Scour Prediction at Bridge Piers in Cohesive Bed Using Gene Expression Programming

Available online at www.sciencedirect.com

ScienceDirect Aquatic Procedia 4 (2015) 789 – 796

INTERNATIONAL CONFERENCE ON WATER RESOURCES, COASTAL AND OCEAN ENGINEERING (ICWRCOE 2015)

Scour Prediction At Bridge Piers In Cohesive Bed Using Gene Expression Programming Mohammad Muzzammila*, Javed Alama and Mohammad Danisha a

Department of Civil Engineering, AMU, Aligarh 202002, India.

Abstract Accurate and reliable estimation of the scour depth at a bridge pier is essential for the safe and economical design of the bridge foundation. The phenomenon of scour at the pier placed on sediments is extremely complex in nature. Only a limited number of studies have been reported on local scour around bridge piers in cohesive sediment mainly due to the fact that scour modeling in cohesive beds is relatively more complex than that in sandy beds. Recent research has made good progress in the development of data-driven technique based on artificial intelligence (AI). It has been reported that AI-based inductive modeling techniques are frequently used to model complex process due to their powerful and non-linear model structures and their increased capabilities to capture the cause and effect relationship of such complex processes. Gene Expression Programming (GEP) is one of the AI techniques that have emerged as a powerful tool in modeling complex phenomenon into simpler chromosomal architecture. This technique has been proved to be more accurate and much simpler than other AI tools. In the present study, an attempt has been made to implement GEP for the development of scour depth prediction model at bridge piers in cohesive sediments using laboratory data available in literature. The present study reveals that the performance of GEP is better than nonlinear regression model for the prediction of scour depth at piers in cohesive beds. © byby Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of ICWRCOE 2015. Peer-review under responsibility of organizing committee of ICWRCOE 2015 Keywords:Gene Expression Programming (GEP); Scour; Cohesive bed; Regression.

* Corresponding author. Tel.: +91-9412878092. E-mail address: [email protected]

2214-241X © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of ICWRCOE 2015 doi:10.1016/j.aqpro.2015.02.098

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Nomenclature C d0, d1, d2, d3 Frp GEP GMDH MAD MPE N R RMSE WC ŷs ߬Ƹ‫ݏ‬

Clay content Random numerical constants Pier Froude number Gene expression programming Group method of data handling Mean Absolute Deviation Mean Percentage Error Total number of observed data Correlation coefficient Root Mean Squared Error Water content Dimensionless maximum equilibrium scour depth Dimensionless shear strength

1. Introduction Scour is the engineering term for the erosion of the soil surrounding a bridge foundation (piers and abutments) caused by water. Due to high flow of water, it erodes and carries away material not only from the bed and banks of streams but also from around the piers and abutments of bridges. However, scour does not always take place at same rate for all the bed materials but it is different for different materials, i.e. courser the material, higher will be the scour rate and vice-versa. Therefore, loose granular soils are rapidly eroded by flowing water, while cohesive or cemented soils are more scour-resistant. It should be noted here that, ultimate scour in cohesive or cemented soils can be as deep as scour in sand-bed streams, which can be explained as, under constant flow conditions, scour will reach maximum depth in sand-gravel bed material in hours whereas for cohesive bed material, it would require days to attain maximum scour. Other materials like glacial till, sandstones, and shale even takes months to reach maximum scour under same condition, while limestone and dense granite would take much longer time in years and in centuries respectively. Moreover, it is a complicated process of determining the magnitude of scour due to cyclic nature of some scour processes and also due to the fact that scour can be deepest near the peak of a flood, but hardly visible as floodwaters recede and scour holes refill with sediment. Scouring does not only depends on the nature of material but also on the type of obstruction, such as scouring process at pier nose is different than that around abutments. Since scour is a very complex phenomenon therefore its modeling is very complex and hence there is a challenge on the researchers to develop new techniques and skills so that this phenomenon can be easily understood and then can be determined precisely. Recent advancement in soft computing techniques has made this job a bit easy and it is more acceptable and reliable than conventional methods of analysis. Various hydraulics engineering problems are now being solved using several Artificial Intelligence (AI) techniques, viz, Artificial Neural Networks (ANN), Genetic Algorithm (GA), Genetic Programming (GP), Gene Expression Programming (GEP), Radial Basis Function (RBF), Group Method of Data Handling (GMDH) etc. The soft computing tool of GEP has recently got recognition over various other tools due its simple modeling, easy coding and fast computations. Several researches from various engineering fields have shown that this is more accurate and feasible than other older techniques. Extensive study on pier scour has been conducted since 1950s, and these studies are based on the laboratory test results in non-cohesive soil. Several investigators have proposed various relationships for scour depth, pier width and correction factors based on laboratory and field experiments (Laursen and Toch 1956, Tison 1961, Larras 1963, Jain and Fischer 1980, Raudkivi 1986, Melville and Sutherland 1988, Melville 1997, Richardson et al. 1995, 2001, Sheppard et al. 2014). Scour depth prediction at bridge pier in cohesive soil was first proposed by Hosny (1995). He conducted flume tests on scour around cylindrical bridge pier, considering different streambeds of unsaturated cohesive soil, saturated cohesive soil and mixed beds of cohesive with non-cohesive soils and found that local scour depth gets affected by

