Prediction of local scour around bridge piers: artificial-intelligence-based modeling versus conventional regression methods
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Abstract
This paper presents the use of two artificial intelligence modeling methods, namely genetic programming (GP) and adaptive neuro-fuzzy inference system (ANFIS), to predict pier scour depth based on clear water conditions of 320 data sets of laboratory and field data measurements. The scour depth was modeled as a function of five main dimensionless parameters: pier width, approaching flow depth, Froude number, standard deviation of grain size distribution, and channel open ratio. A functional relationship was established using the trained GP model, and its performance was verified by comparing the results with those obtained by the ANFIS model and seven conventional regression-based formulas. Numerical tests indicated that the GP model yielded much superior agreement than the ANFIS model or any other empirical equation. The advantage of the GP model was confirmed by applying the derived GP equation to predict the scour depth around the piers of Imbaba Bridge, Egypt.
Keywords
Local scour depth Genetic programming Adaptive neuro-fuzzy Regression methods Bridge piersList of symbols
- B
Channel width (m)
- D
Pier width (m)
- d_{s}
Equilibrium scour depth (m)
- d_{50}
Mean sediment size (m)
- D/d_{50}
Dimensionless pier width
- d_{s}/D
Dimensionless pier scour depth
- F_{r}
Froude number (dimensionless)
- g
Gravitational acceleration (m/s^{2})
- L
Length of pier (m)
- U
Approach flow velocity (m/s)
- R
Correlation coefficient
- Y
Approach flow depth (m)
- Y/D
Dimensionless approach flow depth
- α
Channel open ratio
- θ
Angle of attack (°)
- σ
Standard deviation of grain size distribution
Abbreviations
- ANFIS
Adaptive neuro-fuzzy inference systems
- AI
Artificial intelligence
- ANNs
Artificial neural networks
- GAs
Genetic algorithms
- GP
Genetic programming
- HAD
High Aswan Dam
- HRI
Hydraulics Research Institute
- LGP
Linear genetic programming
- MAPE
Mean absolute percentage error
- MCM
Million cubic meters
- RMSE
Root-mean-square error
Introduction
Bridge scour is the result of the erosive action of flowing water, excavating and carrying away material from the bed and banks of streams and from around the piers and abutments of bridges (Richardson and Davis 2001). The scour is accountable for about 60% of bridge failures (Lagasse et al. 1997), resulting in loss of lives and huge economic losses. Designing the bridge foundation safely needs an accurate estimation of scour depth; underestimation may lead to bridge failure, while overestimation will lead to excessive construction costs (Azamathulla and Ghani 2010).
Pier scour attracted significant research interest for more than a century now, and numerous studies on this subject were published. Much of this research dealt with laboratory model studies of local scour. In this context, several reviews summarized equations for pier scour depths in Breusers et al. (1977), Dey (1997), and Melville and Sutherland (1988). However, these equations are often suitable only for conditions similar to those under which they were developed. Moreover, this empirical approach suffers from its associated simplified conditions and scale effects. When applying the existing empirical equations for predicting bridge pier scour to field cases, the scour depths are overpredicted (Babaeyan-Koopaei and Valentine 1999). This means increased construction and maintenance costs as the foundation levels are required to be deeper than it should be.
Soft computing tools gained importance in many fields as they differ from conventional hard computing in many ways, such as their robustness, low solution cost, and tolerance to imprecision (Chuan-Yi et al. 2013). Artificial intelligence methods, such as artificial neural networks (ANNs), adaptive neuro-fuzzy inference systems (ANFIS), genetic programming (GP), and linear genetic programming (LGP), are now widely used to predict scour around hydraulic structures and bridge piers. The American Society of Civil Engineers Task Committee (2000) reported the application of ANNs in different fields of hydrology. Deo et al. (2008) used GP to predict scour depth downstream of spillways. Azamathulla et al. (2008a, b) used ANNs and GP to determine scour depth downstream of ski-jump buckets. Guven et al. (2009) applied LGP for predicting scour depth at circular piles. ANFIS and genetic expression programming were used by Azamathulla et al. (2009a, b) to estimate scour below flip buckets. For scour below a submerged pipeline, Azamathulla et al. (2011) employed the LGP model. Najafzadeh and Barani (2011) compared the group method of data handling-based GP and the back-propagation system to predict scour depth around bridge piers.
