Explicit expression for effective moment of inertia of RC beams

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 short-term 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.


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, 2010).The stiffnesses of the uncracked and cracked portions are averaged in the springs in parallel approach (Branson, 1965;Al-Zaid et al., 1991;Al-Shaikh and Al-Zaid, 1993;SAA-AS 3600, 1994;TS 500, 2000;CSA-A23.3, 2004;ACI 318, 2005;AASHTO, 2005), whereas the flexibilities of the uncracked and cracked portions are averaged in the springs in series approach (Ghali, 1993;CEN Eurocode 2, 2004;Bischoff, 2005;Bischoff and Scanlon, 2007;Bischoff, 2007).
Considering parallel springs approach first, the following equation of effective moment of inertia e I in terms of fully cracked and uncracked moment of inertia was originally proposed by Branson (1965) for simply supported beams as Eq. ( 1) was derived empirically based on the experimental test results of simply supported rectangular reinforced concrete (RC) uniformly loaded beams with tension reinforcement, t = 1.65% and ratio of moment of inertia of the fully cracked transformed cross section and moment of inertia of the gross cross section, cr g II = 0.45 at maximum applied (elastic) moment equal to 2.5 cr M (Branson, 1965).Eq. ( 1) has been adopted in many international standards and codes (SAA-AS 3600, 1994;TS 500, 2000;CSA-A23.3, 2004;ACI 318, 2005;AASHTO, 2005) to calculate e I and therefore deflection, taking m = 3.Some researchers (Bischoff, 2005;Gilbert, 1999;Scanlon et al., 2001;Gilbert, 2006) found out that Eq. ( 1) with m = 3 calculates effective moment of inertia accurately in case of medium to high tension reinforcement( t > 1%), while it overestimates effective moment of inertia for low tension reinforcement ( t < 1%).Al-Zaid et al. (1991) 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 1.5 .Next, consider the springs in series approach.The models based on this approach (Bischoff, 2005;Bischoff and Scanlon, 2007;Bischoff, 2007) take into account tension stiffening effect in concrete for calculating e I .The deflections obtained by the expression proposed by Bischoff (2005) have been found in good agreement with experimental deflections for lightly reinforced beams ( t < 1%) (Gilbert, 2006;Bischoff and Scanlon, 2007).Kalkan (2010) found out that the expressions given by Eq. (1) and Bischoff (2005) estimate deflections of moderately-reinforced to highly-reinforced concrete beams ( t >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 mid-span 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 Latin A m erican Journal of Solids and Structures 12 (2015) 542-560 prediction of time effects in RC frames (Maru and Nagpal, 2004), prediction of damage detection in RC framed buildings after earthquake (Kanwar et al., 2007), structural health monitoring (Min et al., 2012;Kaloop and Kim, 2014), bending moment and deflection prediction in composite structures (Chaudhary et al., 2007(Chaudhary et al., , 2014;;Pendharkar et al., 2007Pendharkar et al., , 2010Pendharkar et al., , 2011;;Tadesse et al., 2012;Gupta et al., 2013), predicting the creep response of a rotating composite disc operating at elevated temperature (Gupta et al., 2007), optimum design of RC beams subjected to cost (Sarkar and Gupta, 2009), static model identification (Kim et al., 2009), response prediction of offshore floating structure (Uddin et al., 2012), prediction of deflection in high strength selfcompacting concrete deep beams (Mohammadhassani et al., 2013a;2013b) and prediction of energy absorption capability and mechanical properties of fiber reinforced self-compacting concrete containing nano-Silica particles (Tavakoli et al., 2014a;2014b).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) 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.

