femu of ss skewed psc girder

KSCE Journal of Civil Engineering (2013) 17(3):518-529DOI 10.1007/s12205-013-0599-z

− 518 −

www.springer.com/12205

Load Rating and Assessment of Bridges

Finite Element Model Updating of a Simply Supported Skewed PSC I-girder Bridge using Hybrid Genetic Algorithm

Dae-Sung Jung* and Chul-Young Kim**

Received December 3, 2012/Revised January 6, 2013/Accepted January 11, 2013

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Abstract

Hybrid Genetic Algorithm (HGA) which combines the genetic algorithm as a global optimization and the simplex method as alocal optimization is proposed for a finite element model updating of a real prestressed concrete bridge structure. In order to minimizethe updating error between the measurement and the finite element model updating result, objective functions which arecombinations of fitness functions based on the natural frequency, the mode shape and the static displacement are introduced. And aninterface tool is also developed in order to utilize various element library and numerical analysis tools which are provided bycommercial finite element and numerical analysis programs. A simply supported skewed PSC girder bridge which has 30 m spanlength is selected for the verification of the proposed FE model updating algorithm. Static vehicle loading test and forced vibrationtest by traveling vehicle as well as ambient vibration test were carried out to obtain the reference measurement data for numericalupdating. A grillage model is used for the finite element analysis. Effect of the spring element to simulate the realistic supportcondition which is not perfectly free or restrained in real situation as well as that of the objective function on the updating accuracyare studied. From the result of parametric study, it is investigated that the use of spring element for support condition is effective tominimize the updating error for natural frequency and mode shape. Furthermore, including the static displacement fitness functiontogether with those of dynamic properties may improve the global behavior of updated finite element model. It is concluded that thehybrid genetic algorithm proposed in this study is a very effective finite element model updating method to find an accurate result inupdating real bridge structure based on measured data.Keywords: finite element model updating, hybrid genetic algorithm, genetic algorithm, simplex method, PSC girder bridge

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1. Introduction

It is very important to build a refined finite element model forsystem identification, damage detection and assessment of loadcarrying capacity of real structures. Generally, numerical simulationis quite different from measured behavior due to various reasons,such as assumed material properties, construction discrepancy,deterioration and structural damage, etc. In order to reduce thisdifference, many finite element model updating method havebeen proposed. While approaches which update directly massand stiffness matrices were proposed in the past, updatingmethods based on optimization algorithm have been widelystudied recently (Imregun and Visser, 1991; Mottershead andFriswell, 1993; Friswell and Mottershead, 1995; Farhat andHemez, 1993).

Among conventional methods, direct method which directlymodifies system matrices and sensitivity-based parameter updatingmethod based on the sensitivity of parameters may be the mostcommon ones. The direct method, which finds change of massand stiffness matrices by solving the equation of motion, is hardto find an accurate result and cannot guarantee the sparseness,

positive-definiteness and symmetry of updated system matricessince the change of mass and stiffness matrices are alwayscoupled. Furthermore, it often discloses numerical instabilityproblem in reverse analysis due to measurement error and lack ofmeasurement points. Sensitivity-based parameter updating methodhas advantage over direct method since it is efficient to updateparameters which are more sensitive to dynamic properties ofstructures and can solve the numerical instability problem ofdirect method. It is hard to apply this method to complicated realstructures since it requires sensitivity matrices for all parametersto which the dynamic response of real structures does not changeproportionally.

Recently, approaches such as the Genetic Algorithm (GA), theSimulated Annealing (SA) and the trust region Newton methodwhich directly modifies the finite element model based on theoptimization theory have been proposed. These methods cansolve above-mentioned problems and give updated result whichmay have physical meaning. Levin and Lieven (1998) appliedthe genetic algorithm and the simulated annealing independentlyon the updating problem of thin-walled wing structure. Modakand Kundra (2000) proposed an updating method for nonlinear

*Member, Research Professor, Hybrid Structural Testing Center, Myongji University, Gyeonggi-do 449-728, Korea (E-mail: dsjung@ mju.ac.kr)**Member, Professor, Dept. of Civil Engineering, Myongji University, Gyeonggi-do, 449-728, Korea (Corresponding Author, E-mail: [email protected])


