Abstract
Deflection is an important design parameter for structures subjected to service load. This paper provides an explicit expression for effective moment of inertia considering cracking, for uniformly distributed loaded reinforced concrete (RC) beams. The proposed explicit expression can be used for rapid prediction of shortterm deflection at service load. The explicit expression has been obtained from the trained neural network considering concrete cracking, tension stiffening and entire practical range of reinforcement. Three significant structural parameters have been identified that govern the change in effective moment of inertia and therefore deflection. These three parameters are chosen as inputs to train neural network. The training data sets for neural network are generated using finite element software ABAQUS. The explicit expression has been validated for a number of simply supported and continuous beams and it is shown that the predicted deflections have reasonable accuracy for practical purpose. A sensitivity analysis has been performed, which indicates substantial dependence of effective moment of inertia on the selected input parameters.
Keywords:
Concrete cracking; deflection; finite element analysis; moment of inertia; neural network; reinforced concrete; tension stiffening
Nomenclatures
A_{st}, A_{sb} ,area of top and bottom reinforcement, respectively
B, D width and depth of beam
B_{f}, D_{f} width and depth of flange
B_{w}, D_{w} width and depth of web
E_{c}, E_{s} modulus of elasticity of concrete and steel, respectively
I_{e} effective moment of inertia
I_{g}, I_{cr} moment of inertia of gross and fully cracked transformed cross section, respectively
I_{j} ^{jth} input parameter
L length of beam
M_{cr}, M_{e} minimum moment at which the cracking takes place at a crosssection in the beam and applied (elastic) moment, respectively
O_{1} output parameter
bias bias of hidden or output neuron
d_{EXP}, d_{FEM}, d_{NN} midspan deflection from experiments, FEM, and neural network/explicit expression, respectively
d_{t}, d_{b} effective concrete cover at top and bottom. respectively
f_{t} tensile strength of concrete
cylindrical compressive strength of concrete at 28 days
h_{k} ^{kth} hidden neuron
m constant
q number of input parameters
r number of hidden neurons
w, n uniformly distributed load and modular ratio, respectively
w_{cr} minimum load at which the cracking takes place in the beam cracking
weight of the link between I_{j} and h_{k}
weight of the link between h_{k} and O _{1}
εcr, εu cracking strain, and maximum tensile strain of concrete, respectively
ρt, ρc percentage tension and compression reinforcement, respectively
Subscript
j input neuron number
k hidden neuron number or function number
o output neuron number
Superscript
ho connection between hidden and output layers
ih connection between input and hidden layers
1 INTRODUCTION
Deflection is an important parameter to check the serviceability criteria of structure. The short term deflection is generally calculated using effective moment of inertia of entire span at service load. The equations for effective moment of inertia, available in literature, are mainly based on two approaches: (i) springs in parallel and (ii) springs in series (Kalkan, 2010Kalkan, İ., (2010). Deflection prediction for reinforced concrete beams through different effective moment of inertia expressions. International Journal of Engineering Research Development 2(1): 7280.). The stiffnesses of the uncracked and cracked portions are averaged in the springs in parallel approach (Branson, 1965Branson, D.E., (1965). Instantaneous and timedependent deflections of simple and continuous reinforced concrete beams, HPR Report No.7(1), Alabama Highway Department, Bureau of Public Roads, Alabama: 178.; AlZaid et al., 1991AlZaid, R.Z., AlShaikh, A.H., AbuHussein, M.M., (1991). Effect of loading type on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 88(2): 184190.; AlShaikh and AlZaid, 1993AlShaikh, A.H., AlZaid, R.Z., (1993). Effect of reinforcement ratio on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 90(2): 144149.; SAAAS 3600, 1994Standards Association of Australia (SAA) AS 3600, (1994). Australian standard for concrete structures, Australia.; TS 500, 2000Turkish Standards Institute TS 500, (2000). Requirements for design and construction of reinforced concrete structures, Turkey.; CSAA23.3, 2004Canadian Standards Association (CSA) A23.3, (2004). Design of concrete structures, Canada.; ACI 318, 2005American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA.; AASHTO, 2005American Association of State Highway and Transportation Officials (AASHTO), (2005). AASHTO LRFD bridge design specifications (SI units), USA.), whereas the flexibilities of the uncracked and cracked portions are averaged in the springs in series approach (Ghali, 1993Ghali, A., (1993). Deflection of reinforced concrete members: A critical review. ACI Structural Journal 90(4): 364373.; CEN Eurocode 2, 2004European Committee for Standardization (CEN) Eurocode 2 BS EN 199211, (2004). Design of concrete structuresPart 11: General rules and rules for buildings, Belgium.; Bischoff, 2005Bischoff, P.H., (2005). Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars. Journal of Structural Engineering ASCE 131(5): 752762.; Bischoff and Scanlon, 2007Bischoff, P.H., Scanlon, A., (2007). Effective moment of inertia for calculating deflections of concrete members containing steel reinforcement and fiberreinforced polymer reinforcement. ACI Structural Journal 104(1): 6875.; Bischoff, 2007Bischoff, P.H., (2007). Rational model for calculating deflection of reinforced concrete beams and slabs. Canadian Journal of Civil Engineering 34(8): 9921002.).
