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PAGE 1 1 IDENTIFICATION OF TRUCK AT T RIBUTES USING GENETIC ALGORITHM OPTIMIZATION (GA) By GIRISH VALA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUILDING CONSTRUCTION UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Girish Vala PAGE 3 3 To my parents and family for their love and support, without them I would not have the drive and dedication in my education PAGE 4 4 ACKNOWLEDGMENTS I would like to thank Dr. Ian Flood for being a great mentor and without whose thank Dr Shanker and Dr. Obonyo for providing me with valuable inputs and guidance from time to time Finally, I would like to thank Sushmit, Deepak, Sandeep, Rohin, Vishal and Tapan for helping me in my research. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 Background ................................ ................................ ................................ ............. 12 Objective ................................ ................................ ................................ ................. 13 Methodology ................................ ................................ ................................ ........... 14 Overview ................................ ................................ ................................ ................. 15 2 LITERATURE REVIEW ................................ ................................ .......................... 16 Load Identification Method ................................ ................................ ...................... 16 Least Squares Method ................................ ................................ ............................ 16 Neural Network ................................ ................................ ................................ ....... 20 Genetic Algorithms ................................ ................................ ................................ 21 3 BRIDGE SIMULATION ................................ ................................ ........................... 23 Bridge Model ................................ ................................ ................................ ........... 23 Influence Line of Bridge ................................ ................................ .......................... 23 Identification of Load ................................ ................................ ............................... 24 Truck Model ................................ ................................ ................................ ............ 25 Error Calculation ................................ ................................ ................................ ..... 25 Simulation in MATLAB ................................ ................................ ............................ 25 Measurement Noise ................................ ................................ ................................ 26 Dynamic Axle Weight ................................ ................................ .............................. 27 4 GENETIC ALGORITHM ................................ ................................ .......................... 32 Introduction ................................ ................................ ................................ ............. 32 Terminology ................................ ................................ ................................ ............ 32 Fitness Scaling ................................ ................................ ................................ ....... 34 Rank Fitn ess Scaling ................................ ................................ ........................ 34 Proportionate Fitness Scaling ................................ ................................ ........... 34 Shift Linear Fitness Scaling ................................ ................................ .............. 35 Selection ................................ ................................ ................................ ................. 35 PAGE 6 6 Roulette Wheel Selection ................................ ................................ ................. 35 Stochastic Uniform Selection (SUS) ................................ ................................ 36 Tournament Selection ................................ ................................ ...................... 3 6 Crossover ................................ ................................ ................................ ............... 36 Single Point Crossover ................................ ................................ ..................... 37 Two Point Crossover ................................ ................................ ........................ 37 Heuristic Crossover ................................ ................................ .......................... 37 Arithmetic Crossover ................................ ................................ ........................ 37 Scattered Crossover: ................................ ................................ ........................ 37 Intermediate Crossover ................................ ................................ .................... 37 Mutation ................................ ................................ ................................ .................. 38 Outline of Simple Genetic Algorithm ................................ ................................ ....... 38 Multiple Species ................................ ................................ ................................ ...... 39 Migration ................................ ................................ ................................ ................. 39 Migration Direction ................................ ................................ ........................... 39 Migration Interval ................................ ................................ .............................. 40 Migration Fraction ................................ ................................ ............................. 40 Stopping Criteria ................................ ................................ ................................ ..... 40 5 MEASUREMENT OF GA PARAMETER ................................ ................................ 45 Introduction ................................ ................................ ................................ ............. 45 Selection Criteria ................................ ................................ ................................ .... 45 Effect of Fitness Scaling ................................ ................................ ......................... 46 Effect of Selection ................................ ................................ ................................ ... 46 Effect of Crossover ................................ ................................ ................................ 47 Effect of Migration and Multiple species ................................ ................................ .. 47 6 RESULTS ................................ ................................ ................................ ............... 63 Effect of Number of Measuring Location and Noise Level ................................ ...... 63 Computational Time ................................ ................................ ................................ 64 Effect of Time Varying Load ................................ ................................ .................... 64 7 CONCLUSION AND RECOMMENDA TION ................................ ............................ 89 Conclusion ................................ ................................ ................................ .............. 89 Recommendation ................................ ................................ ................................ .... 90 LIST OF REFERENCES ................................ ................................ ............................... 91 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 93 PAGE 7 7 LIST OF TABLES Table page 5 1 GA Operators ................................ ................................ ................................ ..... 56 5 2 Results of 20 run of Genetic Algorithm ................................ ............................... 57 5 3 Average of Result ................................ ................................ ............................... 58 5 4 Error in id entification ................................ ................................ ........................... 58 5 5 Results meeting selection criteria ................................ ................................ ....... 59 5 6 Comparison of Selection Operator ................................ ................................ ..... 60 5 7 Standard Deviation of Identified load and Spacing ................................ ............. 60 5 8 Error in identification after Migration result. ................................ ........................ 61 5 9 Optimal GA operators ................................ ................................ ......................... 62 6 1 Error in the axle load and spacing for noise free bridge response. ..................... 77 6 2 Error in the axle load and spacing at 5% noise. ................................ .................. 78 6 3 Error in the axle load and spacing at 10% noise ................................ ................. 79 6 4 Error in the axle load and spacing at 15% noise ................................ ................. 80 6 5 Error in the axle load and spacing at 20% noise ................................ ................. 81 6 6 Error in the axle load and spacing at 20% noise ................................ ................. 82 6 7 Effect of Time varying load in axle load and axle spacing for noise free bridge response ................................ ................................ ................................ ............. 83 6 8 Effect of Time varying load in axle load and axle spacing for 5% noise ............. 84 6 9 Effect of Time varying load in axle load and axle spacing for 10% noise ........... 85 6 10 Effect of Time varying loa d in axle load and axle spacing for 15% noise ........... 86 6 11 Effect of Time varying load in axle load and axle spacing for 20% noise ........... 87 6 12 Effect of Time varying load in axle load and axle spacing for 25% noise ........... 88 PAGE 8 8 LIST OF FIGURES Figure page 3 1 Bridge Model ................................ ................................ ................................ ...... 28 3 2 ................................ .. 29 3 3 MATLAB Window ................................ ................................ ............................... 30 3 4 Simulated Bridge Stress ................................ ................................ ..................... 30 3 5 Nois e Polluted Bridge Response (Noise Level = 5%) ................................ ......... 31 4 1 Representation of solution A) Binary Coding B) Real Coding ............................. 41 4 2 Mapping raw fitness to expected number of children. A) Score of Individual, B) Expectation of individuals ................................ ................................ .............. 41 4 3 Roulette wheel selection ................................ ................................ ..................... 41 4 4 Stochastic Universal Sampling ................................ ................................ ........... 