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optimization problem with constraints and applied it to a simplysupported reinforced concrete beam based on laboratory testdata. Jaishi and Ren (2006) proposed an updating method usingtrust region Newton method based on sensitivity method anddamage detection algorithm using modal flexibility residual.Rafiq et al. (2005) studied model updating method which usedboth genetic algorithm and regression analysis together forupdating of a masonry wall structure. Zhu and Hao (2006)applied conventional sensitivity-based method, neural networkand genetic algorithm to experimental result of 5 story steelframe structure and compared the result to estimate theefficiency of each method. Zhang et al. (2000) studied FE modelupdating of 1/150 scaled suspension bridge model using theeigenvalue sensitivity-based updating approach based on thefirst-order Taylor-series. In this study, the sum of weightedfrequency error norms and weighted perturbation norms of theparameters was used as an objective function. Sanayei et al.(2012) carried out load rating of a 47 m continuous three-spanconcrete slab on steel stringer bridge by nondestructive loadtesting as well as manual model updating. SAP2000 was used forfinite element analysis and an error function between theexperimental strain and the analysis result was adopted as anobjective function. Yang and Ou (2009) studied model updatingby the laboratory test of a 1/40 scaled model of ShandongBinzhou Yellow River Highway Bridge (cable-stayed bridge)which has two symmetric main spans. Finite element model wasanalyzed by MATLAB and ANSYS, and the residual of bothnatural frequency and mode shape was used as an objectivefunction. Zhao (2011) investigated model updating of YangtzeRiver Bridge which is 1040m long cable-stayed bridge usingANSYS optimization tool. Updating variables were the elasticmodulus, moment of inertia and mass density of main girder,diaphragm, pylon and cable elements. Only the error function ofnatural frequency was used as an objective function. Merce et al.(2007) studied model updating of Clifton Suspension Bridge withan objective function considering MAC value as well as naturalfrequency error function. ANSYS optimization tool was utilizedand manual updating and automatic updating were used together.Updating variables, which were elastic modulus and mass density,were selected by sensitivity analysis using ANSYS gradient tool.

In previous publication of authors, Jung and Kim (2009) proposedHybrid Genetic Algorithm (HGA) which combined the geneticalgorithm as a global optimization together with the simplexmethod as a local optimization. An interface tool which integratedcommercial finite element library and numerical analysis toolboxwas also developed so that the model updating problem oncomplicated structural models can be solved. In addition, forverification purpose, numerical parametric study on the grillagemodel of a 2-span continuous bridge model as well asexperimental study by laboratory test of a simply supportedsmall scale bridge model were carried out and proved its validityand efficiency.

In this study, the proposed HGA method was applied to thefield measurement data of a prestressed concrete girder bridge to

verify its applicability to real bridge structures. Parametric studyon the effect of spring elements for simulating support conditionsand that of fitness function to be considered was also carried out.This paper will start with the introduction of basic theoreticalbackground of the proposed HGA method followed by thedescription about field measurement and the discussion about theresult of parametric study.

2. Theoretical Background

2.1 Genetic AlgorithmThe genetic algorithm is a stochastic random search technique

in global space based on the evolutionary theory such asevolution and natural selection. Since GA can define objectivefunction with various design variables and searches solutionrandomly from multiple stating points in the problem which hasmultiple local minima, it is more flexible and efficient to find theglobal minimum or maximum than other updating methods suchas gradient-based method (SQP - Sequential Quadratic Program)and conjugate gradient method which uses single starting point.

Generally, the genetic algorithm calculates the fitness of objectivefunction regarding each chromosome of randomly generatedparent population and reproduces next population by crossoverand mutation. In reproduction process, usually roulette wheelselection, ranking selection, tournament selection, ranking selectionand elitist preserving selection are used to select the best fitnesschromosome. For crossover process, scattered crossover, singlepoint crossover and 2 points crossover are the most commonones. Mutation process is to change specific gene information ofchromosome randomly in the given solution space and is veryuseful in the sense that it can make up for the missing geneinformation in initially generated parent population. However, thefrequency of mutation, thus the probability of mutation, is verysmall compared with that of crossover.

2.2 Modified Simplex Method by Nelder and MeadThe simplex method used in this study is based on the

algorithm proposed by Nelder and Mead in 1965. This methodconverges fast and is very efficient to find local optimum point,especially when the number of updating variables is relativelysmall. The simplex method falls in the general category of directsearch method and is widely used for unconstrained nonlinearoptimization problem. In this method, updating direction isdecided by comparing function values at nearby points instead ofimproving objective function value by mathematical calculationor approximation of gradient. In the search space with Nvariables, after N+1 initial vertices are defined, iterative calculationis performed to find the optimum vertex which minimizesobjective function by simplex operation.

