]

          B1-0.0F

          5

          18.672

          5

          14.678

          156

          168.232

          6

          5.054

          188.2

          205.910

          13.56

          11.714

          B2-0.0F

          13

          18.672

          12.5

          14.678

          178

          168.232

          9.74

          5.054

          206.4

          205.910

          14.78

          11.714

          B1-0.5F

          15

          18.672

          15

          14.678

          156

          168.232

          3.91

          5.054

          194.8

          205.910

          12.65

          11.714

          B2-0.5F

          17

          18.672

          17.5

          14.678

          178

          168.232

          6.64

          5.054

          198.8

          205.910

          10.82

          11.714

          B1-1.0F

          25

          18.672

          20.1

          14.678

          156

          168.232

          2.94

          5.054

          211.7

          205.910

          9.65

          11.714

          B2-1.0F

          34

          18.672

          23.5

          14.678

          178

          168.232

          4.23

          5.054

          234.4

          205.910

          13.23

          11.714

          Table 4.4 Experiment and ANN results( Bayesian Regularization Algorithm) for Fiber Reinforced Concrete Beams with Incorporating Nano fillers and Micro fillers

          Ref

          Test Specimens

          first crack load (kN)

          Deflection (mm)

          yield load (kN)

          Deflection (mm)

          ultimate load (kN)

          Deflection (mm)

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          [27]

          B1

          19

          21.939

          14.1

          14.812

          0

          34.286

          0

          36.271

          53

          51.106

          21.02

          17.306

          B2

          25

          21.939

          19

          14.812

          6

          34.286

          22

          36.271

          56.8

          51.106

          24.07

          17.306

          B3

          32

          21.939

          20

          14.812

          15

          34.286

          30

          36.271

          62.1

          51.106

          22.02

          17.306

          B4

          31

          21.939

          19.5

          14.812

          25

          34.286

          32

          36.271

          60.2

          51.106

          20.07

          17.306

          B5

          9

          21.939

          10

          14.812

          30

          34.286

          36

          36.271

          43.3

          51.106

          14.49

          17.306

          B6

          27

          21.939

          22

          14.812

          35

          34.286

          39

          36.271

          59.8

          51.106

          20.01

          17.306

          B7

          18

          21.939

          15

          14.812

          40

          34.286

          42

          36.271

          46.3

          51.106

          16.04

          17.306

          [6]

          NF

          36.6

          30.134

          1.12

          1.834

          46

          51.621

          9.15

          13.523

          62.6

          62.479

          94.53

          69.089

          NF

          30.6

          28.136

          1.09

          2.070

          77.9

          76.693

          12.06

          15.506

          97.9

          93.928

          73.03

          74.292

          S13

          26.6

          23.503

          0.75

          2.471

          80.6

          74.579

          11.96

          10.027

          87.3

          89.922

          28.41

          40.241

          S13

          23.3

          21.899

          0.67

          2.836

          109.9

          100.319

          12.73

          11.700

          124.1

          121.691

          20.3

          45.485

          S19.5

          18

          22.408

          0.82

          2.675

          78

          80.663

          11.54

          8.702

          93.3

          96.580

          30.51

          32.107

          S19..5

          16.7

          21.101

          0.63

          3.389

          103.3

          106.129

          12.29

          11.091

          125.2

          127.850

          43.35

          38.849

          S30

          21.3

          22.408

          1.12

          2.675

          79.9

          80.663

          11.33

          8.702

          95.9

          96.580

          30.46

          32.107

          S30

          18.7

          21.101

          0.61

          3.389

          105.3

          106.129

          13.01

          11.091

          124.6

          127.850

          45.28

          38.849

          T30

          18

          17.707

          0.78

          3.157

          77.9

          80.684

          11.03

          11.958

          96.6

          98.685

          36.22

          34.920

          T30

          14.7

          14.640

          0.51

          2.597

          111.9

          106.344

          13.22

          11.258

          133.9

          130.164

          43.64

          36.162

          [18]

          CC

          21

          27.111

          0.68

          1.044

          20

          26.972

          10.5

          12.159

          212

          205.850

          43.82

          33.834

          CC-SN

          27

          28.526

          0.52

          0.116

          35

          37.909

          15.2

          15.367

          218

          218.253

          36.59

          39.629

          CC-ST

          42

          29.979

          0.83

          0.934

          40

          50.969

          20

          18.811

          225

          232.852

          36.81

          45.492

          GG

          23

          32.293

          0.67

          0.894

          50

          42.823

          22.2

          18.395

          207

          218.858

          38.85

          44.118

          GG-SN

          35

          33.150

          0.65

          1.543

          60

          53.105

          21.5

          20.944

          230

          230.532

          50.88

          48.796

          GG-ST

          39

          33.911

          0.79

          2.219

          70

          65.417

          25

          23.475

          259

          244.318

          56.19

          53.066

          [44]

          S0L0

          19.62

          35.174

          3.48

          11.878

          39.24

          77.912

          8.83

          16.434

          49.05

          99.423

          30.25

          21.687

          S0L5

          29.43

          35.174

          3.7

          11.878

          63.77

          77.912

          10.42

          16.434

          88.29

          99.423

          21.22

          21.687

          S1L0

          22

          35.174

          5.82

          11.878

          39.6

          77.912

          11.2

          16.434

          58.8

          99.423

          48.16

          21.687

          S1L5

          49.05

          35.174

          10

          11.878

          93.2

          77.912

          21.1

          16.434

          112.82

          99.423

          34.92

          21.687

          [4]