Different from traditional physically based analytical or empirical approaches, this study investigates the utility of artificial intelligence modeling tools in predicting scour depth around bridge piers. The main objective of this research was to further enhance the available inductive modeling tools for predicting bridge scour by developing ANFIS and GP-based models for pier scour prediction utilizing available laboratory and field data and comparing their performance with several well-known bridge pier regression-based models. The already existing equations used in this study are Modified Laursen by Neill (1964), Shen et al. (1969), The Colorado State University (1975), Jain and Fischer (1979), Kothyari et al. (1992), Modified Froehlich by Fischenich and Landers (1999), and Richardson and Davis (2001). A further objective of this research was to find out which of the existing formulae works for the Nile River and how well the newly developed formula performs. Thus, the applicability of the GP model, provided that it yielded better prediction results, to large-scale models and field data was verified via applying the developed GP model to the case of Imbaba Bridge, Giza, Egypt.
Proposed artificial intelligence networks
Genetic programming (GP) and ANFIS being recently the most widely used branches of soft computing in hydraulic engineering were employed in this research as an alternative tool in the prediction of local scour around bridge piers.
Adaptive neuro-fuzzy inference system
The ANFIS is a hybrid scheme which uses the learning capability of the ANN to derive the fuzzy if–then rules with appropriate membership functions worked out from the training pairs, leading finally to the inference (Tay and Zhang 1999). The difference between the common neural network and the ANFIS is that while the former captures the underlying dependency in the form of the trained connection weights, the latter does so by establishing the fuzzy language rules (Azamathulla et al. 2009a, b). The input in ANFIS is first converted into fuzzy membership functions, which are combined together and, after following an averaging process, used to obtain the output membership functions and finally the desired output (Mousavi et al. 2007).
The configuration of an adaptive network performs a static node function on its incoming signals to generate a single node output, and each node function is a parameterized function with modifiable parameters (Navneet et al. 2015). Thus, a trial-and-error method, where a range of different shapes, numbers, and types of membership functions, as well as various parameters used as input data, should be followed toward identifying the optimal ANFIS architecture.
Genetic programming
- 1.
Generate an initial population of random compositions of the functions and terminals of the problem (computer programs).
- 2.
Execute each program in the population and assign it a fitness value according to how well it solves the problem.
- 3.
Create a new population of computer programs.
- (a)
Copy the best existing programs.
- (b)
Create new computer programs by mutation.
- (c)
Create new computer programs by crossover.
- 4.
The best computer program that appeared in any generation, the best-so-far solution, is designated as the result of genetic programming (Koza 1992).
Based on the natural selection obtained by way of the evolutionary process, GP produces an optimal function set (formula). The use of this flexible coding system allows the algorithm to perform structural optimization (Chuan-Yi et al. 2013). This can be useful in solving many engineering problems. In the development of the GP model, the terminal set, functional set, fitness function, algorithm control parameters, and termination criterion are defined (Koza 1992). The first three components determine the algorithm search space, whereas the last two components affect the quality and speed of the search.
Pier scour parameters
Conventional regression models
Most of the pier scour prediction formulae available in the literature are based on conventional regression methods and most overpredict pier scour, resulting in an uneconomical bridge foundation design. In this section, a list is made of the bridge pier scour equations used in this study. In all the formulae listed below, it was assumed that the flow angle of the attack is negligible and that the pier shape is rectangular. Chang (1988) reported that the scour depth of circular piers is 90% of that for rectangular piers and 80% of that for sharp-nosed piers. It was also assumed that the effect of the flow angle of attack and the circular shape of the pier will mutually cancel each other. This is done because of the lack of information about these factors in the data.
The Modified Laursen by Neill (1964) equation
Shen et al. (1969) formula
The Colorado State University (CSU) formula (1975)
Jain and Fischer (1979) equations
Kothyari et al. (1992) formula
Modified Froelich by Fischenich and Landers (1999) formula
Richardson and Davis (2001) formula
Development of ANFIS and GP models
Ranges of data used in the development of ANFIS and GP models
Variables | Data used in training and testing the ANFIS and GP models | Data used in verifying the ANFIS and GP models by Mueller and Wagner (2005) |
---|---|---|
Pier width (D) | 0.028–0.25 m | 0.29–4.27 m |
Approaching flow depth (Y) | 0.021–0.70 m | 0.12–12.62 m |
Approaching flow velocity (U) | 0.17–1.28 m/s | 0.15–4.48 m/s |
Median sediment size (d_{50}) | 0.0002–0.008 m | 0.001–0.108 m |
Channel open ratio (α) | 0.82–0.97 | 0.83–0.95 |
Froude number (F_{r}) | 0.036–0.75 | 0.038–0.83 |
Equilibrium scour depth (d_{s}) | 0.11–0.26 m | 0.00–7.65 m |
Dimensionless pier width (D/d_{50}) | 3.66–750 | 2.68–4270 |
Dimensionless approaching flow depth (Y/D) | 0.11–24.35 | 0.03–43.52 |
Dimensionless pier scour depth (d_{s}/D) | 0.22–2.53 | 0.20–2.36 |
ANFIS model
The ANFIS model was established using the MATLAB fuzzy logic toolbox. First, all data and input parameters were utilized in search of the best performing ANFIS structure. This involves running models (22 models) with various types of membership functions and the number of membership functions for each input parameter.