The weighting factors of simplex operators proposed byBarton and Ivey (1996) were used in this study. The values ofweight factors on simplex operators such as reflection, expansion,contraction and shrinkage as in Fig. 1 are 1.0, 2.0, -0.5 and 0.5,respectively. The stopping condition in iterative operation as was

Dae-Sung Jung and Chul-Young Kim

− 520 − KSCE Journal of Civil Engineering

proposed by Dennis and Woods (1987) was used as in Eq. (1).

(1)

where, (2a)

(2b)

where is the relative difference between the evaluation valueof the kth simplex and k-1th simplex, and is the convergencetolerance. indicates the Euclidian norm of the kth worstpoint on the simplex x.

2.3 Hybrid Genetic AlgorithmHybrid model updating algorithm uses global and local

optimization consecutively to find the best optimum solution. Inthis study, the hybrid genetic algorithm which adopts the geneticalgorithm as a global optimization method and the modifiedsimplex method proposed by Nelder and Mead as a localoptimization method is proposed. In order to solve complicatedmodel updating problem of real bridge structures systematically,an interface tool for integrating a commercially developed finiteelement analysis program which has a versatile element libraryand a numerical analysis toolbox which has very accurate solverroutines is also developed. ABAQUS (Simulia Dassault Systems,2007) and gads toolbox of MatLAB (The Mathworks, Inc., 2006)are integrated by the developed program code. Model updatingprocedure by the proposed HGA method is shown in Fig. 2.

2.4 Objective Function and Fitness FunctionEquation (3) shows the objective function proposed in this

study, which is a linear combination of fitness functions withrespect to natural frequency, mode shape and static displacement.The fitness functions in Eqs. (4a)~(4c) are error functions inwhich measured data and analysis result are one-to-onecorrespondence. The fitness function with respect to naturalfrequency is the proportion of the difference between measuredand calculated values. Normalized Modal Difference (NMD)which represents the difference between normalized modeshapes was used as the fitness function w.r.t. mode shape. Inorder to consider the phase information during updating process,absolute value was not taken for the denominator of ModalAssurance Criterion (MAC) which is an index of correlation ofmode shapes. This helps to distinguish bending and torsionalmode so that the order of modes is also taken into consideration.To prevent numerical difficulties when the denominator becomes

zero or very small value, summation of displacements at allnodes was taken for the fitness function w.r.t. static displacement.

(3)

, (num. of modes) (4a)

, (num. of modes) (4b)

,

Ψk φk φk 1– εs<–=

φk1∆k----- xi

k xwk–

i 1 … n 1+, ,=

xik xw

k≠

lim= max

∆k max 1 xwk,( )=

Ψk

εs

xwk

fmin fitness1 f( ) fitness2 φ( ) fitness3 u( )+ +=

fitness1 f( ) 1m---- αi

f ie f i

a–f i

e--------------⋅

i 1=

m

∑= i 1 … m, ,=

fitness2 φ( ) 1m---- βi NMD⋅

i 1=

m

∑= i 1 … m, ,=

fitness3 u( ) 1N---- γp

uje uj

a–j 1=

n

∑

uje

j 1=

n

∑

-----------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

p 1=

N

∑= j 1 … n, ,=

Fig. 1. Simplex Operators (N=3): (a) Initial Simplex, (b) Reflection, (c) Expansion, (d) Contraction, (e) Shrinkage

Fig. 2. Flowchart of the HGA-based FE Model Updating Method


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(num. of measuring pts.), (load cases) (4c)

In Eqs. (4a)~(4c), fitness1(f), fitness2(φ) and fitness3(u) representthe fitness functions w.r.t. natural frequency, mode shape andstatic deflection. The superscripts a and e stands for the analyticaland experimental result of the updating model, respectively. Inaddition, αi, βi and γp are the weighting factors for the fitnessfunction at the ith mode and pth load case, which are all set to 1.0in this study. NMD and MAC can be defined as follows:

(5)

(6)

where, and are ith mode shape vector from experimentand analysis, respectively.

3. Dynamic Properties of Simply Supported PSCGirder Bridge

3.1 Keum Dang BridgeKeum Dang Bridge is a simply supported bridge which has

post-tensioning prestressed concrete girders and is skewed by 15degree. This bridge is located on the test road in Chung-bu inlandexpressway, where many newly developed technologies areadopted for try-out purpose and normal traffic does not passthrough over the bridge. It is 5 m apart from the mainexpressway so that ambient vibration is caused by the traffic onthe main expressway even though the magnitude is very small.The first northbound span was tested, which is 30 meters longand 12.6 meters wide and has 4 PSC-I type girders as is shown inFig. 3.