FINITE ELEMENT MODEL AND ITS VALIDATION
The finite element model (FEM) has been developed using the ABAQUS (2011) software.The beam has been modelled using B21 elements (2-node linear Timoshenko beam element).Under service load, the stress-strain 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 post-failure stress-strain data proposed by Gilbert and Warner (1978).A high shear stiffness has been assumed to neglect the shear deformations.
The results of FEM have been compared with the experimental results (mid-span deflections of the beam under increasing uniformly distributed load, , w after the cracking of the concrete) reported by Al-Zaid et al. (1991) for a simply supported beam (VB) with 2.5 m clear span (effective span = 2.62 m) and cross-sectional dimensions BD200 200 mm (Figure 2).The other properties considered are: cylindrical compressive strength of concrete at 28 days, Solids and Structures 12 (2015) 542-560 N/mm 2 ; modulus of elasticity of concrete, c E = 2.96 10 4 N/mm 2 ; modulus of elasticity of steel, s E = 2 10 5 N/mm 2 ; tensile strength of concrete,   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, cr is taken as 0.00012 and in view of low/moderate tensile reinforcement, sb A = 402.12mm 2 (= 1.2%), the plastic strain is u cr taken as 0.0004.For convergence, about 16 elements are required when cracking is considered (Patel et al., 2014).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) for four sets of rectangular cross-sectional (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 cross-sectional properties, material properties, span lengths and uniform  1.The finite element models can therefore be used for generation of data sets.

SAMPLING POINTS AND DATA SETS
For development of neural network, significant parameters need to be identified.Eq. ( 1

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 feed-forward 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).Sigmoid function (logsig) is used as an activation function and the Levenberg-Marquardt 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, 2004;Kanwar et al., 2007;Gupta et al., 2007;Pendharkar et al., 2007;2010;2011;Chaudhary et al., 2007;2014;Sarkar and Gupta, 2009;Gupta and Sarkar, 2009;Min et al., 2012;Tadesse et al., 2012;Mohammadhassani et al., 2013a;Gupta et al., 2013) and is considered as one of the efficient algorithms in engineering applications (Hsu et al., 1993).One hidden layer is chosen and the number of neurons in the layer is decided in the learning process by trial and error.coefficient (R 2 ) and coefficient of variation (COV) (Sozen et al., 2004;Azmathullah et al., 2005) of training, validating and testing data sets are shown in Table 3

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., 2012;Gupta et al., 2013).As stated earlier, the sigmoid function (logsig) has been used as the activation function.The output 1 O (Figure 5) may therefore be obtained as below (Tadesse et al., 2012;Gupta et al., 2013): (2) where, q and r are the number of input parameters and the number of hidden neurons respectively;  H may be obtained from Eqs. ( 5)-( 10) by using the weights and biases (Table 4

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 (mid-span 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.

Simply supported beams
First, the results have been compared with experimental results reported by Washa and Fluck (1952) for sets of simply supported beams with rectangular cross-section (Figure 2): A2,A5; B2,B5; C2,C5; D2,D5 subjected to uniformly distributed loads, and designated, here, as VB1-VB4, respectively.Two beams in a set are identical.The details of the beams are given in Next, the results have been compared with experimental results reported by Yu and Winter (1960) for simply supported beams with T cross-section (Figure 7): A-1; B-1; C-1; D-1; E-1; F-1 subjected to uniformly distributed loads, and designated, here, as VB5-VB10, respectively.The cross-sectional and material properties of the beams are given in Table 6.The mid-span deflections obtained from the proposed explicit expression  6 along with the reported experimental mid-span deflections EXP d .Again, the values obtained from the proposed explicit expression are in reasonable agreement with the reported experimental values of mid-span deflections.
The results (mid-span deflections) obtained from the proposed neural network/explicit expression need to be compared with the finite element results for lightly reinforced simply supported beams 1%    Consider another simply supported beam VB12 subjected to uniformly distributed load with the same cross-sectional (Figure 2) and material properties as that of beam VB11 except sb A .The value of sb A is now assumed as 900 mm 2 .The close agreement is observed between the mid-span deflections obtained from the proposed explicit expression, FEM and ACI 318 (2005) as shown in Figure 9.

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)   The mid-span 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) at 5 cr w .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 sb A and st A are increased to 200 mm 2 .The close agreement is observed between the mid-span deflections obtained from the proposed explicit expression, FEM and ACI 318 (2005) as shown in Figure 11.

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 t , MM constant as 0.5.Rich influence of t on eg II is seen in Figure 12.Though, the effect is significant for all values of t , the effect of lower values of t is more significant in case of higher  MM up to 1.00.The effect is more for higher value of t .MM is kept constant as 0.5.The value of eg II is found to increase with the increase in value of c .A significant variation is observed in case of higher value of t .MM is kept constant as 0.5.The nature of plot changes from concave to convex with increase in t .