          BY-0-2.5

          0

          40.469

          0

          22.152

          41.46

          55.440

          6.99

          8.188

          48.45

          61.693

          31.2

          19.944

          BY-1-2.5

          10

          40.457

          5

          22.150

          43.52

          55.411

          5.98

          8.171

          50.2

          61.660

          14

          19.907

          BY-2-2.5

          20

          40.562

          15

          22.176

          47.67

          55.667

          5.93

          8.321

          50.8

          61.962

          18.95

          20.238

          BX-1-2.5

          30

          40.457

          20

          22.150

          57.5

          55.411

          7.85

          8.171

          63.6

          61.660

          12.87

          19.907

          BX-2-2.5

          40

          40.562

          20.2

          22.176

          59.82

          55.667

          6.82

          8.321

          67.9

          61.962

          21.08

          20.238

          BX-1-2.5- 360

          50

          40.457

          25

          22.150

          58.95

          55.411

          9.31

          8.171

          64.95

          61.660

          20.64

          19.907

          BX-2-2.5- 360

          60

          40.562

          30

          22.176

          61.57

          55.667

          8.87

          8.321

          70.2

          61.962

          21.55

          20.238

          BX-1-2.5- 180

          70

          40.457

          35

          22.150

          61.37

          55.411

          6.77

          8.171

          70.4

          61.660

          19.7

          19.907

          BX-1-2.5- 180

          80

          40.562

          41

          22.176

          64.46

          55.667

          8.38

          8.321

          74.4

          61.962

          24.2

          20.238

          BY-0-3.5

          15

          31.724

          10

          19.335

          26.41

          32.939

          9.8

          11.978

          30.8

          38.149

          25.73

          27.361

          BY-1-3.5

          20

          31.698

          12

          19.330

          26.2

          32.882

          10.05

          11.954

          30.2

          38.088

          24.75

          27.311

          BY-2-3.5

          25

          31.925

          30

          19.382

          26.99

          33.389

          7.48

          12.167

          30.9

          38.642

          16.15

          27.764

          BX-1-3.5

          30

          31.698

          22

          19.330

          37.96

          32.882

          10.47

          11.954

          41.8

          38.088

          16.82

          27.311

          BX-2-3.5

          35

          31.925

          22

          19.382

          38.2

          33.389

          10.01

          12.167

          42.7

          38.642

          20.8

          27.764

          BX-1-2.5- 360

          40

          31.698

          28

          19.330

          39.84

          32.882

          10.64

          11.954

          44.1

          38.088

          18.55

          27.311

          BX-2-2.5- 360

          45

          31.925

          30

          19.382

          41.19

          33.389

          10.77

          12.167

          45.55

          38.642

          48.64

          27.764

          BX-1-3.5- 180

          50

          31.698

          32

          19.330

          42.22

          32.882

          10.9

          11.954

          45.95

          38.088

          17.1

          27.311

          BX-2-3.5- 180

          60

          31.925

          33.3

          19.382

          42.27

          33.389

          11.03

          12.167

          46.9

          38.642

          27.1

          27.764

          [46]

          M60

          3.5

          5.276

          3.63

          3.307

          6

          7.429

          8.46

          6.665

          8.25

          8.308

          13.44

          17.985

          M60-0.5%

          3.75

          5.276

          3.58

          3.307

          6.25

          7.429

          8.29

          6.665

          8.29

          8.308

          14.81

          17.985

          M60-1.0%

          4.25

          5.276

          3.44

          3.307

          7

          7.429

          7.75

          6.665

          7.75

          8.308

          15.5

          17.985

          M60-1.5%

          4.25

          5.276

          3.55

          3.307

          6.75

          7.429

          6.75

          6.665

          8.06

          8.308

          16.1

          17.985

          [40]

          RA

          14.39

          21.515

          1.26

          4.247

          29.42

          39.277

          7.91

          9.036

          41.68

          62.363

          21.05

          43.816

          RAC3

          16.52

          21.515

          1.41

          4.247

          36.77

          39.277

          9.02

          9.036

          51.48

          62.363

          33.46

          43.816

          RAC5

          21.28

          21.515

          3.67

          4.247

          46.58

          39.277

          10.1

          9.036

          66.19

          62.363

          46.81

          43.816

          RAU3

          32.94

          21.515

          7.98

          4.247

          51.48

          39.277

          11.42

          9.036

          71.09

          62.363

          53.26

          43.816

          RAU5

          36.81

          21.515

          9.23

          4.247

          53.7

          39.277

          10.74

          9.036

          78.45

          62.363

          57.21

          43.816

          RB

          28.32

          33.481

          3.68

          5.944

          39.22

          54.370

          8.11

          14.443

          53.93

          73.328

          31.28

          50.668

          RBC3

          30.95

          33.481

          4.71

          5.944

          51.48

          54.370

          11.35

          14.443

          61.29

          73.328

          36.23

          50.668

          RBC5

          32.17

          33.481

          4.97

          5.944

          53.24

          54.370

          12.41

          14.443

          63.74

          73.328

          56.91

          50.668

          RBU3

          33.69

          33.481

          9.35

          5.944

          58.8

          54.370

          12.85

          14.443

          88.25

          73.328

          61.04

          50.668

          RBU5

          39.41

          33.481

          11.14

          5.944

          63

          54.370

          12.69

          14.443

          100.51

          73.328

          65.59

          50.668

          [28]

