Department of Civil Engineering Sydney NSW 2006 AUSTRALIA.

Transcription

1 Sydney NSW 2006 AUSTRALIA Environmental Fluids/Wind Group Neural Network assessment for scour depth around bridge piers D-S Jeng, BE ME PhD S. M. Bateni, BE ME E. Lockett, BE November 2005

2 Environmental Fluids/Wind Group Neural Network assessment for scour depth around bridge piers D-S Jeng, BE ME PhD S. M. Bateni, BE ME E. Lockett, BE November 2005 Abstract: The mechanism of flow around a pier structure is so complicated that, it is difficult to establish a general empirical model to provide accurate estimation for scour. Interestingly, each of the proposed empirical formula yields good results for a particular data set. In this study, an alternative approach, artificial neural networks (ANN), is proposed to estimate the equilibrium and timedependent scour depth with numerous reliable data base. Numerous ANN models, multi-layer perceptron using back propagation algorithm (MLP/BP) and radial basis using orthogonal least-squares algorithm (RBF/OLS), Bayesian neural Network (BNN) and single artificial Neural Network (SANN) were used. The equilibrium scour depth was modeled as a function of five variables; flow depth, mean velocity, critical flow velocity, mean grain diameter and pier diameter. The time variation of scour depth was also modeled in terms of equilibrium scour depth, equilibrium scour time, scour time, mean flow velocity and critical flow velocity. The training and testing data are selected from the experimental data of several valuable references. Keywords: Neural networks; Bridge pier; Back propagation algorithm; Orthogonal least square algorithm, Scour depth

3 Copyright Notice, Research Report R855 Neural Network assessment for scour depth around bridge piers 2005 D-S Jeng, S. M. Bateni and E. Lockett This publication may be redistributed freely in its entirety and in its original form without the consent of the copyright owner. Use of material contained in this publication in any other published works must be appropriately referenced, and, if necessary, permission sought from the author. Published by: The University of Sydney Sydney NSW 2006 AUSTRALIA November 2005 This report and other Research Reports published by The Department of Civil Engineering are available on the Internet: 2

4 Contents 1. Introduction Literature Review Experimental approaches Theoretical approaches Artificial Neural Network (ANN) Local Scour Depth Single Artificial Neural Network Model (SANN) Principle of Artificial Neural Network (ANN) Single Artificial Neural Network (SANN) SANN Model for Scour around a Pier Results and discussions Training procedure Test of SANN Model Summary Multi Artificial Neural Network (MANN) Multi layer perception network (MLP/BP) Radial basis network (RBF/OLS) MANN model for scour around piers Development of ANN Models Data presentation Results and discussion Equilibrium scour depth prediction using the original data set Equilibrium scour depth prediction using non-dimensional data set Time-dependent scour depth prediction Comparison MLP/BP with existing equilibrium scour depth prediction equations Sensitivity analysis Summary Bayesian Neural Network (BNN) Neural Network Framework Training of Neural Network Selecting Parameters Development of the Models Numerical Results Comparison of the analyses using the non-dimensional and original data sets as inputs Committee Model Comparison of Bayesian network model with existing scour prediction equations Sensitivity Analysis Summary Conclusions...68 References

5 1. Introduction Scour is defined as the process whereby underwater sediment is removed from the base or toe of a structure by waves and currents. The Shoreline Protection Manual (2001) replaces the term erosion with scour to distinguish the process caused by the presence of a structure in the marine environment. (a) (b) Figure Examples of scour around piers (a) scour and bed degradation at Kaoping Bridge, Taiwan, (b) failure of Kaoping Bridge, Taiwan due to combination of general and local scour (Lin, 1998) Placing a hydraulic structure in either a river or marine environment will alter flow patterns in the vicinity of the structure. The changes to the flow pattern cause an increase in sediment movement leading to the phenomenon of scour. The oldest and most comprehensive experimental study on bridge pier scour was conducted by Chabert and Engeldinger (1956). Understanding the phenomenon of bridge pier scour is of paramount concern to the hydraulics engineering profession as without this detailed knowledge bridge failures can occur, resulting in loss of life and devastating destruction (Figure 1.1). From a purely economic standpoint, businesses of all sizes depend on major interstates, city streets and rural roads to move products and services. Therefore, where roads and bridges are temporarily or permanently closed due to damage sustained because of scour, the economy will suffer. Given the importance of understanding the stability of hydraulic structures exposed to scour, extensive research has been conducted on the mechanisms and dynamics of scour and scour patterns around different objects, Herbich et al. (1984) and Sumer et al. (2001). 4