Considering parallel springs approach first, the following equation of effective moment of inertia (I_{e} ) in terms of fully cracked and uncracked moment of inertia was originally proposed by Branson (1965)Branson, D.E., (1965). Instantaneous and timedependent deflections of simple and continuous reinforced concrete beams, HPR Report No.7(1), Alabama Highway Department, Bureau of Public Roads, Alabama: 178. for simply supported beams as
where, M_{cr} = minimum moment at which the cracking takes place at a crosssection in the beam; M_{e} = applied (elastic) moment along the span; I_{g} = moment of inertia of the gross cross section; I_{cr} = moment of inertia of the fully cracked transformed cross section and m = constant.
Eq. (1) was derived empirically based on the experimental test results of simply supported rectangular reinforced concrete (RC) uniformly loaded beams with tension reinforcement, ρt = 1.65% and ratio of moment of inertia of the fully cracked transformed cross section and moment of inertia of the gross cross section, I_{cr} /I_{g} = 0.45 at maximum applied (elastic) moment equal to 2.5M_{cr} (Branson, 1965Branson, D.E., (1965). Instantaneous and timedependent deflections of simple and continuous reinforced concrete beams, HPR Report No.7(1), Alabama Highway Department, Bureau of Public Roads, Alabama: 178.).
Eq. (1) has been adopted in many international standards and codes (SAAAS 3600, 1994Standards Association of Australia (SAA) AS 3600, (1994). Australian standard for concrete structures, Australia.; TS 500, 2000Turkish Standards Institute TS 500, (2000). Requirements for design and construction of reinforced concrete structures, Turkey.; CSAA23.3, 2004Canadian Standards Association (CSA) A23.3, (2004). Design of concrete structures, Canada.; ACI 318, 2005American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA.; AASHTO, 2005American Association of State Highway and Transportation Officials (AASHTO), (2005). AASHTO LRFD bridge design specifications (SI units), USA.) to calculate I_{e} and therefore deflection, taking m = 3. Some researchers (Bischoff, 2005Bischoff, P.H., (2005). Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars. Journal of Structural Engineering ASCE 131(5): 752762.; Gilbert, 1999Gilbert, R.I., (1999). Deflection calculation for reinforced concrete Structures Why we sometimes get it wrong. ACI Structural Journal 96(6): 10271033.; Scanlon et al., 2001Scanlon, A., Orsak, D.R.C., Buettner, D.R., (2001). ACI code requirements for deflection control: A critical review. ACI Special publications 20301: 114.; Gilbert, 2006Gilbert, R.I., (2006). Discussion of 'Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars' by P.H. Bischoff. Journal of Structural Engineering ASCE 132(8): 13281330.) found out that Eq. (1) with m = 3 calculates effective moment of inertia accurately in case of medium to high tension reinforcement(ρt > 1%), while it overestimates effective moment of inertia for low tension reinforcement (ρt < 1%).
AlZaid et al. (1991)AlZaid, R.Z., AlShaikh, A.H., AbuHussein, M.M., (1991). Effect of loading type on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 88(2): 184190.experimentally proved that the value of m in Eq. (1)depends on the loading configurations and suggested m = 2.8 (in Eq. (1)) for uniformly distributed load when M_{c} > 1.5M_{cr} . The value of m was found to change from about 3 to 4.3 for moderatelyreinforced concrete beams (ρt = 1.2%, I_{cr} /I_{g} = 0.34) in the range of M_{cr} < M_{c} < 1.5M_{cr} . AlShaikh and AlZaid (1993)AlShaikh, A.H., AlZaid, R.Z., (1993). Effect of reinforcement ratio on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 90(2): 144149. performed experiments on midspan point loaded beams with varying reinforcement. The values of m was found to vary from about 1.8 to 2.5 for lightly reinforced beams (ρt = 0.8%, I_{cr} /I_{g} = 0.22) in the range of 1.5M_{cr} < M_{c} < 4M_{cr} , while for the heavily reinforced beams (ρt = 2%, I_{cr} /I_{g} = 0.44), m varied in a range of 0.9 to 1.3. They also suggested m = 3  0.8ρt incorporating reinforcement effect in Eq. (1)for point loaded beams. AlZaid et al. (1991)AlZaid, R.Z., AlShaikh, A.H., AbuHussein, M.M., (1991). Effect of loading type on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 88(2): 184190.; AlShaikh and AlZaid (1993)AlShaikh, A.H., AlZaid, R.Z., (1993). Effect of reinforcement ratio on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 90(2): 144149. also proposed to calculate I_{e} based on cracked length incorporating reinforcement and loading effects respectively.
Next, consider the springs in series approach. The models based on this approach (Bischoff, 2005Bischoff, P.H., (2005). Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars. Journal of Structural Engineering ASCE 131(5): 752762.; Bischoff and Scanlon, 2007Bischoff, P.H., Scanlon, A., (2007). Effective moment of inertia for calculating deflections of concrete members containing steel reinforcement and fiberreinforced polymer reinforcement. ACI Structural Journal 104(1): 6875.; Bischoff, 2007Bischoff, P.H., (2007). Rational model for calculating deflection of reinforced concrete beams and slabs. Canadian Journal of Civil Engineering 34(8): 9921002.) take into account tension stiffening effect in concrete for calculating I_{e} . The deflections obtained by the expression proposed by Bischoff (2005)Bischoff, P.H., (2005). Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars. Journal of Structural Engineering ASCE 131(5): 752762. have been found in good agreement with experimental deflections for lightly reinforced beams (ρt < 1%) (Gilbert, 2006Gilbert, R.I., (2006). Discussion of 'Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars' by P.H. Bischoff. Journal of Structural Engineering ASCE 132(8): 13281330.; Bischoff and Scanlon, 2007Bischoff, P.H., Scanlon, A., (2007). Effective moment of inertia for calculating deflections of concrete members containing steel reinforcement and fiberreinforced polymer reinforcement. ACI Structural Journal 104(1): 6875.).