3.2 Evaluation of Dynamic Properties by Vehicle LoadingTest

Input parameters for FEM analysis were evaluated fromvarious field tests such as static vehicle loading test, forcedvibration test by traveling vehicle and ambient vibration test.Loading vehicle and its specification is shown in Fig. 4 andTable 1.

Sensor locations are shown in Fig. 5. Displacement sensorswere located at L/2 point of each girder and accelerometers wereplaced at L/4, L/2 and 3L/4 point at each side of slab. Onlyvertical component was measured for both displacement andacceleration.

Three static load cases are shown in Fig. 6 and 7 and speeds forvehicle loading test are 10, 20, 30, 40, 50, 60, 40, 80 and 100 km/hr.

p 1 … N, ,=

NMD φie{ } φi

a{ },( ) 1 MACi–MACi

----------------------=

MACiφi

e{ }T φi

a{ }( )2

φie{ }

T φie{ }( ) φi

a{ }T φi

a{ }( )----------------------------------------------------------=

φie{ } φi

a{ }

Fig. 3. Test Span of Keum Dang Bridge

Fig. 4. Loading Vehicle

Table 1. Dimension and Weight of Loading VehicleDimension (m) Weight (kN)

L1 L2 L3 L4 Front Middle Rear Total3.2 1.3 2.0 1.85 70.66 98.98 91.73 261.37

Fig. 5. Sensor Location

Fig. 6. Static Vehicle Loading

Fig. 7. Load Cases of Static Vehicle Loading Test: (a) Load Case 1, (b) Load Case 2, (c) Load Case 3



3.3 Test ResultStatic deflections for each load case are listed in Table 2, where

averaged values are used for input parameters on model updating.Typical acceleration time histories by loading vehicle, that of

ambient vibration and their corresponding power spectral densitiesare shown in Figs. 8 and 9, respectively. Natural frequencies inTable 3 were estimated from peaks of Averaged NormalizedPower Spectral Density (ANPSD) functions. In addition, modeshapes in Fig. 10 were derived from transfer functions and phaseangle information. The first mode is a bending mode and thesecond is a torsional mode. The third mode, even though it lookslike the first bending mode due to the lack of measuring points, is atransverse bending mode coupled with longitudinal bending. Thismay be caused by the fact that this bridge is skewed. Dynamicproperty resulted from ambient vibration test was used for thereference at model updating since more refined result could beacquired by averaging large number of sampled signals.

4. Finite Element Model Updating

4.1 FE Model and Updating variablesKeum Dang Bridge was modeled as a grillage composed of

Table 2. Static Displacement (unit: mm)

GirderLC1 LC2 LC3

1st 2nd AVE. 1st 2nd AVE. 1st 2nd AVE.G1 -1.72 -1.62 -1.67 -0.88 -0.91 -0.90 -0.4 -0.44 -0.42G2 -0.96 -0.94 -0.95 -1.12 -1.12 -1.12 -0.85 -0.86 -0.86G3 -0.23 -0.25 -0.24 -0.64 -0.65 -0.65 -1.04 -1.00 -1.02G4 0.00 0.01 0.01 -0.24 -0.22 -0.23 -0.52 -0.50 -0.51

Fig. 8. Acceleration Time History: (a) Vehicle Loading Test (ACC-2, v=40 km/h), (b) Ambient Vibration (ACC-2)

Fig. 9. ANPSD: (a) Vehicle Loading Test (ACC-2, v=40 km/h), (b) Ambient Vibration (ACC-2)

Table 3. Natural FrequencyNatural frequency (Hz)

Testing Method 1st mode 2nd mode 3rd mode

AVT (ambient vibration test) 5.88 7.15 9.99

FVT(traveling

vehicle test)

10 km/h1st 5.96 7.15 9.892nd 6.03 7.15 9.89

20 km/h1st 6.05 7.15 9.962nd 6.23 7.03 9.89

30 km/h1st 5.91 7.18 9.892nd 5.96 7.28 9.91

40 km/h1st 6.18 7.13 9.892nd 5.93 7.10 9.81

50 km/h1st 5.93 7.10 9.862nd 5.83 7.13 9.94

60 km/h1st 5.86 7.25 9.942nd 5.91 7.25 9.84

70 km/h1st 5.83 7.20 9.842nd 5.98 7.15 9.84

80 km/h1st 5.83 7.18 9.812nd 6.03 7.18 9.94

100 km/h1st 5.83 7.08 9.962nd 5.83 7.10 9.74

AVE. 5.95 7.16 9.88


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319 beam elements and 24 spring elements which were as shownin Fig. 11 and was analyzed by commercial FEA program,ABAQUS. Model updating using commercial FEA program hasan advantage of utilizing versatile element and material propertylibrary. On the other hand, modeling a fine mesh with a largenumber of elements may lead to a very long analysis time. In thisstudy, in order to reduce analysis time, the number of updatingvariables was reduced by classifying the section property ofgirders and cross beams into 6 groups (SECT01 through SECT06)according to tendon profiles and sectional properties. Slabtogether with cross beams was divided into many strips tosimulate transverse load distribution.