6 Blodgett (1978) studied 383 bridge failures caused by catastrophic floods. Approximately half of these failures were caused by local scour. Although some of the scour was attributed to the increased local and contraction scour, due to accumulation of ice and debris, a large portion resulted from erroneous prediction of scour depth during engineering design. Among these, 86% of the 577,000 bridges in the National Bridge Registry (NBI) of America are built over waterways. More than 26,000 of these bridges have been found to be scour critical, meaning that the stability of the bridge foundation has been, or could be, affected by the removal of bed material. Some of the different hydraulic structures that have been investigated for scour include: - Piles - Piers - Pipelines - Abutments Research to determine the scour depths around these structures continues due to its significance to the engineering profession. Understanding scour is essential in the design of both foundations of structures and 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. It is for this reason that extensive experimental investigation has been conducted in an attempt to understand the complex process of scour and to determine a method of predicting scour depth for various pier situations. To date, no generic formula has been developed that can be applied to all pier cases to determine the extent of scour that will develop. Numerous empirical formulae have been presented to estimate equilibrium scour depth at bridge piers, including Laursen and Toch (1956), Shen (1971), Hancu (1971), Breuser et al. (1977), US DOT (1993), Melville and Sutherland (1988) and Chiew (1999). These approaches are summarised in Table 2.1. 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. It is the lack of knowledge in predicting scour depth for all pier conditions that has led to the undertaking of this thesis project. In this study, an alterative approach, Artificial Neural Network models, will be established to estimate the scour depth around piers. Numerous neural network models will 5

7 be outlined and some numerical examples will be used to demonstrate the capacity of NN models. 6

8 2. Literature Review The phenomenon of scour around hydraulic structures has been studied for a few decades. Evidence of its effects has been observed since hydraulic structures were first placed into the marine environment. These early studies led to the development of a large database of information relating to the experimental, theoretical and numerical methods used in the area for a range of different hydraulic structures. The aim of this literature review is to outline the techniques that are currently employed to determine scour around a pier placed in the marine environment and to put the work of this thesis into context. 2.1 Experimental approaches Typical investigation into scour depth development has been through various methods of experimentation. Experimental relationships may be inadequate because of the large number of parameters affecting scour. The depth of local scour is a function of a number of different parameters, many of which are interrelated. For a single circular cylinder in an erodible bed, the factors influencing local scour can be divided into those (a) describing the fluid, (b) describing the sediment, (c) describing the flow and (d) describing the structure. Research of the available literature on pier scour experiments highlights the many possible scenarios for the development of scour depth. The oldest and most complete experimental study on bridge pier scour was conducted by Chabert and Engeldinger (1956). One test channel had a width of 0.80 m with four different pier diameters ranging from 0.05 to 0.15 m, four uniform sediments with a grain size between 0.26 and 3.2 mm, and three flow depths 0.10m, 0.20m, and 0.35 m. The second channel was 3.0 m wide containing uniform sediment of 3 mm. In addition to the 0.150m circular cylinder, six other pier shapes were tested and additional experiments conducted aimed at finding optimum arrangements for scour protection. The total study involved around 300 tests ranging from a few hours to days. Maximum scour resulted at the transition between the clear-water and the level-bed regimes. Despite the large data set, no general scour relation was established. Kothyari et al. (1972) approximated shear stress at the bottom of a scour hole by assuming that the shear stress under the vortex is a function of the cross-sectional area of the vortex and that the vortex expanded to fit the scour hole. Kothyari et al. (1972) gave the following procedure for finding the scour depth as a function of time: 7

9 Compute τ p,t, probabilistic shear stress, and d s =0 with A0 2 τ p, t = 4. 0τ u = ρu *, t A ; (2.1) t Calculate t 0, the time required to remove one particle, with C d t = 1 0 p ; u 0, t *, t Increment d s by one grain diameter over time t 0 ; Increment the time by t 0 recomputed τ p,t for the incremented d s and repeat the procedure. Scour ends when τ p,t τ c, the critical shear stress Ettema (1980) developed a model that modeled only the upstream half of the scour hole. He found that the width of the erosion area at the bottom of the scour hole varied according to D/d, where D is the pier diameter and d the water depth. Ettema (1980) derived an equation for the time rate of scour as d s D t = k k1 d + 3 2k4D 2 1 cot φ I cotφ p (2.2) where k 1 is a volume constant, k 2 and k 4 are area constants and φ is the sediment submerged angle of repose. Ettema (1980) did not attempt to solve for the scour depth as a function of time and so he did not attempt to evaluate I p, the integral of the probability function for the entrainment of particles within the scour hole. He did note that I p can be considered independent of particle size for similar values of d s /D, flow intensity based on shear velocities (U/U c ) and normalized sediment size (D/d). Melville (1997) presented an integrated approach to estimate local scour depth at bridge piers similar to Ettema (1980) and Chiew (1984). The method proposed is based on empirical relations, termed K-factors that account for the effects of flow depth, foundation size, flow intensity, sediment characteristics, foundation type, shape and alignment and 8