Kalkan (2010)Kalkan, İ., (2010). Deflection prediction for reinforced concrete beams through different effective moment of inertia expressions. International Journal of Engineering Research Development 2(1): 7280. found out that the expressions given by Eq. (1) and Bischoff (2005)Bischoff, P.H., (2005). Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber reinforced polymer bars. Journal of Structural Engineering ASCE 131(5): 752762. estimate deflections of moderatelyreinforced to highlyreinforced concrete beams (ρt > 1%) accurately on using the experimental value of cracking moment which, however, is difficult to obtain for each and every case.
It is observed from the review that no single approach or model is directly applicable for the entire range of practical reinforcement. Therefore, development of an approach for rapid estimation of the midspan deflections in uniformly distributed loaded RC beams considering entire practical range of reinforcement at service load is desirable. The approach should be simple to use requiring a minimal computational effort but must give accuracy that is acceptable for practical applications. The application of neural network can be such an alternate approach. For generation of training data for neural networks, finite element technique may be used.
Nowadays, neural networks are being extensively applied in the field of structural engineering. Some of the recent applications of neural networks in the field of structural engineering include prediction of time effects in RC frames (Maru and Nagpal, 2004Maru, S., Nagpal, A.K., (2004). Neural network for creep and shrinkage deflections in reinforced concrete frames. Journal of Computing in Civil Engineering ASCE 18(4): 350359.), prediction of damage detection in RC framed buildings after earthquake (Kanwar et al., 2007Kanwar, V., Kwatra, N., Aggarwal, P., (2007). Damage detection for framed RCC buildings using ANN modelling. International Journal of Damage Mechanics 16(4): 457472.), structural health monitoring (Min et al., 2012Min, J., Park, S., Yun, C.B., Lee, C.G, Lee, C., (2012). Impedancebased structural health monitoring incorporating neural network technique for identification of damage type and severity. Engineering Structures 39: 210220.; Kaloop and Kim, 2014Kaloop, M.R., Kim, D.K., (2014). GPSstructural health monitoring of a long span bridge using neural network adaptive filter. Survey Review 16(334): 714.), bending moment and deflection prediction in composite structures (Chaudhary et al., 2007Chaudhary, S., Pendharkar, U., Nagpal, A.K., (2007). Bending moment prediction for continuous composite beams by neural networks. Advances in Structural Engineering 10(4): 439454., 2014Chaudhary, S., Pendharkar, U., Patel, K.A., Nagpal, A.K., (2014). Neural networks for deflections in continuous composite beams considering concrete cracking. Iranian Journal of Science and Technology: Transactions of Civil Engineering 38(C1+): 205221.; Pendharkar et al., 2007Pendharkar, U., Chaudhary, S., Nagpal, A.K., (2007). Neural network for bending moment in continuous composite beams considering cracking and time effects in concrete. Engineering Structures 29(9): 20692079., 2010Pendharkar, U., Chaudhary, S., Nagpal, A.K., (2010). Neural networks for inelastic midspan deflections in continuous composite beams. Structural Engineering and Mechanics 36(2): 165179., 2011Pendharkar, U., Chaudhary, S., Nagpal, A.K., (2011). Prediction of moments in composite frames considering cracking and time effects using neural network models. Structural Engineering and Mechanics 39(2): 267285.; Tadesse et al., 2012Tadesse, Z., Patel, K.A., Chaudhary, S., Nagpal, A.K., (2012). Neural networks for prediction of deflection in composite bridges. Journal of Constructional Steel Research 68(1): 138149.; Gupta et al., 2013Gupta, R.K., Patel, K.A., Chaudhary, S., Nagpal, A. K., (2013). Closed form solution for deflection of flexible composite bridges. Procedia Engineering 51: 7583.), predicting the creep response of a rotating composite disc operating at elevated temperature (Gupta et al., 2007Gupta, V.K., Kwatra, N., Ray, S., (2007). Artificial neural network modeling of creep behavior in a rotating composite disc. Engineering Computations 24(2): 151164.), optimum design of RC beams subjected to cost (Sarkar and Gupta, 2009Sarkar, K., Gupta, M.K., (2009). Comparative study of optimum design approaches and artificial neural network based optimum design of a singly reinforced concrete beam. Journal of Structural Engineering SERC 36(5): 235242.), static model identification (Kim et al., 2009Kim, D.K., Kim, D.H., Cui, J., Seo, H.Y., Lee, Y.H., (2009). Iterative neural network strategy for static model identification of an FRP deck. Steel and Composite Structures 9(5): 445455.), response prediction of offshore floating structure (Uddin et al., 2012Uddin, M.A., Jameel, M., Razak, H.A., Islam, A.B.M., (2012). Response prediction of offshore floating structure using artificial neural network. Advanced Science Letters 14(1): 186189.), prediction of deflection in high strength selfcompacting concrete deep beams (Mohammadhassani et al., 2013aMohammadhassani, M., NezamabadiPour, H., Jumaat, M.Z., Jameel, M., Arumugam A.M.S., (2013). Application of artificial neural networks (ANNs) and linear regressions (LR) to predict the deflection of concrete deep beams. Computers and Concrete 11(3): 237252.; 2013bMohammadhassani, M., NezamabadiPour, H., Jumaat, M.Z., Jameel, M., Hakim, S.J.S., Zargar, M., (2013). Application of the ANFIS model in deflection prediction of concrete deep beam. Structural Engineering and Mechanics 45(3): 319332.) and prediction of energy absorption capability and mechanical properties of fiber reinforced selfcompacting concrete containing nanoSilica particles (Tavakoli et al., 2014aTavakoli, H.R., Omran, O.L., Kutanaei, S.S., Shiade, M.F., (2014). Prediction of energy absorption capability in fiber reinforced selfcompacting concrete containing nanosilica particles using artificial neural network. Latin American Journal of Solids and Structures 11(6): 966979.; 2014bTavakoli, H.R., Omran, O.L., Shiade, M.F., Kutanaei, S.S., (2014). Prediction of combined effects of fibers and nanosilica on the mechanical properties of selfcompacting concrete using artificial neural network. Latin American Journal of Solids and Structures 11(11): 19061923.). These studies reveal the strength of neural networks in predicting the solutions of different structural engineering problems.
This paper presents an alternative approach for estimating effective moment of inertia which is neither spring in parallel nor spring in series approach. Neural network model is developed, at service load, for predicting effective moment of inertia (and deflection), in a RC beam considering entire practical range of tension and compression reinforcement, tension stiffening and flexural concrete cracking. The data sets for training, validating and testing are generated using finite element models. The finite element models have been developed in ABAQUS (2011)ABAQUS 6.11, Standard user's manuals, USA, 2011. software and validated with the experimental results available in literature. Explicit expression has been obtained based on developed neural network model which can be used in design offices by practicing engineers. The proposed neural network/explicit expression has been validated for a number of simply supported and continuous RC beams. Sensitivity analysis has been performed to understand the influence of relevant parameters on effective moment of inertia.
2 FINITE ELEMENT MODEL AND ITS VALIDATION
The finite element model (FEM) has been developed using the ABAQUS (2011)ABAQUS 6.11, Standard user's manuals, USA, 2011. software. The beam has been modelled using B21 elements (2node linear Timoshenko beam element). Under service load, the stressstrain relationship of concrete is assumed to be linear in compression. Concrete has been considered as an elastic material in tension before cracking and softening behaviour is assumed after cracking (Figure 1). Further, at service load, the stress in reinforcement is assumed to be in the linear range. The steel reinforcement has been embedded into the concrete using "REBAR" option in which a perfect bond is considered between steel reinforcement and concrete. In order to consider cracking and tension stiffening, the smeared crack model has been used. Tension stiffening has been defined using postfailure stressstrain data proposed by Gilbert and Warner (1978)Gilbert, R.I., Warner, R.F., (1978). Tension stiffening in reinforced concrete slabs. Journal of Structural Division ASCE 104(12): 18851900.. A high shear stiffness has been assumed to neglect the shear deformations.
The results of FEM have been compared with the experimental results (midspan deflections of the beam under increasing uniformly distributed load, w, after the cracking of the concrete) reported by AlZaid et al. (1991)AlZaid, R.Z., AlShaikh, A.H., AbuHussein, M.M., (1991). Effect of loading type on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 88(2): 184190. for a simply supported beam (VB) with 2.5 m clear span (effective span = 2.62 m) and crosssectional dimensions B × D = 200 × 200 mm (Figure 2). The other properties considered are: cylindrical compressive strength of concrete at 28 days, = 38.2 N/mm^{2}; modulus of elasticity of concrete, E_{c} = 2.96 × 10^{4} N/mm^{2}; modulus of elasticity of steel, E_{s} = 2 × 10^{5}N/mm^{2}; tensile strength of concrete, f_{t} = 3.47 N/mm^{2}; cracking moment, M_{cr} = 5.2 kNm; area of top reinforcement, A_{st} = 78.54 mm^{2} and area of bottom reinforcement, A_{sb} = 402.12 mm^{2}. The effective concrete cover at top (d_{t} ) and at bottom (d_{b} ) have been taken as 30 mm and 33 mm respectively.
In order to define the smeared crack model, the absolute value of the ratio of uniaxial tensile stress at failure to the uniaxial compressive stress at failure is taken as 0.09. The strain at cracking, εcr is taken as 0.00012 and in view of low/moderate tensile reinforcement, A_{sb} = 402.12 mm^{2} (= 1.2%), the plastic strain is (εu  εcr) taken as 0.0004. For convergence, about 16 elements are required when cracking is considered (Patel et al., 2014Patel, K.A., Chaudhary, S., Nagpal, A.K., (2014). Analyticalnumerical procedure incorporating cracking in RC beams. Engineering Computations 31(5): 9861010.). Results from the developed FEM and experiments are compared in Figure 3. Close agreement is observed between the results from FEM and experiments.