          B1

          11.18

          8.963

          20

          12.792

          11.6

          17.624

          15

          20.154

          100.68

          96.242

          36.9

          41.471

          B2

          10.13

          8.769

          7

          12.536

          11.5

          15.759

          15.1

          19.045

          97.55

          93.962

          38

          39.266

          B3

          8.6

          8.654

          5

          12.186

          11.5

          13.317

          14

          17.504

          90.7

          90.970

          37.2

          36.174

          B4

          7.02

          8.725

          5.6

          11.855

          11.7

          11.129

          14.1

          16.001

          84.26

          88.275

          34.1

          33.116

          B5

          11.39

          8.784

          22

          12.561

          11.8

          15.939

          15.3

          19.154

          101.64

          94.182

          41.65

          39.484

          B6

          9.49

          8.657

          10

          12.224

          12

          13.573

          20.2

          17.671

          94.51

          91.283

          40.4

          36.511

          B7

          7.44

          8.697

          11

          11.917

          11.7

          11.528

          16

          16.286

          86.8

          88.767

          31.4

          33.700

          B8

          6.16

          8.938

          10

          11.584

          12.7

          9.447

          21

          14.726

          78.63

          86.191

          22.5

          30.479

          B9

          6.79

          8.894

          9.95

          11.626

          12.5

          9.701

          22.2

          14.927

          82.66

          86.507

          28

          30.899

          B10

          8.2

          8.857

          14

          11.666

          12.8

          9.942

          19

          15.116

          89.45

          86.807

          38.75

          31.291

          B11

          6.65

          8.906

          11

          11.614

          12.7

          9.628

          14.5

          14.870

          82.11

          86.416

          27.2

          30.779

          B12

          7.91

          8.882

          13

          11.639

          12.9

          9.778

          13

          14.987

          86.68

          86.602

          36.95

          31.024

          [29]

          CB1

          12

          0.646

          1.15

          7.861

          43

          46.986

          12.1

          7.560

          48

          48.829

          31.5

          35.649

          CB2

          13

          17.169

          1.29

          8.912

          90.5

          87.933

          13.2

          22.955

          94.5

          97.957

          25.2

          34.172

          CB3

          20

          21.488

          1.38

          3.353

          117.5

          115.494

          12.2

          12.799

          117.5

          124.763

          12.2

          5.021

          HT1

          35

          35.950

          1.73

          7.296

          124

          109.995

          10.8

          2.269

          130

          129.339

          17.1

          28.888

          HT2

          45

          40.866

          2.23

          1.961

          135

          146.867

          9.6

          11.018

          176

          172.406

          14.7

          20.465

          HT3

          45

          44.369

          1.63

          2.165

          178

          166.808

          13.2

          7.806

          194

          189.523

          17.4

          6.069

          HC1

          11

          14.058

          1.01

          10.796

          40

          41.088

          9.5

          12.527

          53

          49.343

          74

          56.226

          HC2

          15

          18.455

          0.88

          2.940

          90.5

          92.309

          12.6

          9.259

          103

          105.080

          40.9

          46.998

          HC3

          15

          35.271

          0.74

          2.670

          112.5

          113.562

          12.7

          18.697

          126

          124.321

          40.9

          60.487

          RC1

          10

          16.853

          0.74

          6.727

          43

          48.208

          11.1

          15.946

          59

          58.587

          86.7

          72.835

          RC2

          10

          11.189

          0.34

          5.157

          90

          89.828

          10.1

          9.438

          104

          107.551

          39.1

          36.074

          RC3

          15

          17.130

          0.75

          6.177

          110

          113.002

          10.6

          12.720

          131

          127.085

          27.3

          33.534

          [13]

          CCO.20

          0

          2.703

          0

          11.804

          0

          15.685

          0

          10.095

          11.81

          10.784

          110.9

          84.748

          SFRCO.20

          5

          2.212

          3

          11.920

          10

          16.146

          5

          10.921

          18.72

          11.433

          64.5

          86.272

          CCO.30

          10.5

          11.401

          12.5

          14.816

          20

          25.399

          18

          17.915

          18.48

          18.390

          109.5

          97.225

          SFRCO.30

          13

          10.282

          14

          15.079

          30

          26.590

          11

          19.888

          21.29

          20.117

          86.2

          100.766

          CCO.40

          20

          22.816

          20.3

          18.348

          40

          36.103

          31

          26.348

          25.39

          26.243

          107.9

          110.272

          SFRCO.40

          26

          23.306

          22

          18.254

          50

          35.732

          35

          25.525

          27.54

          25.643

          95.3

          108.757

          CCO.53

          32

          35.658

          24

          22.532

          55

          48.416

          42

          37.226

          31.56

          35.625

          104..3

          127.103

          SFRCO.53

          40.5

          34.952

          39.9

          22.628

          60

          48.962

          59

          38.334

          34.85

          36.590

          157

          129.040

          CCO.81

          55

          66.340

          41

          30.170

          65

          70.859

          48

          53.867

          47.9

          50.637

          92.1

          149.788

          SFRCO.81

          68

          66.773

          40

          30.210

          70.5

          70.964

          53

          53.525

          47.01

          50.536

          176.5

          149.145

          CC1.06

          74

          84.416

          35

          32.243

          76

          84.615

          65

          60.252

          58.11

          61.472

          56.7

          153.803

          SFRC1.06

          83

          81.450

          22

          31.286

          80

          82.995

          78

          60.054

          61.77

          61.226

          175.7

          153.334

          CCI.60

          97

          97.982

          39

          22.983

          84

          96.760

          36

          43.324

          81.35

          76.719

          41.5

          105.573

          SFRC1.60

          102

          99.681

          10.5

          23.521

          99

          99.333

          21

          43.514

          82.4

          78.612

          164.2

          105.667

          CC2.02

          107

          111.181

          15

          21.930

          101

          112.142

          17

          42.985

          87.85

          96.543

          59.4

          89.765

          SFRC2.02

          110

          110.153

          25

          21.795

          105

          109.277

          14

          43.482

          97.3

          93.989

          126

          91.241

          CC2.13

          113

          112.157

          34

          22.111

          110

          109.800

          70

          45.839

          93.74

          96.253

          70.7

          92.380

          SFRC2.13

          118

          110.488

          28

          21.960

          117

          104.912

          63

          46.847

          100.2

          91.927

          101.2

          95.169

          CC2.50

          120

          118.539

          14

          24.038

          115

          113.669

          58

          54.710

          100.82

          105.635

          70

          98.098

          SFRC2.50

          122

          119.240

          15.9

          24.091

          125

          116.034

          78.5

          54.283

          108.15

          107.850

          133

          96.734

          [8]