10 approach channel geometry on scour depth. Melville developed the following equation to determine equilibrium scour depth d s = K K K K K K (2.3) yw l d s θ G where the K-factors are defined by envelope curves developed from valid laboratory data. The method proposed by Melville is incapable of predicting the lesser scour depths that can occur if the design flood duration is shorter than that required to develop the equilibrium scour depth. The expressions envelop the available laboratory data making them fundamentally conservative. Yanmaz and Altinbilek (1991) conducted clear-water bridge pier experiments for almost uniform sediments with grain sizes of 0.84 and 1.07 mm in a 0.67 m wide rectangular flume. Circular and square shaped piers between 47 and 67 mm were tested under zero angle of attack. A differential equation for the progress of scour hole depth was established and solved numerically. Yanmaz and Altinbilek (1991) discussed the shortcomings of their model. The model only applies in the range of experimental values on which the coefficients were based. The non-dimensional representation of time involving parameters of geometry and granulometry is notable. Oliveto and Hager (2002) used a rectangular channel one metre wide and eleven metres long to conduct clear water pier scour experiments for 3 uniform sediments of grain size 0.55, 3.3 and 4.8mm and 3 non-uniform sediments. Piers between 1-60% of channel width were tested with flow depths ranging from 1-40% of channel width. A general scour equation was developed based on similarity analysis and detailed hydraulic experimentation. This equation is useful in predicting scour depth providing that the experimental criteria are met. Sheppard et al. (2004) conducted 14 large diameter pier scour experiments in a 6.1m wide, 6.4m deep and 38.4m long flume. Clear water scour tests were performed with three different diameter circular piers (0.114, and 0.914m), three different uniform cohesionless sediment diameters (0.22, 0.80 and 2.90mm) and a range of water depths and flow velocities. Two different mathematical functions were found to fit local scour time history in the clear water range: 1 1 ( ) ( ) d s ( t) = a 1 + c 1 (2.4) 1+ abt 1+ cdt 9

11 d s [ 1 exp( bt) ] + c[ 1 ( dt) ] ( t) = a exp (2.5) where a, b and c are curve fitting coefficients and t is the time elapsed. The duration of the test must be at least that used in the experiments in order to produce accurate estimates of equilibrium scour depths. The discovery of the sensitivity of clear-water equilibrium scour depth and scour rate on suspended sediment concentration is also significant. This might help explain some of the scatter in published data and the differences in data obtained by different researchers. It might also help explain why laboratory data results often over predict clearwater scour values observed in the field. This equation had been found to fit experimental time series reasonably, Gosselin (1997). Bertoldi and Jones (1998) developed this same equation and determined the coefficients of the equation by fitting a long-term experiment with sediment of the same size as that of the data set. Melville and Chiew (1999) conducted experiments in four different flumes to consider the temporal development of clear-water local scour depth at cylindrical bridge piers in uniform sands. To provide a broader range of data, Ettema s (1980) and Graf s (1995)) results of local scour depth where included. Melville and Chiew (1999) drew the following conclusions from the study - Equilibrium scour depth is approached asymptotically under clear-water conditions (Figure 2.1) - After 10% of the time to equilibrium, scour depths vary between about 50% and 80% of equilibrium depth depending on flow intensity (Figure 2.1) - Equilibrium time scale for development of the scour hole is a function of flow intensity (V/Vc), flow shallowness (y/d) and sediment coarseness (D/d50). - The equations developed allow estimation of local scour depth. The relations provided by Melville and Chiew (1999) have not been confirmed at prototype scale and thus are unproven. Knowledge of the design flood hydrograph is necessary to assess the time of scouring. 10

12 Figure 2.1 Temporal Development of Scour Depth (Melville and Chiew, 1999) The above papers have dealt with prediction of scour depth in an experimental way. The formulas developed are useful in the situations that they relate to however, they may not be appropriate for predicting scour depth in a variety of other pier situations. The existing methods of predicting scour depth are summarised in Table

13 Table 2.1 Different Approaches for Scour Depth Prediction Reference Proposed Theories Lauesen and Toch (1956) d se = 1.35D Y Shen (1971) UD d se = v Hancu (1971) d 2 se U U c = D U c gd Breusers et al. (1977) d se D = f U U c 2 tanh Y D Melville and Sutherland (1988) dse D = K K K K K where K I, K d, K y K α and I d y α s K s are respectively flow intensity, sediment size, flow depth, pier shape and alignment coefficients. US DOT (1993) dse Y 3 U = K in which K D D 3 is a gy factor that accounts for the state of bed mobility and for the related presence of dunes on approach bed; it varies from 1.1 to 1.3 Melville and Chiew (1999) d = K K K where the K factors are flow se yd l d depth-pier width, flow intensity and sediment size coefficients. 12

14 2.2 Theoretical approaches One of the major inadequacies that can be observed from the experimental work just reviewed is that there is no appropriate design method to estimate depth of local scour at piers in different environmental conditions. There are various formulas available however, these formulas seem to have no similarities in their appearance or their predictions. A theoretical approach, which is a more analytical approach than experimentation, may be a possible way of predicting local scour depth for a number of different environmental conditions. Melville (1997), mentioned above, followed an analytical approach to develop envelope curves based on the available data for a variety of different parameters. From the envelope curves he was able to empirically determine expressions for the various factors that can affect local scour depth. These expressions were than combined into a single equation that can be used to estimate local scour depth, d s = K yw K l K d K s KK θ K G. Using the scour depth equation and the flowchart (Figure 2.2) scour depth can be determined for design purposes. The local scour depth equation takes into consideration a variety of factors. It envelopes extensive experimental data and thus provides a broad base on which to estimate scour depth. The disadvantage of this expression is that it is inherently conservative and thus may result in over design. 13

15 Figure 2.2 Flowchart of method for scour depth estimation (Melville, 1997) Adopting a theoretical approach to develop a model to estimate local scour depth is still in its infancy however, it will be a valuable tool when more research is conducted and data is available for modelling. Although experimentation provides a broad idea of what scour depth levels may be in the environment, large discrepancies exist between estimated and actual scour depth. This is due to field conditions being less regular than idealized conditions in the lab. Hence adopting a theoretical approach to estimate local scour depth may be more accurate. 14