42 4 5 Single Point crossover ................................ ................................ ........................ 42 4 6 Two Point crossover ................................ ................................ ........................... 42 4 7 Scattered crossover ................................ ................................ ............................ 42 4 8 Genetic Algorithm Operators ................................ ................................ .............. 43 4 9 Flowchart of Genetic Algorithm ................................ ................................ ........... 44 5 1 GA Toolbox in MATLAB ................................ ................................ ..................... 49 5 2 Effect of Fitness Scaling Operator ................................ ................................ ...... 49 5 3 Comparison of Rate of Convergence for Fitness Scaling operator A) Proportionate scaling B) Rank scaling ................................ ................................ 50 5 4 Rate of convergence for selection A) Roulette selection B) Stochastic Uniform selection C) Tournament selection ................................ ........................ 51 5 5 Effect of Selection Operator ................................ ................................ ............... 52 5 6 Effect of Crossover ................................ ................................ ............................. 53 PAGE 9 9 5 7 Comparison of Rate of Convergence for crossover A) Arithmetic crossover B) Heuristic crossover C) Inte rmediate crossover D) Singlepoint crossover E) Scattered crossover ................................ ................................ ............................ 53 6 1 Error in axle load and spacing for noise free data at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ................................ ...................... 65 6 2 Error in axle load and spacing for 5% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ................................ ................................ ........... 66 6 3 Error in axle load and spacing for 10% noise at vari ous sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ................................ ................................ ....... 67 6 4 Error in axle load and spacing for 15% noise a t various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ................................ ................................ ....... 68 6 5 Error in axle load and spacing for 20% noise at vari ous sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ................................ ................................ ....... 69 6 6 Error in axle load and spacing for 25% noise at various sa mpling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ................................ ................................ ....... 70 6 7 Effect of Time Varying load on axle load and axle spacing for no ise free response at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. ...... 71 6 8 Effect of Time Varying load on a xle load and axle spacing for 5% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. .......................... 72 6 9 Effect of Time Varying load on axle load and axle spacing for 10% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. .......................... 73 6 10 Effect of Time Varying load on axle load and axle spacing for 15% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. .......................... 74 6 11 Effect of Time Varying load on axle load and axle spacing for 20% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C ) 400 Hz. .......................... 75 6 12 Effect of Time Varying load on axle load and axle spacing for 25% noise at various sampling frequency A ) 100 Hz. B) 200 Hz. C) 400 Hz. .......................... 76 PAGE 10 10 Abstract of Thesis P resented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science in Building Construction IDENTIFIC ATION OF TRUCK ATTRIBUTES USING GENETIC ALGORITHMS By Girish Vala December 2010 Chair: Ian Flood Cochair: Esther Obonyo Major: Building Construction The ability to accurately determine the loading attributes of a truck (namely the axle configuration, the spacing between the axles, and the load imposed by e ach axle) while it is in motion is an important function for the design and structural health monitoring of bridges, and highways. Truck weigh in motion (WIM) as it is termed is an inverse problem whe re the load is identified from the observed response of the structure over which it is travelling. The problem has been reasonably well solved using neural network techniques, but there is still significant room for improvement in terms of reducing the num ber of misclassifications of trucks and increasing the precision of the axle spacing and load estimates. The problem can be formulated as an optimization problem. Genetic algorithms (GAs) are proven robust and efficient search optimization techniques. The potential of the GA approach for reverse identification of axle configuration and loading from bridge girder stress envelopes has been investigated and compared to an ex isting neural network solution. The investigation is a pilot study that considers a sim ply supported steel girder bridge with a concrete deck. The bending stresses of the bridge are PAGE 11 11 simulated numerically and are used as the input for reverse modeling. The identification procedure is carried out using GAs by minimizing error between the measu red bridge response and reconstructed bridge response. The performance of the GA depends on the tuning of genetic operators, hence different operator settings are considered and tuned for optimality. Advance strategies such as migration and multiple specie s with real coded representation variable s are adopted to improve the performance. The effect of measurement parameters such as sampling frequency (50 400 Hz), levels of noise (5 25%), time varying load and measuring sections on accuracy of identification are also investigated. The performance of the GA approach is found to outperform the exi sting neural network solution. The significance of this is that, unlike the neural network approach, the GA solution can be applied to any bridge configuration for whic h a reasonable stress model exists. Moreover, the computational time for the GA is found to be on average 3 4 seconds which, although is several orders of magnitude slower than the neural network solution, it is well within what could be considered an acce ptable delay for generating a solution. PAGE 12 12 CHAPTER 1 INTRODUCTION Background Moving load identification is a type of inverse problem where parameter s of model are identified from the observed data. Identification of axle load of truck travelling on the brid ge is a type of an inverse problem. Traditionally, truck weights are measured at weight station which used to penalize the overweight trucks Also, the load history obtained from the WIM station is used for the design of bridges, pavement surface infrastru ctural design and planning However, this system is expensive and causes the traffic to slow down as each truck takes a considerable amount of time for weighing. Hence advanced WIM technology which calculates the weight from the responses of bridge have be en developed and analyzed by many researchers. Yu and Chan (2007) have reviewed and compared the various method of load identification. Tradi ti onal WIM system has developed from the earlier work done by Moses (1978) The method involves the inverse of the s ystem matrix and is solved by least square methods. However, m ethod based o n the inversion of matrix is computationally expensive; hence the pseudo inverse or singular value decomposition method is used to reduce the computational load This method shows h igh fluctuation in the error due to presence of measurement of error and ill posed condition. In order to tackle these regularization techniques with least square error method is used This technique performs better in comparison with the without regulariz ation method. But the problem involved in the se techniques is to find the optimal value of the regularization parameter and it is difficult for real problem to find the optimal parameter. Many methods requires optimization algorithm to update the parameter s to minimize the error. Sequential quadratic PAGE 13 13 programming, dynamic programming generally have been used. Howe ver, the se algorithms are based on finding the zero gradient from the provided auxiliary information. The algorithm requires good formulation of th e equation containing gradient information. It is difficult to form the equation for nonlinear constrained problem. Also t raditional optimization algorithm finds it difficult to escape from the local minima. Hence an algorithm is required to tackle that pr oblem. On the other side, Genetic Algorithm is a type of optimization technique which does not require the gradient or any auxiliary information about the system and it is capable escap ing from the local minima. It can find the global optimal solution of t he problem. Since GA works on many solutions at a time, it converges to the single solution faster. All these quality make GA capable of searching for a global solution.GA has been used successfully to other field. However, very few studies have been imple mented in moving load identification. The present study has considered the advanced Genetic Algorithm strategy to minimize error between measured response and reconstructed response from the approximated load and spacing. The proposed study has effectivel y designed the architecture of GA by investigating different algorithm strategy and analyzing the error in results. Performance of GA in identification is evaluated. Effect of measurement error, number of location of measurement, dynamic effect of the truc k and sampling frequency of measured data on performance of the GA have been investigated. Objective The primary objective of the study is to measure the accuracy of GA in identifying the axle load and spacing for different level of noise, sampling freq uency of recorded data, time varying load and number of measuring sections along the beam where the response of bridge is recorded. Other objectives of the study are described below: PAGE 14 14 To model the bridge and truck for simulating the behavior of bridge durin g travelling of truck with constant magnitude of axle load. To model the measurement noise present in the real bridge response. To model the axle load considering the dynamic nature of the truck. To select the optimal genetic operators for fine tuning of GA Methodology In this study, the bridge has been modeled as a simply supported steel girder with concrete on top of it. The t ruck has been modeled as a set of three consta nt moving load s. One front and two rear axle load with certain spacing. Bridge r esponse is simulated in MATLAB, using the superposition of static influence line of bending moment at measuring sections along the beam. Bridge response considered in the study is bending stress produced at the measuring section along the beam. Measured br idge responses for known axle loads are used as the input to the system. White noise is added to the bridge response to create an artificially contaminated data. Error between measured and simulated stress data is minimized using GA. Different genetic oper ators such as