          B1 0 -6

          12

          25.172

          14.1

          24.398

          4.8

          6.870

          5.9

          7.113

          4.14

          2.833

          57.8

          30.349

          B2 0-8

          15

          25.205

          15

          24.174

          7.1

          7.929

          9.1

          6.921

          6.05

          4.190

          38

          29.985

          B3 0-10

          18

          27.771

          15.9

          24.439

          9.6

          12.927

          11.1

          8.627

          6.48

          9.110

          31.8

          34.852

          B4-0.5/45- 10

          20

          27.863

          19

          23.974

          11.6

          14.908

          11.5

          8.535

          4.78

          13.031

          19.8

          34.706

          B5-1/45-6

          25

          27.806

          20.6

          24.358

          13.2

          13.331

          12.3

          8.635

          7.82

          9.902

          26.9

          34.871

          B6-2/45-6

          29

          25.166

          28

          24.302

          10.1

          7.254

          6.5

          7.108

          4.48

          3.711

          23.2

          30.298

          B7-2/45-8

          34

          25.181

          31

          24.055

          11.2

          8.341

          7.9

          6.869

          4.21

          5.107

          24.9

          29.849

          B8-2/45-10

          36

          27.814

          33

          24.334

          14.2

          13.445

          13.4

          8.635

          7.71

          10.126

          24.8

          34.872

          B9-2/80-10

          40

          27.814

          37

          24.334

          14.9

          13.445

          14.2

          8.635

          7.85

          10.126

          16.3

          34.872

          [30]

          B1-0.0F

          5

          19.915

          5

          15.008

          156

          172.684

          6

          7.208

          188.2

          207.946

          13.56

          12.465

          B2-0.0F

          13

          19.915

          12.5

          15.008

          178

          172.684

          9.74

          7.208

          206.4

          207.946

          14.78

          12.465

          B1-0.5F

          15

          19.915

          15

          15.008

          156

          172.684

          3.91

          7.208

          194.8

          207.946

          12.65

          12.465

          B2-0.5F

          17

          19.915

          17.5

          15.008

          178

          172.684

          6.64

          7.208

          198.8

          207.946

          10.82

          12.465

          B1-1.0F

          25

          19.915

          20.1

          15.008

          156

          172.684

          2.94

          7.208

          211.7

          207.946

          9.65

          12.465

          B2-1.0F

          34

          19.915

          23.5

          15.008

          178

          172.684

          4.23

          7.208

          234.4

          207.946

          13.23

          12.465

          Table 4.5 Experiment and ANN results (Scaled Conjugate Gradient Algorithm) for Fiber Reinforced Concrete Beams with Incorporating Nano fillers and Micro fillers

          Ref

          Test Specimens

          first crack load (kN)

          Deflection (mm)

          yield load (kN)

          Deflection (mm)

          ultimate load (kN)

          Deflection (mm)

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          Exp

          ANN

          [27]

          B1

          19

          25.660

          14.1

          15.751

          0

          45.794

          0

          2.233

          53

          52.413

          21.02

          22.090

          B2

          25

          25.660

          19

          15.751

          6

          45.794

          22

          2.233

          56.8

          52.413

          24.07

          22.090

          B3

          32

          25.660

          20

          15.751

          15

          45.794

          30

          2.233

          62.1

          52.413

          22.02

          22.090

          B4

          31

          25.660

          19.5

          15.751

          25

          45.794

          32

          2.233

          60.2

          52.413

          20.07

          22.090

          B5

          9

          25.660

          10

          15.751

          30

          45.794

          36

          2.233

          43.3

          52.413

          14.49

          22.090

          B6

          27

          25.660

          22

          15.751

          35

          45.794

          39

          2.233

          59.8

          52.413

          20.01

          22.090

          B7

          18

          25.660

          15

          15.751

          40

          45.794

          42

          2.233

          46.3

          52.413

          16.04

          22.090

          [6]

          NF

          36.6

          14.248

          1.12

          7.106

          46

          62.002

          9.15

          12.301

          62.6

          89.956

          94.53

          36.561

          NF

          30.6

          15.903

          1.09

          5.246

          77.9

          65.032

          12.06

          10.565

          97.9

          88.121

          73.03

          36.948

          S13

          26.6

          19.859

          0.75

          5.931

          80.6

          83.133

          11.96

          16.522

          87.3

          99.441

          28.41

          49.563

          S13

          23.3

          21.027

          0.67

          4.425

          109.9

          84.214

          12.73

          14.402

          124.1

          96.780

          20.3

          49.271

          S19.5

          18

          20.685

          0.82

          5.311

          78

          85.170

          11.54

          16.800

          93.3

          102.346

          30.51

          51.194

          S19..5

          16.7

          21.794

          0.63

          3.879

          103.3

          85.632

          12.29

          14.528

          125.2

          99.582

          43.35

          50.755

          S30

          21.3

          20.685

          1.12

          5.311

          79.9

          85.170

          11.33

          16.800

          95.9

          102.346

          30.46

          51.194

          S30

          18.7

          21.794

          0.61

          3.879

          105.3

          85.632

          13.01

          14.528

          124.6

          99.582

          45.28

          50.755

          T30

          18

          22.357

          0.78

          6.787

          77.9

          95.950

          11.03

          18.736

          96.6

          97.752

          36.22

          56.472

          T30

          14.7

          23.120

          0.51

          5.497

          111.9

          96.893

          13.22

          16.778

          133.9

          94.586

          43.64

          55.909

          [18]