16 2.3 Artificial Neural Network (ANN) It has been highlighted that various design methods and formulas for the estimation of local scour depth around bridge piers have been proposed. The main problem with these formulas is that the existing equations are based on laboratory data. They do not accurately predict environmental conditions and thus, tend to give conservative estimates. An alternative method to overcome the variations involved with experimental and theoretical estimates is Artificial Neural Network (ANN). ANN s act as universal function approximators, this making them useful in modelling problems in which the relationship between dependent and independent variables is poorly understood. Recently, ANN has been widely applied in various areas of hydrology and water resource engineering. Among these, Grubert (1995) used ANN to predict the flow conditions when interfacial mixing in stratified estuaries commences. The neural network results were compared to the semi-theoretical solution based on a combination of results from inviscid flow theory, turbulent flow theory and interfacial friction experiments. Although neither of the two solutions was perfect in every respect, they were sufficiently close to one another. Engineers can now compute the critical velocity at which interfacial mixing commences at a particular location in a stratified estuary or fjord. Kambekar and Deo (2003) carried out scour data analysis using neural networks. Different networks were developed to predict the scour depth based on the input parameters of wave height, wave period, water depth, pile diameter, Reynold s number, maximum wave particle velocity, maximum shear velocity, Shield s parameter and Keulegan-Carpenter number. The neural network was able to provide a better alternative to the statistical curve fitting with a weight matrix developed to predict non-dimensional scour depth from the input of wave height, wave period, water depth and pile diameter. Birikundavyi et al. (2002) investigated the performance of neural networks as potential models capable of forecasting daily streamflows. An appropriate model was identified and a comparison approach was used to evaluate it against a conceptual model presently in use. It was found that the neural networks outperform the deterministic model for up to 5-day-ahead forecasts. It was also found that the results obtained with the neural network were far superior to the ones obtained with the classic model. 15

17 Nagy et al. (2002) used an ANN model to estimate the natural sediment discharge in rivers in terms of sediment concentration. Several trials were done to design a suitable architecture of the network. The model was trained with measured field data of variables selected on the basis of fluid and sediment dynamics. Model verification was done with a large number of data from several rivers. The results indicated that a neural network approach estimates sediment concentration well compared to conventional methods. Coppola et al. (2003) demonstrate the feasibility of training an ANN for accurately predicting transient water levels in a complex multilayered ground-water system under variable state, pumping, and climate conditions. The ANN was trained to predict transient water levels in response to changing pumping and climate conditions. The trained ANN was validated with ten sequential seven-day periods and the results compared against both measured and numerically simulated ground-water levels. The results indicate that the ANN technology has the potential to serve as a powerful prediction and management tool for many types of ground-water problems. ANN models are attractive in the area of estimation of local scour around bridge piers in this study. This is because of their adaptive nature where learning by example replaces programming or making functions in solving problems. This feature renders computational models very appealing in domains, where one has little or incomplete understanding of the problem to be solved but where training data examples are available. In the reviewed papers it was concluded that ANN provided a higher level of accuracy in solving a particular problem when compared to experimental and theoretical results. ANN may therefore be a viable alternative in the estimation of local scour depth around bridge piers, provided a reliable database is available. 16

18 3. Local Scour Depth Equilibrium scour depth around a circular pier in a steady flow over a bed of uniform, spherical and cohesionless sediment depends on numerous groups of variables such as; flow, sediment characters, and pier geometry. This scour situation involves an approach flow over a loose bed and a complex three dimensional flow field at the pier. The basic similitude requirements for hydraulically modeling the simplest of pier-scour situations are difficult to satisfy. Scour depth at a pier as in Figure 3.1, depends on variables characterizing the fluid, flow, bed sediment, and pier. Thus, the following functional relationship can describe scour depth (Ettema et al. 1998). d s e = f ( ρ, μ, U, Y, g, d50, U, D), (3.1) c in which ρ = fluid density; μ = fluid dynamic viscosity; U = average velocity of approach flow; Y = flow depth; g = gravitational acceleration; d = particle mean diameter; U = critical value of U associated with initiation of motion of particles on bed surface; D = diameter of the pier and d se = equilibrium scour depth. 50 c Figure 3.1: Flow and local scour around a circular pier. The eight independent variables in (3.1) are reducible to a set of five non-dimensional parameters. If ρ,u, and D are chosen as repeating variables, the following functional relationship describes scour depth normalized with pier diameter: 17