selection, crossover, mutation, migration and multiple species are considered. Effect of each operator on accuracy of identification is evaluated and optimal GA operators are chosen for further investigation. Nine measuring sections each with an equal distance of L/7 from the midpoint are considered, where L is length of the bridge. Noise free response data is added with certain percentage of white noise level of 5%, 10%, 15%, 20%, and 25%. Sampling frequency ranging from 50 Hz to 400 Hz is co nsidered. Effect of noise level, sampling frequency and number of measuring section on identification are investigated. Time varying axle load is considered to investigate the dynamic effect of axle load. PAGE 15 15 Overview The entire study is divided in six chapter s. The first chapter gives the intro duction objective and methodology of propose d topic. The second chapter review s the development of b ridge weigh in motion. It also describe s the various method s used for load identification from bridge response data. Ch apter three describe s the bridge model developed for simulating the bridge response. Chapter four gives an introduction to GA, working of GA and different operator s considered to tune the GA. Chapter five d escribe s the d ata analysis of simulation trial car ried out to tune the GA. It also describes the statistical step s taken to determine the optimal settings for the GA. Chapter s ix analyze s the results of the effect of measurement parameter s like sampling frequency, number of measuring po int and noise on lo ad identification. C hapter seven form s the conclusion based on the analyses of result s from the chapter six. PAGE 16 16 CHAPTER 2 LITERATURE REVIEW Load Identification Method WIM is an inverse problem as the Axle load is to be determined using the measured response of the bridge during the travelling of the truck. Various load identification methods are proposed by many researchers. Comprehensive review s of load identification methods such as interpretive method, Time domain method, frequency domain method and compa rison analysis among them have been investigated by Yu and Chan (2007). The authors discussed the effect of vehicle speed and Axle Spacing to Span Ratio (ASSR) on identification methods. The findings from this study showed that TDM does not have any effect on load identification due to vehicle speed and ASSR. Effect of sampling frequency is least for TDM. Hence the proposed study is conducted under time domain method. Least Square s Method Least square method is traditionally employed in all load identificat ion method. I nterpretive method converts the bridge response such as vertical displacement and bending moment into modal values and then these values are differentiated to modal velocity and modal acceleration. The problem is broken down to a set of linear algebraic equation relating load and modal values. Time domain method formulate the deflection response to integral equation and considering discrete form of problem having step function in small time interval. It is broken down into linear algebraic equa tion. Frequency domain method convert time history of bridge response into frequency response using Fourier transform and set of linear equation in terms of frequency and force is formed. PAGE 17 17 Each method breaks down the problem into a set of linear algebraic e quation of the form below and the solution is obtained by least squares method. (2.1) Where is coefficient matrix of bridge and truck system which is known for the problem is unknown force vector and is time series vector of measured bridge response. Solution to the equation 2.1 can be writt en in following form as (2.2) Where is inverse of matrix Earlier work on WIM was developed by Moses (1979) The author use d the static bridge response using influence line of bending moment. The system predict ed the individual axle load and gross weight of a truck in motion. A bridge was instrumented with strain gauges to record the strain produced during the truck motion. However inversion of large matrices requires long computational time. To overcome this problem Pseudo Inverse (PI) technique has been adopted in many studies. In PI technique the equation 2.2 can be fo rmulated as (2.3) Where is called a s pseudo inverse of matrix A. Equation can be solved by least square methods. PI t echnique calculates the inverse matrix having some property of inverse of Matrix but not necessarily all of them. Popular method to find the pseudo inverse is Singular Value Decomposition (SVD) method SVD is carried out when the matrix A is rank deficient matrix and Exact PI solution is not available because it requires full rank matrix. Usually, SVD process is carried out in two step s At the first step, matrix A is reduced to upper bi diagonal matrix using household information. At PAGE 18 18 second step, matrix A i s reduced to diagonal matrix using QR algorithm. QR algorithm is based on calculation of Eigen value and Eigen vectors of matrix. However, PI solution and SVD based solution sho w s large amount of fluctuation in calculating the error and also the methods a re based on the inversion of the matrices, therefore identification takes longer computational time ( Pinkaew 2006) The method is sensitive to measurement error due to ill condition inverse problem. In order to overcome this problem least square error with smoothening term called regularization is adopted. (Law et al 1997 and 1999) The problem can be formulated as. (2.4) Where = e rror weighting matrix usually taken as inverse of covariance of the measured data. = smoothing matrix, which is generally taken as identity matrix. = residual error The Langrangian expression of the reg ularization problem is obtained as (2.4) Where = regularization parameter and load vector x can be obtained from the eq uatio n 2.4 as (2.5) Matrix can be divided into smaller matrix by breaking down the problem into subproblem which will reduce the computational time (Zhu and Law 2002) In order to produce the accurate result using regularization, it requires the optimal value of regularization parameter which is approximated by the generalized cross validation, L curve method. Pinkaew (2006) proposed the updated static component (USC) PAGE 19 19 technique with regularization technique to improve the result of the regularization method. U SC technique extract the static component f rom the identified axle load leaving dynamic component s and using the iterative technique the dy n amic c omponent is updated. The iterative technique used in the study is sequential quadratic programming (SQP) (Leming and Stalford, 2003) SQP is used to minimize the error between measured response and reconstructed response considering the nonlinear constrain ed equations as ( 2.6 ) SQP assume the objective function to be quadratic in nature while constrains are in the form of a linear equation. It solves the series of quadratic equation using the Boyden, Fletcher, Goldfarb and Shanno (BFGS) formula to update the Hessian of Lagrangian, (2.7) Where = Lagrange multiplier, = constraint of variable including lower and upper bound. studied by La w and Fang (2001). This technique formulates the recursive function to determine the optimal solution. Kalman Filter method is also used for overcoming the noise and measurement error present in the data (Koh and Perry, 2010 & Gagarin, 1991). These techniq ues calculate the original value of the measured data filtering the noise and error in the data and the associated calculated value. It measures the amount PAGE 20 20 of uncertainty associated with predicted value and perform the weighted average of the predicted and measured value. Neural Network Neural netwo r k approach is effectively implemented in load identification due to its robust nature to tackle the measurement noises (Gagarin and Flood, 1994) The method s by combining the layer o f neurons using weighted links. The network can be trained which means it can adjust to the weight on the link to match the input/output pattern. The neural network gets trained using Generalized Delta Rule (GDR). The study by Gag arin showed that for finding each axle weight using GDR is tedious and slow. In order to overcome inefficiency of GDR, Gagarin and Flood (1994) further proposed the artificial neural network using the radial Gaussian incremental learning approach to estima te truck load. Along with axle weight detection, the proposed network also classifies the type of truck from the strain data of the bridge. Yan et. al. (2007) has proposed the method of neural network with the use of Principle Component Analysis (PCA) tech nique for vehicle classification. PCA extracts the feature from time series data hence it represents the data of a reduced length which is the input of Neural Network system. The main drawback of the approach is that it requires the large amount of data to train the network and lack of some pattern in the data leads to the incorrect identification. Each method of load identification performs the optimization routine for minimizing the least square error between observed bridge response and simulated bridge response by updating the truck parameter. Updating of parameter require s a good optimization algorithm that search es for a better solution. However traditional method employed in the algorithms are based on the gradient search for which exact PAGE 21 21 formulation o f the differential equation is to be provided which is difficult for nonlinear system and constrained variable The nature of solution in load identification is such that it has many local optimum points which are difficult to escape for a local optimizati on technique which has been used traditionally (Hashemi and Kargarnovin, 2007) In order to overcome this problem use of global optimization techniques provide a better solution. Genetic Algorithm is one of the global optimization techniques which are inv estigated in the study. Genetic Algorithms Genetic algorithm (GA) is a stochastic optimization technique. It is inspired by the has used the genetic algorithm for optim ization minimizing the error between measured acceleration and reconstructed acceleration. The author used the Simple Genetic algorithm techniques such as fitness scaling, elitism, etc. Also, to reduce the search space, multistage optimization has been pro posed by the author. Simple GA has premature convergence problem. Variation in GA operators like crossover, selection, mutation, change in population size, mutation rate, and crossover rate can prevent the premature con vergence. In past few years much adv ancement in the architecture of GA has been achieved to use it in nonlinear constrained problem, advanced strategy such as different coding scheme, advance mutation, selection parameter has been proposed in order to increase the performance and overcome th e probl ems of simple Genetic Algorithm (Monti. et. al, 2009). However these advanced strategies has not been implemented in load identification method. The proposed study has used the advanced techni ques for load identification. Also, the performance of th e PAGE 22 22 algorithm in identifying accurate results considering measurement noise and dynamic nature of truck is investigated. PAGE 23 23 CHAPTER 3 BRIDGE SIMULATION Bridge Model A Bridge can be modeled as beam element model. Where, beam