          CC

          21

          21.529

          0.68

          6.705

          20

          54.435

          10.5

          4.720

          212

          213.236

          43.82

          48.941

          CC-SN

          27

          21.841

          0.52

          6.544

          35

          55.695

          15.2

          4.983

          218

          214.760

          36.59

          50.464

          CC-ST

          42

          22.081

          0.83

          6.407

          40

          58.013

          20

          5.562

          225

          216.027

          36.81

          51.976

          GG

          23

          20.936

          0.67

          6.407

          50

          54.148

          22.2

          4.689

          207

          211.617

          38.85

          45.367

          GG-SN

          35

          21.201

          0.65

          6.145

          60

          54.938

          21.5

          4.928

          230

          212.991

          50.88

          47.206

          GG-ST

          39

          21.403

          0.79

          5.906

          70

          56.946

          25

          5.554

          259

          214.141

          56.19

          49.083

          [44]

          S0L0

          19.62

          17.217

          3.48

          16.280

          39.24

          12.870

          8.83

          12.473

          49.05

          27.432

          30.25

          29.092

          S0L5

          29.43

          17.217

          3.7

          16.280

          63.77

          12.870

          10.42

          12.473

          88.29

          27.432

          21.22

          29.092

          S1L0

          22

          17.217

          5.82

          16.280

          39.6

          12.870

          11.2

          12.473

          58.8

          27.432

          48.16

          29.092

          S1L5

          49.05

          17.217

          10

          16.280

          93.2

          12.870

          21.1

          12.473

          112.82

          27.432

          34.92

          29.092

          [4]

          BY-0-2.5

          0

          25.806

          0

          15.045

          41.46

          32.113

          6.99

          8.622

          48.45

          44.304

          31.2

          25.501

          BY-1-2.5

          10

          25.808

          5

          15.046

          43.52

          32.135

          5.98

          8.639

          50.2

          44.340

          14

          25.507

          BY-2-2.5

          20

          25.783

          15

          15.039

          47.67

          31.932

          5.93

          8.488

          50.8

          44.014

          18.95

          25.450

          BX-1-2.5

          30

          25.808

          20

          15.046

          57.5

          32.135

          7.85

          8.639

          63.6

          44.340

          12.87

          25.507

          BX-2-2.5

          40

          25.783

          20.2

          15.039

          59.82

          31.932

          6.82

          8.488

          67.9

          44.014

          21.08

          25.450

          BX-1-2.5- 360

          50

          25.808

          25

          15.046

          58.95

          32.135

          9.31

          8.639

          64.95

          44.340

          20.64

          25.507

          BX-2-2.5- 360

          60

          25.783

          30

          15.039

          61.57

          31.932

          8.87

          8.488

          70.2

          44.014

          21.55

          25.450

          BX-1-2.5- 180

          70

          25.808

          35

          15.046

          61.37

          32.135

          6.77

          8.639

          70.4

          44.340

          19.7

          25.507

          BX-1-2.5- 180

          80

          25.783

          41

          15.039

          64.46

          31.932

          8.38

          8.488

          74.4

          44.014

          24.2

          25.450

          BY-0-3.5

          15

          28.044

          10

          13.536

          26.41

          36.798

          9.8

          7.575

          30.8

          54.890

          25.73

          27.210

          BY-1-3.5

          20

          28.041

          12

          13.539

          26.2

          36.817

          10.05

          7.587

          30.2

          54.924

          24.75

          27.208

          BY-2-3.5

          25

          28.067

          30

          13.513

          26.99

          36.654

          7.48

          7.483

          30.9

          54.620

          16.15

          27.225

          BX-1-3.5

          30

          28.041

          22

          13.539

          37.96

          36.817

          10.47

          7.587

          41.8

          54.924

          16.82

          27.208

          BX-2-3.5

          35

          28.067

          22

          13.513

          38.2

          36.654

          10.01

          7.483

          42.7

          54.620

          20.8

          27.225

          BX-1-2.5- 360

          40

          28.041

          28

          13.539

          39.84

          36.817

          10.64

          7.587

          44.1

          54.924

          18.55

          27.208

          BX-2-2.5- 360

          45

          28.067

          30

          13.513

          41.19

          36.654

          10.77

          7.483

          45.55

          54.620

          48.64

          27.225

          BX-1-3.5- 180

          50

          28.041

          32

          13.539

          42.22

          36.817

          10.9

          7.587

          45.95

          54.924

          17.1

          27.208

          BX-2-3.5- 180

          60

          28.067

          33.3

          13.513

          42.27

          36.654

          11.03

          7.483

          46.9

          54.620

          27.1

          27.225

          [46]

          M60

          3.5

          13.560

          3.63

          6.504

          6

          5.284

          8.46

          13.611

          8.25

          27.399

          13.44

          6.507

          M60-0.5%

          3.75

          13.560

          3.58

          6.504

          6.25

          5.284

          8.29

          13.611

          8.29

          27.399

          14.81

          6.507

          M60-1.0%

          4.25

          13.560

          3.44

          6.504

          7

          5.284

          7.75

          13.611

          7.75

          27.399

          15.5

          6.507

          M60-1.5%

          4.25

          13.560

          3.55

          6.504

          6.75

          5.284

          6.75

          13.611

          8.06

          27.399

          16.1

          6.507

          [40]