19 d se D U U Y D ρud = Ψ(,,,, ). (3.2) U gy D d c 50 μ A choice of other repeating variables would result in a somewhat different set of nondimensional parameters. However, study of literature (Ettema et al. 1998) showed that the non-dimensional parameters in (3.2) mainly control scour process around bridge piers. Therefore, non-dimensional variables in (3.2) are used in this study. Although the parameters which can affect scour depth, as shown in (3.1) and (3.2), have been selected but, other parameters such as flow direction, geometry of pier can also be included later (Melville and Sutherland 1988). Also, the process of local scour at bridge piers is time dependent. Peak flood flows may last only a number of hours or a few days in the field, while have insufficient time to generate equilibrium depth. Thus, according to Melville and Chiew (1999), the relation between the depth of local scour at a bridge pier ( d s ) at a particular time (t ) in a steady flow can be written d s = f ρ, μ, U, Y, D, g, d, U, t, t ), (3.3) ( 50 c e where t e is time for equilibrium depth of local scour to develop. Based on (3.3), Melville and Chiew (1999) presented the following formula to predict local scour depth ( d s ) d d s se exp 0.03 U c t ln U t = e 1.6. (3.4) According to (3.4), the relationship between d s and its dependent parameters can be written d = f d, U, U, t, t ). (3.5) s ( se c e Now, (3.5) can further be written in the following non-dimensional form d s U t = f (, ). (3.6) d U t se c e 18

20 4. Single Artificial Neural Network Model (SANN) Principle of Artificial Neural Network (ANN) Artificial Neural Networks (ANNs) are computational networks that attempt to simulate the networks of nerve cells of the human or animal central nervous system. They are collections of simple, highly connected processing elements that respond (or learn ) according to sets of inputs. As such they are capable of realizing a greater variety of non-linear relationships of considerable complexity between input and output data sets. The brain is composed of over 100 different kinds of special cells called neurons. The number of neurons in the brain is estimated to range from 50 billion to over 100 billion. These neurons are divided into interconnected groups called networks and provide specialized functions. Each group contains several thousand neurons that are highly interconnected with each other. Thus the brain can be viewed as a collection of neural networks. Figure 4.1 Single Neural Network Structure (Birikundavyi et al., 2002) An Artificial Neural Network (ANN) is a model that emulates the neural network of the biological brain. An ANN is composed of basic units called neurons that are the processing 1 Part of this section forms the manuscript, Jeng (2006): Comparisons of ANN models for local scour around a pier. The Fifth International Conference on Engineering Computational Technology, Las Palmas de Gran Canaria, Spain, September

21 elements in a network. Each neuron receives input data, processes it, and delivers a single output (Figure 4.1). The input can be raw data or output of other processing elements. The output can be the final product or it can be an input to another neuron. An ANN is composed of a collection of interconnected neurons that are often grouped in layers. The two basic layered architects are (a) two layers: input and output and (b) three layers: input, intermediate (called hidden) and output. The input layer receives data from the outside world and sends signals to the subsequent layers. The outside layer interprets signals from the previous layer to produce a result that is transmitted to the outside world as the network s understanding of the input data. Each input corresponds to a single attribute of a pattern or other data in the external world. The network can be designed to accept sets of input values that are either binaryvalued or continuously valued. The output of the network is the solution to the particular problem. The initial output is usually incorrect thus the network has to be trained until it gives the proper output. The output data is usually rescaled by the so-called connection weights, one for each wire coming to a neuron from another one. Thus if the i th neuron receives a signal from the j th, and the connection weight for this wire from neuron j to neuron i has the value w ij then the activity received by the i th neuron will have the amount w ij. The total activity received by the i th neuron will be A w u (4.1) i = i ij j where u j is the activity of the j th neuron, being 1 if the j th neuron is active and 0 if inactive. The i th neuron responds with a signal which depends on the value of its activity at that time. The weights represent the relative strengths of the various connections that transfer data from layer to layer. The objective in training a neural network is to find a set of weights that will correctly interpret all the sets of input values that are of interest for a particular problem. 4.2 Single Artificial Neural Network (SANN) The parameters describing the fluid are: the density of the fluid (ρ), fluid viscosity (μ) and kinematic viscosity (ν = μ/ ρ). Those describing the sediment are: grain diameter which may be median grain diameter (d 50 ) or an equivalent grain diameter (d e ), standard deviation of grain size distribution (σ g ), density of the sediment (ρ s ) and fall velocity (w f ). The parameters describing the flow are: depth of the approach flow (y = b), average velocity of approach 20

22 flow (U), roughness of the approach flow (k s ), energy slope of the flow (S 0 ), bed slope (S b ) and bed shear stress (τ). The parameters describing the structure in the case of a pier are: pier diameter (D), pier surface roughness (k c ), critical friction or shear velocity (u*) and the critical depth averaged velocity of the approach flow (U c ). The functional relationship describes scour depth is given by (3.3). In this study, the combinations of parameters chosen to represent scour depth and provide the input variables for the ANN model are shown in equation (4.2). The other parameters shown in equation (3.3) could also be included however, based on the database of experimental results, the selected parameters were chosen to ensure that all the available data was used and similar for each experiment. The output pattern was the equilibrium scour depth (d se ) recorded in the database shown in (Figure 4.2). ( d D, U, y ) d =, (4.2) se f 50, U c Figure 4.2 Input and output parameters used in ANN This study used a database consisting of eleven sets of data (See Appendix A): Observations reported by Kothyari (1972), Melville (1997), Chabert and Engeldinger (1956), Chiew (1984), Ettema (1980), Graf (1995), Hancu (1971), Jain and Fischer (1980) and 21