is an assembly of lumped mass inter connected by massless elastic beam elements. Other type of the bridge is modeled as continues beam model For current study, the bridge is modeled as Timoshenko beam with constant cross section and constant mass per unit length. Damping and inertial effect of the bridge is neglected for the current study. Bridge considered under the study is formulated as simply supported single span steel girder with concrete deck as shown in Figure 3.1. P roperties of the bridge are described below Span Length= 30 m Concre te deck= 7 inch Girder = W33 X 130 = 17268 inch ^4 During the passage of the truck, stresses developed at the bottom of the girder have been simulated. Only static response of the bridge has been considered in the study. The truck is modeled as the constant magnitude of moving axle load. Stress at the bottom of the girder can be calculated from the e quation 3.1 (3.1) W here = Stress in the girder = Bending moment in the girder = Section modulus of girder Influence Line of Bridge Bending moment developed during the p assage of truck can be calculated with influence line. Influence line for bending moment of unit moving load can be calculated by the triangular function as represented by e quation 3.2. PAGE 24 24 (3.2) Where = influence line of bending moment at section at is distance of moving load from left support at time = length of the bridge, is distance of measuring section where Bending Moment is calculated. Considering number of Axle of truck and Axle Spacing Total bending moment as shown in Figure 3.2 is formulated from superposition of e quation 3.2. BM at any point along the length of the beam at time t is given by e quation 3.3 = 1 + 1 2 + 1 2 3 + + 2 + 1 2 . (3.3) Identif ication of Load Load Identification is an inverse problem and requires optimization to minimize the error between the estimated bridge response and measured response. Measured response is assumed to be the numerically simulated response. Error function is defined in e Optimization toolbox is used to perform the search using GA. In GA, error function serves as the Objective Function. (3.4) PAGE 25 25 w here = Sum of Squared Error = Number of Measuring Section = Number of data points in time series = Mea sured Bending Moment = Estimated Bending Moment = Variables (axle load and spacing) Truck Model Configuration of the test truck considered in the study is as follow. Axle 1 (Front Axle) = 15KN Axle 2 (Rear Axle) = 60 KN Axle 3 (Rear Axle) = 60 KN Spacing between Axle1 and 2 = 4 .5 m Spacing between Axle 2 and 3 = 1.22m. Error Calculation In order to check the accuracy of the Algorithm, percentage of error in identification is calculated by the following equation. (3.5) Simulation in MATLAB MATLAB is used for numerical computation. Figure 3.3 is a screenshot of software. Code for simulation of bridge response is written in the code window. File is generat ed with an extension .m which is run to simulat e the response. The response is stored in the variable window. The response is plotted in figure windows as shown in Figure 3.4. It shows the bridge response generated at the midpoint of bridge for test truck at a sampling frequency 40 Hz. The velocity of truck is 20 m/s. Vertical axes are stress in K si and Horizontal axes represents the time in seconds. Bridge response shown above is assumed to be a measured response as truck configuration is known. This respo nse is used as the input to model the identification of axle load and axle spacing. Axle spacing between rear axles is assumed to be known. PAGE 26 26 Hence, the number of parameters to be identified is 4 which includes three axle weight and axle spacing between the front and rear axle. Range of load of each axle and axle spacing is taken from the study by Gagarin (1991) considering the truck is Type 3S. The range of the load and spacing forms the search domain for the GA. Axle 1 = 13.3 to 53.4 kN Axle 2 = 8.8 to 80.1 kN Axle 3 = 8.8 to 80.1 kN Spacing 1 2 = 2.74 to 6.1 m Spacing 2 3 = 1.22 m Range of load and axle space forms the search domain for the GA. GA search for possible combination which gives the minimum cumulative squared error as per e quation 3.4. The work ing of GA is described in next chapter Measurement Noise WIM data is associated with measurement error due to various factors such as acceleration, environment condition like tem perature and water (Prozzi and Hong, 2007). Hence the response recorded by the sensor contains noise which has joint effect of the entire factors. Typically, error has two components as random and systematic in nature. The random error typically occurs due to incapability of device to record the exact response. This error can be termed as Noise. Nature of noise follows the Gaussian distribution which is uncorrelated having zero mean. It can be called as White Gaussian noise. White Gaussian noise has been us ed in various researches. Systematic error occurs mostly due to inadequate calibration (Prozzi and Hong, 2007). Assuming the properly calibrated, smooth road surface, normal environmental condition, only random noise is expected in the data. For current st udy, random noise is PAGE 27 27 considered. I n order to investigate the effect of noise on accuracy of identification, noise is added to the noise free data simulated using static influence line of bridge moment. It is for mulated as White Gaussian noise. Noise vector having zero mean and unit standard deviation and uncorrelated is added to noise free s imulated bridge response vector Noise is added by equation 3.6 Level of noise added to the bridge response data are 5%, 10%, 15%, 20%, 25%. Fig. 3.5 shows the example of noise pol l uted bridge response. (3.6) W here = Noise polluted Stress response of the Bridge = Noise f ree Stress response of the Bridge = Root mean Square Value = Level of Noise = Random noise vector with zero mean and one standard Deviation Dynamic Axle Weight Considering the dynamic effect of truck, load on the axle varies with the time due to the roughness of surface, impact of load on the road and other factors. To investigate the accuracy of the algorithm under dynamic effect, time varying load used in the study by Zhu and Law (2002) is considered. Figure 3.6 shows the bridge response with dynami c effect of axle load. e quation s 3.7 and 3.8 calculate the time varying front load and rear load. (3.7) (3. 8) PAGE 28 28 Figure 3 1. Bridge Model PAGE 29 29 Figure 3 PAGE 30 30 Figure 3 3. MATLAB Window Figure 3 4. Simulated Bridge Stress File Browser Command Window Variable Window Cod e window PAGE 31 31 Figure 3 5. Noise Polluted Bridge Response (Noise Level = 5%) Figure 3 6. Bending Stress due to Time Varying Load PAGE 32 32 CHAPTER 4 GENETIC ALGORITHM Introduction GA theory of natural evolution based on principle of eory for complex optimization problems. GA works on the population of individual s Each individual is a potential solution of the problem. The fitness function which is also known as the Objective Function in traditional optimization algorithm is formulate d. Objective function is defined in equation 3.4 and GA search es for the optimal solution within the constrained domain for which the value of the function is minimum GA is an iterative process where at each iteration each solution is evaluated based on the fitness value and through some stochastic operation; better solution is produced for the next iteration. In every iteration, numbers of solutions are evaluated together and form a population. New population is created using GA operators such as selecti on, crossover and mutation. The process of regeneration continues till GA converges to the best solution. Flowchart of GA considered in the study is described in figure 4.9. Terminology and various operators consi dered are described in this chapter. Termin ology Individual: Individual is a single solution. A size of an Individual is equal to the number of variable to be identified. Fitness : The fitness of Individual is the value of its objective function. In present study it is the error between simulated and measured response. Fitness of individual is used to evaluate the performance in the population. Better the fitness, greater are the PAGE 33 33 chances of survival. In other words better the fitness, the individual is close to optimal solution. Score : It is th e fitness value returned by fitness function. Population : It is a collection of Individuals. Two important aspects of population in GA are the initial population and size of population. Generally, the initial population is randomly generated. The populat ion of larger size contains more diversity and it is easier to explore the search space. But larger the size of population size, larger is the computational cost, time and memory (Sivanandam and Deepa, 2010). Generations : At each iteration, GA performs a series of computation on current population and creates new populations. This new population is called new generation. Parents and Children: In order to create the new generations, GA selects the best fit individuals from current population. These individ uals are called Parents. These parents are used to produce the individuals, who are called children. Survival Rate : Survival rate is the ratio of the expectation of individual to the average expectation of the population. Binary Coding : Each individual is called Chromosome that is encoded in binary string. Figure 4.1 shows an example of binary coding. However, binary coding has many problems. Binary coding finds it difficult to jump between values that are commonly known as hamming cliffs (Sivanandam an d Deepa 2008). Real Coding : In Real Coding every individual is a combination of Real/Integer value as shown in Figure 4.1 These are the values of parameter for identification. Hence it does not require coding and decoding process like binary coding whic h reduces the computational time PAGE 34 34 Fitness Scaling Fitness scaling is performed to prevent premature convergence. It is a process of mapping raw fitness value to the expected number of children for each individual. If the expectation of population has a wid er range than the best individuals will reproduce too quickly and GA will be unable to search for other area of domain space. If the range is not enough, then every individual will reproduce equally which slows down the process. Figure 4.2a shows an exampl e of the scaling where individuals are sorted according to its fitness score and corresponding expectation is shown in Figure 4.2b. The five types of fitness scaling functions being investigated are as follow s Rank Fitness Scaling Rank scaling sorts the i ndividual in the ascending order of the fitness value. Each individual is scaled based on its rank. The best fit individual has a rank 1; next most fit Sum of the sc aled value of entire population is equal to number of parents to be generated in next generation. Proportionate Fitness Scaling In proportionate scaling method an individual is scaled proportionate to its raw score. In order to yield higher expectation for minimum scoring individual, the scores are rotated around the mean. Scores are then normalized in order to get the sum of the scores to one using equation 4.1 (4.1 ) PAGE 35 35 Shift Linear Fitness Scaling This method calculates the expectation using scores, number of parents