Next, the results have been compared with experimental results reported by Washa and Fluck (1952)Washa, G.W., Fluck, P.G., (1952). Effect of compressive reinforcement on the plastic flow of reinforced concrete beams. ACI Journal 49(10): 89108. for four sets of rectangular crosssectional (Figure 2) simply supported beams: A1,A4; B1,B4; C1,C4; D1,D4 subjected to uniformly distributed loads at service load. Two beams in a set are identical. The crosssectional properties, material properties, span lengths and uniform distributed loads of all four beams have been given in Table 1. Additionally, E_{s} has been assumed as 2 × 10^{5}N/mm^{2}. E_{c} and f_{t} are taken in accordance with ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA.. The midspan deflections obtained from the FEM (d_{FEM}) are in close agreement with the reported experimental deflections (d_{EXP}) as shown in Table 1. The finite element models can therefore be used for generation of data sets.
Properties of simply supported beams with rectangular crosssections, considered for validation of FE model.
3 SAMPLING POINTS AND DATA SETS
For development of neural network, significant parameters need to be identified. Eq. (1) shows that I_{e} /I_{g} explicitly depends on I_{cr} /I_{g} and M_{cr} /M_{a} . It is assumed that I_{e} /I_{g} depends on ρt and ρc(percentage compression reinforcement) also. The value of ρc however depends on ρt and ranges from 0.0 to ρt(n  1)/n, where, n = modular ratio. The value of I_{cr} /I_{g} in turn also depends on the combinations of ρt and ρc. Consider a typical beam crosssection as shown in Figure 2 (B = 300 mm; D = 700 mm; E_{c} = 2.73 × 10^{4} N/mm ^{2}; E_{s} = 2.00 × 10^{5} N/mm^{2}; d_{t} = 30 mm and d_{b} = 33 mm). For this beam, the variations of ρc and I_{cr} /I_{g} with ρt are shown in Figure 4. The parameter M_{cr} /M_{a} depends on the load and moment only.
Taking the above observations into account, ρt is also considered as an input parameter alongwith I_{cr} /I_{g} and M_{cr} /M_{e} . The sampling points of the parameters considered for data generation are shown in Table 2. It may be noted that the combinations of sampling points take into account the different values of the ρc (fourth and left out parameter), corresponding to each value of ρt Considering the equation d_{FEM} = 5_{w}L ^{4}/384E_{c}I_{e} , the output parameter I_{e} /I_{g} is obtained as 5wL ^{4}/384E_{c}d_{FEM}I_{g} .
4 TRAINING OF NEURAL NETWORK
Neural network has been developed for the prediction of effective moment of inertia in RC beams. The neural network chosen is a set of multilayered feedforward networks with neurons in all the layers fully connected in the feed forward manner (Figure 5). The training is carried out using the MATLAB Neural Network toolbox (2009)MATLAB 7.8, (2009). Neural networks toolbox user's guide, USA.. Sigmoid function (logsig) is used as an activation function and the LevenbergMarquardt back propagation learning algorithm (trainlm) is used for training. The back propagation algorithm has been used successfully for many structural engineering applications (Maru and Nagpal, 2004Maru, S., Nagpal, A.K., (2004). Neural network for creep and shrinkage deflections in reinforced concrete frames. Journal of Computing in Civil Engineering ASCE 18(4): 350359.; Kanwar et al., 2007Kanwar, V., Kwatra, N., Aggarwal, P., (2007). Damage detection for framed RCC buildings using ANN modelling. International Journal of Damage Mechanics 16(4): 457472.; Gupta et al., 2007Gupta, V.K., Kwatra, N., Ray, S., (2007). Artificial neural network modeling of creep behavior in a rotating composite disc. Engineering Computations 24(2): 151164.; Pendharkar et al., 2007Pendharkar, U., Chaudhary, S., Nagpal, A.K., (2007). Neural network for bending moment in continuous composite beams considering cracking and time effects in concrete. Engineering Structures 29(9): 20692079.; 2010Pendharkar, U., Chaudhary, S., Nagpal, A.K., (2010). Neural networks for inelastic midspan deflections in continuous composite beams. Structural Engineering and Mechanics 36(2): 165179.; 2011Pendharkar, U., Chaudhary, S., Nagpal, A.K., (2011). Prediction of moments in composite frames considering cracking and time effects using neural network models. Structural Engineering and Mechanics 39(2): 267285.; Chaudhary et al., 2007Chaudhary, S., Pendharkar, U., Nagpal, A.K., (2007). Bending moment prediction for continuous composite beams by neural networks. Advances in Structural Engineering 10(4): 439454.; 2014Chaudhary, S., Pendharkar, U., Patel, K.A., Nagpal, A.K., (2014). Neural networks for deflections in continuous composite beams considering concrete cracking. Iranian Journal of Science and Technology: Transactions of Civil Engineering 38(C1+): 205221.; Sarkar and Gupta, 2009Sarkar, K., Gupta, M.K., (2009). Comparative study of optimum design approaches and artificial neural network based optimum design of a singly reinforced concrete beam. Journal of Structural Engineering SERC 36(5): 235242.; Gupta and Sarkar, 2009Gupta, M.K., Sarkar, K., (2009).Modeling of section forces in a continuous beam using artificial neural. Journal of Structural Engineering SERC 35(6): 416422.; Min et al., 2012Min, J., Park, S., Yun, C.B., Lee, C.G, Lee, C., (2012). Impedancebased structural health monitoring incorporating neural network technique for identification of damage type and severity. Engineering Structures 39: 210220.; Tadesse