23 Oliveto amd Hanger (1999) have been used for training and testing. The whole data set consisting of 262 data points was divided into two parts; a training set consisting of 145 data points and a testing or validation set consisting of 117 data points. All data values were normalised to fall in the range 0-1. This normalization is not essential to the neural network approach, but allows the network to be trained more effectively. 4.3 SANN Model for Scour around a Pier In this study a single type of Bayesian model was developed which was based on a single hidden-layer neural network model with a back propagation learning algorithm. The program is modified from SANN model within MATLAB environment proposed by Jeng et al. (2004). An example of the training and testing programs implemented in this thesis is shown below: % Training clc; clear; close all; Table 4.1: An example of training program fid1=fopen('tryagain4j.txt','w'); net=newff([0 1; 0 1; 0 1; 0 1; 0 1;],[2,1],{'logsig','logsig'},'trainrp'); net=init(net); in='newin2.tra'; p=load(in); p=transpose(p); tr='newout2.tra'; t=load(tr); t=transpose(t); net.trainparam.epochs=2000; net.trainparam.show=10; net.trainparam.lr=0.4; net.trainparam.mc=0.3; net.trainparam.goal=1e-3; net=train(net,p,t); result=sim(net,p); fprintf(fid1,'%-5.6f\n',result); fclose('all'); save tryagain4j.mat net; 22

24 % Testing clc; clear; close all; fid1=fopen('test24j.txt','w'); in='testingdata2in.tes'; p=load(in); p=transpose(p); tr='testingdata2out.tes'; t=load(tr); t=transpose(t); disp('test1'); load tryagain4j.mat; out1=sim(net, p); fprintf(fid1,'%-5.6f\n',out1); fclose('all'); Table 4.2: An example of testing program Identifying the best network configuration, and, determining the most favourable value for neurons, epochs, learning rate (lr) and momentum factor (mc) was found by minimizing the difference among the neural network predicted values and the desired outputs. This was done based on a trial-and-error method with the performance of each case calculated based on the root mean square error (RMS) and correlation coefficient (CC) as follows; RMSE = 1 n n k = 1 ( x k y k ) 2 (4.3) n ( y y )( x x ) k k k k CC = k = 1 n n 2 2 ( yk yk ) ( xk xk ) (4.4) k = 1 k = 1 where x and y are target and network output for the k-th output and n is the total number of events considered. The optimal configuration, based upon minimizing the difference among the neural network predicted values and the desired outputs, corresponds to the minimum value of RMS and the optimum value of CC. In total, 187 cases were tested (See Appendix B) with each case configuration as shown in Table 4.3. From the 187 cases, four were 23

25 selected based on having the optimum combination of values of the two statistical parameters just discussed (See Appendix C). Table 4.3 Case architecture Case 1 Altering Neurons EPOCHS NEURONS LEARNING RATE MOMENTUM FUNCTION 1a b 3 1c 4 1d 5 1e 6 1f 7 1g - 1n to o - 1v to w - 1ad to ae - 1al to am -1at to au - 1bb to bc - 1bj to bk - 1br to Case 2 Altering Momentum Function 2a b 0.2 2c 0.3 2d 0.4 2e 0.5 2f 0.6 2g 0.7 2h - 2n o - 2u v - 2ab ac - 2ai aj - 2ap aq - 2aw ax - 2bd be - 2bk Case 3 Altering Learning Rate 3a b 0.4 3c 0.5 3d 0.6 3e 0.7 3f 0.8 3g -3l m - 3r s - 3x y - 3ad ae - 3aj ak - 3ap aq - 3av aw - 3bb

26 4.4 Results and discussions The training and testing results obtained will now be used to form an ANN model that can be implemented to estimate local scour depth for a variety of pier situations. The training results will be presented and the four best-case architectures discussed and plotted. The testing results are made known and to provide further validation of the ANN s accuracy comparisons made with currently used methods of predicting local scour depth Training procedure As previously stated, in developing the most accurate training model architecture, the individual cases were first ranked according to the magnitude of RMS and CC, the best individual model having the minimum RMS and the maximum CC. From the 187 cases a shortlist of four cases was created based on these criteria (Table 4.4). Table 4.4 Case architecture shortlist neurons Epochs Learning Momentum RMS CC rate function Case Case Case Case The four cases had very small RMS during training, ranging from to and consistently good correlation ranging from to Case 2, that included one hidden layer with 4 neurons within that layer, epochs equal to 9000, learning rate of 0.4 and a momentum function of 0.2 was selected as the optimum model with the RMS value of and the CC value of In selecting the optimum case architecture it should be noted that the reliability of the forecasted values do not only depend on the ANN structure, which needs to be carefully chosen through the 25

27 training validation process, but also on the input data. For the results to be reliable the input data needs to be trustworthy. As such the input data used for training the ANN model in this study has been obtained from observations done through controlled experimentation. It is therefore most reliable. Selecting the two optimum models, Case 2 and Case 4, the ANN results were plotted along with the known scour depths. Referring to Figure 4.3 and 4.4, the measured scour depth and that predicted by the trained network, showed reasonably good agreement with most within a range of ±15%. Figure 4.3 Case 2 comparison of ANN results and the experimental results used in training 26