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soil compaction and its initial water content. His results also indicated that the final scour depth gets reduced due to the existence of cohesive soil whereas mixed soils attain the maximum scour depth a little faster than that of saturated cohesive soils. He also proposed some equations for the estimation of pier scour depth in cohesive soils based on regression and dimensional analysis. Since then various researchers have contributed and have proposed various relationships for scour depth at pier and correction factors in cohesive soil (Annandale 1995, Gudavalli 1997, Wei et al. 1997, Briaud et al. 1999, Ivarson 1999, Molinas et al. 1999, Day 2000, Kwak 2000, Li 2002, Ansari et al. 2002, Briaud et al. 2005, Brandimarte et al. 2006, Seung 2009, Debnath and Chaudhuri 2010, Larsen et al. 2011). The recent experimental study conducted by Kothyari et al. (2014) revealed that maximum depth of scour occurs in the wake zone of piers. They also concluded that the fraction of clay and unconfined compressive strength of the sediment is the most significant variable that affects depth of scour in the wake zone of the pier. Moreover, they found that the depth of scour decreased with the increase in clay fraction and it also decreased with an increase in the value of unconfined compressive strength. Past researchers have developed the scour depth equation by dimensional analysis followed by nonlinear regression analysis but this approach is less precise and involves tedious calculations and hence, become less trendy in the new advance world where soft computational skills have emerged with the artificial intelligence techniques where modeling can be easily done with precision by applying less effort. Guven et al. (2008) and Azamathulla et al. (2010) have applied GEP for the scour depth prediction around bridge pier and compared their results with that of other regression techniques and found that the GEP is the best modeling technique for scour depth prediction among other tools. It has been observed from the literature survey that the computational analysis of scour depth based on AI techniques in general and GEP in particular has not been extensively done and there is an immediate need to carry out work in this regard. In the present study, GEP has been used to predict scour depth at bridge pier in cohesive soil using experimental data obtained from literature. 2. Dataset for scour parameters and conventional scour prediction models The experimental data collected by Debnath and Chaudhuri (2010) has been used in the present study. The range of various parameters and their statistics are given in Table 1. Table 1 Range and statistics of the various parameters

Parameters

Data range

Data statistics

Minimum

Maximum

Mean

COV

C

0.050

0.350

0.1929

0.1024

WC

0.192

0.387

0.2198

0.0500

Frp

0.215

0.515

0.2909

0.0563

ŷs

0.183

1.566

0.9018

0.3750

ɒො•

17.308

168.883

71.1686

40.1801

Debnath and Chaudhuri (2010) investigated the effect of clay content, water content, and sand size on the local scour at circular bridge piers embedded in a clay-sand mixed bed. They also developed regression based equation for the estimation of non-dimensional maximum scour depth with a function of pier Froude number, clay content, water content and bed shear strength as given below: ‫ݕ‬ො‫ ݏ‬ൌ ͺǤʹ‫ ܥ‬െͲǤʹͺ ܹܿ ͲǤͳͷ ‫Ͳ ݌ݎܨ‬Ǥ͹ͻ ߬Ƹ‫ ݏ‬െͲǤ͵ͺ

(1)

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A nonlinear regression method in the MATLAB environment for the same dataset in the present study was also implemented to get the scour depth prediction equation. It leads to the following equation for the estimation of scour depth at the bridge pier embedded in the bed of the clay-sand mixture. ‫ݕ‬ො‫ ݏ‬ൌ ͶǤ͹͹‫ ܥ‬െͲǤʹ͵ ܹܿ ͲǤʹͺ ‫Ͳ ݌ݎܨ‬Ǥͷ͵ ߬Ƹ‫ ݏ‬െͲǤʹͷ

(2)