Support conditions were simulated by springs. While verticalrestraints were taken into account by discarding correspondingdegrees of freedom, translational springs in both x and ydirections as well as rotational springs about y-axis wereattached at all supports regardless of their restraint conditions.And then, large spring constant was assigned to actuallyrestrained DOF and small value to unrestrained DOF. Thisapproach takes into consideration the fact that even though theshoes and bearings are supposed to move freely alongunrestrained DOF it does have a certain amount of friction

especially in old bridges. This affects overall structural stiffnessand behavior and eventually estimated structural properties aswell. From experiences of previous test cases, it has beeninvestigated that this affects dynamic properties, especiallyestimated from small magnitude vibration, a lot more thanvariation of stiffness and mass distribution of main structuralcomponents do. Thus, if restraint condition is not considered asupdating variable, the model updating may lead to erroneousresult with unrealistic stiffness and mass distribution for mainstructural members.

Updating variables and their initial values for HGA modelupdating are listed in Table 4. The initial values for springconstants used in this study were decided by preliminaryanalysis. Since these initial values are assumed and lack ofphysical meaning, the updated values may be quite differentfrom the initial ones. Thus the range between upper and lowerbounds were set relatively larger than that of other updatingvariables such as the moment of inertia.

4.2 Parametric StudyParametric study was carried out in order to verify the

applicability of the proposed updating method to a real bridgestructure and to make an optimal FE model for the PSC girderbridge based on field measurement data. This parametric studymainly focuses on two subjects: the effect of updating springelements being used to simulate support restraint condition andthe effect of using static displacement fitness function in additionto those of dynamic properties such as natural frequency andmode shape. Eq. (7) and (8) show two objective functions wherefirst one has fitness function w.r.t. static displacement in addition

Fig. 10. Mode Shape (AVT)

Fig. 11. Finite Element Model of Keum Dang Bridge

Table 4. Updating Variable and Initial ValueMember Group Updating Variable Initial Value Description

SECT01 Iyy 1.003E+12 mm4

Moment of inertia






Springsupports

Kx,fixed 1.000E+08 N/mm Restrained along x-dir.Kx,free 1.000E+04 N/mm Unrestrained along x-dir.Ky,fixed 1.000E+08 N/mm Restrained alongy-dir.Ky,free 1.000E+04 N/mm Unrestrained along y-dir.Kr 1.000E+04 N·mm/rad Rotational spring abouty-axis (unrestrained)



to those w.r.t. dynamic properties.

(7)

(8)

The size of population was set to 10 times the number ofupdating variables and generation size equal to it as analysisconditions. For evolution to next generation, 4 elite chromosomeswhich showed the best fitness among population were used. Forcrossover and mutation function, scattered function and uniformfunction with 75% and 25% probability were used, respectively.Scattered crossover function is a traditional crossover functionwhich exchanges mutually related single or multi-gene informationwhen two parent chromosomes crosses over to childrenchromosomes. On the other hand, uniform crossover functionuniformly changes specific genetic information of childrenchromosomes. When each updating variable is normalized as1.0, the range of population which is lower and upper bounds ofupdating variables was set from 0.1 to 2.0 for beam elements andfrom 0.01 to 10.0 for spring elements. For Simplex method, theinitial value was set to the final result of the GA method and 5%of each updating variable was considered as initial mesh size.

4.3 Updating ResultFigures 12 and 13 show the converging process of the updating

variable, the fitness function and the objective function for case 3and the final updated result of objective functions. Updatingvariables are listed in Table 6 and 7. For genetic algorithmanalysis, only the smallest objective function values out of 100generations are shown. This genetic algorithm has slowconvergence speed and takes long analysis time compared withthe simplex method since fixed number of iterations should becarried out for every generation. On the other hand it gives resultwhich is very close to global optimum based on the fact that thevariation of updating variables is minimal during the simplexoptimization process.