For imparting faster training and adjusting the network parameters to the above network, a two-step process is used. In the first step, the premise parameters are kept fixed and the information is propagated forward in the network to layer 4, where a least-squares estimator identifies the consequent parameters. In the second step, the backward pass, the consequent parameters are held fixed while the error is propagated, and the premise parameters are modified using the gradient descent.
GP model
Results and discussion
Statistical parameters of pier scour models
Equation | RMSE | MAPE % | R |
---|---|---|---|
Training and testing with laboratory data | |||
ANFIS (training) | 0.31 | 21.40 | 0.94 |
ANFIS (testing) | 0.40 | 22.81 | 0.89 |
ANFIS (all data) | 0.33 | 20.92 | 0.85 |
GP (training) | 0.28 | 20.30 | 0.97 |
GP (testing) | 0.36 | 21.20 | 0.91 |
GP (all data) | 0.29 | 19.85 | 0.89 |
The Modified Laursen by Neill (1964) | 0.65 | 60.00 | 0.48 |
Shen et al. (1969) | 0.72 | 59.30 | 0.53 |
The Colorado State University (1975) | 0.89 | 55.20 | 0.41 |
Jain and Fischer (1979) | 0.61 | 47.50 | 0.51 |
Kothyari et al. (1992) | 0.51 | 30.82 | 0.69 |
Modified Froelich by Fischenich and Landers (1999) | 0.60 | 51.36 | 0.57 |
Richardson and Davis (2001) | 1.21 | 65.23 | 0.35 |
Verification with field data of Mueller and Wagner (2005) | |||
ANFIS | 0.28 | 22.25 | 0.87 |
GP | 0.25 | 21.35 | 0.90 |
A comparison between the proposed GP equation (Eq. 15), ANFIS model results, and all other pier scour equations (Table 2) for different ranges of D/d_{50} and Y/D was carried out. For all ranges of D/d_{50} and Y/D, the proposed GP performance gives the best results that are quantitatively reflected in all statistical parameters, i.e., RMSE, MAPE, and R. GP followed by ANFIS outperforms in high-value predictions for the conditions of D/d_{50} > 100, D/d_{50}≦ 40, Y/D > 2, Y/D ≦ 1, compared to all other traditional equations. It should be noted that GP is more effective at extreme ranges of D/d_{50} and Y/D.
The results confirmed that none of the conventional regression equations give acceptable results, as reflected in higher RMSE and MAPE and lower R for D/d_{50} ≦ 100. At D/d_{50} > 100, only the equation by Kothyari et al. (1992) gave good results. Also, for dimensionless approaching flow depth Y/D < 1, the equation of Kothyari et al. (1992) performed well, as reflected in lower RMSE and MAPE. However, at 1 < Y/D, all of them gave reasonable results because the depth was difficult to measure for this range. The comparison of ANFIS and GP performance with other empirical equations presented in Figs. 4, 5, 6, 7, 8, 9, and 10 illustrates that the pier scour equations of Modified Froelich by Fischenich and Landers (1999) and Richardson and Davis (2001) over-estimated scour depth (Figs. 9, 10) because these formulas are based on high safety factors and envelop curves to data. Therefore, the correlation coefficient R for these two equations is lower in some selected ranges of D/d_{50} and Y/D, indicating poor performance. However, the equation of Shen et al. (1969) overpredicted the scour depth to some extent (Fig. 5) but performed well under the conditions of D/d_{50} > 100, 0 < Y/D ≤ 1. Contrary to this, the equations of Jain and Fischer (1979) under-predicted the scour depth at some ranges (Fig. 7). Furthermore, the equation of Kothyari et al. (1992) has an advantage over the other equations, as it is based on a large data range, which was used for regression analysis, but with minimal R in some selected ranges. The correlation coefficient R is lower, showing that there is a wide variation in the prediction of scour depth.
