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Radial Basis Function to Predict Scour Depth Around Bridge Abutment

Shahin Ara Begum, Abul Kashim Md. Fujail Department of Computer Science

Assam University, Silchar, Assam, India {shahin_begum@yahoo.com, abul_fujail@yahoo.com}

Abdul Karim Barbhuiya Department of Civil Engineering National Institute of Technology

Silchar, Assam, India akbarbhuiya@yahoo.com

Abstract - Local scour around bridge abutment is a time-dependent complex phenomenon encountered world-wide. It is difficult to establish a general empirical model that can be applied to all abutment conditions. In this paper, Radial basis function (RBF) Network has been used to predict the maximum scour depth around bridge abutment. An appropriate model is identified using experimental data from literature. The efficiency of the developed model has been evaluated using Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Correlation Coefficient (CC). Experimental results show the suitability and reliability of the application of Artificial Neural Network for scour depth prediction around bridge abutments.

Keyword – artificial neural network; radial basis function; scour depth prediction

I. INTRODUCTION Scour is the erosion caused by water of the soil around

obstruction. The magnitude of the scour is multiplied when the natural flow is disturbed due to the presence of some obstructions like bridge pier, bridge abutment, spur etc. Failure of bridges due to local scour at their foundation is a common occurrence and each year a large amount is spent to repair or replace bridges whose foundations have been under-cut by the scouring action of stream flow [1, 2]. Bridge foundation consists of abutments and piers. Probably the number of existing bridge abutments is much more than the numbers of bridge piers as most of the bridges are of single span. In a report, published by the Department of Scientific and Industrial Research (DSIR) of New Zealand [3], it is mentioned that almost 50% of total expenditure was made to repair and maintain bridge damage, out of which 70% was spent to repair abutment scour. Thus, scour around bridge abutment is a severe hazard to the performance of bridges. Considerable investigations on pier scour have been carried out and a reliable design method is now available [4]. However, evaluating scour around abutment is in preliminary stage. It is essential to understand the scour in the design of foundations of structures as well as scour protection work. Without a detailed understanding of scour, failures are more likely to occur. The depth of scour is an important parameter for determining the minimum depth of foundations as it reduces the lateral capacity of the foundation. Extensive

experimental investigation has been conducted to understand the complex process of scour and to determine a method of predicting scour depth for various abutment situations but no generic formula has been developed yet that can be applied to all abutment conditions to determine the extent of scour that will develop. Although, numerous empirical formulae have been presented to estimate equilibrium scour depth at bridge abutment [5-11], each varies significantly, highlighting the fact that there is a lack of knowledge in predicting scour depth and that a more universal solution would be beneficial. We can achieve this with the help of soft computing models. Artificial Neural Networks (ANNs) are parallel computational models comprised of densely interconnected adaptive processing units. The main characteristic of ANNs is their ability to learn. The learning process is achieved by adjusting the weight of the interconnections according to some applied learning algorithms. Neural networks have been widely studied and have been used effectively in a number of applications including prediction, diagnostics, designing, classification and decision-making [12-13]. Different researchers [14-22] have applied the ANN for analyzing civil engineering problems. In this paper, Radial Basis Function (RBF) Network is established to estimate the scour depth around abutments using scour data from experimental investigation. The remainder of this paper is organized as follows: A brief introduction about maximum equilibrium scour depth is given in section 2. In section 3, we introduce the RBF model for scour around abutment. Section 4 reports our experimental results and finally section 5 concludes this paper.

II. MAXIMUM EQUILIBRIUM SCOUR DEPTH Maximum equilibrium scour depth at an abutment in uniform non-cohesive bed sediments depend on variables characterizing the fluid properties, flow conditions, bed sediment properties and shape of the abutment. The following functional relationship describes the maximum equilibrium scour depth [23]:

dse = f 1(U, ρ, ρs, g, l, ν, h, d50) (1)

where, U = average velocity of approach flow, ρ = mass density of the fluid, ρs = mass density of the sediment, g = acceleration due to gravity, l = length of abutment, ν = kinematic viscosity, h = approaching flow depth, d50 = median grain size, dse = equilibrium scour depth.