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 ( t , cr g II , cr e MM ) has been identified that govern the change in eg II 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 MM .The effect of c and n is found to be less significant and can be incorporated through 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.
of gross and fully cracked transformed cross section, respectively which the cracking takes place at a cross-section in the beam and applied (elastic) moment, respectively deflection from experiments, FEM, and neural network/explicit expression, respectively , erican Journal of Solids and Structures 12 (2015) 542-560 where, cr M = minimum moment at which the cracking takes place at a cross-section in the beam; e M = applied (elastic) moment along the span; g I = moment of inertia of the gross cross section; cr I = moment of inertia of the fully cracked transformed cross section and m =constant.
m was found to change from about 3 to 4.3 for moderately-reinforced concrete beams ( t = 1.2%, -Shaikh and Al-Zaid (1993) performed experiments on mid-span 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 ( in Eq. (1) for point loaded beams.Al-Zaid et al. (1991); Al-Shaikh and Al-Zaid (1993) also proposed to calculate e I based on cracked length incorporating reinforcement and loading effects respectively.
of top reinforcement, st A = 78.54mm 2 and area of bottom reinforcement, sb A = 402.12mm 2 .The effective concrete cover at top t d and at bottom b d have been taken as 30 mm and 33 mm respectively.

Figure 3 :
Figure 3: Comparison of mid-span deflections of beam VB.

Figure 5 :
Figure 5: Configuration of a typical neural network.
Response of neural network model in predicting eg II: (a) training; (b) validating; and (c) testing.

O
respectively.The weights of the links and biases of the output neurons for NN are listed in

t
also.Consider a 2.625 m long simply supported beam VB11 with rectangular cross-section (Figure 2) subjected to uniformly distributed load.The other properties are: B = 200 mm; D = 500 mm; ' c f = 27.9N/mm 2 ; s E = 2.05 10 5 N/mm 2 ; st A = 400 mm 2 ; sb A = Latin A m erican Journal of Solids and Structures 12 (2015) 542-560 700 mm 2 ; tb dd = 35 mm.c E and t f are taken in accordance with ACI318 (2005).Mid-span deflections, for beam VB11 are obtained from the proposed explicit expression,FEM and ACI 318  (2005)  for varying magnitude of uniformly distributed loads, w and shown in Figure8.The mid- span 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 betweenFEM and ACI 318 (2005)  at 4
results for a 12.2 m two equal span uniformly distributed loaded continuous beam (VB13) with rectangular cross-section (Figure2).The other properties are:
. Only one parameter (the parameter under consideration) is varied at a time, keeping the other parameters constant.7.1 Effect oft As stated earlier, t has been considered as the input parameter in the present study.Figure 12 shows the variation of eg II with respect to t for various values of cr g II , keeping the value of

Figure
Figure 13: Variation of

c
Figure15shows the variation of

Figure
Figure 15: Variation of erican Journal of Solids and Structures 12 (2015) 542-560 using the proposed explicit expression .The lower values of t are found to have more significant effect on eg II.
Figure 16: Variation of Latin A m erican Journal ofSolids and Structures 12 (2015) 542-560distributed loads of all four beams have been given in Table1.Additionally, s E has been assumed as 2 10 5 N/mm 2 .

Table 1 :
Properties of simply supported beams with rectangular cross-sections, considered for validation of FE model.

Table 2 :
MM .The sampling points of the parameters considered for data generation are shown in Table2.It may be noted that the combinations of sampling points take the different values of the c (fourth and left out parameter), corresponding to each value of t .Considering the equation Input parameters and sampling points.
cr a MM depends on the load and moment only.Taking the above observations into account, t is also considered as an input parameter alongwith Mid-span Deflection (mm) FEM Experimental Latin A m erican Journal of Solids and Structures 12 (2015) 542-560 into account

Table 3 :
. All the parameters indicate a good agreement.Statistical parameters of neural network.

Table 4 :
Weight values and biases of neural network.
eg II is equal to de-normalized output 1 O .The effective moment of inertia

Table 5
in accordance with ACI 318(2005).The mid-span deflections obtained from the proposed explicit expression c E and t f are taken

Table 5 :
Properties of simply supported beams with rectangular cross-sections, considered for validation of the explicit expression.

Table 6 :
Properties of simply supported beams with T cross-sections, considered for validation of the explicit expression.