As is shown in Table 6, from the result of initial analysis for

both FE model with and without support spring, the degree ofconvergence of the model with support spring is much betterthan the model without spring for dynamic property fitnessfunctions, while there is no difference for static displacementfitness function. Objective function considering fitness functionw.r.t. static displacement as well results in better convergence.Especially in case 4, the fitness function w.r.t. natural frequencybecomes zero so that the updated value exactly coincides withthe measurement. If the error in static displacement is also takeninto consideration even for case 2 and case 4, the overall degreeof convergence is the best in case 3 where support restraints aremodeled with spring elements and static displacement fitnessfunction is included in the objective function.

From the converged values of updating variables in Table 7, itcan be seen that overall stiffness of girders increases up to 70%from initial values although there are little differences from caseto case. This might be resulted from the fact that for the initialmodel reinforcing steels and tendon forces were not taken intoaccount and the strength of concrete were assumed as the design

Obj. f1 min, fitness1 f( ) fitness2 φ( ) fitness3 u( )+ +=

Obj. f2 min, fitness1 f( ) fitness2 φ( )+=

Table 5. Cases of Parameter Study

CasesSupport Condition Objective Function

w/o spring w/ spring f1,min f2,min

Case1 x xCase2 x xCase3 x xCase4 x x

Fig. 12. Convergence of Updating Variables (Case 3)

Fig. 13. Convergence of Objective Function (Case 3)

Table 6. Value of Fitness and Objective Function

Fit. & Obj. functionw/o spring w/ spring

Initial Case1 Case2 Initial Case3 Case40.275 0.168 0.169 0.165 0.049 0.0000.599 0.083 0.092 0.136 0.127 0.1230.509 0.109 (0.150) 0.509 0.114 (0.183)

fmin(sum) 1.383 0.360 0.261 0.810 0.290 0.123

fitness1 f( )fitness2 φ( )

finess3 u( )


Vol. 17, No. 3 / April 2013 − 525 −

value lower than the real one. Especially for case 3, the stiffnessof the middle section of outside girder increases 75%. Thisoccurs because the barrier at the side of the bridge deckcontributes considerably to the stiffness of outside girder. Ingeneral, the stiffness of beam elements contributes to convergenceof displacement in a large degree while support springs contributemuch to that of dynamic properties. It should be noted that largedifferences between initial and updated values for springconstants can be obtained since the initial values does not haveany physical meaning but are completely assumed values.

Natural frequencies are listed in Table 8. Mostly, models withsupport springs give better convergence and especially case 4 inwhich only the fitness functions w.r.t. dynamic properties areconsidered shows almost perfect convergence in all the 3 modesunder consideration. Case 3 which includes the displacementfitness function in addition, also shows good convergence. Modeshapes for initial analysis, case 1 and case 3 are shown in Fig. 15.The 3rd mode turned out to be the lateral mode as in case 1 for themodel without support spring, and the results of all cases weresimilar to that of case 3.

As it can be seen in Table 10, the same displacement wasresulted in for both cases since the degrees of freedom wereeliminated instead of using spring elements for vertical restraints

of the model with spring elements. The displacement of thegirder which has relatively large displacement at each load casehas improved better since the static displacement fitness functionis the ratio of the sum of errors to the sum of displacements at allmeasurement points. The error ratios for G1 at LC1, G2 at LC2and G3 at LC3, which have relatively large displacement at eachload case, are 51.1%, 22.5% and 30.2% for the initial model,-0.3%, -5.5% and 0.4% for case 1, -2.5%, -16.4% and -11.0% forcase 2, -1.3%, -7.4% and -1.4% for case 3 and -22.5%, -2.4%and 4.7% for case 4, respectively. In general, as it can beexpected, case 1 and case 3 in which the static displacementfitness function is considered in addition to the dynamic propertyfitness function show better updating result. Regarding the lateralstiffness distribution and its resulting lateral displacement asshown in Fig. 16, case 3 has the best updating result with themaximum error ratio of 5.6%.

Table 7.Value of Updating Variable

Updating var. Initial Valuew/o spring w/ spring

Initial Case1 Case2 Initial Case3 Case4SECT01 1.003E+12 mm4 1.0 1.088 2.074 1.0 1.243 1.178SECT02 9.411E+11 mm4 1.0 1.389 1.277 1.0 1.249 0.723SECT03 1.139E+11 mm4 1.0 0.645 1.946 1.0 1.055 1.445SECT04 1.640E+09 mm4 1.0 1.441 1.965 1.0 1.314 0.676SECT05 1.033E+12 mm4 1.0 1.117 1.007 1.0 1.251 1.990SECT06 9.666E+11 mm4 1.0 1.747 1.784 1.0 1.756 2.572Kx,fixed 1.000E+08 N/mm - - - 1.0 0.149 4.510Kx,free 1.000E+04 N/mm - - - 1.0 9.416 6.546Ky,fixed 1.000E+08 N/mm - - - 1.0 3.767 3.425Ky,free 1.000E+04 N/mm - - - 1.0 2.907 8.993Kr 1.000E+04 N·mm/rad - - - 1.0 1.733 2.079