For two phase flow phenomenon involving sediment - water mixture the term g, ρ and ρs should not appear as independent parameters in (1). However a better representation is Δg (Δ = s-1, s = relative density of sediment particles i.e. ρs/ ρ). Hence, (1) becomes

dse = f 2(U, Δg, l, ν, h, d50) (2)

Using Buckingham π - theorem and selecting the parameters U and l as repeating variables, yields

)~,ˆ,,(ˆ3 lhRFfd erse � (3)

where Fr = abutment Froude number, that is U/(Δgl)0.5; Re =

abutment Reynolds number, that is Ul/ν; h = h/l ; l~ = l/d50. The influence of Reynolds number Re is considered negligible under a fully turbulent flow over a rough bed [7], [11]. Therefore, (3) reduces to:

)~,ˆ,(ˆ3 lhFfd rse � (4)

The above non-dimensional parametric representation may be justified as

� The term Fr is a measure of the ratio of approaching flow velocity U to (�gl)0.5. It represents the mobility of the sediment particles in the vicinity of abutments during scouring. For no scour, the magnitude of U is less than the value that causes a velocity less than the critical velocity at the abutments corresponds to the bed materials.

� The term h refers to the effect of approaching flow depth h on scour depth.

� The term l~ represents the influence of bed sediment size d50 on scour depth.

According to (4), the relationship between dse and its dependent variables can be expressed as:

dse = f 5(l, d50, h, U) (5)

III. RADIAL BASIS FUNCTION NETWORK In this study, Radial Basis Function Network has been

used and implemented in MATLAB 7.3 environment. RBF Network is a general regression technique and is suitable for both function mapping and pattern recognition problems. Fig. 1 shows a schematic diagram of a general RBF network with m, c, and l nodes in the input, hidden and output layers,

respectively [24, 25]. It shows the m-dimensional input patterns [X] being mapped to l-dimensional output, with nodes in the adjacent layers exhaustively connected. In the figure Фj is the RBF output from the jth hidden node and wkj is the weight between jth node in the hidden layer and kth output node.

Fig. 1. Radial basis function Network.

The RBF function of the jth hidden node used here is a Gaussian function, which is defined as

where σj and μj are the radius (spread) and centre of the jth unit receptive field, respectively, and the norm is the Euclidean norm.

The weights between the hidden and output layers, wkj are modified through the minimization of the error function, which is given as

where, tkp and yk

p are the target output and the calculated output of the kth output node for the pth training pattern, respectively. N is the size of the training data set and l is the number of output nodes.

The output of the network is given by the following expression

where, wkj is the weight between jth hidden node and kth output node and Φj is the output from the jth hidden node.

The RBF model implemented in this study includes Gaussian transfer function and linear transfer function in the hidden layer and output layer respectively.

For the development of the RBF model for Scour around an abutment, the combination of parameters chosen to represent scour depth, as shown in (5), provide the input variables for the ANN model. The output pattern is the equilibrium scour depth (dse). To train and test the ANN model, an experimental database for vertical-wall abutments is taken from Barbhuiya [26]. The experiment was conducted in a 20 m long, 0.9 m wide and 0.7 m deep horizontal flume with uniform non-cohesive sediment as bed materials. The ranges of all the parameters used in the experiment are given in the Table 1. The whole data set consisting of 99 data points which has been divided into a training set consisting of 79 data points and a testing or validation set consisting of 20 data points. All data values were normalized to fall in the range 0.1-0.9. This normalization is done to allow the network to be trained effectively.

The best network configuration is identified by minimizing the difference among the neural network predicted values and the desired outputs. This was done by changing the number of neurons and spread constant on a trial-and-error basis. The performance of each case was calculated based on the mean absolute error (MAE), root mean square error (RMSE) and Correlation Coefficient (CC) as follows:

��

��n

kkk ty

nMAE

1

1

(9)

� �� �

� � � ���

�

��

�

��

���

n

kk

n

kk

k

n

kk

ttyy

ttyyCC

1

2

1

2

1 (11)

where, tk and yk are target and network output for the kth output and n is the total number of events considered. The optimal configuration corresponds to the minimum value of RMSE and the maximum value of CC.

IV. EXPERIMENTAL RESULTS The training and testing results obtained are used to

form the RBF model that can be implemented to estimate local scour depth for a variety of abutment situations. The testing

results are presented in this section and the three RBF architectures corresponding to three best test-cases are discussed and two of the best cases are plotted.