and a Maximum Survival Rate (MSR). MSR considered in the study is two. The fitness scaling should fulfill two criteria. Average of fitness after scaling s hould be equal to average of fitness before scaling To prevent dominance by fit individuals, the number of the copies assigned to them is to be controlled. Selection It is a process by which the individuals are selected to be the parents for next generatio n. Individuals are picked up depending upon their fitness value. More fit individual s will have more chances to be selected and hence the process takes the search towards the best individuals. Five types of selection scheme are considered in the present st udy Roulette Wheel Selection Roulette Wheel method selection is based on the fitness of Individual s proportional to total fitness of population. The probability of the individual to be selected is given by equation 4.2 Consider 4 individuals selected out of 10. On each trial, a random number is generated between 0 to1. Figure 4.3 shows the selected individual. They are 1, 2, 4 and 9 Probability of Selection of Individual= ( 4.2) W he re = F itness of individual = to t al fitness of population PAGE 36 36 Stochastic Uniform Selection (SUS) SUS is a further development in Roulette Wheel Selection method. It introduc es the zero bias and minimum spread. In SUS, each individual has a size of slot equal to its expectation value exactly as in Roulette wheel. However, wheel is step through the equal size of step, so as to cover the entire wheel in parent steps. Distance be tween two steps is 1/NPointer. NPointer is the number of individuals to be selected. Consider 4 individuals are to be selected out of 10 individuals. Distance between two steps is equal to A random number is generated from [0 to 0.167] = 0.15, where fir st pointer falls. Hence, the individuals selected are 1, 2, 3, 4, 6, and 8 as shown in Figure 4.4. Tournament Selection In order to tune GA, selection pressure and population diversity should be adjusted. Tournament selection pressure provides pressure by holding tournament competition among the desired number of individuals ( Sivanandam and Deepa 2008). This method selects the parents by selecting the best players which is equal to the size of tournament out of expectation and number of parent. The best ind ividual is then selected out of that selection set. Crossover Crossover is the process of combining two parents to produce children. It recombines already existing information. Parents are selected randomly and the information is exchanged at a random posi tion of the value. Six types of crossover operator have been investigated. PAGE 37 37 Single Point Crossover This operator makes a cut at random position of two individuals and information contained after section is exchanged between them as shown in the Figure 4.5. Two Point Crossover In a two point crossover, two points of crossover are selected and information between them is exchanged. More crossover points can search the space thoroughly but can also reduce the performance of the GA Heuristic Crossover This oper ator is used for real value coding of individual. It uses the fitness value of the parents in order to produce offspring. Offspring is created using Equation 4. 3 Children= parent 1 + ratio*(parent 2 parent 1) if score of parent 2 > parent 1 (4. 3 ). Ratio in above equation ranges from 0 to 2. Arithmetic Crossover This operator produces the children which is arithmetic mean of two parents. Equation of Arithmetic Crossover operator is as follow: Children= *parent 1 + (1 ) parent 2. Where = random number Scattered Crossover: This operator randomly selects the information from the parents and creates the offspring. Figure 4.7 shown below shows the process. Intermediate Crossover In termediate crossover works similar to heuristic crossover. e quation 4. 4 describes the process of generating children from two parents. Children= parent 1 + scale (parent 2 parent 1) (4. 4 ) PAGE 38 38 The scale is calculated by multiply ing the ratio and random number generated for each value of the individual. Mutation Mutation is performed after the crossover operation to avoid the fast convergence. While crossover tries to exploit the current solution, mutation performs the exploratio n for a different solution. Mutation is done by changing the value of a variable randomly by certain mutation parameters using equation 4.5 In Equation 4. 5 is a parent and is a mutated child. is a function which uses the different parameters defined in and parent to form the mutated child. Each variable in individual is chosen one by one and a random number is generated for each value. If the value is below the migration rate then mutation does not take place and value remains unchanged and if the number is higher than mutation rate then mutation takes place and it alters the value within search space defined by each variable s domain. For the current study, adaptive mutation provided in MATLAB toolbox is used which is defined for the problems having bounds on variable. (4.5) where is variable of individual i s mutated value of variable. Outline of Simple Genetic Algorithm Algorithm creates the random initial population Algorithm creates the n ew population from the current populations by series of operations as follow; a. Calculate the score of every individual in the population. PAGE 39 39 b. Perform the fitness scaling c. Select the parents based on their fitness value. d. Perform the children by cross over or mutation. e. Creates the next generation by replacing the current population with children The algorithm repeats the above steps till the stopping criteria are met. Multiple Species Simple GA deals with single population. However, population can b e divided in species. Species is a subpopulation that is subset of main population within which the individuals are treated with selection, crossover and mutation operators. In a multiple species strategy, use of distinct choices is possible and desirable. Also, migration operator can be used to facilitate the exchange between the subpopulation (Monti et al., 2010). Migration Migration operator exchanges the information between subpopulation. During migration, the best individual from one subpopulation repl aces the worst individual in another subpopulation. Migration is controlled by Migration Fraction, Migration Interval and Migration Direction explained below. Migration Direction Migration can be in forward or both directions. In forward migration, an Indi vidual from nth subpopulation migrates to nth+1 subpopulation. While in m igratio n for both direction, an individual from nth subpopulation migrates to both (n th +1) and (n th 1) subpopulation. PAGE 40 40 Migration Interval It controls the generation at which migration will occur. For example, if an interval is set to 40, migration occurs at interval of 40 generations. Migration Fraction Migration Fraction controls how many individuals migrate between the subpopulation. For example, size of subpopulation is 20 and if the fraction is set to 0.6, then number of individuals that would migrate from a subpopulation is 0.5 20 =1 Stopping Criteria GA uses the criteria for terminating the process and produces the result. Five stopping criteria have been considered. Generation: The GA will stop if a specified number of generations have been reached Time: The GA will stop when a specified amount of time has elapsed Fitness limit: When the best fitness value in current generation is less than or equal to the specified value Sta ll Generation: If there is no improvement over a specified number of stall generation GA will stop Stall Time limit : If the algorithm will run until the specified amount of time of no improvement in fitness function is observed. PAGE 41 41 A B Figure 4 1 Representation of solution A ) Binary Coding B) Real Coding A B Figure 4 2. Mapping raw fitness to expected number of children. A) Score of Individual, B) Expectation of individuals Figure 4 3 Roulette wheel selection PAGE 42 42 Figure 4 4. Stochastic Universal Sampling Figure 4 5. Single Point crossover Figure 4 6. Two Point crossover Figure 4 7. Scattered crossover PAGE 43 43 Figure 4 8. Genetic Algorithm Operators GA OPERATOR Fitness Scaling Rank Proportionate Shift Linear Selection Roulette Wheel Stochastic Uniform Selection Tournament Crossover Single Point Two Point Heuristic Arithmatic Scattered Intermediate Mutation Adaptive Feasible Migration Migration Direction Migration Fraction Migration Interval Multiple Species PAGE 44 44 Figure 4 9. Flowchart of Genetic Algorithm PAGE 45 45 CHAPTER 5 MEASUREMENT OF GA PA RAMETER Introduc tion Performance of GA depends on the tuning of GA operators. In order to get the best results out of GA, it is a good practice to try different combinations of operators and analyze their impact on results. Table 5.1 below describes the different setting s of GA that have been investigated. GA gives different results every time due to its stochastic nature. Therefore, in order to analyze the result, algorithm is run for 20 times and results are stored. Average, standard deviation and variance are calculate d based on the results. Average of result is taken as the identified value of load and spacing. Error is calculated by equation for every result. Figure 5.1 shows the GA toolbox window used for the optimization where the objective function file is defined in the fitness function option. Options will take into consideration the different GA operators. Result window shows the identified axle loads and spacing returned by GA. Selection Criteria Results of simulation from MATLAB are exported and analyzed in Mic rosoft Excel. Table 5.2 shows the Excel Spreadsheet developed for analysis of result. It shows 20 simulation runs for a particular combination of GA operators. Each row includes the identified value of axle load, axle spacing and GA operator used. Table 5. 