jurnal jembatan 1(closed form solution for deflection of flexible composite bridges)

Procedia Engineering 51 ( 2013 ) 75 – 83

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of Institute of Technology, Nirma University, Ahmedabad.doi: 10.1016/j.proeng.2013.01.013

Chemical, Civil and Mechanical Engineering Tracks of 3rd Nirma University International Conference

Closed Form Solution for Deflection of Flexible Composite Bridges

R. K. Guptaa, K. A. Patela,*, Sandeep Chaudharyb, A. K. Nagpala aDepartment of Civil Engineering, Indian Institute of Technology(IIT) Delhi, Hauz Khas, New Delhi, 110016, India

bDepartment of Civil Engineering, Malaviya National Institute of Technology(MNIT), Jaipur, 302017, India

Abstract

Simply supported steel-concrete composite beams are widely used in bridge construction. Deflection is the significant parameter for serviceability limit state of bridges. A lot of computational effort is required for finite element analysis of bridges considering flexibility of shear connectors and shear leg effects. Neural network is presented for prediction of deflections, at service load, in simply supported steel-concrete composite bridges incorporating flexibility of shear connectors and shear lag effect. The training, testing and validation data sets for neural network are generated using finite element models. The finite element models have been developed using ABAQUS software. These models have been validated with available experimental results. Closed form solution is also proposed based on the developed neural network. The use of the neural network requires a computational effort almost equal to that required for the simple beam analysis (neglecting flexibility of shear connectors and shear lag effect). The neural network has been validated for number of bridges and the errors are found to be small. The network/closed form solution can be used for rapid prediction of deflection for everyday design. © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Institute of Technology Nirma University, Ahmedabad.

Keywords: Neural network; Finite element analysis; Composite bridges; Sensitivity anlysis; Deflection.

1. Introduction

Composite beams (Fig. 1) are widely used in the bridge construction. Now days, simply supported composite bridges are mostly used in flyovers and railway over bridges. Shear connectors are provided to connect steel beam and concrete slab. Slip would occur between the concrete slab and steel beam due to flexibility of shear connectors, thereby increasing the deflection. Deflection is the significant parameter for serviceability limit state of bridges consisting of high strength materials. A simple but sufficiently accurate method is therefore desirable for the prediction of deflection of bridges with flexible shear connectors and shear leg effect. Such method would be useful in design office.

Various mechanical connectors are used in composite construction. But, the headed shear connectors are most widely used to connect the steel beam and the concrete slab. The behaviour of composite bridges depends on the behaviour of shear connectors. The shear connector load-slip curve is obtained from various types of ‘push’ or ‘push-out’ test. The load-slip curve is generally non-linear in nature but as an approximation different equivalent linear relationships [2,3] have been proposed. These linear relationships may be assumed at service load.

Some researchers [4-11] have proposed analytical methods for flexible connected composite beams with linear load slip relationship. Some of these analytical methods are simple but applicable only for symmetrical structures considering zero slip at the mid-span [4,6,7]. Also, these methods do not account for the shear lag phenomenon.

* K. A. Patel. Tel.:+91-11-26591234;fax:+91-11-26591234 Email address:[email protected]

Available online at www.sciencedirect.com

© 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of Institute of Technology, Nirma University, Ahmedabad.

76 R. K. Gupta et. al / Procedia Engineering 51 ( 2013 ) 75 – 83

Fig. 1. Composite cross-section. [1]

Neural network has also been developed [1] for prediction of mid-span deflection, at service load, taking account of flexibility of shear connectors and shear lag phenomena for simply supported bridges. Finite element models have been used for generation of training, testing and validation data sets. But these finite element models are complex, tedious to make and require a lot computational efforts.

In this paper, a simplified finite element model has been developed using ABAQUS [12] to analyse flexible composite bridge with minimal computational effort. Further, neural network model has been proposed for prediction of mid-span deflection of simply supported bridges at service load using the data generated from the FEM model. This neural network takes into account flexibility of shear connectors and shear lag effect. The data sets for training, testing and validation are generated using finite element models. The network requires very less computational efforts. Closed form solution has been also derived based on neural network model.

2. Finite element model

In finite element modeling, concrete slab is modeled using S4R shell element and girder is modeled by B31 beam element as shown in Fig. 2. The accuracy achieved by elements is sufficient for composite bridges of usual dimensions. A linear load-slip relationship has been assumed for the shear connectors at service loads. For this purpose slide plane (Basic type) connectors are opted in modeling. It provides connection with flexibility in two perpendicular directions. A finite stiffness value, K is provided along the span of bridge and very high stiffness in other direction. At service load, the stress- strain relationship for structural steel and reinforcement is assumed to be linear. For concrete, the stress-strain relationship is assumed to be linear in compression, which is generally applicable at service load.