Table 8. Initial and Updated Natural Frequency (Hz)

Mode No. Exp.w/o spring w/ spring

Initial Case1 Case2 Initial Case3 Case41 5.884 4.892 5.896 5.908 4.892 5.842 5.8842 7.153 4.990 6.094 6.086 4.990 6.145 7.1533 9.985 6.445 6.445 6.445 9.753 9.984 9.985

Table 9. Errors for Updated Natural Frequency (%)

Mode No.w/o spring w/ spring

Initial Case1 Case2 Initial Case3 Case41 -16.9 0.2 0.4 -16.9 -0.7 0.02 -30.2 -14.8 -14.9 -30.2 -14.1 0.03 -35.5 -35.5 -35.5 -2.3 0.0 0.0

Note) Err. %( ) f ie f i

a–( ) f ie⁄{ } 100%×=

Fig. 14. Comparison of Natural Frequency



4.4 Modal and Static correlationIn Table 11 and 12, correlation results about mode shape and

static displacement are listed.

As it can be seen in Table 11, high correlation was resulted frommodel updating with MAC values greater than 0.972 in all modes.The degree of correlation for static displacement was investigated

Fig. 15. Mode Shapes: (a) Initial Model without Spring, (b) Case1 without Spring, (c) Initial Model with Spring, (d) Case3 with Spring

Table 10. Static Displacement (mm)

Load case Girder Exp.w/o spring w/ spring

Initial Case1 Case2 Initial Case3 Case4

LC1

G1 -1.67 -2.52 -1.67 -1.63 -2.52 -1.65 -1.29G2 -0.95 -1.41 -0.91 -0.96 -1.41 -0.95 -0.88G3 -0.24 -0.47 -0.30 -0.35 -0.47 -0.34 -0.43G4 0.01 0.28 0.14 0.17 0.28 0.14 0.01

LC2

G1 -0.9 -1.17 -0.70 -0.80 -1.17 -0.76 -0.69G2 -1.12 -1.37 -1.06 -0.94 -1.37 -1.04 -1.09G3 -0.65 -1.08 -0.83 -0.75 -1.08 -0.83 -0.92G4 -0.23 -0.51 -0.28 -0.38 -0.51 -0.34 -0.38

LC3

G1 -0.42 -0.61 -0.35 -0.44 -0.61 -0.40 -0.43G2 -0.86 -1.12 -0.87 -0.78 -1.12 -0.86 -0.95G3 -1.02 -1.33 -1.02 -0.91 -1.33 -1.01 -1.07G4 -0.51 -1.07 -0.64 -0.74 -1.07 -0.70 -0.65

Fig. 16. Comparison of Static Displacement: (a) Load Case 1, (b) Load Case 2, (c) Load Case 3


Vol. 17, No. 3 / April 2013 − 527 −

based on DAC (Displacement Assurance Criterion) and NDD(Normalized Displacement Difference) in Eq. (9a) and (9b). Theabsolute error, percent error, scale error and correlation coefficient inTable 13 as proposed in the reference (BRIDGE DIAGNOSTICS,Inc., 2001) were also analyzed and the result is listed in Table 14.It is suggested that the updating result can be acceptable whenthe percent error is less than 10% and the correlation coefficientis greater than 0.9.

(9a)

(9b)

Based on DAC values in Table 12, the objective functionincluding displacement fitness function gives more correlatedresult in general. Since the static deformed shape by loadingvehicle is quite similar with the first bending mode shape, strongcorrelation is resulted in for both initial and updated model. InTable 14, it is clearly shown that error function values are greatlyreduced after updating for all analysis cases especially for case 1through 4 where percent errors are less than 4.1% and correlationcoefficient larger than 0.979.

From all the previous discussions, it can be summarized thatconsidering support restraints by support spring improvesupdating result for natural frequency and mode shape as well asstatic displacement. In case 4 where only the dynamic propertyfitness functions are considered in the objective function,although perfect correlation is resulted in for natural frequenciesof the first 3 modes, relatively poor correlation is acquired forstatic displacement. Furthermore, the 3rd mode shape of theinitial model appeared differently from the measurement resultwhen support spring is not considered. However, the updatingresults for mode shape as well as natural frequency can beimproved by considering it.