28 Figure 4.4 Case 4 comparison of ANN results and experimental results used in training Test of SANN Model In the first steps of the SANN development process, the data is divided into training and testing categories. With the training performed and four models with the optimum architecture selected, it is necessary to test the networks. The testing phase examines the performance of the four models using the derived weights and measures the ability to classify the data correctly. The results of the testing, as tabulated in Table 4.5, shows that the neural network model is capable of predicting scour depth to a level of accuracy not previously met with the experimental approaches. During training the RMS value ranged from to and the CC value from to To further evaluate the accuracy of the neural network models in predicting equilibrium scour depth, a comparison has been made between the testing results and results based on six of the existing formula previously discussed in Chapter 2 using the data set implemented in training (Table 4.5). As shown in Table 4.5, it can be seen that all four models provide improved prediction of scour depth. For the best 27

29 existing equation CC = compared to Case 3 where CC = Corresponding values of RMS are and respectively. This clearly concludes that the proposed ANN model provides a better prediction of the scour depth around a pier. Table 4.5 Comparison of Neural Network with existing formulae RMS CC SANN Case SANN Case SANN Case SANN Case Laursen & Toch Shen Hancu Breusers U.S.DOT Melville & Chiew

30 4.5 Summary This chapter has outlined the basic principles of Single Artificial Neural Networks, the scour depth parameters and database used for this study and the development of the ANN training and testing models designed for this thesis. The training and testing results obtained from these models has been analysed and an accurate model that can be implemented to predict local scour depth around a bridge pier in a river environment has been produced. These results contribute to the development of understanding local scour and provide engineers with a way of determining scour depth for a variety of pier situations. 29

31 5. Multi Artificial Neural Network (MANN) Multi layer perception network (MLP/BP) A typical configuration for a multilayer perceptron, a special class of artificial neural network that will be used in this research, is shown in Figure 5.1. It resembles a model, where a set of data ( x,...) is first fed directly in the network through the input layer, and subsequently,, x 1 2 the multi layer perceptron produces an expected result y in the output layer. The number of hidden layers establishes the complexity of the network, because a greater number of hidden layers increases the number of connections in the ANN. The issue of determining the correct number of hidden layers required to solve a specific task remains an open problem. The number of nodes in each layer is evaluated by trial and error. Figure 5.1 Structure of typical MLP model. In summary, each node multiplies every input by its interconnection weight, sums the product, and then passes the sum through a transfer function to produce its result. This transfer function is usually a steadily increasing S-shape curve called a sigmoid function. Under this threshold function, the output y j from the j-th neuron in a layer is 1 y j = f ( wij xi ) = (5.1) + ( w ) 1 ij x e i 2 This section is part of the manuscript: Bateni, S. M, Borghei, S. M. And Jeng,D-S. (200X): Neural Network assessments for scour depth around bridge piers. Engineering Applications of Artificial Intelligence (submitted). 30

32 where wij = weight of the connection joining the j-th neuron in a layer with the i-th neuron in the previous layer and x i = values of the i-th neuron in the previous layer. The ANNs are trained with a training set of input and known output data. Many learning examples are repeatedly presented to the network, and the process is terminated when either this difference is less than a specified value or the number of training epochs excesses the specified epoch number. At this stage ANN is considered as trained. The back propagation algorithm based upon the generalized delta rule proposed by Rumelhart et al. (1986) was used to train the ANN in this study. 5.2 Radial basis network (RBF/OLS) The RBF network is similar in topology to the MLP network (Fernando and Jayawardena 1998). Figure 5.2 shows a schematic diagram of a general RBF network with N, L, and M nodes in the input, hidden and output layers, respectively. It shows the N -dimensional input patterns [X] is being mapped to M-dimensional outputs [Z], with nodes in the adjacent layers exhaustively connected. The nodes in the hidden layer are each specified by a transfer function f, which transforms the incoming signals. For the j-th input pattern response of the j-th hidden node y j is of the form J X p, the y j p X U j = f { } (5.2) 2 2σ j where =Euclidian norm; U j = center of the j-th radial basis function f ; and σ = spread of the RBF that is indicative of the radial basis from the RBF center within which the function value is significantly different from zero. The network output is given by a linear weighted summation of the hidden node responses at each node in the output layer. The output for k-th node on the output layer z pk is computed as z pk = L j= 1 y j w kj (5.3) 31

33 where w kj = weight connection between hidden and output nodes. From several possible radial basis functions, the most common choice is the Gaussian. The Gaussian RBF center of the j-th hidden node can be specified by the mean U and the deviationσ. j j f U1, 1 W11 1 Zp Xp 1 Xp J f UJ, J W KJ Zp K Xp N W L1 Input Layer f UL, f : Transfer function U, RBF parameters WkJ: Weight of output layer connection p referes to the pth pattern (p =1,2...,N) where N is the number of patterns in the training set L Hidden Layer W ML Output Layer Zp M Figure 5.2 Radial basis function. Training an RBF involves two stages: (1) determining the basis functions on the hidden layer nodes and (2) the output layer weights. Fitting the RBF function involves finding suitable RBF centers and spreads. A variety of techniques have been evolved to optimize the number of RBF centers. The present study employed the minimum description length algorithm (Leonardis and Bischof 1998) to optimize the parameters of the RBF networks. 5.3 MANN model for scour around piers Development of ANN Models Two MANN models namely MLP/BP and RBF/OLS were developed using the same input variables. The current study used thirteen sets of data to predict equilibrium scour depth: Chabert and Engeldinger (1956), Hancu (1971), Ettema (1980), Jain and Fischer (1980), Chee (1982), Chiew (1984), Kothyari et al. (1992), Yanmaz and Altinbilek (1991), Graf (1995), Melville (1997), Melville and Chiew (1999), Oliveto and Hager (2002) and unpublished data from the University of Auckland. The whole data set consisting of 263 data 32