et al., 2012Tadesse, Z., Patel, K.A., Chaudhary, S., Nagpal, A.K., (2012). Neural networks for prediction of deflection in composite bridges. Journal of Constructional Steel Research 68(1): 138149.; Mohammadhassani et al., 2013aMohammadhassani, M., NezamabadiPour, H., Jumaat, M.Z., Jameel, M., Arumugam A.M.S., (2013). Application of artificial neural networks (ANNs) and linear regressions (LR) to predict the deflection of concrete deep beams. Computers and Concrete 11(3): 237252.; Gupta et al., 2013Gupta, R.K., Patel, K.A., Chaudhary, S., Nagpal, A. K., (2013). Closed form solution for deflection of flexible composite bridges. Procedia Engineering 51: 7583.) and is considered as one of the efficient algorithms in engineering applications (Hsu et al., 1993Hsu, D.S., Yeh, I.C., Lian, W.T., (1993). Artificial neural damage detection of existing structure, in: Proc. 3rd ROC and Japan Seminar on Natural Hazards Mitigation Conference.). One hidden layer is chosen and the number of neurons in the layer is decided in the learning process by trial and error.
Different combinations of sampling points of the input parameters and the resulting values of the output parameters are considered in order to train the neural network. Each such combination of the input parameters and the resulting output parameters comprises a data set. The total number of data sets considered for the training, validating and testing of the network are 3444.
Normalisation factors are applied to input and output parameter to bring and well distribute them in the range. No bias is applied to the input and output parameters. Normalisation factors of 4, 2, 7 and 3 are applied to input parameters ρ_{t}, I_{cr}/I_{g}, M_{cr} /M_{g} and output parameter I_{e} /I_{g} respectively.
70% data sets are used for training and the remaining data sets are divided equally in the validating and testing sets. For the training, several trials are carried out with different numbers of neurons in the hidden layer starting with a small number of neurons in the hidden layer and progressively increasing it, and checking the mean square errors (MSE) for the training, validating and testing. The number of neurons in the hidden layer is decided on the basis of the least mean square errors (MSE) for the training as well as validating and testing. Care is taken that the mean square error for test results should not increase with the number of neurons in hidden layer or epochs (overtraining). The final configuration (number of input parameters  number of neurons in the hidden layer  number of output parameters) of NN is 361. The responses of proposed neural network model to predict effective moment of inertia for training, validating, and testing are shown in Figures 6 (a)(c) respectively. The proposed neural network model achieved good performance as the testing data points are mostly on equity line (Figure 6 (c)). The statistical parameters i.e. mean square error (MSE), root mean square error (RMSE), mean absolute percentage error (MAPE), average absolute deviation (AAD), correlation coefficient (R^{2)} and coefficient of variation (COV) (Sozen et al., 2004Sozen, A., Arcaklioglu, E., Ozalp, M., Kanit, E.G., (2004). Use of artificial neural networks for mapping of solar potential in Turkey. Applied Energy 77: 273286.; Azmathullah et al., 2005Azmathullah, M.C., Deo, M.C., Deolalikar, P.B., (2005). Neural networks for estimation of scour downstream of a skijump bucket. Journal of Hydraulic Engineering ASCE 131(10): 898908.) of training, validating and testing data sets are shown in Table 3. All the parameters indicate a good agreement.
Response of neural network model in predicting I_{e} /I_{g} : (a) training; (b) validating; and (c) testing.
5 EXPLICIT EXPRESSION FOR PREDICTION OF EFFECTIVE MOMENT OF INERTIA
For the ease of practicing engineers and users, simplified explicit expression can be developed for the prediction of effective moment of inertia. The explicit expression requires the values of inputs, weights of the links between the neurons in different layers, and biases of output neurons (Tadesse et al., 2012Tadesse, Z., Patel, K.A., Chaudhary, S., Nagpal, A.K., (2012). Neural networks for prediction of deflection in composite bridges. Journal of Constructional Steel Research 68(1): 138149.; Gupta et al., 2013Gupta, R.K., Patel, K.A., Chaudhary, S., Nagpal, A. K., (2013). Closed form solution for deflection of flexible composite bridges. Procedia Engineering 51: 7583.).
As stated earlier, the sigmoid function (logsig) has been used as the activation function. The output O _{1} (Figure 5) may therefore be obtained as below (Tadesse et al., 2012Tadesse, Z., Patel, K.A., Chaudhary, S., Nagpal, A.K., (2012). Neural networks for prediction of deflection in composite bridges. Journal of Constructional Steel Research 68(1): 138149.; Gupta et al., 2013Gupta, R.K., Patel, K.A., Chaudhary, S., Nagpal, A. K., (2013). Closed form solution for deflection of flexible composite bridges. Procedia Engineering 51: 7583.):
where, q and r are the number of input parameters and the number of hidden neurons respectively; bias _{k} and bias _{o} are the bias of k_{th} hidden neuron (h_{k} ) and the bias of output neuron respectively; and are the weight of the link between I_{j} and h_{k} and the weight of the link between h_{k} and O _{1} respectively. The weights of the links and biases of the output neurons for NN are listed in Table 4.