A. Training Procedure The data presented for training the ANN is an important

aspect of designing ANN model. The best configuration is determined by trial and error method. The four parameters (l, d50, h, and U) in (5) are the input pattern and the equilibrium scour depth (dse) is the output pattern for the RBF model. In the first step of the RBF development process, the data is divided randomly into a training set and a testing set. The training of the RBF models were performed with the training dataset and the models with optimum architecture were selected.

The reliability of the predicted values not only depends on the ANN structure, but also on the input data. To get the reliable results, the input data needs to be trustworthy. The input data used for training the RBF models in the present study has been obtained from observations done through controlled experimentation and thus are assumed to be reliable.

B. Testing of RBF Model Once the training of the RBF Network is done with the

training data, it is necessary to test the developed networks. In developing the most accurate training model architecture, the individual cases were first ranked according to the magnitude of MAE/RMSE and CC, the best individual model having the minimum MAE/RMSE and the maximum CC were selected. From 140 training and testing cases a shortlist of 49 cases for each is presented in Table 2 and Table 3, respectively. Table 3 includes the RBF configurations with best testing results and their nearer architectures, whereas, Table 2 includes the corresponding training results. From these 49 testing cases, three cases are selected based on the magnitude of MAE/RMSE and CC and their corresponding training cases are tabulated in Table 4. The training and testing results of two selected best cases are plotted in Fig. 2 - Fig. 5.

The three cases of RBF Network had very small RMSE during testing ranging from 0.0192 to 0.0194 and consistently good coefficient of correlation ranging from 0.9968 to 0.9970. During the training, the RMSE value ranged from 0.0142 to 0.0147 and the CC value from 0.9968 to 0.9970. Case-2s in Table 3, with RBF configuration having 15 Neurons in hidden layer and spread constant 0.9 was selected as the optimum model with RMSE value of 0.0192 and the CC value of 0.9970.

On comparing the value of MAE/RMSE and CC obtained for the developed RBF models and existing empirical formulas [5-8] it is seen from the Table 5 that the developed RBF models give better results as compared to the existing empirical formulas.

The results of the testing of RBF, as tabulated in Table 5, show that the neural network model is capable of predicting scour depth to a very good level of accuracy.

C. Sensitivity Analysis To determine the relative significance of each of the

independent parameters (input neurons) on the scour depth (output), sensitivity tests were conducted with the trained neural network. Sensitivity analysis was done for each of the parameters in (5) with experimental input data with best RBF Network configuration (Table 3, case 2s). This has been done by removing one of the independent parameter in each case, and the results are tabulated in Table 6. From the results it can be concluded that among the parameters in (5) abutment length (l) is most sensitive to scour depth than other parameters. Whereas, height of the flow (h) is least sensitive to scour depth.

V. CONCLUSION In the present study, RBF network has been used to

predict the maximum scour depth around bridge abutment. Experimental result demonstrates the suitability of ANN for prediction of scour depth around bridge abutment. The RBF model (Table 3, case 2s) gives minimum MAE/RMSE and maximum CC than other configurations with test data sets and is considered to be the best.

Sensitivity analysis with the trained neural network shows the relative influence of various independent parameters. From the analysis, it is found that among the independent parameters, abutment length (l) is most sensitive and height of the flow (h) is the least sensitive to maximum scour depth.

The present study has been carried out using RBF Network. Further experimentation can be performed with other neural network models as well as with Neuro-Fuzzy and Neuro-Genetic models with the same data sets. More studies regarding suitability of other ANN models like Multi-layer Perceptron, Bayesian Network can be carried out and compared to the results obtained with the RBF models developed in the present study.

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[16] H. M. Nagy, K. Watanabe, M. Hirano, Prediction of sediment load concentration in river using artificial neural network model. J. Hydraul. Eng., ASCE, vol. 128, No. 6, pp. 588-595, 2002.

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[21] M. Soliman, Artificial neural network prediction of maximum scour hole downstream hydraulic structures, Eleventh International Water Technology Conference, IWTC11 2007 Sharm El-Sheikh, Egypt, pp. 769-777, 2007.