          RA

          14.39

          25.987

          1.26

          16.793

          29.42

          50.414

          7.91

          23.195

          41.68

          55.944

          21.05

          51.147

          RAC3

          16.52

          25.987

          1.41

          16.793

          36.77

          50.414

          9.02

          23.195

          51.48

          55.944

          33.46

          51.147

          RAC5

          21.28

          25.987

          3.67

          16.793

          46.58

          50.414

          10.1

          23.195

          66.19

          55.944

          46.81

          51.147

          RAU3

          32.94

          25.987

          7.98

          16.793

          51.48

          50.414

          11.42

          23.195

          71.09

          55.944

          53.26

          51.147

          RAU5

          36.81

          25.987

          9.23

          16.793

          53.7

          50.414

          10.74

          23.195

          78.45

          55.944

          57.21

          51.147

          RB

          28.32

          32.465

          3.68

          17.541

          39.22

          55.794

          8.11

          26.456

          53.93

          53.934

          31.28

          54.256

          RBC3

          30.95

          32.465

          4.71

          17.541

          51.48

          55.794

          11.35

          26.456

          61.29

          53.934

          36.23

          54.256

          RBC5

          32.17

          32.465

          4.97

          17.541

          53.24

          55.794

          12.41

          26.456

          63.74

          53.934

          56.91

          54.256

          RBU3

          33.69

          32.465

          9.35

          17.541

          58.8

          55.794

          12.85

          26.456

          88.25

          53.934

          61.04

          54.256

          RBU5

          39.41

          32.465

          11.14

          17.541

          63

          55.794

          12.69

          26.456

          100.51

          53.934

          65.59

          54.256

          [28]

          B1

          11.18

          13.134

          20

          4.652

          11.6

          34.482

          15

          18.825

          100.68

          104.791

          36.9

          33.809

          B2

          10.13

          12.185

          7

          4.322

          11.5

          26.666

          15.1

          17.065

          97.55

          98.845

          38

          34.279

          B3

          8.6

          10.884

          5

          3.844

          11.5

          16.437

          14

          14.862

          90.7

          90.730

          37.2

          34.990

          B4

          7.02

          9.640

          5.6

          3.362

          11.7

          7.144

          14.1

          12.965

          84.26

          83.013

          34.1

          35.727

          B5

          11.39

          12.278

          22

          4.355

          11.8

          27.419

          15.3

          17.232

          101.64

          99.428

          41.65

          34.231

          B6

          9.49

          11.024

          10

          3.897

          12

          17.510

          20.2

          15.088

          94.51

          91.600

          40.4

          34.910

          B7

          7.44

          9.874

          11

          3.454

          11.7

          8.857

          16

          13.307

          86.8

          84.460

          31.4

          35.585

          B8

          6.16

          8.610

          10

          2.943

          12.7

          -0.234

          21

          11.532

          78.63

          76.644

          22.5

          36.371

          B9

          6.79

          8.771

          9.95

          3.010

          12.5

          0.901

          22.2

          11.748

          82.66

          77.638

          28

          36.269

          B10

          8.2

          8.922

          14

          3.072

          12.8

          1.974

          19

          11.954

          89.45

          78.573

          38.75

          36.173

          B11

          6.65

          8.725

          11

          2.991

          12.7

          0.577

          14.5

          11.686

          82.11

          77.355

          27.2

          36.298

          B12

          7.91

          8.819

          13

          3.030

          12.9

          1.244

          13

          11.814

          86.68

          77.937

          36.95

          36.238

          [29]

          CB1

          12

          29.190

          1.15

          16.677

          43

          62.162

          12.1

          20.034

          48

          88.746

          31.5

          34.427

          CB2

          13

          37.154

          1.29

          21.403

          90.5

          107.443

          13.2

          22.909

          94.5

          127.361

          25.2

          27.980

          CB3

          20

          44.592

          1.38

          25.343

          117.5

          140.232

          12.2

          28.671

          117.5

          150.377

          12.2

          24.976

          HT1

          35

          28.449

          1.73

          14.792

          124

          61.036

          10.8

          13.421

          130

          82.848

          17.1

          30.296

          HT2

          45

          34.265

          2.23

          17.699

          135

          96.980

          9.6

          18.043

          176

          117.410

          14.7

          28.174

          HT3

          45

          40.768

          1.63

          20.757

          178

          125.658

          13.2

          24.782

          194

          140.420

          17.4

          27.713

          HC1

          11

          25.110

          1.01

          14.595

          40

          64.868

          9.5

          8.228

          53

          85.807

          74

          21.733

          HC2

          15

          28.634

          0.88

          16.873

          90.5

          92.389

          12.6

          16.156

          103

          117.092

          40.9

          22.355

          HC3

          15

          34.212

          0.74

          19.013

          112.5

          113.013

          12.7

          23.623

          126

          139.770

          40.9

          25.037

          RC1

          10

          31.526

          0.74

          10.370

          43

          65.624

          11.1

          15.053

          59

          67.618

          86.7

          40.068

          RC2

          10

          39.709

          0.34

          10.195

          90

          101.327

          10.1

          22.385

          104

          103.882

          39.1

          47.002

          RC3

          15

          46.905

          0.75

          11.733

          110

          126.622

          10.6

          29.875

          131

          128.406

          27.3

          51.524

          [13]