There is a slight change in the exponents in the input parameters. However, the performance of both the prediction equations is almost identical as evident in Table 3. 2. Gene Expression Programming (GEP) Gene-Expression Programming (GEP) is a new evolutionary Artificial Intelligence technique developed by Ferreira (2001). This technique is an extension of genetic programming (GP). The genome is encoded as linear chromosomes of fixed length, as in Genetic Algorithm (GA); however, in GEP the genes are then expressed as a phenotype in the form of expression trees. GEP combines the advantages of both its predecessors, genetic algorithm (GA) and GP, and removes their limitations. GEP is a full-fledged genotype/phenotype system in which both are dealt with separately, whereas GP is a simple replicator system. As a consequence of this difference, the complete genotype/phenotype GEP system surpasses the older GP system by a factor of 100.In GEP, just like in other evolutionary methods, the process starts with the random generation of an initial population consisting of individual chromosomes of fixed length. The chromosomes may contain one or more than one genes. Each individual chromosome in the initial population is then expressed, and its fitness is evaluated using one of the fitness function equations available in the literature. These chromosomes are then selected based on their fitness values using a roulette wheel selection process. Fitter chromosomes have greater chances of selection for passage to the next generation. After selection, these are reproduced with some modifications performed by the genetic operators. In Gene Expression Programming, genetic operators such as mutation, inversion, transposition and recombination are used for these modifications. Mutation is the most efficient genetic operator, and it is sometime used as the only means of modification. The new individuals are then subjected to the same process of modification, and the process continues until the maximum number of generations is reached or the required accuracy is achieved. Because a random numerical constant (RNC) is a crucial part of any mathematical model, it must be taken into account; however, Gene Expression Programming has the ability to handle RNCs efficiently. In GEP, an extra terminal ‘?’ and an extra domain Dc after tail of the each gene is introduced to handle RNCs (Azmathulla et al. 2011). 3. GEP modeling for scour depth prediction at the bridge pier in a bed of mixture of clay and sand In the present study a new approach has been adopted for scour depth prediction model using Gene Expression Programming (GEP), that was developed by Candida Ferreira in 1999 (Ferreira 2001). In GEP, the individuals are encoded as linear strings of fixed length (the genome or chromosomes) which are afterwards expressed as nonlinear entities of different sizes and shapes (i.e. simple diagram representations or expression trees). The great insight of GEP consisted in the invention of chromosomes capable of representing any expression tree. For that Ferreira (2001) created a new language (which she named as Karva language) to read and express the information of GEP chromosomes. Furthermore, the structure of chromosomes was designed to allow the creation of multiple genes, each encoding a sub-expression tree. The genes are structurally organized in a head and a tail, and it is this structural and functional organization of GEP genes that always guarantees the production of valid programs, no matter how much or how profoundly we modify the chromosomes. There are five major steps to use gene expression programming (Ferreira 2001). The first major step is to select the fitness function and initial population. For the present problem, the fitness fi of an individual program i is measured by the following expression: ‫ܥ‬

‫ݐ‬ ݂݅ ൌ σ݅ൌͳ ൫‫ ܯ‬െ ห‫ܥ‬ሺ݅ǡ݆ ሻ െ ݆ܶ ห൯

(3)

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where M is the range of selection, C(i,j) is the value returned by the individual chromosome i for fitness case j (out of Ct fitness cases) and Tj is the target value for fitness case j. Now, if ห‫ܥ‬ሺ݅ǡ݆ ሻ െ ݆ܶ ห (the precision) less or equal to 0.01, then the precision is equal to zero, and fi= fmax = Ct.M. In this case, M = 100 is used and, therefore, fmax = 1000. The advantage of this kind of fitness function is that the system can find the optimal solution for itself (Ferreira 2001). The second major step consists in choosing the set of terminals T and the set of functions F to create the chromosomes. In this, the terminal set consists obviously of the independent variable(s), (see table 2) but the choice of the appropriate function set is not so obvious, but a good guess can always be done in order to include all the necessary functions. The third major step is to choose the chromosomal architecture, i.e., the length of the head and the number of genes. A single gene and two head lengths were used initially and then, the number of genes and heads were increased by one at a time during each run until the most appropriate fit was obtained. It was observed that more than 4 genes and a head length greater than ten did not significantly improve the performance of GEP model. Thus, the head length, h = 10, and 4 genes per chromosome were employed for the GEP model in the present study. The fourth major step is to choose the linking function. In this study, addition was used as a linking function and the final step is to choose the set of genetic operators that cause variation and their rates. A combination of all genetic operators (mutation, transposition and crossover) is used for this purpose (Table 2). Table 2 Summary of GEP parameters S. No.

GEP parameters

Description

1.

Population Size

50

2.

Genes per chromosome

4

3.

Gene head length

10

4.

Functions

+ - × ÷ √ and ^

5.

Gene tail length

11

6.

Mutation rate

0.044

7.

Inversion rate

0.1

8.

Gene transposition rate

0.1

9.

One point recombination rate

0.3

10.

Two point recombination rate

0.3

11.

Gene recombination rate

0.1

12.