[22] S. M. Bateni, D. S. Jeng, B.W. Melville, Bayesian neural networks for prediction of equilibrium and time-dependent scour depth around bridge piers, Engineering Applications of Artificial Intelligence, vol. 38, No. 2, pp. 102-111, February 2007.

[23] A. K. Barbhuiya, S. Dey, Time variation of scour at abutments, J. Hydraul. Eng., ASCE, vol. 131, No. 1, pp. 11-23, 2005

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[25] K. Patra, S. K. Pal, Bhattacharyya K., Fuzzy Radial Basis Function (FRBF) Network Based Tool Condition Monitoring System using Vibration Signal, Machining Science and Technology, Taylor & Francis Group, vol. 14, No. 2, pp. 280-300, August 2010.

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Fig. 2. Comparison of RBF Network predicted and the experimental

equilibrium scour depth dse in training case-1s

Fig. 3. Comparison of RBF Network predicted and the experimental equilibrium scour depth dse in training case-1ah

Fig. 4. Comparison of RBF Network predicted and the experimental equilibrium scour depth dse in testing case-2s

Fig. 5. Comparison of RBF Network predicted and the experimental equilibrium scour depth dse in testing case-2ah

TABLE I. SUMMARY OF EXPERIMENTAL DATA RANGES

Parameter Unit Ranges Abutment length (l) cm 4 - 12 (4, 6, 8, 10 & 12)

Sediment size (d50) mm 0.26 - 3.10 (0.26, 0.52, 0.91, 1.86 & 3.10)

Approach flow velocity (U) cm/sec 21.90 - 67.00

Approach flow depth (h) cm 5.80 - 25.00

TABLE II. RBF TRAINING RESULTS

TABLE III. RBF TESTING RESULTS

Case No. of Neurons

Spread Constant MAE RMSE CC

la

13

0.6 0.0135 0.0173 0.9956 lb 0.7 0.0143 0.0180 0.9954 lc 0.8 0.0138 0.0175 0.9954 Id 0.9 0.0133 0.0170 0.9960 le 1.0 0.0119 0.0158 0.9963 If 1.1 0.0132 0.0170 0.9959 lg 1.2 0.0136 0.0178 0.9956 lh

14

0.6 0.0135 0.0170 0.9958 li 0.7 0.0143 0.0173 0.9957 lj 0.8 0.0130 0.0166 0.9960 1k 0.9 0.0124 0.0159 0.9964 lm 1.0 0.0110 0.0141 0.9969 1n 1.1 0.0126 0.0162 0.9963 lo 1.2 0.0123 0.0163 0.9962 1p

15

0.6 0.0134 0.0168 0.9959 1q 0.7 0.0143 0.0173 0.9957 lr 0.8 0.0126 0.0158 0.9963 1s 0.9 0.0115 0.0147 0.9968 It 1.0 0.0111 0.0140 0.9970 lv 1.1 0.0115 0.0148 0.9967 lw 1.2 0.0122 0.0162 0.9963 lx

16

0.6 0.0132 0.0167 0.9960 1y 0.7 0.0137 0.0167 0.9959 lz 0.8 0.0114 0.0149 0.9966 laa 0.9 0.0114 0.0143 0.9970 lab 1.0 0.0107 0.0135 0.9971 lac 1.1 0.0115 0.0148 0.9967 lad 1.2 0.0108 0.0144 0.9968 lae

17

0.6 0.0123 0.0155 0.9965 laf 0.7 0.0137 0.0167 0.9959 lag 0.8 0.0113 0.0145 0.9967 lah 0.9 0.0114 0.0142 0.9970 lai 1.0 0.0106 0.0134 0.9972 laj 1.1 0.0113 0.0147 0.9968 lak 1.2 0.0102 0.0135 0.9971 lak

18

0.6 0.0120 0.0151 0.9966 lam 0.7 0.0124 0.0155 0.9965 lan 0.8 0.0106 0.0139 0.9969 lao 0.9 0.0114 0.0142 0.9970 lap 1.0 0.0102 0.0131 0.9973 laq 1.1 0.0111 0.0145 0.9968 lar 1.2 0.0103 0.0133 0.9972 las

19

0.6 0.0115 0.0146 0.9968 lat 0.7 0.0100 0.0130 0.9974 lau 0.8 0.0103 0.0133 0.9972 lav 0.9 0.0109 0.0137 0.9971 1aw 1.0 0.0102 0.0130 0.9974 1ax 1.1 0.0108 0.0142 0.9970 1ay 1.2 0.0103 0.0133 0.9972