          CCO.20

          0

          24.685

          0

          11.288

          0

          16.004

          0

          25.816

          11.81

          29.630

          110.9

          91.140

          SFRCO.20

          5

          24.921

          3

          10.987

          10

          17.736

          5

          26.422

          18.72

          31.393

          64.5

          91.734

          CCO.30

          10.5

          28.095

          12.5

          11.392

          20

          20.922

          18

          27.808

          18.48

          31.773

          109.5

          93.248

          SFRCO.30

          13

          28.591

          14

          10.617

          30

          25.014

          11

          29.342

          21.29

          35.698

          86.2

          94.811

          CCO.40

          20

          32.273

          20.3

          12.015

          40

          26.084

          31

          29.679

          25.39

          33.312

          107.9

          94.703

          SFRCO.40

          26

          32.119

          22

          12.379

          50

          24.720

          35

          29.084

          27.54

          31.911

          95.3

          93.934

          CCO.53

          32

          37.845

          24

          12.295

          55

          35.815

          42

          33.115

          31.56

          38.390

          104..3

          97.299

          SFRCO.53

          40.5

          38.022

          39.9

          11.727

          60

          37.835

          59

          34.064

          34.85

          40.227

          157

          98.513

          CCO.81

          55

          51.716

          41

          15.880

          65

          55.736

          48

          37.989

          47.9

          47.300

          92.1

          96.425

          SFRCO.81

          68

          51.701

          40

          16.181

          70.5

          55.163

          53

          37.583

          47.01

          46.785

          176.5

          95.720

          CC1.06

          74

          65.502

          35

          17.798

          76

          78.647

          65

          44.502

          58.11

          60.894

          56.7

          97.819

          SFRC1.06

          83

          65.493

          22

          16.324

          80

          82.002

          78

          46.789

          61.77

          63.014

          175.7

          101.423

          CCI.60

          97

          94.528

          39

          21.101

          84

          111.583

          36

          53.762

          81.35

          84.692

          41.5

          98.321

          SFRC1.60

          102

          94.423

          10.5

          21.664

          99

          108.373

          21

          52.166

          82.4

          83.391

          164.2

          96.471

          CC2.02

          107

          112.715

          15

          22.184

          101

          102.206

          17

          50.498

          87.85

          89.509

          59.4

          93.336

          SFRC2.02

          110

          112.910

          25

          21.918

          105

          105.421

          14

          51.879

          97.3

          90.702

          126

          94.677

          CC2.13

          113

          116.883

          34

          21.384

          110

          104.229

          70

          52.246

          93.74

          92.221

          70.7

          95.130

          SFRC2.13

          118

          117.261

          28

          20.982

          117

          109.807

          63

          54.602

          100.2

          94.374

          101.2

          97.403

          CC2.50

          120

          127.867

          14

          19.030

          115

          92.161

          58

          51.402

          100.82

          94.415

          70

          95.192

          SFRC2.50

          122

          127.652

          15.9

          19.130

          125

          89.799

          78.5

          50.432

          108.15

          93.446

          133

          94.251

          [8]

          B1 0 -6

          12

          22.453

          14.1

          18.945

          4.8

          15.138

          5.9

          12.361

          4.14

          17.574

          57.8

          26.963

          B2 0-8

          15

          23.022

          15

          18.906

          7.1

          15.471

          9.1

          12.446

          6.05

          19.534

          38

          27.806

          B3 0-10

          18

          21.961

          15.9

          19.406

          9.6

          16.646

          11.1

          13.177

          6.48

          16.480

          31.8

          26.875

          B4-0.5/45- 10

          20

          21.336

          19

          20.133

          11.6

          17.094

          11.5

          12.333

          4.78

          17.499

          19.8

          26.422

          B5-1/45-6

          25

          21.791

          20.6

          19.564

          13.2

          16.677

          12.3

          12.943

          7.82

          16.584

          26.9

          26.701

          B6-2/45-6

          29

          22.243

          28

          19.170

          10.1

          15.246

          6.5

          12.079

          4.48

          17.776

          23.2

          26.759

          B7-2/45-8

          34

          22.832

          31

          19.136

          11.2

          15.602

          7.9

          12.180

          4.21

          19.805

          24.9

          27.642

          B8-2/45-10

          36

          21.746

          33

          19.607

          14.2

          16.692

          13.4

          12.883

          7.71

          16.624

          24.8

          26.660

          B9-2/80-10

          40

          21.746

          37

          19.607

          14.9

          16.692

          14.2

          12.883

          7.85

          16.624

          16.3

          26.660

          [30]

          B1-0.0F

          5

          24.841

          5

          29.126

          156

          162.619

          6

          20.701

          188.2

          208.545

          13.56

          7.183

          B2-0.0F

          13

          24.841

          12.5

          29.126

          178

          162.619

          9.74

          20.701

          206.4

          208.545

          14.78

          7.183

          B1-0.5F

          15

          24.841

          15

          29.126

          156

          162.619

          3.91

          20.701

          194.8

          208.545

          12.65

          7.183

          B2-0.5F

          17

          24.841

          17.5

          29.126

          178

          162.619

          6.64

          20.701

          198.8

          208.545

          10.82

          7.183

          B1-1.0F

          25

          24.841

          20.1

          29.126

          156

          162.619

          2.94

          20.701

          211.7

          208.545

          9.65

          7.183

          B2-1.0F

          34

          24.841

          23.5

          29.126

          178

          162.619

          4.23

          20.701

          234.4

          208.545

          13.23

          7.183

        2. Proposed Regression Equations for Fibre Reinforced Concrete Beams Incorporated Micro Fillers and Nano Fillers are presented in table 9

      Table 9. Regression Equation

      PARAMETERS

      REGRESSION EQUATIONS

      FIRST CRACK LOAD

      41.538+0.27b-0.09d+0.00l+0.06fck-0.07fy-0.03Ast-0.10Asv

      DEFLECTION AT FIRST CRACK LOAD

      88.97-0.122b+0.44d-0.02l+0.17fck-0.10fy+0.03Ast-1.35Asv

      YIELD LOAD

      76.40-0.37b+0.46d-0.01l+0.22fck-0.10fy+0.03Ast-0.26Asv

      DEFLECTION AT YIELD LOAD

      6.41-0.05b+0.02d+0.00l+0.01fck+0.00fy+0.01Ast-0.09Asv

      ULTIMATE LOAD

      65.166-0.117b+0.44d-0.02l+0.23fck-0.14fy+0.01Ast+0.91Asv

      DEFLECTION AT ULTIMATE LOAD

      27.161+0.22b-0.08d+0.02l+0.09fck-0.08fy-0.00Ast-0.37Asv

    5. RESULTS AND DISCUSSION

For predicting the performance of fibre reinforced concrete beams incorporating nano fillers and micro fillers, regression analysis using origin pro has been used in this study. The proposed regression equations performed well for the first crack load, deflection at first crack load, yield load, deflection at yield load, ultimate load and deflections at ultimate load for fibre reinforced concrete beams incorporating nano fillers and micro fillers and the comparison of experimental and regression results for fibre reinforced concrete beams incorporating nano fillers and micro fillers are shown in table 9. To evaluate the accuracy of the models, scatter plots were drawn between the experimental and predicted results are shown in Fig13 and Fig 14. illustrate that experimental and predicted values of performance of fibre reinforced concrete beams incorporating nano fillers and micro fillers correlate well and the representative points in the scatter plots correlate to the line of equality.