3 shows the average results of a few combinations of GA operators. Table 5.4 shows the error associated with identified result mentioned in Table 5.3. In order to reduce the computational load, migration and multiple species are not considered to tune simp le GA operator initially. Migration and multiple species parameter s are evaluated after fixing other GA parameters. To select the optimal settings of GA operator, error in the PAGE 46 46 identified parameter is calculated. Initially, combination for which each identi fied parameter is within 10% error is selected and the rest is neglected. Table 5.5 shows the results which meet the selection criteria of 10% error limit. Few of the results within 10% error are having crossover of fraction 1, which means the search is ca rried out without mutation. As mutation is required to maintain the diversity in population, it is not a good practice to use crossover without mutation. Therefore results with crossover fraction 1 are not taken for analysis. Table 5.5 shows the final sele ction of GA setting. Further sorting is done by finding the cumulative error for sorted selection. Further selection if required is done by comparing the standard deviation since the identified result is the average result. Effect of the GA operator on the identification is described as follow Effect of Fitness Scaling Both, the rank scaling and proportionate scaling performed well compared to the shift linear scaling as shown in Figure 5.2. Rank scaling has a consistent error in all the identified paramet ers and also within the selection criteria. Hence, rank scaling is considered as the optimal scaling operator for the study. Figure 5.3 shows the rate of convergence for the rank proportionate. Figure 5.2 shows that rank scaling has a better convergence ov er the proportionate and also finds the best fitness as compared to proportionate Effect of Selection Stochastic Uniform, Roulette and uniform selection have produced results which meet the selection criteria. Figure 5.5 shows the effect of selection oper ator on identification. In order to select one of them as the optimal operator, minimum cumulative error associated with each operator is considered. Table 5.6 shows the cumulative error. Since the cumulative error does not show much difference, standard PAGE 47 47 d eviation is considered. The lowest standard deviation is associated with Stochastic Uniform selection operator, hence it is considered as the optimal selection operator for the study. Effect of Crossover It can be seen from the Table 5.5 that only crossove r operator which met the selection criteria is Heuristic Crossover with Heuristic Crossover ratio as 1.8. Hence, Heuristic Crossover is chosen as the optimal crossover operator for further analysis. Figure 5.6 shows the different crossover operator conside red and their respective errors. Figure 5.7 shows that rate of convergence for each selection operator. Effect of Migration and Multiple species Migration and multiple species increase the accuracy of the identification. To select the optimal setting of su bpopulation size and migration rate, results are sorted with selection criteria. The numbers of the identified results are large. Hence, further sorting was done with cumulative error. Table 5.8 shows the filtered result. It has listed the number of trials that met the selection criteria. From Table 5.7, it can be seen that optimal setting of migration parameter producing the least cumulative error is for the selection No.11 Also, No. 6 has a good accura cy. Standard deviation associated with both selections is tabulated below. Selection 11 has the least cumulative error. Hence, migration setting and multiple species setting adopted are as follow Subpopulation Size = 20 x 3 Migration Interval = 40 Migratio n Fraction= 0.6 Migration Direction= forward PAGE 48 48 After analysis, the effect of different GA operators, optimal GA setting adopted is described in Table 5. 9 This optimal setting is used for identification and is kept unchanged for all further analysis. The ef fect of number of measuring sections, sampling frequency, time varying load and noise is evaluated in the next chapter PAGE 49 49 Figure 5 1. GA Toolbox in MATLAB Figure 5 2. Effect of Fitness Scaling Operator PAGE 50 50 A B Figure 5 3. Comparison of Rate of Converge nce for Fitness Scaling operator A) Proportionate scaling B) Rank scaling PAGE 51 51 A B Figure 5 4. Rate of convergence for selection A) Roulette selection B) Stochastic Uniform s election C) Tournament selection PAGE 52 52 C Figure 5 4. Continued. Figure 5 5. Effect of Selection Operator 0 5 10 15 20 25 30 35 40 Stochastic Uniform Uniform Roulette Wheel selectiontournament % Error Axle1 Axle2 Axle3 Spacing PAGE 53 53 Figure 5 6. Effect of Crossover A Fig ure 5 7. Comparison of Rate of Convergence for crossover A) Arithmetic crossover B) Heuristic crossover C) Inte rmediate c rossover D) Singlepoint crossover E) Scattered crossover 0 5 10 15 20 25 30 35 40 % Error Axle 1 Axle 2 Axle 3 Spacing PAGE 54 54 B C Figure 5 7. Continued. PAGE 55 55 D E Figure 5 7. Continued. PAGE 56 56 Table 5 1. GA Operators Operator Options Trial 1 Trial 2 Fitness Scaling Proportionate, Rank, Shift Linear Rank Proportionate Selection Roulette, Stochastic Uniform, Tournament, Uniform, Remainder Uniform Remainder Crossover Scattered Single Point, Two Point, Intermediate, Heuristic (ratio: 0 to 2), A rithmetic Scattered Heuristic( 0.8) Crossover Fraction 0 to 1 0.4 0.8 Number of Subpopulation 3 3 3 Subpopulation Size 5 to 20 5 : 5 : 10 10:10: 15 Migration Direction Forward, Both Forward Both Migration Interval 5 to 60 40 45 PAGE 57 57 Table 5 2. Result s of 20 run of Genetic Algorithm Identified parameter GA operator Run Axle1 Axle2 Axle3 Spacing Crossover Fraction Heuristic Ratio Crossover Fitness Scaling S election 1 17.58 74.38 25.91 4.86 0.8 1.6 Heuristic Proportionate Stochastic 2 17.58 74.39 25.91 4.76 0.8 1.6 Heuristic Proportionate Stochastic 3 17.56 74.37 25.94 4.67 0.8 1.6 Heuristic Proportionate Stochastic 4 15.04 49.98 49.98 4.19 0.8 1.6 Heuristic Proportionate Stochastic 5 14.99 50.06 49.94 4.03 0.8 1.6 Heuristic Proportionate Stochastic 6 17.26 76.33 24.18 4.74 0.8 1.6 Heuristic Proportionate Stochastic 7 17.67 73.34 26.95 4.82 0.8 1.6 Heuristic Proportionate Stochastic 8 14.99 50.10 49.91 4.12 0.8 1.6 Heuristic Proportionate Stochastic 9 14.99 49.98 50.03 4.21 0.8 1.6 Heuristic Proportionate Stochastic 10 15.06 49.47 50.53 4.24 0.8 1.6 Heuristic Pr oportionate Stochastic 11 17.58 74.37 25.92 4.74 0.8 1.6 Heuristic Proportionate Stochastic 12 17.58 74.48 25.82 4.75 0.8 1.6 Heuristic Proportionate Stochastic 13 17.58 74.38 25.91 4.82 0.8 1.6 Heuristic Proportionate Stochastic 14 15.01 49.98 50. 02 4.07 0.8 1.6 Heuristic Proportionate Stochastic 15 17.60 74.20 26.05 4.66 0.8 1.6 Heuristic Proportionate Stochastic 16 17.57 74.40 25.90 4.56 0.8 1.6 Heuristic Proportionate Stochastic 17 14.98 50.15 49.85 4.13 0.8 1.6 Heuristic Proportionate Sto chastic 18 17.56 74.41 25.90 4.69 0.8 1.6 Heuristic Proportionate Stochastic 19 17.64 74.40 25.82 4.69 0.8 1.6 Heuristic Proportionate Stochastic 20 15.01 50.00 50.02 4.20 0.8 1.6 Heuristic Proportionate Stochastic PAGE 58 58 Table 5 3. Average of Res ult Average of results GA operator Trial Axle1 Axle2 Axle3 Spacing Crossover fraction Heuristic ratio Crossover Fitness scaling S election 1 22.15 61.88 36.23 4.81 0.2 0 Scattered Proportionate Stochastic 2 21.76 49.70 48.16 4.67 0.2 0 Ari thmetic Proportionate Stochastic 3 21.74 53.15 45.05 4.66 0.4 0 Intermediate Proportionate Stochastic 4 20.41 59.37 40.04 4.85 0.4 0 Intermediate Rank Roulette 5 16.99 61.36 38.36 4.49 0.4 0.8 Heuristic Rank Stochastic 6 17.50 52.12 46.57 4.22 1 0.8 He uristic Rank Roulette 7 16.10 59.73 39.95 4.26 0.6 0.6 Heuristic Rank Uniform 8 18.77 64.82 34.67 4.74 0.6 0 Two point Rank Stochastic 9 18.21 51.41 47.44 4.44 0.8 0 Two point Rank Roulette Table 5 4. Error in identification Error ( %) GA operato r Trial Axle1 Axle2 Axle3 Spacing Crossover f raction Heuristic r atio Crossover Fitness scaling S election 1 47.64 23.77 27.55 20.16 0.2 0 Scattered Proportionate Stochastic 2 45.04 0.60 3.67 16.80 0.2 0 Arithmetic Proportionate Stochastic 3 44.94 6.31 9.90 16.41 0.4 0 Intermediate Proportionate Stochastic 4 36.05 18.73 19.93 21.22 0.4 0 Intermediate Rank Roulette 5 13.27 22.72 23.27 12.26 0.4 0.8 Heuristic Rank Stochastic 6 16.66 4.24 6.86 5.57 1 0.8 Heuristic Rank Roulette 7 7.35 19. 46 20.11 6.57 0.6 0.6 Heuristic Rank Uniform 8 25.15 29.64 30.66 18.57 0.6 0 Two point Rank Stochastic 9 21.42 2.82 5.11 11.02 0.8 0 Two point Rank Roulette PAGE 59 59 Table 5 5. Results meeting selection criteria Error below (10%) GA operator Select ion Axle1 Axle2 Axle3 Spacing Crossover Fraction Heuristi c r atio Crossover Fitness Scaling S election 1 6.67 4.92 5.36 6.58 0.4 1.8 Heuristic Rank Stochastic u niform 2 9.23 8.60 9.21 7.56 0.4 1.8 Heuristic Rank Stochastic u niform 3 9.2 5 6.22 6.97 7.36 0.6 2 Heuristic Proportionate Stochastic u niform 4 6.19 9.39 9.59 7.92 0.8 2 Heuristic Rank Roulette 5 1.93 8.63 8.38 5.76 0.8 1.8 Heuristic Proportionate Stochastic u niform 6 6.68 8.93 9.22 7.67 0.8 2 Heuristic Rank Stochastic u niform 7 7.63 4.10 4.52 6.95 0.8 1.8 Heuristic Proportionate Roulette 8 8.29 7.85 8.57 6.39 0.8 1.6 Heuristic Proportionate Stochastic u niform 9 8.71 8.77 9.48 8.25 0.8 1.6 Heuristic Proportionate Uniform 10 2.92 9.76 9.90 4.95 0.8 1.8 Heurist ic Proportionate Roulette 11 5.16 8.16 8.42 6.84 1 1.6 Heuristic Proportionate Uniform 12 7.45 5.45 6.61 4.76 1 1.6 Heuristic Rank Uniform 13 5.76 6.07 6.29 6.19 1 1.6 Heuristic Proportionate Uniform 14 7.65 8.61 9.11 7.13 1 1.8 Heuristic Proportionate Roulette 15 9.56 4.08 5.07 5.63 1 2 Heuristic Proportionate Uniform 16 6.54 8.76 9.38 6.73 1 2 Heuristic Proportionate Stochastic u niform 17 9.34 8.57 9.42 6.75 1 1.8 Heuristic Proportionate Uniform 18 8.96 5.35 6.66 4.20 1 2 Heuristic Proportion ate Uniform 19 4.12 9.54 9.62 6.60 1 2 Heuristic Proportionate Stochastic u niform PAGE 60 60 Table 5 6. Compariso n of Selection Operator Axle 1 A x le 2 Axle 3 Spacing Cumulative Stochastic Uniform Error (%) 1.93 8.63 8.38 5.76 24.70 deviation 1.05 8.80 8.70 0.22 18.77 Roulette Error (%) 7.63 4.10 4.52 6.95 23.20 deviation 2.07 10.9 10.8 0.24 24.08 Uni form Error (%) 7.45 5.4 3 6.6 0 4.75 24.26 deviation 1.34 14.4 13.9 0.28 29.92 Table 5 7. Standard Deviation of Identified load and Spacing Stand a rd Deviation sub population size Migration Selection Axle 1 Axle 2 Axle 3 Spacing Cum. error 1 2 3 Interval Fraction Direction 6 0.17 0.6 0.5 0.06 1.33 10 15 20 35 0.4 Both 11 0.08 0.36 0.3 0.06 0.8 20 20 20 40 0.6 Forward PAGE 61 61 Table 5 8. Error in identification after Migration result. Error (%) Subpopulation Size Migration Selection Axle1 Axle2 Axle3 Spacing Cum error 1 2 3 Int erval Fraction Direction 1 1.01 2.39 2.38 4.39 10.17 10 20 20 50 0.2 both 2 1.46 1.48 1.53 4.67 9.13 15 15 20 50 0.2 both 3 1.17 1.95 2.00 4.94 10.06 20 20 20 35 0.2 both 4 1.29 1.93 1.99 4.29 9.50 20 20 20 40 0.2 both 5 4.25 1.95 1.38 4.07 11.65 10 1 0 15 50 0.4 both 6 0.30 0.35 0.28 3.99 4.92 10 15 20 35 0.4 both 7 1.06 2.21 2.23 4.08 9.58 20 20 20 30 0.4 both 8 0.11 2.71 2.86 3.00 8.68 20 20 20 50 0.4 both 9 1.23 2.06 2.08 4.01 9.39 15 20 20 25 0.6 forward 10 0.72 2.61 2.56 4.30 10.20 15 20 20 3 5 0.6 forward 11 0.01 0.08 0.09 3.29 3.47 20 20 20 40 0.6 forward 12 2.57 3.64 3.74 4.14 14.09 10 15 15 35 0.8 forward 13 2.00 1.12 1.34 4.70 9.16 10 15 15 45 0.8 forward 14 1.00 2.32 2.32 4.67 10.30 10 15 20 35 0.8 forward 15 1.10 2.14 2.15 4.35 9.74 10 20 20 35 0.8 forward 16 0.92 2.51 2.44 4.31 10.17 15 15 20 35 0.8 forward 17 1.22 2.05 2.06 4.84 10.17 20 20 20 40 0.8 forward 18 0.51 2.82 2.74 3.96 10.02 20 20 20 45 0.8 forward 19 2.09 1.08 1.27 4.61 9.05 5 20 20 45 1 forward 20 7.61 0.01 0.79 5.93 14.33 10 10 10 10 1 forward PAGE 62 62 Table 5 9 Optimal GA operators Operator Type Population Size 20 x 3 Fitness Scaling Rank Selection Stochastic Uniform Cr ossover Crossover Fraction Migration Direction Migration Fraction Migration Interval Heuristic 0.8 Forward 0.6 40 PAGE 63 63 CHAPTER 6 RESULTS Bridge response has been simulated with an increase in the measuring section, increase in sampling frequency and increase in noise level. Minimum Sampling frequency considered is 100 Hz in order to cap ture the minimum 100 data point of the bridge response profile as recommended ( Yu and Chan 2007). The results are tabulated in Table 6.1 and graphically presented in figure 6.1. in Figure 6.1, the spacing refers to the axle spacing between front and rear a xle load. Spacing between rear axle load s is assumed to b e known. From the results, it is found that front axle load is subjected to more fluctuation in error compared to rear