The value of I_{e} /I_{g} is equal to denormalized output O _{1}. The effective moment of inertia I_{e} may be obtained from Eq. (2) by putting the values of from Table 4 as
where, H _{1}, H _{2}, H _{3}, H _{4}, H _{5} and H _{6} may be obtained from Eqs. (5)(10) by using the weights and biases (Table 4) as
6 VALIDATION OF NEURAL NETWORK/EXPLICIT EXPRESSION
The developed neural network/explicit expression is validated for a number of simply supported and continuous beams with a wide variation of input parameters. The results (midspan deflections), obtained from the proposed neural network/explicit expression are compared with the experimental results for simply supported beams available in literature and with the FEM results for continuous beams.
6.1 Simply supported beams
First, the results have been compared with experimental results reported by Washa and Fluck (1952)Washa, G.W., Fluck, P.G., (1952). Effect of compressive reinforcement on the plastic flow of reinforced concrete beams. ACI Journal 49(10): 89108. for sets of simply supported beams with rectangular crosssection (Figure 2): A2,A5; B2,B5; C2,C5; D2,D5 subjected to uniformly distributed loads, and designated, here, as VB1VB4, respectively. Two beams in a set are identical. The details of the beams are given in Table 5. Additionally, E_{s} is assumed as 2.05 × 10^{5}N/mm^{2}. E_{c} and f_{t} are taken in accordance with ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA.. The midspan deflections obtained from the proposed explicit expression (d_{NN} ) are shown in Table 5 along with the reported experimental midspan deflections (d_{EXP} ). The values obtained from the proposed explicit expression are in reasonable agreement with the reported experimental values of midspan deflections.
Next, the results have been compared with experimental results reported by Yu and Winter (1960)Yu, W.W., Winter, G., (1960). Instantaneous and longterm deflection of reinforced concrete beams under working loads. ACI Journal 57(1): 2950. for simply supported beams with T crosssection (Figure 7): A1; B1; C1; D1; E1; F1 subjected to uniformly distributed loads, and designated, here, as VB5VB10, respectively. The crosssectional and material properties of the beams are given in Table 6. The midspan deflections obtained from the proposed explicit expression (d_{NN} ) are shown in Table 6 along with the reported experimental midspan deflections (d_{EXP} ). Again, the values obtained from the proposed explicit expression are in reasonable agreement with the reported experimental values of midspan deflections.
Properties of simply supported beams with rectangular crosssections, considered for validation of the explicit expression.
Properties of simply supported beams with T crosssections, considered for validation of the explicit expression.
The results (midspan deflections) obtained from the proposed neural network/explicit expression need to be compared with the finite element results for lightly reinforced simply supported beams (ρt ≤ 1%) also. Consider a 2.625 m long simply supported beam VB11 with rectangular crosssection (Figure 2) subjected to uniformly distributed load. The other properties are: B = 200 mm; D = 500 mm; = 27.9 N/mm^{2}; E_{s} = 2.05 × 10^{5 }N/mm^{2}; A_{st} = 400 mm^{2}; A_{sb} = 700 mm^{2}; d_{t} = d_{b} = 35 mm. E_{c} and f_{t} are taken in accordance with ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA.. Midspan deflections, for beam VB11 are obtained from the proposed explicit expression, FEM and ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA. for varying magnitude of uniformly distributed loads, w and shown in Figure 8. The midspan deflections obtained from the proposed explicit expression and FEM are close for the range of the load considered. The difference between FEM and proposed explicit expression is 2.91% as compared to 16.81% difference between FEM and ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA. at 4w_{cr} ,(w_{cr} = cracking uniformly distributed load).
Consider another simply supported beam VB12 subjected to uniformly distributed load with the same crosssectional (Figure 2) and material properties as that of beam VB11 except A_{sb} . The value of A_{sb} is now assumed as 900 mm^{2}. The close agreement is observed between the midspan deflections obtained from the proposed explicit expression, FEM and ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA. as shown in Figure 9.
6.2 Continuous beams
In order to validate the proposed explicit expression for a continuous beam, results from the explicit expression are also compared with FEM and ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA. results for a 12.2 m two equal span uniformly distributed loaded continuous beam (VB13) with rectangular crosssection (Figure 2). The other properties are: B = 152.4 mm; D = 203.2 mm; = 24.1 N/mm^{2}; E_{s} = 2.07x10^{5} N/mm^{2}; A_{st} = A_{sb} = 112 mm^{2}; d_{t} = d_{b} = 25 mm. E_{c} and f_{t} are taken in accordance with ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA..
The midspan deflections obtained from the proposed explicit expression and FEM are close for the range of the load considered (Figure 10). The difference between FEM and proposed explicit expression is 5.34% as compared to 28.25% difference between FEM and ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA. at 5w_{cr} .