Figure 13. Scatter plots of first crack load, deflection at first crack load and yield load.

Figure 14. Scatter plot of Deflection at yield load, Ultimate load and Deflection at Ultimate load.

Table 10 Statistical Indicators

OUTPUT PARAMETERS

Fibre reinforced concrete beams incorporating nano fillers and micro fillers

RMSE

R2

MAPE

First Crack load

4.56

0.97

11.73

Deflection at First Crack load

3.17

0.97

12.37

Yield load

10.89

0.98

10.27

Deflection at Yield load

3.12

0.98

12.03

Ultimate load

12.57

0.97

12.70

Deflection at Ultimate load

4.99

0.98

12.77

6.CONCLUSION

  1. Fiber reinforced concrete beams incorporating nano fillers and micro fillers exhibit improved performance in terms of load and deformation capacity.

  2. ANN model constituted with Bayesian Regularization Algorithm gives better results than the other two algorithm.

  3. ANN modeling has proved to be a reliable tool for predicting the performance of FRC beams incorporating nano fillers and micro fillers.

REFERENCES

[1] Anwar Mohammed (2014), Influence of Nano Materials on Flexure Behavior and Compressive Strength of Concrete, Housing and Building Research Journal, Vol.11, pp 113-128.

[2] Alberto Meda et al (2012),Flexural Behavior of RC Beams in Fiber Reinforced Concrete, Elsevier Composites : Part B , Vol.43, pp 2930-2937. [3] By Byung Hwan Oh , Flexural Analysis of Reinforced Concrete Beams Containing Steel Fibers, Journal of Structural Engineering , Vol.10, pp :

2821-2835.

[4] Barzin Mobasher, Yiming Yao, Chote Sornakom (2015), Analytical Solutions for Flexural Design of Hybrid Steel Fiber Reinforced Concrete Beams, Engineering Structures, Vol.100,pp 167-177.

[5] Bimal Babu Adhikary, Hiroshi Mutsuyoshi (2006), Prediction of Shear Strength of Steel Fiber RC Beams Using Neural Networks, Construction and Building Materials, Vol.20,pp 801-811.

[6] Benmokrane.B, Chaallal.O,Masmoudi.R 1995, Glass Fibre Reinforced Plastic Rebars for Concrete Structures, Construction and Building Materials, Vol.9, pp 353-364.

[7] Balaji.E, Senthil Selvan.S, Vishnu Prasad .P.R,An Experimental Study on the Effect of Nano Silica on Strength and Durability of Concrete,

International Journal of Civil Engineering and Technology, Vol.8, pp 1182-1188.

[8] By Byung Hwan Oh (1992), Flexural Analysis of Reinforced Concrete Beams Containing Steel Fibers, Journal for Structural Engineering, Vol.10,pp 2821-2835.

[9] Chao Xu,Ping Zhou Cao,Kai Wu,Shi Qi Lin,Ding Guo Yang (2019),Experimental Investigation of the Behavior Composite Steel Concrete Composite Beams Containing Different Amounts of Steel Fibers and Conventional Reinforcement, Construction and Building Materials, Vol.202,pp 23-26.

[10] Dancygier A N and Savir Z. (2011), Effects of Steel Fibres on Shear Behaviour of High- Strength Reinforced Concrete Beams, Advances in Structural Engineering, Vol 14.

[11] Doo Yeol Yoo,Young Soo Yoon (2015), Structural Performance of Ultra High Performance Concrete Beams With Different Steel Fibers,Engineering Structures, Vol.102,pp 409-423.

[12] Doo Yeol Yoo, Hyun Oh Shin, Jun Mo Yang, Young Soo Yoon 2014, Material and Bond Properties of Ultra High Performance Fibre Reinforced Concrete with Micro Steel Fibres, Elsevier Composite Part B, Vol 58, pp 122-133.

[13] Deepthi Dennison (2014), Effect of Metakaolin on the Structural Behavior of Normal and Steel Fiber Reinforced Concrete Beams, International Journal of Science and Engineering Research, Vol.5.

[14] Daniel C.T.Cardoso,Gabriel B.S.Pereira,Flavio A.Silva,JULIO J.H.Silva Filho (2019),Influence of Steel Fibers on the Flexural Behavior of RC Beams with Low Reinforcing Ratios, Composite Structures, Vol.222,pp 110-926.

[15] Delsye .C.L.Teo, Md.Abdul Mannan, John V.K uriann (2006), Flexural Behavior of Reinforced Lightweight Concrete Beams Made with Oil Palm Shell, Journal of Advanced Concrete Technology, Vol.4, pp 1-10.

[16] Ehsan Nikbakht, Amin Al Fakih, Chieng Chew Hui (2019), An Experimental Investigation on the Shear and Flexural Behavior of Steel Reinforced HPSCC Beams, Structures, Vol.19,pp 286-295.

[17] Frank Kusel, Elsabe Kearsley.( 2019), Effect of Steel Fibres in Combination with Different Reinforcing Ratios on the Performance of Continuous Beams, Construction and Building Materials, Vol 227, pp 116-533.

[18] Farid Hasan Nattaj, Mahdi Nematzadeh (2017), The Effect of Forta Ferro and Steel Fibers on Mechanical Properties of High Strength Concrete with and without Silica Fume and Nano silicaConstruction and Building Materials, Vol.137,pp 557-572.

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