axle load. For noise free data all the results are obtained within 3% of error a t all sampling frequency. From the data it can be seen that even single measuring section at midspan is sufficient to produce the accurate result. The range of each measurement is as follow Sampling Frequency = 100 to 400 Hz Number Measuring Section= 1 to 9 Level of Noise = 5 to 25 % Effect of Number of Measuring L ocation and Noise Level Figure 6.1 to 6.3 show the effect of measuring point and noise level from low to high sampling frequency rate. Bridge responses were simulated at 1, 3, 5, 7, 9 points a long the length of the bridge. Points are placed at equal distance of 1/8th of the length of span from the midpoint. Levels of noise added to noise free data are 5%, 10%, 15%, 20%, 25%. For noise free bridge response, number of measuring point does not sho w any significant effect on accuracy. However, as the noise level increases, error in identification increases. Improvement in the result is not consistent as increment of measuring location. However, error is reduced within 10% considering minimum 3 PAGE 64 64 mea suring location. The front axle load is more sensitive to error than the rear axle loads. Error in rear axle load s remained within 5% at low sampling frequency and for higher frequencies; the error is within 1% for all level of noise. Error in front axle l oad is high for polluted noise data at low sampling frequency but keeping sampling frequency high as 400 Hz error is reduced within 5 10%. Axle spacing identification shows randomness in error. However, at higher frequency, the error can be seen within 5% except for certain levels of noise. Computational Time The computational time taken by algorithm for identification is also investigated. Time taken for estimation is listed in table. For the study, personal computer with Intel core i5 2.67 GHz with 4GB R AM is used. CPU time is taken considering the 100 data points of bridge response as the input. Algorithm takes from 3 4 second to compute the parameter. However, the size of population influences the computing time. Increase in the population size increase the fitness evaluation time Effect of Time Varying Load Effect of dynamic nature of axle load on investigation is considered. Fig ure 6 7. to 6 1 2 shows the influence of sampling frequency and measuring point on the accuracy of estimation of axle load an d axle spacing. The result shows that influence of frequency and measuring section on front is significant while rear axle remains within 3% error for all level of noise and measuring section. For low frequency, error in spacing is greater but for higher f requency the error is reduced within 5%. PAGE 65 65 A B C Figure 6 1 Error in a xle load and s pacing for noise free data at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 66 66 A B C Figure 6 2. Error in axle load and spacing for 5% noise at vario us sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 67 67 A B C Figure 6 3. Error in axle load and spacing for 10% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 68 68 A B C Figure 6 4. Error in axle load and spacing for 15% noise a t various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 69 69 A B B Figure 6 5. Error in axle load and spacing for 20% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 70 70 A B C Figure 6 6. Error in axle load and spacing for 25% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 71 71 A B C Figure 6 7. Effect of Time Varying load on axle load and axle spacing for noise free response at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 72 72 A B C Figure 6 8 Effect of Time Varying load on axle load and axle spacing for 5% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 73 73 A B C Figure 6 9 Effect of Time Varying load on axle load and axle spacing for 10 % noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 74 74 A B C Figure 6 10. Effect of Time Varying load on axle load and axle spacing for 15% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 75 75 A B C Figure 6 11. Effect of Time Varying load on axle load and axle spacing for 20% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 76 76 A B C Figure 6 1 2 Effect of Time Varying load on axle load and axle spacing for 25% noise at various sampling frequency A) 100 Hz. B) 200 Hz. C) 400 Hz. PAGE 77 77 Table 6 1. Error in the axle load and spacing for noise free bridge response. Measuring Location s Sampling Frequency Axle1 ( %) Axle2 ( %) Axle3 ( %) Spacing ( %) 1 100 1.31 1.77 1.17 2.49 1 150 0.96 1.20 1.00 3.17 1 200 0.63 0.60 0.70 1.51 1 250 0.52 0.20 0.70 1.62 1 300 0.05 0.03 0.03 0.65 1 350 0.37 0.30 0.50 1.90 1 400 0.09 0.20 0.19 0.54 3 100 0.38 0.25 0.14 0.63 3 150 0.98 0.14 0.19 3.48 3 200 0.01 0.02 0.02 1.08 3 250 0.54 0.20 0.38 2.79 3 300 0.01 0.00 0.00 0.80 3 350 0.37 0.20 0.20 1.24 3 400 0.00 0.00 0.00 0.47 5 100 0.02 0.09 0.07 2.30 5 150 1.01 0.40 0.10 4.58 5 200 0.02 0.02 0.01 0.94 5 250 0.55 0.19 0.20 2.47 5 300 0.01 0.02 0.01 0.46 5 350 0.38 0.10 0.12 1.03 5 400 0.02 0.03 0.02 0.5 2 7 100 0.00 0.00 0.00 2.44 7 150 1.07 0.21 0.23 4.83 7 200 0.03 0.06 0.05 0.78 7 250 0.59 0.20 0.50 2.46 7 300 0.01 0.02 0.01 0.55 7 350 0.40 0.13 0.23 1.79 7 400 0.02 0.05 0.04 0.53 9 100 0.04 0.06 0.03 2.39 9 150 1.10 0.20 0.13 4.01 9 200 0.01 0.02 0.01 0.75 9 250 0.61 0.10 0.20 2.42 9 300 0.00 0.02 0.02 0.74 9 350 0.42 0.10 0.12 0.92 9 400 0.01 0.02 0.02 0.23 PAGE 78 78 Table 6 2 Error in the axle load and spacing at 5% noise. Measuring Location Sampling Frequency Axle1 ( %) Axle2 ( %) Axle3 ( %) Spacing ( %) 1 100 5.93 2.17 3.09 2.48 1 150 1.98 0.24 0.62 0.42 1 200 0.95 0.85 0.70 0.28 1 250 0.19 3.53 3.32 1.59 1 300 1.52 1.77 1.35 0.05 1 350 0.53 1.74 1.50 0.67 1 400 1.40 0.86 0.46 0.37 3 100 2.00 1.91 1.46 0.45 3 150 1.54 1.14 0.86 0.67 3 200 0.71 0.41 0.27 0.04 3 250 1.67 1.16 1.09 0.55 3 300 0.69 1.50 1.30 0.96 3 350 0.71 0.57 0.40 0.36 3 400 0.60 0.79 0.64 0.20 5 100 1.01 0.43 0.33 0.25 5 150 0.24 0.48 0.41 0.25 5 200 0.55 0.57 0.66 0.54 5 250 0.58 0.3 1 0.18 0.47 5 300 1.24 1.70 1.62 0.92 5 350 0.66 0.25 0.13 0.08 5 400 0.36 0.57 0.48 0.08 7 100 0.19 0.15 0.09 0.13 7 150 0.28 0.80 0.80 0.12 7 200 0.28 0.06 0.05 0.19 7 250 1.06 0.85 0.60 0.06 7 300 0.99 0.14 0.38 0.53 7 350 0.12 0.25 0.19 0.01 7 400 0.10 0.24 0.24 0.03 9 100 0.18 0.47 0.37 0.94 9 150 0.35 0.19 0.08 0.71 9 200 0.11 0.30 0.28 0.49 9 250 0.03 0.26 0.28 0.04 9 300 0.09 0.22 0.24 0.28 9 350 0.17 0.18 0.17 0.31 9 400 0.80 0.38 0.20 0.26 PAGE 79 79 Table 6 3 Error in the axle load and spacing at 10% noise Measuring Location s Sampling Frequency Axle1 ( %) Axle2 ( %) Axle3 ( %) Spacing ( %) 1 100 11.51 6.28 6.91 5.13 1 150 7.57 0.84 2.59 1.56 1 200 2.46 7.98 6.65 1.43 1 250 4.49 4.22 4.63 2.18 1 300 1.96 7.42 6.5 2 1.08 1 350 1.62 1.40 1.44 0.93 1 400 7.26 6.68 5.16 0.51 3 100 2.59 5.40 5.17 2.03 3 150 0.95 2.78 3.01 1.55 3 200 2.77 0.81 1.69 1.62 3 250 2.78 0.61 0.95 1.40 3 300 0.53 0.56 0.91 0.26 3 350 2.01 0.82 1.40 0.91 3 400 2.18 0.45 0.21 0.36 5 100 2.13 1.44 0.97 0.60 5 150 0.39 1.35 1.67 0.28 5 200 0.53 0.27 0.49 0.24 5 250 3.59 3.46 4.00 1.58 5 300 3.20 0.94 0.57 1.04 5 350 0.40 0.91 0.84 0.61 5 400 2.08 5.25 4.37 1.45 7 100 7.08 1.68 2.87 1.29 7 150 2.41 1.04 0.60 0.16 7 200 0.74 4.36 3. 94 1.57 7 250 3.00 2.21 1.58 0.31 7 300 2.48 0.69 1.06 0.43 7 350 2.52 1.39 0.83 0.17 7 400 0.03 0.37 0.23 0.28 9 100 3.23 0.86 0.01 0.97 9 150 0.09 0.22 0.12 0.04 9 200 0.50 0.68 0.91 0.04 9 250 0.77 2.96 2.76 0.91 9 300 0.91 1.42 1.14 0.20 9 35 0 0.12 1.06 0.73 0.75 9 400 0.13 0.08 0.08 0.02 PAGE 80 80 Table 6 4. Error in the axle load and spacing at 15% noise Measuring Location Sampling Frequency Axle1 ( %) Axle2 ( %) Axle3 ( %) Spacing ( %) 1 100 25.43 30.67 25.63 1.16 1 150 13 .27 16.43 14.77 1.46 1 200 12.27 1.16 0.55 4.33 1 250 9.47 6.60 8.58 5.65 1 300 12.06 6.33 4.70 2.94 1 350 11.61 11.62 13.16 6.18 1 400 2.74 5.53 5.34 0.78 3 100 5.90 10.62 8.04 2.64 3 150 1.43 4.63 4.11 1.41 3 200 2.27 3.50 2.69 1.67 3 250 5.80 6 .78 5.33 0.61 3 300 3.95 5.93 5.19 0.04 3 350 1.04 1.56 1.70 0.48 3 400 3.99 3.79 2.46 0.56 5 100 0.74 2.74 3.20 0.23 5 150 1.48 0.44 0.04 0.13 5 200 0.51 3.52 3.53 0.78 5 250 3.82 0.02 0.65 0.89 5 300 3.23 3.28 2.61 0.18 5 350 0.39 0.49 0.88 0.55 5 400 1.35 1.27 1.83 1.17 7 100 1.62 6.13 5.76 0.85 7 150 7.94 1.83 0.96 2.22 7 200 2.61 3.78 3.49 0.10 7 250 0.00 0.08 0.38 0.86 7 300 2.73 0.13 0.24 0.95 7 350 0.66 0.32 0.59 0.56 7 400 1.92 1.53 1.17 0.01 9 100 1.30 0.67 0.36 0.09 9 150 6.49 0.11 0.78 0.96 9 200 0.06 2.78 2.77 0.55 9 250 0.78 1.33 1.45 0.11 9 300 3.85 2.91 2.19 0.12 9 350 1.30 4.21 4.53 2.05 9 400 0.61 2.76 2.49 0.85 PAGE 81 81 Table 6 5. Error in the axle load and spacing at 20% noise Measuring Locations Sampling Frequ ency Axle1 ( %) Axle2 ( %) Axle3 ( %) Spacing ( %) 1 100 4.27 4.85 3.31 1.26 1 150 0.13 15.13 14.22 3.88 1 200 10.39 3.41 0.50 2.66 1 250 4.19 13.37 12.14 6.04 1 300 17.47 14.46 11.48 0.42 1 350 10.65 8.83 7.23 1.50 1 400 2.77 0.56 0.71 1.80 3 100 9.58 1.66 0.50 2.63 3 150 5.14 0.53 1.49 2.03 3 200 2.44 6.58 6.16 1.44 3 250 4.20 11.12 9.97 1.57 3 300 1.42 1.24 0.88 0.64 3 350 5.38 8.01 6.42 1.35 3 400 4.04 2.09 2.62 2.58 5 100 3.77 7.25 7.84 2.60 5 150 3.24 3.00 3.12 1.09 5 200 8.04 1.27 2.65 2.59 5 250 0.84 0.89 0.14 0.52 5 300 3.57 3.21 4.12 2.34 5 350 3.85 2.88 3.97 1.07 5 400 0.81 2.31 2.21 0.33 7 100 4.16 6.87 6.32 0.89 7 150 8.90 1.28 0.42 2.17 7 200 2.92 6.97 6.02 1.05 7 250 0.73 7.43 7.46 2.53 7 300 1.65 1.93 2. 18 0.03 7 350 1.31 2.22 2.05 0.11 7 400 3.84 0.32 0.47 0.91 9 100 3.93 1.00 2.01 0.98 9 150 8.92 5.34 3.23 0.31 9 200 3.85 10.81 8.96 3.00 9 250 0.10 1.24 0.94 0.39 9 300 0.50 3.64 3.15 0.65 9 350 3.63 1.48 2.12 0.87 9 400 2.20 0.32 0.94 1.16 PAGE 82 82 Table 6 6. Error in the axle load and spacing at 20% noise Measuring Locations Sampling Frequency Axle1 ( %) Axle2 ( %) Axle3 ( %) Spacing ( %) 1 100 17.00 15.63 10.96 1.93 1 150 11.33 0.84 1.17 1.02 1 200 16.72 9.43 5.72 1.98 1 25 0 15.55 2.83 5.10 7.32 1 300 5.60 23.54 22.25 4.94 1 350 8.20 4.13 3.06 0.63 1 400 9.25 6.22 7.00 6.17 3 100 11.33 0.04 2.43 2.04 3 150 14.77 8.68 6.15 0.13 3 200 1.48 2.36 2.26 0.14 3 250 6.33 5.74 4.39 0.39 3 300 3.58 1.45 1.52 0.59 3 350 3.92 1 .50 2.30 0.06 3 400 3.71 9.96 10.39 3.66 5 100 4.17 6.32 5.60 1.29 5 150 6.48 4.16 4.76 2.69 5 200 2.19 3.10 3.91 0.24 5 250 5.92 5.74 4.43 0.97 5 300 1.68 7.83 7.52 1.94 5 350 5.18 0.35 0.70 1.43 5 400 2.79 1.29 0.75 0.54 7 100 2.47 4.87 4.47 1.9 2 7 150 2.46 1.72 2.31 0.01 7 200 1.42 2.21 1.25 0.46 7 250 0.61 4.15 3.77 1.06 7 300 0.43 3.12 2.89 0.75 7 350 5.59 3.72 2.03 0.33 7 400 1.70 3.28 3.05 0.37 9 100 3.06 0.03 0.68 1.13 9 150 9.04 0.80 2.76 2.29 9 200 7.39 3.36 1.82 0.16 9 250 0.20 5.00 4.45 2.39 9 300 1.36 1.35 1.27 1.14 9 350 0.30 5.57 5.24 1.69 9 400 4.64 0.29 1.08 0.43 PAGE 83 83 Table 6 7. Effect of Time varying load in axle load and axle spacing for noise free bridge response Measuring Locations Sampling Frequency Axle1 (%) Axle2 (%) Axle3 (%) Spacing (%) 1 100 0.06 8.16 7.61 3.64 1 150 1.05 6.75 6.07 1.14 1 200 1.48 5.94 5.23 1.84 1 250 0.19 11.12 10.62 3.62 1 300 2.03 12.47 12.19 4.90 1 350 2.61 12.92 12.67 5.12 1 400 4.94 14.95 14.96 6.66 3 100 0 .99 1.07 0.86 0.28 3 150 0.97 1.05 0.85 0.33 3 200 0.96 1.04 0.84 0.54 3 250 0.27 2.99 2.70 1.10 3 300 0.05 2.71 2.40 1.11 3 350 1.32 0.65 0.29 0.49 3 400 0.76 0.37 0.12 0.44 5 100 0.24 0.26 0.21 0.79 5 150 0.24 0.25 0.21 0.36 5 200 0.24 0.25 0.20 0.27 5 250 0.59 0.41 0.60 0.33 5 300 0.55 0.27 0.47 0.60 5 350 0.43 0.27 0.45 0.60 5 400 0.41 0.19 0.37 0.42 7 100 0.61 0.65 0.52 0.70 7 150 0.60 0.64 0.52 0.26 7 200 0.58 0.63 0.51 0.22 7 250 1.10 0.13 0.17 0.70 7 300 0.96 0.16 0.11 0.57 7 350 0.92 0.26 0.02 0.44 7 400 0.83 0.27 0.01 0.35 9 100 0.13 0.12 0.09 0.42 9 150 0.13 0.11 0.09 0.55 9 200 1.95 3.51 4.02 1.48 9 250 0.50 0.53 0.70 0.61 9 300 0.47 0.38 0.55 0.55 9 350 0.38 0.34 0.51 0.36 9 400 0.36 0.25 0.43 0.55 PAGE 84 84 Table 6 8. Effe ct of Time varying load in axle load and axle spacing for 5% noise Measuring Locations Sampling Frequency Axle1 (%) Axle2 (%) Axle3 (%) Spacing (%) 1 100 1.82 9.66 8.62 1.53 1 150 0.75 5.55 5.01 1.07 1 200 3.43 6.76 5.58 0.76 1 250 1.70 13.35 12.48 3.74 1 300 0.26 9.44 8.86 2.91 1 350 3.33 12.82 12.78 4.92 1 400 3.96 16.15 15.94 6.50 3 100 0.46 0.79 0.68 1.17 3 150 2.50 2.23 1.68 0.48 3 200 0.54 0.21 0.29 0.57 3 250 1.02 0.63 0.76 0.62 3 300 2.29 6.92 6.98 2.76 3 350 0. 