34 points which was divided into two parts randomly a training or calibration set consisting 180 data points and a validation or testing set consisting of 83 data points. The data reported by Melville and Chiew (1999), Kothyari et al. (1992) and Oliveto and Hager (2002) were used to predict scour depth at a particular time t. The whole data set consisting of 1700 data points was divided into two parts randomly; a training set consisting of 1138 data points, and a validation or testing set consisting of 562 data points. The ranges of different parameters involved in this study are given in Table 5.1. Table 5.1a. Range of different input - output parameters used for the estimation of equilibrium scour depth Parameters Range Flow depth (Y ) (m) Flow mean velocity (U ) (m/s) Grain mean diameter (mm) Critical flow velocity (m/s) Pier diameter ( D ) (m) Equilibrium scour depth (m) ( d ) Table 5.1b. Range of different input - output parameters used for the estimation of timedependent scour depth Parameters Range Scour time (t ) (min) Equilibrium scour time (min) Flow mean velocity (U ) (m/s) Critical flow velocity ( U c ) (m/s) Equilibrium scour depth (m) Scour depth ( d s ) (m) The performance of all ANN configurations was assessed based on calculating the mean absolute error (MAE), and the root mean square error (RMSE). The coefficient of 2 determination, R, of linear regression line between the predicted values from the neural network model and the desired output was also used as a measure of performance. The three statistical parameters used to compare the performance of the various ANN configurations are: 33

35 MAE = 1 N N i= 1 O i t i, (5.4) RMSE = N i= 1 ( O t ) i N i 2, (5.5) R 2 N 2 ( Oi ti ) 1, (5.6) 2 ( O O ) i= 1 = N i= 1 i i where O i and t i are target and network output for the ith output, and target outputs, and N is the total number of events considered. O i is the average of The ANN configuration that minimized the two error measures described in the previous section (and optimum R 2 ) was selected as the optimum. The whole analysis was repeated several times. In this study, two types of MLP/BP models were developed- (1) single hidden-layer ANN models consisting of only one hidden layer; and (2) multiple hidden-layer ANN models consisting of two hidden layers. The task of identifying the number of neurons in the input and output layers is normally simple, as it is dictated by the input and output variables considered to model the physical process. But as mentioned, the number of neurons in the hidden layer(s) can be determined through the use of trial-and-error procedure (Eberhart and Dobbins 1990). The optimal architecture was determined by varying the number of hidden neurons (from 1 to 20), and the best structure was selected. The training of the ANN models was stopped when either the acceptable level of error was achieved or when the number of iterations exceeded a prescribed maximum of The learning rate of 0.05 was also used Data presentation How the data are presented for training is one of the most important aspects of neural network method. Often this can be done in more than one way. The best configuration being determined by trial and error methodology. It can also be beneficial to examine the input/output patterns or data sets that the network finds difficult to learn. Therefore, two combinations of data were considered as inputs. Three of eight parameters namely fluid 34

36 density, fluid dynamic viscosity, and gravitational acceleration are constant in all experiments. Therefore, the first combination involves just five of the eight parameters as the input pattern and the equilibrium scour depth ( d se ) as the output pattern and the second combination includes the five non-dimensional parameters, and normalized equilibrium scour depth ( d se / D ) as the input and output patterns. Both of the mentioned combinations of inputs have been used for the two ANN types. This enables a comparison of the performance of MLP and RBF models for these two combinations of data. Also, two combinations of data were used to predict time-dependent scour depth. The first combination involves five parameters as the input pattern and the time-dependent scour depth ( d s ) as the output pattern and the second combination includes two non-dimensional d parameters and the relative scour depth ( d s se ) as the input and output patterns respectively. 5.4 Results and discussion Equilibrium scour depth prediction using the original data set In this section, first original data is used to establish MANN models. The results of MAE and RMSE of two ANN models are presented in Figure 4. The MLP models had very small RMSE during training [ranging from m to m]. However, the value was slightly higher during validation ( m to m). The models showed consistently good correlation throughout the training and testing (> 0.7 for all models). The MLP configuration that included one hidden layer and 16 neurons within that layer gave the minimum error, and was selected as the optimum model. (Figure 5.3 (a) & (b)). In the RBF model, the center selection process found an appropriate tolerance value of and the radial basis spread of 1. For this value of tolerance to be achieved, 50 significant regressors were required, and they were appropriately and automatically selected by the algorithm in sequence. Thus, the catchment model based on the RBF network was composed of 50 nodes in its hidden layer [Figure 5.3(a) & (b)]. As it is illustrated in Figure 5.3, increase of number of hidden nodes, intensifies significantly the ability of ANN to predict values of interest. It must also be noticed that the reliability of forecasted values does not only depend on the ANN structure (which need to be carefully chosen through the training validation 35