Fig. 2. Finite element model.

cD

sDc sD


3. Validation of finite element model

In order to validate model, the results obtained from finite element model have been compared with experimental results reported for two simply supported beam U4 and beam E1 by Chapman and Balakrishnan [13]. Beam U4 and beam E1 were subjected to uniformly distributed load and mid-span point load respectively. The actual load-slip curve (obtained from experiments) has been given as input in slide plane connector option available in ABAQUS. Fig. 3 shows the comparison between deflections obtained from proposed finite element model and experiments for both beams. The deflections are also compared with finite element model by Tadesse et al. [1]. The deflections are found in close agreement for both beams.

0

25

50

75

100

125

150

175

0 5 10 15 20 25 30 35 40 45

Load

(kN/

m)

Mid-span Deflection (mm)

Experimental (Chapman and Balakrishnan 1964)

FEM (Tadesse et al. 2012)

FEM (Proposed)

(a)

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30 35 40

Load

(kN

)

Mid-span Deflection (mm)

Experimental (Chapman and Balakrishnan 1964)

FEM (Tadesse et al. 2012)

FEM (Proposed)

(b)

Fig. 3. Comparison of mid-span deflection of (a) beam U4 and (b) beam E1.

4. Selection of input parameters and sensitivity analysis

Significance of input parameters is studied by carrying out sensitivity analysis for development of neural network. The sensitivity analysis used to define the different combinations of input parameters required to form the training, testing and


validation data sets. The mid-span deflection of simply supported composite bridge may be obtained from /r fd d , where rd is mid-span deflection of a composite bridge obtained from simple beam analysis (neglecting flexibility of shear

connectors and shear lag effect) and fd is mid-span deflection of the composite bridge considering flexibility of shear connectors and shear lag effect. The value of rd may be obtained from analytical equation or any readily available software. The value of fd is obtained from finite element analysis using ABAQUS software.

The shear lag phenomenon affects the deflection of composite beams [14] and is significantly influenced by b/L [15, 16], where b is width of slab and L is length of span. As stated earlier, mid-span deflections of composite bridges increase due to flexibility of shear connectors. Girahammar and Pan [17] and Nie and Cai [11] observed that the flexibility of connectors can be defined by the two flexibility parameters L and . The parameters are given as (Fig. 1)

22

1 s o

KL( L) =A E I p

(1)

02

s cs

cs

E A G pKL D

(2)

0

1 20 0 c

AA

I A d (3)

0.s c

s c

A AA

nA A (4)

0c

sI

I In

(5)

where, K is the stiffness of shear connectors; p is the longitudinal spacing (pitch) of shear connectors; s cn=E /E is the modular ratio ; cE and sE are modulus of elasticity of concrete and steel respectively; cA and sA are cross sectional area of concrete slab and steel beam respectively; cI and sI are second moment of area (about centroid) of concrete slab and steel beam respectively; csG is the distance between centroid of concrete slab ( cG ) and steel section ( sG ); and csD is the total depth of composite section. As can be seen in Eqs. (1) and (2) these parameters depend upon cross sectional properties, span length, pitch and stiffness of shear connectors. The K, stiffness of shear connectors depends on flexibility parameters

L and . Three parameters which may affect /r fd d are tentatively chosen as: (i) b/L , (shear leg effect) (ii) L and (iii) (for

flexibility of shear connectors). The practical range of L is taken as 3 to 14 and b/L is that of 0.1 to 0.33 (1/10 to 1/3). In sensitivity studies, only one parameter is varied in turn, keeping the other two parameters constant. A finite element

model of simply supported bridge subjected to a uniformly distributed load applied throughout the span of the beam has been considered to perform sensitivity analysis. The properties of bridge are: L = 8m, Dc = 100 mm, '

cf = 36 N/mm2,

Es=205000 N/mm2, p = 125 mm and UB 762 267 147 steel section; where 'cf is characteristics concrete cylinder strength

at 28 days. To obtain the required value of L and , K has been varied and for variation of b/L , b has been given three different values as b = 800 mm, 1600 mm, 2640 mm.

4.1. Effect of b L

As stated earlier, the ratio b/L is an important parameter affecting the shear lag phenomena. This parameter is not taken into account in the simple elastic analysis. The other parameters have been kept constant (e.g. L= 8). Fig. 4 shows the variation of r fd /d with b/L . It is observed that the variation is significant and can be represented fairly accurately by considering three sampling points: b/L = 0.1, 0.2 and 0.33.