95 3.06 2.54 0.47 3 400 2.56 2.40 1.76 0.55 5 100 0.70 1.08 1.00 1.08 5 150 0.09 0.20 0.02 0.06 5 200 1.96 1.15 0.65 0.27 5 250 0.24 0.21 0.21 0.26 5 300 1.28 0.57 0.20 0.54 5 350 1.01 0.51 0.19 0.55 5 400 0.20 0.53 0.64 0.49 7 100 0.07 0.00 0.14 0.01 7 150 0.46 0.35 0.20 0.55 7 200 0.62 0.89 0.67 0.33 7 250 1.08 0.43 0.11 0.86 7 300 0.93 0.29 0.02 0.30 7 350 0.36 2.02 1.76 1.00 7 400 0.19 0.61 0.73 0.36 9 100 1.11 1.28 1.05 0.70 9 150 0.16 0.20 0.17 0.62 9 200 0.11 0.03 0.00 0.41 9 250 0 .35 1.12 1.12 0.69 9 300 0.67 0.10 0.34 0.71 9 350 0.57 0.23 0.46 0.37 9 400 0.56 0.26 0.46 0.38 PAGE 85 85 Table 6 9. Effect of Time varying load in axle load and axle spacing for 10% noise Measuring Locations Sampling Frequency Axle1 (%) Axle2 (%) Axle3 (%) Spacing (%) 1 100 9.33 10.23 10.85 5.86 1 150 5.85 0.49 0.41 0.14 1 200 3.01 6.26 4.76 0.84 1 250 3.77 16.16 14.75 3.57 1 300 0.58 9.76 9.35 2.92 1 350 1.56 1.90 1.33 0.39 1 400 6.02 13.49 13.90 6.42 3 100 0.15 1.78 2.01 0 .82 3 150 2.62 7.82 6.71 1.18 3 200 0.65 1.32 1.19 0.38 3 250 1.86 1.51 1.06 0.71 3 300 4.89 6.47 7.20 3.02 3 350 4.67 5.18 4.04 0.54 3 400 3.24 4.14 3.10 0.50 5 100 1.75 0.06 0.30 0.76 5 150 6.68 4.99 3.56 0.64 5 200 6.85 2.04 0.22 0.80 5 250 0. 59 0.70 0.39 0.44 5 300 1.04 1.32 1.59 0.87 5 350 3.21 3.58 3.01 0.44 5 400 0.57 1.17 0.94 0.56 7 100 0.02 4.03 4.18 1.04 7 150 1.51 5.63 4.85 1.75 7 200 2.09 0.48 0.17 0.51 7 250 5.38 4.08 2.72 0.41 7 300 1.59 1.97 1.53 0.30 7 350 0.18 0.53 0.45 0.34 7 400 3.05 0.58 0.24 0.49 9 100 1.51 3.07 2.53 0.16 9 150 2.08 2.03 1.32 0.16 9 200 0.99 3.55 3.46 1.48 9 250 2.78 3.76 3.36 0.59 9 300 0.75 0.28 0.17 0.43 9 350 0.18 1.07 1.04 0.42 9 400 2.82 0.97 0.30 0.55 PAGE 86 8 6 Table 6 10. Effect of Time var ying load in axle load and axle spacing for 15% noise Measuring Locations Sampling Frequency Axle1 (%) Axle2 (%) Axle3 (%) Spacing (%) 1 100 0.02 34.40 33.37 8.35 1 150 15.59 7.76 9.63 7.37 1 200 11.33 20.08 21.02 11.28 1 250 9 .48 6.90 8.17 7.31 1 300 16.39 26.03 27.24 14.59 1 350 9.02 1.43 0.02 3.17 1 400 7.80 12.80 13.65 6.61 3 100 3.10 2.73 1.78 0.73 3 150 9.17 3.03 5.19 2.54 3 200 1.89 5.08 5.41 0.91 3 250 1.71 6.92 6.65 1.85 3 300 3.90 4.36 3.95 0.89 3 350 1.06 4.0 0 3.87 2.18 3 400 1.51 3.92 3.53 1.11 5 100 1.34 1.54 1.53 0.54 5 150 4.04 2.51 1.26 0.22 5 200 1.63 5.27 5.17 1.49 5 250 0.93 1.89 1.98 1.36 5 300 1.13 2.22 2.49 0.88 5 350 0.76 1.80 1.32 0.91 5 400 0.19 6.70 6.29 1.68 7 100 11.33 8.98 6.67 0.55 7 150 3.69 0.45 0.07 0.35 7 200 3.13 3.52 3.05 0.60 7 250 0.46 5.98 5.45 2.28 7 300 1.55 6.18 6.36 3.22 7 350 2.41 5.64 4.93 0.57 7 400 0.40 1.13 1.02 1.17 9 100 7.97 0.97 0.84 2.31 9 150 4.88 5.51 4.23 0.14 9 200 1.43 2.69 2.95 1.44 9 250 0.31 1 .08 1.14 0.18 9 300 3.20 5.24 4.46 0.06 9 350 6.00 0.05 1.20 1.65 9 400 3.83 1.20 0.19 0.36 PAGE 87 87 Table 6 11 Effect of Time varying load in axle load and axle spacing for 20 % noise Measuring Locations Sampling Frequency Axle1 (%) Axle2 (%) Axle3 (%) Spacing (%) 1 100 21.55 3.55 0.78 5.22 1 150 0.67 13.16 13.46 2.69 1 200 0.46 19.06 18.16 5.69 1 250 11.33 20.84 18.14 3.31 1 300 14.03 12.76 16.01 7.28 1 350 11.31 8.04 9.98 5.84 1 400 11.73 13.13 14.95 7.60 3 100 7.38 13.46 10.95 3.49 3 150 6.19 3.66 3.96 1.20 3 200 7.77 1.20 0.66 0.76 3 250 1.21 8.48 8.07 2.52 3 300 0.28 0.21 0.26 0.53 3 350 1.30 3.25 2.49 1.39 3 400 3.49 6.16 5.72 0.46 5 100 1.13 4.25 3.82 2.42 5 150 2.90 3.62 4.43 1.46 5 200 3.41 3.78 3.22 0.08 5 250 0.58 4.55 3.83 1.41 5 300 1.18 10.23 9.07 2.28 5 350 4.97 4.97 3.95 1.69 5 400 4.40 4.64 3.47 0.37 7 100 1.61 3.38 3.20 0.19 7 150 4.02 0.59 0.14 0.04 7 200 2.28 4.78 4.23 1.12 7 250 7.71 1.84 3.06 3.50 7 300 0.38 7.61 7.39 3.19 7 350 0.94 0. 64 1.09 0.84 7 400 4.43 2.03 0.95 0.54 9 100 2.85 0.34 0.52 0.05 9 150 1.95 0.64 0.29 0.15 9 200 1.21 0.88 0.44 0.04 9 250 2.04 0.52 0.97 1.22 9 300 1.06 1.34 0.92 0.43 9 350 2.81 1.81 1.17 0.38 9 400 1.76 2.37 1.87 0.47 PAGE 88 88 Table 6 12. Effect of Time varying load in axle load and axle spacing for 25% noise Measuring Locations Sampling Frequency Axle1 (%) Axle2 (%) Axle3 (%) Spacing (%) 1 100 16.62 28.84 30.81 10.95 1 150 7.96 17.52 18.31 7.02 1 200 8.43 11.23 8.72 1.40 1 250 13.76 7.88 9.53 5.36 1 300 8.28 18.17 19.21 7.07 1 350 12.99 30.90 31.67 14.93 1 400 12.21 17.59 18.88 10.77 3 100 10.43 15.82 13.34 3.39 3 150 4.69 0.74 2.53 2.90 3 200 5.01 0.83 1.68 1.09 3 250 1.17 0.15 0.17 0.14 3 300 2.33 1.56 1.11 0.67 3 350 1.49 4.52 3.84 2.13 3 400 7.14 3.69 2.38 0.38 5 100 3.48 4.14 4.88 1.01 5 150 2.57 8.50 8.19 4.86 5 200 3.20 4.54 5.27 1.77 5 250 0.67 0.53 0.07 0.47 5 300 2.96 3.53 3.00 1.43 5 350 2.07 8.15 7.12 2.15 5 400 3.18 5.31 4.87 0.61 7 100 6.89 10.47 8.70 3.68 7 150 3.34 1.58 0.46 1.52 7 200 8.22 5.37 3.67 1.09 7 250 1.85 15.24 13.80 3.39 7 300 1.36 3.89 4.12 1.19 7 350 2.84 1.03 0.47 0.76 7 400 4.44 3.18 1.91 0.38 9 100 7.17 3.02 3.84 1.26 9 150 8.97 3.27 4.18 4.57 9 200 6.37 0.59 0.72 1.51 9 250 3.95 6.96 6.12 1.34 9 300 1.79 0.33 0.32 1.02 9 350 2.42 1.04 1.51 1.35 9 400 0.30 2.07 2.20 0.50 PAGE 89 89 CHAPTER 7 CONCLUSION AND RECOMMENDATION Conclusion The Performance of GA in prediction o f loading attributes of a truck and axle spacing is studied. The bending stresses of the bridge are numerically simulated and used as input for prediction. The effect of measurement parameters such as sampling frequency, noise, Time varying load and measuring sections on accuracy of identification have als o been also investigated. Implementation of the GA optimization has improved the accuracy of the static WIM algorithm. For noise free bridge response, truck attributes can be found within the accuracy of 1% considering the bridge response recorded only at the midspan However, presence of the dynamic effect of the truck and measurement noise affects the accuracy. Noise is always present due to incapability of sensor which can be modeled as white noise to simulate the nature of noise present in the data. Dyn amic nature of axle occurs due to rough surface of the road profile, high weight ratio which has been modeled as time varying load of sin and cosine function. As compared to noise free data, single measuring location is not adequate to produce the better r esult. Increasing in the number of measured location increases the accuracy. However, sampling frequency increase d the accuracy noticeably. Increase in the sampling frequency gives the better resolution of response which in tur n gives mor e detail about the bridge response. Hence, it helps to overcome the effect of dynamic load and noise to identify the static weight of the truck. PAGE 90 90 Recommendation Application of GA in identification of static axle weight and axle spacing from the stress measurement of the bri dge is found to be accurate However, there has been more interest in finding the dynamic characteristic of the Truck. Application of proposed GA s to find the dynamics of the Truck can be the future scope of the study. Also, bridge and truck can be modeled using finite element method to simulate accurate behavior of the bridge response. PAGE 91 91 LIST OF REFERENCES Au, F. T. K., Jiang, R. J., and Cheung, Y. K. (2004). "Parameter identification of vehicles moving on continuous bridges." J.Sound Vibrat., 269(1 2) 91 111. Deesomsuk, T., and Pinkaew, T. (2009). "Effectiveness of Vehicle Weight Estimation from Bridge Weigh in Motion." Advances in Civil Engineering, 2009. Gagarin, N. (1991). "Advances in weigh in motion with pattern recognition and prediction of fa tigue life of highway bridges." PhD thesis, University of Maryland at College Park, MD Gagarin, N., Flood, I., and Albrecht, P. (1994). "Computing Truck Attributes with Artificial Neural Networks." J.Comp.in Civ.Engrg., 8(2), 179 200. Hashemi, R., and Ka Force Using Genetic World Acad. Of Sci, Engrg and Tech., 36, 147 153. Koh, C. G., and Perry, M. J. (2010). Structural Identification and Damage Detection using Genetic Algorithm s. Law, S. S J. Sound Vib., 201, 1 22 Law, S. S AS ME J. Dyn. Syst., Meas., Control, 12, 394 401. Law, S. S., Bu, J. Q., Zhu, X. Q., and Chan, S. L. (2004). "Vehicle axle loads identification using finite element method." Eng.Struct., 26(8), 1143. Law, S. S., and Zhu, X. Q. (2004). "Dynamic behavior of d amaged concrete bridge structures under moving vehicular loads." Eng.Struct., 26(9), 1279. Leming, S. K., and Stalford, H. L. (2003). "Bridge Weigh in Motion System Development Using Superposition of Dynamic Truck/Static Bridge Interaction." Proceedings o f the American Control Conference, Monti, G., Quaranta, G., and Marano, G. C. (2010). "Genetic Algorithm Based Strategies for Dynamic Identification of Nonlinear Systems with Noise Corrupted Response." J.Comp.in Civ.Engrg., 24(2), 173 187. Pinkaew, T. (2006). "Identification of Vehicle Axle Loads from Bridge Response using Updated Static Component Technique." Engineering Structures, 28(11), 1599 1608. PAGE 92 92 in Motion System Measurement Errors on Load Pavement J.Trans. Engrg., 133(1), 1 10. Sivanan dam, S. and Deepa, S. (2008). Introduction to Genetic Algorithms. Springer, NY The Mathworks. (2004). "Genetic Algorithm and Direct Search Toolbox user guide for MATLAB." Yan, L., Fraser, M., El gamal, A., Fountain, T., and Oliver, K. (2008). "Neural Networks and Principal Components Analysis for Strain Based Vehicle Classification." J.Comp.in Civ.Engrg., 22(2), 123 132. Yu, L., and Chan, T. H. T. (2007). "Recent Research on Identification of Mov ing Loads on Bridges." J.Sound Vibrat., 305(1 2), 3 21. Zhu, X. Q., and Law, S. S. (2002). "Moving Loads Identification Through Regularization." J.Engrg.Mech., 128(9), 989 1000. PAGE 93 93 BIOGRAPHICAL SKETCH Girish was born and brought up in Mumbai, India; a place where construction is always booming. Growing up in the family involved in the business of construction, he always desired a career in this area. He decided to pursue a career in civil engineering and com pleted his schooling from the V eermata J ijabai T echnological I nstitute one of he decided to do specialization in Building construction management to learn the current practices in the construction industry. After p assing through the screening exams, he secured admission in the M.E. Rinker Sr. School of Building Construction to fulfill his endeavors. The valuable knowledge and skill that he gained from the Rinker school will surely he lp him to excel in his objectives and goals 