Fitness function

Root relative squared error

The explicit formulation of the GEP for the scour depth prediction at the bridge pier in the cohesive sediments has been obtained as: ‫ݕ‬ො‫ ݏ‬ൌ ൬ቀʹ‫ ݌ݎܨ‬ቁ െ ͵‫ܥ‬൰ ൅ ͲǤ͸ͷ͸ͳͲͶ ൅ ܹܿ ൅

ͳ ߬ො ‫ݏ‬

(4)

The expression trees for the above GEP formulation are shown in Figs. 1 and 2. In these figures, d0 = 2, d1 = 3, d2 = 0.656104 and d3 = 1. A simplified form of Eq. 4 is as follows: ‫ݕ‬ො‫ ݏ‬ൌ ͲǤ͸ͷ͸ ൅ ʹ‫ ݌ݎܨ‬െ ͵‫ ܥ‬൅ ܹܿ ൅

ͳ ߬ො ‫ݏ‬

(5)

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Fig.1 Sub ET-1 for GEP formulation

Fig.2 Sub ET-2 for GEP formulation

4. Performance of scour depth prediction models at bridge piers in cohesive sediments In order to assess the performance of the various scour depth prediction models at bridge piers in cohesive sediments under considerations, The commonly used performance parameters such as Root Mean Squared Error (RMSE), Mean Percentage Error (MPE), coefficient of correlation (R) and Mean Absolute Deviation (MAD) are adopted in the present study. The definition of these parameters is provided in Appendix. Table 3 Performance evaluation of models Performance parameters

Debnath and Chaudhuri (2010)

Nonlinear regression (Present study)

GEP

R

0.84

0.86

0.93

MPE

-10.06

-11.21

-3.97

MAD

0.18

0.16

0.11

RMSE

0.36

0.41

0.23

The performance parameters for the GEP and the regression models for the same set of scour data are shown in Table 3. It may be observed from this table that the correlation coefficient of Debnath and Chaudhuri (2010) equation is 0.84 and that of nonlinear regression of the present study is 0.86, whereas the GEP has the correlation coefficient of 0.93. The MPE, MAD and RMSE of Debnath’s equation and present regression equation are almost same but those of GEP are the least. Hence, it may be concluded that the performance of GEP is the among all scour prediction equation in the present study. The scatter diagram between observed and predicted relative scour depth has been shown in Fig.3. This figure also indicates that the gene expression programming is least scattered from the line of perfect agreement than that of the Debnath’s equation and nonlinear regression equation.

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Fig. 3 Plot of observed v/s predicted

5. Conclusion The Gene expression programming (GEP) was implemented as an alternative tool for modeling of scour depth prediction at bridge pier embedded in cohesive sediments and its performance over that of the conventional regression prediction model. It was found that the performance of GEP is more encouraging and better than that of the conventional regression model for the prediction of scour depth at bridge pier in cohesive beds. It is observed that the equation obtained by GEP (R=0.93 and RMSE=0.23) is much simpler and far better than the regression equation proposed by Debnath and Chaudhuri (2010) (R=0.84 and RMSE=0.36). Hence, it can be said that not only GEP can suitably accounts for complexity and nonlinearity behavior of scouring in cohesive sediments but also it can reduce the complex model into a simple equation. Appendix ܴൌ

ത ത σܰ ݅ൌͳ൫ܻ ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ െܻ ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ ൯Ǥ൫ܻ ݅ሺ݉‫ ݈݁݀݋‬ሻ െܻ ݅ሺ݉‫ ݈݁݀݋‬ሻ ൯ ʹ ܰ ʹ ത ത ටσܰ ݅ൌͳ൫ܻ ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ െܻ ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ ൯ Ǥσ݅ൌͳ൫ܻ ݅ሺ݉‫ ݈݁݀݋‬ሻ െܻ ݅ሺ݉‫ ݈݁݀݋‬ሻ ൯

ܻ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻെܻ݅ሺ݉‫ ݈݁݀݋‬ሻ ൰൨ ൈ ܻ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ

ͳ

‫ ܧܲܯ‬ൌ  ൤σܰ ݅ൌͳ ൬ ܰ

ͳ

‫ ܦܣܯ‬ൌ  ൤σܰ ݅ൌͳ ฬ ܰ

ͳ

ܻ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ െܻ݅ሺ݉‫ ݈݁݀݋‬ሻ ฬ൨ ܻ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ

ܻ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻെܻ݅ሺ݉‫ ݈݁݀݋‬ሻ ൰ ܻ݅ሺ‫ ݀݁ݒݎ݁ݏܾ݋‬ሻ

ܴ‫ ܧܵܯ‬ൌ ඨ ൤σܰ ݅ൌͳ ൬ ܰ

ͳͲͲ

(A1)

(A2) (A3)

ʹ

൨

(A4)

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