The relatively inferior performance of the regression-based models further strengthens the notion that such models are not always suitable for effectively predicting bridge pier scour depth.
GP model application to Imbaba Bridge, Egypt
According to the results discussed above, the GP model proved to provide a better prediction of the local scour depth around bridge piers other than the ANFIS model and other empirical formulas. Hence, it is used to investigate the local scour around bridge piers for the Nile River conditions. In Egypt, many investigators have worked on this important subject, but most have built their findings on laboratory data that have simplified conditions and scale effects. Therefore, the developed GP equation was employed to predict the local scour around the piers of the Imbaba Bridge. The results of the GP equation were then compared to those revealed by the seven regression equations illustrated earlier in "Conventional regression models" section.
Field data of local scour depth at Imbaba Bridge, Egypt (HRI 1992)
Pier no. | Distance from the left bank (m) | Local bed level (m) | Local water depth (m) | Local velocity (m/s) | Local F_{r} | Measured local scour depth (m) |
---|---|---|---|---|---|---|
1 | 69 | 15.00 | 5.00 | 0.60 | 0.27 | 3.90 |
2 | 104 | 13.00 | 7.00 | 0.65 | 0.26 | 8.05 |
3 | 139 | 12.00 | 8.00 | 0.72 | 0.25 | 3.65 |
4 | 209 | 10.50 | 9.50 | 0.75 | 0.25 | 3.25 |
5 | 279 | 9.00 | 11.00 | 0.78 | 0.24 | 3.10 |
6 | 350 | 7.50 | 12.50 | 0.80 | 0.24 | 2.60 |
Predicted local scour at Imbaba Bridge, Egypt
Pier no. | Scour depth (m) | |||||||
---|---|---|---|---|---|---|---|---|
Modified Laursen by Neill (1964) | Shen et al. (1969) | Colorado State University (1975) | Jain and Fischer (1979) | Kothyari et al. (1992) | Modified Froelich by Fischenich and Landers (1999) | Richardson and Davis (2001) | GP | |
1 | 5.65 | 5.12 | 5.00 | 5.48 | 4.35 | 4.71 | 4.73 | 4.15 |
2 | 10.32 | 9.98 | 9.76 | 10.12 | 8.86 | 9.53 | 9.91 | 8.20 |
3 | 5.13 | 4.88 | 4.65 | 5.25 | 4.11 | 4.43 | 4.62 | 3.85 |
4 | 4.98 | 4.37 | 4.08 | 4.86 | 3.87 | 4.17 | 4.35 | 3.47 |
5 | 4.68 | 4.16 | 4.03 | 4.51 | 3.72 | 3.97 | 4.30 | 3.25 |
6 | 4.44 | 3.93 | 3.82 | 4.23 | 3.18 | 3.81 | 4.12 | 2.94 |
Accuracy of GP equation for Nile River conditions at Imbaba Bridge, Egypt
Pier no. | Distance from the left bank (m) | Measured local scour depth (m) (HRI 1992) | Predicted local scour depth by GP (m) | % Error |
---|---|---|---|---|
1 | 69 | 3.90 | 4.15 | 6 |
2 | 104 | 8.05 | 8.20 | 2 |
3 | 139 | 3.65 | 3.85 | 5 |
4 | 209 | 3.25 | 3.47 | 7 |
5 | 279 | 3.10 | 3.25 | 5 |
6 | 350 | 2.60 | 2.84 | 9 |
Conclusions
The developed GP formula as given in Eq. (15) for the prediction of pier scour depth showed better agreement with experimental results than did the ANFIS model and the other regression equations considered in this study.
The proposed GP formula has a higher and more stable accuracy (smaller errors and greater R) in all ranges of pier scour parameters than the other empirical equations. The other equations work well only in some selected ranges of these conditions.
The study also validates the promise of ANFIS and GP as effective modeling tools for applications in hydraulic modeling.
The developed GP equation herein, Eq. (15), yielded very good agreement (6% average error) with field data than other existing empirical equations when applied to Imbaba Bridge Giza, Egypt. Thus, the GP equation is quite convenient for scour prediction in the Nile River conditions. For further studies, it is recommended to test the developed GP equation with several field data of Egyptian bridges such as El-Tahrir Bridge, El-Menyia Bridge, and Kafr El-Zayat Bridge.
Notes
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