4.2 Effect of L

The ratio L has been considered as an important parameter since as stated earlier, it is required to define the flexibility of a composite bridge and is therefore chosen as a parameter affecting r fd /d . For this b/L is kept constant as 0.2. As L and


are two interdependent parameters so value of has not been kept constant for changing value of L [Eqs. (1) and (2)]. The variation of r fd /d with L has been shown in Fig. 5 and it is observed that the variation is significant. Therefore it can be represented by five sampling points: L = 3, 6, 9, 12 and 14.

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.05 0.1 0.15 0.2 0.25 0.3 0.35

dr /df

b/L

Fig. 4. Variation of dr/df with b/L.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15

dr /df

L

Fig. 5. Variation of dr/df with L.

4.3Effect of

The parameter is also significant parameter affecting the deflection of flexible composite bridges and is therefore chosen as a parameter for sensitivity analysis. The value of b/L has been kept constant equal to 0.22. Fig. 6 shows the variation of r fd /d with and observed that the variation is significant but almost linear. Therefore, the variation can be represented fairly accurately by chosen sampling points of L .


0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.02 0.04 0.06 0.08 0.1 0.12

dr /df

Fig. 6. Variation of dr/df with .

5. Training of Neural Network

Three input parameters (i) b/L , (ii) L ,(iii) and an output parameter /r fd d have been finalised to train the neural network. Therefore, multilayered feed-forward neural network has been chosen for training. The neural network with neurons in all the layers fully connected in the feed forward manner (Fig. 7). The training, testing and validation are carried out using the MATLAB [18]. Levenberg–Marquardt (LM) back propagation learning algorithm has been used for training. Sigmoid function (logsig) is used as an activation function. 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. Each such combination of the input parameters and the resulting output parameters comprises a data set. Normalisation factors applied to output parameter to bring and well distribute them in the range. 0.62 has been multiplied to output parameter for this purpose.

Fig. 7. Configuration of multilayered feed-forward network.

70% data sets are used for training. For testing and validation, the remaining data sets are equally divided. Several trials are carried out with different numbers of neurons in the hidden layer for training. Each training starting with a small number of neurons in the hidden layer and progressively increasing it. Simultaneously, the root mean square errors (RMSE) for the training, validation and testing are checked. The number of neurons in the hidden layer is decided on the basis of the least root mean square errors (RMSE) for the training, testing and validation. Four neurons in a hidden neuron have been


finalised. The final configuration (number of input parameters-number of hidden neurons in hidden layer-number of output parameters) of neural network is shown in Fig. 7. This configuration (3-4-1) gives root mean square error (RMSE) equal to 0.0045 which indicate a good performance of neural network.

6. Closed form of solution for prediction of deflection

For the ease of design engineers and users, simplified closed form solution can be obtained from trained neural network for the prediction of deflections. The values of inputs, weights of the links between the neurons in different layers, and biases of output neurons are required to predict mid-span deflection. As stated earlier, the sigmoid function has been used as the activation function. The output O1 (Fig. 7) may therefore be obtained as below [1]:

,10

1

1

1

1

1

hork

H kk

wbias

e

O

e

(6)

,

,1

qi h

j kk j kj

I biaswH (7)

where, q is the number of input parameters; r is the number of hidden neurons; kbias is the bias of thk hidden neuron( kh );

obias is the bias of output neuron; ,ihj kw is the weight of the link between jI and kh ; ,1

hokw is the weight of the link between

kh and 1O . The weights and biases for neural network are listed in Table 1.

Table 1. Weight values and biases for neural network.

Connection Weight/bias

Number of hidden layer neuron 1 2 3 4

Input to Hidden

1,

ih

kw -3.7131 21.0394 32.2915 2.9099

2,

ih

kw -5.1664 -6.8610 2.8743 -3.7225

3,

ih

kw -8.7564 -5.2205 2.1747 -3.3297 biask 13.584 -2.3326 -9.1017 0.7534

Hidden to Output ,1hokw -2.2607 0.4022 -0.3687 -1.0846

The value of /r fd d is equal to de-normalized output O1. The values of fd can be obtained from Eq. 6 using ,1

hokw values

from Table 1 as

0 1 2 3 4

2.2607 0.4022 0.3687 1.08461 1 11

1 10.62

1 H H H H

r

fbias

e e ee

dd

e (8)

rd can be obtained from analytical equation or any readily available software and H1, H2, H3, H4 can be calculated from Eq.

7 by using weights and biases values from Table 1 where, bias0 is equal to 2.9068.