In cases of FE models considering support spring, the mostcorrelated result is acquired in case 3 where spring element isused to simulate realistic restraint condition and static displacementfitness function is also considered in objective function. For this

DAC um ua,( ) um{ }T ua{ }

2

ua{ }T ua{ }( ) um{ }

T um{ }( )------------------------------------------------------------=

NDD ukm{ } uk

a{ },( ) 1 DACkk–DACkk

-----------------------=

Table 11. Comparison of NMD and MAC

Mode No.w/o spring w/ spring


MAC1 0.970 0.977 0.976 0.971 0.979 0.9832 0.996 0.996 0.996 0.996 0.996 0.9963 0.291 0.999 0.997 0.972 0.972 0.972

NMD1 0.175 0.155 0.158 0.175 0.145 0.1322 0.063 0.065 0.065 0.063 0.065 0.0683 1.560 0.029 0.052 0.170 0.170 0.169

Table 12. Comparison of DAC and NDD

Load Casew/o spring w/ spring


DACLC1 0.991 0.994 0.990 0.991 0.993 0.974LC2 0.978 0.969 0.973 0.978 0.972 0.947LC3 0.968 0.991 0.967 0.968 0.986 0.996

NDDLC1 0.097 0.076 0.102 0.097 0.085 0.164LC2 0.150 0.179 0.166 0.150 0.170 0.237LC3 0.182 0.096 0.184 0.182 0.117 0.065

Table 13. Error FunctionsError Function Equation

Absolute Error

Percent Error

Scale Error

Correlation Coefficient

δm δa–∑δm δa–( )2 δm( )∑⁄

2∑

δm δa–max gage δmmax gage∑⁄∑

δm δm–( ) δa δa–( ) δm δm–( )2 δa δa–( )

2∑⁄∑

Table 14. Error Functions for Static Displacement

Error functionw/o spring w/ spring

Initial Case1 Case2 Initial Case3 Case4Absolute error 4.368 0.937 1.290 4.368 0.980 1.572Percent error 0.233 0.014 0.022 0.233 0.016 0.041Scale error 0.485 0.120 0.150 0.485 0.129 0.206Correlation coefficient 0.993 0.992 0.992 0.993 0.994 0.979



model updating study of Keum Dang Bridge, updated results ofupdating variables show reasonable values compared with designvalues.

5. Conclusions

In this study, hybrid genetic algorithm was proposed, whichuses sequentially the genetic algorithm as a global search methodand the simplex method as a local search method and itsverification of applicability was carried out with field measurementdata of a prestressed concrete bridge. The objective functionincluding static displacement fitness function as well as fitnessfunction with respect to natural frequency and mode shape wasalso investigated. A modified MAC without absolute value wasused to consider the phase information.

Parametric study was performed to investigate the effect ofspring element which is used to simulate realistic supportrestraint condition. It could be seen that the use of spring elementgreatly improves updating result especially for dynamic properties,while there is no big improvement for static displacement. Also,the effect of the fitness function was also studied by includingstatic displacement fitness function in addition to the fitnessfunctions with respect to natural frequency and mode shape. Inthe case only dynamic property fitness functions are consideredwith the spring element, updated natural frequency convergedexactly the same as the reference value up to the 3rd mode underconsideration, but it was less effective for the static displacement. Inthis case, percent error was improved from initial value of 23.3%to 4.1%and DAC and correlation coefficient became 0.947 and0.979, respectively. On the other hand, in the case the staticdisplacement fitness function was also included and the springelement is used, the displacements showed good agreement with1.6% percent error and DAC and correlation coefficient greaterthan 0.972 and 0.994, respectively, while natural frequencieswere updated within 1% error except the 2nd mode. Overall, forthe grillage model studied in this paper, the use of spring elementand the static displacement fitness function gives better updatingresult. In case 3, among the updating variables, the moments ofinertia of girder elements increased about 25% except at themiddle portion of outside girder. This can be deduced from thefact that the initial percent error was 23.3% for displacement.

From all the investigation and discussion in this study, it can beconcluded that the proposed hybrid genetic algorithm can beapplied for the model updating of real bridge structures based onthe field measurement and can be used as a good basis for furthersystem identification, assessment and load rating.

Acknowledgements

This research was supported by Korea Expressway Corporationand a grant (10TIB01-Modular Bridge Research Centre) fromConstruction Technological Innovation Programme funded byMinistry of Land, Transport and Maritime Affairs of Koreangovernment. The authors’ sincere appreciation goes to Korea

Institute of Science and Technology Information (KISTI) on theusage of Korea Research Environment Open NETwork(KREONET).

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