Similarly, another 12.2 m two equal span continuous beam VB14 with rectangular crosssection (Figure 2) subjected to uniformly distributed load has been considered. The crosssectional and material properties are taken same as that of beam VB13 and only A_{sb} and A_{st} are increased to 200 mm^{2}. The close agreement is observed between the midspan deflections obtained from the proposed explicit expression, FEM and ACI 318 (2005)American Concrete Institute (ACI) 318, (2005). Building code requirements for structural concrete (ACI 31805) and commentary (ACI 318R05), USA. as shown in Figure 11.
7 SENSITIVITY ANALYSIS
The proposed explicit expression shows satisfactory performance on validation with experimental results available in literature and FEM results. A sensitivity analysis is carried out next to capture the influence of individual input parameters on output parameter using the proposed explicit expression. The effect of input parameters ρ_{t}, I_{cr}/I_{g}, M_{cr} /M_{e} along with additional parameters ρ_{c}, n on output parameter I_{e} /I_{g} is studied. Only one parameter (the parameter under consideration) is varied at a time, keeping the other parameters constant.
7.1 Effect of ρt
As stated earlier, ρt has been considered as the input parameter in the present study. Figure 12 shows the variation of I_{e} /I_{g} with respect to ρt for various values of I_{cr} /I_{g} , keeping the value of M_{cr} /M_{e} constant as 0.5. Rich influence of ρt on I_{e} /I_{g} is seen in Figure 12. Though, the effect is significant for all values of ρt, the effect of lower values of ρt is more significant in case of higher I_{cr} /I_{g} .
7.2 Effect of I cr/I g
The variation of I_{e} /I_{g} with respect to I_{cr} /I_{g} for various values of ρt is shown in Figure 13. The value of M_{cr} /M_{e} is kept constant as 0.5. The effect is significant only for lower values of I_{cr} /I_{g} in case of low ρt. However, the effect extends of the range considered for I_{cr} /I_{g} in case of higher values of ρt. The effect of I_{cr} /I_{g} is significant for all values of ρt.
7.3 Effect of M cr/M e
As stated earlier, M_{cr} /M_{e} has been considered as the input parameter affecting I_{e} /I_{g} . The variation of I_{e} /I_{g} with respect to M_{cr} /M_{e} for different values of ρtis shown in Figure 14 (a). The value of I_{cr} /I_{g} is kept constant as 0.5. Similarly, Figure 14 (b)shows the variation of the ratio I_{e} /I_{g} with respect to M_{cr} /M_{e} for different values of I_{cr} /I_{g} . The value of ρt is kept constant as 1.5. As expected, the effect of M_{cr} /M_{e} is significant during cracking (M_{cr} /M_{e} < 1) and the value of I_{e} /I_{g} increases with increase in value of M_{cr} /M_{e} up to 1.00. The effect is more for higher value of ρt.
Variation of I_{e} /I_{g} with respect to M_{cr} /M_{e} for different (a) ρtvalues, and (b) I_{cr} /I_{g} values.
7.4 Effect of ρc
Figure 15 shows the variation of I_{e} /I_{g} with respect to ρc for different values of ρt. The value of M_{cr} /M_{e} is kept constant as 0.5. The value of I_{e} /I_{g} is found to increase with the increase in value of ρc. A significant variation is observed in case of higher value of ρt.
7.5 Effect of n
The variation of I_{e} /I_{g} with respect to n for different values of ρt is shown in Figures 16 (a)(b) for ρc = 0 and ρc = ρt(n  1)/n respectively. The value of M_{cr} /M_{e} is kept constant as 0.5. The nature of plot changes from concave to convex with increase in ρt.
8 CONCLUSIONS
An explicit expression has been proposed for the prediction of effective moment of inertia (and deflection) considering concrete cracking, tension stiffening and entire practical range of reinforcement at service load. A set of three parameters (ρ_{t}, I_{cr}/I_{g}, M_{cr} /M_{e} ) has been identified that govern the change in I_{e} /I_{g} and therefore deflection. Using the sampling points of these parameters and the validated FEM, the data sets have been generated for training, validating and testing of neural network. The explicit expression has been developed from the trained neural network. The proposed explicit expression has been validated for a number of simply supported and continuous beams and it is found that the predicted deflections have reasonable accuracy for practical purpose. Sensitivity analysis has been carried out to capture the influence of individual input parameters on output parameter. The effect of the input parameters ρ_{t}, I_{cr}/I_{g}, M_{cr} /M_{e} on output parameter I_{e} /I_{g} is studied using the proposed explicit expression . The lower values of ρt are found to have more significant effect on I_{e} /I_{g} . The effect of M_{cr} /M_{e} is found to be significant during cracking (M_{cr} /M_{e} < 1) and the value of I_{e} /I_{g} is found to increase up to 10.0 with increase in value of M_{cr} /M_{e} . The effect of ρc and n is found to be less significant and can be incorporated through I_{cr} /I_{g} .
The methodology presented herein can be further developed for beams with point loads. The effect of shear deformation may be incorporated in future studies by considering span to depth ratio of beam as an input parameter. Similarly, age of loading and characteristic compressive strength of concrete can also be considered as input parameters to account for shrinkage cracking in future studies.
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Publication Dates

Publication in this collection
Mar 2015
History

Received
05 Apr 2014 
Reviewed
21 July 2014 
Accepted
18 Aug 2014