3.7131 5.1664 8.7564 13.5841H b/L L + (9)

21.0394 6.861 5.2205 2.33262H b/L L (10)

32.2915 2.8743 2.1747 9.10173H b/L+ L+ (11)

2.9099 3.7225 3.3297 0.75344H b/L L + (12)


7. Validation of neural network

Validation of the proposed neural network/closed form solution is essential to conclude whether the same could be applied to a real problem. The trained neural network is hence validated with a wide variation of input values. Six example bridges (EB1-EB6) are considered for validation of neural network. The properties of all bridges are given in Table 2. None of these combinations of inputs has been considered in the training, testing and validation. The deflections obtained from the developed neural network and finite element predictions are compared. The percentage errors of the values obtained from proposed closed form solution the ANN when compared with ABAQUS results are also shown in the Table.3.

Table 2. Properties of example bridges.

Bridge Steel section L (m)

cD (mm)

b (mm)

p (mm)

'cf

(N/mm2)sE

(N/mm2) EB1 UB 914 419 343 7.0 100 1050 125 36 205000 EB2 UB 914 419 343 7.0 100 1050 125 36 205000 EB3 UB 914 419 343 7.0 100 1050 125 36 205000 EB4 UB 914 305 289 6.0 100 1500 125 36 205000 EB5 UB 914 305 289 6.0 100 1500 125 36 205000 EB6 UB 914 305 289 6.0 100 1500 125 36 205000

Table 3. Comparisons of deflection obtained from proposed neural network and FEM.

Bridge /b L L f

d (FEM)#

(mm)

f

d (NN)*

(mm)

r

d (mm)

Error (%)

EB1 0.15 4.5 0.03617 1.917 1.939 1.462 -1.1477 EB2 0.15 8.5 0.01014 1.702 1.619 1.462 4.8766 EB3 0.15 12.5 0.00469 1.472 1.500 1.462 -1.9022 EB4 0.25 4.5 0.04370 1.319 1.362 0.923 -3.2600 EB5 0.25 8.5 0.01225 1.178 1.143 0.923 2.9711 EB6 0.25 12.5 0.00565 1.076 1.069 0.923 0.6506

#FEM = Finite Element Method, *NN = Neural Network These comparisons show a good correlation between the deflection results predicted by proposed close form solution and

finite element method (ABAQUS). The maximum absolute error is found to be 4.88 % for the bridges considered, which is a small value for practical purposes.

8. Conclusions

(i) A simplified finite element model has been proposed to analyze simply supported composite bridge considering flexibility of shear connectors and shear leg effect with minimal computational effort at service load.

(ii) Three parameters that govern the mid-span deflections are identified. Sensitivity analysis is carried out for these input parameters to arrive at the sampling points.

(iii) A neural network has been developed for prediction of mid-span deflection of simply supported bridges at service load.

(iv) Closed form solution has been also derived based on neural network model for the prediction of deflection. The closed form solution requires very less computational efforts.

(v) The methodology presented herein can be further extended for multi-span bridges.

Acknowledgement

This work was supported by department of Science and Technology, Ministry of Science and Technology, Government of India (Project No. SR/S3/MERC/0023/2008). The authors express their gratitude for the support.


References

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[2] Johnson, R.P., Molenstra I.N., 1991. Partial shear connection in composite beams for buildings, Proceeding Institution of Civil Engineers 91(2), p. 679.

[3] Wang Y.C., 1998. Deflection of steel-concrete composite beams with partial shear interaction, Journal of Structural Engineering 124(10), p. 1159. [4] Dezi, L., Ianni, C., Tarantino, A.M., 1993. Simplified creep analysis of composite beams with flexible connectors, Journal of Structural Engineering

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1265. [6] Jasim, N.A., 1999. Deflections of partially composite beams with linear connector density, Journal of Constructional Steel Research 49(3), p. 241. [7] Jasim, N.A., Atalla, A., 1999. Deflections of partially composite continuous beams: a simple approach, Journal of Constructional Steel Research

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stiffness matrix and applications, Computers & Structures 80(11), p. 1001. [11] Nie, J., Cai, C.S., 2003. Steel-concrete composite beams considering shear slip effects, Journal of Structural Engineering 129(4), p. 495. [12] ABAQUS. ABAQUS theory manual. Pawtucket (RI): Hibbit. Karlsson & Soresen, Inc; 2009. [13] Chapman, J.C., Balakrishnan, S., 1964. Experiments on composite beams, Structural Engineer 42(11), p. 369. [14] Chiewanichakorn, M., Aref, A.J., Chen, S.S., Ahn, S.L., 2004. Effective flange width definition for steel-concrete composite bridge girder, Journal

of Structural Engineering 130(12), p. 2016. [15] Sedlacek, G., Bild, S., 1993. A simplified method for the determination of the effective width due to shear lag effects, Journal of Constructional Steel

Research 24, p. 155. [16] BSI Part 5. Code of practice for design of composite bridges. BS 5400 steel, concrete and composite bridges. London: British Standards Institution;

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jurnal jembatan 1(closed form solution for deflection of flexible composite bridges)

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