Case No. of Neurons

Spread Constant MAE RMSE CC

2a

13

0.6 0.0193 0.0249 0.9945 2b 0.7 0.0180 0.0237 0.9949 2c 0.8 0.0225 0.0252 0.9944 2d 0.9 0.0160 0.0204 0.9964 2e 1.0 0.0171 0.0217 0.9962 2f 1.1 0.0174 0.0203 0.9964 2g 1.2 0.0191 0.0239 0.9944 2h

14

0.6 0.0189 0.0246 0.9948 2i 0.7 0.0205 0.0259 0.9944 2j 0.8 0.0178 0.0217 0.9960 2k 0.9 0.0151 0.0204 0.9965 2m 1.0 0.0166 0.0216 0.9965 2n 1.1 0.0178 0.0223 0.9955 2o 1.2 0.0216 0.0281 0.9953 2p

15

0.6 0.0183 0.0238 0.9951 2q 0.7 0.0206 0.0259 0.9944 2r 0.8 0.0166 0.0211 0.9963 2s 0.9 0.0144 0.0192 0.9970 2t 1.0 0.0160 0.0215 0.9965 2v 1.1 0.0241 0.0284 0.9927 2w 1.2 0.0275 0.0357 0.9897 2x

16

0.6 0.0176 0.0232 0.9952 2y 0.7 0.0202 0.0268 0.9939 2z 0.8 0.0172 0.0224 0.9965 2aa 0.9 0.0143 0.0194 0.9967 2ab 1.0 0.0168 0.0234 0.9956 2ac 1.1 0.0201 0.0241 0.9955 2ad 1.2 0.0192 0.0229 0.9960 2ae

17

0.6 0.0183 0.0232 0.9947 2af 0.7 0.0198 0.0262 0.9940 2ag 0.8 0.0171 0.0215 0.9966 2ah 0.9 0.0143 0.0194 0.9968 2ai 1.0 0.0170 0.0236 0.9957 2aj 1.1 0.0220 0.0276 0.9937 2ak 1.2 0.0181 0.0228 0.9959 2al

18

0.6 0.0197 0.0247 0.9950 2am 0.7 0.0161 0.0227 0.9959 2an 0.8 0.0173 0.0216 0.9961 2ao 0.9 0.0142 0.0194 0.9968 2ap 1.0 0.0162 0.0220 0.9960 2aq 1.1 0.0229 0.0296 0.9928 2ar 1.2 0.0194 0.0242 0.9960 2as

19

0.6 0.0219 0.0280 0.9941 2at 0.7 0.0184 0.0263 0.9953 2au 0.8 0.0168 0.0216 0.9962 2av 0.9 0.0149 0.0203 0.9964 2aw 1.0 0.0151 0.0215 0.9963 2ax 1.1 0.0230 0.0296 0.9929 2ay 1.2 0.0191 0.0241 0.9959

TABLE IV. THREE BEST CASES OF RBF MODELS

TABLE V. COMPARISON OF RBF NETWORK WITH EXISTING FORMULAE

Method RMSE CC

Case-1s 0.0147 0.9968

Case-1aa 0.0143 0.9970

Case-1ah 0.0142 0.9970

Liu et al. (1961) [5] 0.0807 0.7619

Gill (1972) [6] 0.1149 0.6657

Froehlich (1989) [8] 0.2371 0.6026

Melville (1992) [7] 0.0978 0.7225

TABLE VI. SENSITIVITY ANALYSIS RESULTS FOR THE PARAMETERS IN (5)

Case No. of Neurons

Spread Constant MAE RMSE CC

1s 15 0.9 0.0115 0.0147 0.9968 1aa 16 0.9 0.0114 0.0143 0.9970 1ah 17 0.9 0.0114 0.0142 0.9970

Method MAE RMSE CC RBF Network with l, d50, h, and U

0.0115 0.0147 0.9968

RBF Network without l 0.1015 0.1226 0.8152

RBF Network without d50

0.0146 0.0189 0.9949

RBF Network without h 0.0102 0.0131 0.9975

RBF Network without U 0.0126 0.0160 0.9964